An effective forces study of polarized asymmetric nuclear matter
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MASTER THESIS An effective forces study of polarized asymmetric nuclear matter Author: Alejandro ROMERO ROS Supervisors: Dr. Artur POLLS MARTÍ Dr. Bruno JULIÁ-DÍAZ Dep. Física Quàntica i Astrofísica Facultat de Física Universitat de Barcelona August 2, 2017
iii Acknowledgements I would like to sincerely thank my advisors, Dr. Artur Polls and Dr. Bruno Juliá-Díaz, for their constant help and support on the fulfilment of this project. Also, I would like to thank my friends and family for their companion and enlivenment.
v Contents Acknowledgements iii 1 Effective Forces 1 1.1 The Skyrme interaction . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 The Gogny interaction . . . . . . . . . . . . . . . . . . . . . . . 3 2 Hartree-Fock Formalism 5 2.1 Mean field description: Hartree-Fock . . . . . . . . . . . . . . . 6 2.2 Two-body matrix elements . . . . . . . . . . . . . . . . . . . . . 8 3 Energy Contributions 11 3.1 Single-Particle Potential . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Kinetic and Potential Energies . . . . . . . . . . . . . . . . . . . 15 3.2.1 Potential Energy using the Slater Function . . . . . . . 20 3.3 TotalEnergy............................. 23 3.3.1 Isospin symmetry energy . . . . . . . . . . . . . . . . . 26 3.3.2 Spin symmetry energy . . . . . . . . . . . . . . . . . . . 27 4 Spin-Isospin Channel Contributions 31 4.1 Partial Wave Decomposition . . . . . . . . . . . . . . . . . . . . 34 5 Interaction Potential Parametrization 39 6 Magnetic Susceptibility 45 Summary and Conclusions 49 A Integrals and Formulae 51 A.1 Usefulintegrals ........................... 51 A.2 Skyrme kinetic and potential energy . . . . . . . . . . . . . . . 53 A.3 Slater results for the Gogny interaction . . . . . . . . . . . . . . 55 A.4 Spin-isospin third component channel contributions . . . . . . 56 A.5 Magnetic susceptibility matrix elements for the Skyrme interaction................................. 61 Bibliography 63
1 Chapter 1 Effective Forces The properties of homogeneous nuclear and neutron matter at high densities play a crucial role in the determination of the structure of neutron star interiors [1, 2, 3]. Terrestrial nuclei do not provide enough information to constrain an equation of state (EoS) under the extreme conditions of density and isospin asymmetry, and also a possible spin polarization that we can find inside neutron stars. Thus, a potentially safe way to obtain the EoS relies on microscopic many-body calculations based on realistic nucleon-nucleon (NN) interactions [4]. These realistic NN interactions should fulfil a minimum set of requirements. In particular, realistic potentials are build to reproduce the Nijmegen database [5], which contains a full set of NN elastic scattering phase-shifts up to energies of about 350 MeV and also to reproduce the properties of the only two nucleon bound state: the deuteron, which is bound by 2.22 MeV in the partial wave 3S1−3D1. In principle, the realistic interactions are the apppropiate ones to be used in the so called ab initio many-body calculations, which aim at providing a first-principles description of the EoS of symmetric nuclear matter and neutron matter. The NN interactions fulfilling these minimum requirements have a notorious complexity and depend on the spin and isospin degrees of freedom of the NN system. In particular, the tensor dependence, which is the direct responsible of the coupling between the 3S1and 3D1partial waves in the deuteron, increases the complexity of the problem. Several many-body techniques have been developed to deal with these realistic interactions. Due to the strong repulsion present in any realistic interaction, they require the use of correlated wave functions or the sum of infinite perturbative diagrams (ladder diagrams) [6]. The large complexity of the calculations and the NN interactions themselves make very convenient the use of effective NN interactions, which are commonly implemented in the Hartree-Fock approximation and, therefore, easier to treat. Although effective forces have successfully been used in both finite nuclei and nuclear matter, they do not reproduce the phase-shifts nor the properties of the deuteron. By construction, most of the effective forces used in the literature are well behaved around the saturation density of nuclear matter and for moderate isospin asymmetries. In general, these effective interactions depend on a set
2Chapter 1. Effective Forces of parameters that are determined by reproducing basic properties of symmetric nuclear matter as the saturation density ρ0, the binding energy e0(ρ0), the incompressibility modulus κ(ρ0)and the symmetry energy esym(ρ0)together with the binding energy of some double magic nuclei [7]. Once a parametrization is determined, one can, for instance, study nuclei and nuclear matter under conditions far from those used to define the force, in quite a simple way. When doing so, however, one must always keep in mind the particular limitations of the parametrization that has been using. In this work, we will use two types of effective interactions: the Skyrme interaction [8, 9], that has a zero-range character and a simple spin-isospin dependence, and the Gogny interaction [10], which besides a zero-range term has finite-range terms of Gaussian shape along with a spin-isospin dependence. The main objective will be to build the Hartree-Fock framework to calculate single-particle potentials and total energy per particle of polarized asymmetric nuclear matter and explore the possible spin instabilities of nuclear and neutron matter with effective forces constructed using conditions of zero spin polarization. 1.1 The Skyrme interaction The Skyrme forces where introduced by T. H. R. Skyrme [8] around 1950 and, since the pioneering work of Vautherin and Brink [9], there has been an intensive and successful use of Skyrme effective NN interactions [11, 12]. They are characterized by a zero-range interaction (contact interactions) that leads to simple analytical expressions for basic properties of nuclear matter such as the ones we have already mentioned (binding energy, incompressibility modulus, saturation density and symmetry energy). A Skyrme force has the following structure: ˆvSk(r) =t0(1 + x0ˆ Pσ)δ(r) +1 6t3(1 + x3ˆ Pσ)[ρ(R)]γδ(r) +1 2t1(1 + x1ˆ Pσ)hδ(r)k02+k2δ(r)i +1 2t4(1 + x4ˆ Pσ)hδ(r)ρ(R)k02+k2ρ(R)δ(r)i +t2(1 + x2ˆ Pσ)k0δ(r)k+t5(1 + x5ˆ Pσ)k0ρ(R)δ(r)k +iW0(σ+σ0)·k0×δ(r)k, (1.1) where r=r1−r2,R= (r1+r2)/2,k= (∇1−∇2)/2iis the relative momentum acting on the right and k0is its conjugate acting on the left, ˆ Pσ= (1+σ·σ0)/2is the spin exchange operator and γ, tiand xiare the parameters of the force. The contact terms provide the S-wave contributions while momentum dependent ones (gradient terms) give the P-wave components. The
1.2. The Gogny interaction 3 spin-orbit term (W0) is irrelevant for infinite nuclear and neutron matter and we will not consider it in our calculations. The main advantage of these forces comes from their analytical character which makes it very useful to get a physical insight into problems where the fully microscopic calculations are either very time-consuming or not yet possible to implement. Notice that not all the Skyrme parameters can be completely well determined through the fits of given sets of data and only certain combinations related to the basic properties mentioned before are really empirically determined [13]. This leads to a scenario where, for instance, different Skyrme forces produce similar EoS for symmetric nuclear matter but very different results for neutron matter. This is easily understood if one considers that neutron matter or, equivalently, systems with large isospin asymmetry are not part of the common input data employed to determine the parameters of the interaction. This feature should be corrected if Skyrme-type forces are to be used in conditions of large neutron to proton ratios such as neutron matter in neutron stars or nuclei near the drip line. Recently, an extensive and systematic study has tested the capabilities of almost 90 existing Skyrme parametrizations to provide good neutron-star properties [14]. It was found that only 27 of these forces passed the restrictive tests imposed, the key being the behaviour (increasing) of the symmetry energy Esym(ρ)with density. A very convenient set of Skyrme forces are the ones provided by the Lyon group [15, 16], which also take into account variational results for neutron matter obtained with realistic interactions to fix the parameters of the force [17]. In particular, we have chosen SLy4 for our calculations. The parameters which define SLy4 are reported in table 1.1. The properties of symmetric nuclear matter provided by SLy4 are shown in table 1.3. t0=−2488.9MeVfm3t1= 486.8MeVfm5t2=−546.4MeVfm5t3= 13 777 MeVfm7/2 x0= 0.83 x1=−0.34 x2=−1x3= 1.35 TABLE 1.1: Parameters of the Skyrme SLy4 interaction. Not shown parameters are zero. γ= 1/6. 1.2 The Gogny interaction The Gogny interaction, proposed in 1980 by D. Gogny [10], is a well known and extensively used effective nuclear interaction. Unlike the Skyrme force, the Gogny interaction has a build-in finite-range which brings the Gogny force closer to a realistic interaction.
