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2013 43 Irene Calvo Almazán Molecular diffusion on surfaces: the diffusive behavior of aromatic compounds absorbed on graphitic surfaces studied with QuasiElastic-Neutron Scattering (QENS) Departamento Director/es Física de la Materia Condensada García Vinuesa, Luis Miguel Fouquet, Peter Miret Artés, Salvador Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Irene Calvo Almazán MOLECULAR DIFFUSION ON SURFACES: THE DIFFUSIVE BEHAVIOR OF AROMATIC COMPOUNDS ABSORBED ON GRAPHITIC SURFACES STUDIED WITH QUASI-ELASTICNEUTRON SCATTERING (QENS) Director/es Física de la Materia Condensada García Vinuesa, Luis Miguel Fouquet, Peter Miret Artés, Salvador Tesis Doctoral Autor 2013 Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Departamento Director/es Director/es Tesis Doctoral Autor Repositorio de la Universidad de Zaragoza – Zaguan http://zaguan.unizar.es UNIVERSIDAD DE ZARAGOZA Molecular diffusion on surfaces – The diffusive behavior of aromatic compounds adsorbed on graphitic surfaces studied with quasi-elastic neutron scattering (QENS) Molecular diffusion on surfaces – The diffusive behavior of aromatic compounds adsorbed on graphitic surfaces studied with quasi-elastic neutron scattering (QENS) Irene Calvo Almazán A mis padres Índice 5 of coverage and temperature in the diffusive behavior of benzene molecules on the basal plane of graphite. For this purpose we have perform several measurements of neutron quasi-elastic spectrum of benzene adsorbed on graphite, at different coverages (from the very low coverage regime 0.1 ML to the complete monolayer 1.0 ML) and in a wide range of temperatures, from 60K to 140K. We use the neutron spectroscopy technique of time-of-flight and we combine measurements on hydrogenated C6H6and deuterated benzene C6D6. Since neutrons can be coherent or incoherently scattered, the resulting quasi-elastic spectra arising from C6H6or C6D6displays differences which can be attributed to correlated dynamics in time and space. We obtain a complementary view on the diffusion process which helps us to improve the understanding of the physical mechanisms governing the diffusion, and in particular, to identify the main source of friction in this system. The dissertation is organized as follows: Chapter 1 introduces the main experimental technique, neutron time-of-flight spectroscopy, which is compared to surface specific techniques. We aim to highlight the unique place of neutrons in surface studies, and the important role played by substrates like exfoliated graphite which display a large surface/volume rate. In Chapter 2 we review the theoretical framework within which surface diffusion models can be developed. We also provide a succinct description of molecular dynamics calculations, since it is a useful tool for the interpretation of neutron quasielastic data, even though in this work we do not present any MD simulation result. Chapter 3 contains the description of the samples and the experimental conditions in which we operate with the time of flight spectroscope IN6 (ILL, Grenoble, France). We also summarize the measured results (the scattering functions) making emphasis in their change with coverage and temperature. Chapter 4 deals with the analysis of the scattering functions. We test several diffusion models to fit the quasi-elastic energy profile. Chapter 5 discusses the results of the fits, and values for the diffusion coefficient and friction parameter are extracted. We drive our attention to the dependence of these parameters on temperature and coverage, since this carries the fingerprint of the physical mechanisms governing the diffusion of the molecules (phononic coupling between molecules and the substrate or collisions between molecules). We also compare the results obtained for deuterated and hydrogenated benzene with the measurements published in Ref. [45]. Finally, Chapter 6 is a summary of the main conclusions that we can draw from the neutron study of the diffusion of benzene adsorbed on the basal plane of graphite. We also give an outlook of possible perspectives for a future development of the present study. Chapter 1 Quasi-elastic neutron scattering, applied to diffusion of adsorbed species on the surface This chapter overviews the experimental aspects of neutron studies on surface diffusion. It is divided into two sections. The first section aims to underline the place of quasi-elastic neutron scattering, QENS, in surface diffusion studies. The comparison of QENS to other surface specific techniques highlights the kind of information about diffusion processes revealed by neutrons, its limits and the type of systems on which neutrons are especially powerfull. The second section focuses on the neutron time-of-flight technique, which is the main technique in use in this work. A detailed description of their modus operandi, the physical magnitudes which are measured and their dynamical window is provided. 1.1 QENS spectroscopy applied to surface diffusion Surface diffusion is a complex and ubiquitous phenomenon: it can take place in a myriad of different substrates and involves manifold physical mechanisms. Accordingly there exists a large portfolio of surface specific techniques among which, one finds quasi-elastic neutron scattering. We start this section reviewing the main surface diffusion techniques in use. Then, we focus on the best suited systems to test the power of neutrons for probing surface diffusion. The combination of hydrogenated compounds adsorbed on substrates characterized by a large effective surface area for adsorption constitute excellent samples for the application of neutrons to surface studies. 8 Chapter 1. Quasi-elastic neutron scattering, applied to diffusion of adsorbed species on the surface 1.1.1 QENS techniques vs surface specific techniques A complete description of the diffusion process demands to relate the thermodynamical properties (like the melting and other phase transitions temperature, the coexistence of different phases, the vapor pressure) with the structure of the layer, its chemical composition and the dynamics of the adsorbed particles within the layer. Hence, a detailed study of diffusion process requires a multifaceted experimental approach. Comprehensive reviews discuss the characteristics, advantages and disadvantages of the available methods [5,6,46–49]. This section focuses on the techniques revealing similar aspects of diffusion than QENS techniques. The comparison of the time and distance scales, together with the type of substrates, the kind of diffusion (tracer or chemical diffusion) and the physical information (diffusion coefficients, friction parameters and structural information) affordable for each technique underlines the specific place that QENS occupies in surface diffusion studies. Nevertheless, the lack of overlap in the time and spatial scale between the different techniques [48] renders the comparison of experimental results arising from different measurement methods difficult. Therefore, there is no technique which can give a complete description of diffusion and the interplay between different experimental approach is mandatory. There are many criterions to classify surface specific techniques: the physical mechanism underlying the measurement method or the way in which information released. For instance, Ref. [48] proposes to distinguish between techniques inducing inhomogeneities in the density of the adlayer which evolve in time according to the Fick’s law or techniques measuring the thermal fluctuations of the system in its true thermal equilibrium state. The fluctuationdissipation theorem allows, in the latter case, to extract the information about diffusion [20]. In general, the surface specific techniques surveying the dynamics and structure of adsorbed layers belong to the second group of techniques and require the sample to be in thermal equilibrium. They can be further classified into two broad groups. First of all, there is the group of microscopic techniques which image the diffusion of individual particles or clusters in real space and time [5,6,47]. On the other side, lie the diffraction and spectroscopy techniques where the dynamical and structural information is released in the Fourier transformed space of momentum and energy transfer. Amongst the latter we will discuss with detail the helium and neutron scattering techniques. The main properties of the experimental methods (resolution, time and length scales, remarks about the kind of samples) are summarized in Tab. 1.1.1 and displayed in Fig. 1.1. 1.1. QENS spectroscopy applied to surface diffusion 9 spectrometer, enabling the rate of refilling and hence D c to be obtained. More recently, laser optical diffraction (LOD) has been developed to enable more delicate measurements. In this case, an optical grating pattern is desorbed from the surface, which decays as the overlayer diffuses. An important point to consider when analysing data from any laser based technique is the potential for surface damage, as has recently been observed [41]. Electron based microscopic and spectroscopic techniques are also susceptible to inducing surface damage, ranging from electron stimulated desorption of weakly bound species to the serious forms of damage often seen in high power electron microscopes. Fig. 2 gives a graphical comparison of the length and timescales over which a selection of techniques are sensitive to diffusion. In general terms, imaging techniques occupy the longer timescales while scattering methods offer greater potential in the short timescale region. Techniques, such as laser optical diffraction (LOD), are an intermediate case in that they follow the evolution of a macroscopic, non-equilibrium profile and thus provide important information on macroscopic processes, but rather restricted information about processes on the microscopic scale. Experimental results have therefore been compared with microscopic theory only rather indirectly, usually through macroscopic diffusion constants. Equilibrium methods generally involve monitoring atomic diffusion or other larger scale fluctuations of in the surface configuration with time. Typical techniques include field ion microscopy (FIM), scanning tunneling microscopy (STM), field emission microscopy (FEM) [7] and fluorescence correlation spectroscopy (FCS) [42]. FEM and FCS provide fluctuation measurements as isolated particles move in and out of a defined measurement volume. They operate on intermediate length-scales, defined by the measurement volume, which enables diffusion constants to be extracted. Both FIM and Fig. 2. Graphical comparison of the principal experimental techniques which have been applied to surface diffusion, in terms of the approximate lengthand timescales to which the measurement process corresponds. Labels indicate primarily microscopic imaging based techniques (M), optical or laser based techniques (O) and scattering based methods (S). Increasing these ranges of applicability is still the subject of ongoing work. Note that simple diffusion processes can be measured using several different techniques over either short lengths and times, or equally over longer lengths and times. However, microscopic phenomena are only evident on atomic length-scales and fast times. Currently, QHAS is the only surface technique which simultaneously allows microscopic length-scales and picosecond to nanosecond timescales to be studied, while the system is in true thermal equilibrium. The figure is constructed using information from [7,6] with additional information on STM [43], FEM [10], FCS [42], LOD/SHD [45,46], PEEM (either limited by phosphor decay/ CCD framing rates [47] or as a pulse synchronised measurement [48,49]). Quasi-elastic neutron scattering (QENS) [50] is only weakly surface sensitive, but is included for comparison with QHAS. A.P. Jardine et al. / Progress in Surface Science 84 (2009) 323–379 329 NMR (spec) Figure 1.1: Comparison between the length and time scale covered by the most relevant surface specific technique. (M) means microscopic techniques, (O) refers to optical or laser based techniques and (S) refers to scattering techniques. The physical process and the structures observable at the different distances and times scales are also included. The graph has been taken from Ref. [1]. We incorporate the band in time covered by NMR spectroscopy, which does not have spatial resolution. 10 Chapter 1. Quasi-elastic neutron scattering, applied to diffusion of adsorbed species on the surface Thermodynamical techniques Well established thermodynamical techniques such as isotherm measurement can determine the area occupied by adsorbed particles and the adsorption dynamics. Calorimetry is an effective method for finding the temperatures of phase transitions and for determining the order of the transition. This techniques often surveys the macroscopic properties of surfaces like the melting temperature or the surface effective area for adsorption of materials. Optical and laser techniques The idea underlying the application of laser techniques for studying diffusion is that desorption is sensitive to the lateral interaction between adsorbed species [50]. In other words, desorption techniques give also information about the lateral diffusion energetic barrier of the adsorbates and can measure chemical diffusion coefficients Dc(ensuing from the second Fick’s law [1]). In the case of laser induced thermal desorption, LITD, a powerful laser pulse irradiates a small spot on the surface, rising the local temperature and desorbing the adparticles lying within the illuminated area. A second pulse is emitted after a lapse of time allowing for the refilling of the depleted region and provoking a new desorption process. In this way the rate of refilling is measured and can be related to the chemical diffusion coefficient (diffusion due to the chemical potential arising from inhomogeneities in the density of adatoms on the surface). Another related technique is the linear (also laser) optical diffraction (LOD) which monitors the smearing out with diffusion of an optical grating pattern [1,51]. LOD covers a very large range of diffusivities, although always over macroscopic distances and time scales. Finally, we include in this section the photoemission electron microscopy, PEEM, which measured the spreading of an adsorbed layer when a coverage gradient is created (by the means of LITD or deposition mask) [5]. On the side of equilibrium methods, we should mention the fluorescence correlation spectroscopy (FCS) which measures the spontaneous fluctuations of molecule concentration in a well defined area of the surface [1]. The diffusion coefficient can be extracted from the decay with time of the fluctuation autocorrelation function [52]. We can approach FCS to the spin-echo technique, as we shall see later. Microscopic techniques One of the most widely used microscopic procedure for surface studies is the scanning tunneling microscopy, STM, based on the concept of quantum tunneling. STM instruments are made of a sharp and conducting tip whose position is controlled by a piezoelectric motor. When the distance between the tip and a conducting surface is within ∼1 nm, a tunneling current starts flowing. This current is kept constant by varying the position of the tip with respect to the surface: i.e applying a certain volt- 1.1. QENS spectroscopy applied to surface diffusion 11 age to the piezoelectric motor which regulates the movement of the tip. If the voltage is recorded while the tip sweeps the surface, the topography of the surface is imaged. STM is an extremely powerful method for analyzing tracer diffusion on conducting surfaces [5]. It is able to follow an adsorbed atom in its migration on the surface with a resolution of around 10 Å [5]. It is possible to extract hopping rates from a statistical analysis of series of images taken with the microscope [5, 47] and diffusion coefficients of the order 10−19 ∼10−16 cm2.s−1are measured [5]. It is worthwhile to mention as weel the atomic force microscopy, AFM: it is a modified STM microscope which pushes the adsorbed species with its tip and measures the force required to move the particle from one adsorption site to the neighboring site across the energy barrier [47]. Forces of the order of pico and nano-Newtons are accessible [53]. Magnitudes like the activation energy for diffusion or the binding energy between the adsorbate and the substrate are deduced from the integration of the forces over the diffusing path [47]. The friction parameter and a sketch of the potential energy surface can also been survey thanks to this method [46]. Other microscopic techniques like the field emission microscopy, FEM, measure the work function of a well localized small area of the surface: the density of electrons (in the form of an electric current) promoted from the conducting band of the substrate to the free state of the microscope metal tip under the influence of a high electric field [5]. Fluctuations in the density of adatoms related to diffusive processes affect the work function of the substrate and, hence, can be observed with the FEM method [47]. A similar idea is applied in field ion microscopy (FIM) which achieves images of surface with a resolution in the range of 3 ∼50 Å [5,47, 48] and measures diffusion coefficients of single metal adatoms of the order of 10−17 ∼10−15 cm2.s−1[48]. To summarize, microscopic techniques provide a wealth of information about adsorption processes on the surface with an atomic resolution. They can follow a tracer in their migration on the surface, describe their diffusive behavior and measure their diffusion coefficient and their mean square displacement. They also are sensitive to the binding energy of the adsorbate to the surface and to the activation barrier for lateral motion (translations). Nevertheless their applicability is still reduced to small adsorbates concentrations and slow diffusivities [5]: the typical atomic motion occurs at the picosecond scale which is orders of magnitude faster than the time windows affordable for microscopy techniques [16]. Furthermore, most of them require conducting surfaces to work properly. Finally, an important disadvantage is that the measurement process influences strongly the diffusion in the system: the equilibrium state of the adsorbed layer is seriously disturbed [5,16]. 12 Chapter 1. Quasi-elastic neutron scattering, applied to diffusion of adsorbed species on the surface Diffraction techniques The structure of the adsorbed layers, is studied by the means of diffraction techniques. Among them we can find the low energy electron diffraction (LEED). However, care must be taken with the study of physisorbed layers with LEED since the not very high current electron beam (∼1µA) can desorb weakly adsorbed particles [46]. Other techniques such as X-ray diffraction and neutron diffraction image the structure of the adsorbed layer in the Fourier reciprocal space of momentum transfer: the diffraction pattern. Information about the chemical composition of the surface can also be obtained. All this techniques reach perfectly the atomistic resolution (very high momentum transfer range). As a complement we can mention X-ray and neutron reflectometry supplying interesting details such as the surface roughness, morphology and growth processes [46]. Spectroscopy techniques Spectroscopy techniques yields information about the dynamics of adsorbates on the substrates. For instance, nuclear magnetic resonance (NMR) monitor the diffusion of ad-particles by measuring the spinlattice relaxation spectra [5]. The position of the peaks and their broadening give valuable information about the translational and rotational dynamics of adsorbed species [37, 54, 55]. On the other side, infrared spectroscopy (IR) allows to study in detail the internal degrees of freedom of molecules. In particular it encourages and supplies experimental data for the development of models on the coupling between the vibration-orientation-translation modes of adsorbed molecules [46]. Helium and neutron scattering Quasi-elastic helium scattering and neutron scattering are closely related techniques whose main exprimental function is the scattering function, as it will be explained later, in section 1.2.1. In both cases, the dynamics and the structural information about adsorbates on a surface is probed in the Fourier transformed space of energy/momentum transfer (except for helium and neutron spin-echo which can access the time/momentum transfer domain [16, 46, 56, 57]). In particular, the quasi-elastic broadening arising from the energy exchange between the probes and the diffusing particles, depends on the diffusive regime of the latter. These techniques are unique to survey the diffusive motion reaching the atomic resolution and covering the picosecond-nanosecond time window . Diffusion coefficients above 10−6cm2.s−1and over atomic distances can be measured [5]. In addition, we underline the non-destructive approach of these techniques: the equilibrium state of the adsorbed layer is only weakly perturbed by the scattering. It is precisely the fluctuations restoring the thermal equilibrium in the sample, which give rise to the energy distribution of helium or neutron scattered probes. The main distinction between helium and neutron probes stands in 1.1. QENS spectroscopy applied to surface diffusion 13 their very different interaction with matter. While neutrons are scattered by the nuclei [56, 58–60], helium atoms are sensitive to the electronic cloud of atoms. This is reflected in the interaction potential which takes the simple form of the Fermi pseudo-potential [58] (see section 1.2.1 and in Eq. 1.3) for neutrons and differs from the helium interaction potential with the atoms on the surface or in a molecule. Therefore, neutrons can distinguish between different atoms in a molecule, while helium atoms feel the electronic cloud of the whole molecule. Conversely, helium atoms are a highly surface sensitive probe while neutrons are well-known bulk probes. Nevertheless, this disadvantage of neutron probes is overcome with the use of large effective surface area substrates as exfoliated graphite (see the next section 1.1.2). A further disparity between neutron and helium scattering arising from the nucleus-neutron interaction lies in the nucleus scattering length bfor neutrons which is an operator acting on the neutron and nucleus spin [58, 61]. Thereby the scattering of neutrons by a collection of atoms has a twofold nature: the coherent scattering only changes the energy and the wave vector of the neutron whereas the incoherent scattering flips the neutron spin as well. The consequences of this peculiar feature of neutron-matter scattering endow neutrons with the capability of distinguishing between collective and individual diffusion. Conversely, helium scattering corresponds only to the coherent scattering of neutrons [1]. The theoretical models for diffusion are, in general, developed for the incoherent scattering (because of its easier interpretation in terms of individual atoms diffusive behavior) but can be extend to the coherent case [1,2]. Table 1.1: Comparison between the principal surface techniques and the neutron scattering techniques. Technique Time window Length resolution Observable processes Range of D [cm2.s−1] Remarks LITD ms< 100-1000 µm Collective or chemical diffusion 10−8-10−5Well suited for studying physisorption, adsorption of gases or large molecules on single crystals [5] Provoke surface damage [1]. Requires UHV conditions. LOD s - 100 s 1 µm Collective or chemical diffusion 10−15-10−7Applicable to a large number of adsorbate-substrate systems. Measures down to the monolayer coverage. STM ms< 10 Å Motion of individual atoms and molecules 10−19-10−16 Requires conducting surfaces. The tip-surface interaction affects the equilibrium state of the adsorbed elements. Ultra High Vacuum, UHV, conditions [48]. Direct space imaging. 14 Chapter 1. Quasi-elastic neutron scattering, applied to diffusion of adsorbed species on the surface FIM ms< 3-50 Å Tracking of trajectories of isolated diffusing species and small clusters [6] 10−17-10−15 Metal atoms and refractory metals [48]. Adsorbate/substrate with high binding energies [47]. Materials resisting very high electrical fields ∼ 108V.cm−1[6]. FEM (fluctuation mode) ps <100Å Adatom density fluctuations. 10−14-10−9Application of high electric field ∼107V.cm−1[6]. Diffusion and jumping of individual atoms NMR ps - 0.1 ms Local density of electronic states. Weak signal from surfaces [55]. Identification of specific adsorption sites on a substrate Requires metallic surfaces or nuclei with magnetic moment. Diffusion coefficients. QHAS ps - 10 ns 10 - 100 Å Structure of adsorbed layer and dynamics of adsorbed species >10−6Information in the Fourier domain of (Q,~ω) External (frustrated rotations and translations) of the adsorbed species Only measures coherent scattering. Requires ultra-high vacuum conditions. High surface sensitivity. QENS ps - 0.1 ms 10Å Structure of adsorbed layer and dynamics of adsorbed species. >10−6Information in the Fourier domain of (Q,~ω) Tunneling, internal (vibrations) and external (frustrated rotations and translations) modes of the adsorbed species Low surface sensitivity: substrates with high effective surface for adsorption. Flexible sample environment: do not require UHV. Extremely sensitive to hydrogen atoms. 