Collective excitations of even axial nuclei

Nuclear Physics 30 (1962) 442---451, ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprxnt or mlerofdm without written permission from the pubhsher

COLLECTIVE E X C I T A T I O N S OF EVEN AXIAL NUCLEI V. I. B E L Y A K and D

A. Z A I K I N

P. M. Lebedev Physical Inst,tute o~ the U S S R Academy o[ Sciences, Moscow, U S S R Received 8 May 1961 The collectwe excited states of axial even nuclei are considered with the interaction of the r o t a t i o n w i t h fl- and 7-oscillations taken into account. The expression for the w a v e functions and energy levels are obtained as expansions in inverse powers of the rigidity p a r a m e t e r s of fl- and ~,-oselllatlons The reduced probabihtles of E2 transitions between these states are calclated The results are c o m p a r e d w i t h experimental d a t a and the t h e o r y of non-axial nuclei. The possibility of e s t a b h s h m g experimentally the n a t u r e of excited 0 + states is discussed.

Abstract:

1. Collective Motion Equations The collective motion H a m i l t o n i a n for the even nucleus in the adiabatic a p p r o x i m a t i o n is o b t a i n e d b y averaging the H a m i l t o n i a n of the nucleus o v e r the functions of nucleon individual motion, and can be r e p r e s e n t e d as

~2

4

32 ~r =-----

9?5

+3c°tg3

7

(1.1)

~'

3

V(fl' 7) = ½C~8--fl°)2+flD(7--V°)9--~-2B e =

BC/~*,

q :

e ~3--fl°)'+ ~ flo q(7--7°)2'

2DBflo3/t~2.

F o r axial nuclei to which we shall confine our discussion 70 ---- 0. In the equation of collective motion ( ~ - - E ) F ( f l , 7, 0,) -~ 0 it is c o n v e n i e n t to go over to the function ku = fl2F, satisfying the equation r, 0,) = 0, 442

COLLECTIVE

443

EXCITATIONS

in w h i c h / ~ differs f r o m / ~ (1.1) in that the term --(41fl)~l~fl is replaced by 2/fl 2. Under the assumption that the nuclear surface oscillations are small [(fl--fl0)2<< rio*, ~2<< 1], i.e., the nuclear surface rigidity is large [p½ = e½ × fl02>> 1, q½>> 1], we keep only the principal terms in the expansion of the Hamiltonian ~ in the powers fl--flo, ~. After introducing the variables y = q¼~ and x = ei(fl--flo ) we represent the Hamiltonian in the form /~ = i ~a~

0(x) +

(l_]_p_¼x)2_ ],

~o(X) --

o ----

~xu -[- x ~,

(1.4)

~ (v, o,) -- ~o(V, o,) + ~ l (V, o,), ^ •~o(Y, 0,) --

~2 ~Y*

1 ~ ]3 2 ~ - ½k-~J / l f 2 --~JZ 1 ~ ~i, y ay + y2 + __4y2 -~- ~/

(1.5)

~al (r, 0 t) ~- -- q--I 2y ( j 1 2 - J 2 2 ) / 3 V 3 . It should be borne in mind that additional terms will appear in (1.3) if all terms of the same order are taken into account consecutively in the expansion of the operator/~. The oscillations being anharmonic, there arises the undivided part of the Hamiltonian incorporating the terms of the form p-~ y2x and p-½y2 x*, while ~o(X) incorporates terms oc p - i x 3 and oc p-½x4, and a~t~(y, 0~) the terms q ~ (const.+y ~) and q-~ y~/~y. In 3¢~(y, 0,) the terms of the type q - l j , and q-13ta* are not written out. The effect of all these additional terms can be investigated b y perturbation theory. This analysis shows that taking account of the additional terms is not essential since the results of interest, i.e. the ratio of electromagnetic transition probabilities and the corrections to the rule of intervals for energy in a definite rotational band, do not depend on these terms in the present approximation. Thus, if the nuclear surface oscillations are small, we have

T ( x , y, 0,) =/(x)qb(y, 0,), A

[ ~ ( y , 0 , ) - ~ ] ~ ( y , o,) = 0, ~ o ( ~ ) + (1 + p - ~ ) 2

~ /(~) -- 0,

(1.6)

