Studies of the nuclear inertia in fission and heavy-ion reactions

1 .D.2 : 1 .E.6

Nudear Physks A296 (1978) 289-306 ; © North-Xollaed Pr~lirhinp Co., Mrstendam xot to be reproduoed by Ph~P~t ~ mia~o8lm without writtm pa~misaioa ftom the publlshar

STUDIFS OF THE NUCLEAR INERTIA IN FLSSION AND HEAVY-ION REACTIONS P. MÖLLER f and J . R HIX Theoretical Dinfsion, Los dlmnos Scient{~ic Laboratory, Unhxrsity o,J Ca1{Jornia, Los Alantaf, New Mexico 8733 ff Received 5 September 1977 Abstract : On the basis of the non-self-0onsistent cranking model we study some aspects of the nuclear inertia of interest in fission and heavy~ion reactions. First, we consider in the adiabatic limit the inertia for a doubly closed-shell nucleus in a deformed spheroidal harmonio~oscillator singlo-particle potential plus a small pérturbation . When expressed in terms of a coordinate that describes the deformation of the nuclear matter distribution, the inertia for small oscillations about a spherical shape is exactly equal to the incompressible, irrotational vah~e. For large distortions it deviates from the incompressible, irrotational value by up to about f 1 ~ away from level crossings . Second, in order to study the dependence of the inertia upon a level crossing, we consider in detail two levels of the above system. This is done both in the adiabatic limit and for large collective velocities . At level crossings the adiabatic inertia relative to the deformation of the matter distribution diverges ~ 1/ayh where Idyl ~ the mag~~ of the perturbation . However, for large collective velocities the contribution to the inertia from a level crossing is leas than 41d yl/~~, where f, is the - collective velocity of the matter distribution . Although we have not oonsiderod the effect of large velocities on the remaining levels of the many-body system or the effect of a statistical ensemble of states, some of our reauhs suggest that for high excitation energies and moderately large collective velocities the nuclear inertia approaches approximately the irrotational vahre .

1. Introduction In developing a theory o~ fission and heavy-ion reactions one may use one of two general approaches. The first approach is a fully microscopic self-consistent calculation in which one solves, for a given force between the nucleons, the resulting many-body equations in some approximation, such as the time-dependent IiartreeFock approximation ' - 3 ) . The second approach, which we follow here, is a description of the properties of nuclei in terms of a few collective coordinates. In such an approach the equations of motion contain three terms: the potential energy, the kinetic energy and a dissipative term. A useful starting point in a study,of fission and heavy-ion reactions is a knowledge of the potential energy as a function of the most important collective shape degrees of frcedom . During the past ten years, great progress has been made towards understanding this part of the pTOblem in terms of the macroscopio-microscopic method `-'). There the energy is considered to be the sum of a smoothly varying f fr

Present address : Department of Mathematical Physics, University of Lund, Lund 7, Sweden. This work was supported by the US Energy Research end Development Administration. 289

290

P. MÖLLER AND J . R NIX

macroscopic part and fluctuating microscopic shell and pairing corrections. The great progress was made possible by use of Strutinsky's method for calculating the microscopic shell corrections. The agreement between calculated properties of the potential~nergy surface and experimental data is impressive 6''). Alternatively, one may calculate the potential energy by use of self-consistent microscopic methods s . 9). However, because of their complexity, these methods have not been extensively explored up to now: For the two remaining terms in the equations of motion - the kinetic and dissipative terms - the situation is quite different. Both are essential for an understanding ofnuclear dynamics, but only recently have various models forthe dissipative term been studied and calculated results compared with experimental data 10-1 z), The comparisons do not differentiate between the models used, and at present very little is known about the dissipative term. The kinetic term and the corresponding inertial parameter have been studied somewhat more exténsively than the dissipative term. However, the models for the nuclear inertia are also poorly understood 13) and in many cases poorly explored because of computational difficulties . One frequently uses a macroscopic model for the nuclear inertia and calculates it in terms of incompressible, irrotational hydrodynamicalflow 1a) . For a classical system this provides a rigorous lower limit to the inertia, as first proved by Kelvin's). The most commonly used microscopic model for the nuclear inertia is the adiabatic cranking model of Inglis' e -1 s) . However, many objections have been raised against this model 19-21 ), and several improvements have been suggested. These include the self-consistent cranking model proposed for rotations by Thouless and Valatin ss), a similar self-consistent model applied ~to the pairing-plus-quadrupole force by Baranger and Kumar s 1) and the'self-cranked generator-coordinate model of Haff and Wilets 23 - 2s), The last model leads to different expressions for the massparameter in the classically allowed region above the barrier and in the classically forbidden region below the barrier. However, in some numerical calculations performed with this model the calculated inertia is much lower than the incompressible, irrotational value sa). These models and their relationship to the cranking model are discussed extensively in refs. 26 : Z'). Because oftheir complexity, these models have not yet been applied to actual calculations of fission or heavy-ion reactions. Instead, for spontaneous fission the inertia is usually calculated either by means of the adiabatic cranking model s" se-3o) or semiempirical models 31 .32 whereas for induced fission or heavy-ion reactions the inertia is usually calculated for incompressible, nearly inotational hydrodynamical flow 10-12). In spontaneous fission the internal excitation energy is low and the collective velocities are small. But in the later stages of fission and in heavy-ion reactions these two conditions are no longer satisfied. One's goal should therefore be to develop microscopic models that are.also valid for large collective velocities and high internal excitation energies.

