Level densities of excited systems with pairing interactions

Nuclear Physics A92 (1967) 345--352; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

LEVEL DENSITIES OF EXCITED SYSTEMS W I T H PAIRING INTERACTIONS J. N I ~ M E T H

Institute of Theoretical Physics, E6tvb's University, Budapest Received 15 A u g u s t 1966 Abstract: T h e energy a n d e n t r o p y o f excited systems are determined with the help o f the corrected BCS a p p r o x i m a t i o n . T h e level density is obtained for low t e m p e r a t u r e s as a function o f the

excitation energy a n d the energy gap. T h e level densities o f even a n d o d d - m a s s systems are compared.

1. In~oducfion

In the last years it turned out, that in the case of atomic nuclei the pairing interactions play an important role 1). Their effect is significant first of all for ground states and low excitations, but the level density difference between even- and odd-mass nuclei shows, that the effect of pairing interactions cannot be neglected even for higher excitations. If we want to determine the level densities of nuclei, we have to take them into account. The pairing correlation can be taken into account with the help of the BCS approximation 2). This approximation, however, is not applicable directly for systems with low particle number. In these cases we can get better results applying correction methods. One of the simplest correction methods was suggested by Lipkin 3). It seems that it gives better results than the blocking method 4). In the following we shall determine the level density of excited systems as a function of the excitation energy for a simple and mathematically easily treatable model, and compare the level densities of even and odd-mass systems. In this way conclusions can be drawn about the level densities of the real nuclei. 2. The ground state energy of even- and odd-mass systems

In the following we shall consider a system with constant pairing interaction. The model Hamiltonian is the following: tI : Z v(a +÷a ++aLav ) - v

~ a ~ ++a v -+a ¢ - a v , + ,

(2.1)

where e~ are the one-particle energies, V is the matrix element of the pairing interaction, a +, a~ are the creation and absorption operators and v characterizes the quantum states of the particles. For the sake of simplicity we assumed that the pairing 345

346

j. N~MEXH

interaction is constant if v and v' are between vl and v2, and zero otherwise. This very simple model describes rather well certain proprieties of atomic nuclei ~' 3). To obtain the eigenvalues of (2.1), a Bogolyubov transformation should be performed 5). Let us introduce the following operators: +

~+ = u~av+-v~av-, c~v- : uvav- +way++,

(2.2)

u2+v~ : 1.

(2.3)

where

It is very difficult to determine an exact eigenfunction of (2.1), but it is easy to get an approximative solution. The difficulty with this approximation lies in the fact that the trial wave function is not an eigenfunction of the particle number operator. Once a Bogolyubov transformation is performed, it is hard to remove the particle number fluctuation from the wave function, however we can suppress its effect on the energy. For this aim let us consider the following Hamiltonian: JdY : H - 21 N - 25 N 2,

(2.4)

where N is the particle number operator, 21 and 22 are two free parameters to be determined by the condition that the mean value of N in the ground state equals the number of particles : n, (2.5) and the average of ~ : be independent of n as much as possible. The corrected energy can be obtained as E~ = E + 2 t n + 2 2 n2, (2.6) where E = . Introducing the transformation (2.2) into (2.4), the Hamiltonian can be written in the form jgo :

(2.7)

U+H11 + H 2 0 + H 2 2 +H31 + / / 4 0 ,

where U n,,

~.

2 ~ e v v v 2_ _ A z~ _ _ l / ~ v _ 241 n _ d , V

zn2_422~

2

II v Uv2 ,

+ + = Z [ ( u ~ - v 2v ) ( e _ , ~,+ ( 4 2 z _ v)vZ)+d,z+2auvvv](~v+o:~++o:v_~v_),

H2o = Z [2u~'v~(e~-2+( 4 2 2 - V ) v z ) - A ( u z - v ~ ) ] ( ~ v + c~+-+~v-%+)'

(2.7a)

(2.7b) (2.7c)

A = V Z u~v~,

(2.8)

2 = 21 + 2 2 2 ( n + 1),

(2.9)

and in //22, H31 and //4o there are four quasipartMe creation and absorption operators. F r o m the condition H2o =- 0 we get the parameters v~ the following

PAIRING INTERACTIONS

347

equation

(2.10)

2u, v~(e, + ~v 2) = A (u ~ - v~),

where the notations 422 - V

= ~,

gv-2

= ev

are introduced. Supposing that c~<< A or c~<< e~, we get for v~ in second order

u,

~ 1+

+ ~ ~

2 = 1 (- 2 1 - Vv

1-

,

::)1A2"(1-e~)3 4

(2.11a) (2.11b)

Ev

where e~ = x / ~ + A 2. To obtain the energy we have to solve (2.5) and (2.8), taking into account (2.11). To get these solutions let us suppose that the one-particle levels are almost continuous and the level density distribution is homogeneous in the energy region e2-81. For strongly deformed heavy nuclei this is almost the realistic case 6). If this assumption is valid, we can substitute the summations with integrations Y, --+ Po

d/3.

