Nucleon correlations and nuclear level densities

Nuclear Physics 42 (1963) 353--366; ~ ) North-Holland Publishing Co., Amsterdam Not

to be reproduced by photoprint or microfilm without written permission from the publisher

NUCLEON

CORRELATIONS

AND NUCLEAR LEVEL DENSITIES

D. W. L A N G

Research School of Physical Sciences, Australian National University, Canberra ACT and

Bartol Research Foundation of the Franklin Institute, Swarthmore, Pennsylvania t

Received 8 October 1962 Abstract: The high energy behaviour of two nuclear models with pairing correlations is studied. It is found that level densities very closely resemble the Fermi gas forms at all energies above the neutron binding energy. For a model with a constant pairing energy between particles in doubly occupied pair states, the moment of inertia is lower than the rigid body value up to energies considerably higher than the neutron binding energy. In the second model, which is an analogue of the superconducting state of a metal, the moment of inertia reaches the limiting value below the neutron binding energy. It is suggested that for the latter model the Gorter-Casimir description of a superconductor gives a sufficient approximation to the level density.

1. Introduction T h e level density a n d o t h e r statistical p r o p e r t i e s o f a n i n d e p e n d e n t particle m o d e l o f a nucleus a r e well k n o w n 1). I n this m o d e l , for simplicity, the nucleus is described as a F e r m i gas o f N n e u t r o n s a n d Z p r o t o n s c o n t a i n e d in a p o t e n t i a l well. T h e states available to a single particle in the well are a s s u m e d to d e p e n d solely on the p o t e n t i a l a n d to be i n d e p e n d e n t o f the o c c u p a t i o n n u m b e r o f a n y o t h e r state. The single particle states are a s s u m e d u n i f o r m l y spaced in energy. States o f the nucleus are c o m p l e t e l y specified b y e n u m e r a t i n g which single particle levels are occupied. Statistical t h e r m o d y n a m i c s can then be used to derive such features as a level density at a given excitation U. T h e t h e r m o d y n a m i c a r g u m e n t s i n t r o d u c e the t e m p e r a t u r e t a n d the f o r m u l a e evolved specify c o n t i n u o u s v a r i a t i o n for quantities which are in fact discrete. Such f o r m u l a e are valid o n l y for excitations c o n s i d e r a b l y larger t h a n the level spacing. T h e excitation U a n d the t h e r m o d y n a m i c t e m p e r a t u r e t are then linked by the e q u a t i o n U = at2--t.

(1.1)

T h e p a r a m e t e r a d e p e n d s on the densities gn, gp o f single particle levels for n e u t r o n s a n d p r o t o n s near the F e r m i level: a = ~lt2(g,+gp) = ~2g. Work supported by the U.S. Atomic Energy Commission. April 1963

353

(1.2)

354

D.W.

LANO

The density of levels of the nucleus of spin quantum number J, and a particular parity, at the energy U is to(U, N, Z, J) = g½(2J+ 1)n exp [2x/~-U-(J+½)2/2ct]

(1.3)

( U + t)296x/3c i

The magnitude of c determines the distribution of J values of levels. If the single particle level densities are constant near the Fermi level, (1.4)

c = rn2g.

The magnetic quantum number m is the projection of J on some selected axis of the nucleus. In eq. (1.4), m 2 is the mean square of m for single particle states of energy close to the Fermi level. For any Fermi-Thomas description of a nucleus, in which the density of nuclear matter at a point depends only on the depth of the potential well there, the quantity ch 2 is equal to the moment of inertia of the nucleus rotating as a solid body 2, 3) and in such a model is independent of excitation. Of interest experimentally is the energy quantity z defined by O. Iogto(U . . . ) tgu .

1 r

~

VU

2 ( u + t)

.

t l.5)

It follows that U ~ avZ-4r.

