Excited states of odd-mass nuclei with small non-axiality

_ 1.D.4 _ l

Nuclear Physics 37 (1962) 106--118; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprlnt or microfilm without written permission from the publisher

EXCITED STATES OF ODD-MASS N U C L E I W I T H SMALL NON-AXIALITY A. S. D A V Y D O V A N D R. A. S A R D A R Y A N

Nuclear Research Institute of the Moscow State University, Moscow, USSR Received 1 J a n u a r y 1962 Abstract: T h e sequence o f the spins a n d energy ratios o f the excited states o f o d d - m a s s nuclei is calculated w i t h o u t differentiating between one-particle a n d collective excitations. T h e results o f the theory are c o m p a r e d with experiments.

1. Introduction

It is customary to divide the excited states of atomic nuclei into one-particle and collective excitations. The latter are in turn divided into vibrational (fl and ~ oscillations) and rotational. The division of excitations into one-particle and collective ones applies comparatively well to even atomic nuclei since the spectrum of one-particle excitations is separated from the ground state by a "gap" of the order of 1 to 2 MeV. in odd-mass atomic nuclei the energy of one-particle excitations is of the same order as the rotation and oscillations of the nuclear surface, and so ~the isolation of oneparticle excitations from the general excitation of the nucleus is hardly justifiable in some cases. Davydov 1) showed that in even nuclei whose equilibrium form is close to the axial-symmetrical, rotation cannot be regarded independently of y-oscillations. Thus, it should be expected that small excited states of odd-mass atomic nuclei have a highly involved nature which does not permit the simple isolation of the rotational, oscillatory and one-particle excitation energies. These excited states correspond to definite values of parity and total angular momentum. This investigation is aimed at studying the excited states of odd-mass nuclei with small non-axiality on the basis of a simple nuclear model making it possible to study such excitations from a single viewpoint. As the nuclear model we take a system consisting of the core, the equilibrium form of which corresponds in the ground state to an ellipsoid o f revolution, and one outer nucleon. The nuclear surface can make small oscillations ]~ and 7 with respect to the equilibrium values/~0 # 0 and 70 = 0. The nuclei with the ground states spin equal to ½ and { call for a special study (see refs. 2, 3) for example). Therefore this paper is concerned with the nuclei with the ground state spin equal to { and ~-. The formulae obtained make it possible to calculate the sequence o f the spins and energy ratios o f the excited states of odd-mass nuclei through a certain number of parameters. In the adiabatic approximation in nuclear surface ]~-oscillations the formulae become much simpler: which enables us to deter106

EXCITED STATES OF ODD-MASS NUCLEI

107

mine the sequence of the spins and energy ratios of excited states through one parameter for the ground one-particle-rotational band and through two parameters for the first anomalous band. The results of the theory are compared with the experimental values for heavy atomic nuclei.

2. Principal Equations of the Model When the deviations from the equilibrium values flo and 7o are small the excited states of the system under consideration consisting of the core and an outer nucleon are determined by the Schroedinger equation

(Hv-t-Hrot-PHp(x)-t-I-Iint-e)~ = 0,

(2.1)

where Hp + Hi,t is the operator of the Hamiltonian of the outer nucleon in the nuclear core field: Hp(x) takes into account the central-symmetrical part of the field, and I'1int

=

--T(r)fl{cos7(3)2--32)+~/3 sin 201/,2--)2)2

(2.2)

is the operator taking into account the non-spherical part of the nuclear core field. Here r is the distance between the nucleon and the nuclear centre, x are the spacial and spin coordinates of the nucleon, )1, )2 and J3 are the operators of the projections of the outer nucleon angular momentum on the axes of the coordinate system connected with the nucleus. Further,

