Relative phase of the M1 and E2 radiations in the 2+ → 2+ transitions in even nuclei

Nuclea~ Phys,cs 30 (1962) 579--598; (~) North-Holland Pubhshing Co., Amsterdam N o t to be reproduced b y photoprint or mtcrohlm without written pernnssxon from the pubhsher

R E L A T I V E P H A S E O F T H E M 1 A N D E2 R A D I A T I O N S I N T H E 2 + --., 2 + T R A N S I T I O N S I N E V E N N U C L E I TARO TAMURA AND HIROSHI YOSHIDA Department of Physics Tokyo University of Education, Tokyo, Japan Received 14 August 1961 Abstract: The relative phase of the M1 and E2 radiations which are mixed in the electromagnetic

transition between the first and the second excited 2+ states in even nuclei lying in the vibrational and rotational regions is calculated theoretically. Models are chosen in which the states due to quasi-particle excitation are coupled to the orgmally pure collective states. It is then found that the relative phase is primarily determined by the sign of the magnetic moments of the admixed quasi-particle states, and that m this way agreement, with experiment is obtained in most of the cases considered. It thus seems that the configuration of the lower lying quasi-particle states ~s very important m determining above relative phase.

1. I n ~ o d u c f i o n

Several models 1- 5) were proposed to describe nuclei which b e l o n g to the vibrat i o n a l region a n d were f o u n d to explain to some extent the general features o f the experimentally k n o w n spectra. The q u a n t i t a t i v e agreement with experiments, however, is n o t necessarily satisfactory t a n d we certainly need deeper investigations. T h e purpose o f the present paper is to discuss, i n order to clarify the n a t u r e o f these nuclei, a feature o f the electromagnetic t r a n s i t i o n s between the first a n d the second excited 2 + states. As is well k n o w n , this t r a n s i t i o n is a mixture o f M1 a n d the E2 radiations, a n d o u r m a i n c o n c e r n is the ratio 6 or rather the relative phase g/16l o f these two radiations. The models so far p r o p o s e d m a y be classified i n t o two categories; i n one o f them 3 - s) o n l y the collective m o d e o f excitation is considered, while i n the other 1, 2) the singleparticle m o d e o f excitation is considered to be coupled to the collective one. I n the former case n o M1 r a d i a t i o n is admixed with the d o m i n a n t E2 radiation, because there can exist n o precession o f the magnetic m o m e n t a r o u n d the total a n g u l a r momentum. M1 r a d i a t i o n is certainly admixed, o n the other h a n d , i n the second model, a n d i n fact e.g. Raz 2) m a d e a fairly detailed c o n s i d e r a t i o n o n it. His calculation is, however, t Perhaps the recent work of Davydov and Chaben ~) might be an exception ip this sense, as it explains very well many known experimental data. It is the opinion of the present authors, however, that even this theory needs further improvement probably by taking into account the y-vibrations. 579

580

T A R O T A M U R A A N D HIROSHI YOSHIDA

limited to a highly idealized case and thus does not seem to be precise enough to discuss the relative phase mentioned above which varies from nucleus to nucleus. The calculation of the present paper will be based on the model proposed some time ago by Komai and one of the present authors (T. T.) 4). This model was applied to the spectrum of Cd H4 and fairly good agreement with experiment was obtained. Therefore, although it is not meant that this model is a very conclusive one, it may be worthwhile to base our argument on this model.

-05

~8~5

t

W1B2

NIO01

R~X~(~Pd~°° jz )se7o

x ~ ( ~ Ta2° ' x2 2°Y

50

I

8(

~2s

s~

V

Sm~5, ~ 0 90 o~6° 100

Os

0190 (~,jpt196 T

194

0S186

N

RJOO

05

~74,

~g8

Fig. 1. Experimental values o f the m i x t u r e ratio 6 o f the E2 a n d the M 1 r a & a t l o n s m the transltion between the first a n d t h e second excited 2 + states in even-even nucleL T h e ordinate is r/ = (~/16[) (1 + ]6[) -1, while t h e abscissa is the n e u t r o n n u m b e r N. This figure is prepared by Sakal 7).

This model belongs to the above first category and thus, as it stands, does not result in any M1 admixture. We are thus forced to take into account the coupling of the single-particle modes to the collective one. The merit o f starting from T K lies, however, in the fact that the main features of the excitation spectra, e.g. the deviation from the simple harmonic spectrum, is already explained within the framework of the collective excitation and thus the effect of the single-particle model could be taken into account by a simple perturbation calculation, which makes the argument fairly easy. After summarizing the experimental data in sect. 2, the calculation of 6 for nuclei which belong to the vibrational region will be given in sect. 3.1. As is seen in sect. 2

RELATIVE PHASE OF THE M1 AND E2 RADIATIONS

581

there exists a fair amount of experimental data on 6 also for nuclei which belong to the rotational (and the transition) regions. Discussion of these cases is given in sect. 3.2. based on the model considered by T a m u r a and Udagawa 6) for the description of rotational nuclei, the basic idea of which is similar to that of TK. Comparison with experiments is made in sect. 4, and some discussion is given in sect. 5. 2. Experimental Data The ratio 6 of the probability amplitudes of the E2 and the M1 transitions between the first and the second 2 + states in even nuclei is defined as c5

x/3 (~) = 10 '

(1)

where hco is the energy difference of the two 2 + states. The experimental data are recently summarized by Sakai 7), and the result is reproduced in fig. 1. In fig. 1 the abscissa is the neutron number N, while the ordinate is taken as t/ = (~/1~1)/(1 + 101L following the convention of Sakai who found that this way of plotting makes experimental situation transparent. Because of the definition of 6 in (1), t/is close to zero when the M1 transition is weak, while it tends to +__1 when the M1 transition predominates over the E2 transition. The sign oft/, which we denote by e(i.e, e = ,#It/I) coincides of course with that of 6(i.e. e = 6/[61), and it will be worthy to note, as is seen from fig. 1, that the number of elements which give positive e exceeds the number of elements which give negative e. This feature will be explained m sects. 3 and 4.

