A model for describing the structure of highly excited states in deformed nuclei (I)

Nuclear Physics A196 (1972) 433 -45 1; @ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

A MODEL FOR DESCRIBING THE STRUCTURE OF HIGHLY EXCITED STATES IN DEFORMED NUCLEI (I) V. G. SOLOVIEV

and L. A. MALOV

Joint Institute for Nuclear Research, Dubna, USSR Received 5 May 1972 Abstract: A model for describing states of intermediate and high excitation energy in deformed nuclei which is based on the quasiparticle-phonon interaction is suggested. A secular equation for determining the energy and the wave functions is obtained by means of the variational principle. The neglect of the non-coherent terms leads to the appearance of extraneous states. A method for eliminating them is developed. On the basis of the calculations performed for ‘j9U, at an energy close to the neutron binding energy, the average spacing between the $+ states is found to be 33 eV which is twice as large as the experimental value. The magnitude of the one-quasiparticle components is between IO-’ and IO- ll. The increase of the density of the states with increasing excitation energy is demonstrated. It is concluded that the quasiparticlephonon interaction is responsible for both the complication of the state structure with increasing excitation energy and the fragmentation of the one-particle states over many nuclear levels.

1. Introduction The density of the low-lying states is not large in medium and heavy nuclei. The average spacing Do between the low-lying levels with definite spin I and parity n is several MeV. The structure of the ground and low-lying states is rather well described in the framework of the semi-microscopic approach by means of quasiparticles and phonons. The density of states grows with increasing excitation energy. When the excitation energy is equal to the neutron binding energy B,, the average spacing between the levels with definite I” in medium and heavy nuclei takes the values from 1 eV to 10 keV. The neutron binding energy B, for nuclei lying inside the B-stability region or near it varies from 4 to 10 MeV. At such an excitation energy the density increases by a factor of 105-106. The structure of highly excited states, i.e. states with neutron binding energy B, and higher, is very complicated. The properties of these states are described by the statistical model. Thus, the properties of the low-lying and highly excited states are described by the models of quite a different character. It is interesting to clarify what mechanism is responsible for the fragmentation of one-, two-, three- and higher particle states over many nuclear levels. It is also important to understand how the density increases with increasing excitation energy. In ref. ‘) it was indicated that the interaction of quasiparticles with phonons may be one 433 December

1972

434

V. G. SOLOVIEV

AND

L. A. MALOV

of the mechanisms which is responsible for the fragmentation of the one-particle states. The fragmentation and density-increase processes were studied there by means of the model which had been employed in ref. “) for the calculation of the energy and the structure of non-rotational states of odd-A deformed nuclei. The density of the 3’ states in 239U at an energy & = B, was thus found to be larger by a factor of 100 than that calculated by the independent quasiparticle model. The former model was used to demonstrate the fragmentation of the one-particle states over many nuclear levels. It was also pointed out ‘) that it is necessary to modify this model by adding terms involving a quasiparticle plus two and more phonons to the wave function so that it can be used for the study of the highly excited states. In the present paper a model is suggested for describing the structure of the excited states of odd-A deformed nuclei. A secular equation and expressions for the coeficients of the wave functions of excited states are derived. Some possible applications of the model to the description of the density and the structure of highly excited states are demonstrated by the example of 239U. 2. Formulation of the model The model is formulated in the framework of the semi-microscopic approach where the average field, superconducting pairing interactions and interactions between quasiparticles are taken into account. The Hamiltonian describing the interactions between the nucleons in the nucleus is written as H =

Hav+Hpair+HQ,

(0

where Ha, is the average nuclear field of the neutron and proton systems, Hpair are the interactions leading to superconducting pairing correlations and HP is the multipole interaction. Spin-multipole interaction terms, Gamov-Teller and other terms may also be added to this Hamiltonian. We formulate the model for the case of an odd-A deformed nucleus. A generalization to the case of the spherical nucleus will be performed elsewhere. We consider the interaction of quasiparticles with phonons in an odd-A deformed nucleus [see refs. 3, “)I. In the odd-A nucleus there is one quasiparticle, in addition to the quasiparticles and phonons of the doubly even nucleus. The presence of quasiparticles leads to some change in phonons, but as a rule this change is not large and is neglected here. We assume that in the odd-A nucleus the phonons are the same as in the doubly even A - 1 nucleus. The constants of the multipole-multipole interactions are fixed in the calculation of the phonon energy in doubly even nuclei. Therefore in the calculation of the interaction of quasiparticles with phonons there is not a single free parameter. Taking into consideration the secular equations for determining the phonon energies on (where g implies &j, and 4 = &, j is the number of the root of the secular

HIGHLY

EXCITED

STATES

435

equation), the appropriate part of the Hamiltonian (1) can be written 3S“) in the form

+B(v, Here Q, is the phonon operator relation function, L the chemical quantum numbers (VO) denotes %w = u,v,, +z+D~, where u,, v, matrix elements of the multipole

