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Fermi liquid theory and spectrum of excitations of the Pb208 nucleus
1.C: 1.E.1
Nuclear Physics 70 (1965) 545--552; (~) North-Holland Pubhshin9 Co., Amsterdam Not to be reproduced by photoprmt or microfilm without written permission from the pubhsher
F E R M I L I Q U I D T H E O R Y AND S P E C T R U M OF EXCITATIONS OF T H E Pb 2°s NUCLEUS V. N. G U M A N and B. L. B I R B R A I R
Phystcal-Techmcal Institute of the USSR Academy of Scwnces, Lenmyrad Received 17 August 1964 Abstract: The Pb ~°s excitation spectrum is calculated in terms of L a n d a u ' s Fermi hquld theory. It is s h o w n that the region o f decrease of nuclear density is essential in this problem, and the parameters of the effechve forward scattering amphtude are found. The calculated excitation spect r u m is in agreement with experiment
A phenomenological approach to the theory of the nucleus based on Landau's Fermi liquid theory has recently been developed by Migdal et al. 1). In the framework of this approach the interaction between quasi-particles is described by the quantity F~(pl, p2) given by the relation
P°m* a2F°(pl, P2) = ~ PL(COS 012)[fL+OZal "a2+Zl" Zz(f~+9'Lax" a2)], ~2
L
(1)
which is an'analogue of the forward scattering amplitude. The authors of ref. 1) suppose that F '° must not differ appreciably from the scattering amplitude in nuclear matter (an infinite medium with a density corresponding to the central region of nuclei) and hence the expansion coefficients F ~' in the Legendre polynomials PL are connected with the general properties of the nucleus by the well-known relations of the Fermi liquid theory 2, 3). It is assumed furthermore that at low excitation energies the most essential role in nuclear processes belongs to the zero harmonics of the amplitude which has the form, in the coordinate representation,
F~'
--
4%
3po a 2
5(r,-r2)[f-k-go" 1 • O ' 2 - { - ' t "
1 •
zz(f'+g'a1"
a2)],
(2)
where a is the Green function renormalization, Po the density at the centre of the nucleus and e o the Fermi energy. Migdal et al. 4) showed that the magnetic moments of nuclei and the giant dipole resonance can be described in satisfactory agreement with experiment in terms of the above approach. However, at low-excitation energies the diffuseness of the nuclear surface plays an essential role in nuclear processes, and therefore a rigorous solution of the problem must take into account the dependence of F ~' on the density p(r). The sim545
546
V. N. G U M A N A N D B. L . B I R B R A I R
plest formula satisfying the condition /.O,(p) = r ~ , , r~',
p = po p = 0,
(Po is the density at the centre of the nucleus, F,~ the amplitude in nuclear matter and F[° the amplitude in vacuo) has the form
(1 ~[his dependence, however, is actually unknown and may be expressed by a more complicated formula. Nevertheless, it can be supposed that if U ° is preserved for determining eqs. (1) and (2), I.e., If expressions valid for an infinite medium are used, properties of F '° will prove intermediate between F~mand F~'. The parameters f ' , 9 and 9' obtained in ref. 4), from the data on the magnetic moments and giant resonances do not contradict the assumption of the near-equality of F °~ and F~. However, the parameter f cannot be determined from these data and hence the need for investigating further the properties of F °'. One of the sources of information about F °' in the nucleus may be the spectra of the excited states of particle-hole nuclei. The parameters of F ° are determined in this paper from the data on the spectrum of excitation of the nucleus Pb z°8. The choice of this nucleus is due to the fact that correlation effects like Cooper pairing are inessential in it so that the spectrum of excitations at low energies is determined by F °~ alone. Besides, the problem of describing the spectrum of the nucleus Pb z°s in terms of the quintal many-body problem is of independent interest. In no work on Pb 2°8 known to us has an excitation spectrum agreeing with experiment been obtained by introducing a single set of inter-nucleon interaction parameters