NUCLEAR INERTIA

29l

Attempts in this direction are being made. For example, expressions for the nuclear inertia at high excitation energies are given by Schütte and Wilets s3) and by Hofmann and SlemenS 34, 33). When the system is started mainly in a single excited state, Schütte and Wilets conclude from their expression that the inertia should decrease as the system becomes excited. From their expression it appears that even a negative inertia is possible . In the derivation of Hofmann and Siemens, the initial state of the system is not a pure state but is instead a statistical ensemble described by a temperature T. They make no statement about the dependence ofthe nuclear inertia upon T. Expressions for the nuclear inertia at low excitation energies and large collective velocities have also been derived sa, as' 3,), mainly by extending the perturbation methods used in deriving the adiabatic expression to higher order. Unfortunately, the results are somewhat contradictory. For example, Schütte and Wilets 33) find that the inertia détresses at higher velocities, whereas Liran et al. ae) and trhaûm and Dickmann s') find that the inertia increases at higher velocities . One may therefore still agree with Rowe 13) that the cranking model itself is poorly understood. In view of the sometimes contradictory and incomplete results of some of the above calculations, it is not clear whether the cranking model is applicable to the large velocities and high excitation energies of interest in many reactions. To gain some additional understanding of the cranking model and to explore what it tells us about the true nuclear inertia in the adiabatic limit and in other limits of interest in fission and heavy-ion reactions, we apply it to two special cases. First, we consider in the adiabatic limit the inertia for a many-body system in a deformed spheroidal harmonic-oscillator single-particle potential plus a small perturbation . Second, in order to study the dependence of the inertia upon a level crossi~ we consider in detail two levels of the above system ; this is done both in the adiabatic limit and for large collective velocities. Although we have not considered the effect of large velocities on the remaining levels of the many-body system, some of our results suggest that for high excitation energies and moderately large collective velocities the nuclear inertia approaches approximately the irrotational valûe. 2. Kelvin's theorem When interpreting results of model calculations of the nuclear inertia it is useful to know that, according to Kelvin i'), the irrotational inertia constitutes a lower limit to the inertia for a classical incompressible liquid . Expressed in another way, the irrotational motion of a liquid has less kinetic energy than any other type of motion with the same normal velocity of the boundary. To prove this, we start with the kinetic energy for irrotational motion, T~ =

ip I v~d3r.

(2 .1)

292

P. MÔLLER AND J . R. NIX

ü ¢ is the velocity potential, then From the oquation of continuity we also know that o,~ satisfies V ' Di~ = 0.

(2.3)

Now, suppose that o is the velocity associated with any other motion that has the same normal velocity at the boundary. One can then always write where because of the boundary condition, and where because of the equation of continuity and eq. (2.3). The total kinetic energy is then T = zP asd3r

J

V¢d3r,

(2.~

where Tl = ip

J

vid3r.

'Because of a vector identity and eq . (2.6) it follows that O . (~D i ) = oi ' V~+~V' o i = oi ' V~. We therefore have, by virtue of Gauss' theorem and eq. (2.5~

This leads to

J

o f ' V~d3r = ~ fin' ti,dS = 0. T = T~+TI .

(2.8)

(2 .9) (2.10) (2.11)

Because T, is positive we have proved that T,< < T, which constitutes Kelvin's theorem. .

