a Xi :nucl- h/0111013 1 6 No 2001
Anha monic ib a ions in nuclei
M. Fallo a), Ph. Chomazb), M.V. And ´esc), F. Ca a ad), E. G. Lanzad), J. A. Sca pacia)
a)Ins i u de Physique Nucl´eai e, IN2P3-CNRS, F-91406 O say Cedex, F ance
b)GANIL, B.P. 5027, F-14076 CAEN Cedex 5, F ance
c)Depa amen o de F´ısica A ´omica, Molecula y Nuclea , Uni e sidad de Se illa, Apdo 1065, E-41080 Se illa, Spain
d)Dipa imen o di Fisica Uni e si ´a di Ca ania and INFN, Sezione di Ca ania, I-95129 Ca ania, I aly
In his le e , we show ha he non-linea i ies o la ge ampli ude mo ions in a omic nuclei induce
gian quad upole and monopole ib a ions. As a consequence, he main sou ce o anha monici y is
he coupling wi h con igu a ions including one o hese wo gian esonances on op o any s a e.
Two-phonon ene gies a e o en lowe ed by one o wo MeV because o he la ge ma ix elemen s
wi h such h ee phonon con igu a ions. These e ec s a e s udied in wo nuclei, 40Ca and 208Pb.
PACS numbe s : 21.60E , 21.10Re, 21.60Jz, 24.30Cz
Many-body e mionic sys ems possess collec i e ib a-
ional s a es which a e well desc ibed as bosonic modes
(phonons). The exis ence in a omic nuclei o such s a es,
bo h low-lying and Gian Resonances (GR) is now well
es ablished up o he second quan um [1,2]. Howe e ,
hei p ope ies such as ene gy and exci a ion p obabil-
i y a e s ill open ques ions. F om he expe imen al poin
o iew he s ong exci a ion c oss sec ion o wo phonon
s a es calls o he p esence o la ge anha monici ies bu
up o now, all he heo e ical es ima es we e poin ing
o weak de ia ions om a ha monic spec um. To ou
knowledge, so a only he mixing o one- and wo-phonon
s a es has been conside ed in mic oscopic calcula ions,
wi h wo excep ions. In e . [3] he coupling o some spe-
ci ic h ee-phonon con igu a ions has been included as a
mechanism gene a ing he damping wid h o he Double
Gian Dipole Resonance. In e . [4] he agmen a ion
o he doubly exci ed low lying oc upole s a es in 208Pb
has been s udied by allowing he coupling o one- and
h ee-phonon con igu a ions wi h a low ene gy cu -o in-
oduced o educe he diagonaliza ion space. Fo his
eason monopole, (GMR) and quad upole (GQR) con-
ibu ions which, as we will see, play an impo an ole,
we e neglec ed.
In he p esen pape we show ha a co ec desc ip ion
o he s a es o which he main componen is a wo-
phonon con igu a ion equi es he inclusion o one- and
h ee-phonon ones. We s ess he essen ial ole played by
he b ea hing mode in he nuclea anha monici y as an
impo an no el y o he p esen analysis since olume
modes a e usually no conside ed in damping o coupling
mechanisms. Mo eo e , we will show ha he e y col-
lec i e GQR plays also an impo an ole.
The s a ing poin o ou calcula ion is a mapping o
he e mion pa icle-hole ope a o s a†
pahin o boson op-
e a o s B†
ph as o example he one p oposed in e . [5]
a†
pah→B†
ph + (1 −√2) X
p′h′
B†
p′h′B†
p′hBph′+.... (1)
a†
pap′→X
h
B†
phBp′h, aha†
h′→X
p
B†
phBph′(2)
whe e a†(a) c ea es (annihila es) one nucleon in an oc-
cupied (h) o unoccupied (p) single pa icle s a e. The
second e m on he igh hand side o eq.(1) is a co -
ec ion ha akes ca e o he Pauli p inciple. Then we
cons uc a boson image o he Hamil onian, unca ed a
he ou h o de in he B†and Bope a o s. In oducing
he Bogoliubo ans o ma ion o bosons:
Q†
ν≡X
p,h
(Xν
phB†
ph −Yν
phBph) (3)
and imposing ha he quad a ic pa o he boson Hamil-
onian in he new ope a o s is diagonal, we ob ain he
usual Random Phase App oxima ion (RPA) equa ions
o he Xand Yampli udes.