3. Formulation of the Calculation 3.1. V I B R A T I O N A L

NUCLEI

As was explained in the introduction we base our calculation on T K for the vibrational nuclei. This model starts from the model of W J, in which the collectwe states are specified by three quantum numbers: the seniority number as defined by Rakavy s) A; the number of nodes in the fl part of the wave function n and the spin L (We consider exclusively states with positive parity and thus the parity will usually not be specified in the following.) Thus any WJ state is described as I(An)I), and in particular the ground, the first and the second 2 + states, are described as I(00)0), 1(10)2) and 1(20)2). In TK, as well as in WJ, there is no M1 transition between the two 2 + states, and to get a finite M1 contribution one has to couple the single-particle excited states to the collective one. We treat this coupling in the lowest order perturbation calculation, where the single-(quasi-) particle motion is described in the fashion of Kisslinger and Sorensen 9).

582

TARO TAMURA AND HIROSHI YOSHIDA

The Hamiltonian which describes the coupling is given by Hin t

= --

~-~

2~,

^ QM, * QM

(2) A

where QM is the collective quadrupole operator, while the relevant part of QM is given as QM =

~

jmj'ra"

M + + qjm;j,ra,(UjVj,"]-Uj, l).l)(O~jm~j,m,"~-J~jraO~j,ra, ).

(3)

Here qjm; M j'=" = Qm[ r2 Y2MIj'm'), while otj+ and/~j+ (~j a n d / ~ ) are creation (annihilation) operators of a quasi-particle in a state lj>. The quantities u~ and v~ are defined by

uj2 =

v, = ½(1

-

(4)

with Ej =

2.

(5)

ej is the energy of a free single-particle stateL while 2 and A are the Fermi energy and half the energy gap. Each state of T K is a linear combination of WJ states and (An) are no longer good quantum numbers. As the anharmonic term introduced in T K is considered, however, to be weak, the T K states can still approximately be specified by these quantum numbers, and in the following the states [(An)I) are understood to mean TK, instead of WJ states. Consider a state with spin /, I(jlj2)2: (a'n')I' ; IM),

(6)

which is composed of a collective T K state J(A'n')I') and a state with spin 2 of a pair of quasi-particles in orbitals IJa) and ]J2). Because of the Hamiltonian (2), these kinds of states are coupled to each purely collective T K state and the resulting state ](An)I)' may be written as

I(An)I)' = I(An)I) + ~_, b((An)I ; (A'n')I' ; (jtj2)2)l(j,j2)2; (A'n')I' ; I).

(7)

jlJ2F

The expansion coefficient b, which is considered to be small compared to unity, is given by

b((An)I, (a'n')I' ; (jr j2)2) = (_)x+r x((A'n')I'llQll(An)I)QxllQllJ2)(uj~v,~+uj~v~,), x/20(2I + 1)(E((A'n')I') + E jl + E,2- E((An)I))

(8)

where E((An)I) is the energy of a T K state ](An)I). Because of a term E((A'n')I') in the denominator, b will be a small quantity if this E((A'n')I') is very large. In the following numerical calculation, in which we are

RELATIVE

PHASE

OF

THE

MI

AND

E2

583

RADIATIONS

interested only in the lowest two 2 v states, the state [(A'n')I') may be limited to the ground state t, i.e. we set I ( A ' n ' ) I ' ) =- 1(00)0). Then the wave function of these two 2 + states, 121) and 122), can be written as IZa) = 1(10)2)+ Y'. b((10)2, (00)0; (Jlj2)Z)lOljz)2; (00)0; 2), JU2

(9.1)

122) = 1(20)2)+ ~] b((20)2; (00)0; (jlj2)2)l(jxj2)2; (00)0; 2).

(9.2)

JlJ2

In calculating the matrix element of the operator ~J~(M1) between the two states (9.1) and (9.2), we first note that the matrix element between the first terms on the right-hand side of these two expressions vanish, because this is nothing but the M1 matrix element between two T K states. Further the terms which come from the first term of (9.1) and the second term of (9.2) (and vice versa) vanish because of the following reason. ~ ( M 1 ) consists of two parts; one is collective and the other is of quasiparticle, and they are respectively zero and two quasi-particle operators. As the first term o f (9.1) is a zero quasi-particle state while the second term of (9.2) is a two quasiparticle state, the zero quasi-particle part of the operator gives no contribution to the matrix element. Further, it is easily seen that the two quasi-particle part also contributes nothing because of the angular momentum selection rule. In the term which comes from the second terms of (9.1) and (9.2), we first note that the contribution from the collective part of ~J~(M 1) vanishes, because it is proportional to the magnetic moment of the T K ground state which vanishes itself. We are thus left only with the contribution from the quasi-particle part of ~ ( M 1 ) , and using the explicit from of b of (8), we finally get the desired matrix element as (22[l~(M1)[[Zt>

=

E

b((20)2, (00)0; (j'xj'2)2)b((lO)2, (00)0; (jxj2)2)((j'xj'z)2ll~(M1)ll(jtj2)2)

JlJ2fllJ'2

=: (Z2/IO)((20)211QII(OO)OX(IO)211QII(OO)O)(k~)~'(eh/2Mc)Bj~,2;jIj~,

(10)

with •t

Bj'lJ'2; JlJ2

=

~

.t

.

A

.