10 Chapter 2. Hartree-Fock Formalism -1500 -1000 -500 0 500 1000 1500 2000 2500 Skyrme SLy4 Ω<kστ,k’σ’τ’|v|kστ,k’σ’τ’>A [MeVfm3] σ = σ’;τ = τ’ σ = σ’;τ ≠ τ’ σ ≠ σ’;τ = τ’ σ ≠ σ’;τ ≠ τ’ -1500 -1000 -500 0 500 1000 1500 2000 2500 0 0.5 1 1.5 2 2.5 Gogny D1 k’ [fm-1] Ω<kστ,k’σ’τ’|v|kστ,k’σ’τ’>A [MeVfm3] FIGURE 2.1: Matrix elements of the Skyrme SLy4 (letf) and Gogny D1 (right) interactions times volume as functions of the momentum k0at saturation density ρ0= 0.16 fm−3with k= 0 fm−1.
11 Chapter 3 Energy Contributions After presenting the two families of effective forces (Skyrme and Gogny) considered in Chapter 1, and with the framework developed in Chapter 2, in this chapter we present the calculations of the total energy of polarized asymmetric nuclear matter and the single-particle potential that feels a nucleon with momentum kand a given third component of the spin (σ) and isospin (τ) due to the interaction with the other nucleons. We calculate first the single-particle potential followed by the calculation of the kinetic and potential energies, which define the total energy of the system. The calculation of the potential energy is performed by summing all the two-body matrix elements. After that, we present an alternative calculation for the potential energy using the Slater functions, which is equivalent to the calculation of the distribution function of the free Fermi sea. Finally, we calculate the isospin and spin symmetry energies. 3.1 Single-Particle Potential The single-particle potential, defined in Eq. (2.10), is Uστ (k) = X σ0τ0X k0≤k0 F σ0τ0 hkστ, k0σ0τ0|ˆv|kστ, k0σ0τ0iA.(3.1) In the case of the Skyrme interaction, the single-particle potential for a given σand τ, calculated using Eq. (2.19) and Eq. (3.1), has the following expression: USk στ (k) = X σ0τ0t0+1 6t3ργ+1 4k2+3 5k0 Fσ0τ0 2(t1+t2)ρσ0τ0 +t0x0+1 6t3x3ργ+1 4k2+3 5k0 Fσ0τ0 2(t1x1+t2x2)ρσ0τ0δσσ0 −t0+1 6t3ργ+1 4k2+3 5k0 Fσ0τ0 2(t1−t2)ρσ0τ0δσσ0δττ0 −t0x0+1 6t3x3ργ+1 4k2+3 5k0 Fσ0τ0 2(t1x1−t2x2)ρσ0τ0δττ0, (3.2)
12 Chapter 3. Energy Contributions being k0 Fσ0τ0the Fermi momentum for the σ0τ0component, Eq. (2.16). Notice that the single-particle spectrum is parabolic in kand, therefore, the effective mass is momentum independent. For the Gogny force, the single-particle potential for a given στ, calculated using Eq. (2.20) and Eq. (3.1), reads UG στ (k) = X σ0τ0ργt0[(1 −δσσ0δττ0) + x0(δσσ0−δττ0)]ρσ0τ0 +π3/2X i µ3 i[Wi+Biδσσ0−Hiδττ0−Miδσσ0δττ0]ρσ0τ0 −1 √πX i [Wiδσσ0δττ0+Biδττ0−Hiδσσ0−Mi] ×1 µike−q2 +−e−q2 −+√π 2erf(q−, q+), (3.3) where q±=µi 2(k±k0 Fσ0τ0)and erf(a, b) = 2 √πRb adx e−x2is the error function. Notice that the last term of the previous equation should be carefully computed when k= 0. In this case UG στ (k= 0) reads UG στ (0) = X σ0τ0ργt0[(1 −δσσ0δττ0) + x0(δσσ0−δττ0)]ρσ0τ0 +π3/2X i µ3 i[Wi+Biδσσ0−Hiδττ0−Miδσσ0δττ0]ρσ0τ0 −1 √πX i [Wiδσσ0δττ0+Biδττ0−Hiδσσ0−Mi] ×√πerf µik0 Fσ0τ0 2−µik0 Fσ0τ0e−µi 4k0 F σ0τ0. (3.4) In Figs. 3.1 and 3.2, we show the results of the single-particle potential for the Skyrme SLy4 interaction and the Gogny D1 interaction, respectively. Both figures show the neutron (Un↑(k),Un↓(k)) (upper panels) and proton (Up↑(k), Up↓(k)) (lower panels) single-particle potentials for different asymmetries (β) and spin polarizations (∆n,∆p) at saturation density ρ0= 0.16 fm−3. The results for symmetric nuclear matter (β= 0) are shown in the left and central columns, whereas the results for asymmetric nuclear matter with β= 0.5 are shown in the right column. Solid black lines show the results of nonpolarized matter (∆n= ∆p= 0) as a reference. Dotted red and green lines show the results of spin-polarized matter. The main difference between the Skyrme SLy4 and the Gogny D1 interactions is that the single-particle potential for the Skyrme SLy4 interaction shows a parabolic behaviour with k, see Eq. (3.2), whereas the single-particle potential for the Gogny D1 interaction does not show a global parabolic behaviour in the full range of k, see Eq. (3.3), showing a saturation behaviour for large values of k. This effect is caused because the Skyrme interaction is a
3.1. Single-Particle Potential 13 Skyrme SLy4 -150 -100 -50 0 50 100 150 200 USk nσ(k) [MeV] β=0, ∆n=0.75, ∆p=0 β=0, ∆n=0.75, ∆p=0.5 n↓ n↑ -150 -100 -50 0 50 100 150 200 β=0.5, ∆n=0.75, ∆p=0.5 -150 -100 -50 0 50 100 150 200 0 1 2 3 4 5 6 7 USk pσ(k) [MeV] k [fm-1] 0 1 2 3 4 5 6 7 k [fm-1] p↓ p↑ 0 1 2 3 4 5 6 7 -150 -100 -50 0 50 100 150 200 k [fm-1] FIGURE 3.1: Skyrme SLy4 interaction single-particle potentials. Upper panels show the neutron single-particle potentials and lower panels show the proton single-particle potentials as a function of momentum kfor different asymmetries and spin polarizations at saturation density of symmetric nuclear matter ρ0= 0.16 fm−3. Left and central panels show results for symmetric nuclear matter whereas right panels shows results for asymmetric matter. Solid black lines show the results for nonpolarized matter (∆n= ∆p= 0) as a reference, whereas dotted red and green lines show the results for spin-polarized matter.