(2.12)

NUCLEAR INERTIA

293

3. CYa~cieg model for the nuclear h~ertia

The cranking model was originally introduced by Inglis for calculating the rotational moment ofinertia 1 s. ie). The model has since bcen widely used also for calculating ground-state vibrational inertial and in fission. To derive the cranking-model expression for the nuclear inertia Ba associated with the collective coordinate a, one starts by specifying a model many-body Hamiltonian H(a). The eigenvalue equation H(a~i = E,(a~/r,.

defines the static solution to the many-body problem. Next one assumes that a is a prescribed function of time, a = a(t~

(3.2)

corresponding to an externally cranked field, and solves the time~ependent Schrôdinger equation The nuclear inertia is then determined from the expression To solve the above problem it is usually necessary to make some approximations. One often treats the cranking model in the adiabatic limit, in which a is a slowly varying function of time . This implies that eq. (3.3), after a suitable ansatz is made for SP', can be solved by perturbation theory and that eq. (3.2) can be written as a

ao +ât,

(3.5)

where the collective velocity â is small. We rewrite eq. (3.4) in the form <~`IH(a)I~) = Eo(a)+iB~(a~c2 +higher-order terms

(3.6)

~ II Z B°d a = 2h2 t#o Ek-Eo

(3.7)

and use second-order perturbation theory to obtain se)

which is the adiabatic cranking-model expression for the nuclear inertia. In a single-particle model for the nucleus, eq. (3.7) reduces to B"ad = 2~2 ~ II Z = 2~Z ~ II Z i .J

EI -Ei

i.1

(E1 -E !) 3

(3.8)

where h is the single-particle Hamiltonian and ei and e1 are thesingle-particle energies ; i and j refer to occupied and unoccupied levels, respectively. The derivation leading to eq. (3.7) should be valid also if 10) is an excited state. Then Ek - Eo is sometimes negative and one would expect from this model a

294

P . MÖLLER AND J. R NIX

decrease as) or even a negative value for the inertia for an excited state. Some groups sa. se. 3') have suggested extensions of the adiabatic cranking model by use of higher-order perturbation expansions to solve eq. (3.3). The results obtained in this way are somewhat contradictory. Other suggestions concerned a method se) for calculating a(t), as well as the additional models mentioned in the introduction. 4. Calculated results We. now apply the cranking model to two special cases. First, we consider in the adiabatic limit the inertia for a many-body system in a deformed harmonic-oscillator single-particle potential plus a small perturbation. By omitting the spin-orbit term and by specializing to a doubly closed-shell nucleus we are led to particularly simple results that are usually given in closed form . Second, in order to study the increase of the inertia in the vicinity of a level crossing, we consider in detail two levels of the above system . This two-level system is treated both in the adiabatic limit and for large velocities. As is customarily done, we initially calculate the inertia corresponding to a coordinate that describes the deformation of the single-particle potential. However, because of the non-self-consistency inherent in the cranking model, the deformation of the nuclear matter distribution does not follow identically the deformation of the single-particle potential. When comparing with results of a macroscopic model, we therefore transform to a coordinate that describes the deformation of the nuclear matter distribution. Under coordinate transformations the kinetic energy is invariant. This means that, for a one-dimensional sequence of shapes described by either a coordinate a or a coordinate r, Ba(da/dt)Z = B,(dr/dt)z.

(4.1)

Therefore, the inertia B, with respect to r is related to the inertia BQ with respect to a by Our model is defined by the single-particle Hamiltonian where

z P ho = ~ + lZl1l~Wx(XZ+y2)+WzZZ~,

and where dh is a small perturbation such that

(4.4)

NUCLEAR INERTIA

295

for i * j. Here i and j label the eigenfunctions of the unperturbed Hamiltonian of eq. (4.4). The condition of volume conservation is fulfilled by requiring that msco = = c~ô.

(4.6)

a = c~o/u~=.