(1.7) (1.s)

444

V. I.

BELYAK

AND

D. A.

ZAIKIN

2. 13-Oscillations In eq. 0.8) describing fl-oscillations,

^ = A x~l(x)

[(1 + p-ix)2 1

1] "mA(--21b-¼x+3p-'½x2)

can be regarded as a perturbation since p½ >> 1. The solutions of the unperturbed problem tk [ x ~ 0 ( x ) - ~ 2 ] / 0 ( ~ ) = 0, *2 = e - - A , (2.1) with the boundary condition ]°(--ibi ) : 0 are expressed through the Hermitian f u n c t i o n s / ° ( x ) ----exp(--~!x2)H.(x). The relevant eigenvalues e~° = 2v+ 1 are determined from the equation

H.(--#)

= 0.

(2.2)

Since p½ >> 1 we can use the asymptotic form of the Hermitian function in finding the roots of eq. (2.2) and obtain % ---- n+A,,, where n is an integer, and A~ ~ 2~-½(n !)-lp tin'+1) exp(--p½) is a negligibly small correction. Thus within the accuracy of the present discussion we have

/nO(X):

e-iX~

UniX) -~-An \ ~ ] . = n l

~

ezO(n) :-- 2 ( n + z J , ) + l m 2 n + l .

(2.4)

Taking account of ~ l ( x ) b y the perturbation method leads to the following results:

I~(~)

=

I.°(~)

+ Ap-¼ ~-

[V,~/o+~(x)_~n~_~(x)]

~, (n) = 2 n + 1 +p-½[{-(2n+ 1)A--A2].

(2.5) (2.6)

The anharmonicity of fl-oscillations leads, generally speaking, to the corrections of the sanle order as that obtained in eqs. (2.5) and (2.6). Yet the results of interest to us do not, as was indicated in sect. 1, depend in the approximation under consideration on taking into account the anharmonicity of fl-oscillations.

3. y - O s c i l l a t i o n s and Rotation In eq. (1;7) describing ?-oscillations and rotation, where ~f(y, 0,) is given b y (1.5), Ydl(Y, 0,) oc q-t can be regarded as perturbation since q½ >> 1. The solutions of the unperturbed problem [5@0(y, 0,)--A°]9(y, 0,) = 0.

(3.1)

COLLECTIVE EXCITATIONS

445

are the functions ]M %,~K = gn~,~:(Y)AYMK(O,),

(3.2)

corresponding to the proper values

20 Kj ~- 4nv+g+2+q-½[½J(J+ 1)--~-K2],

(3.3)

where K is the projection of the total moment on the symmetry axis of the nucleus and n v is a non-negative integer. In eq. (3.2), A~M are the Wignerfunctions normalised and symmetrized in the usual manner: g AYM~(0,) =

2J+l 16~2(1+6~0)

and the normalised functions

[DYM~(O*)+(--1)YDIM-K(O')]'

g%K(Y) are of the form

q¼ V2F(½K+nT+I) g,#c(V) -- F ( ½ K + I ) ~ ~ e-½V~vIKF(--n,½ K + I , V2), where F (~,/3, Z) is a confluent hypergeometric function. A similar approximation is used in refs. 1,2). In this approximation, K is a constant of motion. Taking into account ¢ the perturbation ~tt~1 leads to the admixture of states relating to other K. Yet, owing to the considerable rigidity of V-oscillations these admixtures are not large and K can be regarded as an approximate quantum number. The concrete form of wave functions in the first approximation of perturbation theory is ]M ~ra'(hIM._ 9YMK+q_t(3~/3)_l{~/l+ (__ 1)]6g° ~+(j, K)[~f~n~¢%_l.K+ 2 1 -JM + VnT+~K-~ 1 9% ,~+2] -- ~/1 + (-- 1)7 0K2 0~_(J, K)

--JM [~/% + ~1 K 9.~,~-2

J~ + ¢%+19~+1,~:-2]},

(3.4)