14 Chapter 3. Energy Contributions Gogny D1 -150 -100 -50 0 50 UG nσ(k) [MeV] β=0, ∆n=0.75, ∆p=0 β=0, ∆n=0.75, ∆p=0.5 n↓ n↑ -150 -100 -50 0 50 β=0.5, ∆n=0.75, ∆p=0.5 -150 -100 -50 0 50 0 1 2 3 4 5 6 7 UG pσ(k) [MeV] k [fm-1] 0 1 2 3 4 5 6 7 k [fm-1] p↓ p↑ 0 1 2 3 4 5 6 7 -150 -100 -50 0 50 k [fm-1] FIGURE 3.2: Gogny D1 interaction single-particle potentials. Upper panels show the neutron single-particle potentials and lower panels show the proton single-particle potentials as a function of momentum kfor different asymmetries and spin polarizations at saturation density of symmetric nuclear matter ρ0= 0.16 fm−3. Left and central panels show results for symmetric nuclear matter whereas right panels shows results for asymmetric matter. Solid black lines show the results for nonpolarized matter (∆n= ∆p= 0) as a reference, whereas dotted red and green lines show the results for spin-polarized matter.
3.2. Kinetic and Potential Energies 15 zero-range force while the Gogny interaction has a finite-range. Notice that the saturation behaviour of the single-particle potential for the Gogny interaction for large kis a consequence of the fact that the exchange term of the interaction goes to zero. Thus, the contribution for large kis provided by the direct term, which has no momentum dependence. One can point out that for non-polarized asymmetric nuclear matter the single-particle spectrum for protons becomes more attractive than the one for neutrons. This is true for both the Skyrme SLy4 and the Gogny D1 interactions. It can be seen by comparing the solid black lines of the upper and lower right column panels of both previous figures. Concerning the dependence of the single-particle potentials of the species with the spin-polarization, we observe an opposite behaviour for the polarized neutrons. In fact, the single-particle potential of the less abundant component (n↓) becomes more attractive than that of the most abundant component (n↑) in the case of the Gogny D1 interaction. The opposite is true for the Skyrme SLy4 interaction. For the proton single-particle spectrum, we observe a similar behaviour for both forces, being, in general, the less abundant component (p↓) more attractive, for both interactions. 3.2 Kinetic and Potential Energies We have already calculated the single-particle spectrum of the different species for different spin polarizations and isospin asymmetries. The next step will be to calculate the kinetic and the potential energy contributions to the total energy. The kinetic energy per particle1, in a fermionic system composed by different species, is the following: 1 NhΨ|ˆ T|Ψi=1 NX στ X k<kF στ hkστ|ˆ k|kστi=3 5 ~2 2mn (k2 Fn↑+k2 Fn↓) + 3 5 ~2 2mp (k2 Fp↑+k2 Fp↓) =~2 2mn 1 ρ(ξn↑+ξn↓) + ~2 2mp 1 ρ(ξp↑+ξp↓), (3.5) where ξτσ are related to the average kinetic energy of the Fermi model of nuclear matter [22], ξτσ =3 5k2 Fστ ρτσ =3 5(6π2ρτσ)2/3ρτσ =3 203 2π2ρ2/3 ρ(1 ±β)5/3(1 ±∆τ)5/3. (3.6) 1The kinetic energy per particle as a function of the total density ρ, isospin asymmetry β and neutron and proton spin polarizations ∆nand ∆pis provided in the Eq. (A.10)
16 Chapter 3. Energy Contributions The potential energy per particle can be calculated by performing the sum over the spin and isospin components and the momentum kup to the Fermi momentum kFστ of the single-particle potential of nucleons with momentum kspin σand isospin τ: 1 NhΨFS|ˆ V|ΨFSi=1 2NX στ X k≤kF στ Uστ (k).(3.7) Equivalently, using Eq.(3.1), the previous expression can be written as hVi N=1 2NX σσ0ττ0 Ω2 (2π)6ZkF στ Zk0 F σ0τ0 d3kd3k0hkστ, k0σ0τ0|ˆv|kστ, k0σ0τ0iA, (3.8) where the integrals over momentum can be expressed in terms of a function Φ(k,k0)with a k, k0dependence, can be found in the appendix A.1 for both the Skyrme and Gogny interactions. I[Φ(k,k0); kFστ , k0 Fσ0τ0]≡1 2N Ω2 (2π)6ZkF στ Zk0 F σ0τ0 d3kd3k0Φ(k,k0).(3.9) For the Skyrme interaction, the potential energy per particle2reads VSk N=1 4ρ[2t2(1 + x2)][ξn↑ρn↑+ξn↓ρn↓+ξp↑ρp↑+ξp↓ρp↓] +1 4ρ[t1(1 −x1) + t2(1 + x2)][ξn↑ρn↓+ξn↓ρn↑+ξp↑ρp↓+ξp↓ρp↑] +1 4ρ[t1(1 + x1) + t2(1 + x2)][ξn↑ρp↑+ξn↓ρp↓+ξp↑ρn↑+ξp↓ρn↓] +1 4ρ[t1+t2][ξn↑ρp↓+ξn↓ρp↑+ξp↑ρn↓+ξp↓ρn↑] +1 ρt0+1 6t3ργρnρp+1 ρt0x0+1 6t3x3ργ[ρn↑ρp↑+ρn↓ρp↓] +1 ρt0(1 −x0) + 1 6t3(1 −x3)ργ[ρn↑ρn↓+ρp↑ρp↓], (3.10) 2The Skyrme potential energy per particle as a function of the total density ρ, isospin asymmetry βand neutron and proton spin polarizations ∆nand ∆pis provided in the Eq. (A.11)
3.2. Kinetic and Potential Energies 17 while the potential energy per particle for the Gogny interaction reads VG N=ργ−1t0(1 −x0)(ρn↑ρn↓+ρp↑ρp↓) + ργ−1t0ρnρp+ργ−1t0x0(ρn↑ρp↑+ρn↓ρp↓) +π3/2 2ρX i µ3 iWiρ2+Bi(ρn↑+ρp↑)2+ (ρn↓+ρp↓)2 −Hi(ρ2 n+ρ2 p)−Mi(ρ2 n↑+ρ2 n↓+ρ2 p↑+ρ2 p↓) −1 4ρ√πX i(Wi+Bi−Hi−Mi)gi(kFn↑, kF n↑) + gi(kF n↓, kFn↓) +gi(kFp↑, kF p↑) + gi(kF p↓, kFp↓) + (Bi−Mi)gi(kFn↑, kF n↓) + gi(kF n↓, kFn↑) +gi(kFp↑, kF p↓) + gi(kF p↓, kFp↑) −(Hi+Mi)gi(kFn↑, kF p↑) + gi(kF n↓, kFp↓) +gi(kFp↑, kF n↑) + gi(kF p↓, kFn↓) −Migi(kFn↑, kF p↓) + gi(kF n↓, kFp↑) +gi(kFp↑, kF n↓) + gi(kF p↓, kFn↑), (3.11) where gi(kFστ , k0 Fσ0τ0) = 1 3π2µ3 iµ2 ik2 Fστ −µikF στ µik0 Fσ0τ0+µ2 ik02 Fσ0τ0−2e−µ2 i 4(kF στ +k0 F σ0τ0)2 −µ2 ik2 Fστ +µikF στ µik0 Fσ0τ0+µ2 ik02 Fσ0τ0−2e−µ2 i 4(kF στ −k0 F σ0τ0)2 −√πµ3 i 2k3 Fστ −k03 Fσ0τ0erf µi 2(kFστ −k0 Fσ0τ0) +√πµ3 i 2k3 Fστ +k03 Fσ0τ0erf µi 2(kFστ +k0 Fσ0τ0). (3.12) Notice that gi(kFστ ,0) = gi(0, k0 Fσ0τ0) = gi(0,0) = 0. In Fig. 3.3 we show the kinetic energy (upper row) and the potential energy per particle for the Skyrme SLy4 interaction (middle row) and for the Gogny D1 interaction (lower row) for non-polarized (∆n= ∆p= 0), totally polarized (∆n= ∆p=±1) and anti-polarized (∆n=−∆p=±1) nuclear matter. The case of symmetric nuclear matter (β= 0) is shown in the left column, asymmetric nuclear matter with β= 0.5is shown in the central column and neutron matter (β= 1) is presented in the right column. First of all, we observe that the kinetic energy increases with the isospin
18 Chapter 3. Energy Contributions 0 20 40 60 80 100 120 T/N [MeV] β=0 β=0.5 0 20 40 60 80 100 120 β=1 -60 -40 -20 0 20 40 60 80 VSk/N [MeV] -60 -40 -20 0 20 40 60 80 -60 -40 -20 0 20 40 60 80 0 0.1 0.2 0.3 0.4 VG/N [MeV] ρ [fm-3] 0 0.1 0.2 0.3 0.4 ρ [fm-3] ∆n=∆p=0 ∆n=∆p=±1 ∆n=-∆p=±1 0 0.1 0.2 0.3 0.4 -60 -40 -20 0 20 40 60 80 ρ [fm-3] FIGURE 3.3: Kinetic (upper row) and potential (middle and lower row) energy per particle for the Skyrme SLy4 interaction (middle row) and for the Gogny D1 interaction (lower row) as functions of the total density for symmetric nuclear matter (left column), asymmetric nuclear matter (central column) and neutron matter (right column) and non-polarized (∆n= ∆p= 0), totally polarized (∆n= ∆p=±1) and antipolarized (∆n=−∆p=±1) nuclear matter.