for all states of this nucleus. Thus, in ref. s) the spacing of all levels has been obtained in agreement with experiment with the exception of the first excitation 3- : its position proved to be 1 MeV higher. Evidently, one cause of the discrepancy of the results of ref. s) with experiment is that the authors neglected the excitations of the core and reduced the problem to the solution of the SchrOdinger equation. To describe the situation of the level 3-, the octupole-octupole interaction was introduced in ref. 6). The situation of other levels was not obtained in this paper. Brown 7) first correctly stated the problem of the excitation spectrum of the particle-hole nucleus. There is an essential difference in our approach; in the investigation of Brown the particle-hole interaction is described by a potential and the equation of motion he derived is linear in the interaction (i.e., coincides with the time-dependent equation of the Hartree-Fock method). In this paper, we describe the interaction by the amplitude of scattering of quasi-particles through zero angle (in momentum space), in which the interaction between nucleons is taken into account in all orders of perturbation theory. Let us note that owing to this circumstance there is no need
FERMI LIQUID THEORY
547
of anti-symmetrizing the matrix elements (2) since all the properties of symmetry are already included in the definition of F ~. The equation for excitation energies of the partmle-hole type is similar to the zero sound equation and has the form (8}'1 -- 8)-2 -- (~)(~'1~'2 = (n'~l -- ~.~2) E (a2/~°3)).1~-2,).3~.4 (~.3~.4 • 23).4
(3)
In our case 2 = nljm, where n is the radial quantum number, l the orbital angular momentum, j the total angular momentum and m its projection. Since the panties of the states of particles and holes in Pb 2°8 are opposite, the excitations at low energies have a negative parity. Calculating the matrix elements (2) we obtain, after all possible simplifications, for even angular momenta of pair particle-hole ~(832 -- 831) 2
--('02]¢jIJ2 =
2F(jl J2 J3 J¢)4[Jll[J21[J31FJ4]
Z
./3.
[J]
× ( [ 0 + g ) 6'~ t , + ( g - g ) (' 1 - 6 ~ r ) ] C j,o, _ ~ , ~ Cja_~j4~ ~o
+ [ ( g + g ),at~,+(g_g,)(l_a~,)](_l)h+,3+./,+./3+l
(4) Ji
s~
and for odd angular momenta Yz J3 J4)4[Jl][J2][J3][J4] ./3./, L t
,
x {[ ( f + f )aw + ( f - f ) ( 1
[J] JO
JO
-- aw)] C j, _ ~./2~C./3 - ~,,~(8:~ - 8./,)
-~-[(g-q-~")azt,.+(g-g')(l-art,)] ( - I ) / ' +la+./l+.13+IC ./,=./~ Jl , c ./~-j,,~t Jl . ,i 8 ./,,-8./3)}
+ Z 4F(jl J2 J5 j6)F(j3 j,, J5 J6)[Jd[J6]x/[JI][J2][y3][J,d ../o
[J]=
x [(f+f')a~,, + ( f - f ' ) ( 1 -- 6,,,)] [(g + g')at,, + (g -- g')(l -- ate)] Jo-~rJ,½ C.,.o .i, C"'./5~./6½~ t- l)h +t'+./'+./'+ 1] ¢J3'." x C./3 .15-~./6~ C./~t..,2~
0)
Here we have written [j] = 2 j + l , and
F ( j , J2 J3 J4) - .,480- 2 f( R,,,t,(r jt)R,ut,(r2) a(rl- r2) R,~,3( r ,)R,,,,,(r2)r21 r 2 dr, dr 2 . apoa d d " 47rr~ r 2 In eqs. (4) and (5) J i s the angular momentum of the pair, jlj3 > F, a n d j 2 j , < F, where F is the Fermi level. The expansion of the amplitude in the angular momenta of the particle-hole pair for scalar terms coincides with the expansion of a(r~2) in the Legendre polynomials Pz(cos 0~2), and in the case of negative parity it contains only odd J. Therefore the problem of excitations with even J includes only the spin terms of F°L This makes it
548
V
N,
G U M A N A N D B. L . B I R B R A I R
possible to determine 9 and 9' from the experimental data on the energies of excited states with even J. The diagonalization of the matrices, determination of the matrix elements and the radial integrals they contain were performed with a rapid-action computer BESM-2 of the Computer Centre of the USSR Academy of Sciences. The radial functions tabulated in ref. 8) are used. The accuracy of our investigation is determined by the accuracy of slngle-parucle energies and functions in this last paper. IVleV 40
35
30
26
i~
'
'
'
;o
.
.
.
.
~'o 3~
_-5
Fig. 1. D e p e n d e n c e o f the p o s i t i o n o f the level J = 3 - o n the n u m b e r o f states o f the p a r t i c l e - h o l e pair t a k e n into a c c o u n t .