~2(J K) = I ~ / ( J T K ) (JTK-- 1) (J-I-K+ 1 ) ( J ± K + 2 ) . In particular the wave functions of the ground state and the state with J -- 3 coincide with the nnpertrubed: ~ = 9~ and #80M~ 902, aM. tWO lower states with J = 2 are described b y the functions (~oM :

2 ~ -"~ ~2q 900

-~ 902 2M'

~b022M ----- 902 2M - - ~2' q --t ( 9 02M 0 - ~ 9 12M 0 )"

The eigenvalues of ~'~(V, 0,) taking account of ~ ( y , 0,) yield 0 2%g.r = ,,ln~,gj-q_~ Zl (y, K),

(3.5)

A (J, K) ----~ { [ 1 + (-- 1)]aKo] (K+2)o%2(JK)- (K--2)(1--6g0)m 2(JK)}. t T h e i n t e r a c t i o n of ~ - o s c i l l a t m n s w i t h t h e r o t a t m n s h a s b e e n t a k e n into a c c o u n t in reI. s) for a H a m i l t o m a n d i f f e r e n t f r o m t h a t for t h e collective model.

446

v.I.

BELYAK AND D. A. ZAIKIN

Using eqs. (1.6), (2.5) and (3.4) the wave functions of collective motion ~,, ,b,~:fu(x, y, 0,) can be constructed. The energy values corresponding to them are obtained from (1.9), (2.6), (3.3), and (3.5) and can be represented as

E(n B, n z, K, J) -= }hoo[ep+ % + erot], e~ = 2np+l,

(;)'

e~, =

[-

(4n~+K+2),

(3.6)

A (JK)

erot = P - ½ k_ 1v1 J ( J + 1)

q

and in particular * I

The terms oc p-½ coming from the anharmonicity of the oscillations and taking account of j~2/4 sin2~, in the higher orders (see sect. 1 and eq (2.6)) are dropped in eB and e~. These terms lead to the same shift (of the order of rotational energy) of all levels of a definite rotational band. The term oc p-1 J ( J q - 1 ) is dropped in erot. Taking account of this term leads to the change of the effective moment of inertia b y the magnitude ocp-½ and is inessential in considering deviations from the interval rule. From eq. (3.6) it follows that the ratio of the first fl-vibrational level 0+ to the first rotational level 2+ is E100o __ Eo~

p½,

(3.7)

Eoo02 E21 while the ratio of the first two levels 2+ is

E22 (3.s) E00o2 - - E21 Eqs. (3.7) and (3.8) furnish a simple method for determining the parameters p and q experimentally. E00~2

4. P r o b a b i l i t i e s

-

of E l e c t r i c

Quadrupole

Transitions

The reduced probability of the electric quadrupole transition between the states npn~,KJ and n pn'~K'J', expressed in the units 9Z2e2R2flo2/80~2 is b (E2, n~n~KJ -+ n'pn'~K'J')

I;o

2 g 42~ = D~0 cos ~,+ (D~z+Dj,_~) sin y/V2. * T h e e f f e c t of f l - o s c i l l a t x o n - r o t a t l o n i n t e r a c t i o n o n t h e r o t a t i o n a l l e v e l s p a c i n g w a s c o n m d e r e d b y A S D a v y d o v el a l . * , ~ , 5 ) .

COLLECTIVE

EXCITATIONS

447

Using eq. (4.1) and the collective motion wave functions obtained in the preceding sections let us calculate the ratios of E2-transition probabilities, first of all between states w i t h the same vibrational q u a n t u m numbers np == 0. We shall see t h a t for the E2-transition probability ratios within a rot a t i o n a l band, for b(E2, oo0J --~ 0002 -+ 0000), for example, taking account of the interaction of the rotation with fl and y -oscillations will lead to corrections of the order p-1 and q-1. For the transition probabilities between the levels corresponding to the rotational bands with different K (K = 0 and K = 2) the interaction of the rotation with fl-oscillations also leads to corrections of the order p - i ; while the rotation-y-oscillation interaction is more essential in this case and taking it into account leads to corrections of the order q-½;

n;,

O00J')/b(E2,

b(E2, 002J --~ oooJ')

2J'+ 1 = 5 (2J'20IJ2) 2 b(E2, 0022 -+ 0000) 2J+l × {l+2q-½ [ § +

(4.2)