3.2. Kinetic and Potential Energies 19 -55 -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 VG/N [MeV] ρ [fm-3] Gogny D1 β=0 β=0.50 β=0.75 β=0.90 β=1 FIGURE 3.4: Gogny D1 potential energy per particle of nonpolarized matter as a function of the total density for diffetent isospin asymmetry values. asymmetry regardless of the spin polarization of the system. However, this statement is not true for the potential energy, for neither the Skyrme SLy4 nor the Gogny D1 interactions. For the Skyrme SLy4, the potential energy of non-polarized and antipolarized matter becomes more repulsive as the isospin asymmetry increases, whereas the potential energy of totally polarized matter becomes more attractive. Similarly, the Gogny D1 potential energy of anti-polarized matter becomes more repulsive as the isospin asymmetry increases and the opposite happens to the potential energy of totally polarized matter. It is worth to point out the behaviour of the potential energy of nonpolarized matter with the Gogny D1 interaction, showed in Fig. 3.4. There is a density value around ρ∼0.365 fm−3at which the potential energy per particle is almost independent of the isospin asymmetry. Therefore, the potential energy for non-polarized matter becomes more repulsive for density values ρ≤0.365 fm−3and becomes more attractive for density values ρ≥0.365 fm−3 as the isospin asymmetry increases. Let us now analyse the kinetic energy dependence on the spin polarization. We observe that the kinetic energy of non-polarized matter is smaller than that of totally and anti-polarized matter for any isospin asymmetry. This is a direct consequence of the Fermi character of the nucleon. In a similar way, the potential energy of non-polarized matter is more
26 Chapter 3. Energy Contributions state is preferred. The value of ¯ρincreases with the asymmetry and it is always larger for the Gogny D1 interaction. We oserve that the energy per particle of non-polarized symmetric matter for both the Skyrme SLy4 and Gogny D1 interactions is very similar and becomes more repulsive for large isospin asymmetries. However, they completly differ for non-polarized neutron matter. Notice that for the case of totally polarized neutron matter both interactions predict similar results because, as we have already showed in Fig. 2.1 and Fig. 3.3, it can be approximated by a free Fermi gas. 3.3.1 Isospin symmetry energy Since the energy per particle of non-polarized asymmetric nuclear matter is a function of the total density and the isospin asymmetry, it can generally be expressed as a Taylor expansion around β= 0. That is, e(ρ, β) = e(ρ, 0) + esym(ρ)β2+e4(ρ)β4+O(β6).(3.19) Assuming charge symmetry of the nuclear forces, the interaction is symmetric under neutron and proton exchange and only even powers of βappear in the expansion. esym(ρ)is the so called symmetry energy and, if we only take the expansion up to second order, it can be approximated by, esym(ρ)∼e(ρ, 1) −e(ρ, 0) ,(3.20) where e(ρ, 1) and e(ρ, 0) are the energy per particle of non-polarized neutron matter (β= 1) and for non-polarized symmetric nuclear matter (β= 0), respectively. Notice that a change of sign of esym(ρ)would mean that there is a phase transition from nuclear to neutron matter. In Fig. 3.7 we show the symmetry energy (solid line) for the Skyrme SLy4 interaction (left panel) and the Gogny D1 interaction (right panel) as functions of the total density. The total energy per particle (dotted lines) of nonpolarized (∆n= ∆p= 0) symmetric nuclear matter (β= 0,∆n= ∆p= 0) and neutron matter (β= 1) are also plotted in the same figure. The Skyrme SLy4 interaction shows no phase transition from nuclear matter to neutron matter at any value of the density. However, the Gogny D1 interaction does show a phase transition at ρt∼0.52 fm−3. This means that, for non-polarized matter, for ρ > 0.52 fm−3the ground state ceases to be nuclear matter to become neutron matter. The study of the symmetry energy of nuclear matter has also generated many theoretical works to analyse its density dependence around the saturation density and to understand the physical constraints imposed by experimental data [25, 26, 27, 28, 29].
3.3. Total Energy 27 -30 -20 -10 0 10 20 30 40 50 60 0 0.1 0.2 0.3 0.4 0.5 0.6 Energy per particle [MeV] ρ [fm-3] Skyrme SLy4 0 0.1 0.2 0.3 0.4 0.5 0.6 -30 -20 -10 0 10 20 30 40 50 60 ρ [fm-3] Gogny D1 e(β=0) e(β=1) esym FIGURE 3.7: Isospin symmetry energy per particle (solid line) and total energy per particle (dotted lines) of non-polarized (∆n= ∆p= 0) symmetric nuclear matter (β= 0) and neutron matter (β= 1) for the Skyrme SLy4 interaction (left panel) and the Gogny D1 interaction (right panel) as functions of the total density. 3.3.2 Spin symmetry energy The energy per particle of spin-polarized neutron matter does not change when a global flip of the spins is performed. Therefore, it can be expanded on the spin polarization ∆nas e(ρ, ∆n) = e0(ρ, 0) + Ssym(ρ)∆2 n+S4(ρ)∆4 n+O(∆6 n),(3.21) where e0(ρ, 0) is the energy per particle of non-polarized neutron matter (∆n= 0) and Ssym(ρ)is the so called spin symmetry energy, Ssym(ρ) = 1 2 ∂2e(ρ, ∆n) ∂∆2 n∆n=0 .(3.22) If the dependence of the energy on the spin polarization is practically parabolic then the contributions from O(4) and higher orders can be neglected. Therefore, in this case, the spin symmetry energy can be estimated as the difference between the energy per particle of totally polarized (∆n= 1) and non-polarized neutron matter (∆n= 0), Ssym(ρ)∼e(ρ, 1) −e(ρ, 0) .(3.23) A change of sign of Ssym(ρ)would indicate a ferromagnetic phase transition from non-polarized to totally polarized neutron matter. In Fig. 3.8 we present the spin symmetry energy (solid line) for the Skyrme SLy4 interaction (left panel) and the Gogny D1 interaction (right panel) as a
28 Chapter 3. Energy Contributions -50 0 50 100 150 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Energy per particle [MeV] ρ [fm-3] Skyrme SLy4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -50 0 50 100 150 200 ρ [fm-3] Gogny D1 e(∆n=0) e(∆n=1) Ssym FIGURE 3.8: Spin symmetry energy per particle (solid line) and total energy per particle (dotted lines) of non-polarized (∆n= 0) and totally polarized (∆n= 1) neutron matter (β= 1) for the Skyrme SLy4 interaction (left panel) and the Gogny D1 interaction (right panel) as functions of the total density. 0 50 100 150 200 250 300 -1 -0.5 0 0.5 1 E/N [MeV] ∆n Skyrme SLy4 -1 -0.5 0 0.5 1 0 50 100 150 200 250 300 ∆n Gogny D1 ρ=0.48 fm-3 ρ=0.64 fm-3 ρ=0.80 fm-3 ρ=0.96 fm-3 FIGURE 3.9: Total energy per particle of neutron matter (β= 1) at different densities for the Skyrme SLy4 interaction (left panel) and the Gogny D1 interaction (right panel) as a function of the neutron spin polarization.