It is clear from eqs. (4) and (5) that the problem of finding eigenvalues essentially differs from the similar problem in the solution of the Schr6dinger equation when it reduces to the dlagonalization of a symmetrical matrix with respect to co. In our case the problem reduces to the diagonalization of an unsymmetrical matrix with respect to 0)2, and this immediately restricts the region of permissible values of the parameters for which the equations have real roots. It turns out in this case that the allowed values of the parameters lie within rather narrows limits. For example, the allowed values of the parameters f and f ' for which eq. (5) has real roots lie within -1.1
-0.5,
-0.5
< 0.5.
(6)
Outside this region co2 is either negative or complex. A similar region of allowed values exists for g and 9' as well.
FERMI LIQUID THEORY
549
The quantum states of quasi-particles and quasi-holes located in adjacent main shells were taken into account in the calculations. It can readily be seen that the quantum states lying in alternate shells do not enter into the problem since they coincide in parity (it will be recalled that all the lower levels of Pb 2°8 have a negative parity). The quantum states lying in each third shell and more remote ones are neglected; on the one hand, between them there is a large energy gap of the order of 20 MeV and on the other hand the corresponding radial functions differ by the number of nodes so that their contribution to the energies of lower levels prove to be negligible. Note that these states can be taken into account rigorously by the renormahzation of F'". The dependence of the situation of the level 3- on the order of the matrix was investigated. The result represented in fig. 1 points to an essential dependence; saturation is only reached at the 26th order. This circumstance compelled us to diagonalize for each level the matrix of the highest order possible for the given single-particle states. For example, the order of the matrix for the 3- level is 32, for the 6- level it is 22, etc. Let us note, however, that the position of the 3- level is most sensitive to the order of the matrix since the interaction in this state is the largest compared with states with higher J, as a result of which the level 3- is the first excited state of Pb 2°8. The constants 9 and 9' are determined from the data on the levels 4 - (3.47 MeV) and 6 - (3.96 MeV). The values obtained he within 0.3<9,
9' < 0 . 7 .
These values do not contradict the data of ref. 4) on the magnetic moments, according to which 9 = 9' = 0.5. For these values of 9, 9' eq. (4) gives E ( 4 - ) = 3.4 MeV,
E ( 6 - ) = 4 MeV.
Substituting the values of 9 and 9' obtained into eq. (5) we can d e t e r m i n e f a n d f ' by the well-known 9) five levels with odd J. Thus d e t e r m i n e d f a n d f ' lie on a straight line represented in fig. 2. The coordinates of the extreme points of this straight line are f = -0.9,
f' = -0.15;
f = -0.7,
f ' = 0.2.
We cannot d e t e r m l n e f a n d f ' more accurately since the accuracy of our calculations does not exceed 0.1 MeV. Fig. 3 gives a comparison of the results yielded by the different points of the straight line of fig. 2 with experimental data. It is clear from fig. 3 that the results of the calculations are in satisfactory agreement with the experiment for all levels with known values of the spins. For the level with an energy 3.37 MeV the calculations give J = 5- and for the level corresponding to 3.6 MeV two close levels with spins 3- and 4 - . Besides, at E = 3.9 MeV the calculations give a level with spin 5 - . Thus, in the framework of the Fermi liquid theory we can describe the Pb 2°8 excitation spectrum in satisfactory agreement with experiment by describing the interaction between quasi-particles by a single set of parameters for all states.
550
V.N.
GUMAN
AND
B. L . B I R B R A I R
Let us now c o m p a r e o u r v a l u e s f , f ' , g a n d g ' with their values in vacuo. A c c o r d i n g to ref. 4)
f}
!
fl = -1.41,
= -0.01,
t
g l = 0.36,
gy = 0.40. !
i
--
- i
i
i
v.~
-0.7
-o~
Fig. 2. Sahsfactory a g r e e m e n t with experiment o f f a n d f ' located on a given straight hne.
Mev
40,
6
-
m
5
"
~
6~
5"
s"
#:3"
4"
5~ 6"
3.0
3 - . - -
3
"
-
-
20 e2tpeztrrt
eeveCs
jo-0.75 j'. o~5
J o-os6 J'=-oo5 °8"- 0.5
Fig, 3. C o m p a r i s o n o f the experimental s p e c t r u m o f the Pb ~°s nucleus with two variants o f the calculated spectrum.