(2)

v’)+ P~(vv’)B(v, v’))}.

and E(V) = J[C* + (E(v)-il)2], where C is the corpotential and E(v) the one-particle energy; the set of the one-particle state, cr = + 1; v,,, = u,u,, - v,v,~, are the Bogolubov transformation coefficients. The moment operator with q = Ap are denoted “) as:

fq(v1v2),

if if

K,+P K,+K,

= K2, = +p,

where K is the projection of the angular momentum on the nuclear symmetry axis. Next, B(v, v’) = 1 6u%‘o 9 0

qv, v’) = c cTayf_aCly’a, 0

where mV:is the quasiparticle creation operator; y = c cl

Ew)>2d”~%w>+ 44) [(E(V)+E(v’))2 -ci$J’

Y,Y’

(3b)

,

the sum being taken over the single-particle levels of the neutron and the proton systems. The wave function of an odd-N nucleus which describes the states with a given K” is written as

where Y. is the wave function of the ground state of a doubly even nucleus, (PO) denotes the set of quantum numbers of a one-particle state with a given K”, and i is the number of the state. The normalization condition for the wave function (4) has the form (Y;(K”)Yj(K”))

= 1 = (Cf)“{l+

(Dg,)’ B vf7 + c

cc

c (F$)2+

c

c (R:?i)2}.

(5)

V. G. SOLOVIEV

436

AND L. A. MALOV

Now we calculate the average value of HDqover the state (4), with the result

(y’:(K”Poqyi(K”))

=

(ci)” (&(P)+ _)T z (E(V) + og)(D~~)2

c (E(V)+ wg+ Og2)(qcy+ c c (E(V)+ wg+ cog2+ og,)(R;y)2

+c 8.82

8.82.93

y.2

v-2

The energies vi of the non-rotational states and the functions Cj, Of&, Fit:’ and Rrp’ are determined by means of the variational principle

q(y*(K”)H”, WY -ni[(C;)‘(l++~(D$,)2+ WO

1 c(F$)2+ 8.92 Y(I

1 ~(RE3i)2)-11) 9.92.83 v.9

After performing a variational procedure and some transformations, following system of equations:

(E(V)+ cog+

cog, -

I]JFrgi = Q c

= 0.

(7)

we obtain the

2!z!z

“’ JY, Yg*

x fyvv’)“f~(pv’) - 0342(vv’)f!,4pv’) +fq(vv’)f:2(pv’) - cp(vv’)f9Z,(pv’) [ 1 E(V')+ Wg-qi E(V’)+mg2-Ili r(f”‘(vv3)fq’(vz v3) +f4’(vv3)343(v~ v3))Fz +4fqz(w)3g3(v, %) - .fq2(vv,)fqyv, Vj))Fg$o c c VW %Y3 _ I gayz,vs JY,,JC2(E(v3)+Wg-Vi)

+$

(f”(vv,)fq3(v2

%)

+ +

+

f%“‘,)fq”(v2

h))F;;z

4f4(vv3)3q3(v, 4 - 3w3)fq3(V* %))q:$n I JY,(e(vJ) +Og2 - Vi)

HIGHLY EXCITED STATES V YY,

+tCL c E(V3)+ Og+ Jr,, 83

Vt9 V3

V”,“)

+a*, -vi

Og,

J- L-(fq3(%)fq3(v2 v3) tJYg,

“‘f,4(pv)+2CCk &?(vv’)F$Jy 82 v’ 82 DC, = @!J E(V)+ oe - ‘li RBBzu3i _ Pa _-

1

437

@(VV’)F$#) 1

(10)

1

243 4~) + 0, + wg2+ me3- 47 [f”(vv’)F$;

- cTf”(VV’)F$J

(11)

+ ...

In P$?!Fi there are two more terms obtained from the first one by a cyclic interchange of indices g, g2, g3. We separate the coherent terms of the r.h.s. of eq. (9):

f’“(vv’)f,“(pv’)- up(vv’)fQnJ’)

x

[

+ fyvV’)f,ypv’)

&(V’> + Og2-Vi

E(V')+COg-tji

Y3

[

+u(fq2(vV3)fq3(V~ v3)- 3q*(VV3)fq’(V2 v,))qt$0 +

09 -

vi)

(f4bGl-*‘(v2 v3)+f”(vv3)3”‘(vzv3))q% 4Eea ~(fq(w)fq3(v2v3)-3q(vv3)fq3(v2

z%

+c

J Y,, Y,,(~vJ

83

83h32

Y2fY

1

(fq2&)fq3(vZ 4 + f4’(VV,)f4’(V, v3)E%

+ac c VW3VW3 c Y1

- ujyvv’)f!2,(pv’)

+

JGyg, (E(v3)+ogz-Vi)