3.3. Total Energy 29 functions of the total density. The total energy per particle (dotted lines) of non-polarized (∆n= 0) and totally polarized (∆n= 1) neutron matter (β= 1) are also plotted. The Skyrme SLy4 interaction predicts a spin instability at ρs∼0.72 fm−3. Hence, non-polarized neutron matter is no longer the ground state of the system. The Gogny D1 interaction predicts no spin instability at any density. Therefore, the ground state is always non-polarized neutron matter. In Fig. 3.9 we show the total energy per particle of neutron matter at different density values for the Skyrme SLy4 interaction (left panel) and the Gogny D1 interaction (right panel) as a function of the neutron spin polarization. We can see that for higher densities than a critical density ρc= 0.6fm−3, provided by the magnetic susceptibility criteria (see Chapter 6), the Skyrme SLy4 interaction presents a spontaneous spin symmetry breaking where the non-polarized state is no longer stable, driving the system to a nonzero polarized state. Thus, the parabolic approximation is no longer valid [22]. A further increase of the density drives the system to a ferromagnetic state, i.e., totally polarized. The Gogny D1 interaction keeps the parabolic behaviour as a function of the spin polarization in all the range of densities considered [30]. In Chapter 6 we discuss the spin instability of nuclear matter analyzing the magnetic susceptibility.
31 Chapter 4 Spin-Isospin Channel Contributions The spin-isospin structure of the nucleon-nucleon interactions plays a very important role in describing the richness of the nucleon-nucleon scattering data and the structure of nuclei. To get further insight into the difference between the two forces we are considering, we discuss in this chapter the contribution to the potential energy of the different spin-isospin channels of the two-particle states. The diagonal two-body matrix elements of the Skyrme interaction in terms of the spin-isospin channels are given by hkk0STMSMT|ˆvSk(r)|kk0STMSMTiA= =1 Ωt0+1 6t3ργ+1 4(k−k0)2(t1+t2) +t0x0+1 6t3x3ργ+1 4(k−k0)2(t1x1+t2x2)(−1)S+1 −t0+1 6t3ργ+1 4(k−k0)2(t1−t2)(−1)S+T −t0x0+1 6t3x3ργ+1 4(k−k0)2(t1x1−t2x2)(−1)T+1, (4.1) which lead to the following contributions of each spin-isospin channel to the potential energy per particle hVST i) for polarized asymmetric nuclear matter: VSk 11 N=1 41 4ρ[2t2(1 + x2)]h(ξn↑+ξn↓+ξp↑+ξp↓)ρ+ (ξn↑+ξn↓)ρn+ (ξp↑+ξp↓)ρp + (ξn↑+ξp↑)(ρn↑+ρp↑)+(ξn↓+ξp↓)(ρn↓+ρp↓) +ξn↑ρn↑+ξn↓ρn↓+ξp↑ρp↑+ξp↓ρp↓i; (4.2)
32 Chapter 4. Spin-Isospin Channel Contributions VSk 10 N=1 21 ρt0(1 + x0) + 1 6t3(1 + x3)ργ(ρnρp+ρn↑ρp↑+ρn↓ρp↓) +1 4ρ[t1(1 + x1)]hξn↑ρp↑+ξn↓ρp↓+ξp↑ρn↑+ξp↓ρn↓ + (ξn↑+ξn↓)ρp+ (ξp↑+ξp↓)ρni; (4.3) VSk 01 N=1 21 ρt0(1 −x0) + 1 6t3(1 −x3)ργh(ρn↑+ρp↑)(ρn↓+ρp↓) + ρn↑ρn↓+ρp↑ρp↓i +1 4ρ[t1(1 + x1)]hξn↑ρn↓+ξn↓ρn↑+ξp↑ρp↓+ξp↓ρp↑ + (ξn↑+ξp↑)(ρn↓+ρp↓)+(ξn↓+ξp↓)(ρn↑+ρp↑)i; (4.4) VSk 00 N=1 41 4ρ[t2(1 −x2)] (ξn↑ρp↓+ξn↓ρp↑+ξp↑ρn↓+ξp↓ρn↑).(4.5) Adding up together all the spin-isospin channel contributions in Eqs. (4.2)- (4.5) we recover the potential energy contribution in Eq. (3.10). For the Gogny interaction, the spin-isospin structure of the diagonal twobody matrix elements of the Gogny reads hkk0STMSMT|ˆvG(r)|kk0STMSMTiA= =1 Ωργt01−(−1)S+T+x0(−1)S+1 −(−1)T+1 +π3/2X i µ3 iWi+Bi(−1)S+1 −Hi(−1)T+1 −Mi(−1)S+T −π3/2X i µ3 ie−µ2 i 4|q|2(−1)S+TWi+ (−1)T+1Bi−(−1)S+1Hi−Mi. (4.6)
Chapter 4. Spin-Isospin Channel Contributions 33 Then, the potential energy per particle in terms of the spin and isospin channels (hVST i) is given by VG 11 N=π3/2 2ρX i µ3 i[Wi+Bi−Hi−Mi] ×ρ2 n↑+ρ2 n↓+ρ2 p↑+ρ2 p↓+ρn↑ρn↓+ρp↑ρp↓+1 2(ρn↑ρp↑+ρn↓ρp↓+ρnρp) −1 4ρ√πX i [Wi+Bi−Hi−Mi] ×gi(kFn↑, kF n↑) + gi(kF n↓, kFn↓) + gi(kF p↑, kFp↑) + gi(kF p↓, kF p↓) +1 2gi(kFn↑, kF p↑) + gi(kF n↓, kFp↓) + gi(kF p↑, kFn↑) + gi(kF p↓, kF n↓) +gi(kFn↑, kF n↓) + gi(kF n↓, kFn↑) + gi(kF p↑, kFp↓) + gi(kF p↓, kF p↑) +1 4gi(kFn↑, kF p↓) + gi(kF n↓, kFp↑) + gi(kF p↑, kFn↓) + gi(kF p↓, kF n↑); (4.7) VG 10 N=1 2ργ−1t0[1 + x0][ρn↑ρp↑+ρn↓ρp↓+ρnρp] +1 2 π3/2 2ρX i µ3 i[Wi+Bi+Hi+Mi](ρn↑ρp↑+ρn↓ρp↓+ρnρp) −1 4ρ√πX i [−Wi−Bi−Hi−Mi] ×1 2gi(kFn↑, kF p↑) + gi(kF n↓, kFp↓) + gi(kF p↑, kFn↑) + gi(kF p↓, kF n↓) +1 4gi(kFn↑, kF p↓) + gi(kF n↓, kFp↑) + gi(kF p↑, kFn↓) + gi(kF p↓, kF n↑); (4.8) VG 01 N=1 2ργ−1t0[1 −x0]hρn↑ρn↓+ρp↑ρp↓+ (ρn↑+ρp↑)(ρn↓+ρp↓)i +1 2 π3/2 2ρX i µ3 i[Wi−Bi−Hi+Mi]hρn↑ρn↓+ρp↑ρp↓+ (ρn↑+ρp↑)(ρn↓+ρp↓)i −1 4ρ√πX i [−Wi+Bi+Hi−Mi] ×1 2gi(kFn↑, kF n↓) + gi(kF n↓, kFn↑) + gi(kF p↑, kFp↓) + gi(kF p↓, kF p↑) +1 4gi(kFn↑, kF p↓) + gi(kF n↓, kFp↑) + gi(kF p↑, kFn↓) + gi(kF p↓, kF n↑); (4.9)
34 Chapter 4. Spin-Isospin Channel Contributions VG 00 N=1 2 π3/2 2ρX i µ3 i[Wi−Bi+Hi−Mi][ρn↑ρp↓+ρn↓ρp↑] −1 4 1 4ρ√πX i [Wi−Bi+Hi−Mi] ×hgi(kFn↑, kF p↓) + gi(kF n↓, kFp↑) + gi(kF p↑, kFn↓) + gi(kF p↓, kF n↑)i. (4.10) Again, adding up together all spin-isospin channel contributions in Eqs. (4.7)- (4.10) we recover Eq. (3.11). One can still perform an additional decomposition by analysing the contribution to the potential energy of the different third components of the spinisospin two-body states. The expressions can be found in appendix A.4. In Fig.4.1 and Fig. 4.2 are shown the contributions of each channel (dotted lines) to the potential energy per particle (solid line) for different values of the isospin asymmetry (β) and the spin polarizations (∆n,∆p) for both the Skyrme and Gogny interactions, respectively. Looking at Eqs. (4.2)-(4.5), Eqs. (4.7)-(4.10) and Fig. 4.1 and Fig. 4.2, one can see that the contribution of the channel hV11iis zero by construction for the Skyrme SLy4 as imposed in the parametrization of the force. For the Gogny D1 interaction hV11iis not strictly zero, but really small compared to the other contributing channels. The contribution of T= 0 channels, hV10iand hV00i, decrease when the asymmetry increases and become zero for neutron matter. Similarly, the S= 0channels, hV01iand hV00i, do not contribute for totally polarized matter. hV00ionly contributes for nuclear matter when nucleons have different spin polarizations. It might seem zero for the Gogny D1 interaction, but it does contribute within the proper conditions, just as hV11i. 4.1 Partial Wave Decomposition A deeper insight in the understanding on the interaction can be obtained by performing a partial wave decomposition of the interaction energy. This can be useful to compare different interactions and enlighten the differences with realistic interactions. It is convenient to express the two-body states in terms of the center of mass and the relative motion. |KCM krSTMSMTi=1 ΩeiKCM ReikrrχS MSχT MT =1 ΩeiKCM R4πX `m i`j`(kr)Y∗ `m(ˆ k)Y`m(ˆr)χS MSχT MT,(4.11) where r=r1−r2is the relative coordinate between, R= (r1+r2)/2is the center of mass coordinate, kr=k−k0is the relative momentum, KCM = (k+k0)/2is the center of mass momentum. The plane wave associated to the