1.1.2 Systems for a neutron surface study: carbon based molecules physisorbed on exfoliated graphite One of the strongest limitation for the application of neutrons to surface study is the lack of signal arising from the atoms in the surface or in the edges of the system [1]. Neutrons are well known bulk probes: their weak interaction with matter enhances their penetration depth (of the order of cm) [62]. As a result, surfaces are invisible to neutrons. This has some advantages and provides neutrons with the capability of accessing buried phases when coexistence between an adsorbed and a liquid phase exists [63]. But, it is certainly not the best suited set-up for surface diffusion studies. The advent of materials with a large specific area for adsorption, such as exfoliated graphite or 1.2. QENS spectroscopy and spectrometers 21 Detector � ki � kf Sample � Q Figure 1.4: Representation of the scattering of a neutron by a collection of atoms. The incoming neutron is represented by a plane wave characterized by a well defined wave vector ki, or equivalently by a well defined wavelength λi. The interaction between the neutron and the nuclei in the sample gives rise to spherical waves: the scattered neutron density of states is isotropic, in a first approximation. However, in the asymptotic limit, the spherical scattered waves describing the outgoing neutron can be approximated by plane waves with well defined wave vectors kf, depending on the momentum and energy exchange with the target system. The momentum transfer vector, Q, is the difference between the neutron wave vectors before and after the scattering (see Eq. 1.7). 22 Chapter 1. Quasi-elastic neutron scattering, applied to diffusion of adsorbed species on the surface Furthermore, it is worth to realize that the energy and the momentum transfer are related by the dispersion relation of the free neutron (Eq. 1.5) [80]: ~2 2mQ2= 2E0+~ω−2p(~ω+E0)E0cos φ. (1.8) Eq. 1.8 implies that neutron scattering is restricted to a region of the energymomentum transfer space (Q,~ω)(see Fig. 1.5), which depends on the the incident energy of the neutrons E0(or in its initial wavelength). Volume 98, Number 1, January-February 1993 Journal of Research of the National Institute of Standards and Technology Q2= ko 2+ k2-2kokcos(). (5) Converting wave-vectors to energies (see Appendix A), and using Eq. (3), Eq. (5) may be rewritten as follows: h2Q2/2m =2EMh -2VEo(Eo-hw) cos(qS). (6) In Fig. 17 we show accessible regions for two choices of the incident energy. Detectors are assumed to fill the angular range from 50 to 140°. The accessible region gets smaller as Eo is decreased. In particular the accessible range of Q for elastic scattering (hw = 0) is reduced since in this case Eq. (6) simplifies to the well-known (Bragg) relationship Q =2ko sin(k/2). (7) and the chopper. The chopper contribution is independent of distance but the wavelength spread from the monochromator translates into a time spread which increases with distance. The net result is that the width of the time distribution in the unscattered beam, at a point distantx from the chopper, may be written as follows: (8) where or, measures the burst time of the chopper and A, is the spread in the reciprocal velocity distribution of neutrons in the beam. The time distribution for a two-chopper spectrometer may be similarly written [30]. In this case the time spread is an appropriate average of the burst times a, and o2 for the two choppers: A decrease in Eo also means a proportionate reduction in the maximum possible energy transfer in neutron energy loss. The improved resolution that results as Eo is decreased, i.e., as Au is increased, is a direct consequence of the contraction in accessible (Q,w) space. -6 -4 -2 0 2 (meV) Fig. 17. Plots of the accessible region in (Q,co) space for neutrons of wavelength 5 and 10 A (energy 3.272 and 0.818 meV, respectively). The minimum and maximum scattering angles are 50 and 1400. There is no (theoretical) limit to the energy transfer in neutron energy gain. 5.2 Resolution and Intensity An important contribution to the overall energy resolution of a TOF spectrometer arises from the spread in the time distribution of neutrons in the incident beam. In the FCS there are two independent sources of incident beam time spread: the monochromator 2 = 2 O22/(O,2 + (72). (9) The reciprocal velocity spread o,. is given by =(a2 = + 0J2)/L, 22(10) where L12is the distance between the two choppers, and the distance x is measured from an "effective chopping point" which is located between the choppers at a distance L1o=L 12£72/(a72+ £22) (11) from the first chopper. These expressions may be used to calculate the time distribution in the incident beam of the DCS. Important additional contributions to the overall energy resolution of a TOF spectrometer stem from flight path uncertainties due to the size of the sample and the geometry of the detectors [5]. The count-rate in a TOF experiment is determined by the intensity of neutrons at the sample position, the scattering properties of the sample, and the detecting efficiency of the array of detectors. To some extent the individual experimenter can select the scattering properties of the sample such as its size and shape, and how it is contained. The efficiency of the detector array can generally be increased, without degrading resolution, simply by purchasing more detectors; thus it largely depends on the available budget..The principal challenge in designing a TOF spectrometer is therefore to optimize the "front end" of the instrument, and indeed considerable efforts have been devoted to this task for both of the CNRF spectrometers. 84 0,2(X) = of 2 +x2 O"r 2, Figure 1.5: Attainable (Q,~ω)areas for incoming neutrons of 5 Å and 10 Å wavelength respectively. Graph taken from Ref. [80]. The scattering function and the intermediate scattering function Finally, inserting the Fermi pseudopotential (Eq. 1.3) into Fermi’s golden rule (Eq. 1.2) leads to the differential scattering cross-section as function of the momentum and energy transfer, which are measured by the standard spectroscopic instruments [58,60,81]: d2σ dΩdω =k k0 1 2π~ N X j,j0=1 b∗ jbj0Z∞ −∞ dt exp(−iωt)exp(−iQ·rj0(t)) exp(iQ·rj(0)), (1.9) For a classical system whose equilibrium state follows the Boltzmann statistics [56,59,75], the averages over the thermally distributed spatial states and over the spin states can be performed independently [56,58,81]. It follows that the differential scattering cross section reads as the product of two terms. The first term is the product of the scattering lengths of nuclei jand j0averaged 1.2. QENS spectroscopy and spectrometers 23 over the neutron and the target system nuclear spin states. It contains all the information about the neutron-matter interaction: hb∗ jbj0i=X µX s PµPshs|hµ|b∗ jbj0|µi|si.(1.10) The second term is the Fourier transform in time of a function averaged only over the thermal distribution of the translational-rotational states4of the target system, the so-called correlation function [58]: Yj,j0(Q, t) = hexp(−iQ·rj0(t)) exp(iQ·rj(0))i =X λ Pλhλ|exp(−iQ·rj0(t)) exp(iQ·rj(0))|λi.(1.13) The factorization of the differential cross-section stems from the weak interaction between neutrons and matter and, in particular, from the impulsive scattering due to the neutron-nuclear interaction [58,75]. From this perspective, the scattering can be considered as a weak perturbation to the equilibrium state of the target, described by Pλ[56, 58]. Thus, the differential neutron scattering cross-section can be studied in the framework of the linear response theory: it measures the response of the target system, i.e. the thermal fluctuations restoring the equilibrium after the perturbation caused by the scattering of the neutron [56,58,75]. In the neutron scattering terminology, the response function is called the scattering function or scattering law [58]: S(Q, ω) = 1 NX j,j0Z∞ −∞ dt exp(−iωt)Yj,j0(Q, t).(1.14) It is the main function measured with backscattering or time of flight spectroscopy. It portrays the thermal fluctuations of the target system, its structure and dynamics, in the Fourier transformed space of the momentum transfer Q and the energy transfer ~ω[58]. From the point of view of diffusion, it contains valuable information about the diffusive behavior of the elements adsorbed on 4Note that rj0(t)is the Heisemberg operator yielding the position of the j0-th nucleus obtained through the Hamiltonian Hof the target system [58,60]: exp −it ~Hexp(−iQ·rj0) exp it ~H= exp(−iQ·rj0(t)).(1.11) When t= 0, we have: exp −it ~Hexp(−iQ·rj) exp it ~H|t=0 = exp(−iQ·rj(0)) (1.12) 24 Chapter 1. Quasi-elastic neutron scattering, applied to diffusion of adsorbed species on the surface the surface [1,16,57]. The scattering function reads as the time Fourier transform of the intermediate scattering function (ISF) [60]: I(Q, t) = 1 N N X j,j0=1 Yj,j0(Q, t),(1.15) which is the summation of the correlation functions for all the elements in the system, as defined in Eq. 1.13. It is also an experimental function but should be measured by means of special spectroscopy techniques such as neutron spin-echo spectroscopy [1,16]. In contrast to the scattering function, it draws a picture of the dynamics of the system in time, even if the spatial information is still in the reciprocal space Q. 1.2.2 QENS dynamical and spatial range QENS techniques survey an energy-time window of meV-µeV/10−10-10−12s and a space windows reaching the atomic scale (from 0.1 to 10 Å). [56,59,82]. Hence, the dynamical region (dynamics and structure) revealed by QENS techniques matches the typical times and length scales for molecules diffusing on surfaces at technically interesting temperatures (between 100K and 300K) [13, 56]. Diffusing particles in the sample exchange small amounts of energy with neutron, lying in the quasi-elastic range of ±2meV [56]. As a result, the elastic peak broadens and the continuous distribution of energies arising from this "quasi-elastic" scattering is characterized by its the half width half maximum (HWHM): the so-called quasi-elastic broadening (see Fig. 1.6). Equivalently, the broadening of the elastic peak in the momentum/energy transfer domain (Q,~ω)translates into a decay with time of the intermediate scattering function lying in the momentum transfer/real time phase space (Q, t). The time constant characterizing this time decay is the Fourier transformed in time of the quasi-elastic broadening in energies. As a result, the dependency on the momentum transfer of the QENS broadening, or analogously of the time decay, is the experimental signature of the diffusive processes taking place in the system [1, 57,83–85]. The major part of the models aim to predict a determinate shape for the quasi-elastic profile given a particular type of diffusion. 1.2.3 QENS spectrometers As we have now briefly reviewed the basic theory for neutron scattering and we have defined the main physical quantities: energy and momentum transfer, 1.2. QENS spectroscopy and spectrometers 25 2 meV -4 meV 2!/a Q !E Cut at constant Q 10-5 10-4 10-3 S(!E) [arb. units] -4 -3 -2 -1 0 1 2 !E [meV] Constant Q QENS-broadening: !(Q) I(Q1,t) I(Q2,t) I(Q3,t) I(Q4,t) Time constant: !-1(Q)=" #(Q) Figure 1.6: Left panel: typical QENS scattering law, S(Q,∆E), where we plot intensity versus momentum transfer Qand energy transfer ∆E=~ω. Right panel: cut at constant Qof the scattering function. The quasi-elastic energy profile is highlighted by the green solid line Bottom panel: intermediate scattering function at different Q values. The so-called quasi-elastic window (left panel) is the green area around the elastic band, at ∆E= 0, in red (the most intense signal). The shape of the quasielastic region is the characteristic fingerprint of a determined diffusive regime. This translates into a Qdependent time constant of the ISF (bottom panel). The form of the scattering function reproduces the energy/momentum transfer domain (Q,~ω) which is attainable with ToF spectroscopy for a given initial wave length (see Fig. 1.5 and Eq. 1.8). 26 Chapter 1. Quasi-elastic neutron scattering, applied to diffusion of adsorbed species on the surface together with the chief experimental functions, we can undertake the description of the QENS techniques. Above all we will explain the way in which the physical quantities are measured in every technique and the parameters determining the accessible dynamical window. There are two basic parameters shaping the accessible phase space (Q,~ω). First of all, the instrumental resolution sets the lower limit of the time/energy window: i.e. the slowest or less energetic processes visible by the technique in particular experimental conditions. Conversely, the neutron incoming wavelength gives the upper limit of the spectral range (see Eq. 1.8): i.e. the fastest or most energetic processes observable under given experimental conditions. We will start with the description of time of flight and backscattering spectroscopy, which correspond to standard spectroscopic techniques probing the energy/momentum transfer phase space. We will finish with the neutron spin-echo technique whose physical principle is completely different from the latter techniques, giving access to the time/momentum transfer domain. Time-of-flight spectroscopy Time-of-flight (ToF) spectroscopy is a powerful technique for investigating dynamical processes in condensed matter [80]. Its principal strength lies in the wide domain of (Q,~ω)that ToF instruments can access [80,86]. There are two main reasons which open the access to a large spectral range. First of all, the resolution covers a wide spread of energies from 10 µeV up to 1000 µeV [87]. The second reason is that ToF instruments can work under any initial wavelength condition (from cold neutrons of wavelength 30 Å ∼3 Å to very hot neutrons of wavelength 1 Å ∼0.2 Å ), provided that they are equipped with the appropriate mechanical devices for chopping the beam [86]. As a result, ToF spectroscopy is a very flexible technique. It can, in one side, reach the time domain of magnetic and phononic low energy excitations or tunneling processes (in the range of 2.5 meV up to 10 meV), which falls out of the energy range of high energy triple axis spectrometers [80]. But, on the other side, it also covers perfectly the so-called quasi-elastic energy range [80], where translational and rotational dynamics of atoms and molecules in condensed matter takes place (with characteristic diffusion coefficients for translations between 10−12m2.s−1and 10−8m2.s−1[87], or residence times shorter than 10−9s for reorientations [27]) . Hence ToF spectroscopy is one of the best suited techniques for exploring surface diffusion with neutrons. Time-of-flight principle Neutrons extracted from a moderator source, are lead in most cases to the instruments by neutron guides. If the slits that the neutron beam encounters along its path are large if compared with the 1.2. QENS spectroscopy and spectrometers 27 neutron wavelength, then, neutrons can be treated as classical particles and travel with a well defined velocity [86]. The ToF method relies in the behavior of neutrons as classical particles for defining the energy of the incident neutron beam (the monochromatization of the incident beam) and for measuring the energy transfer between the neutron and the sample. A neutron of mass mn with a well defined velocity, needs a certain time ∆tto cover a distance L:the time of flight [56,86]. The connection between the neutron wavelength and its velocity, the De Broglie relation mnv=h/λ [78], yields: ∆t=L v=Lmnλi h.(1.16) Eq. 1.16 states the key for extracting a monochromatic beam out of the neutron white beam traveling in the guide by means of a ToF energy selector made of a pulsing and a monochromator chopper system. The principle is illustrated in the top panel of Fig. 1.7 and 1.8: the distance between the chopper devices, Lch, and their opening frequencies are synchronized in order to achieve a precise initial energy of the neutron pulses: if Lch and ∆tch are settled fix, then, only the neutrons traveling with the appropriated wavelength λiwill pass through the pulsing and the monochromator chopper (see Fig 1.8). The outgoing pulse will only contain neutrons with the desired wavelength, i.e. traveling with the same velocity. However, some ToF instruments, like IN5 (ILL) [88] make use of a single crystal monochromator for selecting the incoming wave length, and chop the incoming beam afterwards with a Fermi chopper (see bottom panel of Fig. 1.7). On the other side, the link between the time-of-flight of a neutron and its kinetic energy is straightforward: E=1 2mnv2=~2k2 2mn =h2 2mnλ2=1 2mnL ∆t2 .(1.17) The exchange of energy during the scattering is inferred from the scattered neutron time of flight: a change in the neutron wavelength due to inelastic scattering will be immediately translated into a change of time-of-flight and will be recorded [80]. The ToF instruments which apply the time of flight principle for selecting the initial wavelength and measuring the energy transfer are called direct geometry ToF spectrometers (see the top panel in Fig. 1.8). In contrast, there exist ToF instruments where single crystals analyze the final energy of the neutrons according to Bragg law (see the lower panel in Fig. 1.8). These are the indirect geometry ToF spectrometers. They can attain a very large dynamical window (very large energy transfer up to 500 meV [27]) and are particularly suited for performing molecular vibrational spectroscopy with very high resolution [86]. 28 Chapter 1. Quasi-elastic neutron scattering, applied to diffusion of adsorbed species on the surface Sample 24 Tasso Springer: J J / f? SC / D ,,p-o N M Fig. 7. "90~ '' (at FRJ-2 Dido-reactor). R reflector crystal; N neutron conducting guides; M monochromator crystal with piston motor for producing a Doppler shift of energy E0; S sample; SH shielding; SC arrangement of selector crystals; D detectors. Bragg angle of selector and analyzer approximately 90 ~ (From Alefeld [29]) Z( t ) Ir~ quasielastic II line A inelastic / I \ /~ "~ / (~= u~ J.~ _L" A'tp Axp Fig. 8. Diagram for a time-of-flight spectrometer. Azp pulse length; ~. pulse repetition interval; v=v o corresponds to elastic scattering. Z(t) time-of-flight spectrum. Shaded: Region of frame-overlap 24 Tasso Springer: J J / f? SC / D ,,p-o N M Fig. 7. "90~ '' (at FRJ-2 Dido-reactor). R reflector crystal; N neutron conducting guides; M monochromator crystal with piston motor for producing a Doppler shift of energy E0; S sample; SH shielding; SC arrangement of selector crystals; D detectors. Bragg angle of selector and analyzer approximately 90 ~ (From Alefeld [29]) Z( t ) Ir~ quasielastic II line A inelastic / I \ /~ "~ / (~= u~ J.~ _L" A'tp Axp Fig. 8. Diagram for a time-of-flight spectrometer. Azp pulse length; ~. pulse repetition interval; v=v o corresponds to elastic scattering. Z(t) time-of-flight spectrum. Shaded: Region of frame-overlap 24 Tasso Springer: J J / f? SC / D ,,p-o N M Fig. 7. "90~ '' (at FRJ-2 Dido-reactor). R reflector crystal; N neutron conducting guides; M monochromator crystal with piston motor for producing a Doppler shift of energy E0; S sample; SH shielding; SC arrangement of selector crystals; D detectors. Bragg angle of selector and analyzer approximately 90 ~ (From Alefeld [29]) Z( t ) Ir~ quasielastic II line A inelastic / I \ /~ "~ / (~= u~ J.~ _L" A'tp Axp Fig. 8. Diagram for a time-of-flight spectrometer. Azp pulse length; ~. pulse repetition interval; v=v o corresponds to elastic scattering. Z(t) time-of-flight spectrum. Shaded: Region of frame-overlap 24 Tasso Springer: J J / f? SC / D ,,p-o N M Fig. 7. "90~ '' (at FRJ-2 Dido-reactor). R reflector crystal; N neutron conducting guides; M monochromator crystal with piston motor for producing a Doppler shift of energy E0; S sample; SH shielding; SC arrangement of selector crystals; D detectors. Bragg angle of selector and analyzer approximately 90 ~ (From Alefeld [29]) Z( t ) Ir~ quasielastic II line A inelastic / I \ /~ "~ / (~= u~ J.~ _L" A'tp Axp Fig. 8. Diagram for a time-of-flight spectrometer. Azp pulse length; ~. pulse repetition interval; v=v o corresponds to elastic scattering. Z(t) time-of-flight spectrum. Shaded: Region of frame-overlap 24 Tasso Springer: J J / f? SC / D ,,p-o N M Fig. 7. "90~ '' (at FRJ-2 Dido-reactor). R reflector crystal; N neutron conducting guides; M monochromator crystal with piston motor for producing a Doppler shift of energy E0; S sample; SH shielding; SC arrangement of selector crystals; D detectors. Bragg angle of selector and analyzer approximately 90 ~ (From Alefeld [29]) Z( t ) Ir~ quasielastic II line A inelastic / I \ /~ "~ / (~= u~ J.~ _L" A'tp Axp Fig. 8. Diagram for a time-of-flight spectrometer. Azp pulse length; ~. pulse repetition interval; v=v o corresponds to elastic scattering. Z(t) time-of-flight spectrum. Shaded: Region of frame-overlap 24 Tasso Springer: J J / f? SC / D ,,p-o N M Fig. 7. "90~ '' (at FRJ-2 Dido-reactor). R reflector crystal; N neutron conducting guides; M monochromator crystal with piston motor for producing a Doppler shift of energy E0; S sample; SH shielding; SC arrangement of selector crystals; D detectors. Bragg angle of selector and analyzer approximately 90 ~ (From Alefeld [29]) Z( t ) Ir~ quasielastic II line A inelastic / I \ /~ "~ / (~= u~ J.~ _L" A'tp Axp Fig. 8. Diagram for a time-of-flight spectrometer. Azp pulse length; ~. pulse repetition interval; v=v o corresponds to elastic scattering. Z(t) time-of-flight spectrum. Shaded: Region of frame-overlap Time Distance Time-of-flight/Energy Intensity/ Neutron counts Pulsing Chopper Frame overlap Chopper Monochromatizing Chopper Sample Elastic line Inelastic spectrum Quasielastic line Detector Continuous neutron flux Detector Fermi Chopper 24 Tasso Springer: J J / f? SC / D ,,p-o N M Fig. 7. "90~ '' (at FRJ-2 Dido-reactor). R reflector crystal; N neutron conducting guides; M monochromator crystal with piston motor for producing a Doppler shift of energy E0; S sample; SH shielding; SC arrangement of selector crystals; D detectors. Bragg angle of selector and analyzer approximately 90 ~ (From Alefeld [29]) Z( t ) Ir~ quasielastic II line A inelastic / I \ /~ "~ / (~= u~ J.~ _L" A'tp Axp Fig. 8. Diagram for a time-of-flight spectrometer. Azp pulse length; ~. pulse repetition interval; v=v o corresponds to elastic scattering. Z(t) time-of-flight spectrum. Shaded: Region of frame-overlap 24 Tasso Springer: J J / f? SC / D ,,p-o N M Fig. 7. "90~ '' (at FRJ-2 Dido-reactor). R reflector crystal; N neutron conducting guides; M monochromator crystal with piston motor for producing a Doppler shift of energy E0; S sample; SH shielding; SC arrangement of selector crystals; D detectors. Bragg angle of selector and analyzer approximately 90 ~ (From Alefeld [29]) Z( t ) Ir~ quasielastic II line A inelastic / I \ /~ "~ / (~= u~ J.~ _L" A'tp Axp Fig. 8. Diagram for a time-of-flight spectrometer. Azp pulse length; ~. pulse repetition interval; v=v o corresponds to elastic scattering. Z(t) time-of-flight spectrum. Shaded: Region of frame-overlap 24 Tasso Springer: J J / f? SC / D ,,p-o N M Fig. 7. "90~ '' (at FRJ-2 Dido-reactor). R reflector crystal; N neutron conducting guides; M monochromator crystal with piston motor for producing a Doppler shift of energy E0; S sample; SH shielding; SC arrangement of selector crystals; D detectors. Bragg angle of selector and analyzer approximately 90 ~ (From Alefeld [29]) Z( t ) Ir~ quasielastic II line A inelastic / I \ /~ "~ / (~= u~ J.~ _L" A'tp Axp Fig. 8. Diagram for a time-of-flight spectrometer. Azp pulse length; ~. pulse repetition interval; v=v o corresponds to elastic scattering. Z(t) time-of-flight spectrum. Shaded: Region of frame-overlap open close Monochromator Time-of-flight/Energy Intensity/ Neutron counts Sample Time Distance Figure 1.7: Time-distance diagram. The distance traversed by neutrons is plotted against the time. The slope is given by the velocity of the neutron. Top panel: The pulsing chopper cuts the continuous neutron flux into neutron pulses. Since the pulsing chopper has a finite opening time, there exists a certain wavelength dispersion in the neutron pulse since not all the neutrons travel with exactly the same velocity (different slopes in the distance-time diagram). This initial dispersion in energies gives rise to the instrumental resolution. Next, the overlap frame chopper prevents the slowest neutrons from one pulse to mix at the detector with the fastest neutrons of next pulse (dotted red lines). Later, the monochromator chopper, rotating in synchrony with the pulsing chopper, allows the passage only to the neutrons with the desired initial energy while it absorbs the rest of neutrons traveling faster or slower. Bottom panel: The initial wavelength is selected with a monochromator. Afterwards, beam is pulsed with a Fermi chopper. In any case, a monochromatic neutron pulse reaches and is scattered by the sample. Finally, the detector counts the number of neutrons arriving at a certain time: the time-of-flight spectrum. Elastically scattered neutrons constitute the so-called elastic line. Neutrons winning energy from the target will arrive before the elastic neutrons while neutrons loosing energy in the target will arrive later. 1.2. QENS spectroscopy and spectrometers 29 Lch Ls Ld Pulsing Chopper Monochromating Chopper Sample Detector Bank Time Distance ∆tch ∆ts ∆td � kf � kf � kf � kf � ki � ki � ki � ki � ki � ki � kf −� ki � ki � Q � kf −� ki � ki � Q � kf −� ki Energy Ei=1 2mn�Ls ∆ts�2 Ef=1 2mn�Ld ∆td�2 Ls Ld Monochromator Fermi Chopper Sample Detector Bank Time Distance ∆ts ∆td � kf � kf � kf � kf � ki � ki � ki � kf −� ki � ki � Q � kf −� ki � ki � Q � kf −� ki Energy Ef=1 2mn�Ld ∆td�2 Continuous white beam λi=2dsin(2θ 2) Ei=h2 2mnλ2 i Pulsing Chopper Monochromator Chopper Sample Detector Bank Analyzer crystals Lch Ls Time Distance ∆tch ∆ts � kf � kf � kf � ki � ki � ki � ki � ki � ki � ki � kf � Q −� ki � ki � kf � Q −� ki � ki � kf � Q −� ki Figure 1.8: Top and medium panel: direct ToF spectrometer. In the first case the time of flight principle is at work for selecting the incoming wavelength. In the second case a single crystal is used to obtain a monochromatic beam. Bottom panel: indirect geometry ToF spectrometer. The first part devoted to the selection of the initial neutron wavelength is identical to the direct ToF spectrometer. Conversely, analyzer crystals are used for estimating the kinetic energy of the scattered neutrons. 