By in e ing eq.(3) we can exp ess HBin e ms o he
collec i e Q†and Qope a o s:
HB=H11Q†Q+ (H21Q†Q†Q+h.c.) + H22Q†Q†QQ
+(H30Q†Q†Q†+h.c.) + (H31Q†Q†Q†Q+h.c.)
+(H40Q†Q†Q†Q†+h.c.) + ... (4)
wi h Hνν′
11 =Eνδνν′.The Hma ices a e exp essed in
e ms o he X and Y o ans o ma ion (3). The con-
ibu ions o eq.(4) coming om he high o de e ms o
he expansion (1) appea o be educed by he numbe o
con igu a ions in ol ed in he collec i e s a es and he e-
o e can be neglec ed. In he case o closed shell nuclei,
he RPA co ela ions a e mode a e. The e o e, he Y/X
a ios a e small. In eq.(4) we will neglec he H e ms
con aining a leas one Y ampli ude. These wo app ox-
ima ions lea e una ec ed only he i s h ee e ms o
eq.(4) [6]. We will compa e he spec a o 40Ca and 208Pb
ob ained by he diagonaliza ion in he spaces con aining
up o wo-phonon s a es and up o h ee-phonon s a es,
espec i ely. In e . [7] a simila analysis was done in he
wo le el Lipkin model. I was ound ha his app oxi-
ma ion is well jus i ied and one ge s good esul s in he
la ge space o he eigens a es which main componen
is a wo-phonon con igu a ion.
All calcula ions ha e been pe o med by using he SGII
Sky me in e ac ion [8]. We include all na u al pa i y
1
RPA collec i e one-phonon s a es wi h angula momen-
um J≤3 which exhaus a leas 5% o he EWSR
and all wo- and h ee-phonon con igu a ions buil wi h
hem, wi hou any ene gy cu -o , wi h bo h na u al and
unna u al pa i y.
Le us s a looking a he esul s o 40Ca. In able I
we show he one-phonon s a es aken in o accoun . In
able II we show some esul s o he diagonaliza ion o a
selec ed se o s a es. The ene gies ob ained in he space
up o wo-phonons (see e . [9]) a e epo ed he e o
compa ison. The so-calcula ed anha monici y was lim-
i ed o a ew hund ed keV. Le us now s udy he mo e
comple e calcula ion including he h ee phonon s a es.
As a gene al commen , one can say ha he shi in-
duced by he coupling o h ee-phonon s a es is ai ly
la ge, being in almos all he cases mo e han 1 MeV,
and always downwa d. This can be unde s ood in second
o de pe u ba ion which, as can be seen om he able,
gi es a good es ima e o he ene gies in mos cases. In
second o de pe u ba ion he co ec ion o he ene gy is
gi en by
∆Ei=< ϕi|V|ϕi>+X
j6=i
|< ϕj|V|ϕi>|2
E0
i−E0
j
(5)
whe e |ϕi>is he conside ed unpe u bed s a e, |ϕj>
all he o he s a es and E0 he co esponding unpe -
u bed ene gies. Since he diagonal, i s o de , con i-
bu ion is small in mos cases, he sign o he shi is ha
o he denomina o in he second o de e m. The e o e,
i |ϕi>is a wo-phonon s a e, he con ibu ions om
h ee-phonon con igu a ions a e nega i e in mos cases
since mos o he h ee phonon s a es lye abo e he wo
phonon ones. Mo eo e , whene e a GMR is added on
op o any s a e, he H21 e ms (see eq.4) a e la ge , o
he o de o 1 o 2 MeV in 40Ca. The speci ic alues
can be ound in he las h ee columns o able I. Indeed,
in he coupling leading o he addi ion o one GMR on
op o any s a e, he esidual in e ac ion be ween he un-
de lying e mions is no unca ed by conse a ion laws,
because he pa icles and he holes in ol ed in he GMR
ca y iden ical pa i y and spin quan um numbe s (c . e .
[10]). Phenomenologically, his s ong coupling o all col-
lec i e ib a ions wi h he b ea hing mode comes om
he ac ha in a small nucleus like he 40Ca any la ge
ampli ude mo ion a ec s he cen al densi y. The e o e,
su ace modes canno be decoupled om a densi y a ia-
ion in he whole olume as clea ly seen in ecen TDHF
simula ions e . [11].