4.1. Partial Wave Decomposition 35 Skyrme SLy4 -40 -20 0 20 40 ∆n=∆p=0 ∆n=∆p=±1 β=0 ∆n=-∆p=±1 -40 -20 0 20 40 VSk ST/N [MeV] 0 0.1 0.2 0.3 0.4 β=0.5 ρ [fm-3] total v00 v01 v10 -40 -20 0 20 40 0 0.1 0.2 0.3 0.4 ρ [fm-3] 0 0.1 0.2 0.3 0.4 β=1 ρ [fm-3] FIGURE 4.1: Skyrme SLy4 interaction potential energy per particle spin-isospin channel contributions for symmetric nuclear matter (upper row), asymmetric nuclear matter with β= 0.5(middle row) and neutron matter (lower row) and nonpolarized (left column), totally polarized (center column) and anti-polarized (right column) matter as functions of total density. Solid line shows the total potential energy per particle. Dotted lines show the spin-isospin channel contributions. Noncontributing channels are not plotted. hV11inever appears because it is set to zero by SLy4.
42 Chapter 5. Interaction Potential Parametrization -10 0 10 20 30 40 ESk/N [MeV] β=0.25 ∆n=0 ∆n=0.5 ∆n=0.75 ∆n=1 Parametrization -10 0 10 20 30 40 β=0.5 -10 0 10 20 30 40 -1 -0.5 0 0.5 1 EG/N [MeV] ∆p -1 -0.5 0 0.5 1 -10 0 10 20 30 40 ∆p FIGURE 5.2: Total energy per particle as function of proton spin polarization (∆p) for Skyrme SLy4 interaction (upper row) and Gogny D1 interaction (lower row) for different values of the neutron spin polarization (∆n= 0,0.5,0.75,1) and different isospin asymmetries (β= 0.25,0.5) at saturation density ρ0= 0.16 fm−3. Symbols show the results for the exact calculations. Solid line shows the results for the parametrization in Eq. (5.5).
Chapter 5. Interaction Potential Parametrization 43 -30 -20 -10 0 10 20 30 40 0 0.1 0.2 0.3 0.4 E/N [MeV] ρ [fm-3] Skyrme SLy4 β=0.25,∆n=0.3,∆p=0.4 β=0.5,∆n=0.5,∆p=0.25 β=0.8,∆n=0.5,∆p=0.2 Parametrization 0 0.1 0.2 0.3 0.4 -30 -20 -10 0 10 20 30 40 ρ [fm-3] Gogny D1 FIGURE 5.3: Total energy per particle as function of density for Skyrme SLy4 interaction (top) and Gogny D1 interaction (bottom) for different values of the neutron and proton spin polarizations (∆n,∆p) and different isospin asymmetries (β) at saturation density ρ0= 0.16 fm−3. Symbols show the results for the exact calculations. Solid lines show the results for the parametrization in Eq. (5.5).
45 Chapter 6 Magnetic Susceptibility In the astrophysical context, pulsars are known to be rapidly rotating neutron stars with an intense magnetic field [32, 33, 34, 35]. Therefore, the possibility of phase transition to a ferromagnetic state in neutron matter has been a field of study that still presents some contradictory results. This transition could have important consequences for the evolution of protoneutron stars, in particular for the spin correlations in the medium that strongly affect the neutrino cross-sections and mean free paths inside the star [36]. Some studies done with microscopic calculations show that there is no possible ferromagnetic phase transition [31, 37, 38, 39, 40, 41, 42, 43, 44, 45], and so do others using effectives forces [30, 46]. On the other hand, calculations using different effective forces do find a ferromagnetic phase transition [22, 47]. Some of the results are, however, merely academic since they cannot be reached by protoneutron stars. The magnetic susceptibility χ(ρ)characterizes the response of the system to an external magnetic field and gives a measure of the energy required to produce a net spin alignment in the direction of the field. For nuclear matter, it is defined as a 2×2matrix 1 χ=1/χnn 1/χnp 1/χpn 1/χpp,(6.1) where the matrix elements 1/χij are given by 1 χij =∂Hi ∂Mj i, j =n, p. (6.2) Here Mjis the magnetization of the species jper unit volume, Mj=µj(ρj↑−ρj↓) = µjρj∆j,(6.3) where µjis the magnetic moment of the species j, µn=−1.913µN, µp= 2.792µN, µN=e~ 2mp = 0.105 efm,(6.4)
46 Chapter 6. Magnetic Susceptibility and Hiis the magnetic field induced by the magnetization of the species i, which can be obtained from Hi=ρ∂e ∂Mi ,(6.5) where e=e(ρ, β, ∆n,∆p)is the total energy per particle. Using Eqs. (6.3) and (6.5), the matrix elements 1/χij1can be written as 1 χij =ρ µiρiµjρj ∂2e ∂∆i∂∆j ,(6.6) where the second derivatives can be taken at ∆i= ∆j= 0 if the field is assumed to be small. The stability of matter against spin fluctuations is guaranteed if det(1/χ)> 0. A change of sign of the determinant indicates the existence of a spin instability and the onset of a phase transition towards a new spin ordering. It is convenient in order to highlight the role of the nucleon-nucleon interaction to study the magnetic susceptibility of the system in terms of the ratio det(1/χ)/det(1/χF), where χFis the magnetic susceptibility of the two component free Fermi gas, 1 χF =ρ µiρiµjρj ∂2eF ∂∆i∂∆j ,(6.7) where the free Fermi gas energy per particle (eF) is equal to the kinetic energy per particle, see (3.5). Thus, the two components read 1 χFnn =ρ µ2 nρ2 n 10 9 ~2ξ 2mnρ(1 + β)5/3(1 + ∆n)−1/3+ (1 −∆p)−1/3, 1 χFpp =ρ µ2 pρ2 p 10 9 ~2ξ 2mpρ(1 + β)5/3(1 + ∆p)−1/3+ (1 −∆p)−1/3. (6.8) Expressing the total energy per particle as the sum of the kinetic and potential energy per particle, one can write the ratio det(1/χ)/det(1/χF)as follows: det(1/χ) det(1/χF)= 1 + ∂2(V/N) ∂2∆n ∂2(T/N) ∂2∆n 1 + ∂2(V/N) ∂2∆p ∂2(T/N) ∂2∆p − ∂2(V/N) ∂∆n∂∆p ∂2(T/N) ∂2∆n ∂2(V/N) ∂∆p∂∆n ∂2(T/N) ∂2∆p . (6.9) 1Analytic expressions of the magnetic susceptibility matrix elements for the Skyrme interaction are reported in the appendix A.5.