(u~ ~o, 2 + v j , u, ~)(u~ o~ + v jr u , 3 0 ~ IlallJz)(JallallJz) ~/(1 + Oj,,,,2)(1 + (~JIJ2)(Ej,, ..~ E j. 2 - E((20)2))(Ej, + E,2 - E((10)2)) + ( _ )Jl +,'2{o(j~jx) W ( . j l j , 2 2 ; lj2)(us,uj, + vj~vj.,)3j2,, 2 . . . . . W[.JiJ2 . . . . . 22 , lJl)(ui2uj,, + v j , vj,,)(Sj,~,~ -{'gt.J1J2)

(11)

+ gO'2ja)W(j'2jx 22; lj2)(uj, uj. z + v,, Vj,2)(~j2j, l + OO'2j2)W(j'2j2 22; ljt)(u,~ u j,2 + vj2 vj,2)3,1i,1}, 0 0 7 ) = (J'll~ff~(M1)l[J)/((¼~)~(eh/ZMc)). t On this point see a further argument given in sect. 5.

(12)

584

TARO TAMURA AND HIROSHI YOSHIDA

In eq. (11), E((00)0) is put equal to zero. In evaluating the matrix elements <(00)011QII (10)2> and <(00)011QI1(20)2>, the T K states I(00)0>, 1(10)2> and 1(20)2> are expanded as linear combinations of WJ states I(An)0> and [(An)2> with N < 5, where N = A+2n. This mixing is caused because in T K a new anharmonic term

kl(fl-fll)

cos 3~,

(13)

is introduced, where k 1 is taken to be positive to ensure that a prolate deformation is favoured over the oblate one. I f the magnitudes of various parameters which appear in the original WJ Hamiltonian, as well as in (12), are taken so as to give the best fit to the spectrum of Cd 11., it is found that ((10)2IIQI[(00)0> = 2.35/K,

<(20)2IIQII(00)0> =

-0.303kl/K 2,

(14)

with K = (BC/h2) ÷, B and C being defined in WJ. Of course Cd ~ 4 is a special case of vibrational nuclei, and the right-hand side of (14) may vary from nucleus to nucleus. Nevertheless we hope that (14) will give a correct order of magnitude in those cases too, and particularly that the sign given by (14) will be quite reliable. In evaluating the matrix element of ~ ( E 2 ) between the states (9.1) and (9.2), we note that the matrix element between the first terms on the right-hand side of these two expressions is non-vanishing and compared to it the contributions from the second terms are negligible. Then

(¼n)ZeRZo<(20)21lQll(lO)2> = _0.730ZeR2o/K,

((20)2119X(E2)I[(10)2> =

(15)

where the same T K wave functions 1(10)2) and 1(20)2), as used in obtaining (14), are used. Using (10), (14) and (15), the ratio 6 eq. (1) is now obtained as

J=C1/

Z

BJ'I~'2,J,~.

(16)

J'lJ'2JlJ2

In (16) the constant

C1-14"55 (K) 2 Mczh°gZR~ kl

(17)

(hc) 2

is positive definite, and so the sign of 6 is the same as the sign of the series ~Bj.,j,2, ~,J2. The quasi-particle pair state I(jlj2)2), m a y be classified into two groups, one in w h i c h j l = J2, and the other in which j l # J2. Because of the terms Ej, +Ej2, in the energy denominator of Bj.I~,~;j,j2, and because of the geometrical factors in the numerator, it is found that the states which belong to the first category usually give the most important contribution to the above summation. In particular the contribution from the terms with j l = J2 = J'~ = J ' z , is 4(ujlv~,)

2

Il • " • ,2

~j,J,, Jl~, = (2e~,- E((20)2))(2Ej,-e((10)2))

V

10j~(jl

3 " " +l)(2jl+l)g(J'J1)"

(18)

R E L A T I V E P H A S E OF T H E M 1

AND E2 RADIATIONS

585

So long as 2E 1 > E((20)2), as is usually the case, the sign of (18) is the same as the sign of g(JlJl), and as is seen from the definition (13), g ( i l j l ) is proportional to the magnetm m o m e n t (Schmidt value) of a nucleon m a orbital [J) in the j-j coupling shell model. As is well known the Schmidt value is positive for approximately three quarters o f all the possible cases, i.e. it is positive if [j) is a proton, except for the case with [j) = ]p½), and it is positive if [J) is a neutron w i t h j = l - ½ and is negative if j = I + ½. This fact may explain why the number with positive 6 exceeds that of negative 6, as is seen from fig. 1. The terms in the s u m m a t m n which are o f next importance are those with j~, = j , and J2' = J2 ( b u t J l ~ J2)" Their contribution is given as

(u ,, v j=+ u ,=v,,)2 ]l z BjIJ2;

j132

(Es~ + Ej~ - E((E0)2))(Es, + Es~ - E((10)2))

(19 )

+ 1 ) # ( J l J , ) + Jz(Ja + l ) + 6 - J l ( J , + 1)#(jzj2)t" / ~ 2x/30j,(j, 7 ~ + 1)(2j, + 1) 2x/30j202 + 1)(2j= + 1)

x [ j , ( j ~ z

These terms are o f a nature, similar to that ofBj,j~ s,~ as far as the sign is concerned, although it is not so definite as in (18). This fact may further support the above conclusion concerning the sign of 6. The terms other than (18) and (19) are also included in the numerical calculations discussed in sect. 4.2, but they are found to be quite small in most of the cases. In some cases like Sn a16 and Ni 6° in which the second 2 + state might better be considered as a two-quasi-particle state, rather than a m e m b e r of the two p h o n o n states, we should describe 122) as 122) = I(j~j2)2)

(20)

instead of (9.2), where [(jlj2)2) means a state in which Ejl + Ei= is the smallest among all the possible two quasi-particle states. In this case 121) is still expressed by (9.1), and after some calculation we get (22119~(M1) 112,) = y, b((10)2, (00)0; (j,jz)2)((jlja)2[I~(M1)N(j'~j'2)2) J'lJ'2 .t

.-x

,!