I he s a e is a wo-phonon one, hen he ma ix el-
emen s coupling i o he s a e ob ained by exci ing a
b ea hing mode on op o i a e abou 3 MeV (up o 5.5
MeV) when he less (mo e) collec i e componen o he
40Ca GMR is conside ed. E en la ge ma ix elemen s
a e ob ained, when he s a es connec ed by H21 in ol e
se e al GMR. In ha case a Bose enhancemen ac o
appea s and no Clebsch-Go dan coe icien s en e in he
calcula ion. Thus he ma ix elemen be ween he dou-
ble and he iple GMR loca ed a 18.25 MeV, M1, is
√6 imes la ge han be ween he single and he double
M1. Tha gi es a ma ix elemen o -5.22 MeV, gi ing
a con ibu ion o -1.49 MeV o he second o de ene gy
co ec ion o he double M1s a e. An e en la ge alue
comes ou in he case o he double GMR loca ed a
22.47 MeV, M2, and he iple M2, namely a ma ix el-
emen o -9.69 MeV gi ing a -4.18 MeV con ibu ion o
he ene gy shi o he double M2. This is due o he ac
ha M2is mo e collec i e han M1in 40Ca.
Some hing simila , bu less s ong, happens also o
he ma ix elemen s connec ing some s a e wi h ha
buil by adding one GQR phonon. We quo e wo ex-
amples. The low-lying componen o he Gian Dipole
Resonance |D1>has a ma ix elemen o he esidual in-
e ac ion wi h he s a es |D1⊗M1>,|D1⊗M2>and
|D1⊗Q1>equal o -1.38 MeV, -2.12 MeV and -1.25
MeV espec i ely. Ano he example, wi h o al J=1,
is gi en by he ma ix elemen s be ween |D1⊗Q1>
and |(M1⊗D1)1⊗Q1>,|(M2⊗D1)1⊗Q1>and
|(Q1)2
2⊗D1>equal o -2.74 MeV, -4.61 MeV and -1.41
MeV espec i ely.
These indings clea ly indica e ha la ge ampli ude
mo ions a e s ongly coupled bo h o su ace and olume
oscilla ions, he la e being mo e impo an in 40Ca. I is
wo hwhile s essing ha such la ge co ec ions o he en-
e gy o wo-phonon s a es a e ob ained despi e he qui e
la ge absolu e alues o he ene gy di e ence be ween he
coupled s a es. The e o e, in oducing an ene gy cu -o
in he h ee-phonon s a es included in he calcula ion
may lead o e oneous esul s. Le us conside o exam-
ple he case o he 0+membe o he mul iple o double
low-lying oc upole s a es. A i s o de pe u ba ion,
i is shi ed up by 2.24 MeV. The second o de co ec-
ion coming om he single GMR s a es is -0.93 MeV.
These wo con ibu ions, leading o a o al shi o +1.31
MeV, domina e he e ec s o he coupling wi h one- and
wo-phonon s a es as con i med by he esul o he diag-
onaliza ion in his subspace. When h ee-phonon s a es
a e included, one ge s a u he shi down o 1.86 MeV
coming om he con igu a ion including a GMR on op
o he wo oc upoles. This con ibu ion is absen in e .
[4] because he ene gy cu -o in oduced he e in o de
o educe he numbe o h ee-phonon con igu a ions was
oo low. The same happens o he o he membe s o he
mul iple as well as o he double D1o D2, he double
Q1and he D1o D2⊗Q1s a es.
The esul s o 208Pb a e shown in ables III and IV.
The same gene al ema ks al eady made o 40Ca apply
also in his case. The mos ele an di e ence is ha he
ole played by he GMR and he GQR in 40Ca is now in-
e ed, he la e being dominan in 208Pb. This educed
impo ance o he GMR may come om he ac ha
in la ge nuclei he su ace ib a ions can occu wi hou
2
changing he olume. Concluding abou he ene gy o he
wo-phonon s a es one can see ha he inclusion o he
h ee phonon con igu a ions induces an anha monici y o
mo e han 1 MeV in 40Ca bu only o a ew hund ed keV
in 208Pb. Because o he loca ion a high ene gy o he
h ee phonon s a es, he obse ed shi is sys ema ically
downwa d. I is impo an o s ess ha he conside ed
esidual in e ac ion only couples s a es wi h a numbe o
phonon a ying a maximum by one uni . The e o e, he
ene gy a ia ion o he wo-phonon spec um induced by
inclusion o ou and mo e phonon s a es would be small
since i co esponds o a hi d o de pe u ba ion in ol -
ing wo la ge ene gy di e ences in he denomina o .