(1.6)

It will be noted that the derivative quantity ~ is close in magnitude to the thermodynamic temperature t when t is large and for this reason it is sometimes referred to as a nuclear temperature. The usefulness of the independent particle approximation depends on the state o f nuclear matter 4). It is required that any repulsive potential between the individual particles should be considerably smaller in range than the mean distance between particles in the aggregate; and that the kinetic energy at the Fermi level in the assembly be a higher energy than that involved in the interaction between individual particles. Under these conditions the mean free path of a nucleon in nuclear matter is long (about 25 fm at the lower edge of the continuum) and the two-body interaction of nucleons give rise to a potential which is approximately independent of configuration. Because the assumption of independence is an approximation the actual states of the nucleus are mixtures of the independent particle model states and the properties of individual states differ from the predictions of the model. Thus the odd-even effect in nuclear binding energies can be described by assuming that interactions among an even number of neutrons or of protons lead to a condensed state with all particles paired; but that the condensation does not occur, or occurs with considerably less condensation energy, in an odd number of particles of one type. Regularities of low energy excitation spectra of nuclei fit with a description of a nucleus either rotating or vibrating as a liquid drop.

NUCLEAR CORRELATIONS

355

Further discussion is then required of properties which depend on the statistical distribution of energy states and the variations, if any, from the independent particle description. Two methods of incorporating the residual interactions as extra interactions in time reversed pairs of the independent particle model are discussed in this paper. The model of sect. 2, in which the interaction effects each pair independently, is already well known 4.7) and formulae quoted are an essentially unchanged summary of previous work 7) on the model. Sects. 3, 4 and 5 are concerned with adapting formulae, known to describe the behaviour of a superconductor, to describe a nucleus. 2. Pairing of Nucleons For a nuclear well with an axis of symmetry, e.g., in the Nilsson 5) model, single particle states are degenerate in pairs. The particles of a pair have all their quantum numbers equal in magnitude and differ only in having opposite sign for the projection of the angular momentum on the spin axis. A pair interaction independent of other pairs may be introduced 6, 7) by postulating that with both states of the r th pair filled, the pair is bound to the nucleus with an energy 2E,, but with either state empty, a particle in the other is bound only by E , - ½ A , . The odd-even effect on the binding energy of nuclei follows. The quantity A, can vary considerably for pairs of neighbouring energies but this dependence is ignored here and the subscript will be omitted. The trend of A for an unpaired particle in the ground state of a nucleus of mass A is s) A ~ 3.36-0.0084A MeV for A > 40. (2.1) The consequences of the model may be studied by extension of the methods used for the simpler model, i.e., a Fermi gas without pairing energy, and the resultant formulae are exactly those of the simpler case in thelimit A ~ 0. At sufficiently high energies the formula (1.1) can be replaced by U = at 2 _ t - iJ-~gA 2 _ SeA,

(2.2)

where ~ takes the values 0, 1, 2, for even, odd-mass and odd nuclei, respectively. The formula (2.2) is a high energy approximation. It is sufficient at all energies above that given by t ~ ]A. This temperature corresponds to an excitation U < ~-~aA 2, which is always less than 6 MeV. All the formulae taken for this section from ref. 7) have been compared numerically with the more complex and accurate forms given there. The accuracy seems sufficient, even at much lower energies, for comparison with any experiment in which enough levels are involved for statistical assumptions to be valid. The level density at high energies is co(U, N, Z, ,/, A) = a½(2J+ 1)n exp [ 2 x / a ( U + t - ~ g A 2 + ½ e A ) - ½ g A --(J"l( U + t + t-~gA 2 + ½~A)296x/~(c,,)t

½)2/2C"1'] (2.3)

356

D.w. ta~r~G

This is simplified by introduction of the quantity U' defined by

U'= U+~gA2+½~A;

(2.4)

then co(U, N, Z, J, A) = g~(2J+ 1)n exp [ 2 ~ ( a U ' ) - ½ g A - ( J + ½ ) 2 / 2 c " t ]

(2.5)

( U' + t)2964~c '') The "moment of inertia" c of eq. (1.4) has been replaced by c" given by

c" = (c')t(½c + ½c')t,

(2.6)

c' t = ct exp( - 0.44fiA ) + em 2 [1 - exp( - 0.44fiA)],

(2.7)

/~t = 1.

(2.8)

The value of c" can be very small at low energies and exhibits a slow trend toward c at high energies. Near the neutron binding energy c" ~ ½c. Detailed discussion in ref. 7) showed that the model is capable of describing the angular momentum expected in low energy states but does not predict any of the observed regularities in energy. Eq. (2.5) differs from eq. (1.3) in that c" replaces c, U' replaces U and there is a subtraction term ½yA in the exponential part of the formula. Near and above the neutron binding energy the variation of c" with energy is negligible by comparison with that of ~/aU' the leading term of the exponential. The difference of U and U' and the term ½gA are both temperature independent. Thus, from eq. (2.4) and eq. (2.5) the analogue ofeq. (1.5) is d --

~U

log (co) =

(9 log (co)

~U'

= -1 ~

z

V~.