= ,Z((10)211QII(00)0) Y. (JlllQllJ2)(u'"%=+v"lu'") (_y',+J2 s',s'z Es, , + Es, =- E((10)2)

(21)

x ( ( j r []9~(M 1)[]j~ ) W O , j ' ~22; lj'2)(uj. ` us, + vs, , vsl)6~=.c= "

+ (Jl II~(MI)IIJ~)WOlj'2 22; 1j~)(us, ~us, + vs,=vs,) s2J', + (je[19~(M1)[lj'~)W(j~j'~ 22; ljz)(Uj, uj,, +vj=vs,,)aa~a, 2

+ The

•

quantity

II

.!

•

.v

,

,v

1)1112>W(s s 22, L, ,)(uj,2 uj2 + vj,

3.

(2dl~(E2)l121) is obtained (from (21)) simply by replacing

586

TARO TAMURA AND HIROSH1 YOSHIDA

(jxIIgX(M1)IIj'a) by , W(jlj'~22; [J'2)etc. by -- W ( j , j ' t 2 2 ; 2j'2) etc., and (uc~uy ~+ vy,,v~) etc. by (u.r ~u j, x - V.r,V.~) etc. 3.2. ROTATIONAL NUCLEI In the rotational region the 12~+) state is considered as the second member of the ground rotational band, while the [22+) state is considered as the ground state of the ~-vibrational band. These states cannot, however, be pure states, if there exists the so-called rotation-vibration interaction, some arguments about which have been given by Tamura and Udagawa 6). It is found there that, if the above pure states are written as ~b(21) and ~b(22), then the TU states 121) and [22> are expressed as 121) = ¢ ( 2 , ) + W ~ a p ~1b ( 2 2 ) , --

1

122) = ~,(22)--~/2a p - ~ ~b(21),

(22.1) (22.2)

where p = E(22)/E(21), while a is a positive constant of the order of unity. It is easy to see that there can exist no M1 transition between two states (22.1) and (22.2), and thus we are again forced to admix the quasi-particle excitation mode to (22). Here, too, the admixture is caused by the interaction H i . , = --½Z Z Q,*Q,.

(23)

This time, however, Q, and 0 , are considered to refer to the intrinsic coordinate sysA tern, and so Q, is expressed as Q, = Z ,t~l /.

? fib= t + fla, =u=), v~, + va, u~=)(otu,

(24)

Dl~2

I2z specifying Nilsson states zo), while q[,l al~e~2

:

(~llr2y2,l[22).

(25)

The calculation of matrix elements of ~O~(M1) and ~J~(E2) can be performed in essentially the same fashion as for the vibrational nuclei, and thus we show only the final result. Denoting by [2'1) and 12'2) the two 2 + states after the single particle motion is admixed, the matrix element for ~ff~(M1) becomes (2~[19~(M1)II2~) = 3~/3 ~/5 aZ2 ~ p

(g2lO2lo°)2 ~ C~'1~"; ~ ' ~ '

(26)

where j_,

°

~n[21

n[2]

= (uw~ va,~ + va,, ua.~)(u m va~ 7-,,m,,o~j~u.,a.~,tm~ (t2~. t2[l~Y~(M1)it2x Q2)(E(a[ ga[)-E(22))(E(ta, a2)-E(21))

(27)

RELATIVE PHASE OF THE M1 AND E2 RADIATIONS

587

Here E(21) and E(22) are energies of the TU states [21) and 122), while E(O102) denotes the energy of a state 1t2102) in which the collective motion stays in its ground state, while a two quasi-particle state withO 1 + 0 2 = 2 is excited. #o and #2 means the fl-y part of the wave function ~k(21) and ~b(22), respectively. The matrix element of ~IR(E2)between states 12'2) and 12'1) may again be replaced by that between ~/(22) and ~b(2t) and the result is (2~11~(E2)112~)= V]-O -7

3ZeRZo(fl sin ~), 4~r

(28)

which is positive definite. Combining the results (26) and (28), it is seen that the sign of 6 is determined by the sign of the series which appears in (26). Among the summands C~.,a,2, ma~ the dominant term is again those with O't = f21 and f2' 2 = f22, for which we get 2

[2]

2

(ua, va2+valuu2) (qa~a2) (I21 f22[~(M1)lI21 ~'~2). (29) Ca,n,;a,a2 = (E(f21 O2)-- E(22))(E(01 0 2 ) - E(21)) long as E(~1~'-22) > E(22), the sign of Ca,a2; a~a2is the same as that of the last

So factor of the right-hand side of (29) which is again just (except for a positive constant coefficient) the magnetic moment of the quasi-particle state [~21f22). It is expressed as (f2~ f221~(M1)II21 f22) = (g2~lTf~(M1)101)+ (I221~02(M1)102)

~-3 ( e-2--~c)[2a,+(gs-O,){ ~

~

AI "t-S l =*Q1 ) ' 0

la,,a,'2S1+

~

la,~a~12S2}a,±a: =2] •

- - ,'12 -I-Sz = ~'22'>-0

(30) Now the situation is rather clear. In (30) aj,~ is the coefficient of expansion of a Nilsson state 101) in terms of the Mayer-Jensen states I1~A~ S~ ) where A~ and S~ are the Z-components of the orbital and spin angular momenta. Thus ~a~ + S, = a, ~ 0 [a~,a~l2 = 1. On the other hand $1 can be +½ or - ½ and thus the absolute magnitude of the quantity in the curly bracket in (30) can be at most unity and is usually much smaller than that. On the other hand the first term 2g~ in the square bracket of (30) is 2.0 when the configuration is of proton and in that case usually dominates the second term. Therefore we can expect to get positive 6, except for the rare cases in which quasineutron or (unusual) quasi-proton states appear quite low, so that the negative contributions from the above second term predominate. This fact may explain why the experimental 6 given in fig. 1 is always positive for rotational nuclei. (In the actual calculation treated in sect. 4.2, it is found that the contribution from the second term is not necessarily very small, but that the contributions are predominately positive).

4. Comparison with Experiments In this section we discuss only the sign of 6. Some discussion of its magnitude is given in sect. 5.