(4.7)

Following Schütte and Wilets sa). we use a deformation coordinate a defined by

We assume that dh is sufficiently small that it may be neglected everywhere except near a level crossing and that its dependence on a may be neglected when calculating the inertia. The single-particle energy-level spectrum of h o is plotted e) in fig. 1. Expressed in ' units of hmo(e), where s9 )

21~

I

~a

Ratio of axes, c/a 2

~2

3

4

19y2 17/2 I~

\~~

\

" ~~ ~

~w~_ \W 11 ~2

W

~2 5~2

16

8

10 4

2

2

~2

I

i

~,

0.2

-I

Q4

a

L

Q6

i

I

QS

v

1

IA

Deformation E Fig . 1 . Single-particle energy levels of a pure harmonio-oscillator potential for prolate spheroidal deformations 6 ).

296

P. MÖLLER AND J. R. ND{

the levels are straight lines as functions of the coordinate s. The relation between a and e is given by a-

C l+3e l~.

(4.9)

1-3e

In addition to the deformation coordinates a and e, we also use the coordinates rP and rm They give the distance between the c.m. of the two halves of the system for the single-particle potential and the nuclear matter distribution, respectively . Their precise definitions are

Vv(r)5 Y'v

rm

Vp(r)5 YD

2 ~ - zp(rk1 3r/ ~ p(rki3r, s2o =~o

(4.11)

where VP(r) is the deformed harmonic-oscillator potential, ~P is a constant and p(r) is the nuclear matter density. The potential coordinate rP is independent of ~ P if we express rP in units of R1, P, where the radius Ro . P ofthe spherical potential distribution is defined by the expression 40) Vv(r)~ ~v

Vv(r)~ ~v

Similarly, for a diffuse-surface shape we. define the radius R,, m . corresponding to the spherical matter distribution by ~°) (' (' lu. (4.13) R ,m Cs(n+3) p(r)i' +zdr p(r)rzdr . J J J One can then prove that for spheroidal distortions

=

I

(4.14) 4.1 . MANY-BODY SYSTEM

We now use eq. (3.8) to calculate the nuclear inertia for both spherical and deformed shapes for an N = Z nucleus that is doubly magic. The spherical case for filled shells is easily done analytically for the Hamiltonian ho. One finds for the adiabatic cranking model that z ~i~~(a = 1) = (4.15) i(N~), 16 ficn 0

NUCLEAR INERTIA

29 7

where f(N~) = N~+8N~+23N~+28N~+12 .

(4 .16)

Here N~ is the oscillator quantum number of the last filled shell. When eq. (3.8) is used for the spherical and deformed harmonic oscillator, there are no matrix elements of â/ôa between states whose energy levels cross sa). In the spherical case there are only matrix elements between states for which N ~ differs by 2. We wish to compare eq . (4.15) with the inertia for the incompressible, irrotational flow of a liquid, which is a')

where R is the radius of the spherical drop and M = mA is its total mass. One can also prove that fitvo

sRz . ~fi z

f(N~)

mA,

(4.18)

where m is the nucleon mass and A is the mass number. When inserted into eq. (4.15) this yields ~~~~(a = 1) = (4.19) â iôRi.~~~

It therefore appears at first sight that the cranking-model inertia is ~ the irrotational inertia, which classically is the lower limit of the inertia. But, as noted by Wilets az~ in B~°` the coordinate a refers to the deformation of the potential, whereas in B'â it refers to the deformation of the matter distribution (which in this classical case coincides with the potential distribution). We therefore want to transform Bâ'°` to a coordinate related to the matter distribution. To do this we first transform eq. (4.17) to the coordinate rp by use of eqs. (4.2) and (4.14). For the incompressible, irrotational inertia we then obtain (4 .20) .

since for a classical liquid rP = rm. For a diffuse-surface body whose equidensity surfaces are spheroids of equal deformation, eq . (4.20) generalizes to 2

128 (rm/R 1 m) ] R t . m

For the microscopic inertia Bm'°` the transformation is somewhat more complicated. First, we note that for a spherically symmetric diffusé-surface body, rpnr« = (4.22) âR ~ . ~

298

P . MdLLER AND J. R NIX

Second, it is easy to show that for a system that is magic at the spherical shape and that is defined by the deformed harmonic-oscillator potential given by eq . (4.4~ rm = âRi . m( ~ ct~a.