I'1v= - - - -

t4 ~

+-~1

)'~y

+ -2 (fl-fl°)z+Bc°Zf122

is the operator describing the fl and ? oscillations of the nuclear core surface, and

h2

(I--jr) 2 h 2 (~,20V.2 i 2 2~ tc ] ~ ~ 6Bfl t J __)2 Hrot - 8B 2 Z sin (,_ T --

}

3

-2(i0

}

is the operator of the nuclear rotational energy; i, I1, i 2 and i3 are the operators of the total angular momentum and its projections on the axes of the coordinates connected with the nucleus. The operator of the Hamiltonian of eq. (2.1) commutes with the operators of the squares of the total angular momentum. The wave function of eq. (2.1) satisfying the necessary symmetry conditions 2) can be written in the form

~'i = F,i(fl)~,J*(x, 7, 0102 Oa),

(2.3)

where

= E Z q#,(r)10Km>, re

K

[IjKrn> - x/2-I-+l {D~r(O,)~o~~_2re(x) + ( - 1)r-S'D~t, _ r(O,)~o~r+zre(x)}. (2.4) 4=

108

A. S. DAVYDOV AND R. A. SARDARYAN

In eq. (2.4) the K-summation is performed over the values of K = ½, -~, etc. the msummation is performed over integral values of m satisfying the inequality ½ - j < 2m < I+j, the function ~7c being determined for the values of K and rn satisfying the inequality IK-2m[ < j ; the ~p~*(x) are the proper functions of the operators Hp(x) corresponding to the proper values Ej,; the quantum number f2 = K - 2 m is the projection of the angular momentum of the outer nucleon on the nuclear axis (axis 3), and • denotes other quantum numbers (in particular, state parity). Substituting eq. (2.3) into eq. (2.1) and replacing eq. (2.2) by its average value with respect to the state ~o~~ and zero oscillations of the nuclear surface Hi.,

1)],

-

we find a set of two equations

2B ~ 2 - WA(fl)

=

(2.5)

i.e. the equation containing the variable fl in which the potential energy is represented by the term

Wa(fl) = C

hZ(A + 2)

(2.6)

and the equation pertaining to other variables of the set

[£-t-~2-A]eDIj,~(x, 7, 0,) = 0,

(2.7)

where y c~7 ~ ' ~

+D72+¼

~5 +½ ( a - ) a ) ,

g2 = l [j(i+ l ) - 3 j ~ ] + ½ [ I ( I + l ) + j ( j + l ) - i ~ - 3 2 - 2 ( 1*0 ,* +12J2)], * D =

2,

¢ -

6Bflao(T)

(2.8) (2.9) (2.9a)

With fixed values ofjz, eq. (2.7) enables us to calculate the parameter A of the division of the equations for each value of the total angular momentum of the system described by the quantum number I = ½, -~, etc. Using the values of A thus found one can calculate from eq. (2.5) the energy differences E - E j , which will determine the excited states of the system corresponding to the inner state Ej~. The method of solving eq. (2.7) will be indicated in the following section. 3. Calculation of the Parameters A

Substituting the functions (2.4) into eq. (2.7) and bearing in mind that the operators /~ and ~ do not change the absolute value of the quantum number m, we obtain a set

EXCITED STATES OF ODD-MASS NUCLEI

of equations for each absolute value of

109

m:

( £ = - A ) ~ , ( l m l ) + ~ a~,~K(lml)¢~(lml) = 0,

(3.1)

Ka

where m

Irnl

7 g7 Y

+Dy2+ T

'

aK,K(lrnl) ~'~ -- (IjK'o~'lmll f2lljKcqrnl).

(3.3)

From eq. (3.2) it follows that in the states with a definite value of ]ml the operator of the potential energy of y-oscillations is represented by the term

Consequently, the nucleus in the state Im] ¢ 0 has no axis of symmetry. Eqs. (3.1) are a set of differential equations. If the functions Z~'~I(7) are the proper functions of the operator Lm, i.e.,

(Lm-Llmlz)ZlzmI(T)=

0,

(3.4)

the solution of the set (3.1) should be sought in the form ~b~=l(y) = Z)([ml)z/ml(y).