588

TARO TAMURA

AND

HIROSHI YOSHIDA

4.1. VIBRATIONAL REGION 4.1.1. The Pt, H g region We first consider Pt isotopes for which it is k n o w n that 6 is positive for both Pt 194 and Pt 196, as seen in fig. 1. In evaluating 6 theoretically by using (16), we first have to k n o w the energies which appear in the d e n o m i n a t o r o f B~,v:, ' J,J2 as seen in (11). F o r that purpose we first use the result o f the recent calculation by T a m u r a and U d a g a w a 11), and the lower two quasi-particle states thus obtained are shown with dotted lines In fig. 2. These states have spin 2 and the configuration o f each quasi0850

. . . . . . . . . . . (d~s'e) 2 + P

0 730

...........

(d~) 2 2 +

0621

-

2+

536 . . . . . . . . . .

0328

2+

0 000

O+ 7ePt I~4

(f~_)~2+ N

P

0850

. . . . . . . . .

(d~s½)2+

0730

. . . . . . . . . .

(d~)~ 2 ~ P

0 685

2+

0354

2t-

P

01-

0000 7apt 1~e

Fig. 2. Level spectra (a) for Pt194and (b) for Pt 196.Experimentally known collective 2+ states are shown with full lanes, whale the two quasi-particle 2+ states are shown with dotted lines. To each of the latter, there are shown the considered configuration and the label P or N to mean whether it is a quas~proton or -neutron state. Energies are given in MeV. These remarks apply also to all the following figures. particle state is shown on the right-hand side o f the corresponding line. Further the label P or N attached to the level specifies whether this is a quasi-proton or quasineutron pair state. I n fig. 2 are also shown the states 121 +> and 122 + > with full lines which are just the experimental values. To use them for E((10)2) and E((20)2) is certainly not correct, because E((10)2) and E((20)2) are obtained without taking into account the coupling between the collective and qffasi-particle motions. The shift due to this coupling is considered, however, to be small and, as long as we are interested mainlyin the sign o f 8, the error introduced in this way would not be serious. N o w looking at fig. 2(a) for Pt 194 we note that the two quasi-particle states which appear are only p r o t o n states, and by using (16), (17) and (11) it is easy to see that we get 6 > 0 f3r this case in agreement with experiment. I n fig. 2(b) for Pt 196 a twoneutron state appears, in addition to the p r o t o n states previously appeared in Pt 194.

589

RELATIVE PHASE OF THE M1 AND E2 RADIATIONS

This neutron state has, however, a ( f ~ ) 2 configuration for which j l - ½ and thus gives a positive contribution to ~5. This fact shows that we can expect to get ~5 > 0 for Pt 196 too, and further that r/pt,~ > r/pt,9, is also in agreement with experiment. For Hg 19s the [22 +) state appears very much higher than twice the energy of the 121+) state and consequently, it is not clear how to identify this state. We thus would not go into detail for this case. =

4.1.2. The Sn, Te, Xn region In Sn 116 it seems that the 122+) state is to be considered as a two quasi-particle state rather than a member of the two surfon states 9) and in such a case we could use (21) instead of (16). The quasi-particle states as obtained by using the parameters of KS, as well as the experimental 121÷ ) state, are shown in fig. 3, and insertion of these values in (21) proves that Li > 0, in agreement with experiment. 3 113-3016' 2898 2827

{l'l~J') ~

2+

N

"(d_3d_5)2+ N ~ 2

. . . . . . . . . .

~

. . . . . . . . . .

\(s*dS)2~

N

2226

(d~) 2

2~o8

(s~_d~)2+ N

2+ N

1274

2+

O÷

0 000 ~oSn EIO

Fig. 3. Level spectrum for Sn u6.

For Te, it is known experimentally that r/Te12~ > r/r~126 > 0. In fig. 4(a) the relevant states for Te 122 are given in the same fashion as in fig. 2. Noting that the lowest quasi-particle state is of a proton and the second one is of a neutron w i t h j = l - ½ we expect to get positive c5, which is in fact proved by calculation using (16), (17) and (11). In Te 126 the (g~)2 state appears lower than the 122+) one as is seen in fig. 4(b). The parametric values of KS used here, are, however, those appropriate to the Sn isotopes and thus we may expect that this (g~)2p state actually appears somewhat higher than that shown in fig. 4(b), but still lower than the ( h ~ ) 2 N state. Then the positive contribution from the (g~)2 p will exceed the large negative contribution from the ( h ~ ) e N state, resulting in positive ~, but the effect of cancellation would .lead to r/Te,~6 < ~lre,22- Thus the result is quite consistent with experiment.

590

TARO

214 . . . . . . . . . .

TAMURA

AND

HIROSHI

YOSHIDA

(d~)22"1- N (h~)22 + N

2 06

136 . . . . . . . . . .

(g~)22+ p

126

2+

0 57

2+

I 40 I 36

2+ - (g~)22+ P

065

0 O0

O+

s;r~,22

2+

000

0~-

52T,2 ~

Fig. 4. Level spectra (a) for Te lss and (b) for Te 1=8. 213

(g;d~)2÷ N

16o . . . . . . . . . .

(g~72+ P

2 06 . . . . . . . . . . . .

(h~)22 + N

160 . . . . . . . . . . .

(g7)22+ P

og8 086

2+

" 2+

0,40 0 39

2+

2+

O+

000

Fig. 5. Level spectra (a) ('or

O÷

000

54Xe 126

54xel28

X e 126 a n d

(b) for X c t=s.