Chapter 6. Magnetic Susceptibility 47 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 det(1/Χ)/det(1/ΧF) ρ [fm-3] Skyrme SLy4 β=0 β=0.25 β=0.5 β=0.75 β=1 FIGURE 6.1: Ratio det(1/χ)/det(1/χF)nuclear matter with different isospin asymmetries (β= 0,0.25,0.5,0.75,1) as a function of the total density for the Skyrme SLy4 interaction. For neutron matter, the matrix (6.1) reduces to 1/χnn. Using the spin symmetry energy, see Eqs. (3.22) and (3.23), one can write down 1/χnn as 1 χnn =2ρ µ2 nρ2 n Ssym(ρ),(6.10) which can be handy when the interaction is too complex to compute the derivatives. In Fig. 6.1 we present the ratio det(1/χ)/det(1/χF)of asymmetric nuclear matter with β= 0,0.25,0.5,0.75,1as a function of the total density for the Skyrme SLy4 interaction. A phase transition appears at a critical density ρc∼ 0.6fm−3for neutron matter with the Skyrme SLy4 interaction. That is, for ρ > 0.6fm−3appears a spontaneous polarization of the ground state. As β decreases, ρtmoves toward larger density values. In Fig. 6.2 we compare the ratio det(1/χ)/det(1/χF)of neutron matter as a function of the total density between the Skyrme SLy4 and the Gogny D1 interactions. We observe that, for densities ρ < 0.1fm−3, the ratio det(1/χ)/det(1/χF) for both the Skyrme SLy4 and Gogny D1 interactions present a similar behaviour. As density increases further away from ρ= 0.1fm−3, the SLy4 ratio starts decreasing to eventually become negative at the critical density ρc∼0.6fm−3, as we have already mentioned. Contrary, the D1 ratio keeps smoothly increasing with density thus, no spin instability is found at any density.
48 Chapter 6. Magnetic Susceptibility 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 det(1/Χ)/det(1/ΧF) ρ [fm-3] Skyrme SLy4 Gogny D1 FIGURE 6.2: Ratio det(1/χ)/det(1/χF)of neutron matter (β= 1) as a function of the total density for the Skyrme SLy4 interaction and the Gogny D1 interaction. These results of pure neutron matter are consistent with the ones obtained for the spin symmetry energy (see subsection 3.3.2). Notice that, for neutron matter, a change of sign on the spin symmetry energy approximation (3.23) is a sufficient condition for a phase transition, whereas a change of sign in the ratio det(1/χ)/det(1/χF)is a necessary condition. That is why the densities at which the phase transition is predicted are different for the spin symmetry energy (ρs∼0.72 fm−3) and the magnetic susceptibility ρc∼0.6fm−3.
49 Summary and Conclusions In this master thesis we have constructed the formalism to calculate the singleparticle potential and energy per particle of homogeneous nuclear matter with any isospin asymmetry and spin polarization. The calculations have been performed in the framework of the Hartree-Fock formalism using effective interactions. We have considered the Skyrme SLy4 as an example of zero-range interactions and the original Gogny D1 as a characteristic finiterange force. We have prepared a complete framework that allows the computation of: spin-isospin channel contributions, partial wave decompositions and calculations using the Slater function that permit to check the numerical calculations. Once we have obtained the energy per particle, we have also considered the second derivatives of this energy with respect to isospin asymmetry to calculate the isospin symmetry energy, and with respect to the spin polarization to determinate the spin symmetry energy and study the magnetic susceptibility. It has been found, in agreement with the existing literature, that the Skyrme SLy4, whose parametrization has incorporated microscopic results for neutron matter, does not show isospin instability and the symmetry energy is always positive, i.e., neutron matter never becomes the ground state. On the contrary, the symmetry energy for the Gogny interaction becomes zero at a density ρs∼0.52 fm−3. We have also checked that a parametrization of the energy per particle, previously suggested by Vidaña and Bombaci [31] for realistic interactions, adapts well to the effective interactions studied in the thesis, at least for small isospin asymmetries and spin polarizations. This parametrization allows us to express the energy per particle in terms of a few average matrix elements. By computing the magnetic susceptibility, we have found that SLy4 predicts a spin instability at both neutron matter and asymmetric nuclear matter, with a critical density that shifts to smaller densities when increasing the isospin asymmetry. In the Gogny case, we have only performed the calculations for neutron matter and no sign of an instability has been found. Similar results, using other effective interactions, are found in the literature [46]. One should keep in mind that previous microscopic results [37, 31] do not show any spin instability. It would be useful to incorporate these results into the parametrization of the effective interactions [48]. Finally, we would like to extend this analysis to a wider family of Skyrme forces and the recent proposed Gogny forces. Also, we consider doing a similar analysis to explore the spinodal instabilities.