(4.23)

Here the summation is over occupied Arbitals i. The quantity et is proportional to the value of rm for the single-particle state i for the spherical shape. Because of eqs. (4.13) and (4.22 ~Fi = 1 for a completely filled shell, both for the spherical shape and for deformed shapes prior to the first level crossing. Therefore, prior to the first level crossing, and

rin = aRi .bJa,

(4.24)

da _ 32 rin drin 9R1 " m

(4.25)

For the spherical shape this simplifies to da _ 8 dr~ 3R,, m We then use eqs. (4.2), (4.19) and (4:21) to obtain for the spherical shape i Bmicr = -( Z " m l mA. 5 (Ri.,~

(4.26)

(4.27)

By comparing this result with eq. (4.21) evaluated for the spherical shape, we see that for the spherical shape B~'`r = B;T, in agreement with Wilets' result aZ) for small distortions about a sphere . For deformed shapes far from a sphere, Wilets estimates that Bâ~°r = 4Bâr, where a refers to the deformation ofthe potential. In transforming to the matter coordinate, Wilets deduces that because for deformed shapes r~ ~ rp on the average, and because the independent particles will follow the lowest envelope ofparticle energies, then the microscopic inertia is â the irrotational value . However, this conclusion, which would violate Kelvin's minimum-energy theorem, is erroneous, as we show below. For the deformed case, although rm x rP on the average, the derivative is given by drP/drm 2 .(except in the vicinity of a level crossings as is seen in fig. 2. Also, in the case of a pure deformed harmonic oscillator there is no interaction between levels that cross, which means that the single particles do not change their quantum number and consequently do not follow the lowest envelope of particle energies. However, in our calculations we assume that there is added to the Hamiltonian a small perturbation such that all unperturbed levels that would cross are split by a small amount 2~d V~ [see eqs. (4.3), (4.4) and (4.5)] . Then, for adiabatic motion the single particles follow the lowest envelope of particle energies.

NUCLEAR INERTIA

299

m 0 t z ~E

W

K W

Q

0.75

Qb0

Q89

.90 0

0.96

POTENTIAL COORDINATE rp (UNITS OF RI )

L:00

Fig . 2 . l]ependence of the matter coordinate rm upon the potential coordinate rp for 70 protons and 70 neutrons in a pure spheroidal harmonic~oscillator potential plus a small perturbation. The coordinates r~ and rp are the distances between the c.m. of the two halves of the matter distribution and the two halves of the single-particle potential, respectively. The unit R l for the matter coordinate r~ is defined by eq. (4.13).

We again use eq. (3.8) to calculate Bâ'°` and eq. (4.2) to transform to Bin'°r. The results are given in fig. 3. We see that the calculated inertia Bin'°` is very large at level crossings. In fact, as seen from eq.,(3 .8), Bin'°` oc 1/~dV~ 3 . Awây from level crossings, B°''" is approximately â the irrotational inertia B;ô. However, when we relate the calculated microscopic inertia not to the motion of the potential but to the motion of the matter distribution, the situation becomes different, as we now show. We make the transformation from Bin'°` to Bin'°r by use of eqs. (4.2), (4.14) and (4.23) . The quantity ~~Ct depends on the specific levels that are occupied . For each deformation we determine the occupation numbers ofthe various levels and calculate the values of c, by numerical integration. The calculated adiabatic nuclear inertia BT~" is plotted in fig. 4. At a level crossing, matter is transferred from an equatorial orbital with a small value of rm to a polar orbital with a large value of r m Therefore, whereas a level crossing corresponds to a single value of the potential coordinate rP, it corresponds to an interval in terms of the matter coordinate rm . For example, the first level crossing corresponds to r P = 0.8703R,, p and 0.8079R t , m <_ rm 5 0.8435R t .m. It is interesting to compare the behaviour of Bin"' to that of B°m°` at a level crossing . Y Because drp/dr,~ is small near a level crossing (see fig. 2), one expects B, m to be much smaller than B,p near a level crossing . We show later that for one particle in two

300

P . MÖLLER AND J . R . NIX IA Q9 0.8

Z " 70 A " 140

N~N

m 5 w i

DEFORMED HARMONICOSCILLATOR POTENTIAL

d

03F-

o~y

aeo

oss

aso

~ ass ~ ~ Loo

POTENTIAL COORDINATE rp IUNITS OF

RI)

Fig. 3 . Dependence of the nuclear inertia B, upon the potential coordinate rP for 70 protons and 70 neutrons in a pure spheroidal harmonio-oscillator potential plus a small perturbation . The solid curve gives the adiabatic-cranking-model result, the ahan-dashed curve gives the reduced mess, the long-dashod curve gives the incompressible, irrotational result, and the dotted curve gives the limit of this curve for an infinitely long apheroidal noodle. The units R, and Rz are defined by «l . (4.12).