(3.5)

Substituting eq. (3.5) into eq. (3.1) and taking into account eq. (3.4) we obtain the system of algebraic equations = O. (Llmlx--A)A)"(lml)+ E awK(lmt)ZK(Iml) "'~ ~'

(3.6)

K,~

The solution of the differential eq. (3.4) is expressed through a confluent hypergeometric function by using the equality Z/ml(y) = (4Dy2)~ImlF( -,~,, Iml + 1, y2x/D ) exp (-{yZx/D),

(3.7)

where the quantum numbers 2 = 0, 1, 2 etc. The proper value m2 Ll=lz = 2~/D(22+ Iml + 1)+ 12-

(3.8)

corresponds to each function (3.7). For the set of algebraic equations (3.6) to have non-trivial solutions the following condition should be fulfilled: lla~,~K-~aw,~a=,=ll = 0.

(3.9)

110

A. S. DAVYDOV AND R. A. SARDARYAN

The roots of this equation are numbered by the subscript l running through the values 1, 2, 3, etc., so that ~ < e2 < e 3 < . . . . Knowing the roots et of eq. (3.9) we obtain for the parameter A the expression

A

--

A('c,j, I, lml, 2,

l)

--

2~/D(22+ [ml+ 1)+ -rnz +ej'(lm[, I).

(3.10)

12j

Taking into account eqs. (3.4) and (2.4) we can write the wave function (dependent on the 5 variables x, ? and 03 of the state corresponding to eq. (3.10) in the form

qblml.W.,. ~,, v', ?, Of) = X [IjKc4ml>A~tZlmr(?) •

(3.11)

K , cx

4. Calculation of Excited State Energies

The parameter A (3.10) calculated in sect. 3 determines the potential energy of floscillations with the aid ofeq. (2.6). Expanding eq. (2.6) in the value fla corresponding to the minimum Wa(fl) we may write

wa(fl) where

=

Wa(fl a) + ½Ca(fl -- fl~) 2,

[ I+ 3h2(A +2)-] ~_l'

cA=c

(4.1)

fla = flo+ h2(A+2)

Bcfl~

After substituting eq. (4.1) into eq. (2.5) we obtain the equation the solution of which was investigated by Davydov 1). Thus it can be shown that the wave function of eq. (2.5) is expressed through the Hermitian function Hv(z)

(-n,

= [2F(-v)l-'.=o ~

l)nF ( ~ - ~ ) (2z)n

(4.2)

re(z),

(4.3)

with the aid of the relation N

FA(B) = ~ e

_½52

where z --

p(fl-/h) ~ #i fla

,

/z1 = # [1 + 3 ( A + 2 ) (~)*] -+ ,

(p-1)p 3 =(A+2)#4; v is the root of the transcendent equation

(p)

H~ -- ~

= 0;

(4.4)

EXCITED

111

S T A T E S OF O D D - M A S S N U C L E I

N is the normalization factor;

is the "non-adiabaticity parameter". The nuclear energy corresponding to a definite value A and non-adiabaticity parameter # is expressed by the formula

AE~Tl(ll,~v) = E-E~j = hog{(v+½)Vl+3(A+2)(#-t4+ \p/

A+2

3

.

+

(4.5)

2#

The total wave function of the states under consideration is determined by the product o f e q . (3.11) by eq. (4.3). In those cases when # < ½ the value v in eq. (4.4) differs but little from integers, i.e. v~n=0,1,2,.... In this case the functions (4.2) pass into a Hermitian polynomial and eq. (4.5) can be replaced by the approximate expression

AE~71(II2n) = hog(n+½)+ h 2 ( A + 2 ) { l + 3 ~ 2 ( n + ½ ) - # ' * ( A + 2 ) } . 2Bfl~

(4.6)