1

V. G. SOLOVIEV

438

AND

L. A. MALOV

where

+rJ

cvg

(f:3(vv’>>2 . (12) 83v’ y,, E(V’)+Wg+Cog2+Ogp-~i

Now we have to solve the system of equations (9a) for different values of v, g and g2 and insert Fit;’ in eq. (8). As a result, we obtain a secular equation. The solution for the system (9a) can be represented as a ratio, of which the denominator is the determinant of the system and the numerator is the determinant which is derived from the denominator by the replacement of the coefficients for F$,f by the appropriate constant terms of the equation. We get an equation for finding the poles of the secular equation (8) from the condition for the determinant of the system to be equal to zero. However, because of numerous values of v, g and g2, it is very hard to solve exactly the system (9a). Then we pass to an approximate solution of the system (9a). By rejecting the noncoherent terms we have

Q5 ;t

[f~~(vv’)f~~v~~+~~~~‘)fr.(pv’) r, 52

pva -

B 1 + f’(vv’)f,“‘(pv’)-a_P(vv’)fJ!Z,(pv’)

p92i _

&(V’) + %2-vi E(V)+Og+Wg~-fJi-T~g*i

1*

(13)

HIGHLY EXCITED STATES

439

We substitute eq. (13) in eq. (8) and obtain the secular equation in the explicit form

I)

+fyvv’)f,4l(pv’)- a.p(vv’)f!<(pv’) I2

4v’)+ %2 -vi ~(~)+0#+0g*-~i-

= 0.

(14)

TyBuzi

Here ?i are the roots of the secular equation, i = 1,2, 3 . . . . The 1.h.s. of eq. (14) has first-order poles for ylpol= E(V)+ o# + w#~- Ttuzi and has no poles for q = E(V)+ CO# + O#Z+ O#S,and the poles for r] = E(V)+ og are extraneous. The approximate solution for eq. (9a) in the form (13) is close to the exact one since the rejected non-coherent terms are unimportant. But this rejection has led to the appearance of extraneous poles and thereby extraneous solutions for the secular equation (14). Indeed, the poles of eq. (14) are defined as the roots of the equation

The roots of eq. (15) are located between the poles &(V’)+Og-$o’ = 0, &(V’)+ w, + og2 + COgs - vPO’= 0,

(16)

which are independent of v. All roots of eq. (15) are the poles of eq. (14). Eq. (15) is solved for each value of v, and each pole (16) is shown up in eq. (4) as many times as a new value is ascribed to v. Therefore, instead of one pole of eq. (14) corresponding to qpO’from eq. (16), there appear several poles, and thus there appear extraneous solutions for eq. (14). Disregarding the interaction of quasiparticles with phonons, the excitation energies are determined from eq. (16) and the equation E(v’)+Wg+Wg2-~Po’ = 0,

(16a)

i.e. they are equal to the sum of the energies of a quasiparticle and of one, two or three phonons. When the quasiparticle-phonon interaction is taken into account the exci-

V. G. SOLOVIEV

440

AND

L. A. MALOV

tation energies move away from the poles (16) and (16a); however, the number of excitations remains unchanged. There is a correspondence between the poles in eqs. (16) and (16a) and the roots of the secular equation containing non-coherent terms. The calculations show that between each two poles in eqs. (16) and (16a) there are several roots of eq. (14); in this case only one of them is very far from the pole, and the remaining roots are in the vicinity of the pole, The studies have shown that the true solution for (14) corresponding to the solution for the system (9a), taking account of the coherent terms, has an energy much further from the corresponding pole. The remaining roots are superfluous. A similar situation arises when studying the low-lying non-rotational states in odd-A deformed nuclei ’ - 4). In this case, near the pole E(v)+o#- qp“’ = 0 there are as many roots as there are different states p with identical K”. Consideration of the appropriate non-coherent terms results in the disappearance of the extraneous roots, and only one root remains with an energy extremely far from the pole E(V)+ oe - qpO’ = 0. Consideration of the non-coherent terms does not introduce any significant change in the value of the true root. Thus, in addition to the secular equation (14), we use the condition of exclusion of extraneous roots, which is formulated in working out the routine for solving eq. (14) by an electronic computer. We find the functions D$, and RPyce3im . an explicit form. To this end, we insert FBgzi in the form (13) in eqs. (10) and (11). As a result, we have PVC

x

p’(vv*)

f4Yv2

[ +

f”(v2

+

vdfmJ3)-

(

G.$q*(vv2)f4Yv2 f4(v2

“mu+

- dq2(v2

E(V3)+

E(v3)+

-

V3)fXPV3)

( d4(v2

wg-?i

V3)f4YPV3) )

wgz-)li v3)fwv3)

v3)fwvJ

+ dq2(v2

V3)f4(PV3)

E(v3)+o#-?i d4(v2

E(v3) + og2-?i

VJ.eYPV3)

)I) ,

HIGHLY EXCITED STATES

441

+ f4(v2h)fzwJ - cf’(v, 4f%(P3) 1 dvJ)+@g2-Vi V3)f4(P’3) - a343(vv,)f4Yv2df%(PVd +f7fq2(v2 ( E(v3) +og-?i + f4(v2wzr(PV,)+ Em2 VX2(P~d + @g*-