30 Chapter 1. Quasi-elastic neutron scattering, applied to diffusion of adsorbed species on the surface In any case, for both types of spectrometers, it follows from the time-offlight measurement principle that ToF spectroscopy requires pulsed beams to work properly. Neutrons in a continuous and monochromatic incoming beam are indistinguishable [86]. Pulsing the beam permits to label each neutron and assign it a characteristic initial time: by definition, all the neutrons traveling together in a monochromatic pulse have the same initial time. Conversely, they differ in their arrival time to the detector bank, which depends on wether they have gained, lost or conserved their energy when scattered by the sample. In pulsed neutron sources, the emission of neutrons is already performed into packets and the source itself acts as an initial clock reference for the neutrons time of flight [86]. Even though, choppers are still useful in spallation sources: the source acts as the first chopper, but monochromatization requires, at least, a system of two choppers. Conversely, when the neutron source is continuous, like in a nuclear reactor, the incoming neutron beam must be chopped. For this purpose, two different mechanical devices exist: in the first case the pulsing is done by the means of a collimator turning around an axis perpendicular to the direction of the beam, this is a Fermi chopper [80,86]. The beam is already monochromatized before reaching the Fermi chopper with a single crystal in the appropriate orientation for diffracting the desired initial wavelength (see lower panel of Fig. 1.7). In a second case, the beam passes through absorbing disks with transparent slits turning in the plane perpendicular to the beam direction. These are the disks choppers [80,86]. This second methods allows to monochromatize and pulse the beam simultaneously (see top panel of Fig. 1.7 and Fig. 1.8). It turns out that the dispersion in the incoming wavelengths δλ/λ is tuned by the dimensions of the opening αand the rotatinal velocity ωof the chopper. These two parameters tailor the shape of the neutron pulse in time and space. For instance, in the Fermi chopper the time width of the pulse is [86]: τ=α ω,(1.18) which gives rise to a wavelength dispersion at any time tof (see Eq. 1.16): δλ λ=τ t⇒δλ λ=α ωt.(1.19) The typical chopper rotating frequencies for achieving a 1%relative error in the wavelength accuracy are of the order of 6000 rpm [86]. The total resolution of a ToF instrument comprehends the wavelengths dispersion of the incoming neutron pulse and the uncertainty related to the flying path [86]. The latter includes, the beam divergence, the thickness of the sample and the detector size [86,87]. Many improvements on the resolution function have raised the experimental resolution to the order of 10 µeV [56,86]. For instance the addition of choppers avoids the frame overlap (see top panel in Fig. 1.7): i.e the simultaneous arrival to the detector bank of the fastest neutrons of one pulse with 1.2. QENS spectroscopy and spectrometers 37 If the scattering is elastic then neutrons travel through the coils with exactly the same velocity: v1=v2and the full polarization of the beam, P=hcos(∆φ)i should be recovered (P= 1) at the end of the second coil because ∆φ=φ1−φ2=γZL 0 Bdl 1 v1−1 v2.(1.25) Conversely if the scattering is inelastic, then, the velocities of the neutrons in each coil are different: v16=v2. Consequently, a loss of the polarization will be observed (see Fig. 1.12). As a result, NSE measures the polarization of the beam at the end of the second coil and yields the intermediate scattering function I(Q, t)[16,93] thanks to the following relation: P=Rd~ωS(Q,~ω) cos(~ωt) Rd~ωS(Q,~ω)= Re I(Q, t) I(Q,0).(1.26) The use in NSE of Larmor precession as an internal clock of each neutron, not only extends the probing power of neutrons to time scales of picosecond maintaining a sub-µeV resolution [16] but it also decouples the energy resolution from the momentum transfer resolution. The ability of encoding the velocity information of neutrons in the number of Larmor precessions undergone by their magnetic moment of the neutron in its way through the coils, free us to select the initial wavelength. As a result, NSE overcomes the conflict between resolution and intensity thanks to an intelligent use of neutrons properties. The neutron spin-echo spectrometer IN11 at ILL IN11 is one of the spin-echo spectrometers at the ILL and it is located in the ILL7 guide hall at the end of the cold guide H141. It can access a very broad range of incoming wavelengths from 3.1 Å to 12 Å and can achieve a Fourier time range spanning over three orders of magnitude (from 0.2 ps to 3 ns) for an incoming wavelength of 4 Å and using the IN11C configuration (see Fig. fig:IN11). This corresponds to an energy window of 2×10−40.15 meV. The momentum transfer range for the same initial wavelength is of 0.02 - 2.7 Å−1. Finally its resolution in terms of the relative error in the wavelength, ∆λ/λ is of 15%under this conditions. Conclusion: why neutrons? To conclude, we have seen that there exist several experimental techniques for surveying surface diffusion in all the distance and times scales. However, we can not designate one of them to be the perfect and the most complete method, since all of them present limitations, advantages and disadvantages. 38 Chapter 1. Quasi-elastic neutron scattering, applied to diffusion of adsorbed species on the surface Polarizer � B � B 1st coil z xy z x � B x z x 2nd coil �v z z x z z � B Analyser φ(vi) φ(vf=vi) Polarizer � B � B 1st coil z xy z x � B x z x 2nd coil �v z x z z � B Analyser φ(vi) z φ(vf�=vi) Figure 1.12: Illustration of the spin-echo principle. The target is represented by the star. Left panel: elastic process in which the full polarization of the beam is recovered at the end of the second coil. Right panel: inelastic scattering where a loss in polarization is recorded, due to the difference of the velocities of the neutrons in the first and second coil. 1.2. QENS spectroscopy and spectrometers 39 Figure 1.13: Scheme of the IN11 spectrometer at ILL, taken from www.ill.eu. Table 1.2: Summary of the chief characteristics of QENS techniques: the energy resolution and the energy transfer, ~ω, which give rises to a finite time window. We also include the reciprocal range covered and its corresponding distances in the direct space. Finally we enumerate the physical mechanisms which are visible under the experimental conditions. The numerical indications are taken from Refs. [27,56,72, 86,87,89,95]. Technique Energy resolution/ ~ω Q[Å−1]/ Observable Instruments time window distance [Å] processes Direct Time-of-flight 1000 - 10 µeV < 3 meV ≤2.6 Å−1Tunneling rotation IN4, IN6, IN5 10−13 - 10−9s 2 Å <Diffusion coefficients: at ILL 10−810−4cm2.s−1 Molecular rotation Indirect Time-of-flight 16 - 1 µeV <500 meV ≤2.6 Å−1External modes OSIRIS (ISIS, UK) 10−13 - 10−9s 2 Å <of adsorbed layers Vibrational spectroscopy: 10−14-10−12s. Backscattering 1 - 0.1 µeV -30 µeV - 30 µeV 0.1 - 2.0 Å−1Diffusion IN16 (ILL) 100 ps - 0.01 µs 3 - 60 Å Molecular reorientation Spin-echo ∼10 neV 0.2 µeV - 0.15 meV 0.3 - 3 Å−1Translational diffusion IN11 (ILL) 10 ps - 100 ns Rotational diffusion 40 Chapter 1. Quasi-elastic neutron scattering, applied to diffusion of adsorbed species on the surface In our case, we seek for understanding the diffusion of physisorbed aromatic rings compounds (benzene, naphthalene or pyrene) adsorbed on the basal plane of graphite. Despite of their bulk character, neutrons turn to be an extremely well suited probe for this diffusion study. First of all, QENS techniques match perfectly the time and distance scale on which relevant aspects of diffusion occur. Secondly, neutrons can distinguish between atoms in the molecule and they are extremely sensitive to the presence of hydrogen. It is very difficult to spot such a light atom with other techniques such as photons or He atoms. Finally, the use of exfoliated compressed graphite, Papyex, allows to adsorb a sufficient amount of molecule in order to have a reliable quasi-elastic signal arising from the adsorbates. All this reasons encourages us to undergo a neutron study of the diffusion of aromatic compounds adsorbed on the basal plane of graphite. We shall see in the next chapter how theory and MD simulations enables to understand the experimental data with great detail. Chapter 2 Theoretical background for diffusion studies with neutron scattering This chapter deals with the theoretical formalism for understanding neutron scattering data arising from diffusive process of surface adsorbed species. The Van Hove formalism, enlarges the concept of the pair correlation function to space and time. It is a powerful theoretical tool for understanding the neutron differential scattering cross-section in terms of the diffusive regime of the atoms in the sample. Furthermore, it also gives a physical meaning to one of the most striking features of neutron scattering: the coherent and incoherent scattering. With this solid theoretical framework, it is possible to predict the shape of the differential neutron cross-section, and in particular of the quasielastic line, according to the behavior of the diffusing system. However, even if the development of a reliable theoretical support is crucial for the application of neutron scattering to diffusion studies, the recent improvement in computing power play an essential role. The capability of performing molecular dynamics (MD) simulations with large systems and realistic force fields fosters the interpretation of experimental data beyond the traditional analytical models. It contributes to building up new approaches and tackle the underlying complexity in diffusive processes. The chapter is organized as follows. First of all, we introduce the Van Hove formalism and its chief function: the Van Hove correlation function which contains all the dynamical information about the system. Afterwards, we review the most classical models builded based on this formalism, highlighting the fingerprint of the different diffusive regimes in the experimental data. The final 42 Chapter 2. Theoretical background for diffusion studies with neutron scattering section is devoted to the MD simulations: the description of the main methods and the role it plays for the interpretation of neutron scattering experimental data. 2.1 The Van Hove formalism In the previous chapter we saw how neutron scattering dress an atomic description of the dynamical processes and the structural properties of the target. Besides, the weak interaction of neutrons with matter, reduced to the nuclear scattering in the case of non-magnetic systems, allows the factorization of the differential scattering cross-section. In a first approximation, the neutron experimental data are the product of a term containing all the information about the neutron interaction with the nuclei in the system (the scattering length b) and a function related to the dynamics and structure of the system: the scattering function. Neutron scattering perturbs slightly the thermal equilibrium state of the system. Following the linear response theory, and more precisely the fluctuation-dissipation theorem, the scattering function images in the reciprocal space of momentum and energy transfer (Q,~ω)the thermal fluctuations restoring the equilibrium of the system. The Van Hove formalism, supplies the theoretical tools to link the scattering function, and hence the response of the system to a weak perturbation, with the diffusion of atoms or molecules adsorbed on the substrate. 2.1.1 The Van Hove correlation function The main function in the center of the formalism is the Van-Hove generalized correlation function [75], G(r, t), defined as the probability of finding a particle at a position, rat time, t, provided there was a particle (the same or a different one) at the origin, r=~ 0, at t= 0 [58,60,72,75]: G(r, t) = 1 N N X j,j0=1 Zhδ(rj(0) −r0)δ(rj0(t) + r−r0)idr0,(2.1) We would like to stress that in this chapter we only treat classical systems: the Heisemberg operators for the position of the particles jand j0,rj(0) and rj0(t), commute [57,60]. On the contrary, if the system presents a quantum behavior, the position operators at different times would not commute anymore1. As a 1The Baker-Hausdorff or disentangling theorem helps to deal with operations of noncommutative operators [57]. If Aand Bare two operators, then the BH theorem states: exp Aexp B= exp([A, B]/2) exp(A+B). 2.1. The Van Hove formalism 43 result, the Van Hove correlation function would have an imaginary part, arising from the nonzero commutator between rj(0) and rj0(t). A complex correlation function is the signature of a quantum system [57,75]. The delta function properties2link the Van-Hove correlation function with the previously described neutron scattering function and the intermediate scattering function, ISF. Indeed, G(r, t)reads as the space-momentum transfer Fourier transform of the intermediate scattering function, ISF (see Eq. 1.15) [60,75,96]: G(r, t) = 1 (2π)3ZI(Q, t) exp(−iQ·r)dQ,(2.2) or, equivalently, it is related through the Fourier transform in time–energy and space to the scattering law [60]: G(r, t) = ~ (2π)3Z Z exp(−i(Q·r−ωt))S(Q, ω)dQdω. (2.3) If we split the summation running over all the atoms in the system in Eq. 2.1, function G(r, t)can be separated into two contributions [75]: (i) The self correlation function Gs(j=j0) [75]: Gs(r, t) = 1 N N X jZhδ(rj(t)−r0)δ(rj(0) + r−r0)idr0,(2.4) which gives the probability of finding the j-th atom at position rat time tif this atom was at the origin at time t= 0. (ii) The distinct correlation function Gd, where j6=j0[75]: Gd(r, t) = 1 N N X j6=j0Zhδ(rj(t)−r0)δ(rj0(0) + r−r0)idr0,(2.5) probes the probability of finding the j-th atom at position rat time tprovided a different atom, j0, was at the origin at time t= 0. Adding together both functions, we recover the original correlation function: G(r, t) = Gs(r, t) + Gd(r, t).(2.6) and [A, B]is a complex number. In the case of the Van Hove correlation function, the non-commutative operators Aand Bare rj(0) and rj(0). Since [rj(0),rj(0)] is a complex number, the Van Hove correlation function turns into a complex function. 2In particular, the relation between the delta function and the integral over all the space of the exponential function (in three dimensions): δ(r−r0) = 1 (2π)3Zall space exp(iQ·(r−r0))dQ 44 Chapter 2. Theoretical background for diffusion studies with neutron scattering As a result, the Fourier transformations in time and space connect the Van Hove correlation function describing the dynamics of the system, with experimental neutron functions such as the ISF or the scattering law [56,58,60,75,96]. Thereby a neutron spectrum measures the correlations in time and space of atomic positions in the system. We will see in the next section, how the distinction between the self and the distinct correlation function endow with a physical meaning the coherent and incoherent scattering of neutrons by diffusing atoms. 2.1.2 The coherent and incoherent scattering according to the Van Hove formalism The interaction between the neutron probes and the atom nuclei generates two mechanisms of scattering: the so-called coherent and incoherent scattering. The origin of the twofold nature of neutron scattering stems from the the specificity of each nucleus scattering length band, also, from its dependence on the neutron and nuclear spin operators [58,97]. The power of the Van Hove formalism is asserted by its capacity to associate each kind of scattering to the individual (for the incoherent scattering) and to the collective (in the case of coherent scattering) dynamics of the diffusing elements in the sample. Each isotope has a specific scattering length bfor neutrons, since neutron scattering is a nuclear interaction [56,58]. In a collection of Natoms, there is a statistical distribution of chemical species and of isotopes of each specie. As a consequence, when neutrons are scattered by a crystal, even if it is made of identical atoms, the interaction potential is not homogeneous but varies from atom to atom [58]. However, the effects on the neutron scattering of such statistical distribution in space of the system-neutron interaction potential can be addressed by calculating the average potential and its corresponding standard deviation. The coherent neutron scattering We assume that all the atoms in the system have the same scattering length, taken to be the statistical average of the scattering lengths: hbi= N X j cjbj,(2.7) provided we know the relative density of the j-th species, cj(where PN jcj= 1) [58, 60, 61]. The total number of neutrons scattered by this completely homogeneous system is readily found by substituting by hbi, the bjfor jover all the nuclei in the system and integrating the differential scattering cross 2.1. The Van Hove formalism 45 section (Eq. 1.9 in the section 1.2.1 Chap. 1) in all the space and for all the energies: σcoh =Zd2σ dΩdωdΩdω = 4πhbi2.(2.8) This is the so-called bound coherent scattering cross section and it is proportional to the squared average of b[56]. However it should be noted that in such a system, it is impossible to tell which specific atom scatters a given incoming neutron, for all the nuclei have the same scattering label. It is preferably to say that the neutron is scattered by all the set of Natoms: i.e the neutron adopts a wave behavior [97] such that a proper trajectory can not be associated. Therefore the average of the system scattering potential gives rise to interference effects [58,60]: the intensity of neutrons presents a strong anisotropy and intensity peaks only appear for particular directions of the scattered beam in which the Bragg condition is satisfied. The resulting momentum transfer corresponds to the Bravais lattice vectors Gdescribing the arrangement of the elements in the sample in the reciprocal space: k−k0=G.(2.9) Hence, the wave nature of neutron allows to do diffraction experiments and to measure diffraction patterns (the ensemble of intensity peaks at given directions of the space). But coherent scattering scope does not reduce to structural information: it enlightens the dynamics of collective processes and can be fruitfully exploited for the study of diffusion. Indeed, coherently scattered neutron exchange energy and momentum with the sample by exciting or annihilating collective modes: the interaction is always with a collection of atoms and never with single atoms [97]. The incoherent scattering Conversely, the deviations of the scattering potential with respect to its average are randomly distributed in space [58,60]: the isotopes of an atom have not allocated positions, but there exists a certain probability to find a given isotope in a certain position. Such deviations are quantified by the standard deviation of the averaged scattering length [58,60]: b2−hbi2= N X j cj(bj)2−  N X j cjbj  2 ,(2.10) which describes the degree of heterogeneity in the system. Furthermore, the scattering of a neutron by a nucleus depends also on the relative orientation of the neutron and the nuclear spin. Therefore, there exists the possibility of flipping the neutron and the nucleus spin during the scattering process [97], apart from exchanging energy and momentum with the nucleus. Thereby, in 46 Chapter 2. Theoretical background for diffusion studies with neutron scattering contrast with the coherent scattering, we can identify which atom of the system interacts with the neutron [97]. In other words, we can reconstruct the neutron probe trajectory [97] meaning that we deal with a particle rather than with a wave. This kind of contribution to the scattering can not be analyzed in terms of interferences between waves. It corresponds to single neutron - single nucleus interaction and is completely isotropic: this is the so-called incoherent scattering [58,60]. Consequently, the part of neutrons scattered by a statistical distribution of chemical species and isotopes in the system [60] needs to be proportional to the standard deviation of scattering lengths [56]: σinc = 4π(hb2i−hbi2).(2.11) This is the bound incoherent scattering cross-section. To conclude, the coherent/incoherent scattering is nothing else than the manifestation of the wave-particle duality of neutrons [97]. Some systems will enhance the neutron wave behavior while other systems will stress the neutron particle nature. In the so-called coherent systems, the scattering length distribution present negligible deviations with respect to its average. The corresponding incoherent cross section is small if compare with the coherent cross section. Hence, neutron probes tend to interact with these kind of systems acting as waves and the scattered intensity presents strong variations with the momentum transfer, due to the constructive/destructive interferences. A clear example are the carbon based or the deuterated systems (see Tab. 2.1). On σ[barns=10−24cm] H D C σinc 80.27 2.05 0.0005 σcoh 1.76 5.59 5.564 Table 2.1: Incoherent and coherent scattering cross-sections of hydrogen and carbon atoms [58]. the other side, systems containing hydrogen atoms are well-known for their high incoherent cross section [56] and neutrons interact with protons behaving as particles. The scattered intensity will be isotropic [58], the interaction is always between a single neutron and a single nucleus whose spins are coupled and change during the scattering, in such a way that every nucleus gets tagged and becomes recognizable [97]. Finally, the total cross-section σof a system is the total number of neutrons scattered by the system, in all directions and energies. It comprehends both contributions arising from the coherent and incoherent scattering cross sections [56]. It reads as the summation of the 2.2. Traditional models for diffusion in QENS spectroscopy 53 As a consequence, the incoherent ISF, Iinc(Q, t), brings to light the microscopic mechanisms of diffusion, since it measures the time dependence of the MSD or, equivalently, of the velocity correlation function of the particles diffusing in the system vQ(0)vQ(τ)[60,104] in the direction of the momentum transfer Q: Iinc(Q, t)≃exp −Q21 dZt 0 dτ(t−τ)vQ(0)vQ(τ),(2.37) where vQ=Q·vis the projection of a particle velocity on the direction of Q. Even if the MSD and the velocity correlation function bear information about single particle motion, we drop the index jin the notation because these functions are averages over the statistical ensemble associated to the equilibrium state of the system. Hence, they characterize the averaged dynamics of single particles in the system, but, without making any distinction between individual elements. To conclude, we have seen that every diffusive regime is characterized by a particular velocity correlation function, resulting on the way the microscopic mechanisms drive the diffusion (the frequency of collisions for instance). Thereby, the temporal evolution of the MSD changes according to the way particles diffuse [58,60] and determines the temporal evolution of the incoherent ISF and the profile in energies of the corresponding scattering function [60]. Thus, neutrons can extract very detailed information from the atomic point of view about the dynamics of the system and can identify the main diffusive regimes. The models that are described in the following are build on the basis of the Van-Hove formalism and the Gaussian approximation. The Brownian diffusion The Brownian model describes the continuous diffusion of an adparticle colliding randomly with the other adparticles in an energy dissipative environment (high frictional coupling between adsorbate and substrate) [56]. In the long time limit, when the time between two successive collisions becomes negligible, the mass transport is governed by Fick’s second law and described by the Ficks’s equation [60]: ∂n(r, t) ∂t =D∇2 dn(r, t)(2.38) where n(r, t)is the density of atoms at the spatial point rat time t,Dis the diffusion coefficient and ∇2 dis the Laplacian operator for a space of ddimensions5. Here, we consider isotropic diffusion, such that Dis a constant. However, if the diffusion is anysotropic, then Dbecomes a tensor. Consequently, for 5∇2 dis expressed in cartesian, cylindrical or polar coordinates, depending if the particles diffuse in one dimension, on a surface or in a volume respectively. 54 Chapter 2. Theoretical background for diffusion studies with neutron scattering isotropic diffusion, the Van-Hove self-correlation function G(d) s(r,r(0), t), giving the probability of finding a particle at a certain point provided the same particle was at r(0) when t= 0, must fulfill Fick’s equation for diffusion [101]: ∂G(d) s(r,r(0), t) ∂t =D∇2 dG(d) s(r,r(0), t).(2.39) A suitable solution is G(d) s(r,r(0), t)being a Gaussian function, provided that the variance, or mean square displacement, satisfies the following constrain [58,60]: d dtσ2(t) = 1 d d dt ∆r2= 2D⇒∆r2(t)=2dD|t|+c. (2.40) Consequently, in the long time limit, the diffusion coefficient reads as a function of the velocity correlation function, via Eq. 2.35 [101]: D=1 dlim t→∞Zt 01−τ thv(0) ·v(τ)idτ =1 dZ∞ 0hv(0) ·v(τ)idτ. (2.41) Eq. 2.41 is a Green-Kubo formula [21] and belongs to the class of relations stated by the fluctuation-dissipation theorem [20]. These are the bridge between macroscopic magnitudes such as Dwith microscopic functions like the velocity correlation function [20,21]. On the other hand, Einstein’s solution of the equation of motion of a Brownian particle, the well-known Langevin equation, which describes the motion of a particle subject to a random force representing the effect of collisions F(t)and a frictional force −ηm˙ r(t)[101,105]: m¨ r(t) = −γm˙ r(t) + F(t), where h˙ r(t)i= 0 and hF(t)i= 0 ⇒˙ r(t)·F(t0)=h˙ r(t)iF(t0)= 0, (2.42) yields an alternative formula for the diffusion coefficient Dconnected to the friction parameter η. This latter parameter, ηquantifies the exchange of energy between adsorbate and substrate [101]. A solution of Eq. 2.42 for the velocity reads [101,105]: ˙ r(t) = ˙ r(0) exp(−γ|t|) + 1 mZt 0 F(t0) exp h−η m(t−t0)idt0,(2.43) and requires that the velocity correlation function follows an exponential decay of time [101,105]: hv(0) ·v(t)i≡h˙ r(0) ·˙ r(t)i=v(0)2exp(−η|t|).