(2.12)

To perform the calculations we shall assume in the following the validity of the inequalities t Vpo < 1, Apo > 1, (2.13) A / A < 1, V/A < 1, where ~2 is the number of levels in the energy region 8z-/31, and P0 = (2/(82 - /31). Introducing the notations ~ 2 - 2 = b, ,~1--2

=

a~

b + a = B,

(2.14)

gl'~-g2 = 2e,

we obtain for the energy in second order approximation (A.8) E ~ e" = ±T7~'2"±~'"z-±"24-', q 4 ~'' 4" 7 + 1 /32-81 7-1 f2

(2.15)

Eliminating from (2.15) the n-dependence, we get = -V+

y + l g2-/31 7--1 f2

(2.16)

The condition ~ < A is fulfilled whenever A/A~2 = 1]poA < 1. t These inequalities are rather well fulfilled for strongly deformed heavy nuclei, where p o l ~ 300-400 keY, V ~ 70-140 keV, A ~ 600-900 keV, e2-- q = .4 ~ 10-20 MeV (ref. 5)).

348

J. NEMETH

21 can be d e t e r m i n e d f r o m (2.14) a n d (A.5)

2, =

e,

-

1 ~,+1

2 -y~-i ( e 2 - e O - ½ V "

(2.17)

W i t h the help o f (2.16), (2.17) the corrected energy turns out to be 2.n.4,~

O

)' (132--/31)

1-

.

(2.18)

O u r next a i m is to determine the energy o f an o d d - m a s s system. A n o d d - m a s s system can be considered as the first excited state o f an even system. F o r excited states the m e a n value o f the particle n u m b e r o p e r a t o r is not equal to the exact particle n u m b e r any more, a n d H2o, the d a n g e r o u s t e r m in the H a m i l t o n i a n differs f r o m zero. The v~ p a r a m e t e r are the same as in the g r o u n d state a n d 2't , 2~ are d e t e r m i n e d in such a way that the energy is i n d e p e n d e n t o f n. U s i n g eqs. (2.7), the energy o f an o d d system can be o b t a i n e d as E °dd =

Eo+E~o ,

(2.19)

where Eo = + rn2 + r E E~ ° = (u,.2o - V~o)(e~o 2 + ~v~° -FV~o+ 2 6 + ½f'(n + 1 ) ) + 2 i + 2Auvov~ o , 6 = 2,-),'1,

F = 4(22-2~),

(2.19a) (2.19b)

a n d v o characterizes the q u a n t u m state occupied by the o d d particle. The state v o can be d e t e r m i n e d in the g r o u n d state f r o m the c o n d i t i o n t h a t the energy should have a m i n i m u m . F r o m (2.19) it is easy to eliminate again the n-dependence, and the corrected energy turns out to be *

E °aa = E ~ v e " + A + ~ 2 - e ' 40

7+1 7- 1

(2.20)

3. The energy and level density of excited systems I n the case o f an excited system there are two kinds o f excitations: the single particle a n d the p a i r excitations. In the following we shall deal only with single particle excitations, and we consider the pair excitations just as single particle ones. T h e discussion o f p a i r excitations is r a t h e r difficult because o f the s p u r i o n states s), b u t for n o t t o o high excitation energy it is r e a s o n a b l e to assume that their i m p o r t a n c e is n o t t o o significant4). Let us denote by s v the p r o b a b i l i t y o f a single particle excitation in the state v. W i t h this n o t a t i o n the energy o f an excited system can be written as

ES = E o + Z E~l)sv + Z E~2v)(svs¢--Sv6,'~') , * The more detailed calculations are in ref. 7).

(3.1)

349

PAIRING INTERACTIONS

E(2) can be obtained from (2.7). Eliminating the nwhere --v E t~) equals (2.19b) and _~, 3 the corrected energy dependence of (3.1) and neglecting the terms proportional to s~, turns out to be for not too high excitation energy

E~ = g,.-½Vn+ 4

,,,,.(1 7

Q

+ ~ s ~ [ E'+½V+e2-g'2Q evE.+e2--et4f2 A~ 22]

_¼ ~s~s., [e.e¢(~-el).+AzV+ 4A2e. (l_ Ev Ev'

~23] " (3.2)

~'~

To obtain the level density of an excited system we have to determine its entropy. The entropy can be written as

S = ~ [s~ In s~+(1 -s~)in (1 -sv)],

(3.3)

and the s~ parameters can be obtained by minimizing the free energy of the system as a function of sv. The free energy of an excited system is

F = E-- OS,

(3.4)

and s~ turns out to be s~ -

1

(3.5)

1 + e p aV.la~ '

where fl = O -a, and O is the temperature of the excited system. The quantity is easily obtainable from (3.2) dE' -- P ~ + Z R ~ ¢ s ¢ , ds~

dE/ds v (3.6)

where Pv = Ev+½V+ (e2-e')ev + @2-e')A2 , 2t2E~ 4QE2

(3.7)