)I...I.(18) E(v3) + yli

In Ryvpi there are two more terms which are obtained from the first one by a cyclic interchange of the indices g, g2, g3. The quantity (CL)’ defines the contribution of the one-quasiparticle component to the normalization condition (5) of the wave function (4). The quantities +(C~)z~,(D$,S)2, t(C:)2~o(~~‘)2 and $(C~)2~o(R$,f3i)2 define the contribution of the components corresponding to a quasiparticle in the v-state plus one, two and three phonons, respectively. Only one state p with a given K” is taken into account in the wave function (4). As is shown in refs. 2, 4, 5), a noticeable contribution to some low-lying states comes from two single-particle states with identical K”. The mixing of two, three and higher singleparticle states may become more important with increasing excitation energy. However, a simultaneous account of several single-particle states with a given K” does not affect the density of non-rotational states and is not very important for determining the wave function components. Let us give some formulas for a simpler case when the wave function contains no components representing a quasiparticle plus three phonons. Then the wave function has the form

Instead of eqs. (8)-( 11) there are equations

“2)

(fq2(““2)fq2(%

+ fq2(vv2)fq2(v~

+ ‘J(fq2(““z)fq2(v~

X

JY,,(e(v2) (fq*(v~2)fq(~3

+

“2)

v2))&o

“2) - fq2(V%)fq2(% + Og + me2

+ fq2(““2)fq(V3

+ dfq2(v”2)_fq(v, JY,(E(v2)+wg+ug,-Vi)

V2))&,

-Vi)

V2))3$ v2)--

.fq2(vv2)fq(v3

v2))D$3-o

-m

1 (21) 9

442

V. G. SOLOVIEV

1

1 pni = _ Pa

AND L. A. MALOV

4 E(V)4Wg+0g2-qi - afyvv’)D$

fyvvyD$,

x c v YY’ VI

_. + fq(Vv’)DSm - d4(vv’)DB,:f -0

Jr,2

[

Jr,

1.

Taking account of the coherent terms and neglecting the non-coherent eq. (21) yields D$& =

1 !?L 2 Jr,

(22)

-I

f:(P)

ones in

(23)

&(V)+og-~i-S;”

where pi

=

4

u& Ez(VV’))2(l +a,,)

c c

Y

92

v’

Y,,

E(V’)+Wg+0g2-)li’

(24)

The secular equation for determining the excitation energies vi is of the form

v%4)z _ E(p)-qi-$& Yg E(V)+Og-tji-Sfi B

Y

o

.

(25)

This equation contains extraneous roots, the method of elimination of which is the same as for eq. (14). The functions FpBf and Ci are given by pgti

Pa

x

= 1

1

1

8 :Yg Ye2E(“)+mg+mgz-Iii

- up2(vv’)fl~(pv’) + fq(VV’)f:*(pV’)- u.q(vv’)f~z,(pv~) ) c Vpv’%v~ [ f”‘(vv’)f,“(pv’) &(V’)+ sg! &(V') + Wg,- vi - St:’ 1 Y’

OS -

qj -

+ f”(vv’)f,4’(pv’) -

&(V’)+ cog2 3. Semi-microscopic

u_fyVV’)f!~(pv’) 2 tji - Sff’ I) .

-

(27)

description of the structure of highly excited states

The semi-microscopic model suggested in this paper may be thought of as a specification of the general semi-microscopic approach to the study of the structure of highly excited states l* “). We give the main assumptions of this approach.

HIGHLY EXCITED STATES

443

The operator form of the wave function presented as a sum of terms with a different number of quasiparticles underlies the excitation-state studies. There is a hierarchy of the components of the wave function with different numbers of quasiparticles. In the case of neutron resonances, the neutron, radiative and g-widths are expressed in terms of the wave function coefficients. According to refs. ‘3“) t h e wave function of the highly excited state of an odd-A deformed nucleus is of the form Y,(l”) =

c biJv)a,+,Yyo “I3

+

y g

ye I$

. 9

GXva,

v2 a2,

v3 a3)a,+atl,:,,

aZsmJ Qt

YO+ . . . .

The coefficients b’ define the contribution of the corresponding quasiparticle component. The pairing-vibration phonon operators Ql are introduced instead of the operators a:+ a,‘- ; Y,-, is the vacuum wave function for the operators a,, and 52,. The wave function contains no phonon operators, except for Sz,. The phonon operators are written as a superposition of the two-quasiparticle components, and therefore the appropriate terms may be assumed to be included in eq. (28). It is not difficult to relate the coefficients of the wave function (28) to those of the wave function (4). For example, the one-quasiparticle coefficients are connected as follows: lC;l’ = 21b:,(p)12.