(2.44) 2.2. Traditional models for diffusion in QENS spectroscopy 55 If we substitute Eq. 2.44 into the fluctuation-dissipation Eq. 2.41, we obtain6: D=v(0)2 dZ∞ 0 exp(−η|τ|)dτ =1 dv(0)2 γ=kBT mη .(2.45) Combining the Einstein formula with the fluctuation dissipation theorem leads to an expression for the inverse of the friction parameter in terms of the velocity correlation function: 1 η=m kBT 1 dZ∞ 0hv(0)v(τ)idτ. (2.46) In terms of neutron scattering, the linear time dependence of the MSD generates a single exponential decay in the momentum transfer/real time domain measured with the incoherent ISF (substitute σ2(t)in Eq. 2.34 by Eq. 2.40): Iinc(Q, t) = exp −Q2D|t|.(2.47) Correspondingly, the quasi-elastic peak as a function of the energy transfer, (Q, t), has a Lorentzian shape: Sinc(Q,~ω) = 1 2π ~DQ2 (~ω)2+ (~DQ2)2(2.48) where the quasi-elastic broadening Γ(its HWHM) follows a square law of the momentum transfer Q: Γ(Q) = ~DQ2.(2.49) Ballistic diffusion In contrast to the Brownian motion, a different diffusive landscape appears when the adsorbed particles do not feel any significant frictional coupling with the substrate and the collisions with other adsorbates become less frequent. In essence this behavior matches with the perfect gas behavior and is known as the ballistic diffusion. As a result, particles move freely on the surface or in the volume without any force exerting over them. 6In the last step, we substitute v2(0)by its calculated value according to the equipartition theorem. For the general case of ddimensions, the equipartition theorem applied to v2(0)leads to: v2(0)=Zdv(d)v2(0)Φ(d)(v)∝Zd|v|vd+1 exp −β 2mv2⇒v2(0)=d mβ , where dv(d)is the differential volume in ddimensions, Φ(d)(v)is the corresponding Botzmann distribution function of velocities and β= 1/kBT. 56 Chapter 2. Theoretical background for diffusion studies with neutron scattering Quasielastic Broadening G Momentum Transfer Q Ballistic Diffusion Brownian Diffusion Jump Diffusion Figure 2.1: Signatures of the archetypal diffusion processes in plots of quasi-elastic line broadening or inverse relaxation times versus momentum transfer. Lines are shown for ballistic diffusion (solid line), Brownian diffusion (dashed line), and jump diffusion (dashed-dotted line). This gives rise, in the case of classical particles whose dynamics is governed by the second law of Newton, to a rectilinear and uniform motion. Hence, the corresponding MSD is very simply related to the its averaged squared velocity and time [106]: h∆ri=v2(0)t2(2.50) and the correlation function is deduced from Eq. 2.36: hv(0) ·v(t)i=v2(0).(2.51) The constant value of the velocity correlation function is consistent with the absence of forces acting on the particle and producing any accelaration. This reflects the typical behavior of classical particles in a gas: the density is so diluted that they rarely interact with each other. Hence, the frequency of collisions between them decreases significantly, or in other words, the time between collisions becomes longer. Thus, particles spend most of the time in the free particle state, undergoing straight trajectories until they collide with another particle [106]. The incoherent ISF is readily deduced from the MSD of ballistic particles (see Eq. 2.37): Iinc(Q, t) = exp −Q2 2 1 dv2(0)t2= exp −Q2 2 kBT Mt2(2.52) where we make use once again of the equipartition theorem for deducing the value of v2(0)in ddimensions, dkBT/M. The incoherent ISF displays a Gaussian profile with time. Accordingly, the incoherent scattering function is 2.2. Traditional models for diffusion in QENS spectroscopy 57 the Fourier transformed in time–energy of the ISF and has a Gaussian dependence on the energy7: Sinc(Q, ω) = 1 ~rM πkBT 1 Qexp −M 2kBTQ2ω2(2.53) The ballistic scattering function is characterized by a FWHM rising linearly with the momentum transfer, Q[98]: Γ(Q)=2~r2 ln(2)kBT M|Q|(2.54) Mbeing the mass of the particle. Finally, the Van-Hove self correlation function can be inferred from the Fourier transformation in momentum transfer/real space of the incoherent ISF and follows a Gaussian profile of ∆rj. It is interesting to note that in the short time range (between two successive collisions), Brownian particles move freely and ballistically [58,60]. As a result, Eq. 2.40 describes the long time limit behavior of Brownian particles while Eq. 2.50 gives the short time behavior, when Brownian particles behave as free particles and diffuse ballistically [60]. In this sense, a general form of the ISF holding for the short time and the long time behavior of Brownian particles can be found if we introduce directly the velocity autocorrelation function, Eq. 2.44 into Eq. 2.37 and integrate over the time [57]: Iinc(Q, t)≃exp −Q21 dZt 0 dτ(t−τ)v(0)2exp(−η|t|) = exp −χ2(e−ηt +ηt −1)(2.55) 7A more exact derivation of the scattering law arising from ballistic motion and comprehending non classical particles, consists in replacing in the differential cross-section (Eq. 1.2) the translational-rotational state of the target system, |λi, and its corresponding energy eigenvalues Eλby the states and energy eigenvalues of a free particle i.e. planes waves [60]: |nji=1 √Veiξj·rj⇒Enj=~2 2M|ξj|2 ξjbeing the wavevector of the j-th particle, analogously to kbeing the wavevector of the incoming neutron, represented as well by a plane wave. On the condition that the momentum transfer is conserved: ξ0 j=Q+ξj[60], the incoherent scattering function becomes a Gaussian function of the energy transfer [60]: Sinc(Q,~ω) = β 4πEr1/2 exp −β 4Er (~ω−Er)2, centered at Er=~2Q2/2M, the so-called recoil energy due to the linear momentum exchange between the free nucleus and the neutron probe when the scattering takes place [60]. The corresponding ISF and Van Hove correlation function are deduced thanks to the application of the Fourier transformation in time and space on the ballistic scattering function. Nevertheless, the resulting MSD and velocity correlation function match with their classical limit in Eqs. 2.50 and 2.51, respectively. 58 Chapter 2. Theoretical background for diffusion studies with neutron scattering where χ2is the shape parameter [57] defined as: χ2=v(0)2Q2 η2.(2.56) In the very short time scale (below the lapse of time between two collisions), Eq. 2.55 can be Taylor expanded around t= 0 yielding a Gaussian dependence of time [57]: t << 1⇒e−ηt ≃1−ηt +1 2(ηt)2⇒Iinc(Q, t)≃exp −v(0)2Q2t2(2.57) where the friction parameter is cancelled out. This is the ISF arising from the ballistic behavior of Brownian particles between collisions. Conversely, in the very long time limit, the term e−ηt vanishes and the ISF becomes a single exponential decay [57]: t >> 1⇒e−ηt →0⇒Iinc(Q, t)≃exp χ2exp "−v(0)2Q2 ηt#,(2.58) where we recover the diffusion coefficient, D=v(0)2/η, as defined in Eq. 2.45. On the other hand, the value of the friction parameter ηalso affects the time dependence of the ISF in Eq. 2.55 [57]. In situations of very high coverage, molecules are constantly colliding against each other and the part of the friction related to interaction between adsorbates is significant. As a result, the term e−ηt vanishes and Eq. 2.55 is a single decay [57]: η1⇒χ21, χ2η1⇒Iinc(Q, t)≃exp[−2χ2ηt],(2.59) On the contrary, for two dimensional gases where molecules rarely interact with each other, the collisional friction parameter will be close to zero. Then, we Taylor expand e−ηt around η= 0 and obtain a Gaussian function of time [57]: η1⇒χ21, χ2η1⇒Iinc(Q, t)≃exp −1 2hv2 0iQ2t2.(2.60) As a result, Eq. 2.55 delivers a very general form of the ISF which reaches the short and long time behavior of Brownian particles [57], and, in addition which is sensitive to the friction and yields the correct time dependence for a system of colliding particles (Brownian system) or for a gas like system where the interaction between particles is nearly insignificant [57]. The so-called shape parameter monitors the change in the ISF profile with friction [57]. Jump diffusion Finally, the Chudley-Elliott jump diffusion model describes the dynamics of particles when they experience very high energy barriers for 2.2. Traditional models for diffusion in QENS spectroscopy 59 diffusion and little friction. Their motion is no longer continuous but stepwise: the adparticles are trapped in equilibrium positions for a time τwhere they undergo vibrations around the potential energy minimum [56, 106,107]. Spontaneous thermodynamical fluctuations can push them to jump towards another equilibrium site [56,72,96,107]. The probability P(r, t)of finding a particle at position rat the instant tfollows a master equation of the form: ∂ ∂tP(r, t) = 1 nX m 1 τm [P(r+lm, t)−P(r, t)] ,(2.61) where the set of vectors {lm}links together the set of accessible neighboring sites for a particle lying at r. They form a Bravais lattice and τmis the residence time of the particle in the m-th, site and nis the total number of available sites from an initial position. Since the Bravais lattice implies the equivalence between adsorption sites [107–109], τmor equivalently the jump rate from one to another, 1/τm, is the same for all the sites. Hence the jump rate can be taken out of the summation. Furthermore, the Fourier transformation in space, allows to factorize the master equation as the product of a time depending function and a function accounting for the for the spatial distribution of the adsorption sites in the lattice: ∂ ∂t Zdrexp(iQ·r)P(r, t) = 1 n 1 τX m [exp(−iQ·lm)−1] Zdrexp(iQ·r)P(r, t). (2.62) The master equation turns into a simple differential equation in time: ∂ ∂tI(Q, t) = 1 n 1 τX m [exp(−iQ·lm)−1] I(Q, t)(2.63) where we recognize in the space Fourier transform of the probability P(r, t), the intermediate scattering function ISF, I(Q, t).The solution of Eq. 2.63 is straighforward and leads to a single exponential decay on time: I(Q, t) = I(Q,0) exp[−M(Q)t].(2.64) As a result, the hopping of species on a Bravais lattice provokes a single exponential decay of the neutron intensity with time in the time-momentum transfer domain (Q, t)measured by spin-echo or, equivalently, a single Lorentzian shaped spread of energies in the energy-momentum transfer space (Q,~ω) probed with standard spectroscopy neutron techniques: S(Q,~ω) = 1 2π~ M(Q ω2+M(Q)2(2.65) The time constant M(Q)characterizing the ISF time exponential decay or, analogously, the HWHM of the Lorentzian scattering function contains the 60 Chapter 2. Theoretical background for diffusion studies with neutron scattering information about the geometry of the lattice: M(Q) = −1 n 1 τX m [exp(−iQ·lm)−1] .(2.66) In the limits of long times (after many jumps have been performed) the underlying Van-Hove self-correlation function is once again a Gaussian function [107,110]. A detailed description of these models can be found in many general books on neutron scattering (see Refs. [56,58,72,96]) or recent review articles (see for instance Ref. [83,98]). Nevertheless, diffusion processes are often rather complex and cannot always be easily categorized into the previous simple cases. Further analytical developments, which combine different types of diffusion mechanism are required. Beyond the analytical models, it has become widely accepted that molecular dynamics (MD) simulations hold the key for the interpretation of the data. Nowadays, the available computational power and the various advances in software [83,111], have encouraged the use of simulation tools to address complex diffusion problems. 2.3 Molecular dynamics simulations and QENS spectroscopy In general terms, molecular dynamics MD simulations consist in solving numerically the equations of motion for a given system and a given (mostly classical) Hamiltonian. The resulting trajectories for each particle in the system enable to calculate many of the function bearing microscopic information about the dynamics such as the mean square displacement or the Van Hove correlation function [22]. Therefore they supply an extremely useful complement to neutron measurements, since the Fourier transformations in space and time of the simulated correlation function provide simulated ISF and scattering laws to be compared with the experimental data [23]. On the other side, the MD simulations results and all the underlying approximation which allow to run the calculations should be validated. Experimental data play an essential role: they are the benchmark for evaluating the pertinence of the theoretical hypothesis on which simulations are performed and the accuracy of the input parameters [22,112]. Neutrons measurements are specially important because they extract very precise knowledge about the density of states of the target sample in the length and time scale which can be computed with MD simulations [22,112]. As a result, MD simulations and by simulating the atomistic behavior while the MD simulations approximations receive a feedback from the 2.3. Molecular dynamics simulations and QENS spectroscopy 61 comparison with experimental data. 2.3.1 Driving Force Fields technique The driving force fields, technique provides an analytical approach to the potential energy surface, U(r1,...,rN)depending on the coordinates of the N particles in the system and arising from all the possible interactions ruling the dynamics of the system [22]. The trajectories of the elements in the system are the solutions of the classical equations of motion (Newton’s second law of mechanics) [22]: mi d2 dt2ri=fi=−d dri U(r1,r2,...,rj,...,rN).(2.67) A force field FF consists of a simplified and analytical function which substitutes the true interatomic energy potential in the equations of motion [112]. Molecules, for instance, are defined as groups of atoms connected by harmonic forces [22]. The crucial point is that the simplified model playing the role of interatomic potential is only valid in the region and for the system to be simulated. It follows that a universal definition for a FF can not exist [112]. On the contrary, it requires a set of parameters for adapting the exact form of the model function to each system and each problem [22]. The parametrization of the FF is done according to ab-initio calculations or taking experimental data from neutrons, X-rays or electron diffraction and NMR, infrared or X-ray and neutron spectroscopy. Although a standard FF can not be defined, a typical expression may contain the following terms [22]: U(r1,...,rN) = X bonds 1 2k0(r−r0)2+X angles 1 2ka(θ−θ0)2+X torsions Vn 2[1 + cos(nφ −δ)] +X improper Vimp +X LJ 4ij σ12 i,j r12 i,j −σ6 i,j r6 i,j !+X Coulomb qiqj rij . (2.68) where the four first terms describe the intramolecular interactions while the two last summations refer to the repulsive coulombic interaction and the Van der Waal forces (the 12-6 Lennard-Jones potential). Finally, solving the equation of motion, Eq. 2.67, demands initial (the set of positions and velocities of the particles) and boundary conditions. The choice of suitable initial conditions is extremely important since most of the information extracted from MD simulations are averages. Thus, the initial conditions should reproduce a representative configuration of the statistical equilibrium ensemble [112]. A correct initial configuration may be achieved by thermalizing the system: running the simulation at a given temperature where the particles in the system 62 Chapter 2. Theoretical background for diffusion studies with neutron scattering are provide with an initial Maxwellian velocity distribution. After a certain number of simulation steps, the system forgets the initial values of the velocities and adopts a velocity distribution corresponding to the equilibrium state at the simulation temperature. This configuration can be adopted as a valid initial condition on which we can run a longer simulation [22, 112]. On the other side, to find appropriate boundary conditions is essential for a successful simulation. The cyclic boundary conditions in which the box containing the simulated system is repeated with a certain periodicity (characterized by a wavelength λ) fit perfectly to the simulation of ordered systems. However, they introduce an artificial symmetry in disordered systems [22]. In this case, the periodicity should not be smaller than the size of the box, in order to avoid effects of self-interaction between particles [22]. Thereby, systems or situation presenting long-range interaction, like phase transitions are difficult to handle [112]. Finally, we note that the driving force field technique suits well the situations in which chemical bonds are maintained and the thermodynamical conditions (temperature and pressure) do not affect dramatically the electronic structure of the system [112]. Hence, chemical reactions can not be simulated with this technique, but require a different approach. On the other side, they present several advantages like the power of deal with very large systems [112], their versatility to adapt to very precise and different problems [22] or the simplicity with which they describe intra-molecular interactions, for instance. 2.3.2 The Langevin equation The Langevin equation (see paragraph 2.2 in section 2.2) sets the equation of motion for the ith particle of mass mincluding the friction force, γvi, an external force generated by the substrate free energy potential, −∇Veff (x, y) [113], and a summation of random forces reproducing the kicks of phonons on the surface, ξi(t), and the collisions with other adsorbates, Fij(t)[83]: m¨ ri(t) = −∇Veff (x, y)−γm ˙ ri(t) + ξi(t) + X j6=i Fij(t).(2.69) The subsequent trajectories are found by integrating the Langevin equation. The underlying physical basis as potentials and parameters (in particular the friction parameter) can be freely adapted to generate simulation results that fit the experimental findings. We have already see how the Langevin equation yields detailed information on the microscopic nature of diffusion (in the paragraph 2.2 in section 2.2). 3.1. Sample preparation and measurement protocol 69 Table 3.3: Summary of the relevant physical parameters of exfoliated graphite, Papyex. Graphite: Mass of a carbon atom ×10−26[kg] 1.99 Density of Papyex [g.cm−3] [68] 0.95 Eff. surface for adsorption [m2.g−1] 25.5 Scattering cross-sections [barns=10−24 cm2] Incoherent 1×10−3 Coherent 5.54 Absorption 3.50×10−3 Total 5.54 Debye Temperature ΘD[K] along [0001] direction 730 (on the surface) [116] 800 (in the volume) [116] in the [0001] plane [116] 1400 Graphite lattice parameters [Å] [117] a, b 2.462 c 3.34 70 Chapter 3. ToF measurements on benzene adsorbed on the basal plane of exfoliated graphite substrates 3.1.2 Sample preparation The first step in the sample preparation consists in cutting the disks out of the large sheets of commercial exfoliated graphite Papyex (Le Carbone Lorraine) [114]. Afterwards we clean the substrates in a pyrolytic oven: we put the graphite disks inside a quartz tube connected to a turbo system. The furnace temperature is settled at 350˚C and we leave the graphite during approximatively 20 hours for outgassing. Then, the disks are introduced in the sample holder where we also pipette the volume of benzene corresponding to a given coverage. Finally we close and seal hermetically the sample holder with steel knife seal. These two last operations are performed in a non controlled atmosphere (open air) and at room temperature. Thus, there will be air within the sample holder and it is possible that water adsorbs as well on the substrate. However the amount of benzene (see Tab. 3.1) will be much higher than the quantity of water diluted in the air within the volume of the sample holder (roughly estimated to be, at least, two orders of magnitude smaller than the amount of benzene for 0.1 ML)1. Furthermore the scattering cross section of the water molecule (167.6 barns) is five times smaller than the one of benzene (523.3 barns). As a result, most of the scattering will come from the benzene molecules adsorbed on the graphite, rather than from the water occasionally trapped in the sample holder. Once the sample holder is closed hermetically, preventing any contact of the inner volume with open air, we can store the samples until the experiment takes place. 3.1.3 Measurement protocol We have a fixed measurement protocol in order to obtain a comparable set of experimental data. To cover a wide thermal range from 1.5K to 300K we work with an orange standard cryostat constructed and developed at the ILL [88]. The sample holder is screwed to the bottom of a sample stick which is introduced into the main chamber of the cryostat. The temperature and the rate of cooling (the cold valve opening) are controlled through the main computer of the instrument. Orange cryostats work with nitrogen for cooling the temperature down to 70K and continue with helium down to 1.5 K [88]. The 1In order to calculate the amount of water diluted in the air within the sample holder, we assume that there was a relative humidity of 30 % and that the temperature was of 20˚C in the room where we prepared the sample. This conditions give rise to an absolute humidity, AH, of 5.2 g.m−2. If the sample holder (whose volume is of Vsh = 42.4cm3) is completely fill with air, the mass of water trapped inside is: mH2O=AH ×Vsh = 220.5×10−6g. Since the density of water vapor is 0.804 g.cm−3, we estimate that the volume of water inside the sample is of 0.274×10−3cm3. This volume needs to be compared with the volume of benzene needed to obtain a certain coverage, in Tab. 3.1. 3.1. Sample preparation and measurement protocol 71 resolution function is measured at 2K where there should not be any dynamics and molecules are frozen on the surface. Afterwards we measure the different temperatures starting at 40K, 60K, 75K (only for 1.0 ML of h-benzene and 0.9 ML of d-benzene), 100K, 140K and 200K. The desorption of the benzene molecule on exfoliated graphite starts at 150K [28,45]. Thus, the coverage of the surface is no longer constant at 200K. However our hermetic sample holders allow to measure beyond the desorption temperature because the sample volume is kept constant and a thermodynamical equilibrium exists between the gaseous and the adsorbed phases of benzene. As a result, the coverage on the surface is still constant at 200K, but it will be different from the nominal coverage at 140K. All the measurements have been performed with an incoming wavelength of 5.12 Å . In addition, we have also performed short scans using a slightly longer incoming wavelength of 5.9 Å, in order to explore multiple scattering issues. A longer wavelength shifts the momentum transfer range towards smaller values but provides a gain in the energy resolution. The main instrumental parameters such as the incoming wavelength, λi, the energy resolution, ∆Eres, the maximum energy loss of neutrons ∆Emax and the momentum transfer range have been summarized in Tab. 3.4. We complete the experimental data set with an empty cell made only of graphite and a vanadium sample. The former is useful to evaluate the substrate contribution to the total scattering. The latter is necessary for the normalization of the experimental sample. Vanadium is a perfect incoherent elastic scatterer in the energy/momentum transfer range surveyed by the spectrometer in the quasi-elastic configuration (i.e. when the incoming wavelength is 5 Å , see Tab. 3.4). The vanadium spectrum gives the distribution of neutron energies in the incoming flux (the monitor counts). Table 3.4: Summary of the experimental conditions adopted for the measurements in the IN6 ToF spectrometer at the ILL [88]. IN6 Time of Flight spectrometer parameters: Incoming wavelength, λi, [Å] 5.12 5.9 Energy resolution, ∆Eres, [meV] 0.07 0.05 Neutron maximum energy loss ∆Emax =h2/(2λ2 imn)[meV] 3.1 2.4 Momentum transfer range Q= (4π/λi) sin(2θ/2) [Å−1] 0.21, ..., 2.08 0.19, ..., 1.8 72 Chapter 3. ToF measurements on benzene adsorbed on the basal plane of exfoliated graphite substrates 3.1.4 Multiple scattering issues In scattering theory, neutrons are scattered only once by the sample. However this might not be the case when the sample has an important scattering crosssection or large dimensions. Multiple scattering changes the energy profile of the outgoing neutrons and can lead to a misunderstanding of the physical mechanisms at work in the sample system [56]. In time of flight spectroscopy, the energy exchange between the sample and the neutron probe is encoded into the neutron time of flight. If the distance between the chopper and the sample, Ls, and from the sample to the detector bank, Ld, are well known, then, the initial and the final energy of the neutron are perfectly established (see section 1 in Chap. 1 ): Ei=1 2mnLs ∆ts2 Ef=1 2mnLd ∆td2. The multiple scattering modifies the neutron effective flight path and, hence, its time of flight. The energy uncertainty related to the flying path is the following: δE = 2EδL Ld .