I we now analyze he spli ing o he wo-phonon mul-
iple s we can see ha i emains small o gian eso-
nances (abou a ew hund ed keV) while i may go up o
1 MeV o low lying s a es in 40Ca. Compa ing he spli -
ing and he o de ing o he s a es ob ained in i s o de
pe u ba ion and in he ull calcula ion we can see ha
hey emain almos unchanged. The e o e, he diagonal
ma ix elemen s o he esidual in e ac ion a e esponsi-
ble o his spli ing and o de ing.
Le us now in es iga e he mixing induced by he esid-
ual in e ac ion. In ables II and IV he mixing coe icien s
o he wo main componen s in each s a e a e p esen ed.
Fi s we can see ha he e is always one componen ha
emains e y la ge, explaining he success o he pe u -
ba ion app oach. The impo an poin is ha in gene al
we obse e la ge mixing coe icien s, namely abou 0.2
o 0.4 o mo e in 40Ca and 0.15 o 0.3 in 208Pb. This
may ha e e y impo an consequences in he exci a ion
p ocess as we will in es iga e in a o hcoming wo k.
I is wo hwhile men ioning ha , in some cases, a
h ee-phonon componen appea s wi h a la ge ampli ude
in he wa e unc ion o a (mainly) wo-phonon s a e, de-
spi e he ac ha he esidual in e ac ion does no couple
di ec ly hese con igu a ions oge he . In a ew cases, in-
deed, his is he second main componen as can be seen
o 40Ca in able II ( he |(D1)2
0>and |M2⊗Q1>s a es)
and o 208Pb in able IV ( he |(M1)2>s a e). This
happens because he diagonal ma ix elemen s o he
Hamil onian in he wo-phonon and h ee-phonon con-
igu a ions a e close and he ma ix elemen s coupling
he la e wi h o he con igu a ions a e la ge. A simila
si ua ion has been ound in 208Pb o wo one-phonon
(mainly) s a es which ha e a h ee-phonon con igu a ion
as second impo an componen , e en hough ou Hamil-
onian does no couple di ec ly s a es which numbe s o
phonons di e by mo e han one. This is he case o he
s a e which main componen is |M1>, wi h ampli ude
c0=−0.79, and o which he second mos impo an
componen is |(3−)2
2⊗2+>wi h c1= 0.55. Bo h com-
ponen s ha e la ge ma ix elemen s wi h he wo-phonon
s a e |(3−)2
0>. How his mixing o he monopole es-
onance may a ec he monopole esponse, and so he
usual conclusion abou he comp essibili y, is now un-
de s udy. The o he case is he single high ene gy oc-
upole esonance |O > which is s ongly mixed wi h he
s a es |(2+⊗3−)J⊗Q1>. The ene gy o hese s a es
a e, howe e , shi ed by less han 100 keV. This is co-
he en because he s ong mixing is coming om a quasi
degene acy o he conside ed s a es.
Summa izing, he spec um o wo-phonon s a es is
s ongly modi ied by hei coupling o he h ee-phonon
ones. All o he s a es appea mixed wi h he exci a ion
o a GMR and GQR on op o i . This is due o he
ac ha mos o he ma ix elemen s o H21 coupling a
phonon wi h he same phonon plus a GMR o a GQR a e
la ge. Mo eo e , because o he Bose enhancemen ac-
o s, he e ec o H21 be ween he wo and h ee phonon
s a es is e en la ge . I is also o be no ed ha many
o he impo an h ee-phonon s a es a e highe in en-
e gy han he wo-phonon ones. The e o e, hey induce
a sys ema ic shi down o he wo phonon s a es as he
sum o se e al qui e la ge nega i e con ibu ions. This
unexpec ed inding can be unde s ood as a modi ica ion
o he cen al densi y in la ge ampli ude mo ion leading
o an exci a ion o he b ea hing mode. The case o he
GQR seems o be ela ed o he ex eme collec i i y o
his s a e leading o a s ong quad upole esponse o he
quad upole componen o he non-linea i ies o he mean-
ield. We also wan o s ess ha , because o he pe u -
ba i e na u e o he obse ed phenomenon, he possible
in oduc ion o ou -phonon s a es should no modi y he
abo e conclusions abou wo phonon s a es. O cou se,
ou indings imply ha in o de o ge a co ec h ee-
phonon spec um one should u he enla ge he space
up o ou -phonons. This is a o midable ask which is
beyond he scopes o he p esen pape .