~/2°~+(J'°)(2J'°2]J2)-°~+(J'2)(2J'°°[J°)8(J'even!l}, 3V'3

(2J'20]J2)

b(E2, 0022 -+ 0000) _ b(E2, 22 -+ 0) -- q-½. b(E2, 0002 -+ 0000) b(E2, 21 -+ 0)

(4.3)

F r o m eq. (4.2) it follows, in particular, b(E2, 0022 --~ 0002) _ b(E2, 22 -+ 21) _= ~ b(E2, 0022 --> 0000) b(E2, 22 --> 0 ) b(E2, 0022 --~ 0004) _= b(E2, 22 ~ 41) b(E2, 0022 ~ 0002) b(E2, 22 ~ 21)

_

1

(1 + 4q-½),

(4.4)

(1 + 2a--~-q-½),

(4.5)

so

b(E2, 0023 -+ 0004) b(E2, 3 ~ 41) b(E2, 0023-+ 0002) ~--- b(E2, 3 -+ 21) = ~- ( l + - ~ q - ½ ) .

(4.6)

L e t us also give the expressions determining the transition probabilities from the excited fl- and y-vibrational levels: b(E2, lOOJ -+ oooJ') b(E2, o o o J - + o o o J ' ) = ½p-½'

b(E2, OlOJ -+ o02J') b(E2, o o o J ~ o o o J ' ) = q-½

(4.7)

In paticular we have b(E2, 1000 ~ 0002) _ b(E2, 0B -+ 21) = = ~p-½, b(E2, 0002 -+ 0000) b(E2, 21 -+ 0)

(4.8)

b(E2, 0100 -+ 0022) b(E2, 0 r ~ 22) = 5q-½. b(E2, 0002 ~ 0000) -- b(E2, 21 -+ 0)

(4.9)

448

V. I. B E L Y A K A N D D . A. Z A I K I N

The transitions from excited fl-vibrational levels to the rotational band K = 2 are less probable by an order than the transitions to the ground rotational band. Now for excited y-vibrational levels the reverse is the case. In particular we obtain b(E2, 1000 --> 0022) _ b(E2, 0B ~ 22) b(E2, 1000 ~ 0002) b(E2, 0~ ~ 21)

q-½,

b(E2, 0100 ~ 0002) b(E2, Or ~ 21) b(E2, 0100 ~ 0022) ~ b(E2, 0 v ~ 22) ~ ¼q-½"

(4.11)

The ratios (4.10) and (4.1 1) are interesting from the viewpoint of determining the nature of the level 0+ experimentally (see sect. 5). From eqs. (4 2)--(4.11) it is evident that the E2-transition probability ratios depend oD_ two parameters q and p which are simply connected, through eqs. (3.7) and (3.8), with the relative level spacing 2+ and 0+. Having determined q and p for each nucleus from experimental data on the position of these levels one can calculate the E2-transltion probability ratios between the collective states of this nucleus. The results for the ratios (4.3)-- (4.6) and (4.8) are listed in the table. It should be noted that eqs. (4 2)-- (4.9) contain the terms of such an order that t h e y are not affected by taking into account the anharmonicity of fl- and y-oscillations. 5. D i s c u s s i o n of R e s u l t s

The relative energy level spacing depends on the rigidity parameters for the quadrupole oscillations p and q the values of which can be calculated by eqs. (3.7) and (3.8) from experimental data on the levels 0+ and 2 +. In particular, the deviations from the rule of intervals in rotational bands can be calculated using the quantities p and q thus obtained (see eq. (3.6)) and are in good agreement with experiment. Since the levels 0+ have not yet been detected in many nuclei, it is possible, in general, to evaluate, proceeding from the experimentally observed deviation from the rule of intervals in rotational bands and the position of the level E22. the value of the parameter p and indicate the energies/or which the level 0B+ can be expected. The results of this evaluation the accuracy of which is not high, of course, are listed in table i and marked by an asterisk. The ratios of electric quadrupole transition probabilities also depend on the parameters p and q and can be calculated by eqs. (4.2)--(4.11). It is clear from the table that the ratio (4.4) thus calculated is in good agreement with experiment 7). Somewhat worse agreement exists for the ratio (4.3) 7, 8-1o). The transition probabilities from the level 0+ make it possible to establish the nature of these levels which can be both fl- and ~,-vibratlonal. In Er 16s the