deformed harmonio-oscillator levels plus a small perturbation, ,B~'°' oc 1/~d V~. Thus, even for the inertia relative to the matter coordinate, the adiabatic cranking model yields a result that goes to infinity at the level crossing, even though in terms of rm the amount of matter redistribution is the same at the level crossing as away from the level crossing . We see from fig. 4 that away from level crossings B~'" is approximately equal to tire irrotational inertia. Before the first level crossing we find numerically that

We feel that before the first level crossing the microscopic and irrotational inertias are probably identically equal. Although the method for proving this algebraically seems fairly straightforward, it is tedious and we have at this point not done it.

NUCLEAR INERTIA

30 1

Fig. 4. Dependence of the nuclear inertia B,~ upon the matter coordinate r, for 70 protons and 70 neutrons in a pure spheroidal harmoniaascillatar potential ph~s a small perturbation . The solid curve gives the adiabatic-cranking-model result, the short-dashed curve gives the reduced mass, the long~ashed curve gives the incompressible, irrotational result, and the dotted curve gives the limit of this curve for an infinitely long spheroidal needle . The units Rl and R~ are defined by eq . (4.13).

At larger distortions BT'°' differs from B;m by up to about ± 1 ~, with the microscopic nuclear inertia sometimes larger than and sometimes smaller than the irrotational value. 4 .2 . TWO-LEVEL SYSTEM

In the previous paragraph we have seen that the adiabatic~;ranking-model inertia is approximately equal to the irrotational value as long as each particle in the system remains in a particular orbital (Nn~lfl) . Here N is the total number of nodes, n= is the number of nodes in the z-direction, and A and l? are the projections on thenuclear symmetry axis ofthe orbital and total angular momentum, respectively . Because the mechanism that makes the calculated inertia larger than the irrotational value is the change of orbital at a level crossing, we now study this problem in detail . Other studies of level crossings, which, however dif%r from ours in the interpretation of their effect on the nuclear inertia, are, for instance those of refs. 4s, 4a). We consider a system of one particle in two deformed harmonic-oscillator levels

302

P. MÖLLER AND J . R. ND{

W Z W W U

F K Q

a i w U Z

N

DEFORMATION Fig. 5. Dependence of the singlo-particle energy levels upon deformation in the vicinity of a level crossing . The dashed curves give the energy levels of the unperturbed Hamiltonian ho , and the solid curves give the energy levels of the perturbed Hamihonian h .

plus a small perturbation . Fig. 5 illustrates this system near the level crossing. The dashed lines labelledwith the energies E, andEb arethe eigenstates ofthe Hamiltonian defimed by eq. (4.4); they are characterized by the quantum numbers (1Ven=A'Sl°) and (1Vbn~~l bl2b), respectively. The eigenfunctions ly, and ~z ofthe perturbed Hamiltonian defined by eq . (4.3) are related to the eigenfunctions ~a and ~b of the unperturbed Hamiltonian by

where

~ - E. A z = ~+'z z[(Eb -EJz+4dVz]~ .

(4.29)

The dependence upon a of the energy difference Eb - E, is

1 Eb-E, _ ~ (n~-n=)+ Ja(1V6-1V'+n= -ns)] fitc~o. a

(4.30)

By evaluating eq. (3.8) we then find for the contribution to the inertia of one particle in a two-level system z L(~-nsx2la+~a)+(1Vb-N°)1/a]zdVz fl d9~°~ (4.31) . = 2az EJz +4dVz]~ [(Eb -

NUCLEAR INERTIA

303

From this expression it follows that at the level crossing dBâ'°`

oC

1/~d V~ 3 .

(4.32)

To transform to the contribution relative to the matter coordinate rm we first note that where 2d, and 2db are the values ofrm at the spherical shape for the equatorial singleparticle orbital ~e and the polar single-particle orbital fib, respectively . After some algebra we find that 'm

2ftwo az

x where

(yz +4H;~~

Nb- N`+r~-n= - ns-

C

2~/a

az ~J~

-z

(4 .34)

y = (~-EJl üc~ o,

(4.35)

H,b -- d v/~c~ o.

(4.36)

From eq. (4.34) it follows that at the level crossing ~,~ . a 1/~d v~ .