(3.1) Multiple scattering effects should be taken into account when the mean free path of the neutron within the sample is of the same order of magnitude than the size of the sample or when the sample presents a transmission of the direct beam lower than 90 % [56]. In our case, we specify the mean free path lof the neutrons in Tab. 3.1. We also include the scattering density cross section and the corresponding mean free path corresponding to the adsorbed layer for every coverage. Even though these values do not have a physical relevance, when we compare them to the scattering cross section of the substrate, they give an idea of which is the main scatterer in our sample and the most probable source of multiple scattering. Since the scattering cross section of the adsorbed layer is two orders of magnitude lower than the scattering cross section of the whole sample, most of the intensity comes from the substrate. Therefore, the mean free path for different samples do not depends on the coverage but on the mass of graphite contained in the sample. Thereby, we expect that multiple scattering is mostly due to the substrate. All the samples are characterized by a total mean free path around 10 cm which is of the same order of magnitude than the radius, 1.25 cm, and the 3.1. Sample preparation and measurement protocol 73 !E [meV] Q [Å-1]!E= 0 meV !E [meV] Q [Å-1]!E= 0 meV !E [meV] Q [Å-1]!E= 0 meV Sample 2.5 cm of diameter, !=5.9 Å, !Eres=50 µeV; 2 K Sample 2.5 cm of diameter+ Cd shielding, !=5.12 Å; !Eres=70 µeV; 2K Sample 1.0 cm of diameter+ Cd shielding, !=5.12 Å, !Eres=70 µeV; 2 K Q=0.25 Å-1 Q=0.25 Å-1 Q=0.25 Å-1 Q=1.8 Å-1 Q=1.8 Å-1 Q=1.8 Å-1 Figure 3.1: Left panel: scattering function for a sample containing 0.5 ML of hydrogenated benzene. The sample holder has a cylindrical shape. The diameter is of 2.5 cm and the height is of 6 cm. The incoming neutron wavelength is of 5.9 Å and the energy resolution is 50 µeV. Medium panel: scattering function for a sample of the same characteristics than the previous one, except for the presence of cadmium adsorbing disks in between the exfoliated graphite disks. The shielding disks are equidistantly placed and the distance between them is of 1 cm. Right panel: Scattering function measured for a sample of 0.5 ML of h-benzene in a cylindrical sample holder of diameter 1.0 cm, height 6 cm and containing six equidistant cadmium shielding disks. For the two last samples, the incoming wavelength of neutrons is of 5.12 Å and the resolution in energies is of 70 µeV. height, 6 cm, of the sample holder. Thus, we need to take care in identifying the parts of the scattering function S(Q, ∆E)with a high probability of being contaminated. Experimentally, we carry out measurement on samples of different diameters and containing cadmium shielding disks between the exfoliated graphite disks. Varying the diameter of the sample affects the probability of neutrons which are scattered in the plane parallel to the sample to be scattered later. On the other side, the introduction of cadmium absorbing disks, prevent neutrons scattered once out of the plane to be scattered again and reach the detector bank [56]. Fig. 3.1 displays the scattering functions of three different samples containing 0.5 ML but with different diameters and with cadmium shielding. We observe that the introduction of cadmium disks and the reduction of the sample diameter affects the areas of low momentum transfer, around Q= 0.2Å−1, and around the Bragg peak of graphite at 1.80 Å−1. These two regions of the spectrum are dominated by the coherent scattering of the carbon atoms of the substrate. Multiple scattering is often attributed to incoherent systems [56]. Conversely, in our case, the multiple scattering attenuation seems to affect the intensity arising from the coherent contribution of the substrate atoms, rather than the intensity measured in the spectral area containing most of the h-benzene incoherent contribution (between 0.3 Å−1and 1.8 Å−1). The reason can be that, in all of our samples, the quantity of carbon atoms in the substrate is, at least, two orders of mag- 74 Chapter 3. ToF measurements on benzene adsorbed on the basal plane of exfoliated graphite substrates 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 Intensity [arb. units] 0.200.150.100.050.00 !E [meV] Distribution of the extra flying path Diam 2.5 cm Diam 1.0 cm + Cd shielding IN6: !Eres/2=0.035 meV Incoming beam Detector Neutron single scattered Neutron double scattered Figure 3.2: Left panel: Distribution of the additional neutron flying path introduced by the geometry of two cylindrical sample with different diameters and heights. The resolution of IN6 is of 0.07 meV when the incoming wavelength is of 5.12 Å. The broadening due toe the thinner sample with the cadmium shielding is still observable, since we only consider the HWHM of the energy resolution, 0.035 meV. Right panel: scheme of the Monte Carlo simulation. We take a cylindrical and homogeneous sample (the scattering cross section is the one calculated for the 0.5 ML sample). We evaluate the probability of the neutron to be scattered twice within the volume of the sample (the intensity). The distance of the two scatterers projected on the direction of the outgoing flux gives the additional length that the neutron needs to cover before arriving to the detector. Within this framework, the flying path can only get longer when multiple scattering ocurs. This translates into an asymmetry of the scattering function, which is already observable in the resolution function, since it arises from the geometry and the size of the sample and not from the dynamics of the system. nitude greater than the amount of protons in the adsorbed benzene molecule (see Tab. 3.1). Eventually, the density of protons in the sample is extremely diluted and the multiple scattering events from collisions between protons and neutrons are very low, in comparison with the frequency of multiple scattering events of neutrons with carbon atoms of the substrate. Nevertheless, we have also carried out very simple Monte Carlo simulations to evaluate the broadening due to the geometry and the size of the sample in the experimental time of flight spectra on which our study is based. The distribution function of energy broadenings for two different sample is plotted in Fig. 3.2. In that case, the flying path only gets longer when multiple scattering occurs. This translates into an asymmetry of the scattering function, which is clearly observable in the resolution function, since it arises from the geometry and the size of the sample and not from the dynamics of the system. 3.2. Tof measurement on benzene adsorbed on graphite at different coverages and temperatures 75 3.2 Tof measurement on benzene adsorbed on graphite at different coverages and temperatures As was already said, temperature and coverage affect complementary features of the diffusive behavior of benzene adsorbed on graphite. We cover a wide range of temperatures and coverages in the sub-monolayer range, with several measurements of time-of-flight spectra of benzene adsorbed on exfoliated graphite. The whole experimental data set has been measured on the time of flight spectrometer IN6 at the ILL. The forthcoming section gives an overview of the experimental data set and sketches qualitatively the role played by these two control parameters. However, the data treatment included in the following chapter contains the quantitative description of the diffusive process. Hence, no conclusions can be drawn from the direct presentation of the experimental results. This section is organized in three paragraphs. The first one summarizes the experimental measurements on the exfoliated graphite substrate, Papyex, without any adsorbed benzene. The two subsequent paragraphs review the experimental data measured on hydrogenated and deuterated benzene adsorbed on exfoliated graphite. 3.2.1 The Papyex scattering function We have a complete set of measurements devoted to the graphite substrate Papyex in all the interesting thermal range for surface diffusion: from 2K to 140K. The comparison between the scattering function of the graphite with the scattering function of benzene/graphite allows to identify the features arising exclusively from the substrate. We show the scattering function of graphite as a function of momentum transfer Qand energy transfer ∆Eat 2K and 140K in Fig. 3.3. The temperature does not significantly affect the quasielastic spectrum of graphite. Thus, any change of the quasi-elsatic spectrum of benzene/graphite with temperature is related to benzene diffusion. On the other hand, the graphitic character of the sample is attested by the intense spot along the elastic line, at 1.80 Å−1which is attributed to the [002] Bragg peak of the hexagonal and to the [003] Bragg reflexion of the rhombohedral phases of graphite [68]. The phononic spectrum of graphite is hardly observable in the scattering function because of its high energy range with respect to the quasi-elastic region and its weak intensity if compare with the elastic peak in the quasi- 76 Chapter 3. ToF measurements on benzene adsorbed on the basal plane of exfoliated graphite substrates !E [meV] Q [Å-1]!E= 0 meV [002]H/[003]R !E [meV] Q [Å-1]!E= 0 meV Phonons Exfoliated graphite at 2K Exfoliated graphite at 140K Figure 3.3: Scattering cross-section measured for exfoliated graphite at 2 K (left panel) and 140 K (right panel). The intense spot at 180 Å−1is the Bragg reflexion coming from the diffraction of the [002] hexagonal and the [003] rhombohedral families of planes in the graphite. The spectrum of graphite does not present any significant change with temperature in the quasi-elastic area. The beginning of acoustic phononic branches is visible at 140 K, but their intensity is too weak with respect to the Bragg peak, so that their contribution falls in the background. elastic range. Fig. 3.4, plots the non-rebinned spectrum of graphite for 140K: the distribution of measured neutron intensities is not re-scaled when we change the variable from time-of-flight to energy transfer. This is a trick to keep the high energy range part of the spectrum visible. We observe the two phononic branches emerging from the Bragg peak. The comparison of Fig. 3.4 with the phononic dispersion curves in Ref. [118], suggests that these are longitudinal acoustic phonons along the [100] direction. Besides, this indicates that part of the background beyond -6 meV comes from phonons. We should avoid this region in the data analysis and only consider the energy window around the elastic peak where lies the cleanest quasi-elastic signal arising from benzene diffusion. 3.2.2 Measurement on hydrogenated benzene adsorbed on graphite: incoherent scattering We recall that the hydrogenated benzene scattering is strongly dominated by incoherent incoherent (see scattering cross sections in Tab. 3.2). It probes the diffusion of single protons in the molecules. The scattering cross section S(Q,∆E)is the Fourier transform in time and space of the Van Hove self correlation function Gs(r, t)which yields the probability of finding an atom at a certain position rat time tif this very same atom was at the origin at t= 0. Hence, the experimental data in this section measure the statistical average of 3.2. Tof measurement on benzene adsorbed on graphite at different coverages and temperatures 77 !E [meV] Q [Å-1]!E= 0 meV [002]H/[003]R LA in [100] direction Figure 3.4: Non rebinned spectrum of exfoliated graphite at 140K. Inelastic features are still visible because we skip the re-scaling of intensities when we transform the neutron intensity as function of the time-of-flight into a function of the energy transfer. We observe two acoustic phononic branches starting at the Bragg peak and dispersing towards high energies. Comparing the map of intensities with the dispersion curves in Ref [118], suggests that these are the longitudinal acoustic (LA) phonons in the [100] direction. proton trajectories on the surface. Dependence of the diffusion on coverage Figs. 3.5 and 3.6 display the scattering cross section S(Q, ∆E)measured for hydrogenated benzene at 140 K for four coverages spanning from 0.1 ML to 1.0 ML. We can see the distribution of neutron intensity as a function of momentum transfer Qand energy transfer ∆E. The red zone around the zero energy transfer, ∆E= 0 meV, is the elastic peak. It contains the diffraction pattern of the substrate and all the scattering coming from dynamical process which are too slow to be resolved by the energy resolution. The green area around the elastic peak is the quasi-elastic area. It arises from the interaction between neutrons and diffusing molecules. The shape of the quasi-elastic area as a function of the momentum transfer bears the signature of the diffusive behavior of molecules. The blue area is the background. It consists in the scattered intensity by fast motion falling out of the quasi-elastic energy window, like the graphite phonons. Finally, we recall that the shape of momentum / energy transfer space, (Q, ∆E), is due to the dispersion relation of free neutrons (see Fig. in Chap. 1) [80]. In the following sections we focus on the description and the 78 Chapter 3. ToF measurements on benzene adsorbed on the basal plane of exfoliated graphite substrates comparison of the quasi-elastic area (in green) for all the collection of coverages. The goal is to gain a deeper insight into the role of the interaction between adsorbates and with the substrate on their diffusion behavior. !E [meV] Q [Å-1]!E= 0 meV !E [meV] Q [Å-1]!E= 0 meV 10-5 10-4 10-3 10-2 S(Q,!E)Q cst [arb. units] -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 !E [meV] ToF data measured at IN6 (ILL), 0.1 ML, C6H6 at 140 K Q = 0.50 Å-1 Q = 1.05 Å-1 Q = 1.50 Å-1 10-5 10-4 10-3 10-2 S(Q,!E)Q cst [arb. units] -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 !E [meV] ToF data measured at IN6 (ILL), 0.2 ML, C6H6 at 140 K Q = 0.50 Å-1 Q = 1.05 Å-1 Q = 1.50 Å-1 0.2ML of C6H6 at 140K 0.1 ML of C6H6 at 140K Q=1.05 Å-1 Q=0.50 Å-1 Q=1.50 Å-1 Q=1.05 Å-1 Q=0.50 Å-1 Q=1.50 Å-1 Figure 3.5: Top panel: Scattering function S(Q, ∆E=~ω)for the 0.1 ML (left) and 0.2 ML (right) of benzene adsorbed on exfoliated graphite at 140K. Bottom panel: scattering function as a function of the energy transfer ∆Eat three constant values of Q: 0.5Å−1, 1.05 Å−1and 1.50Å−1. The difference between low Qand high Qrange are more appreciable at 0.2 ML. But the fitting to a theoretical model is required to establish the dependence of the HWHM of the energy profile with the momentum transfer. First of all, we observe that the intensity of the quasi-elastic area increases with coverage. This is logical since the density of protons grows with the coverage. Consequently, the adsorbed layer scattering cross section rises and delivers a brighter scattering. We deduce that the effects of diffusion at low coverages are difficult to identify by a simple inspection of the scattering functions, since they can be hidden by the strong elastic peak. Actually the scattering function of the 0.1 ML sample is very similar to the one of exfoliated graphite at 140 K (see the left panel in Fig. 3.3). Only the fit of the data to theoretical models 3.3. Conclusion 85 the scattering function at high Qvalues differs for d-benzene and h-benzene. The h-benzene scattering cross section is very sensitive to temperature variations and displays several gaps (between 20K and 40K and between 100K and 140K) while the thermal behavior of the d-benzene scattering cross section is very homogeneous in the whole thermal range. 10-6 10-5 10-4 10-3 S(Q,!E)Q cst [arb. units] -2 -1 0 1 !E [meV] ToF data measured at IN6 (ILL), Q=0.50 Å-1, 0.9 ML, C6D6 2 K 20 K 40 K 60 K 75 K 100 K 140 K 10-6 10-5 10-4 10-3 10-2 S(Q,!E)Q cst [arb. units] -2 -1 0 1 !E [meV] ToF data measured at IN6 (ILL), Q=0.50 Å-1, 1.0 ML, C6H6 2 K 20 K 40 K 60 K 75 K 100 K 140 K 10-7 10-6 10-5 10-4 10-3 10-2 S(Q,!E)Q cst [arb. units] -6 -4 -2 0 !E [meV] ToF data measured at IN6 (ILL), Q=1.50 Å-1, 0.9 ML, C6D6 2 K 20 K 40 K 60 K 75 K 100 K 140 K 10-6 10-5 10-4 10-3 10-2 S(Q,!E)Q cst [arb. units] -6 -4 -2 0 !E [meV] ToF data measured at IN6 (ILL), Q=1.50 Å-1, 1.0 ML, C6H6 2 K 20 K 40 K 60 K 75 K 100 K 140 K 0.9 ML of C6D61.0 ML of C6H6 Q=1.50 Å-1 Q=0.50 Å-1 Figure 3.10: Cuts on the neutron coherent scattering function for the 0.9 ML of d-benzene (left panel) and on the incoherent scattering function for 1.0 ML of hbenzene (right panel) at two values of Q: 0.50 Å−1and 1.50 Å−1. The thermal behavior of the scattering cross-section changes at high momentum transfer between hydrogenated and deuterated benzene. This hints at a different perspective on the diffusive processes at work. 3.3 Conclusion To briefly summarize this chapter, we have described the experimental procedure leading to the measurements of a whole set of data for different coverages and temperatures of the system: hydrogenated/deuterated benzene adsorbed on graphite. We have extensively surveyed the diffusive behavior of benzene 86 Chapter 3. ToF measurements on benzene adsorbed on the basal plane of exfoliated graphite substrates in the sub-monolayer regime and in a wide thermal range. Time of flight measurements on four samples of h-benzene and two samples of d-benzene give an overview of the action of coverage and temperature on the diffusion of benzene on graphite. In addition we have tested the existence of multiple scattering , due to the large dimensions of our samples: we have carried out measurements on different size samples and including cadmium shielding. We have also performed Monte Carlo simulations to evaluate the values of the energy broadening due to multiple scattered neutrons. It seems that most of the multiple scattered intensity lies in the regions of the scattering function dominated by carbon scattering (the very low Qrange and the elastic Bragg peak of graphite). Finally the graphite phononic contribution has also been identified as contributing to the background intensity. Hence, we need to be careful during the data analysis and avoid the inelastic part of the scattering function in order to obtain a clean quasi-elastic signal only related to benzene diffusion. The benzene diffusion on graphite is sensitive to both: temperature and coverage. However, the thermal behavior of the incoherent scattering crosssection changes with coverage and with the momentum transfer. A particularly interesting momentum transfer range is found above 1.05 Å−1where neutron scattering access single molecule motion and very short length motion. In the next chapter, the fitting of the experimental data to different theoretical models should help to disentangle the different diffusive mechanisms at low and high momentum transfer, and also the differences between the information provided by incoherent and coherent scattering. Chapter 4 Analysis of the experimental data The exposition of the experimental data in chapter 4 gave a qualitative overview of the diffusion of benzene adsorbed on the basal plane of exfoliated graphite. The shape of the quasi-elastic scattering was shown to be sensitive to temperature and coverage. Furthermore, there is a marked contrast between the high and the low momentum transfer range. The goal of this chapter is to find a suitable theoretical model to fit the experimental data and complete the qualitative outline of the diffusive regime with a quantitative description. However, diffusive processes can hardly ever be described with a single model because of the several physical mechanisms underlying the dynamics of adsorbates on the substrate. Thus, the combination of diverse theoretical perspectives holds the key to tackle the complexity and dress a complete picture of the whole process. In the following sections we discuss the fitting of the experimental spectra to a collection of theoretical models. We start with a brief survey on previous studies of the benzene diffusion on graphite, in order to set a context to our present analysis. In the following section we describe the most general theoretical profiles to fit quasi-elastic scattering data. The third section concerns the possible models to deal with the incoherent quasi-elastic scattering measured on the hydrogenated benzene/graphite system. The final section considers the analysis of the coherent quasi-elastic scattering arising from the deuterated benzene/graphite system. 88 Chapter 4. Analysis of the experimental data 4.1 Previous studies on benzene on graphite Due to their central position in industry carbon based compound on graphite, like alkanes or aromatic rings, have been studied extensively by neutron scattering (diffraction, incoherent elastic scattering and QENS) [26]. Concerning aromatic systems such as benzene adsorbed on graphite, numerous thermodynamical studies and an important theoretical effort were performed [29–33]. The determination of the occupied surface area by benzene and its geometry of adsorption were a matter of debate during a long period: NMR experiments [37, 38] and neutron scattering data [39–41, 120] were giving conflicting results about the orientation of benzene rings: either parallel [41] or perpendicular [37,38,120,121] to the graphite surface. Finally neutron scattering [41,122], LEED [44] and Penning ionization electron spectroscopy (PIES) measurements [42] in close collaboration with MD calculations [43] succeeded in revealing the structure of the adlayers of benzene adsorbed parallel to the surface. Careful re-analysis of the original NMR work yielded finally an agreement with the flat lying interpretation [123, 124]. Since then, further MD studies in the dynamics and the thermodynamical properties of the adsorbed layer has been carried out [34–36], helping for the interpretation of previous experimental results and giving a very detailed picture of the physical mechanisms underlying thermodynamical phase transitions such as the adsorbed layer melting process, for instance. Apart from the structural characterization, the diffusive behavior of aromatic rings is a matter of study where quasi-elastic neutron scattering has proved to be a powerful tool. Ref. [120] presents different models to fit the diffusive behavior of benzene adsorbed on graphite. However, in this reference paper, the orientation of adsorbed benzene is still unclear. Hence the final model corresponds to a situation in which molecules are adsorbed perpendicular to the graphite surface. More recent neutron spin-echo, NSE, and helium-3 spin-echo, HeSE, spectroscopy have shown their suitability for the study of the fast dynamics on the picosecond time scale [45]. These measurements show a clear Brownian diffusive behavior of the benzene adsorbed flat on graphite, and it was the first experiment in which the microscopic dynamic friction coefficient could be established with a reasonable error bar [45]. In particular, the diffusive behavior of 0.5 ML of deuterated benzene was measured with NSE while hydrogenated benzene was used for the HeSE data. The intermediate scattering functions could be fitted to single exponential decays in a very wide range of the momentum transfer (0< Q <1.5 Å−1) [45]. Furthermore, no dependence on the coverage of the diffusive regime was observed, and the friction coupling was mainly attributed to phononic processes [45]. However we 4.1. Previous studies on benzene on graphite 89 should not forget that both experiments measure a coherent scattering signal. In the case of NSE, this can be achieved by using fully deuterated benzene molecules [45], even though, the coherent signal decreases with the momentum transfer while the incoherent signal is evenly distributed in all the Qrange and dominates for large momentum transfer values. In the case of HeSE, the scattering is always coherent [1], since Helium particles are scattered by the molecule’s electronic cloud [1]. Finally, it is worth mentioning the theoretical effort for relating the diffusive behavior of benzene molecules on graphite with its internal degrees of freedom [125]. The hypothesis underlying this study is that there exist two different time scales related with the internal vibrations of the benzene molecule and the motion of its center of mass. The coexistence of the very fast dynamic of internal vibrations with the slower motion of the molecule’s center of mass is the source of chaotic noise which affects the diffusion of the adsorbed molecule on the surface [125]. There are also more general theoretical approaches which combine diffusive and vibrational processes of adsorbate on surfaces [84]. In that case, adsorbates undergo continuous diffusion but can get trapped in potential wells, where they perform a vibrational motion (in that case the adsorbate is an atom and the vibrational motion is different from the internal degrees of freedom of a molecule). The question is how the combination of bound and unbound trajectories affects the diffusive regime [84]. In our study, we have oriented the research towards the effect of rotations in the diffusive process. Diffusion is a complex dynamical process and we need to combine complementary perspective to address a complete description of the phenomenon. We have decide to continue the research started in Ref. [45] and we have extensively survey the diffusion of hydrogenated benzene with time of flight, ToF, spectroscopy. Hydrogenated benzene produces an incoherent scattering signal, and ToF spectroscopy access a different dynamical window than the spin-echo technique (lower resolution but larger energy and momentum transfer window). The goal of this chapter is to find a suitable theoretical model to address the open question of the diffusive behavior of benzene. We wonder wether the differences and similarities of our study with Ref [45] are due to the coherent/incoherent nature of the scattering signal, to the spectroscopy technique in use or to a difference in the physical mechanisms underlying the diffusive regime. 