551
FERMI LIQUID THEORY
Obviously, our values for g, g' a n d f ' do not contradict their vacuum values or the data on the magnetic moments and giant resonances. According to ref. 4 ) f , = 0.35, which does not contradict our value 0.2 since the shift of the position and width of the giant resonance with respect to the predictions of the single-particle model is determined in the quasi-classical approximation by the factor x/3fl/e o = (1 + 2 f ' ) ½, where e o = 40 MeV is the Fermi energy, and fl = 22 MeV is the coefficient of the
ere,,) Fig. 4. Dependence of matrix elements on radial distance. 1: (lh~ li~ lh~t li¥) 2: (2d~ lh~ 2d~ lh~) 3: (2f~2g~ lh~ li~) 4: (2f~2g~f~,2g~).
symmetry energy in the Weizs/icker formula. The case is different with f ; this parameter in a medium must satisfy the stability condition 2fm > -- 1. Our values of f (which agree with experiment and are allowed by eq. (6)) do not satisfy this condition, just as in the case of the vacuum value f. This result may mean, of course, that the higher harmonics of eq. (1) disregarded in the calculations also play a role in the problem under consideration. However, taking into account the higher harmonics essentially increases the number of parameters of the theory so that their determination in the problem under study becomes quite difficult. Quite apart from
552
V.N. GUMAN AND B. L. BIRBRAIR
this fact, it ought to be expected that the constant f will prove negative (which corresponds to attraction between quasi-particles); otherwise the existence in atomic nuclei of collective levels with energies below the energy gap in the spectrum of singleparticle excitations would be unintelligible (the level 3- in Pb 2°8 and the levels 2 + in even nuclei which are remote from the magic). Therefore, our assumptions that the properties of F °~ in real nuclei are intermediate between the properties of vacuo and those in a m e d m m seem reasonable. Let us now consider what region of the nucleus is essential in the problem under study. Fig. 4 represents the dependence of the integrand for certain matrix elements/"~ on the distance from the centre of the nucleus. The dashed line in fig. 4 represents the density p(r) of the Pb 2°8 nucleus i0). F r o m fig. 4 it is clear that all integrands have a maximum in the region where the density decreases. Only such matrix elements in which all radial functions have coinciding nodes (the radial quantum number being larger than or equal to 2) have a maximum in the constant densinty region. However, this coincidence of nodes occurs rather rarely and hence is not typical of the integrands of the matrix elements. The position of the maximum of curve 2 is characteristic. Thus, the region of decrease of nuclear density is essential in the problem under study. It is this circumstance that causes the parameters F °~ we have determined to have properties intermediate between Fm ~ and F~'. In general one could try to determine the parameters F m under the assumption that there is a dependence of F "~ on p(r) and F~ is known. Let us note, however, that the relation F = F(p) implies an assumption about local equivalence which is most justified in the case of weakly heterogeneous systems. The parameters F ~ can furthermore be made accurate by analysing the M1 and E2 transitions in nuclei of the type Pb2°8+ 1 nucleon as well as the T12°8 and BI2°8 excitation spectra. The results of these investigations will be publisheds subsequently. The dependence of the spacing of levels with odd J on g, 9' was also investigated. This dependence proved to be weak, which ought to be expected since the spin terms of the amplitude enter the problem of levels with odd J only through the spin-orbit coupling. The authors are grateful to L. A. Sllv, A. B. Migdal and A. I. Larkin for their interest in the work and stimulating discussions. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
A. B. Mlgdal and A. I. Larkm, JETP 45 (1963) 1036 L. P Pltayevsky, JETP 37 (1959) 1794 A. B. Mlgdal, JETP 43 (1962) 1940 A. B. Mlgdal, Programme and theses of reports of the 14th Annual Conf. for Nuclear spectroscopy m Tblhsl (February 14-22, 1964) p. 23 Pmkston, Nuclear Physms 37 (1962) 312 D. G. Choudhury, Phys. Rev. 129 (1963) 1754 G. E. Brown, Nuclear Physms 24 (1961) J. Blomqvlst and S. Wahlborn, Ark. Fys. 16 (1960) 545 B. S. Dzhelepov and L. K. Peker, Schemes of Decay of Radioactive Nuclei. A > 100 Handbuch der Physlk, Vol. 39, fig. 227