(4.37)

We therefore find, as pointed out earlier, that the adiabatic-cranking-model inertia can also diverge at a level crossing when expressed in terms of a matter coordinate, but only as 1/~d V~. However, the above results based on the adiabatic cranking model are not valid for finite velocities â. From a consideration of our two-level system, we can show that the contribution to the total inertia from a level crossing decreases as the velocity â increases. We do this by observing that the solution of the time-dependent Schrödinger equation for the two-level system gives Eia = E~+ZdB,m(rm , i~izm =<
(4.38)

where E, and Ez are the energies of the perturbed Hamiltonian illustrated in fig. 5. At the level crossing, Ez - E, = 2~d V~ and a2 < 1 . It therefore follows from eq .

30 4 (4.38)

P. MÖLLER AND J. R NDt

that dB,m(rm, r~ c 4~d Y~/rm .

(4.39)

This result agrees with, but is more general than, the conclusion of Schûtte and Whets s3), who find that for a simple two-level,system the nuclear inertia is reduced when the next-order term in the perturbation solution is considered . At high excitation energies, the destruction of pairing leads to a decrease in the magnitude of the effective perturbation ~d V ~. This also decreases the contribution dB,m to the inertia given by eq . (4.39). Of course,. at high excitation energies, one must consider the more complicated problem of the inertia corresponding to a statistical ensemble of states rather than the inertia for a pure state.

S. Sammary and caondodions We have calculated the adiabatic-cranking-model inertia for a system of 140 particles in a deformed harmonic-oscillator single-particle potential. When expressed in terms of a matter coordinate rm, the inertia for small oscillations about a spherical shape is exactly equal to the incompressible, irrotational value. For large distortions it deviates from the incompressible, irrotational value by up to about ± 1 ~ away from level crossings. The calculated nuclear inertia was sometimes slightly lower than the incompressible, irrotational value for spheroidal motion . However, this does not necessarily violate Kelvin's theorem because although the potential motion is spheroidal, the nuclear matter flow is somewhat different. The nuclear inertia should therefore be compared with an incompressible, irrotational inertia calculated for the same motion of the boundary. The difference between the potential coordinate rp and the matter coordinate r~ arises because the static Hamiltonian used in our calculations is non-self-consistent. By neglecting the terms in the Hamiltonian that could make it self~onsistent one leaves out contributions to the inertia that may be as important as the calculated terms themselves z 1). The feature that B,~ goes to infinity at a level crossing is probably unphysical for the same reason. ' a level our Z = N nucleus with 140 particles is approxWe find that at crossing imately self-consistent (drp/dr~ .. 1) provided that ~d V ~ 1 MeV. This estimate was made with two particles at a level crossing and the rest of the system defined by eq. (4.4). Because the pairing gap is of this magnitude, the lack of self-consistency is not so serious when calculating the true adiabatic nuclear inertia by means of the cranking model. At high collective velocities we have shown by a general argument that at a level crossing the contribution to the cranking-model inertia from the two crossing levels decreases with increasing velocity. However, as long as the velocity is moderate and ~d V~/moo ~ 1 one may consider the total inertia to be the sum of the contribution

NUCLEAR INERTIA

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from the level crossing and the adiabatic inertia for the llnpertittbed system of A particles. For such parameter values the calculated inertia should then approach the irrotational value as â increases from zero to moderate values. Furthermore, as the excitation energy increases, the magnitude of the effective perturbation ~d V~ decreases, which also decreases the contribution to the inertia at a level crossing. However, at large velocities the many-body wave function will, as the deformation , changes, become very different from the static ground-state wave function. T'herefo ;e, even if the static problem is self-consistent, the time-dependent problem is no longer self-consistent at large velocities and important contributions to the inertia are probably left out s t ). Unfortunately, the calculated nuclear inertia may be compared only very indirectly with experimental data. For example, one may compare calculated and experimental spontaneous-fission half-lives. In this case the calculated half-life involves an integral of a product of the inertia, usually calculated in the adiabatic limit, and the potential energy. However,there is additional experimental informationavailable from induced fission and from heavy-ion reactions. In these cases the nuclear inertia that enters the equations of motion corresponds to large velocities and high excitation energies. By developing models for the nuclear inertia in such limits and by comparing the solutions of the equations of motion to the experimental data, one hopes to gain increased understanding of both the reactions themselves and of the microscopic models for the nuclear inertia. Our limited study may be regarded as a step in this direction. References

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