90 Chapter 4. Analysis of the experimental data 4.2 Theoretical models for fitting QENS spectra Theoretical models aim to describe the quasi-elastic profile on the basis of the physical mechanisms governing the dynamics of atoms or molecules adsorbed on the surface. Neutron scattering probes the loss of spatial correlation with time due to the displacement of atoms with respect to their original position at time t= 0 during the diffusive process. The intermediate scattering function (ISF), I(Q, t), monitors the correlation loss in time and reciprocal space (momentum transfer Q). The decay of I(Q, t)with time reproduces exactly the drop of the probability to find an atom in its initial position. The scattering function (SF), S(Q,∆E), defined as the Fourier transform of the ISF, shows the effect of the correlation loss in the energy ∆Eand momentum transfer Qdomain. Here, diffusing adsorbates exchange kinetic energy with neutrons producing a continuous energy distribution around the elastic peak. This is the so-called quasi-elastic profile or Q-peak [85]. Since IN6 is a time of flight, ToF, spectrometer, the part of the quasi-elastic profile in the positive energy range is related with the loss of kinetic energy of neutrons which translates into a longer flying path in the ToF technique. Thus, it stems from processes like phonon excitations, or energy transfer to the diffusing molecules. Multiple scattering due to a large sample contribute as well to the positive range of the energy profile. Conversely, the negative part of the spectrum arises from processes where neutrons gain energy, like phonon annihilation or energy transfer from the diffusing elements. Nevertheless, it is important to stress that phonon creation/annihilation processes give rise to inelastic features, peaks at the corresponding frequency. Diffusive processes can also produce distinct spectral features in S(Q,~ω). In particular, frustrated lateral motion of atoms or molecules in the plane parallel to the surface gives rise to an inelastic feature: the T-peak [85]. Despite of that, we focus on the Q-peak since the benzene Tpeak has not been observed in the surveyed quasi-elastic energy window. Since the largest part of our experimental data consists in time-of-flight spectra, we seek for the theoretical description of the quasi-elastic scattering function. We are especially interested in two complementary features of the spectrum: the dependence of the shape of the quasi-elastic broadening and of its half-width-halfmaximum, HWHM, as a function of the energy and the momentum transfer. The theoretical framework within which quasi-elastic models are designed, was originally conceived to study incoherent quasi-elastic scattering. Nevertheless, it can be adapted to fit coherent scattering provided a number of assumptions are made [2, 99]. We firstly focus on the incoherent scattering models. Afterwards, we discuss the fitting of the deuterated benzene data. However, we will discuss the differences between the quasi-elastic coherent and incoherent scattering properly in the following chapter where the physical interpretation 4.2. Theoretical models for fitting QENS spectra 91 of the fitting results is developed. The incoherent SF, Sth(Q,~ω), for a system of identical diffusing particles on a surface can be split up into three contributions: Sth(Q,∆E) = [y0+ exp(−u2 LQ2)Ael(Q)δ(∆E) + SQENS(Q,∆E)].(4.1) A flat background y0stems from the lattice phonon scattering. The delta function represents the elastic scattering from the non diffusing elements on the system and it basically reduces to the elastic substrate diffraction. The Debye-Waller factor, exp(−u2 LQ2), which decreases the elastic intensity represents the contribution of substrate atoms vibrations, u2 Lbeing their mean square displacement, due to lattice phonons [56]. The last term contains the quasi-elastic scattering of the diffusing elements, SQENS(Q,~ω)[56]. Its intensity is also affected by the internal vibrations of the molecule which produce a distinct Debye-Waller factor, exp(−u2 VQ2), where u2 Vis the mean square displacement of the atoms vibrating in the molecule [56]. However, we embody this factor into the generic function SQENS(Q,~ω)in order to clearly distinguish between the different contributions to the total SF, Eq. 4.1. Since the elastic intensity is namely due to the scattering from the substrate, while the quasi-elastic intensity can be attributed to the diffusing molecules, they are independent terms. Thus, the normalization of each term in the summation of Eq. 4.1 should be carried out separately. In particular, the incoherent quasi-elastic scattering function needs to satisfy the following requirement [60]: Z∞ −∞ SQENS(Q,∆E)d∆E= 1.(4.2) In addition, the theoretical SF should fulfill a last requirement: the detailed balance principle. A completely classical SF is even in energy and momentum transfer: Scl(Q,∆E) = Scl(−Q,−∆E)[56]. Quantum process break the symmetry in ∆Esince inelastic processes involving neutron energy gain or loss are no longer equivalent [56]. The classical approximation of the SF is valid at long times and for long distances (small energy/momentum transfer), but, quantum effects should be taken into account in the short time-distance regime (high energy-momentum transfer). A good approximation of the actual SF is [56]: S(Q,∆E) = exp −∆E 2kBTScl(Q,∆E).(4.3) We apply the detailed balance correction to the theoretical incoherent SF for diffusion on surfaces obtaining: Sth(Q,∆E) = exp(−u2Q2)×exp −∆E 2kBT[y0+Ael(Q)δ(∆E) + SQENS(Q,∆E)]. (4.4) 92 Chapter 4. Analysis of the experimental data The real fitting function, S(Q,∆E), results from the convolution of the theoretical SF with the resolution function Sres(Q,∆E)of the considered spectrometer: S(Q,∆E) = Sres(Q,∆E)⊗Sth(Q,∆E) =Sres(Q,~ω)⊗exp(−u2Q2) ×exp −∆E 2kBT[y0+Ael(Q)δ(∆E) + SQENS(Q,∆E)]. (4.5) Fourier transforming Eq. 4.5 allows to pass to the time domain. Moreover, the convolution theorem turns the Fourier transformation of the convolution in Eq 4.5 into the product of the Fourier transformed functions [56]. Thus, the corresponding ISF is the product between the resolution function and the theoretical ISF: I(Q, t) = Ires(Q, t)×Ith(Q, t) =Ires(Q, t)×1 ~Z∞ −∞ Sth(Q,∆E) exp i∆Et ~d∆E(4.6) Note that the quantum corrections to the SF, give rise to a complex ISF since the Fourier transform of the detailed balance factor exp (−∆E/2kBT)has an imaginary part. However, for a system made of benzene molecules, we can stay in the classical mechanics fram: Firstly, the De Broglie wavelength of benzene at temperatures above 40 K is ca. 0.08 Å which corresponds in the reciprocal space to 12.5 Å−1. Thus, quantum effects are invisible in the momentum transfer range covered by IN6: the maximum Qvalue is 2 Å−1 when the incoming wavelength is 5.12 Å. Secondly, the temperature should fulfill T∆E/kb[126]. If we restrict the analysis to the quasi-elastic window where the maximum energy transfer is 6 meV, the threshold temperature is 70 K. However, most of the intensity is concentrated in the [-2 meV,2 meV] range, and in this case, the threshold lowers down to 20 K. As a result, for temperatures above 40K and a momentum transfer range below 2Å−1we can adopt a classical description of the benzene diffusion process and the theoretical ISF is real and reads [60]: I(Q, t) = Ires(Q, t)×exp(−u2Q2) [y0δ(t) + Ael(Q) + IQENS(Q, t)] . (4.7) The Fourier transformation of the SF changes constant terms in energy into delta functions of time and vice-versa. For instance, the constant background term in the SF becomes a delta function of the time fixing the initial intensity of the ISF. Conversely the elastic peak of the SF becomes a flat background in 4.3. Experimental data of h-benzene, C6H6, on graphite: incoherent scattering 93 the time domain (compare Eqs. 4.1 and 4.7 ). Finally, the quasi-elastic functions in time and energy are also related through the Fourier transformation: IQENS(Q, t) = ~Z∞ −∞ SQENS(Q,∆E) exp i∆Et ~d∆E(4.8) The normalization condition for the incoherent quasi-elastic SF, Eq. 4.2, translates for the incoherent ISF [60]: IQENS(Q,0) = 1.(4.9) As we have seen in Chap. 2, very simple models exist like Brownian motion, jump motion or ballistic motion, which fit the experimental quasi-elastic data with a single quasi-elastic line. Nevertheless, in most cases, a single quasi-elastic profile is not enough to fit the experimental data in all the energy/momentum transfer or time/momentum transfer domain and for all the experimental conditions (temperature and coverage). Diffusion on surfaces is a complex process because it involves several physical mechanisms like translations, rotations, or clustering which take place simultaneously. Hence, we often need to combine several simple models in order to dress more realistic pictures of the dynamics of the system [1]. In this chapter we will focus on the mathematical aspect of the fitting process. Our goal is to find an explicit form of SQENS(Q,∆E)to fit our ToF experimental data. In the forthcoming sections we compare the suitability of different models for SQENS(Q,∆E), going from the most basic ones like a single Lorentzian or Gaussian function, towards more sophisticated models, which combine summations of Lorentzians and Gaussians. Of course, the choice of the models is not arbitrary and relies on a physical interpretation of the process (an accurate description of the underlying atomic dynamics). 4.3 Experimental data of h-benzene, C6H6, on graphite: incoherent scattering The general form of the incoherent ISF for a system of diffusing atoms, as defined in Chap. 2, reads: IQENS(Q, t) = 1 N N X j=j0=1hexp(−iQ·rj(t)) exp(iQ·rj(0))i.(4.10) From the classical mechanics standpoint, the Heisenberg operators for the single atom positions, rj(t)commute at different times: [rj(t),rj(0)] = 0. If the Van Hove correlation function in space and time, G(r, t), is a Gaussian function 94 Chapter 4. Analysis of the experimental data of variable r, the cumulant expansion of Eq. 4.10 reduces to the quadratic term of Q[60]. This is the well-known Gaussian approximation (see Chap. 2) [104]. It yields the incoherent ISF, IQENS(Q, t), as function of the mean square displacement (MSD) of single atoms on the surface, σ2 Q(t) = D∆r2 QE(t)/2, projected on the direction of the momentum transfer vector [60]: IQENS(Q, t)≃exp −Q2 2σ2 Q(t)(4.11) The corresponding SF is found by Fourier transforming Eq. 4.11: SQENS(Q,∆E=~ω) = 1 2π~Z∞ −∞ exp −Q2 2σ2 Q(t)exp[−iωt]dt (4.12) In conclusion, the decay with time of the incoherent ISF or the profile with energy of the incoherent SF will only depend on the time evolution of the single atom MSD on the surface, D∆r2 QE(t). It was already seen in Chap. 2 that there is a very general form of the ISF, Eq. 2.55, which reproduces correctly the long and the short time behavior of Eq. 4.11, and also the low (or ballistic) and the high (or Brownian) friction regimes [57]. The so-called shape parameter,χ2defined in Eq. 2.56, monitors the continuous change of the ISF time decay profile with the friction parameter η[57]. The very low friction regime provides a Gaussian shaped ISF [127] which is equivalent to the short time behavior of the ISF, as defined in Eqs. 2.57. This means that even in the Brownian regime, particles undergo ballistic motion (rectilinear and uniform displacements and quadratic dependence with time of the mean square displacement) in the lapse of time between two collisions [2]. When the density of adsorbed particles on the surface is very low such that the interactions between them are rare and can be accounted with instantaneous collisions, the friction parameter is very low [127]. As a consequence, the lapse of time between collisions, or equivalently the mean free path (the distance covered by a particle between two collisions), become very large [127] and the ballistic behavior of particles is conserved getting to be observable at experimentally accessible time scales (the pico-second time window for He and neutron ToF spectroscopy techniques). On the contrary, if the friction parameter is very high, the decay of the ISF rapidly converges to a single exponential decay corresponding to the long time behavior of the ISF, described in Eq. 2.58 [57]. In that case molecules display a Brownian diffusive behavior: the mean square displacement is linear with time [127], since the very frequent collisions due to a high density of adsorbed molecules, reduces the mean free path and the mean free time within which ballistic motion dominates. Thus, the range of ballistic diffusion in time and space gets so small that it is no longer observable experimentally. The key point that we want to stress is that ballistic and 4.3. Experimental data of h-benzene, C6H6, on graphite: incoherent scattering 101 Table 4.1: Goodness of the fit , χ2parameter divided by the number of fitted points, resulting from the fitting of all the data set at 140 K to models 1 and 2. Q=0.5 Å−1Q=1.5 Å−1 Model 1 Model 2 Model 1 Model 2 0.1 ML 2.91 5.09 7.21 11.64 0.2 ML 3.87 8.04 12.22 21.86 0.5 ML 10.69 36.88 32.20 86.31 1.0 ML 12.40 47.95 40.64 158.99 Table 4.2: Goodness of the fit , χ2parameter divided by the number of fitted points, resulting from the fitting of all coverages in the whole thermal range to models 1 and 2 for Q= 1.5Å−1. 0.1 ML 0.2 ML 1.0 ML Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 60 K 3.28 5.78 7.45 18.44 11.07 48.15 100 K 4.45 7.54 12.88 25.78 21.08 104.84 140 K 7.21 11.64 12.22 21.86 40.64 158.99 coverage data can be approximated by a single quasi-elastic line. Since a single quasi-elastic line is related to a single diffusive process, the failure of these two elementary models shows that there are probably two or more diffusive mechanisms which become apparent in the high Qrange. Furthermore, a higher coverage seems to foster the multiplicity of diffusion processes: the deviations from single quasi-elastic lines are more pronounced for the high coverage than for the low coverage quasi-elastic profiles. 4.3.2 Multiple quasi-elastic profile models: Rotations and Translations The addition of more than one quasi-elastic lines to fit the data correctly, is related to the existence of different diffusing mechanisms [56]. Furthermore, they do not appear necessarily at the same Qrange, since they can involve different length scales. A clear example are the translations and the rotations of molecules adsorbed on the substrate. When molecules translate, they cover distances of several substrate unit cells and the mean square displacement, MSD, is an increasing function of time. The loss in spacial correlation 102 Chapter 4. Analysis of the experimental data is observable across the entire momentum transfer range and the quasi-elastic broadening increases with the momentum transfer. On the other hand, when molecules rotate, the motion of the atoms within the molecule remains confined in a circle or a sphere (depending if the motion takes place on the surface or in three dimensional space). Their MSD reaches a constant value after a certain time [60] and the loss of spacial correlation is very localized. It becomes measurable at the Qvalues comparable to the size of the molecule. In the following paragraph, we describe the theoretical framework which allows to include molecular translations and rotations in the same model. First, we need to factorize the ISF in Eq. 4.10 in terms of the motion of the center of mass (CoM) molecule and the motion of the atoms within the molecule with respect to the CoM frame. The position of each atom of the α-th molecule can split be into the CoM position, Rα, the position of equilibrium of this atom with respect to the CoM frame rα j, and the displacement of the atoms with respect to its equilibrium position, uα j, due to the molecular vibrations [58]: rj=Rα+rα j+uα j(4.19) If the position of the CoM, the orientation of the molecule and its vibrations are not correlated, the ISF is factorable into a function depending on each one of the terms in Eq. 4.19 [56,58,60]. The independence of each term of the ISF is reliable if the characteristic energies of each kind of motion, (vibrations, translations and rotations) belong to a different dynamical window. The covalent bonding of the carbon atoms forming the aromatic ring provides the molecule with high internal vibration frequencies. The lowest vibrational frequencies are associated with out-of-plane distortions of the molecule [129]. Their energy have been measured with Raman or infrared spectroscopy and their values span in the range between 50.3 meV and 80.3 meV [129]. On the other hand, a typical value for the benzene free rotation around its six-fold symmetry axis can be evaluated from the flipping frequency [130] which is reported for hydrogenated and deuterated benzene in Tab. 3.2 of the Chap. 3. We find a rotational energy of 2.7×10−3meV, which clearly lies in a different energy window than the internal vibration frequencies. We can suppose that rotational and translational energies are closer, since the physical parameter governing translational motion is the mass of the molecule while it is its moment of inertia for rotations. However, the assumption of uncorrelated translations and rotations, also known as the weak hindered approximation [81], applies whenever the orientation dependent inter-molecular forces are weak compared to the isotropic forces or if the center-of-mass of the molecule corresponds to its center of charge [81]. Although the interaction between benzene molecules (quadrupolar interaction [35]) is strongly anisotropic in the bulk, giving rise to 4.3. Experimental data of h-benzene, C6H6, on graphite: incoherent scattering 103 the T-configuration in solid benzene crystals [131], the adsorption process favors a coplanar configuration through the interaction with the substrate which weakens significantly the benzene-benzene interaction energy [35, 131]. The coplanar interaction energy becomes one third of the quadrupolar interaction energy at work in the T-configuration [35,131]. In addition, benzene is a top symmetric molecule and no hydrogen bondings occur in the adsorbed phase. This argument has already been used to support the validity of the weak hindered approximation in liquid methane [81]. Thus, the weak hindered approximation can hold as well for the case of adsorbed benzene on graphite, allowing the uncoupling between translations and rotations. The assumption of uncorrelated vibrations, translations and vibrations draws two major consequences. First of all, the operators Rα,rα jand uα jcommute. Secondly, the averaged product of three independent functions is the product of their averages [81]: hexp[−iQ·(Rα(t)−Rα(0))] exp[iQ·(rα j(t)−rα j(0))] exp[iQ·(uα j(t)−uα j(0))]i= hexp[−iQ·(Rα(t)−Rα(0))]i×hexp[iQ·(rα j(t)−rα j(0))]i×exp[iQ·(uα j(t)−uα j(0))]. (4.20) Thus, neglecting the correlations between translations, rotations and internal vibrations we can approximate the total incoherent ISF as the product of three functions accounting for the molecule translations, IQENS,Tr(Q, t), rotations, IQENS,Rot(Q, t), and internal vibrations IV(Q, t): IQENS(Q, t) = exp[−u2 vQ2]×1 Nmol Nmol X α=1 hexp[−iQ·(Rα(t)−Rα(0))]i× 1 Nat Nat X j=1hexp[iQ·(rα j(t)−rα j(0))]i = exp[−u2 vQ2]×IQENS,Tr(Q, t)×IQENS,Rot(Q, t) (4.21) In the quasi-elastic range, the internal vibration contribution to the scattering is well approximated by the Debye-Waller factor [56]. Nmol is the number of molecules in the system and Nat is the number of atoms per molecule. We can now treat each part of the ISF separately. The next paragraph assumes the approximated form of the ISF and focuses on the description of its rotational part arising from uniaxial rotations of the benzene molecule around its sixfold symmetry axis. There are two different kind of rotations to be discuss: continuous rotations and jump rotations. 104 Chapter 4. Analysis of the experimental data Uniaxial rotations If molecules are performing rotations around a definite symmetry axis, while they lie flat on the scattering plane, the momentum transfer vector Q(see Fig. 4.4) and the atoms positions vectors rjare coplanar. We can, thus, expand α ρ Molecular frame ! Q θ=π 2 !rj φj Figure 4.4: In plane geometry: the momentum transfer Qbetween the incoming and the scattered neutron is contained in the same plane on which molecules are lying flat on the substrate. The in plane geometry is obtained setting the incoming beam direction parallel to the macroscopic surfaces of the graphite disks in the sample holder, and measuring the scattered neutrons in the same plane. the exponential function in terms of cylindrical Bessel functions [132]: exp[−iQ·rj(t)] = exp[−iQρ cos(φj(t)−γQ)] = ∞ X n=−∞ inJn(−Qρ)ein(φj(t)−γQ), (4.22) where γQis the direction of the momentum transfer vector Qwith respect to the laboratory frame, φj(t)is the orientation of the molecule at time tand ρ is the radius of the molecule. Introducing Eq. 4.22 into Eq. 4.10, yields an expression for the rotational incoherent ISF: IQENS,Rot(Q, t) = 1 Nat Nat X j=1 *∞ X n=−∞ inJn(−Qρ)ein(φj(t)−γQ)×∞ X n0=−∞ in0Jn0(−Qρ)ein0(φj(0)−γQ)+. (4.23) We can further integrate Eq. 4.23 over all the possible directions of the momentum transfer vector Q. This last step is useful to obtain the ISF associated to the scattering of rotating adsorbates on a Papyex substrate, which 4.3. Experimental data of h-benzene, C6H6, on graphite: incoherent scattering 105 does not present any preferential orientation of the graphite crystals in the xy plane [56,68]. The integration on the angle γQleads to1[56]: IQENS,Rot(Q, t) = 1 Nat Nat X j=1 ∞ X n=−∞ J2 n(Qρ)Dein∆φj(t))E(4.24) Note that in our case, the polar angle is fixed θ=π/2and the Bessel function depends only on the product of the moduli Qρ.∆φj(t) = φj(t)−φj(0) stands for the angular displacement of atom j. If molecules are subject to random and uncorrelated collisions, their angular velocity is randomized [133]. Moreover, the angular displacement is a random variable, whose change in time is due to a Gaussian stochastic process. Therefore, we can apply the cumulant expansion to the exponential functions of the angular displacement ∆φin Eq. 4.24 [100]. In addition, the angular velocity will be a Gaussian stochastic process [133] and the rotational analogue of the Van Hove self correlation function, G(φ(0), φ(t)), is a Gaussian distribution of the angular displacement ∆φ, according to the Doob theorem [134]. Thus, the Gaussian approximation holds and the cumulant expansion is reduced to the second term. As a result, the rotational ISF is a function of the mean square angular displacement ∆φ2(t): IQENS,Rot(Q, t)≃∞ X n=−∞ J2 n(Qρ) exp −n2 2∆φ2(t).(4.25) Eq. 4.25 yields the analogous dependence of the ISF on the angular MSD, to the one that Eq. 4.11 gives for the translational ISF on the MSD. The angular MSD is related to the angular velocity correlation function in a similar way in which the MSD, |∆r|2, is connected to the translational velocity correlation function hv(t)·v(0)i(see Chap. 2): ∆φj(t) = Zt 0 dt0ωα(t0)⇒∆φ2(t)=*Zt 0 dt0ωα(t0)2+= 2 Zt 0 (t−t0)hωα(0) ·ωα(t)i, (4.26) where ωαis the angular velocity of molecule α. Following the Doob theorem, if the angular velocity is a stochastic process, its correlation function is an exponential decay [57,133,134]: Cω(t0) = hωα(0) ·ωα(t)i h|ω(0)α|2i=|ω(0)α|2exp (−ηrott).(4.27) 1The integration over all the directions of Qgives rise to delta functions, defined in its integral form: Z2π 0 dγQei(n+n0)γQ=δn,−n0. 106 Chapter 4. Analysis of the experimental data where ηrot = 1/τωis the rotational friction parameter [133]. Making use of the same arguments than in the section 2.2 of Chap. 2 and following Ref. [57], we deduce a very general form of the rotational ISF by inserting Eq. 4.26 into Eq. 4.25: IQENS,Rot(Q, t)≃∞ X n=−∞ J2 n(Qρ) exp −n2Zt 0 (t−t0)hωα(0) ·ωα(t)i =∞ X n=−∞ J2 n(Qρ) exp −χ2 n(e−ηrott+ηrott−1), (4.28) where we define a rotational shape parameter equivalent to the shape parameter in Ref. [57], χ2(Q): χ2 n=ω2 η2 rot n2,(4.29) but depending on the index of the expansion nand not on the momentum transfer modulus, Q. As was explained in sec. 2.2 of Chap. 2. and in Ref. [57] for the translational ISF, Eq. 4.28 yields a rotational ISF whose profile in time can be tuned by the value of the rotational friction parameter, ηrot. In the high friction regime, the rotational ISF profile tends to a summation of single exponential decays: IQENS,Rot(Q, t) = ∞ X n=−∞ J2 n(Qρ) exp −t τn;1 τn =χ2 nηrot =ω2n2 ηrot (4.30) The equipartition theorem provides a mean square angular velocity, ω2= 1/Iβ, such that the time constant reads: 1 τn =kBTn2 Iηrot =Drn2,(4.31) where Dris the rotational diffusion coefficient [73] and Iis the moment of inertia. Note that the same ISF is obtained in Refs. [56,120,132], by solving the master equation for uniaxial rotational diffusion [135]. In the energy transfer domain, the corresponding high friction SF is a summation of Lorentzian functions: SQENS,Rot(Q, ∆E) = 1 π ∞ X n=−∞ J2 n(Qρ)~/τn (~/τn)2+ (∆E)2,(4.32) Conversely in the low friction regime, the ISF in Eq. 4.28 becomes a summation 4.3. Experimental data of h-benzene, C6H6, on graphite: incoherent scattering 107 of Gaussian shaped decays: IQENS,Rot(Q, t) = ∞ X n=−∞ J2 n(Qρ) exp "−ω2n2 2t2#; Γn=p2 ln 2 hω2in. (4.33) The resulting SF is a summation of Gaussian functions: SQENS,Rot(Q, ∆E) = ∞ X n=−∞ J2 n(Qρ)r2 ln 2 4π~2Γn exp −∆E2 4~2Γn.