Consequently, when # < ½ one can single out approximately the excitations corresponding to fl-oscillations. In the following we use the approximate formula (4.6). I f # > ½ the calculations should be made using eq. (4.5). The excitations corresponding to the value m = 0 will be referred to as the ground band of excited states. The excitations with [ml = 1, 2 etc. will be called the first, second, etc. anomalous bands of excited states. Substituting the value (3.10) into eq. (4.6) and taking into account eq. (2.9a) we obtain

AE!71(II2n) = hco(n+{)+g~7'(ll; 0 {1+ 3.~2gn+,~,--,la,4 -72Bfl2e~71(I/~.)} ,

(4.7)

where

g~71(I/~,) = ho)~(2,~+ Iml + 1)+

h2 I~J(lml, I)+2+ ~21 .

(4.8)

The energy (4.7) of the nuclear ground state is characterized by the value m = n = 2 = 0 and the quantum number I - - - I o to which corresponds the smallest value e]J(OI) i.e. the smallest value of the root of eq. (3.8).

i12

A. S. DAVYDOV AND R. A. SARDARYAN

When the values ( are positive the smallest value o f the root ofeq. (3.8) corresponds to Io = j. Thus we have (AE)g

....

=

½ho

+e

5

,

where o

h2

E~j = hog~+ 2 ~ ° [2+e~J(0Io)].

(4.9)

By subtracting eq. (4.9) from eq. (4.7) we obtain the energy of the excited state corresponding to different values of the quantum numbers L 2, l, n, [mJ. Thus, in the adiabatic approximation with respect to ri-oscillations (# = 0) the energy of the excited states is determined by the expression AE~71(II2,0 - aegroun,, m 2 = hcon+h~%(22+Iml)+ ~ h2 [Aej(iml, i)+ ]~]

(4.10)

where

Ae~J(lrnl,I)

=

ej(lml, I)-e]J(o, Io).

We are interested only in the ratios of the energies of the excited states to the energy of the first excited state and therefore it is convenient to express all energies in dimensionless units. Having divided eq. (4.10) by rio(T) we obtain

E~l(IIRn)

= n ( + ( 2 2 + l m l ) r / + 3 ~ [ As~J(lm[,I ) + ~rn21 ,

(4.11)

where (

hco -

rio(T)'

h% -

rio(T)"

h2 -

6Bfl~( T)

(4.12)

are three dimensionless parameters of the theory. When m = n = 2 = 0 eq. (4.11) reduces to the equation

E°j(IlO0) =

3¢Ae~2(0, I).

(4.13)

Consequently, in the adiabatic approximation (# = 0) the sequence of the spins and energy ratios of the excited states of the ground band (m = n = 2 = 0) is determined only by one parameter of the theory, viz. 4. The excited states of the type (4.13) were considered by the authors in a previous work 4). These excitations can be named rotational-one-particle excitations since they are due to nuclear rotation as well as to the change of the outer nucleon energy. The rotational-one-particle excitations differing in the values I at l = 1 form the band of excited states known as the "rotational band". The rule o f intervals in this band differs, however, from that given by

EXCITED STATES OF ODD-MASS NUCLEI

113

the adiabatic theory of the rotational states of axial-symmetrical nuclei without taking into account the change of the outer nucleon states. The rotational-one-particle excitations corresponding to the values l > 1 form additional bands of excited states. The position of these bands (see sect. 5) is uniquely determined (in the approximation # = 0) by the value 4 chosen for the description of the "rotational band". In the states n = 2 = 0, Iml # 0 the equilibrium form of the nucleus loses axial symmetry. The levels of this type lie above the first levels of the states with m --- 0, and their position depends on the value of the old parameter ~ and new parameter t/which determines the additional energy ~/Iml in the units rio(T) or h~o~ in the conventional energy units. When the quantum numbers 2 and n differ from zero, 7- and ri oscillations participate explicitly in the excited states. The relativity of this interpretation should be borne in mind, of course. If eqs. (4.7) are used for the calculations, the non-adiabatic corrections can be taken into account using the parameter # < ½. When/~ > ½ eq. (4.5) ought to be used. Another system of levels based on another one-particle state E~,j, can be incorporated into the system of levels under consideration with the same parity and based on the state E~j of the shell model. The built-in system of levels will be shifted with respect to the latter system and may differ substantially from it, characterized as it is by other values of the dimensionless parameters ~, t/and 4. Such a system of levels can be separated from the other comparatively easily if the state E~j differs from the state E~,i, in parity. If, on the other hand, the parities of such states are the same, the separation of two (or more) systems of levels is a highly complicated problem.