0.122 0.123 0089 0 080 0.0867 0.0806 0 0798 O 0931 0.1001 0 1111 0 1225 0 1372 0 155 0 187 0 0575 0.0528 0 0528 0.0475 0.0435 0.0447 0.0442 0 0429

Sm 152 Gd 15' Gd Ise Gd Isa D y 16° E r lea E r 168 H I l~s W 182 W is4 W Ise Os lss Os Iss Os Ig° T h ~a T h ~a° Th ~3~ U ~Sa U a84 U 238 Pu 288 Pu ~40

1.00 1.031 1 02

1.092 0 998 1240 1.182 0 966 0 7874 0.8224 1 48 1.222 0.904 0.733 0 768 0.633 0.557 0.965 1.060 0.790 0.868

E~z (MeV)

~ 0 9* ~1" ~1" ~1" 0 805 ~> 1 5* 0.935 0.940

1.086

0.685 ~1" ~ 1 3* >2* >2* 1.462 ~2" ~2" ~2" ~1.2'

EoZ (MeV)

22 4 23.4 23.8

9.0 8.1 139 14 9 11.2 9.8 10 3 15.9 122 82 60 5.6 4.1 3.0 16.8 20.1 15.0 185

E2._22 E,I

~15' ~20" ~20" ~20" 18 5 >~ 30* 21.1 21.9

7.0

5.6 ,~ 9* ~15" >25* >25* 18 1 ~20" ~25" >25* ~11"

EoB E,I

~0.15" ~ 0 1" ~ 0 1" ~0.1' 0 13 > 0 06* 0 12 0 11

0.36

0 45 ~ O 3* ~02" ~<0.1" < 0 1" 0 14 ~0.1" ~0.1" ~0.1" ~0.2"

b(E2, 0B --->21) b(E2, 21 ---> 0)

b(E2, 22 -+ 0)

0.04 0.04 0.04

0.11 0.12 007 0 07 0.09 0.10 0.10 0.06 0.08 0 12 0 17 0.18 0.25 0.33 0 06 0.05 0 07 005

theor, 01

0 0 3 3 1 0 006

0.07 t 0 03 0 06310.013

0 0 3 8 1 0 009 0 0 4 7 1 0 007

0.04 t 0 . 0 2 0 04 t 0 02

004 t002 0.03 t 0 . 0 2

0.03 i 0

exp.

b (E2, 21 --->0)

1.68 1.67 1.67

2 210.3 18 1.9iO 9

2 07 2 13 1.84 1 81 1.94 2 02 1.98 1.79 I 90 2 13 2 38 2 45 2.83 3 34 1.77 1 71 1 81 1.74

1.5±0.2

2 110 3

1.8:]=0 1 1 8103 2 5-4-0 3 2 7±0.6 2.7±0 2 6.8tl.2 2 510 5

2 0±0 1 2.0:J:0 3 1.8+0 3

exp.

theor

b(E2, 22-+0)

b(E2,22---~21)

0.07 0.07 0 07

0 1O 0 11 0.08 0 O8 0 09 0 I0 0.10 0.08 0 09 011 0 13 0 13 0 16 0.20 0.08 0.07 0 08 0.08

0 57 0.56 0.56

0.82 0.86 0.67 0 65 0 73 0 78 0.76 0.63 0 70 086 1 02 1.06 1.31 1.65 0.62 0.58 0.65 0.60

b(E2,3-+21)

--> 41)

b(E2, 3

b(E2, I~2 --)-41) b(E2 22--+21)

The asterisk designates the estimates of the q u a n t i t m s belonging to the ~-vlbrational level 0 + and obtained f r o m the deviation from the rule of intervals in rotational bands.