Nuclear Physics 70 (1965) 545--552; (~) North-Holland Pubhshin9 Co., Amsterdam Not to be reproduced by photoprmt or microfilm without written permission from the pubhsher
F E R M I L I Q U I D T H E O R Y AND S P E C T R U M OF EXCITATIONS OF T H E Pb 2°s NUCLEUS V. N. G U M A N and B. L. B I R B R A I R
Phystcal-Techmcal Institute of the USSR Academy of Scwnces, Lenmyrad Received 17 August 1964 Abstract: The Pb ~°s excitation spectrum is calculated in terms of L a n d a u ' s Fermi hquld theory. It is s h o w n that the region o f decrease of nuclear density is essential in this problem, and the parameters of the effechve forward scattering amphtude are found. The calculated excitation spect r u m is in agreement with experiment
A phenomenological approach to the theory of the nucleus based on Landau's Fermi liquid theory has recently been developed by Migdal et al. 1). In the framework of this approach the interaction between quasi-particles is described by the quantity F~(pl, p2) given by the relation
P°m* a2F°(pl, P2) = ~ PL(COS 012)[fL+OZal "a2+Zl" Zz(f~+9'Lax" a2)], ~2
L
(1)
which is an'analogue of the forward scattering amplitude. The authors of ref. 1) suppose that F '° must not differ appreciably from the scattering amplitude in nuclear matter (an infinite medium with a density corresponding to the central region of nuclei) and hence the expansion coefficients F ~' in the Legendre polynomials PL are connected with the general properties of the nucleus by the well-known relations of the Fermi liquid theory 2, 3). It is assumed furthermore that at low excitation energies the most essential role in nuclear processes belongs to the zero harmonics of the amplitude which has the form, in the coordinate representation,
F~'
--
4%
3po a 2
5(r,-r2)[f-k-go" 1 • O ' 2 - { - ' t "
1 •
zz(f'+g'a1"
a2)],
(2)
where a is the Green function renormalization, Po the density at the centre of the nucleus and e o the Fermi energy. Migdal et al. 4) showed that the magnetic moments of nuclei and the giant dipole resonance can be described in satisfactory agreement with experiment in terms of the above approach. However, at low-excitation energies the diffuseness of the nuclear surface plays an essential role in nuclear processes, and therefore a rigorous solution of the problem must take into account the dependence of F ~' on the density p(r). The sim545
546
V. N. G U M A N A N D B. L . B I R B R A I R
plest formula satisfying the condition /.O,(p) = r ~ , , r~',
p = po p = 0,
(Po is the density at the centre of the nucleus, F,~ the amplitude in nuclear matter and F[° the amplitude in vacuo) has the form
(1 ~[his dependence, however, is actually unknown and may be expressed by a more complicated formula. Nevertheless, it can be supposed that if U ° is preserved for determining eqs. (1) and (2), I.e., If expressions valid for an infinite medium are used, properties of F '° will prove intermediate between F~mand F~'. The parameters f ' , 9 and 9' obtained in ref. 4), from the data on the magnetic moments and giant resonances do not contradict the assumption of the near-equality of F °~ and F~. However, the parameter f cannot be determined from these data and hence the need for investigating further the properties of F °'. One of the sources of information about F °' in the nucleus may be the spectra of the excited states of particle-hole nuclei. The parameters of F ° are determined in this paper from the data on the spectrum of excitation of the nucleus Pb z°8. The choice of this nucleus is due to the fact that correlation effects like Cooper pairing are inessential in it so that the spectrum of excitations at low energies is determined by F °~ alone. Besides, the problem of describing the spectrum of the nucleus Pb z°s in terms of the quintal many-body problem is of independent interest. In no work on Pb 2°8 known to us has an excitation spectrum agreeing with experiment been obtained by