[1] M.N. Ha akeh and A. an de Woude (2001) Gian Res-
onances (Cla endon P ess, Ox o d).
[2] Ph. Chomaz and N. F asca ia, Phys. Rep. 252 (1995)
275.
[3] V.Yu. Ponoma e , P.F. Bo ignon, R.A. B oglia and
V.V. Vo ono , Z.Phys. A356 (1996) 251.
[4] V.Yu. Ponoma e and P. Von Neumann-Cosel, Phys.
Re . Le . 82 (1999) 501.
[5] M. Hage-Hassan and M. Lambe , Nucl. Phys. A188
(1972) 545.
[6] F. Ca a a, Ph. Chomaz and N. Van Giai, Phys. Re . 48B
(1993)18207.
[7] C. Volpe, Ph. Chomaz, M.V. And ´es, F. Ca a a and E.G.
Lanza, Nucl. Phys. A647 (1999) 246.
[8] N.V. Giai and H. Sagawa, Nucl. Phys. A371 (1981) 1.
[9] E.G. Lanza, M.V. And ´es, F. Ca a a, Ph. Chomaz and
C. Volpe, Nucl.Phys. A613 (1997) 445.
[10] D. Beaumel and Ph. Chomaz, Ann. Phys. (N.Y.) 213
3
(1992) 405.
[11] C. Simenel e al, in p epa a ion.
TABLE I. RPA one-phonon basis o 40Ca. Fo each s a e,
spin, pa i y, isospin, ene gy and pe cen age o he EWSR a e
epo ed. In he ollowing columns, VM1s ands o he ma ix
elemen < ν|V|ν⊗M1>, whe e νis he one phonon in he
1s column, he same o VM2and VQ1.
Phonons JπT E(MeV ) %EW SR VM1(MeV )VM2(MeV )VQ1(MeV )
M10+0 18.25 30 −2.13 −2.36 −
M20+0 22.47 54 −2.03 −3.96 −
D11−1 17.78 56 −1.38 −2.12 −1.25
D21−1 22.03 10 −1.48 −2.16 +0.73
Q12+0 16.91 85 −1.36 −2.49 −0.36
Q22+1 29.59 26 −1.70 −2.85 −0.00
3−3−0 4.94 14 −1.74 −2.60 −0.07
O13−0 9.71 5 −1.42 −2.28 −0.43
O23−0 31.33 25 −1.69 −2.72 −0.31
TABLE II. Resul s o 40Ca. In he i s column, he s a es
a e labelled by hei main componen in he eigen ec o and
hei unpe u bed ene gy (in pa en heses). In he second col-
umn, he ampli ude o he main componen c0. Then o each
o al angula momen um J, we show he esul s o he calcu-
la ion in he basis up o 2 phonon s a es, he p esen esul s
o he basis ex ended o 3 phonon s a es, he co esponding
i s o de pe u ba ion heo y ene gy, and he second o de
one. The las wo columns con ain he 2nd main componen
in he eigens a es and he co esponding ampli ude c1. The
subindex in he wo-phonon con igu a ions deno es J. All en-
e gies a e gi en in MeV.