51 Appendix A Integrals and Formulae A.1 Useful integrals When solving the potential energy per particle, see Eq. (3.8), the integrals over momentum expressed in terms of a function Φ(k,k0)with a k, k0depen- dence, I[Φ(k,k0); kFστ , k0 Fσ0τ0]≡1 2N Ω2 (2π)6ZkF στ Zk0 F σ0τ0 d3kd3k0Φ(k,k0),(A.1) where Ωis the volume of the system and Nthe number of particles, result in Eqs. (A.2), (A.3) and (A.4) for the Skyrme interaction and in Eqs. (A.2) and (A.5) for the Gogny interaction. I[Ω−1;kFστ , k0 Fσ0τ0] = 1 2N Ω (2π)64 3πkF στ 34 3πk0 Fσ0τ0 3=1 2ρρστ ρσ0τ0;(A.2) I[Ω−1(k2+k02); kFστ , k0 Fσ0τ0] = 1 2N Ω (2π)64 3πkF στ 34 5πk0 Fσ0τ0 5 +4 3πk0 Fσ0τ0 34 5πkF στ 5 =1 2ρ(ξσ0τ0ρστ +ξστ ρσ0τ0) ; (A.3) I[Ω−1(k·k0); kFστ , kF σ0τ0] = 0 ; (A.4)
58 Appendix A. Integrals and Formulae For the Gogny interaction, the VG ST MSMT/N contributions read VG 1111 N=π3/2 2ρX i µ3 i[Wi+Bi−Hi−Mi]ρ2 p↑ −1 4ρ√πX i [Wi+Bi−Hi−Mi]gi(kFp↑, kF p↑) ; (A.33) VG 1110 N=π3/2 2ρX i µ3 i[Wi+Bi−Hi−Mi]ρn↑ρp↑ −1 2 1 4ρ√πX i [Wi+Bi−Hi−Mi]gi(kFn↑, kF p↑) + gi(kF p↑, kFn↑); (A.34) VG 111−1 N=π3/2 2ρX i µ3 i[Wi+Bi−Hi−Mi]ρ2 n↑ −1 4ρ√πX i [Wi+Bi−Hi−Mi]gi(kFn↑, kF n↑) ; (A.35) VG 1101 N=π3/2 2ρX i µ3 i[Wi+Bi−Hi−Mi]ρp↑ρp↓ −1 2 1 4ρ√πX i [Wi+Bi−Hi−Mi]gi(kFp↑, kF p↓) + gi(kF p↓, kFp↑); (A.36) VG 1100 N=1 2 π3/2 2ρX i µ3 i[Wi+Bi−Hi−Mi](ρn↑ρp↓+ρn↓ρp↑) −1 4 1 4ρ√πX i [Wi+Bi−Hi−Mi]gi(kFn↑, kF p↓) + gi(kF n↓, kFp↑) +gi(kFp↑, kF n↓) + gi(kF p↓, kFn↑); (A.37) VG 110−1 N=π3/2 2ρX i µ3 i[Wi+Bi−Hi−Mi]ρn↑ρn↓ −1 2 1 4ρ√πX i [Wi+Bi−Hi−Mi]gi(kFn↑, kF n↓) + gi(kF n↓, kFn↑); (A.38)
A.4. Spin-isospin third component channel contributions 59 VG 11−11 N=π3/2 2ρX i µ3 i[Wi+Bi−Hi−Mi]ρ2 p↓ −1 4ρ√πX i [Wi+Bi−Hi−Mi]gi(kFp↓, kF p↓) ; (A.39) VG 11−10 N=π3/2 2ρX i µ3 i[Wi+Bi−Hi−Mi]ρn↓ρp↓ −1 2 1 4ρ√πX i [Wi+Bi−Hi−Mi]gi(kFn↓, kF p↓) + gi(kF p↓, kFn↓); (A.40) VG 11−1−1 N=π3/2 2ρX i µ3 i[Wi+Bi−Hi−Mi]ρ2 n↓ −1 4ρ√πX i [Wi+Bi−Hi−Mi]gi(kFn↓, kF n↓) ; (A.41) VG 1010 N=ργ−1t0[1 + x0]ρn↑ρp↑ +π3/2 2ρX i µ3 i[Wi+Bi+Hi+Mi]ρn↑ρp↑ −1 2 1 4ρ√πX i [−Wi−Bi−Hi−Mi]gi(kFn↑, kF p↑) + gi(kF p↑, kFn↑); (A.42) VG 1000 N=1 2ργ−1t0[1 + x0](ρn↑ρp↓+ρn↓ρp↑) +1 2 π3/2 2ρX i µ3 i[Wi+Bi+Hi+Mi](ρn↑ρp↓+ρn↓ρp↑) −1 4 1 4ρ√πX i [−Wi−Bi−Hi−Mi]gi(kFn↑, kF p↓) + gi(kF n↓, kFp↑) +gi(kFp↑, kF n↓) + gi(kF p↓, kFn↑); (A.43)
60 Appendix A. Integrals and Formulae VG 10−10 N=ργ−1t0[1 + x0]ρn↓ρp↓ +π3/2 2ρX i µ3 i[Wi+Bi+Hi+Mi]ρn↓ρp↓ −1 2 1 4ρ√πX i [−Wi−Bi−Hi−Mi]gi(kFn↓, kF p↓) + gi(kF p↓, kFn↓); (A.44) VG 0101 N=ργ−1t0[1 −x0]ρp↑ρp↓ +π3/2 2ρX i µ3 i[Wi−Bi−Hi+Mi]ρp↑ρp↓ −1 2 1 4ρ√πX i [−Wi+Bi+Hi−Mi]gi(kFp↑, kF p↓) + gi(kF p↓, kFp↑); (A.45) VG 0100 N=1 2ργ−1t0[1 −x0](ρn↑ρp↓+ρn↓ρp↑) +1 2 π3/2 2ρX i µ3 i[Wi−Bi−Hi+Mi](ρn↑ρp↓+ρn↓ρp↑) −1 4 1 4ρ√πX i [−Wi+Bi+Hi−Mi]gi(kFn↑, kF p↓) + gi(kF n↓, kFp↑) +gi(kFp↑, kF n↓) + gi(kF p↓, kFn↑); (A.46) VG 010−1 N=ργ−1t0[1 −x0]ρn↑ρn↓ +π3/2 2ρX i µ3 i[Wi−Bi−Hi+Mi]ρn↑ρn↓ −1 2 1 4ρ√πX i [−Wi+Bi+Hi−Mi]gi(kFn↑, kF n↓) + gi(kF n↓, kFn↑); (A.47)
A.5. Magnetic susceptibility matrix elements for the Skyrme interaction 61 VG 0000 N=1 2 π3/2 2ρX i µ3 i[Wi−Bi+Hi−Mi](ρn↑ρp↓+ρn↓ρp↑) −1 4 1 4ρ√πX i [Wi−Bi+Hi−Mi]gi(kFn↑, kF p↓) + gi(kF n↓, kFp↑) +gi(kFp↑, kF n↓) + gi(kF p↓, kFn↑); (A.48) A.5 Magnetic susceptibility matrix elements for the Skyrme interaction Analytic expressions of the Skyrme interaction magnetic susceptibility matrix elements, see Eq. (6.6), are pesented below: 1 χnn =ρ µ2 nρ2 n(10 9 ~2ξ 2mnρ(1 + β)5/3(1 + ∆n)−1/3+ (1 −∆n)−1/3 +ξ 16 40 9[2t2(1 + x2)](1 + β)8/3(1 + ∆n)2/3+ (1 −∆n)2/3 +ξ 16 10 9[t1(1 −x1) + t2(1 + x2)] ×(1 + β)8/3h(1 + ∆n)−1/3(1 −∆n)−3(1 + ∆n)2/3 + (1 −∆n)−1/3(1 + ∆n)−3(1 −∆n)2/3i +ξ 16 10 9[t1(1 + x1) + t2(1 + x2)] ×(1 + β)5/3(1 −β)(1 + ∆n)−1/3(1 + ∆p) + (1 −∆n)−1/3(1 −∆p) +ξ 16 10 9[t1+t2] ×(1 + β)5/3(1 −β)(1 + ∆n)−1/3(1 −∆p) + (1 −∆n)−1/3(1 + ∆p) −2ρ 16t0(1 −x0) + 1 6t3(1 −x3)ργ(1 + β)2); (A.49)
62 Appendix A. Integrals and Formulae 1 χpp =ρ µ2 pρ2 p(10 9 ~2ξ 2mpρ(1 −β)5/3(1 + ∆p)−1/3+ (1 −∆p)−1/3 +ξ 16 40 9[2t2(1 + x2)](1 −β)8/3(1 + ∆p)2/3+ (1 −∆p)2/3 +ξ 16 10 9[t1(1 −x1) + t2(1 + x2)] ×(1 −β)8/3h(1 + ∆p)−1/3(1 −∆p)−3(1 + ∆p)2/3 + (1 −∆p)−1/3(1 + ∆p)−3(1 −∆p)2/3i +ξ 16 10 9[t1(1 + x1) + t2(1 + x2)] ×(1 −β)5/3(1 + β)(1 + ∆p)−1/3(1 + ∆n) + (1 −∆p)−1/3(1 −∆n) +ξ 16 10 9[t1+t2] ×(1 −β)5/3(1 + β)(1 + ∆p)−1/3(1 −∆n) + (1 −∆p)−1/3(1 + ∆n) −2ρ 16t0(1 −x0) + 1 6t3(1 −x3)ργ(1 −β)2); (A.50) 1 χnp =1 χpn =ρ µnρnµpρp(ξ 16 5 3[t1x1+t2x2] ×h(1 + β)5/3(1 −β)(1 + ∆n)2/3+ (1 −∆n)2/3 + (1 −β)5/3(1 + β)(1 + ∆p)2/3+ (1 −∆p)2/3i +2ρ 16t0x0+1 6t3x3ργ(1 −β2)); (A.51)
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