WU"

2

U'+t

(2.9)

and hence

(2.1o)

U ~, az2-4~--1½gAZ-½eA.

Use of U' in place of U corresponds to a fictitious ground state lying below the actual ground state of the system. In terms of the superconductor model of later sections this is superficially a negative "'condensation energy". The level density corresponding to a given excitation is increased by the substitution; but this is more than offset by the extra term in the exponential, and at all excitation energies the level density is lowered by increasing A.

3. S u p e r c o n d u c t i n g

Metals

and Nuclear

Matter

Bardeen, Cooper and Schrieffer 9), in a paper which will be referred to as BCS, have given a description of the superconducting state of a metal in terms of residual interactions between electrons, leading to a reasonably detailed picture of the low energy excitations of the Fermi gas of electrons in the metal. This model needs few

NUCLEAR CORRELATIONS

357

changes, mostly in terminology, to be applicable to a nucleus ~0). In a metal it is assumed that there are interactions between pairs of electrons which are alike except in the direction of their m o m e n t u m and their spin. Below some critical temperature a condensed state of the metal exists. Excitation of a given particle requires more energy than in the Fermi gas state. The metal as a whole also acquires a condensation energy which at any given temperature separates the energy of the condensed state and that of the Fermi gas. The condensation energy is a maximum for zero temperature, i.e., for the separation of the ground states. In a metal interest centres on the description of the superconductor properties. In a nucleus the electron pairs are replaced by pairs of nucleons, alike except for the projection of their angular momentum on the symmetery axis of the nucleus. The superconductor property is nonexistent and the interest lies in the spectrum of excitations. A study of Belyaev 11) uses the correspondence of electron and nucleon pairs and shows that the same sort of interaction in a system of few particles still leads to the same sort of effect on the spectrum of excitations. In particular the odd-even effect in nuclear binding energies can be related to the " m o m e n t of inertia" or to a "mass parameter" characterizing low lying states of the energy spectrum. Griffin and Rich ~2) have found reasonable agreement between theory and experiment for nuclei in the rare earth region. The Belyaev study also provides means to describe shell effects in the same formalism as odd-even effects. The study is, however, confined to consideration of the first few excited states. It makes no reference to the form of the high energy spectrum and in particular it is difficult to separate the condensation energy due to the extra interaction from the binding energy due to the nuclear well. Information on the magnitude of the condensation energy will prove necessary in order to discuss the high energy level density. To establish the analogy at higher energies, the BCS model will first be examined in order to relate the magnitude of such quantities as the energy gap, the transition temperature and the condensation energy. For the Bloch model of a metal, i.e. without residual interactions, there are a set of single particle levels ek. Thus, excitation of an electron from the Fermi level to the state k will require energy ek. BCS find the excitations in the condensed system to have magnitudes Ek given by E 2 = e~+02,

(3.1)

where 0 is the solution appropriate to the temperature t ( = f l - 1 ) implicit in the equation 1 V

E tgh ½fie k k 2Ek

(3.2)

For small ek, Ek ~ 0 which is much greater than ek" The smallest excitation available is 0, which means that the single particle level spacing at the Fermi level will appear to be much larger than in the Bloch model. Hence 20 is called the "energy gap."

358

D.W.

LANO

The interaction V o f t h e pairs is assumed constant and the summation in the electronic case is over pair states whose energy difference from the Fermi level is less than the phonon energy hco. In the nucleus the summation is over the states in the shell being filled. An expression for the average energy to be associated with a given temperature measured from the ground state of the Bloch system is u = T

2-Eke-B 1 + Ee k(l) kl+k

02

--V.