(2%

The wave function (28) has a very general form, and can be used for describing the intermediate-energy states and the highly excited states. The number of terms in wave function (28) becomes larger and larger with increasing excitation energy. At excitation energies close to the neutron binding energy B,,, the wave functions (28) contain thousands of few- and many-quasiparticle components and possess the properties of the compound states suggested by N. Bohr. In fact, the formation of a highly excited state can occur through some components and the decay through others, so that the decay of this state is independent of the way in which it is formed. Individual few-quasiparticle components are not large, due to the large number of components of the wave function of the highly excited state. This leads to a significant hindrance of the y-decay probability and to an increase in the lifetime compared to the oneparticle states.

444

V. G. SOLOVIEV

AND

L. A. MALOV

The wave functions (28) can be used for the description of excitations from an energy of 2-3 MeV to that at which resonances do not yet overlap, i.e. when the condition r,
where &Ais the resonance energy. The energy factor is associated with a possible emission of the neutron. The reduced neutron width for the capture of a slow neutron by a doubly even deformed nucleus, with subsequent formation of a highly excited state described by the wave function (28) is of the form

where rS+ = 20 A-* MeV. The function u, points out that the v-state must be a particle state. For the s-wave neutron capture the summation in eq. (32) is performed over the single-particle states with K” = +‘. In ref. ‘) the quantities 161’are determined from the experimental data on neutron and radiative widths. They are found to be j612 M 10e9 in deformed nuclei, 151’ E 10-7-10-9 in spherical nuclei and IQ2 M lo- 6 in nuclei around closed shells and in nuclei with A z 40-60. Table 1 gives the reduced neutron widths rfi and the values of the one-quasiparticle components 15’12for neutron resonances with I” = 3’ in 239U. The experimental data are taken from refs. **‘). Averaging over the resonances and taking the average number of terms in (32), it = 2, we obtain 1612M 10m9. The expressions for the partial radiative and a-widths are obtained in the framework of the semi-microscopic approach 6*‘3‘I). C orre 1a t ions between the reduced neutron, radiative and a-widths in neutron resonances are considered in refs. ‘, ‘3lo) and refs. 12,13) and the cases are indicated which are the most favourable for observing them. The correlation of two processes occurring through one and the same state may exist when the main contribution to both processes comes from one and the same component b’ of the highly-excited-state wave function. The general semi-microscopic approach makes it possible to understand the main features of the structure of highly excited states and to extract information about some components of the highly-excited:state wave functions from experimental data. A detailed theoretical study of the structure can be made with the aid of rather simple models one of which is considered in this paper.

HIGHLY

EXCITED

STATES

445

TABLE 1 Reduced

neutron widths and one-quasiparticle components 2JW (n = 2)

8-B,,

(ev) 6.7 21 37 66 81 103 117 145 165 185 208 237 273 291 347 397 410 434 463 518 535 580 595 average values

lV/2 for resonances with In = 3’

r.r”. W)

IW

0.6. 1O-3 1.9. 1o-3 5.2 * 10-S 3.1 . 10-S 0.2. 1o-3 6.6. 1o-3 2.4. 1O-3 0.6. lo-“ 0.2 * 1o-3 1.0. 1o-2 3.9. 1o-3 1.8. 1O-3 1.5 1o-3 1.0. 10-S 4.0. 10-S 0.4. 1o-3 0.9. 10-S 0.4. 10-a 0.2. 1o-3 1.9. 10-S 1.7. 10-S 1.3. 10-S 3.3. 10-a

0.3 . lo- 9 1.0. 1o-9 2.6. 1O-9 1.6. 1O-9 1.0. lo-lo 3.3. 10-g 1.2. 10-g 0.3. 10-10 1.0. lo-lo 5.0. 10-g 2.0 . lo- 9 0.9. 10-g 0.8. 1O-9 0.5. to-9 2.0. 10-g 0.2. 10-g 0.5. 10-g 0.2. 1o-9 0.1 . 10-g 1.0. 1o-9 0.8. 1O-9 0.7. to-9 1.7. 10-Q

2.3. 1O-3

1.2.

in

10-g

4. Details of calculations The isotope 239U is chosen as an example for performing numerical since there is much experimental information on its neutron and radiative and due to the small neutron

binding

energy

calculations widths *p9),

B, = 4.8 MeV the terms involving

a

quasiparticle plus more than three phonons, which are disregarded in eq. (4), are not so important. Indeed the energies of the first one-phonon states in 238U are about 1 MeV, so that the poles corresponding to a quasiparticle plus three phonons are higher than 3.5 MeV, and the poles corresponding to a quasiparticle plus four phonons are higher than 4 MeV. Thus, the components due to a quasiparticle plus four phonons little affect the density of the states and the magnitude of the one-quasiparticle components up to an excitation energy of 5 MeV. The single-particle energies and the wave functions of the Saxon-Woods potential [ref. 14)] for A = 239 with deformation parameters pz = 0.22 and fi4 = 0.08 are used in the calculations. The interaction constants G, and G, leading to superconducting pairing correlations and the quadrupole and octupole interaction constants @) and K(~) are taken from ref. 14).