RELATIVE PHASE OF T H E M I

591

A N D E2 RADIATIONS

The relevant states for Xe .26 and Xe *2s are shown in fig. 5(a) and (b), while experimentally it is known that 6xe,2, > 0 and 6xo,, < 0. That 6x~,2, > 0 is expected from the fact that only proton states appear as lower quasi-particle states. On the other hand if use is made of the energy 1.60 MeV for the (g~)2 p state we get t3x02~ > 0 , in disagreement with experiment. If we use, however, a somewhat higher value for this energy, in accord with the case of Te ~26, its positive contribution will be overwhelmed by the large negative contribution from the ( h ~ ) 2 N state and thus we get 6xe,~, < 0 in agreement with experiment. It is worthy of note that Te 126 and Xe a2a have the same number of neutrons and thus ( h ~ ) 2 N state appears at the same energy in these two nuclei, while the (g4)2 P state appears higher in Xe 12a than in Te 126 when same parametric values are used as in KS. 2 070

(d~g7)2+

1 907

(d)2)22 + N

1 624 . . . . . . . . . .

(g29)~'2+ p

1358

2+

N

2 083 2 066

............ _ - - = : : ~-~_ = :

=::

2 050

0535

(d~) 2 2 + N 2 (g-Td'-u) 2 + N 222 (g~) 2+ N

1624 . . . . . . . . . .

(g~)2 2~" P

1100

2+

0474

2+

2+

0000

O+ 44Ru 1°°

0 000

-0+ 44Ru Io2

Fig. 6. Level spectra (a) for Ru1°° and (b) for Ru ,0:.

For the lightest elements in this mass number region, i.e. for Ru 1°°, R u 102 and Pd 1°6, we give the relevant level spectra in figs. 6(a) and (b) and 7, respectively, and as is expected from these spectra the calculation gives positive 6 for all of them. Experiments show, however, that for R u 1°2 and Pd 1°6 6 is quite large (or [~/[ is quite small) and thus the sign of 6 is not very definite, although the positive sign seems to be somewhat more likely 7) supporting our calculation. On the other hand for Ru ~°° the result given in fig. 1 shows that T/R,Ioo < 0, and thus our calculation disagrees with

TARO TAMURA AND HIROSHI YOSHIDA

592

e x p e r i m e n t t. T h e e x p e r i m e n t o n r/Ru,OO seems, h o w e v e r , to c o n t a i n a fairly large u n c e r t a i n t y a n d the result g i v e n in fig. 1 m i g h t n o t be c o n c l u s i v e 7). 2 11

(~a~)2,

198 . . . . . . . . . .

(g~)= 2+ N

142 . . . . . . . . . .

(9~)22+

112

2+

0 51

2+

0 O0

N

P

O+

4epd ~°e

Fig. 7. Level spectrum for Pd l°e. 2809 2 630 . . . . . . . . . . .

(f})~

2+ P

2442 . . . . . . . . . .

(f~)2

2+N

2270

. . . . . . . . . . . .

(p~f~)2+

2098

. . . . . . . . . . .

(p~)~

1664

N

.

(p,~ p~)2+ N

2 442 . . . . . . . . . .

(f.~)2

2

. . . . . . . . . .

(p~r~)2+ N

270

.

.

.

.

.

.

.

.

.

.

2098 . . . . . . . . . . .

(p~)2

2+ N

2+ N

2-,- N

2+ 1 331

2+

2+

0799

0 000 0 000

O+

O+

aeN t e o

2~Fe ~s

Fig. 8. Level spectrum for Fe ~8.

F~g. 9. Level spectrum for Nt 6°

t It may be noted that the constant C~ of (17) becomes negative tfkl < 0; Le. if oblate deformation is favoured over prolate in the sense of TK.

RELATIVE PHASE OF THE M1 AND E2 RADIATIONS

593

4.1.3. Liohter elements The situation in Fe s6 is similar to that in Hg 198 and we shall not discuss it here. For Fe 58 the relevant states are shown in fig. 8. The lowest (pl)ZN state belong to the j = l + ½ category which gives negative contribution to 6 and thus we may expect that 6 is negative theoretically. In this case, however, several other low lying quasi-particle states appear, and these all give positive contributions. By actual calculation it is found that on the whole we get 6 > 0, in agreement with experiment. For Ni 6° we should perhaps identify the 122+) state with the (p])2 N state and the 6 is to be calculated by using (21) rather than (16). Using the level structure shown in fig. 9, it is found that 6 > 0, which agrees with experiment. Experimentally 6 is negative for all the cases Ge 72, Ge 74 and Se 76. We do not intend to perform very detailed calculation for these cases, because we do not know for certain whether these are vibrational nuclei or not, (except perhaps Se 76) and also because we do not have enough knowledge about the parametric values needed to describe the level scheme of quasi-particle states. Some semi-quantitative consideration shows, however, that if the (g~)2N state which appeared in fig. 10 of KS can be lowered by 1 ~ 2 MeV and extrapolated to higher neutron numbers, then we get negative values for 6 in these nuclei and thus get results which agree with experiments. 4.2. ROTATIONAL REGION In fig. 10(a) we give the level structures analogously as for vibrational nuclei, which were taken into account in our calculation of 6 for Sm 152. As before, the lowest two 2 + states shown by full lines are experimentally known collective states, while the levels shown by dotted lines are two quasi-particle states with f21 +f22 = 2. To each of such levels is attached the configuration of the quasi-particles, where the O-quantum number and the number of the Nilsson orbit are explicitly written. P or N specifies whether the state is of a quasi-proton or a quasi-neutron state. To see that we get positive 6 theoretically for Sm as2, it would be enough to show that the values in the square bracket of (30) are predominantly positive. It is found that the values corresponding to the seven quasi-particle states shown in fig. 10(a) are, from bottom to top, 0.03, 2.00, 2.67, - 1.37, - 0 . 0 4 , 4.48 and 2.02, respectively. Thus we are almost certain to get 6 > 0 in agreement with experiment. Fig. 10(b) through (10h) show relevant level structures for Gd xs4, Dy x6°, W 182, W 184, Os 186, osXa8 and Os 190, respectively. The values of the quantity in the square bracket of (30), for the qausi-particle states read from bottom to top, are as follows: Gd154: Dy16°: W'82: W184: Os~86: Os188: Os~9°:

0.03, 2.01, 2.67, 5.42, - 1.37, - 0 . 0 4 ; 0.03, 2.67, - 1.37, - 0 . 0 4 , 4.20, 2.01, 5.42; - 0 . 0 1 , 0.57, 6.07, 2.09, 1.96, - 0 . 0 1 ; 2.526, -0.422, 0.107, 1.960, 4.99, -0.948, 3.005; -0.422, 0.107, 3.526, -0.948, 3.005, -0.765; -0.422, 3.526, 0.107, -0.948, 3.005, -0.765; -0.422, 3.526, -0.765, -0.948, 3.005, 0.107.