(3.3/

The first summation corresponds to single-particle-like excitations from (3.1) and the remainder is a temperature dependent condensation energy, independent o f precisely which excitations are present. For large values of fl, i.e., near zero temperature,

U = -¼gO 2,

(3.4)

2

(3.5)

where 0o is given by -

V

= E ei

k

I

The first excitation of the system will have energy 0o and this may be used in the nuclear case to establish the energy scale. In the odd-even effect the unpaired particle is equivalent to the presence of one excitation 43). Thus, one excitation is equal to ½A as given in sect. 2. The excitation is Ek of eq. (3.1) and since .0o is considerably larger than ek, Oo .~ Ek "~" ½Zt. In the BCS theory 0 becomes zero for t = tc = 0.570o. At this temperature measurement o f the excitation energy from the actual ground state of the metal gives an excitation energy 3.2 ¼#0~.

(3.6)

Above the transition temperature t¢ the superconductor property vanishes and the electrons behave like a Fermi gas whose ground state is ¼g0o2 above the actual ground state. The nucleons are also expected to behave as a Fermi gas above a certain transition energy. In the study of nuclei it will be of interest to discuss angular momentum. The projected value o f the angular momentum along any axis will be the sum of the projected angular momenta of the occupied single particle states. The mean value of this sum will be zero, and as in the case of the pairing model 7) the mean square of the sum must be calculated. This is the sum of the squares of the individual projected angular momenta, each multiplied by the variance of its occupation number. From BCS, the probability of finding a particular single particle state occupied is fk = (exp fie k + 1)- t.

(3.7)

NUCLEAR CORRELATIONS

359

The required variance is f ~ - f ~ and the expected mean square projection of the angular momentum is then ( M z)

5" m~ exp (~E~) (exp #t~k+ 1)z

(3.8)

As 0 decreases so does Ek--~ and the expression on the right hand side of eq. (3.8) reduces to that for a Fermi gas. This implies that the distribution of angular momentum in levels of a given energy will be that of a Fermi gas for all temperatures above the transition temperature. The calculation of the level density depends J) on the evaluation of four terms. The first, the entropy term S for the grand partition function, appears exponentially in the level density, and is therefore the only one of importance in the electronic case. The other three terms are factors introduced by saddle point integrations required to define the number of neutrons, the number of protons and the total excitation present. The entropy term with residual interactions is given by BCS as S

2

exp (--flEk) E F~Ek I_1 + exp (--fEk)

+log (1 +exp --fEk) 1 .

(3.9)

This form of the entropy and the form for an uncondensed gas differ only in that eq. (3.9) has Ek in all places where ek would appear in the uncondensed case. A decrease in the interaction will transform S continuously into the identical Fermi gas form at the same temperature. The factors introduced by defining the numbers of neutrons and protons are identical and will be designated by P: P = [2n ~ exp flEk/(1 +exp fEk)2] -~. k

(3.10)

These two vary continuously into their Fermi gas forms. In considering the density of states of a definite energy the third factor is introduced as ( - 2ha U/aft)- ~. The model has been constructed for a superconductor, and it is well known that au/af has a discontinuity at the transition point from the superconducting state into the normal state. The BCS theory predicts the discontinuity and the ratio of the magnitudes of aU/cgfl. Existence of a discontinuity in the level density would be disconcerting, but examination of the saddle point integration which introduces the factor suggests that where au/df is varying rapidly with f some average replaces a point value in the integration, and should also do so in the resulting expression. The discontinuity will be effectively smoothed in the level density. Because the ratio of au/af below the transition energy to that above is about 2.8, near the transition energy the increase of levels with excitation energy will be larger than expected from entropy considerations, i.e., the derivative "temperature" z defined in eq. (1.5) will be decreased. This effect would be barely detectable in the nuclear case and such quantities as the

360

D.W. LANG

ratio of specific heats must be considered completely unknown except as a consequence of the BCS theory. The total number of nuclear states per unit energy is then

w(N,Z, U) = p2 \( - 27tOU'l ~-~/ -~ exp S.

(3.11)

Note that in this expression each different projection of the total angular momentum of a single nuclear level on a defined axis is counted as a separate state. 4. Approximations in the Superconductor Model In the Fermi gas model summations similar to those indicated in sect. 3 are replaced by integrals and these can normally be evaluated. The introduction of interactions makes evaluation more complicated 14) and approximation is needed. BCS give an expression for the energy gap 0 at a given temperature in terms of the transition temperature t~:

0 ,~ 3.2tc(l-t/t~) ½.