Main c0Jπ≤2ph ≤3ph 1s 2nd 2ndmain c1
componen o de o de componen
3−⊗3−−0.91 0+10.96 9.27 12.12 9.20 M10.21
( 9.88) −0.96 2+10.63 8.89 10.66 8.75 (3−)2
2⊗M2−0.21
−0.96 4+9.85 8.10 9.86 7.96 (3−)2
4⊗M2−0.21
−0.96 6+10.88 9.12 10.88 8.99 (3−)2
6⊗M2−0.21
D1⊗D1−0.92 0+35.27 33.71 35.25 33.59 (3−)2
0⊗M2−0.22
(35.56) −0.96 2+35.10 33.66 35.06 33.59 (D1)2
2⊗M2−0.17
D1⊗Q10.95 1−34.83 33.35 34.72 33.24 (M2⊗D1)1⊗Q10.19
(34.69) 0.96 2−34.56 33.22 34.56 33.16 (M2⊗D1)1⊗Q10.19
−0.96 3−34.67 33.13 34.67 33.02 (M2⊗D1)1⊗Q1−0.19
Q1⊗Q1−0.87 0+33.88 32.47 33.83 32.27 (Q1⊗3−)3⊗O10.32
(33.82) 0.84 2+33.82 32.47 33.82 32.26 (Q1⊗3−)5⊗O1−0.38
0.90 4+34.02 32.61 34.02 32.44 (Q1⊗3−)5⊗O1−0.32
M2⊗D1−0.89 1−40.26 38.14 40.05 37.65 (M2)2
0⊗D10.26
(40.25)
M2⊗Q1−0.73 2+39.62 37.34 39.35 36.80 (O1)2
2⊗M10.40
(39.38)
M2⊗M20.67 0+45.60 42.76 44.87 41.18 (O1)2
0⊗M2−0.55
(44.94)
TABLE III. Same as able I o he nucleus 208Pb.
Phonons JπT E(MeV ) %EW SR VM1(MeV )VM2(MeV )VQ1(MeV )
M10+0 13.61 61 −1.87 −0.92 −
M20+0 15.02 28 −1.32 −1.16 −
D11−1 12.43 63 −0.79 −0.59 −0.68
D21−1 16.66 17 0.00 0.00 −0.64
2+2+0 5.54 15 −0.11 0.07 −1.18
Q12+0 11.60 76 −0.64 −0.48 −0.74
Q22+1 21.81 45 −0.86 −0.63 −0.55
3−3−0 3.46 21 −1.13 −0.62 −0.90
O3−0 21.30 37 −0.99 −0.74 −0.42
TABLE IV. Same as able II o he 208Pb nucleus.
Main c0Jπ≤2ph ≤3ph 1s 2nd 2ndmain c1
componen o de o de componen
3−⊗3−−0.95 0+7.88 6.96 8.06 6.90 (3−)2
0⊗2+−0.17
( 6.93) −0.92 2+7.31 6.57 7.33 6.52 2+−0.28
−0.98 4+7.16 6.55 7.16 6.51 (3−)2
4⊗M1−0.15
0.97 6+7.43 6.63 7.44 6.56 (3−)2
6⊗M10.15
3−⊗2+−0.94 1−9.20 8.26 9.21 8.02 (2+)2
2⊗3−−0.23
( 9.01) 0.97 2−9.12 8.54 9.12 8.50 (2+)2
2⊗3−0.17
0.96 3−9.17 8.70 9.12 8.56 (3−)3
2−0.17
0.96 4−9.07 8.61 9.07 8.45 (3−)30.19
−0.96 5−9.06 8.33 9.06 8.16 (2+)2
2⊗3−−0.18
2+⊗2+0.92 0+11.23 9.88 11.24 9.46 (2+)3−0.31
(11.09) −0.94 2+11.27 10.78 11.12 10.61 (3−)2
2⊗2+0.24
0.94 4+11.25 10.39 11.25 10.13 (2+)30.24
D1⊗D10.97 0+24.91 24.42 24.90 24.40 (D1)2
0⊗M10.11
(24.87) 0.96 2+24.68 24.29 24.68 24.27 3−⊗O0.19
D1⊗Q1−0.96 1−24.07 23.73 24.02 23.71 (3−)2⊗D10.17
(24.03) 0.98 2−23.97 23.82 23.97 23.80 (3−)2
2⊗D1−0.16
0.96 3−24.03 23.74 24.03 23.71 (2+)2
2⊗D10.18
Q1⊗Q1−0.94 0+23.20 22.92 23.20 22.86 (3−)2
2⊗Q1−0.24
(23.20) 0.95 2+23.23 23.17 23.18 23.14 (3−)2
2⊗Q10.22
−0.95 4+23.26 23.10 23.26 23.07 (3−)2
2⊗Q1−0.22
M1⊗D1−0.94 1−26.05 25.35 26.02 25.28 (M1)2
0⊗D1−0.20
(26.05)
M1⊗Q1−0.92 2+25.25 24.77 25.22 24.66 (3−)2
2⊗M1−0.20
(25.21)
M1⊗M10.74 0+27.52 26.23 27.28 25.95 (2+)2
0⊗M20.54
(27.22)
4