594

TARO TAMURA

22e 219 212 2 11 20S 2O3

. . . . . . . . . . . . . . . . . . . . . . -= =========~2 ....

(~ 19)(~ 26) (~' 22)(-~ 3 0 ) ~. i 31 )(-~ 60) \C~ 57)( ½ 60) -- -- -- "-~ 2z 25)(-~ 57)

P P N N N

\(~- ,s)(-½ 26)P

202

{~

1 09

AND

I-HROSHI Y O S H I D A

2,2

- - -

_z=_=_--=_=~.~ 3,~(-"

221109. . . . . . . . . 206 . . . . . . . . . 2 02

eoN - , ~ ( 2~ 57)( ~ ~O)N 2,)(-~ 29)p \~(-~ 25)(-2~ 27)N -\\\'(~

4e( } 53) N

4E~)( ~ 53)N

2+ 2+

100

012 0 O0

2+

O 12

2+

O+

OOO

O+

o4Gd 154

e2Srnls*

250 . . . . . . . . . .

(27 21)(- ~ 29)P

(~~ 29 . . . . . . . . . . 221--- . . . . . . "(~ . . . . 22 11 38 - --- -~" (~!( ~ 207---_-----------Z2 04-

096

2+

008 0 O0

O+

247 . . . . . . . . . 243 . . . . . . . . . . 238 . . . . . . . . . .

(~- 2 5 ) ( - } 4 2 ) N (~ 27)(-½ 4 3 ) P (~- 33){ ½ 4 3 ) P

57)N

206 . . . . . . . . .

(o 18)(-~- 2 7 ) P

4E~( -~ 5 3 ) N

,94 . . . . . . . . . . 191 . . . . . . . . .

(~- 5s)(-,~ 51 )N (~ 42)( ½ 5 1 ) N

29)( ~ 37)( ~ 31)(-~ 57)( ~ 25)(- ~

34)P 38)P 60)N O0)N

2+

eeDY I~°

c

122

2+

010

2+

000

O+ ~4wlS2

d

RELATIVE

]88 84 82

/(~ ------~

........... --~

115954. . . . . . . . . .

090

3~I(-,~ 52)(~ 3 33}( ~ 27)(-~

PHASE

38) P 53) N 43)P 43)P

(3 4(~)(-~ 53)N .(3 57)(½ eO)N " ' ( ~ 37)(½38)P

OF

ThE

MI

AND

E2

203--

(~- 5 ~ ( - ~ e(~N

18,B. . . . . . . . .

(~ 36)(-½ 38) P (~ 52)( ~ 53) N

f84

155 151

-- =

,59

-

. . . . . .

-----(~ 37]( ~ 38)P :~(} 46}( ~ 53)N ~ ~ 57}( ,} eO}N

2+ 077

Oll

595

RADIATIONS

"

O00

2÷

014

O+

0 O0

74w le4

2+

2+ O+

760s 186

(~ 55)(-~ 60)N (-~ 35)(-~ 38)P (~ 52)( ½53)N

1 91 . . . . . . . . . . I 85:-=::===== 11 87 39 - - . . . . . . . . .

(-~ 45)( ½ 5 3 ) N (~ 35)(-½ 3 8 ) P (~2 52)(-~ 5 3 ) N

164 ..........

(~ 55)(-I~ 50)N (~ 37)( 1 3 8 ) P

1 64

(} 46)( ½53)N (~ 3 7 ) ( ½ 3 8 ) P

I 51

(~ 57)( ~ 50)N

1 47 . . . . . . . . .

(~ 57)( ~ e0)N

lg2 . . . . . . . . . . . I 05

...........

I 82 170

0 63

2~

0 16

2+

056

2+

0 19

2*

0 O0 0 00

. 0+

g

O+ 7505190

h

70Cs Ie~

Fig. 10. Level spectra for the rotational nuclei: (a) Sm 15=, (b) GdIS% (c) D y 16°, (d) W z82, (e) W zS~, (f) OsZ8% (g) Os 18s and (h) Os 19°.

596

TARO

TAMURA

AND

HIROSHI YOSH1DA

These are values calculated by assuming that the equilibrium deformation of these nuclei correspond to/~o = 0.3 for the former three cases and//o = 0.2 for the latter four cases. Actually the magnitude of/~o varies slightly from nucleus to nucleus, but the above result would be enough to convince us that our present calculation gives positive 6 for all these nuclei, and thus explains the experiment. 5. Discussions