(4.1)

This equation is unfortunately very limited in its range and it is preferable to use a value read directly from fig. 1 of BCS. The condensation energy terms of eq. (3.3) at a given temperature where the gap has value 0 has the value (after simplification) ¼g(0Z(1 - l o g

02102)).

(4.2)

The approximation (4.2) neglects second order terms but is accurate at the maximum and minimum values of 0, and good in between. At this given temperature and consequent 0 the excitation energy measured from the actual ground state will be

v=6glr2 e-P°(l + 2flO

+

2(f10)2)(1 + 2flO)- ½- C,

(4.3)

where c

=

¼g(o 2 - o 2

log [02/0o -

The consequent value of the entropy is given by S 2 -- 44~ 2 e-P ° 6 (l+2f10) ½(U+C).

(4.4)

The summations in eq. (3.10) both give rise to the approximation y

e pr~

2

~ g~ (1 +

2flO)½,

(4.5)

where g~ is the density of single particle states appropriate for uncondensed neutrons or protons.

NUCLEAR

361

CORRELATIONS

This approximation is also used for eq. (3.8) with the additional assumption (made also for a Fermi gas) that the squared angular momentum projection m 2 of each single particle state is the same, i.e., is equal to (m2). Then the moment of inertia from eq. (1.4) is replaced by

c'"t = ct 2(1 +j0) t + e ( m 2 ) ( 1 - e x p (-flO)). (1 +exp flO)

(4.6)

As in eq. (2.4) the second term is included since an odd mass or odd nucleus has unpaired nucleons at all energies. The BCS model is isotropic, corresponding to spherical symmetry of the nucleus, and it is of interest that the moment of inertia becomes arbitrarily small close to the ground state for such an even nucleus. The approximations for OU/Ofl persist into the uncondensed region due to the averaging over energy. About the limit of complexity acceptable would be the use of a linear combination of the values of OU/Ofl just above and just below the discontinuity, e.g., for U c - E < U < Uc+ E, _

"-d

v

~

+ uo+E-u o~+E - 2 E

uc-r"

(4.7)

The energy E required to make the average in eq. (4.7) extend over a reasonable range is about two MeV. For simple calculations above the transition no notice need be taken of any effect of the averaging process. At these excitations using eq. (4.3) and that 0o ~ ½A formula (I.1) is replaced by

U = atZ-t+lA-syA2-½eA.

(4.8)

The moment of inertia reaches the Fermi gas value at the transition point. Thus the expression (1.3) may be used for the level density provided U is replaced by

U" = U-~-~gA2 +½eA.

(4.9)

The term ~ g d 2 due to the pair interactions is the condensation energy of the ground state of an even nucleus below the independent particle model ground state. The term ½cA appears when there are unpaired particles which disrupt the condensed state. If A = T~sgA 2 there will be no condensation energy in the ground state of an odd nucleus. Since the level density is expected to be a smoothly varying quantity, the nuclear equation of state U" = a z 2 - 4 ~

(4.10)

equivalent to eq. (1.6) will not only apply above the transition but will also be within experimental error for some range of energies below it.

362

D.W. LANG

5. The Gorter-Casimir Approximation for Nuclei T h e a r g u m e n t f o r t h e a p p l i c a t i o n o f t h e B C S t h e o r y to a n u c l e u s leads f r o m t h e e x i s t e n c e o f a g a p in t h e e l e m e n t a r y e x c i t a t i o n s o f nuclei to a d e s c r i p t i o n i n v o l v i n g a c o n d e n s a t i o n e n e r g y . T h e B C S t h e o r y gives a q u a n t i t a t i v e r e l a t i o n b e t w e e n t h e m . A p p l i e d t o nuclei it p r e d i c t s the l o w e r i n g o f the m o m e n t o f i n e r t i a at l o w e x c i t a t i o n a n d d i s r u p t i o n o f all c o h e r e n c e d u e to the i n t e r a c t i o n s at a n e x c i t a t i o n w h i c h is less t h a n t h e n e u t r o n b i n d i n g energy. E v e n w h e r e t h e r e m a i n i n g r e f i n e m e n t o f the t h e o r y c a n be p r o g r a m m e d i n t o a c o m p u t e r it is o p e n to q u e s t i o n w h e t h e r s u c h a p r o c e d u r e w o u l d be j u s t i f i e d by p r e s e n t n u c l e a r d a t a . TABLE 1