V. G. SOLOVIEV

446

AND

L. A. MALOV

As far as our mathematical apparatus is adjusted to working with phonons, we take into account the phonons with 1 = 4 and 1 = 5, although the majority of the appropriate states are not collective. In so doing, we make no mistake, however, since when the root of the secular equation for the one-phonon states comes nearer the pole, the wave function of the one-phonon state turns to that of the two-quasiparticle state. The addition of the I = 4 and II = 5 phonons to the i = 2 and L = 3 phonons widens the region of spin values for two quasiparticles, so that the spins and parities run over all values from O- to 5- and from O+ to 4+. The constant K(~) is taken the same as for rare-earth nuclei, taking into account the A-dependence, and K(~’ is smaller than Kc4) by a factor o f 10. The results of calculations depend weakly on the lcC4)and rcc5) values. All roots of the secular equations for the one-phonon states in 238U are taken into account up to an energy of 5 MeV. For a number of Lp values, e.g. for il = 2, P = 2 and 1 = 3, p = 1, one takes 70 roots, i.e. j,,, = 70. In other cases, e.g. for 1 = 3, p = 0, j,,, = 50. 5. Density of the states in 23gU In ref. ‘) the density of the 4’ states in 23gU is calculated by the model the wave function of which is of the form yi(K”) = Jfci

C {a,’ + C C Dz, a,‘, Q,‘> PO 0

B

*

(33)

v

Taking into account the 3, = 2 and iE = 3 phonons at B = B,, the average spacing D between the 3 levels is found to be 10 keV. In this paper the density is determined from the calculation of the position of the following poles: quasiparticle plus one, two and three phonons. Table 2 gives the average spacing D between the I” = +’ levels in 239U at Q = B, = 4.8 MeV. It is seen from the table that passing from the wave function (33) to the wave function (19) results in the increase of the density by more than two orders of magnitude. This TABLE 2

Average spacing between levels with 1” = 4’ in 239U at 8 = B, = 4.8 MeV jl

_&lax

D (eV) calculated

2,394, 2, 3,4 2, 3 2, 3,4, 2, 394, 2, 3,4,

5

5 5 5

70 70 70 35 20 9

with the wave functions

(4)

(19)

33.2 42.2 96.2 33.3 35.5 47.4

77.5 90.9 178.6 78.1 90.1 163.9

HIGHLY EXCITED STATES

447

fact indicates that at the excitation energy 4.8 MeV in 239U, the wave function components quasiparticle-plus-two-phonons are very important. Passing from the wave function (19) to the wave function (4) we are led to the density being increased by a factor of 2-2.5, therefore the role of the components quasiparticle-plus-three-phonons in the wave function at the energy 4.8 MeV is not very important. The addition of the terms quasiparticle-plus-four-phonons to the wave function (4) leads only to a slight increase of the 4’ level density. In this case D = 30.6 eV at 8 = B, so that the terms quasiparticle-plus-four-phonons may be disregarded in the study of the excited states of 239U up to 5 MeV. In table 2 it is shown how important are the terms with I = 4 and I = 5 and how the density depends on the number of the roots j,,, of the secular equations for phonons. It is seen from the table that the inclusion of the terms with I = 4 and L = 5 leads to an increase in the density of the 3’ states by a factor of 2-3. This indicates that accounting for the multipole interaction terms with ;1 = 4 and I = 5 is very important. Any decrease in the number of roots noticeably affects the +’ state density when j,,,,, < 20. We consider the 3’ state density in 239U as a function of the excitation energy. The results of calculations are given in tables 3 and 4. At excitation energies of 1 and 2 MeV the results of calculations with the wave functions (4) (19) and (33) coincide. At 4 MeV the +’ density calculated with the wave function (4) is larger by a factor of 1.5 than that calculated with the wave function (19) and at 5 MeV this factor is 2.5. It is seen from table 3 that the role of the terms of the wave functions (4) and (19) TABLE 3 Average spacing D between levels with In = 3’ in 239U for j,,,,, = 70 for different excitation energies

8 WW

1.0

1.0 1.0 2.0 2.0 2.0 3.0 3.0 3.0 4.0 4.0 4.0 5.0 5.0 5.0

1

D (ev) calculated with the wave functions (19) (4)

2,3,4, 5 2,394 2, 3 2, 334, 5 2,3,4 2,3 2,3,4,5 2,3,4 2,3 2,3,4,5 2,3,4 2,3 2, 3,4, 5 2,3,4 2, 3

. 103 76.9 * 10’ . 103 143 9.4 . 103 11.4 * lo3 17.9 * 103 1.43 * 103 1.64 * lo3 3.23 * lo3 165 203 454 21 35 48 66.7

66.1 . 103 76.9 * lo3 143 . 103 9.4 . 103 11.4 * 103 17.9 . 103 1.70. 103 1.89 f 103 4.17. 103 246 289 568 66 78 185

448

V. G. SOLOVIEV

AND

L. A. MALOV

TABLE4 Average spacing D between levels with I* = 4’ in 239U 8 (MeV)