introducing a single set of inter-nucleon interaction parameters for all states of this nucleus. Thus, in ref. s) the spacing of all levels has been obtained in agreement with experiment with the exception of the first excitation 3- : its position proved to be 1 MeV higher. Evidently, one cause of the discrepancy of the results of ref. s) with experiment is that the authors neglected the excitations of the core and reduced the problem to the solution of the SchrOdinger equation. To describe the situation of the level 3-, the octupole-octupole interaction was introduced in ref. 6). The situation of other levels was not obtained in this paper. Brown 7) first correctly stated the problem of the excitation spectrum of the particle-hole nucleus. There is an essential difference in our approach; in the investigation of Brown the particle-hole interaction is described by a potential and the equation of motion he derived is linear in the interaction (i.e., coincides with the time-dependent equation of the Hartree-Fock method). In this paper, we describe the interaction by the amplitude of scattering of quasi-particles through zero angle (in momentum space), in which the interaction between nucleons is taken into account in all orders of perturbation theory. Let us note that owing to this circumstance there is no need
FERMI LIQUID THEORY
547
of anti-symmetrizing the matrix elements (2) since all the properties of symmetry are already included in the definition of F ~. The equation for excitation energies of the partmle-hole type is similar to the zero sound equation and has the form (8}'1 -- 8)-2 -- (~)(~'1~'2 = (n'~l -- ~.~2) E (a2/~°3)).1~-2,).3~.4 (~.3~.4 • 23).4
(3)
In our case 2 = nljm, where n is the radial quantum number, l the orbital angular momentum, j the total angular momentum and m its projection. Since the panties of the states of particles and holes in Pb 2°8 are opposite, the excitations at low energies have a negative parity. Calculating the matrix elements (2) we obtain, after all possible simplifications, for even angular momenta of pair particle-hole ~(832 -- 831) 2
--('02]¢jIJ2 =
2F(jl J2 J3 J¢)4[Jll[J21[J31FJ4]
Z
./3.
[J]
× ( [ 0 + g ) 6'~ t , + ( g - g ) (' 1 - 6 ~ r ) ] C j,o, _ ~ , ~ Cja_~j4~ ~o
+ [ ( g + g ),at~,+(g_g,)(l_a~,)](_l)h+,3+./,+./3+l
(4) Ji
s~
and for odd angular momenta Yz J3 J4)4[Jl][J2][J3][J4] ./3./, L t
,
x {[ ( f + f )aw + ( f - f ) ( 1
[J] JO
JO
-- aw)] C j, _ ~./2~C./3 - ~,,~(8:~ - 8./,)
-~-[(g-q-~")azt,.+(g-g')(l-art,)] ( - I ) / ' +la+./l+.13+IC ./,=./~ Jl , c ./~-j,,~t Jl . ,i 8 ./,,-8./3)}
+ Z 4F(jl J2 J5 j6)F(j3 j,, J5 J6)[Jd[J6]x/[JI][J2][y3][J,d ../o
[J]=
x [(f+f')a~,, + ( f - f ' ) ( 1 -- 6,,,)] [(g + g')at,, + (g -- g')(l -- ate)] Jo-~rJ,½ C.,.o .i, C"'./5~./6½~ t- l)h +t'+./'+./'+ 1] ¢J3'." x C./3 .15-~./6~ C./~t..,2~
0)
Here we have written [j] = 2 j + l , and
F ( j , J2 J3 J4) - .,480- 2 f( R,,,t,(r jt)R,ut,(r2) a(rl- r2) R,~,3( r ,)R,,,,,(r2)r21 r 2 dr, dr 2 . apoa d d " 47rr~ r 2 In eqs. (4) and (5) J i s the angular momentum of the pair, jlj3 > F, a n d j 2 j , < F, where F is the Fermi level. The expansion of the amplitude in the angular momenta of the particle-hole pair for scalar terms coincides with the expansion of a(r~2) in the Legendre polynomials Pz(cos 0~2), and in the case of negative parity it contains only odd J. Therefore the problem of excitations with even J includes only the spin terms of F°L This makes it
548
V
N,
G U M A N A N D B. L . B I R B R A I R
possible to determine 9 and 9' from the experimental data on the energies of excited states with even J. The diagonalization of the matrices, determination of the matrix elements and the radial integrals they contain were performed with a rapid-action computer BESM-2 of the Computer Centre of the USSR Academy of Sciences. The radial functions tabulated in ref. 8) are used. The accuracy of our investigation is determined by the accuracy of slngle-parucle energies and functions in this last paper. IVleV 40
35
30
26
i~
'
'
'
;o
.
.
.
.
~'o 3~
_-5
Fig. 1. D e p e n d e n c e o f the p o s i t i o n o f the level J = 3 - o n the n u m b e r o f states o f the p a r t i c l e - h o l e pair t a k e n into a c c o u n t .