Comparison of thermodynamic variables for a Fermi gas and a Gorter-Casimir description Thermodynamic quantity ")

Fermi gas all temps,

Symbol

Excitation energy Entropy Dispersion of energy e)

Gorter-Casimir Gorter-Casimir B) below te above to

U

at 2

] a t ( / t , ~"

S

2at

2at3/te 2

b U/bfl

-- 2at s

- - 6at6 / te 2

at a + ~ate 2 2at -- 2at a

All quantities are for systems with a large number of particles. b) The system is like a Fermi gas whose ground state lies tare g above the actual ground state i.e., ate i below the transition energy. The quantity ½atet is thus the condensation energy C. c) b U / b f l is discontinuous, with a change by a factor ½, at the transition. This factor can be varied together with tbe condensation energy by altering the power of t in the variation in the condensed region. •)

iO

~

-05 ................

O0

05 i

05

IO

1.0

1.5

/

!

//

O0 O0

0.,5

U/at~ Fig. 1. Gorter-Casimir (solid curve) and Fermi gas (dashed curve) values of the entropy S, divided by the transition value, in a system of many particles at an excitation energy U less than the transition energy ~ateL The upper energy scale (Fermi) and the lower (G-C) differ by the condensation energy ½ate 2. The curves are convergent well below the transition. Thus the G-C entropy above the transition, of which the Fermi gas curve drawn is a continuation, gives useful information at much lower energies. The G-C entropy is soon to have a more nearly constant slope (constant nuclear temperature) than the Fermi gas. F o r s i m p l i c i t y i t is c o n v e n i e n t t o a s s u m e t h e c o n n e c t i o n b e t w e e n t h e t h e r m o dynamics of a superconductor and of a nucleus and to consider a more tractable

363

NUCLEAR CORRELATIONS

description of a superconductor given by Gorter and Casimir xs). This assumed that below the transition temperature the electronic excitation energy varies as the fourth power of the temperature. A condensation energy separating the actual ground state from the Fermi gas ground state is then a natural consequence of the continuity of some of the thermodynamic variables at the transition. Even for a superconducting metal considerable refinement of experimental technique is required to find deviations from this simple semi-empirical description. Until the nuclear experiments can discriminate clearly between a nuclear model with the same energy spectrum as a superconductor and one with the more independent pairing properties o f sect. 2, the BCS model is needlessly complicated. It is required mostly for the discussion ofangular momentum and to derive an expression such as eq. (3.4) linking the odd even effect and the condensation energy. The high energy behaviour has already been shown to be that well known for a Fermi gas but any simplification of the low energy description would be welcome. A comparison of values of thermodynamic variables at temperature t in the Fermi gas and in the Gorter-Casimir description appropriate to the temperature is given in table 1. In fig. 1 the GorterCasimir form for the entropy S in the condensed state is compared with the Fermi form. These are drawn for a system of a large number of particles in which the entropy dominates the level density. It is illustrated that the energy variation of the two forms is not too different over a considerable region near the transition and hence that the Fermi form for the energy region above the transition may apply considerably below the transition. The experimental evidence is consistent with the assumption that both the level density and its derivative with respect to U are continuous in U. If the level density is of the form e x p ( S - r log t + k) below the transition and exp ( S - ] l o g t + k x) above the transition then these continuity conditions can be fulfilled. The quantity k t and the number ] are known for a Fermi gas. The assumption of a similar form for the condensed state requires the arbitrary constants k and r. The entropy S is continuous. This relates the constants k and kt:

(5.1)

k = k l + ( r - - ~ ) l o g to.

By definition OS_I OU t

and hence

r(O~uu) b e l o w = 3--(O~uu) above. t

(5.2)

2t

The ratio of specific heats at the transition in superconductors is of order 2.8. This ratio substituted in eq. (5.2) would give r m, 4.2. The derivative temperature T in the condensed phase is given by 1_

1

4.2_1(1

t

4U

t

0.7 atilt,)3

) .