1 Fe~e¢(e2-el)+A2V+ 4A2ev(l_~2)l R~v, =

- ~ t_

f2e~Ev,

E~v~,

K2E~Ev,

'

and sv can be obtained from (3.5) and (3.6) by successive approximation. In first order dE - Pv. (3.6a) dsv Substituting (3.6a) into (3.5), (3.2) can be resolved. For low excitations the integrals can be calculated with the saddle-point method. As a result we can get the excitation

350

J. NgME*H

energy as a function of the temperature: E:ve . = E~ _ E o = ce_eC [21r~ },

(3.8)

LFdJ where A2

A2

C = ½V-t-- 4 A 2-Jr-1/4p 2 + 4po(Ag+-l/4p2) , A2

3A 2

(3.9)

A2

d - ( A Z + l / 4 p Z ) ~ + 4po(A 2 2 + 1/4po)2 4 - - 2po(A2+ l/4po) 2- '

(3.10)

From (3.8) the temperature can be expressed as a function of the excitation energy: /

(3.11)

1~ = I In C_po,:2=/[~q C

Ecx c

From (3.3) and (3.5) we can get the entropy of the system: S

E[x~ [ln ( ]cP°\/'2~/fid'-~x~ + i ]

(3.12)

The entropy of an odd-mass system can be determined in a similar way, but we have to take into account that the ground state of the system now differs slightly from the previous case: --exc = E - - L 0

= poce

+1/2;

-b,

!3.13)

where b = A+ ~ 2 - ~ _ A . 4f2 2f2 2

(3.14)

With the help of (3.3), (3.5), (3.13) and (3.14) the entropy of an odd-mass system can be written as /

According to the statistical model the level density of excited systems can be expressed as p(E) ~ K e s(s), (3.16) where we neglect the energy dependence of the factor K. It is easily seen from (3.12) and (3.15), that the entropy is proportional with 1/A, and the same is valid l\)r the level densities. The low energy spectra of excited nuclei shows, that the energy gap between the ground state and the first excited state is much less for strongly deformed

PAIRING INTERACTIONS

351

nuclei than for the almost closed shell nuclei. This means, that the level densities o f nearly closed shell nuclei are m u c h less than for strongly deformed ones, according to the above obtained results. This fact is in g o o d agreement with the experiments. F r o m (3.12), (3.15) and (3.16) we can determine the level density relations o f oddmass and even systems: flodd

exp

[Soda(E)--S . . . . (E)]

Peven

"~ exp I~ {ln (Cp-°x/'2zr/fld] Eexc+b ] +l} - Eex~ln c ( Eex ~ e c~+ b

(3.17)

F o r the experimental data given by ref. 6), w e get for c ~ 0.7 M e V for d ~ 0.9 MeV, for b ~ 0.65 MeV and for Podd/P.... ~ 2, which shows that the results of our very simplificated calculations are in rather g o o d agreement with the experiments. F o r more exact iesults we have to consider a more realistic model and to perform our calculations with higher accuracy.

Appendix Substituting (2.11) into (2.8), the energy gap can be obtained from the equation

(A.1) Ev

Ev

Performing the integrations, the energy gap turns out to be A 2 --

A2-c

B2/:

2

(A.2)

(z+l) 2,

where z = exp [ - 2 x - ~c] = 7(1 - c~c),

x -

A

(A.3)

V~2

2 /b

A~

w/

+ 4A arctg A 2 + ab

b 41 b 2 + A 2

a

a 2

-TA 2

)

(A.4) "

The value o f B can be determined f r o m (2.5) B = z + 1 ( D A - c~M),

(A.5)

"c-1

where

,)

a2;A

2

•

(A.6)

352

J. NEMETH

Substituting (A.6) into (A.2) A can be written as Az_

z

( z - 1)z

[A2 (AD_~M)2].

(A.2a)

The ground state energy of an even system can be obtained from (2.7a) E .... = u = ¼Vn2+¼c~n2+½~n+½AZpo~c-¼c~Apo

AA

arctg

AZ+ab

- Ipo ~ + 1 [A(1 - D ) + ~M] ~. (A.7) z--1

With the help of (2.13) this can be written in the form E = ¼Vn2+k~n 2

A2 -

17+lA 47-1

n2

f2

? (A2(1 - D 2 ) - 2 ~ A ) . (~-1) 2

,

(A.8)

(A.9)

Literature 1) A. Bohr, B. R. Mottelson and D. Pines, Phys. Rev. 110 (1958) 936; S. T. Beljaev, Mat. Fys. Medd. Dan. Vid. Selsk. 31, No. 11 (1959) 2) J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108 (1957) 1175 3) H. J. Lipkin, Ann. of Phys. 9 (1960) 272; Y. Nogami, Phys. Rev. 134 (1964) 313 4) J. N6meth, to be published 5) N. N. Bogolyubov, JETP (Soviet Phys.) 7 (1958) 141 6) S. G. Nilsson, preprint 7) J. N6meth, Thesis, Budapest, 1965 8) A. Bohr and B. R. Mottelson, Nuclear structure and energy spectra, Lecture notes, Copenhagen, 1963