D (eV) 1.0

2.0 3.0 4.0 4.8 5.0 6.0 8.0

66.7 . lo3 9.4 . 103 I .43 . 103 165 33.2 27 7.1 0.5

containing phonons with 1 = 4 and i = 5 grows slowly with increasing excitation energy. In table 4 it is demonstrated how the average spacing between the 3’ states decreases with increasing energy. The calculations are performed up to an energy of 8 MeV, and a continuous decrease of D is observed up to this energy. Thus, our configuration space is rather large and may serve as a good basis for calculations. It is clear that, due to the limited configuration space, at a certain excitation energy D should stop decreasing, and should start to grow with increasing excitation energy. According to the experimental data [see e.g. ref. ‘“)I the average spacing between the 3’ levels in 239U is De_,= 18 eV at d = B, = 4.8 MeV. Our calculations give D = 33 eV. That is, in our model, the density of the +’ levels is half the experimental value. Such a small difference is evidence that the wave function used is in its complexity close to the wave functions of neutron resonances. Thus, the present model TABLE5 Average spacing between levels with given Kn at an energy I = B,, in *39tJ K”

8’ g!J+ 2i+ %5’ HLI+ ;Y’ 12 p+ 12 %-

D @VI

33.2 33.1 34.7 32.1 39.5 41.3 40.2 48.5 47.6 50.0 57.9 53.2 66.2 13.6 75.8

HIGHLY

EXCITED

449

STATES

may serve as a good basis for studying the properties of the intermediate and highexcitation-energy states. It is worth noting that only a part of the residual interactions has been taken into account in our model. The addition of other residual interactions, like the GamowTeller interaction or multipole interaction terms with 1 > 5, leads to an increase in the density of the states under consideration. The description of the I > $ states in deformed nuclei encounters some difficulties. It is unknown how the rotation of the nucleus proceeds in the highly excited state. In our model, K is a good quantum number which worsens the description of the I > 4 excitations of deformed nuclei. Taking into consideration this remark we study the density of the states as a function of K” values. Table 5 presents the average spacing between levels with given K” in 239U at 8 = B,. It is seen that the density of the i- states is close to that of the 3’ states. The spacing grows slowly with increasing K. 6. Conclusion The large density of the states at excitation energies close to B, follows from the independent quasiparticle model. It is also important that the wave functions of these states should contain many components, and among them non-zero few-quasiparticle components. The main merit of the model suggested is that the wave function of the highly excited state contains a large number of components, namely a one-quasiparticle component and many three-, five- and seven-quasiparticle components. We have obtained several solutions for the secular equation (25) near the neutron TABLE 6

The structure

ofstates

with Kn = 4' in 239U near B, = 4.8 MeV Structure

-0.2629

0.3364

11.114

0.3007

6314 lO-9% 76lJ+Qs(31)

622t+Qs

0.02 %

6314 4.8. lo-’ % 63lt+Q2(22) 0.01 %

6314 2.6. 1O-6 % 604$+Qz(43) 4.2 %

6114 lo-‘% 613t+Qz(43)

0.01 %

(3l)+Q4 (33) 752t+Qs (3l)+Q2 (22) 62Of+Qs (3l)+Q, (30) 633$+Qz (22)+Q7 (44) 743++Ql,(32)+Qt (22) 752r++Qs (3l)+Q2 (22) 752tfQz (22)+Qs (54) 624&+Qz (43)+Q12(20) 620tfQz (43)+Q1 (44) 606J.+Qz (43)+Q1 (22) 622$+Qz (43)+Q1 (22) 734t+Q2 (43)+(212(31)

89.6 7.4 1.7 68.4 30.5 0.9 0.1 62.3 5.4 4.2 2.8 2.8

624&+Q, W+Q,zW)

99.8 % 0.1 %

5OlS+Q,

(4l)+Q1

(31)

% % % % % % % % % % % %

450

V. G. SOLOVIEV

AND

L. A. MALOV

TABLE 7

The structure

of states with Kn = 4’ in 239U near B,, = 4.8 MeV Structure

c*-B* &eV) -0.2629

0.3364

11.114

0.3007

n631J n761J. n761J n761J n631.1 n631J

10e9 %; n752f n624$ 0.015 %; p642+ p521f 0.003 %; p642f ~5324 0.002%; 4.8 * lo-’ %; n622t n631f 0.01 %;

n631J n604J n604$

2.6. low6 %; n624$ n63lJ 4.15%; n632f n631$ 0.05 %;

n611$ n613T

10W5%; n624$ n631$

0.01 %;

n622f n622f n622t

n752t n752+ n7521

n631J, n631$

n624J. p642t

n752$ p521t

69.7 % 14.3%

n631J p642t p532J. lO.O”/,

n633$ n631$ n622+ ~5334 ~6324 52.4 y0 n743t n631$ n622t p4OOt p521t 17.9”/, n633J n631J n622f ~5234 p521T 11.8% n743t n743t n633J, n743t n624$ n624J. n620t n624J. n624$ n624J. n624J.