(5.3)

In table 2 values are given of ,/% for various values of t/t¢ and at,. The right hand

364

D.W. LANG

c o l u m n gives a measure of the sort of value of z expected by extrapolation o f the Fermi gas with a large n u m b e r of particles. This overemphasizes a trend to exhibit a " c o n stant t e m p e r a t u r e " . TABLE 2 Values of r/Ze for a Gorter-Casimir ~) description of the condensed state of a nuclear system in terms of ate and t/te at c

7

10.5

14

21

28

42

U/Uc

0.88 0.89 0.94 ! .00

0.83 0.89 0.93 1.00

0.81 0.85 0.92 1.00

0.78 0.83 0.91 1.00

0.78 0.82 0.91 1.00

0.77 0.82 0.90 1.00

0.317 0.40 0.656 1.00

-- ½

t/te

0.75 0.80 0.90 i .00

0.34 0.70 1.00

• ) The final two columns of the table permit comparison with a Fermi gas. The second to last gives the excitation energy of the system at the given temperature. The final column gives the value ofr/ze which would follow at this energy if the Fermi gas equation appropriate to the uncondensed many particle system with the same values of a and of te also applied below the transition energy. Comparison of the final column with the other values of z/re shows that there is a strong tendency in the model to exhibit a constant nuclear temperature in the condensed state. The entire state of the Gorter-Casimir system considered can be specified by the two dimensionless quantities a t e and t/te. A further parameter would be required for a system obeying eqs. (4.8) or (5.). Inclusion of this would increase the first two values given in the final column of the table.

A n expression for the level density at low energies can now be derived by comb i n i n g eqs. (4.3), (5.1) a n d (5.2) but it is as simple in numerical work to use separate equations. T h e equivalent of eq. (1.1), which describes a relatively small n u m b e r o f particles, will apply at energies above the t r a n s i t i o n energy, a n d is U = at2-t-C+t¢.

(5.5)

6. D i s c u s s i o n and E x p e r i m e n t a l Consequences

The two models i n c o r p o r a t i n g pair interactions make several qualitative predictions. Both predict that the nucleus will be described at a n y sufficient excitation by a Fermi gas model with a n a p p r o p r i a t e a p p a r e n t g r o u n d state, n o t the actual g r o u n d state. Numerically "sufficient excitation" appears to lie well below the n e u t r o n b i n d i n g energy. It is thus reasonable that the level densities at the n e u t r o n b i n d i n g energy should fit the F e r m i gas model, a n d that the q u a n t i t y • when evaluated from the least a m b i g u o u s of the experimental data, i.e., the (n, n ' ) experiments, should vary with excitation energy in a m a n n e r n o t inconsistent with the Fermi gas model 16). At very low energies both models will exhibit level densities more consistent with a c o n s t a n t nuclear temperature t h a n with extrapolation of their high energy forms. T h e s u p e r c o n d u c t o r model shows a m o r e m a r k e d trend to a c o n s t a n t nuclear temperature than does the simple pairing model. Analysis ~7) of actual level density counts

365

NUCLEAR CORRELATIONS

made in this region of excitation has favoured a "constant temperature" approximation but it is still dubious whether all levels have been observed in such counts and more complex behaviour of the level density cannot be excluded. Comparison of eqs. (2.10) and (4.10) shows that they both predict that a straight line graph can be produced from a plot of U+4"r+½ed against ~2. These straight lines have markedly different intercepts on the energy axis. Sufficient experimental points to define the slope and intercept of such a line would thus discriminate between the models. The points would have to be obtained in the region where both formulae are expected to represent their models accurately, i.e., above any constant temperature region. The available experimental evidence is, however, clearly insufficient 16). In fig. 2 the effect of the extra temperature independent terms appearing in the high energy formulae (2.4), (2.5) and (4.9) on the level density is illustrated.