E2I (MeV)

Nucleus

TABLE. 1 Calculational results

s~

o

t~ N

t~

o

450

V. I

BELYAK

AND

D. A

ZAIKIN

first level 0+ (1.462 MeV), for example, lies roughly twice as high as the second level 2+ i.e. where there should be the 0~+ level. Therefore it is quite likely that this level is y-vibrational. We have a similar case for Os 18s. The question on the nature of the 0+-state can be solved experimentally: first, by measuring the E0-transition probability from this state to the ground state 1) and second, by measuring tile E2-transition probabilities to the states with spin 2+. As is clear from eqs. (4.10) and (4.11) in case of the fl-vibrational nature of the 0+state the E2-transition probability from it to the state J = 2, K = 2 is by an order smaller than to the state J = 2, K = 0, while in case of the 7-vibrational nature the opposite is true. It is worthwhile to compare the results of the present paper with those given by the non-axial rotator model 6) with small non-axialy (~0 < 15°) since the latter explains well the experimental data on the low-lying levels of nuclei with E22[E21>> 1. In case of small deviations from axial symmetry, the expressions for energies and transition probabilities can be expanded in the powers of ~0; if only the initial terms of these expansions are taken we have E22 1 E21 ~ ff ~ 2702,

Edl E~ ~ ~

(1-~o

(5.1)

~) ~ ~

(17- )~

b(E2, 22 -+ 21) ~( b(E2, 22 - + o) ~ ~-°(l+sr°~) b (E2, 22 - + 0) 1 b(E2, 21 -+ 0) ~ )J°2 m 2if' etc.

,

(5.2) ~),

1+

(5.3) (5.4)

Eq. (5.3) coincides with (4.4). The explanation of this identity of the results of the present work and the non-axial rotator model is that axial nuclei give rise to the dynamic deviations from axial symmetry 72=/= 0 already in the ground state owing to 7-oscillations. Yet in several cases the present investigation leads to results different from the corresponding results predicted by the non-axial rotator model. In particular, the corrections to the rule of intervals in rotational bands due to the 7-oscillation-rotation interaction differ from the corrections arising as a result of small static deviations from axialsymmetry. Indeed, from eq. (3.6) it follows/or example, that

E41

-V

~),

(1-- 14

which does not coincide with eq. (5.2). Furthermore, the reduced E2 transition probabilities between the levels of different rotational bands for axial nuclei are

COLLECTIVE

EXCITATIONS

451

in the first order in 1/# twice as large as for non-axial nuclei. This can be exemplified by the ratio b(E2, 22 -+ O)/b(E2, 21 -+ 0) (see eqs. (4.3) and (5.4)). It should be noted that the experimental data (rather scarce, it must be admitted, for the region of nuclei under consideration) available for this ratio are explained better in terms of the theory of non-axial nuclei. In conclusion we shall emphasise that the results of the present investigations are obtained under the assumption of the large rigidity for quadrupole oscillations q~ ~ E2jE21 >> 1, p½ ~ Eop/E~l >> 1, and in cases when E2~/E2, and Eop/E2a are not large good agreement with experiment cannot be expected. There are also other restrictions of the present discussion, such as the hydrodynamic dependence of nuclear moments of inertia on fl and 7. The authors are grateful to Prof. A. S. Davydov for his interest in the investigation and stimulating discussions. References S. Davydov, Nuclear Physics 24 (1961) 682

2) B L Birbralr, L K Peker and L. A Shy, J E T P 36 (1959) 803 a) T Tamura and T Udagawa, Nuclear Physics 16 (1960) 460 4) A A 8) A 6) A A

7) 8) 9) 10)

I) R B O.

S Davydov and G F FIllppov, J E T P 33 (1957) 723, A Chaban and A S Davydov, J E T P 33 (1957) 547 A Chaban and A. S Davydov, Nuclear Physics 20 (1960) 499 S Davydov and G F Flhppov, J E T P 35 (1958) 440, S Davydov, Izv AN SSR, Ser Flz 23 (1959)792 M Van Patter, Nuclear Physics 14 (1959]60) 42 R Shehne and H L Nielsen, Nuclear Physics 16 (1960) 518 Elbek, M C Olesen and O. S Skalbreld, Nuclear Physics 19 (1960) 523 Nathan and V I Popov, Nuclear Physms 21 (1960) 631