5. Comparison of Theory and Experiment Using eq. (4.11) the ratios of the excited energy levels were calculated as functions of the parameter 4 for nuclei with spin ~ in the ground state. Fig. 1 represents the levels of the ground band (m = 0). Fig. 2 indicates the lower levels of the first anomalous band (lml = 1) of the excited state. This band begins at the level with I = ½ separated by the quantity ha~ from the ground level of the nucleus with I --- ~. The positions of the subsequent levels with spins ~, ~ etc. are uniquely determined by the parameter 4, the value of which is calculated from the ratio between the energies of the two levels of the ground band. Fig. 3 indicates the ratios of the energies of the excited states of the ground band (m = 0) as functions of the parameter ~ for nuclei having spin ~ in the ground state. Table 1 indicates theoretical and experimental values o f the spins and energies o f the excited states of the nucleus U 233. The parameter = 0.25 was determined from the ratio of the energies of the two levels of the ground band. The value he% was chosen to equal 400 keV. Table 2 lists the spins and energies of the excited states of the nucleus Th z3 x based on the two following one-particle states of different parity: the ground state ~ and the excited state ~. Tables 3 and 4 list the

114

A, S. DAVYDOV AND R. A. SARDARYAN

data on the nuclei Cm 245 and B k 249 with the ground state spin ~. Table 5 lists the data on the nucleus Np 237. The theoretical values listed in all tables are obtained in the adiabatic approximation p=

0.25

0.50

0.75

1.0

Fig. 1. Sequence of spins and dependence on the parameter ~ of the energy of the ground band (m = 0) of the excited states of atomic nuclei with spin ~ in the ground state.

(~t = 0) with respect to fl-oscillations. For lighter nuclei this approximation is very rough and the levels should be calculated by more involved formulae (4.7). In the conventional interpretation the sequence of levels in odd-mass nuclei is considered as a system of independent rotational bands based on one-particle excitations. As such all levels are accepted with the values of the spins-coming out of the conventional sequence of the rotational levels of the axially symmetrical nucleus. Thus, in the nucleus Bk 249 the level of spin ~ with energy 393 keV is considered as an in-

EXCITED STATES OF ODD-MASS NUCLEI

lo[

0

T

l

T

025

050

075

115

1.O

Fig. 2. Anomalous band (Iml = 1) of the excited states o f the nuclei with spin } in the ground state. The energy is marked off on the vertical axis in the same units as in fig. 1.

0

0,25

0.50

075

1.0

Fig. 3, Sequence of spins and the dependence on the parameter ~eof the energy of the ground band (m = 0) of the excited-states of the atomic nuclei with spin ~ in the ground state.