P

.::

C B

A

/t

3



I

"x~'S

Fig. 2. Curves for the high energy forms o f log of the level density co(U) as a function of the excitation U for a Fermi gas (solid), independent pairing from sect. 2 in an even nucleus (heavy dashes), independent pairing in an odd nucleus (light dashes) and BCS correlations in an even nucleus (dots). All other curves wherever they apply lie below the solid curve, i.e., the level density at a given excitation is always reduced by the interaction. The curves are translated only. The points A, B and C are translated positions o f P in a superconductor model with an even, an odd mass and an odd nucleus, respectively. With independent pairing these are A', B' and C'. The slope o f the curves, and hence the value o f t , are the same at P, A, A' and C' and would be, if drawn, at B, C and B'. The interval PP' = ½g/I, PA =

~ g d ~,

P ' A ' = -~2g~l~ and AB = A ' B ' = BC = B ' C ' -- ~,,1. Note that

PC = l'~ gA'-d but that P'C'= ~gd2-rJ. Both models predict or, more precisely, agree with the low values found experimentally for the moment of inertia for nuclei at low energies. They predict different values for the moment of inertia at the neutron binding energy and above since the superconductor analogue eq. (4.6) gives a return to the rigid body value at much lower energies than the simpler pairing model eq. (2.4). A constraint which applies only to the superconductor model is seen in the ground state of a nucleus with not all nucleons paired. If the condensed ground state is to lie below the independent particle model ground state then from eq. (4.9) ~ g A 2 must

366

D.W.

LANe

be greater than A. The difference appears normally to be small. This is the basis o f the usual method of allowing for the odd even effect in nuclear level densities which assumcs, in effect, that an odd mass nucleus has no condensation energy. The requirement that energy must be gained by passage into the condensed state has another consequence, pointed out by Mottelson and Valatin is). The low momcnt of inertia associated with the condensation leads to a high energy of the rotational spectrum. This cannot extend to states where the extra rotational energy is greater than the condensation energy. There is thus a limit on the number of states in a rotational band. Rotational bands based on an excited state should not extend as far in energy as those based on the ground state. F r o m the success of Griffin and Rich 12), it appears reasonable in calculations to use valucs of the pairing energy derived from actual experimental measurement of masses, e.g., those of Bhanot, Johnson and Nier 19), by use of third order differences to exclude at least second order mass terms 20). It will be noted that neithcr of the forms used in this paper considers any effcct on the level density from an interaction between an unpaired neutron and proton 21). At least one further area of experiment is indicated by the hint contained in the Belyaev study on the possible treatment of shell effect on level densities. Data are still needed in the region of closed shells shedding direct light on the level density, and its variation with energy, to whatever excitation is required to destroy all shell structure. More (n, n') experiments would be useful in closed shell regions and are of course desirable through all mass regions. Thanks are duc to Professor K. J. Lc Coutcur for many helpful discussions and to Dr. J. R. Huizenga for a penetrating qucstion. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) II) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21)

J. M. B. Lang and K. J. Le Couteur, Prec. Phys. Soc. 67 (1954) 585 H. Bethe, Revs. Mod. Phys. 9 (1937) 69 C. Bloch, Phys. Rev. 93 (1954) 1094 V. F. Weisskopf, The many-body problem, Cours Dennis • L'l'~coleD'l~te D e Physique Th~:oriquo (Methuen, London, 1958) p. 317 S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955) T. Ericson, Nuclear Physics 6 (1958) 62 D. W. Lang and K. J. L¢ Couteur, Nuclear Physics 14 (1959) 21 A. Stolovy and J. A. Harvey, Phys. Rev. 108 (1954) 353 Bardoen, Cooper and Schrioffer, Phys. Rev. 108 (1957) 1175 Bohr, Mottclson and Pines, Phys. Rov. If0 (1958) 936 S. T. Belyaev, Mat. Fys. Me.dd. Dan. Vid. Selsk. 31, No. II (1959) J. J. Griffin and M. Rich, Phys. Rev. 118 (1960) 850 B. R. Mottelson, The many-body problem, Cours Donnds .~ L'l~cole D'i~to De Physique Thdo. rique (Methuen, London, 1958) p. 309 K. J. Le Coutcur and D. W. Lang, Nuclear Physics 13 (1959) 32 C. J. Gortcr and H. Casimir, Physica I (1934) 306 D. W. Lang, Nuclear Physics 26 (1961) 434 T. Ericson, Nuclear Physics 11 (1959) 481 B. R. Mottolson and J. G. Valatin, Phys. Rcv. Lett. 5 (1960) 511 Bhanot, Johnson and Nier, Phys. Rev. 120 (1960) 235 S. G. Nilsson and O. Prior, Mat. Fys. Medd. Dan. Vid. Selsk. 32, No. 16 (1961) B. A. Kravstov, J E T P ~6 (1959) 871