(4.34) Eq. 4.34 should be compared with the free rotational SF developed in Ref. [59] . However, we only consider uniaxial rotations, whereas Ref. [59] treats three dimensional rotations. Hence, the partial wave expansion, the Eq. 40 of Ref. [59], is the equivalent to our expansion of the exponential functions in terms of first kind Bessel functions in Eq. 4.23. Furthermore, uniaxial rotations allows us to use the one dimensional Maxwell-Boltzmann distribution of angular velocities which is a single Gaussian function Φ(ω)∝exp(−Iω2/2β). In contrast, three dimensional Maxwell-Boltzmann distribution, Φ(ω)∝ω2exp(−Iω2/2β), is required for rotations in 3-d space [133]. We believe that this explains the main differences between the SF in Eq. 4.34 and the partial SFs in Eqs. 50 and 53 of Ref. [59]. To sum up, the ISF and SF profiles produced by free uniaxial rotational diffusion is an infinite summation of terms. Each one of the elements in the sum is weighted by a squared Bessel function of the first kind and order n, Jn(Qρ), depending on the product of the momentum transfer modulus Qand the radius of the molecule, ρ. Fig. 4.5 displays squared Bessel functions of the first kind, for orders ranging from n= 0 to n= 6. The term associated to n= 0 corresponds to the case in which there are no rotations. The corresponding ISF is a constant and, accordingly, the profile in energy is a delta function, δ(∆E). The remaining indexes n > 0are can be understood as the modulus of the angular momentum, which for uniaxial rotations is parallel to the axis of rotations and is reduced to its zcomponent. We observe that for increasing values of n, the intensity of the corresponding terms in the summation is shifted towards larger Qvalues. This is the mathematical reason explaining why rotations are not visible at low Qbut become the dominant scattering contribution at high values of Q, matching the diameter, 2ρ, of the molecules. We also observe that there is a specific value of the momentum transfer, at Qρ ∼1 Å−1for which the zero order Bessel function reaches its minimum. At this particular value of the momentum transfer, corresponding to the inverse of the molecule radius, only the terms weighted by higher order Bessel function contribute to the scattering profile, and thus, the effect of rotations on the quasi-elastic profile can be directly analyzed. The last comment 108 Chapter 4. Analysis of the experimental data 1.0 0.8 0.6 0.4 0.2 0.0 J2 n(Q*radius) 2.01.51.00.50.0 Q [Å-1] Bessel functions of the first kind, J2 n n=0 n=4 n=1 n=5 n=2 n=6 n=3 Figure 4.5: Bessel functions of first kind and norder squared, as a function of the product between the momentum transfer and the radius of the molecule. The radius of a benzene molecule is 2.5 Å [119]. concerns the normalization condition. Since Eqs. 4.32 and 4.32 to 4.34 should be normalized, the summation of squared Bessel functions should converge: P∞ n=−∞ J2 n(Qρ) = 1. On the contrary, the quasi-elastic broadening, HWHM, of each term is independent of the momentum transfer. It only depends on the index nof the expansion. But, the total SF’s quasi-elastic broadening depends on Q, since SQENS,Rot(Q, ∆E)is a summation of functions with varying HWHM and whose amplitudes, the Bessel functions, are functions of Q[132]. The ideal model would consist in the direct application of Eq. 4.28 allowing the extraction of a rotational friction parameter, without any additional assumption. Unfortunately, this is not possible, because the Fourier transform of Eq. 4.28 is written in terms of complete and uncompleted Gamma functions [57]. Such a SF is very difficult to handle and to use for fitting purposes. Furthermore we often need to take their asymptotic behavior. The following models combine the molecule’s rotational and translational motion in the two extreme cases of the diffusive regime: very high and very low friction. We start from the factorized form of the quasi-elastic ISF in Eq. 4.21, and we adopt single Lorentzian or Gaussian functions for describing the diffusion of the molecular CoM (translations) and summations of Lorentzians or Gaussians (like in Eqs. 4.32 and 4.34 ) to account for the rotations. Nevertheless, the addition of more quasi-elastic lines increases the number of fitting parameters. It is, hence, desirable to link together the new fitting variables in order to min- 4.3. Experimental data of h-benzene, C6H6, on graphite: incoherent scattering 109 imize the number of free parameters during the fitting. Unfortunately, this is not always possible and models can fail because there are too many free fitting variables. Model 3 The third model describes the high friction limit of the ISF, as written in Eq. 4.30, when translations and rotations are dumped because of the frictional coupling of the adsorbates with the substrate or with the neighboring molecules. It assumes a single exponential decays for describing the scattering due to translations and a summation of Lorentzian functions standing for the rotational contribution to the ISF: IQENS,model3(Q, t) = ∞ X n=−∞ J2 n(Qρ) exp −1 τ(Q)+1 τnt(4.35) The term n= 0 indexes the purely translational motion contribution, while the other terms with n > 1mix translations and rotations. The total time constant of the each of the terms in the sum is the summation of the time constant related with translations 1/τ(Q)and a time constant related with rotations, 1/τn, labelled with the index of the summation n. The corresponding quasi-elastic SF is a summation of Lorentzian functions: SQENS,model3(Q, ∆E) = 1 π ∞ X n=−∞ J2 n(Qρ)~(1/τ(Q)+1/τn) (~(1/τ(Q)+1/τn))2+ (∆E)2, (4.36) The total fitting function is: Smodel3(Q, ∆E) = Sres(Q, ∆E)⊗ (y0+Ael(Q)δ(∆E) + AQENS(Q)1 π ∞ X n=−∞ J2 n(Qρ)~(1/τ(Q)+1/τn) (~(1/τ(Q)+1/τn))2+ (∆E)2). (4.37) The free parameters of this model are: the background, the elastic and quasielastic amplitudes and the translational and rotational time constants. The two quasi-elastic broadenings for translations 1/τ(Q)and rotations 1/τnneed to be connected and reduced to a single parameter. We relate them through the model of binary collisions between hard disks (adapted from the original model of hard spheres in Refs. [103, 136]). The rotational and translational friction in a system of hard disks, colliding randomly with each other reads: ηsmooth(θ, T)|λ=0 =d A2π µ1/2 ×θpkBT ηtrans(θ, T) = κ κ+ 1 3 2+1 κηsmooth(θ, T) ηrot(θ, T) = 1 κ+ 1ηsmooth(θ, T) (4.38) 110 Chapter 4. Analysis of the experimental data The first friction parameter, ηsmooth(θ, T)is related to smooth collisions where there is only a transfer in linear momentum between the two hard disks. Its inverse, η−1 smooth =τE, is the Enskog relaxation time [103] whose expression we adapt to the case of hard disks. The detail of the calculations for the friction parameter of hard disks collisions is reported in Annex ?. We find that for disks, ηsmooth =τ−1 Edepends on CoM distance of the two disks when they collide d=ρ1+ρ2, the area of a substrate unit cell A, the reduced mass of the system of the two disks µ=m1m2/(m1+m2), coverage θand temperature T. The same dependence of the smooth friction with coverage and temperature is found in Ref. [57] which deals with the diffusion of atoms on the surface. The actual translational and rotational friction parameters are proportional to the smooth friction and depend on the geometry of the benzene. In particular, the term κ= 2I/µd2contains the moment of inertia - reduced mass ratio which determines the part of kinetic energy converted into angular kinetic energy during the collision. Finally, the HWHM of the Lorentzian functions in Eq. 4.37 is: 1 τ(Q)+1 τn =kBT mηtrans Q2+kBT Iηrot n2=2 3κ+ 2 Q2 m+n2 I(κ+1) kBT ηsmooth(θ, T). (4.39) The two fitting parameters 1/τ(Q)and 1/τnare reduced to a single one: 1/ηsmooth(θ, T). The top panel of Fig. ?? summarizes the fitting to the 0.5 ML data. We also include all the single terms in the summation in Eq. 4.36 at low (left panel) and high (right panel) momentum transfer value. We observe that at low Qonly the Lorentzian functions corresponding the small indexes n= 0 (only translations), n= 1 and n= 2 have a significant contribution. Conversely, in the high Qrange, the Lorentzian functions for n= 3,n= 4 and n= 5 come into play. Finally, the intensity of the Lorentzian profiles for indexes above n > 5is negligible. Hence, the summation in Eq. 4.36 can be safely truncated at n= 5. Model 4 The fourth model considers the low friction regime where translations and rotations are performed free. Hence the quasi-elastic ISF is consists in Gaussian functions of time for translations and rotations: IQENS,model4(Q, t) = ∞ X n=−∞ J2 n(Qρ) exp −(Γ(Q)+Γn)t2,(4.40) which gives rise to a summation of Gaussian functions of the energy transfer: SQENS,model4(Q, ∆E) = ∞ X n=−∞ J2 n(Qρ)s1 4π~2(Γ(Q)+Γn)exp −∆E2 4~2(Γ(Q)+Γn). (4.41) 4.3. Experimental data of h-benzene, C6H6, on graphite: incoherent scattering 117 10-6 10-5 10-4 10-3 S(Q,∆E)Q cst [arb. units] -2 -1 0 1 ∆E [meV] Model 6: Q=0.5Å-1, 140 K, 0.5 MLof C6H6 ToF data (In6, ILL) Model 6 fit func. Elastic profile Quasi-elastic profile: jump rotational model (finite summation of Lorentzian func.) n=1,5 n=6 n=2,4 n=3 10-5 10-4 10-3 S(Q,∆E)Q cst [arb. units] -6 -4 -2 0 ∆E [meV] Model 6: Q=0.95Å-1, 140 K, 0.5 MLof C6H6 ToF data (In6, ILL) Model 6 fit func. Elastic profile Quasi-elastic profile: jump rotational model (finite summation of Lorentzian func.) n=1,5 n=6 n=2,4 n=3 Figure 4.9: Fitting of the 0.5 ML data to model 5. We include the different contributions: the elastic part given by the convolution of the resolution function to a delta function centered at ∆E= 0 meV (black dotted line), the total quasi-elastic profile (green solid line) and the total function (grey solid line). We plot the model and the data at low Qvalue, 0.5 Å−1in the left panel and at high Qvalue in the right panel. Dicussion of models 3, 4 and 5 We discuss now the results of the fitting of the three precedent models. Fig. 4.10 compares the fitting to the models 3, 4, 5 and 6 of the 0.1 ML and 1.0 ML data at140K and for two values of the momentum transfer: 0.5 Å−1and 0.95 Å−1. We recall that at the latter value for the momentum transfer, the effect of translations appears only convoluted with the quasi-elastic profile of translations. The 0.1 ML energy profile can be fitted with the four models, even though at high Qmodels 3 and 6 fit better the low energy transfer range. On the contrary, in the 1.0 ML data the differences between models are more pronounced. Model 4 nor 5 provide a good fitted profile. The deviation between the fitted and the experimental profile are visible at high Q. On the other hand, models 3 and 6 seem better suited and give rise to a very similar fitted profile. The 0.5 ML data are also displayed and fitted to the theoretical models in Figs. 4.6 and 4.9. By far, the best fitting is achieved with models 3 and 6. Thus, in the medium/high coverage regime, rotational motion is better accounted with Lorentzian rather than with Gaussian shaped profiles. Fig. 4.11 summarizes the fitting results for 0.1 ML and 1.0 ML in the thermal range from 60K to 140K. At low temperature, the 0.1 ML data can not be fitted by model 3. But its shape is still ambiguous and only the dependence of the fitted parameters with the momentum transfer can determinate the best model. The 1.0 ML data, displays mainly a Lorentzian line shape and model 3 and 6 give a better fit than models 4 and 5. In conclusion, the rotational and the translational scattering functions in the high coverage regime display a Lorentzian profile. 118 Chapter 4. Analysis of the experimental data 10-5 10-4 10-3 10-2 10-1 S(Q,∆E)Q cst [arb. units] -2 -1 0 1 ∆E [meV] -500 0 500 x10-6 -100 0 100 x10-6 -100 0 100 x10-6 Model 3 vs 4 vs 5 and vs 6 Q=0.5Å-1, 140 K, C6H6 0.1 ML 0.2 ML 1.0 ML Model 3 (inf. sum. of Lorentzian func.) Model 4 (inf. sum. of Gaussian func.) Model 5 (jump rot. model) 1.0 ML Residuals: 0.1 ML Residuals: 0.2 ML Residuals: 10-6 10-5 10-4 10-3 10-2 S(Q,∆E)Q cst [arb. units] -6 -4 -2 0 ∆E [meV] -100 0 100 x10-6 -100 0 100 x10-6 -100 0 100 x10-6 Model 3 vs 4 vs 5 and vs 6 Q=0.95Å-1, 140 K, C6H6 0.1 ML 0.2 ML 1.0 ML Model 3 (inf. sum. of Lorentzian func.) Model 4 (inf. sum. of Gaussian func.) Model 5 (jump rot. model) 1.0 ML residuals: 0.1 ML residuals: 0.2 ML Residuals: Figure 4.10: Fitting of the experimental data of 0.1 ML, 0.2 ML and 1.0 ML h-benzene coverage at low Q(top) and high Q(bottom) value with models 3, 4 and 5. 4.3. Experimental data of h-benzene, C6H6, on graphite: incoherent scattering 119 Table 4.3: Goodness of the fit , χ2parameter divided by the number of fitted points, resulting from the fitting of all the data set at 140 K to models 3, 4 and 5. Q=0.5 Å−1Q=0.95 Å−1 Model 3 Model 4 Model 5 Model 3 Model 4 Model 5 0.1 ML 3.07 2.91 2.62 2.19 3.62 2.79 0.2 ML 2.67 7.40 4.45 3.57 10.33 6.88 0.5 ML 5.95 41.64 7.86 7.37 42.14 32.27 1.0 ML 8.19 62.21 11.82 10.35 78.38 33.47 Table 4.4: Goodness of the fit , χ2parameter divided by the number of fitted points, resulting from the fitting of 0.1 ML and 0.2 ML coverages in the whole thermal range to models 3, 4 and 5 for Q= 0.95 Å−1. 0.1 ML 0.2 ML Model 3 Model 4 Model 5 Model 3 Model 4 Model 5 60 K 2.20 2.53 2.35 3.04 4.96 4.12 100 K 2.52 2.93 2.59 4.33 10.46 7.41 140 K 2.19 3.62 2.79 3.57 10.33 6.88 To sum up, we observe that the addition of a rotational contribution has improved the fitting results at medium and high coverage. This result is in contrast with the study on deuterated benzene reported in Ref. [45], where a single exponential decay gave a good fitting of the 0.5 ML data. Furthermore, their quasi-elastic profiles have a marked Lorentzian character. The question is now wether rotations are performed in a continuous or in a jump stepwise fashion. The analysis of the fitted parameters, the HWHM of the translational term and the jump rate will help to elucidate which is the most realistic description The low coverage data display an ambiguous profile and could already be fitted with a single quasi-elastic profile. All the models except model 3 at low temperatures give a reasonable fit. Hence, we should analyze the fitted parameters in order to decide which is the best theoretical description. On the contrary, 120 Chapter 4. Analysis of the experimental data 10-6 10-5 10-4 10-3 S(Q,∆E)Q cst [arb. units] -6 -4 -2 0 ∆E [meV] -40 0 40 x10-6 -40 0 40 x10-6 Model 3 vs 4 and vs 5 Q=0.95Å-1, 0.1 MLof C6H6 at 60 K 140 K Model 3 (inf. sum. of Lorentzian func.) Model 4 (inf. sum. of Gaussian func.) Model 5 (jump rot. model) 60 K Residuals: 140 K Residuals: 10-6 10-5 10-4 10-3 S(Q,∆E)Q cst [arb. units] -6 -4 -2 0 ∆E [meV] -200 0 200 x10-6 -200 0 200 x10-6 -200 0 200 x10-6 Q=0.95Å-1, 0.2 ML of C6H6, 60 K 100 K 140 K Model 3 (sum. of Lorentzian func.) Model 4 (sum. of Gaussian func.) Model 5 (jump rot. model) 60K Residuals: 100K Residuals: 140K Residuals: 10-5 10-4 10-3 10-2 S(Q,∆E)Q cst [arb. units] -6 -4 -2 0 ∆E [meV] -500 0 500 x10-6 -500 0 500 x10-6 -500 0 500 x10-6 Model 3 vs 4 vs 5 and vs 6 Q=0.95Å-1, 1.0 MLof C6H6 at 60 K 100 K 140 K Model 3 (inf. sum. of Lorentzian func.) Model 4 (inf. sum. of Gaussian func.) Model 5 (jump rot. model) 60K Residuals: 100K Residuals: 140K Residuals: Figure 4.11: Fitting of the experimental data of 0.1 ML (upper, 0.2 ML (middle) and 1.0 ML (lower) h-benzene coverage at Q= 0.95−1with model 3, 4 and 5. 4.4. Experimental data of d-benzene, C6D6, on graphite: coherent scattering 121 Table 4.5: Goodness of the fit , χ2parameter divided by the number of fitted points, resulting from the fitting of 1.0 ML in the whole thermal range to models 3, 4 and 5 for Q= 0.95 Å−1. 1.0 ML Model 3 Model 4 Model 5 60 K 4.05 6.19 8.84 100 K 5.89 35.08 28.19 140 K 10.35 78.38 33.47 4.4 Experimental data of d-benzene, C6D6, on graphite: coherent scattering The fitting of the d-benzene data provides a complementary perspective to the analysis and interpretation of h-benzene data. We have already stress that coherent scattering is sensitive to the relative motion of molecules. In the situations where no correlated motion exists, the coherent and the incoherent scattering functions are identical [75]. But the existence of correlations introduces differences between the coherent and the incoherent scattering [75]. As we have already mentioned, the incoherent scattering models can be applied for the analysis of coherent scattering provided some approximations are done [1]. One of the most useful tools was developed by George Vineyard in Ref. [99]. It allows to relate the coherent and the incoherent scattering function on the basis convolution approximation [99]. It consist in writing the Van Hove correlation function G(r)as the convolution of the radial density function g(r)with the self-correlation function Gs(r)[99]. However, this approximation holds for the low coverage regime and its application in the high coverage regime remains uncertain. One of the most striking differences between the coherent and the incoherent data appears in the sensitivity to jump diffusion. The jump diffusion model according to coherent scattering Coherent scattering does not follows the jump of single atoms, as incoherent scattering does. Hence, we need dynamical variables assigned to the collection of atoms and standing for the molecule: these are the so-called configurations developed by G. Coddens in Refs. [97, 141, 142]. For instance, the configuration of a 122 Chapter 4. Analysis of the experimental data molecule in the µ-th orientation is defined in the reciprocal space: Fµ(Q, t) = 1 √Nexp (iQ·R(t)) N X j exp hiQ·r(µ) j(t)i,(4.59) where the index jruns over all the atoms in the molecule, and the position of the atoms has been split into the position of the molecule’s center of mass (CoM), Rand the atomistic positions with respect to the CoM frame, r(µ) j. We do not include the scattering length bin contrast with the cited Refs. to preserve the scattering function free of the neutron-matter interaction effects. Note as well that in the case of independent molecules performing reorientations, configuration is equivalent to orientation [141]. The coherent intermediate scattering function can be written in terms of the configurations: Icoh(Q, t) = 1 nX νµ F∗ ν(Q)pν(0)P(µ, ν(0), t)Fµ(Q)(4.60) where P(µ, ν(0), t)is the probability of finding the molecule in the µ-th orientation at time t, provided it was at the ν-th orientation at t= 0. The initial orientations are weighted by the probabilities pν(0), and nare the different orientations amongst which the molecule can lie. The case of benzene undergoing 60 degrees jump rotations is easily tractable within this framework. First of all, we note that a π/3rotation does not change the benzene molecule orientation: the position of the atoms in the molecule and the configurations fulfill the following relation, rµ+1 j=rµ j−1⇒ Fµ+1(Q) =   N X j exp iQ·rµ+1 j =  N X j exp iQ·rµ j−1 , (4.61) Thus, the effect of the jump rotation in the configuration is invisible because we can rename the indexes j0=j−1(see Fig. 4.12) obtaining: Fµ+1(Q) =   N X j0 exp iQ·rµ j0 =Fµ(Q).(4.62) As we did for the incoherent jump rotation, we can define a master equation which governs the dependence with time of finding the benzene molecule on the orientation/configuration µ[97,141,142]. However, if benzene is constrained to perform six-fold rotations, it always lies in the same configuration: pµ= pµ+1 =··· =pµ+N. The master equation is equal to zero: ∂ ∂tpµ(t) = X ν 1 τµν [pν(t)−pµ(t)] = 0 ⇒pµ= 1,(4.63) 4.4. Experimental data of d-benzene, C6D6, on graphite: coherent scattering 123 !rj µ µ+1 = µ !rj−1 !rj !rj−1 !rj! !rj!+1 N-fold rotation j!=j−1 !rj!−1 Figure 4.12: Effect of a N-fold rotation on a molecule with a N-fold symmetry axis on the configuration Fcµ. We can see that the equivalence between configurations is easily recovered after a N-fold rotation by relabeling the jindexes of the atoms. and we deduce that {pµ}are all constants equal to the unity. Consequently the conditioned probability in Eq. 4.60 is constant and equals 1, and the coherent ISF is constant as well and equals unity. The corresponding scattering function is a delta function of the energy δ(∆E). In conclusion, coherent scattering is not able to record motions in which the initial and the final state (configurations or orientations in that case) are preserved. This applies to the case of benzene molecule when it performs a jump of n2π/6,nbeing an integer. Conversely, reorientations of 2π/n6would be visible, since in that case the orientation of the molecule changes and, hence, the initial and the final configurations are not the same. Fig. 4.13 plots the data of 0.5 ML and 0.9 ML of d-benzene for 140K together with the fitting resulting from models 1 to 5 at 140K and for values of the momentum transfer of 0.5 Å−1and 0.95 Å−1. We do no longer consider the jump rotation model for six-fold rotation, since the coherent scattering function is unsensitive. First of all, we can discard the Gaussian models 2 and 4, since the profile for both coverages is closer to a Lorentzian shape than to a Gaussian profile. The mismatch is clearly seen at Q= 0.95 Å−1. In addition, we observe that the model 1 delivers a fitted profile which clearly diverges from the 0.9 ML data at high Q. Hence, these models which combine translations and rotations are better suited for the high coverage data . Conversely, the 0.5 ML data is fitted by all the Lorentzian models, which provide very similar fitted profiles. Fig. 4.14 shows the fitting of the 0.9 ML at three temperatures: 60K, 100K and 140K. In the low momentum transfer range (upper panel), all the models give rise to the same fitted energy profile. But at high Qvalues the experimental profile displays a clear Lorentzian profile. However we observe that at low temperatures, 60K, all the fitted profiles are very similar. In that case, the 124 Chapter 4. Analysis of the experimental data 10-5 10-4 10-3 S(Q,∆E)Q cst [arb. units] -2 -1 0 1 ∆E [meV] -100 0 100 x10-6 Q=0.5Å-1, 140 K, C6D6 0.5 ML Model 1 (single Lorentzian func.) Model 2 (single Gaussian func.) Model 3 (inf. sum. of Lorentzian func.) Model 4 (inf. sum. of Gaussian func.) 10-5 10-4 10-3 S(Q,∆E)Q cst [arb. units] -6 -4 -2 0 ∆E [meV] -100 0 100 x10-6 Q=0.95Å-1, 140 K, C6D6 0.5 ML Model 1 (single Lorentzian func.) Model 2 (single Gaussian func.) Model 3 (inf. sum. of Lorentzian func.) Model 4 (inf. sum. of Gaussian func.) 10-5 10-4 10-3 10-2 S(Q,∆E)Q cst [arb. units] -2 -1 0 1 ∆E [meV] -100 0 100 x10-6 Q=0.5Å-1, 140 K, C6D6 0.9 ML Model 1 (single Lorentzian func.) Model 2 (single Gaussian func.) Model 3 (inf. sum. of Lorentzian func.) Model 4 (inf. sum. of Gaussian func.) 10-5 10-4 10-3 S(Q,∆E)Q cst [arb. units] -6 -4 -2 0 ∆E [meV] -100 0 100 x10-6 Q=0.95Å-1, 140 K, C6D6 0.9 ML Model 1 (single Lorentzian func.) Model 2 (single Gaussian func.) Model 3 (inf. sum. of Lorentzian func.) Model 4 (inf. sum. of Gaussian func.) Figure 4.13: Fitting of the experimental data of 0.5 ML (left) and 0.9 ML (right) d-benzene at 140K with models 1 to 4, for two values of the momentum transfer: Q= 0.5Å−1(top) and Q= 0.95 Å−1(bottom). 4.4. Experimental data of d-benzene, C6D6, on graphite: coherent scattering 125 Table 4.6: Goodness of the fit , χ2parameter divided by the number of fitted points, resulting from the fitting of all the deuterated benzene data set at 140 K to models 1, 2, 3 and 4. Q=0.5 Å−1Q=0.95 Å−1 Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 0.5 ML 4.77 13.99 4.23 6.92 4.02 6.12 2.43 5.69 0.9 ML 5.84 12.81 4.85 11.23 7.74 17.51 6.88 21.66 quasi-elastic profile is very close to the resolution function. Furthermore the scattered signal is severely reduced because of the low scattering cross section of deuterated benzene. Hence it is difficult to decide which is the most accurate model. In conclusion, we observe that the deuterated benzene quasi-elastic profile Table 4.7: Goodness of the fit , χ2parameter divided by the number of fitted points, resulting from the fitting of all the deuterated benzene data set at 140 K to models 1, 2, 3 and 4. Q=0.5 Å−1Q=0.95 Å−1 Model 1 Model 2 Model 3 Model 4 Model 1 Model 2 Model 3 Model 4 60 K 3.12 4.18 2.99 3.62 4.49 7.75 4.23 9.35 100 K 3.81 5.70 3.54 4.68 5.03 11.82 4.58 14.67 140 K 5.84 12.81 4.85 11.23 7.74 17.51 6.88 21.66 have a Lorentzian shape. This is similar to the quasi-elastic scattering function of hydrogenated benzene. On the other hand, the 0.9 ML data can not be fitted with a single Lorentzian function, which indicates that molecules undergo rotations. However, coherent scattering can only see continuous rotations or jump rotations of 2π/nN degrees, with n > 1. Thus six-fold reorientations are hidden by the symmetry of the molecule. Conversely the 0.5 ML data can be fitted with all the Lorentzian models, and only the shape of the quasi-elastic broadening with the momentum transfer holds the key for the interpretation of the experimental data. The very low coverage remains unknown since the small scattering cross-section of the d-benzene adsorbed layer prevents us to go further down to 0.1 ML. In what concerns the thermal dependence of the 0.9 ML d-benzene data, all the models converge at low temperature. But the interpretation of the fitted profiles is not easy since the quasi-elastic signal is very close to the resolution function width and its signal is weak. 126 Chapter 4. Analysis of the experimental data 10-5 10-4 10-3 10-2 S(Q,∆E)Q cst [arb. units] -2 -1 0 1 ∆E [meV] -200 0 200 x10-6 -200 0 200 x10-6 -200 0 200 x10-6 60K Residuals: 100K Residuals: 140K Residuals: Model 1 vs Model 2 vs Model 3 and vs Model 4 Q=0.5Å-1, 0.9 ML of C6D6 at 60 K 100 K 140 K Model 1 Model 2 Model 3 Model 4 10-6 10-5 10-4 10-3 S(Q,∆E)Q cst [arb. units] -6 -4 -2 0 ∆E [meV] -200 0 200 x10-6 -200 0 200 x10-6 -200 0 200 x10-6 Model 1 vs Model 2 vs Model 3 and vs Model 4 Q=0.95Å-1, 0.9 ML of C6D6 at 60 K 100 K 140 K Model 1 Model 2 Model 3 Model 4 60K Residuals: 100K Residuals: 140K Residuals: Figure 4.14: Fitting of the experimental data of 0.9 ML d-benzene at 60K, 100K and 140K with model 1 to 4. The upper panel groups the low Qvalue spectra while the bottom panel shows the high Qdata spectra. 