It is clear from eqs. (4) and (5) that the problem of finding eigenvalues essentially differs from the similar problem in the solution of the Schr6dinger equation when it reduces to the dlagonalization of a symmetrical matrix with respect to co. In our case the problem reduces to the diagonalization of an unsymmetrical matrix with respect to 0)2, and this immediately restricts the region of permissible values of the parameters for which the equations have real roots. It turns out in this case that the allowed values of the parameters lie within rather narrows limits. For example, the allowed values of the parameters f and f ' for which eq. (5) has real roots lie within -1.1
-0.5,
-0.5
< 0.5.
(6)
Outside this region co2 is either negative or complex. A similar region of allowed values exists for g and 9' as well.
FERMI LIQUID THEORY
549
The quantum states of quasi-particles and quasi-holes located in adjacent main shells were taken into account in the calculations. It can readily be seen that the quantum states lying in alternate shells do not enter into the problem since they coincide in parity (it will be recalled that all the lower levels of Pb 2°8 have a negative parity). The quantum states lying in each third shell and more remote ones are neglected; on the one hand, between them there is a large energy gap of the order of 20 MeV and on the other hand the corresponding radial functions differ by the number of nodes so that their contribution to the energies of lower levels prove to be negligible. Note that these states can be taken into account rigorously by the renormahzation of F'". The dependence of the situation of the level 3- on the order of the matrix was investigated. The result represented in fig. 1 points to an essential dependence; saturation is only reached at the 26th order. This circumstance compelled us to diagonalize for each level the matrix of the highest order possible for the given single-particle states. For example, the order of the matrix for the 3- level is 32, for the 6- level it is 22, etc. Let us note, however, that the position of the 3- level is most sensitive to the order of the matrix since the interaction in this state is the largest compared with states with higher J, as a result of which the level 3- is the first excited state of Pb 2°8. The constants 9 and 9' are determined from the data on the levels 4 - (3.47 MeV) and 6 - (3.96 MeV). The values obtained he within 0.3<9,
9' < 0 . 7 .
These values do not contradict the data of ref. 4) on the magnetic moments, according to which 9 = 9' = 0.5. For these values of 9, 9' eq. (4) gives E ( 4 - ) = 3.4 MeV,
E ( 6 - ) = 4 MeV.
Substituting the values of 9 and 9' obtained into eq. (5) we can d e t e r m i n e f a n d f ' by the well-known 9) five levels with odd J. Thus d e t e r m i n e d f a n d f ' lie on a straight line represented in fig. 2. The coordinates of the extreme points of this straight line are f = -0.9,
f' = -0.15;
f = -0.7,
f ' = 0.2.
We cannot d e t e r m l n e f a n d f ' more accurately since the accuracy of our calculations does not exceed 0.1 MeV. Fig. 3 gives a comparison of the results yielded by the different points of the straight line of fig. 2 with experimental data. It is clear from fig. 3 that the results of the calculations are in satisfactory agreement with the experiment for all levels with known values of the spins. For the level with an energy 3.37 MeV the calculations give J = 5- and for the level corresponding to 3.6 MeV two close levels with spins 3- and 4 - . Besides, at E = 3.9 MeV the calculations give a level with spin 5 - . Thus, in the framework of the Fermi liquid theory we can describe the Pb 2°8 excitation spectrum in satisfactory agreement with experiment by describing the interaction between quasi-particles by a single set of parameters for all states.
550
V.N.
GUMAN
AND
B. L . B I R B R A I R
Let us now c o m p a r e o u r v a l u e s f , f ' , g a n d g ' with their values in vacuo. A c c o r d i n g to ref. 4)
f}
!
fl = -1.41,
= -0.01,
t
g l = 0.36,
gy = 0.40. !
i
--
- i
i
i
v.~
-0.7
-o~
Fig. 2. Sahsfactory a g r e e m e n t with experiment o f f a n d f ' located on a given straight hne.
Mev
40,
6
-
m
5
"
~
6~
5"
s"
#:3"
4"
5~ 6"
3.0
3 - . - -
3
"
-
-
20 e2tpeztrrt
eeveCs
jo-0.75 j'. o~5
J o-os6 J'=-oo5 °8"- 0.5
Fig, 3. C o m p a r i s o n o f the experimental s p e c t r u m o f the Pb ~°s nucleus with two variants o f the calculated spectrum.