116

A. S. DAVYDOV AND R. A. SARDARYAN

dependent one-particle level. As is shown by the calculations here performed all the levels indicated in table 4 are those of one rotational-one-particle band. Their position is determined by one parameter ~ taking into account the entanglement of rotational and one-particle excitations in odd-mass atomic nuclei. The excited levels of the TABLE 1 Spins a n d energies (keV) o f the U 2aa nucleus T h e o r y (~e = 0.25, he% = 400 keV) Ground band m = 0

Experiment 5)

Anomalous band tmi = 1

0 40 92.5

~ { ~

0 40 92

~ ~

313 341

½ {

400 417

155 230 318 338

388 ½ ]

410 ½

400 410 450

460

TABLE 2 Spins a n d energies (keV) o f T h ~al Positive parity levels

Negative parity levels T h e o r y (~e = 0.375)

-~-

0 42 97 163

~, ~

238

~k

255

~-

308 ½

319 340 383

Experiment 0 42 97 168 228 257

T h e o r y (~ = 0.65)

Experiment 8)

29.2

~

70

~ ½

109 127 145

~)

158

~

½ ~-

177 202 205 215 222 236 262 280 284 30o 333

15+56

½

100+56 127:~56 145+56

~L

178+56 220-t- 56

328

117

EXCITED STATES OF ODD-MASS NUCLEI TABLE 3 Spins and energies of Cm 2~5 Theory (~ = 0.60)

Experiment 7)

0 55 130 205 252 288 299 348 397 405 410 440

~-

~

~z

~ ~ ~ ~ ~ ~

0 55

124 207 257 3ol

~ ~-2

357 422

TABLE 4 Spins and energies (keV) of Bk 2~a Theory (~ = 0.24)

Experiment 8)

0 41.7 93.6 152 220 281 396 432 471 520 578

~-

~ ~ ~-2 (~-) (~-)

0 41.7 93.4 156 230

~ ~ ~ (~)

393 433 485 547

TABLE 5 Spins and energies (keV) of Np 2a7 Negative parity levels Theory ~ = 0.30

4~

½ ½

o 33.2 76.8 129 190 225 239 283 295 324 353 394 440 497

Positive parity levels

Experiment ~ ~ ]

½ ~ ~

o 33.2 76.4

332.3 368.5 371

Theory ~ = 0.45 ~ ~ ~ ~ ~ ~ ~~ ~ ½ ~

o 43.4 98.5 169 212

227 245 294 298 305 398

Experiment ~) ~~A -~-

0+59.6 43.4+59.6 98.9+ 59.6 165.42k 59.6 207.9+59.6 245 +59.6

118

A.S.

DAVYDOV AND R. A. SARDARYAN

nucleus U 233 has a still more involved pattern. In this nucleus the excited states with spin 3 (313 keV) and spin ½ (400 keV) are usually regarded as one-particle excitations. According to our calculations the ½ level (400 keV) is formed through a complex combination of one-particle rotational and 7-oscillation excitation. The transition into this excited state involves the violation of the axial symmetry of the nucleus. It can be expected that the levels of this type will occur more often in the region of readily deformed nuclei where the non-adiabatic conditions for fl and 7-oscillations are essential. Thus we come to the conclusion that a number of excited states of odd-mass nuclei which are regarded in conventional interpretation as one-particle excitations are actually a complex combination of one-particle rotational and oscillatory excitations. References 1) A. S. Davydov, Vestnik Moskovskogo Universiteta Fizika No. 1 (1961) 56, Nuclear Physics 24 (1961) 682 2) A. Bohr, Mat. Fys. Medd. Dan. Vid. Selsk. 26, No. 14 (1952) 3) A. S. Davydov, Nuclear Physics 16 (1960) 597 4) A. S. Davydov and R. A. Sardaryan, JETP, 40 (1961) 1429 5) K. Maack Bisgard, P. Dahl and K. Olesen, Nuclear Physics 12 (1959) 612 6) S. A. Baranov, A. G. Zedenkov and V. M. Kulakov, Izv. A N SSSR XXIV (1960) 1035 7) B. S. D~helepov and L. K. Peker, Joint Institute for Nuclear Research, preprint P-288(1959) 8) P. Asaro, S. G. Thomson, F. S. Stephens and I. Perlman, in Proc. Int. Conf. Nuclear Structure, Kingston, ed. by D. A. Bromley and E. W. Vogt (North Holland Publishing Co., Amsterdam, 1960) p. 581