5.1. Summary of the previous models 133 site, ω0, is small with respect to the Debye frequency ωD[143]. In a similar way, electronic friction is temperature independent if ω0is small if compare with the Fermi energy [143]. The small-amplitude frequency of the benzene at the adsorption site is related to the height of the potential energy surface, U0 at the adsorption site and can be approximated by ω0≃(2π/a)pU0/m [143], where ais the graphite lattice constant, 2.45 Å for graphite [145], and mis the mass of the adsorbate. The graphite surface corrugation is estimated of 17 meV [45], delivering a small amplitude frequency of benzene of the order of 3.72×1012s−1or in terms of energy of 0.24 meV. Thus, the conditions for the thermal independence of electronic and phononic friction are satisfied since the very high Debye temperature of the basal plane of graphite, 1400 K [116] yields a Debye frequency of 120 meV and the Fermi level of graphite is found at 8.6 eV [146]. In addition, we can roughly estimate the phononic friction of graphite damping the small oscillations of a benzene molecule [143]: ηph = 0.12m ρω0 CT2 ω0(5.1) where ρis the density of the adsorbate (2.267×103kg.m−3for graphite), CT is the transversal velocity of sound, which in the [0001] basal plane is of 4040 m.s−1(estimated from the elastic constant C33 = 3.71 ×1011 dyn.cm−2 Ref. [118], CT=pC33/ρ). The resulting phononic friction is 1.98×1010 s−1. Thus, the analysis of the results arising from data at different coverages and temperatures provides enlightening information to identify the main source of friction in the system of benzene adsorbed on the graphite basal plane. Free parameters in the different models One of the problems that models combining translations and rotations encounter is the multiplication of free variables to fit. We have already stressed in the previous chapter the need of theoretical tools to link the translational and the rotational quasi-elastic broadening. However, each model is related to a defined diffusive regime and the linking should be done accordingly. In the case of model 3, the connexion between the translational and the rotational quasi-elastic broadening is based on the relation between the translational and the rotational friction parameters. The collision model provides the theoretical frame to interrelate ηTand ηR[103,136,147] as it is seen in Eq. 4.38 of Chap. 4. Eventually, we reduce the two-folded quasi-elastic broadening to a single quasi-elastic broadening which integrates the contribution of translational and rotational friction (see Eq. 4.39 of Chap. 4 ). On the other hand, if friction has a phononic or electronic nature, we simply assume that the translational and the rotational friction parameters are equal. In model 4 we can easily connect the mean square velocity and 134 Chapter 5. Results and discussions of the experimental data analysis the mean square angular velocities thanks to the equipartition theorem (see Eq. 4.44 of Chap 4). Thus, we deduce a total quasi-elastic broadening which depends only on the mean square velocity (see Eq. 4.45). Finally, model 5 is the most difficult model to handle since we ignorate any possible link between the jump rate for the six-fold reorientations and the translational diffusion coefficient. Thus, we take the values of the translational quasi-elastic broadening extracted from model 3 and we leave as free parameter the jump rate. This is the only way to obtain a stable fit with this model. To conclude, the fitting of the experimental data to the different models listed in Tab. 5.1 allows to extract valuable information about the diffusive process. In particular, the dependence of the quasi-elastic broadening on the momentum transfer allows to identify the diffusive regime of the molecules adsorbed on the graphite. We can also deduce the friction parameter, linked to one or more possible sources (electrons, phonons or adsorbates). In the following sections we will compare the output of the different models for the hydrogenated and deuterated benzene, discussing the similarities and divergences of the extracted parameters. This is essential to decide which model suits better the benzene diffusive behavior under different coverage and thermal regimes. 5.2 Hydrogenated benzene All the theoretical models contain parameters which are not exclusively related to the diffusive regime of the molecule. We recall that the general form of the fitting function is lenghtly discussed in Sec. 4.2 of Chap. 4 and is defined in Eq. 4.5. Here we write a very schematic formulation of the fitting function in use: Sfit(Q,∆E) = Sres(Q,∆E)⊗[y0+Ael(Q)δ(∆E) + AQENS(Q)SQENS(Q,∆E)]. This is the convolution between the experimental resolution function Sres(Q,∆E), measured for every coverage at 2K, and a theoretical function. The latter consists in the summation of various terms like a flat background stemming from the surface phonons whose characteristic frequency falls out of the dynamical range, an elastic part arising namely from the atoms of carbon in the substrate and a quasi-elastic part coming from the adsorbed layer. SQENS(Q,∆E)corresponds to the quasi-elastic line which has been modeled according to the diffusive regime of the benzene molecules (see the different theoretical line shapes in Tab. 5.1). The elastic intensity corresponds to the ratio elastic/total scattered intensity. It is, thus, directly related to the amount of graphite atoms 5.2. Hydrogenated benzene 135 NG(which are the elastic scatterers in the system) with respect to the total number of atoms in the system NG+NC6H6, where NC6H6is the number of atoms belonging to the adsorbed layer. Hence the elastic amplitude displays a dependence on coverage since Ael ∝NG/(NG+NC6H6)and NC6H6depends on the coverage θ:θ=NC6H6AC6H6/Aeff (where AC6H6is the experimental area of a benzene molecule lying flat on graphite, 36.7 Å, and Aeff is the effective surface area for adsorption for Papyex exfoliated graphite, 25.5 m2/g). Inversely, the quasi-elastic amplitude is linked to the ratio quasi-elastic/total scattered intensity and, hence AQENS ∝NC6H6/(NG+NC6H6). Its dependence on Qcan be viewed more as an inverse mirror of the decaying of the elastic amplitude with the momentum transfer rather than a characteristic feature of the diffusive regime. No particular physical information on diffusion can be extracted from both amplitudes, nor from the flat background. 5.2.1 The common fitted parameters: amplitudes and background Fig. 5.1 summarizes the elastic, quasi-elastic amplitudes and the background for the 0.1 ML and the 1.0 ML data, resulting from the fitting of all the models. In general terms, the fitting to Lorentzian shaped models gives very similar fitted parameters, while the Gaussian profile models 2 and 4 give slightly different values, especially for the high coverage data. We also observe that the low coverage fitted parameters display smaller values for the quasi-elastic and the offset parameters than the 1.0 ML fitted parameters. This is due to the smaller cross-section of the adsorbed layer of the 0.1 ML sample if compared with the 1.0 ML adsorbed layer cross-section (there are ten times more benzene molecules in the latter than in the former sample). Conversely, the elastic amplitude of 0.1 ML remains closer to the unity, since the realteive amount of graphite with respect to the quantity of adsorbed molecules is very high and elastic intensity constitute the major part of the scattered intensity. In terms of coverage, the low coverage data shows very similar fitted parameters resulting from the fitting of the different models. The most sensitive parameter to the different models is, logically, the quasi-elastic amplitude, but it does not show a particular trend with Gaussian or Lorentzian shape models. This is congruent with the results of the previous chapter, where we found that the energy profile could be fitted with almost all the models. In contrast, for the 1.0ML, the models made of Gaussian functions (models 2 and 4) provide fitted parameters which differs from the general trend. These results were expected since we saw in the previous chapter that the energy profile of the medium-high coverage data is markedly Lorentzian shaped. Hence, we shall retain only the results extracted from models 1, 3, and 5 which consist in a single or summa- 136 Chapter 5. Results and discussions of the experimental data analysis tion of Lorentzian functions. Finally, note that the summation of the elastic, quasi-elastic amplitudes and 1.0 0.8 0.6 0.4 0.2 0.0 Elastic amplitude [arb. units] 2.01.51.00.50.0 Q [Å-1] Elastic amplitude, at 140K, 0.1 MLof C6H6 Model 1 Model 2 Model 3 Model 4 Model 5 Debye-Waller factor of the basal plane of graphite 1.0 0.8 0.6 0.4 0.2 0.0 Elastic amplitude [arb. units] 2.01.51.00.50.0 Q [Å-1] Elastic amplitude, T=140K, 1.0 MLof C6H6 Model 1 Model 2 Model 3 Model 4 Model 5 Debye-Waller factor of the basal plane of graphite 40x10-3 30 20 10 0 QENS amplitude [arb. units] 2.01.51.00.50.0 Q [Å-1] Quasi-elastic amplitude, at 140K, 0.1 MLof C6H6 Model 1 Model 2 Model 3 Model 4 Model 5 40x10-3 30 20 10 0 QENS amplitude [arb. units] 2.01.51.00.50.0 Q [Å-1] Quasi-elastic amplitude, at 140K, 1.0 MLof C6H6 Model 1 Model 2 Model 3 Model 4 Model 5 1.0x10-3 0.8 0.6 0.4 0.2 0.0 Background [arb. units] 2.01.51.00.50.0 Q [Å-1] Background, at 140K, 0.1 MLof C6H6 Model 1: single Lorentzian func. Model 2: single Gaussian func. Model 3: sum. of Lorentzian func. Model 4: sum. of Gaussian func. Model 6: jump rotatinal model 1.0x10-3 0.8 0.6 0.4 0.2 0.0 Background [arb. units] 2.01.51.00.50.0 Q [Å-1] Background, at 140K, 1.0 MLof C6H6 Model 1: single Lorentzian func. Model 2: single Gaussian func. Model 3: sum. of Lorentzian func. Model 4: sum. of Gaussian func. Model 6: jump rotatinal model Figure 5.1: Fitted parameters extracted from the application of models 1 to 6 to the 0.1 ML (left) and the 1.0 ML (right) data. Top panel: elastic amplitudes. We also include the theoretical graphite Debye-Waller factor. Medium panel: quasi-elastic amplitude. Bottom panel: background. the offset is not equal to unity. This is because, the experimental spectra and the experimental resolution function have been normalized to the incoming flux of neutrons and the vanadium scattering function (which provides the initial neutron energy distribution arising from the instrument scattering). Thus, S(Q,∆E= 0) 6= 1, but it can be easily verify that the summation of all the parameters, multiplied to the intensity of the experimental resolution function 5.2. Hydrogenated benzene 137 delivers the experimental intensity of the SF at ∆E= 0. 5.2.2 The quasi-elastic broadening In this section we discuss the quasi-elastic broadenings obtained with the fitting to the different models of the hydrogenated benzene scattering function. We recall that the quasi-elastic broadening, allows to identify the diffusive regime taking place in the system: Its dependence with the momentum transfer is the experimental signature of the molecule’s diffusive behavior. Furthermore, we can extract valuable information such as the diffusion coefficients and the friction parameters, yielding an understanding of the physical mechanism underlying the diffusive behavior of the molecules within the adsorbed layer. Each theoretical model predicts two important features of the quasi-elastic scattering function SQENS(Q, ∆E): the energy profile and the dependence of the HWHM of such energy profile, the so-called quasi-elastic broadening, on the momentum transfer. Both features never appear separately since they reveal the dynamics at the atomic level characterizing each diffusive regime. Both of them should be fulfilled by the experimental data which display the typical signature of a given diffusive regime. Comparison between the different models Figs. 5.2 and 5.3 summarize the quasi-elastic broadenings related to translations extracted from the fitting to all the theoretical models for the low coverage (0.1 ML and 0.2 ML) and the high coverage (0.5 ML and 1.0 ML) data at 140K. In the case of simple models such as model 1 and 2 the translational quasi-elastic broadening is directly the HWHM of the Lorentzian or the Gaussian function respectively. For models 3 and 4 involving a summation of quasi-elastic lines, the translational quasi-elastic broadening corresponds to the HWHM of the term indexed with n= 0. In the case of low coverage data (see left panel of Fig. 5.2), the Gaussian models, 2 and 4, provide linear quasielastic broadenings for the 0.1 ML data in the range of momentum below 1Å−1, while the quasi-elastic broadenings extracted from Lorentzian shaped models, 1 and 3 (or equivalently 5, where we fix the HWHM of the n=Nterm to the model 3 fitted result), do not follow the expected quadratic law of Q. This difference suggest that molecules undergo ballistic diffusion in this low coverage regime. On the other hand, the right panel of Fig. 5.2 displays the translational quasi-elastic broadenings extracted from the fitting of the 0.2 ML data. The quasi-elastic broadenings for the Gaussian models 2 and 4 behave as predicted theoretically (linear with Q) but the results of the fittings show that 138 Chapter 5. Results and discussions of the experimental data analysis the quasi-elastic energy profile presents a rather Lorentzian shape: the goodness of the fit parameter, χ2, for model 2 (single Gaussian function), 21.86, is almost the double than the χ2related to model 1 (single Lorentzian function) of 12.22 at 140K for a value of Q= 1.4Å−1(see Tab. 4.1 in Chap. 4). An important difference is also found between the χ2characterizing model 3 fitting, 3.57, versus the one arising from model 4 fitting, 10.33 for Q= 0.95 Å−1at 140 K (see 4.3). On the other hand, the Lorentzian models 1 and 3 display a quadratic law of Qin a narrow range of the momentum transfer between 0.5 Å−1and 1.5 Å−1. Important deviations are observable below and above this window. At low Qdeviations can be attributed to the poor quasi-elastic signal (the scattering intensity is dominated by the graphite scattering, see Fig. 5.1). Conversely, at high Q, the deviations from the quadratic law can be due to the raising of the scattering signal related to rotations, while simultaneously the intensity of translations is significantly reduced (we recall that in model 3, the intensity of the translational term, with index n= 0, is weighted by a Bessel function of the first kind and order n= 0). The results for the 0.5 ML and 1.0 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 QENS broadening [meV] 2.01.51.00.50.0 Q [Å-1] Translational quasi-elastic broadening, T=140K, 0.1 ML of C6H6 Model 1 Model 2 Model 3/ Model 5 (n=0) Model 4 (n=0) 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 QENS broadening [meV] 2.01.51.00.50.0 Q [Å-1] Translational quasi-elastic broadening, T=140K, 0.2 ML of C6H6 Model 1 Model 2 Model 3/ Model 5 (n=0) Model 4 (n=0) Figure 5.2: Quasi-elastic broadening extracted from the fitting of the experimental data for all the coverages to the theoretical model at 140K. We have included guides to the eye to underline the linear or quadratic dependence of the quasi-elastic broadening on the momentum transfer. ML data can be found in Fig. 5.3. We only display the quasi-elastic broadening of the Lorentzian shaped models, since we have already seen that their quasi-elastic energy profile has a marked Lorentzian character. As it happens with the 0.2 ML, the quasi-elastic broadenings only satisfies the quadratic law of Qin the range of momentum transfer spanning from 0.5 Å−1and 1.5 Å−1. However, we observe that the high Qrange values for the quasi-elastic broadening, especially for the 1.0 ML, seems to follow a quadratic different law with a lower diffusion coefficient. But this can be simply an effect of the rotational 5.2. Hydrogenated benzene 139 quasi-elastic broadening becoming very important at this range of Qvalues (remember that at Q > 1Å−1we probe distances comparable with the size of the benzene molecule). To finish with the comparison between models, we wish 1.0 0.8 0.6 0.4 0.2 0.0 QENS broadening [meV] 2.01.51.00.50.0 Q [Å-1] Translational quasi-elastic broadening, T=140K, 0.5 ML of C6H6 Model 1 Model 3/Model 5 (n=0) 1.0 0.8 0.6 0.4 0.2 0.0 QENS broadening [meV] 2.01.51.00.50.0 Q [Å-1] Translational quasi-elastic broadening, T=140K, 1.0 ML of C6H6 Model 1 Model 3/Model 5 (n=0) Figure 5.3: Quasi-elastic broadening extracted from the fitting of the experimental data for all the coverages to the theoretical model at 140K. We have included guides to the eye to underline the linear or quadratic dependence of the quasi-elastic broadening on the momentum transfer. to comment the differences between the quasi-elastic broadenings arising from single quasi-elastic and multiple quasi-elastic profiles models. In general terms, the quasi-elastic broadenings of model 1 and 2 display higher values than for models 3 or 4. In the former models there is a single quasi-elastic line and the quasi-elastic broadening is the HWHM of the total quasi-elastic energy profile of the scattering function (see Figs. 4.1, for instance). Conversely, models 3 and 4 are made of a summation of Lorentzian and Gaussian functions arising from the separation between translational and rotational motion (see Eq. 4.21 in the previous chapter). What we call the translational quasi-elastic broadening is the HWHM of the term in the summation with index n= 0, and it only contains a portion of the total HWHM observable in the SF. Accordingly, its values should be much smaller than the ones arising from models where a single quasi-elatic line is considered. Dependence on coverage We have already seen when comparing the different models, that there are significant changes of the diffusive regime with the coverage. In this section we fit the quasi-elastic broadenings to the theoretical dependence with the momentum transfer established by each model. We expect to obtain valuable information about the diffusion process, like the mean square velocity of the 140 Chapter 5. Results and discussions of the experimental data analysis molecules for the very low coverage sample, where ballistic motion is observed, or the friction parameter for higher coverage samples. We display in the left panel of Fig. 5.4 the translational quasi-elastic broadenings (the HWHM of the Gaussian function for model 2 or the HWHM of the Gaussian function with index n= 0 for model 4) extracted from the fitting of the 0.1 ML scattering function to models 2 and 4. In the right panel of the same figure, we show the rotational quasi-elastic broadening (the HWHM of the Gaussian function with index n= 1) arising from the fitting of the 0.2 ML data to model 4. We recall that model 2 is made of a single Gaussian function and does not consider separately translations and rotations, while model 4 which is a summation of Gaussian does. We fit the translational quasi-elastic broadenings to a linear law of the momentum transfer where the slope is proportional to the mean square velocity of the molecules on the surface (as stated theoretically, see Tab. 5.1). Conversely the rotational quasi-elastic broadening should be Q-independent and its value is also related to the mean square angular velocity. The extracted mean square velocity and mean square angular velocity are summarized in Tab. 5.2. 2.0 1.5 1.0 0.5 0.0 QENS broadening [meV] 2.01.51.00.50.0 Q [Å-1] Translational quasi-elastic broadening, T=140K, 0.1 ML of C6H6, Model 2 Model 4, n=0 1.0 0.8 0.6 0.4 0.2 0.0 QENS broadening [meV] 2.01.51.00.50.0 Q [Å-1] Rotational quasi-elastic broadening, T=140K, 0.1 ML C6H6 Model 4 (n=1) Figure 5.4: Translational quasi-elastic broadening extracted from the fitting of the 0.1 ML scattering function to models 2 and 4 at 140K. For model 2, the translational quasi-elastic broadening is directly the HWHM of the quasi-elastic profile. For model 4, this is the HWHM of the term n= 0 in the summation of Gaussians. We have fitted both quasi-elastic broadening to a linear law of Q, as established by the theoretical models. Model 4 provides fitted values for the mean square velocities which are in better agreement with the equipartition theorem than those deduced from model 2. This indicates that molecules not only undergo ballistic translations, but they also perform ballistic rotations. However, we should note that the 5.2. Hydrogenated benzene 141 Table 5.2: Results for the mean square velocity resulting from the fitting of the quasielastic broadening from models 2 and 4 to a linear law of the momentum transfer. We also include the mean square angular velocity extracted from the fitting of the rotational quasi-elastic broadening (the HWHM of the n= 1 term in the summation of Gaussian functions of model 2). We compare the fitted values with the equipartition theorem prediction. Model 2 Model 4 Equipartition theorem v2[Å2.ps−2] 4.87±0.04 2.62±0.02 2.98 ω2[10−1ps−2] low Q: 4.20±0.03/ high Q: 3.07±0.04 4.77 agreement with the equipartition theorem is excellent when we consider the fit for the low Qrange (below 1.0 Å−1), but it deviates at higher Qvalues, especially for the mean square angular velocity. This follows from the fact that the translational quasi-elastic broadening deviates from the linear law for Q > 1.0Å−1, suggesting that in this high Qrange the separation between translations and rotations is not properly accounted by model 2. In this very high momentum transfer range, the contribution of translations is strongly reduced, while the rotational contribution to the scattering becomes dominating. Furthermore, at this high Qvalues, the Gaussian approximation might no longer hold and the effect of non-Gaussian effects (we should consider further terms of the cumulant expansion, see Eq. 2.30 in Chap. 2) can enter into play [102]. We observe a reduction of the 20% between the experimental value of the quasi-elastic broadening at Q= 1.5Å−1(1.10 meV) with respect to the theoretical prediction of the linear law of Q(1.4 meV). A similar effect is described in Ref. [102] where they also found in the liquid argon scattering function a decrease of the HWHM of the scattering function of 20% when they include the non-Gaussian correction terms. We discuss now the results for higher coverage data. We choose the results arising from model 3 rather than model 1 on the basis of the quality of the fitting. We observe that the distinction between translational and rotational motion improves the goodness of the fitting. The left panel of Fig. 5.5 displays the translational quasi-elastic broadenings (the HWHM of the term n= 0 in the summation of Lorentzian function of model 3) extracted from the fitting of 0.2 ML, 0.5 ML and 1.0 ML to model 3. The fittings to a square law of the momentum transfer are also displayed, and the resulting diffusion coef- 142 Chapter 5. Results and discussions of the experimental data analysis ficient and friction parameters are listed in Tab. 5.3. The quadratic law of the momentum transfer is satisfied in a very narrow range of Q, where the translational part of the scattering dominates. At very low Q(below 0.5 Å−1) the substrate scattering is very intense while at high Q, model 3 encounters the problem of a very large rotational contribution in comparison with a tiny translational part. In addition, the interpretation of the high Qrange dependence of the quasi-elastic broadening gets complicated by the possibility that the Gaussian approximation on which the model is based does not hold. In any case, we should stress that the extracted diffusion coefficients and friction parameters display a strong dependence on coverage. This is the fingerprint of collisional friction, since phononic and electronic friction are coverage independent. Furthermore the estimated phononic friction value, 2.2×107s−1, is too small to account for the very high extracted friction. We have also computed the collisional friction related to benzene molecules enduring binary collisions and exchanging translational and rotational momentum, as it is proposed in Refs. [103,136] (see Eqs. 4.38). The resulting values are included in the tables, to provide a theoretical benchmark for comparison with the experimental parameters. We observe that binary collisions between benzene molecules generate a too small friction parameter. Besides, the dependence with coverage is not exactly proportional. A possible improvement consists in consider the collision of single benzene molecule with small clusters. This situation is compatible with the phase diagram of the benzene monolayer adsorbed on graphite where the existence of a solid-liquid coexistence phase is observed at around 130 K [34,41]. Besides, the experimental translational diffusion coefficients are in good agreement with MD simulations studying the melting process of the complete monolayer of benzene on graphite (1.0 ML) [34], were a value of 0.60 Å2.ps−1is reported at 140K. The collisional friction in that situation can still be accounted in the theoretical frame of uncorrelated binary collisions [147], but considering that the system presents a mixture of different kind of particles: single benzene molecules in one hand, and small clusters in the other. In that case, the friction parameter would scale with the mass of the small clusters if we consider it as a rigid disk. In the case of the rotational motion, the quasi-elastic broadening is the HWHM of the term indexed with n= 1 in the Lorentzian function summation of of model 3. The right panel of Fig. 5.5 contains the rotational quasi-elastic broadening. Following model 3, it should be Qindependent, and from its value we can deduce the rotational diffusion coefficient and the rotational friction parameter. The fitted results can be found in Tab. 5.4. In that case, the dependence with coverage is still very strong. Furthermore, the rotational friction parameters present much higher values than the translational ones. However, care must be taken since the rotational quasi-elastic broadening displays a Qdependence, even if theoretically this is not predicted. The resulting values of the rotational diffusion coefficient and