551
FERMI LIQUID THEORY
Obviously, our values for g, g' a n d f ' do not contradict their vacuum values or the data on the magnetic moments and giant resonances. According to ref. 4 ) f , = 0.35, which does not contradict our value 0.2 since the shift of the position and width of the giant resonance with respect to the predictions of the single-particle model is determined in the quasi-classical approximation by the factor x/3fl/e o = (1 + 2 f ' ) ½, where e o = 40 MeV is the Fermi energy, and fl = 22 MeV is the coefficient of the
ere,,) Fig. 4. Dependence of matrix elements on radial distance. 1: (lh~ li~ lh~t li¥) 2: (2d~ lh~ 2d~ lh~) 3: (2f~2g~ lh~ li~) 4: (2f~2g~f~,2g~).
symmetry energy in the Weizs/icker formula. The case is different with f ; this parameter in a medium must satisfy the stability condition 2fm > -- 1. Our values of f (which agree with experiment and are allowed by eq. (6)) do not satisfy this condition, just as in the case of the vacuum value f. This result may mean, of course, that the higher harmonics of eq. (1) disregarded in the calculations also play a role in the problem under consideration. However, taking into account the higher harmonics essentially increases the number of parameters of the theory so that their determination in the problem under study becomes quite difficult. Quite apart from
552
V.N. GUMAN AND B. L. BIRBRAIR
this fact, it ought to be expected that the constant f will prove negative (which corresponds to attraction between quasi-particles); otherwise the existence in atomic nuclei of collective levels with energies below the energy gap in the spectrum of singleparticle excitations would be unintelligible (the level 3- in Pb 2°8 and the levels 2 + in even nuclei which are remote from the magic). Therefore, our assumptions that the properties of F °~ in real nuclei are intermediate between the properties of vacuo and those in a m e d m m seem reasonable. Let us now consider what region of the nucleus is essential in the problem under study. Fig. 4 represents the dependence of the integrand for certain matrix elements/"~ on the distance from the centre of the nucleus. The dashed line in fig. 4 represents the density p(r) of the Pb 2°8 nucleus i0). F r o m fig. 4 it is clear that all integrands have a maximum in the region where the density decreases. Only such matrix elements in which all radial functions have coinciding nodes (the radial quantum number being larger than or equal to 2) have a maximum in the constant densinty region. However, this coincidence of nodes occurs rather rarely and hence is not typical of the integrands of the matrix elements. The position of the maximum of curve 2 is characteristic. Thus, the region of decrease of nuclear density is essential in the problem under study. It is this circumstance that causes the parameters F °~ we have determined to have properties intermediate between Fm ~ and F~'. In general one could try to determine the parameters F m under the assumption that there is a dependence of F "~ on p(r) and F~ is known. Let us note, however, that the relation F = F(p) implies an assumption about local equivalence which is most justified in the case of weakly heterogeneous systems. The parameters F ~ can furthermore be made accurate by analysing the M1 and E2 transitions in nuclei of the type Pb2°8+ 1 nucleon as well as the T12°8 and BI2°8 excitation spectra. The results of these investigations will be publisheds subsequently. The dependence of the spacing of levels with odd J on g, 9' was also investigated. This dependence proved to be weak, which ought to be expected since the spin terms of the amplitude enter the problem of levels with odd J only through the spin-orbit coupling. The authors are grateful to L. A. Sllv, A. B. Migdal and A. I. Larkin for their interest in the work and stimulating discussions. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)
A. B. Mlgdal and A. I. Larkm, JETP 45 (1963) 1036 L. P Pltayevsky, JETP 37 (1959) 1794 A. B. Mlgdal, JETP 43 (1962) 1940 A. B. Mlgdal, Programme and theses of reports of the 14th Annual Conf. for Nuclear spectroscopy m Tblhsl (February 14-22, 1964) p. 23 Pmkston, Nuclear Physms 37 (1962) 312 D. G. Choudhury, Phys. Rev. 129 (1963) 1754 G. E. Brown, Nuclear Physms 24 (1961) J. Blomqvlst and S. Wahlborn, Ark. Fys. 16 (1960) 545 B. S. Dzhelepov and L. K. Peker, Schemes of Decay of Radioactive Nuclei. A > 100 Handbuch der Physlk, Vol. 39, fig. 227