n631J n622t n63lJ n622t n631$ n6221 n631J. n622+ n624J n631$ n624$ n631$ n624J n631$ n624$ n631$ n624j. n631J n624j. n631$ n624$ n631J

n743t n622$ n7251 n624J. n615J. n631J n503J n631J p521f p521t p651T ~4024 n624J n631$ ~5234 ~5234 p633t p633f p4OOt p4OOf n752t n752t

n624$ n624J. n624J. n624J. n624$ n624J n624J

n624$ n624J n624$ n624$ n624J n624J n624J

p521t p651f ~5234 ~6331 p400+ n752+ n620t

n631$ n631J n631$ n631J. n631J n631J, n631$

4.1% 2.8 % 1.3 Td 1.3 % 16.7:/, 11.8 % 6.8 % 4.7 7: 3.5% 3.4 % 3.1 %

p521+ 26.7% ~4024 18.8 % p523J 7.5 % p633t 5.6% p4OOt 5.4 % n752t 4.9% n631& 3.4 y/,

binding energy B,. The one-quasiparticle components and the largest components quasiparticle-plus-one-phonon and quasiparticle-plus-two-phonons are calculated for four form In pole,

roots and given in table 6. In of an expansion with respect table 6 it is shown that (Cj)’ and (Cl)’ z lo-* when the

table 7 the same wave functions are given in the to the quasiparticle components. M 10-l’ when the root of eq. (25) is close to the

root is away from the pole, i.e. there is agreement with the experimental data of table 1. It is seen that the quantity (Ci)” for the p = 63 1-1 state located near the Fermi-surface energy is smaller than that for the p = 611-1 state which has an energy somewhat lower than the neutron binding energy. This confirms the conclusion drawn in ref. ‘) that the magnitude of the one-quasiparticle component of the wave function of a neutron resonance increases as the one-particle neutronstate energy approaches B,,. It may be concluded that we have found the right order of magnitude for the density of neutron resonances and the one-quasiparticle components of the wave functions. Thus there is agreement between our calculations and experiment. It is seen from tables 6 and 7 that the wave functions contain a number of large

HIGHLY

EXCITED

STATES

451

components quasiparticle-plus-two-phonons and, consequently, a number of large five-quasiparticle components. It is quite possible that accounting for the rejected non-coherent terms can reduce the large components, but must not affect the magnitude of the one-quasiparticle components. At present it is still unclear whether the presence of large many-quasiparticle components in the wave functions of highly excited states is a shortcoming of the model or the correct reflection of their structure. On the basis of the investigations performed it is possible to conclude that the interaction of quasiparticles with phonons is, to a large extent, responsible for both the complication of nuclear structure with increasing excitation energy and the fragmentation of one-, three- and higher particle states over many nuclear levels. The investigation of this model is continuing. The study of the component composition of the wave functions of highly excited states and states of intermediate excitation energy as well as the calculation of various characteristics of neutron resonances will be discussed in a subsequent paper. References 1) V. G. Soloviev, Izv. Akad. Nauk SSSR (ser. liz.) 35 (1971) 666 2) L. A. Malov, V. G. Soloviev and U. M. Fainer, Izv. Akad. Nauk SSSR (ser. fiz.) 33 (1969) 1244; L. A. Malov, V. G. Soloviev and S. I. Fedotov, Izv. Akad. Nauk SSSR (ser. fiz.) 35 (1971) 747; A. L. Komov, L. A. Malov and V. G. Soloviev, Izv. Akad. Nauk SSSR (ser. fiz.) 35 (1971) 1550 3) V. G. Soloviev, Prog. Nucl. Phys. 10 (1968) 239; V. G. Soloviev and P. Vogel, Nucl. Phys. A92 (1967) 449 4) V. G. Soloviev, Theory of complex nuclei (Nauka, Moscow, 1971) 5) F. A. Gareev, V. G. Soloviev and S. I. Fedotov, Yad. Fiz. 14 (1971) 1165 6) V. G. Soloviev, Yad. Fiz. 13 (1971) 48 7) V. G. Soloviev, Yad. Fiz. 15 (1972); preprint JINR E4-5711 (1971) 8) J. R. Stehn et al., Neutron cross section, BNL-325, 2nd ed., vol. 3 (1965) 9) 0. A. Wasson, R. E. Chrien, G. G. Slaughter and J. A. Harvey, Phys. Rev. C4 (1971) 900 10) V. G. Soloviev, Phys. Lett. 35B (1971) 109; 36B (1971) 199 11) V. V. Voronov and V. G. Soloviev, communication JINR P4-5562 (1971) 12) V. G. Soloviev, ZhETF Pisma 14 (1971) 194 13) V. G. Soloviev, preprint JINR P4-6293 (1972); Phys. Lett., to be published 14j F. A. Gareev, S. P. Ivanova, L. A. Malov and V. G. Soloviev, Nucl. Phys. A171 (1971) 134