It is shown that the experimental results concerning the relative phase o f the M1 and the E2 transitions between the first and the second excited states in even nuclei could be explained reasonably well in most cases, by using very simplified models both in vibrational and rotational regions. This fact may show that the calculations made in the present paper are an appropriate way of describing the actual nature o f the nuclei considered. There remains of course, several points to be discussed, and the first one of these concerns the use of the phenomenological description of the collective states, particularly for the vibrational nuclei. It is now known that the method of Sawada and others 12) can be applied successfully to these nuclei in describing at least their first excited vibrational states, in terms of the quasi-particle pair excitation modes 11,13). Such a method could further be extended so as to describe also the second excited 2 + state. In that event the collective states obtained will no longer be coupled to the quasiparticle 2 + state and the y-transition beween the two collective 2 + states will contain an M1 component from the beginning, allowing the whole description to be done in a completely unified way. Such an extention is now under way by Udagawa and Tamura and will be reported shortly. Nevertheless, we hope that the idea on which the present calculation is based is correct to a good extent and thus will serve as a useful guide for such an improved calculation. The next point to be discussed concerns the absolute magnitude of 6 which we did not consider thus far. As the formulation given in sect. 3 sfiows, this quantity is directly proportional to the strength parameter X of the Q-Q interaction given in (2) (and (23)). It is known that, if the coupling between the particle and the surface motions is described a s / ~ u Y2M(0g0),as was assumedin the original paper by Bohr14), /c is of the order of 40 MeV. Our coupling is given as (X/2)r2~ccM Y*M(O~o),and the comparison of (Z/2)F2 with/c gives X = 130 MeV • R o-2, Ro being the nuclear radius. Using this value of Z, 6 can be evaluated and we get e.g. for X e 126 ~ = 2.8, which is rather small compared with the experimental value 6 > 20. It should be noted, however, that in T K new (collective) potential energy terms are added to Bohr's, and it might be thought that main part of the particle-surface coupling terms is replaced by these new terms. It would thus be reasonable to assume a value for X weaker than in above, and we arbitrarily take X = 40 M e V . Ro -2, which is about one third of the above value. This gives 6 = 27, which agrees with experiment within the experimental uncertainty. To see whether this new X gives agreement in other

RELATIVE PHASE OF THE M I

AND E2 RADIATIONS

597

cases too, we take T e 122 a s an example, and obtain 6 = 2.7, which is in accord t with the experimental value 6 = 4 + 2 . A similar calculation was also performed for Sn 116, using, however, (21) instead of (16), and it was found that 6 = 0.06. Although there is a large uncertainty in the experiment, as is seen in fig. 1, 6 does not seem to be less than 0.25. This discrepancy may mean that some of the admixture of the collective con tribution to the E2 radiation has to be considered. Concerning the absolute value of 6 for the rotational nuclei, the following approximate evaluation is made. As is seen from (26) and (28), the main task in the evaluation of 6 consists in the summation ~ C~,1~.2; ~1~2, other factors being evaluated easily. (We use again the value of Z as determined above.) In this summation, however, it is known, as discussed above, that the main contribution comes from the diagonal elements C~1~, axu~. If the contribution from the non-diagonal C is in fact neglected, the reduced matrix elements (f21f221~(M1)lf21 0 2 ) in the remaining terms can be rewritten as in (30). If, further, the quantity in the sqaure bracket of this expression is replaced by its appropriate (algebraic) average, and similarly the energy denominators (E(O t O 2 ) - E ( 2 2 ) ) and (E(f21 f22)-E(21) ) in C are also replaced by their respective averages, then the above summation is reduced to a constant times a new but very simple summation ~ 1 ~ 1 (f221I 2 Y221I21)l 2. This summation is, however, nothing but the one which appears in the cranking formula 6) for By, the mass parameter of the ~-vibration, if similar avearaging of the energy denominator is made in this formula. It is thus seen that the magnitude of 6 can be estimated, if that of By is known experimentally. By taking h2/Br = 0.03 MeV, as suggested e.g. by the work of McGowan and Stelson 15) for W is* and other neighbouring nuclei, it is found that 3 - 10, which is just the order of magnitude of the experimental 6, as is seen from fig. 1. In the numerical calculation of sect. 4 there a limited number of quasi-particle 2 + states were considered, and the criteria used for the limitation are the following. For vibrational nuclei all the quasi-particle states are considered which lie within the energy of the first excited (collective) 2 + state above the lowest possible quasi-particle 2 + state. This is consistent with the approximation to take only the ground T K state in the expansion (7), which resulted in (9). On the other hand for the rotational nuclei the quasi-particle states are those which lie within 500 keV above the lowest possible one, the choice of 500 keV being quite arbitrary. In both cases the inclusion o f the higher states might change the magnitude of 6 to some extent, but we hope that their contribution is so small that our conclusion concerning particularly the sign of 6 is unchanged. Our final remark concerns the use of T K and T U among other possible phenomenological models. I f e.g. the model WJ were employed we would have obtained 1/6 = 0 within the approximation used here, as is seen from the fact that kl = 0 in (17), * This fact means that our calculation also correctly predicts the hindrance factor of the absolute probability of the M1 transition compared to that of the single-particletransition, a point which w a s discussed by Sakai 7).

598

TARO TAMURA A N D HIROSHI YOSHIDA

and to this extent the discussion on 6 may serve in discriminating between various models for collectwe states. Even WJ may give non-vanishing 6, however, if higher order calculations are performed, and in this way m a n y other models might as well be used for the calculation of the kind reported here. We would like to emphasize here again, however, that irrespective of what models are used, the avalysis of 6 seems to give us good information on how the lower-lying quasi-particle states are occupied, and in this sense accumulation of more data would be quite valuable. We are very much indebted to Professor M. Sakai for many useful discussions. In fact this work was motivated by his kindness in showing us the experimental data, reproduced in fig. 1 of the present paper. We also thank Mr. T. Udagawa for valuable comments and discussions. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) I1) 12) 13)

14) 15)

G. Scharff-Goldhaber and J. Weneser, Phys. Rev. 98 (1955) 212 B J. Raz, Phys. Rev. 114 (1959) 1116 L. Wdets and M. Jean, Phys. Rev. 102 (1956) 788; referred to as WJ T. Tamura and L. G Komai, Phys. Rev. Letters, 3 (1959) 344; referred to as TK A. S. DavldOv and G. F. Fdlppov, Nuclear Physics 8 (1958) 237; A S Davldov and A A. Chaban, Nuclear Physics 20 (1960) 499 T. Tamura and T. Udagawa, Nuclear Physics 16 (1960) 460; referred to as TU. M. Sakal, private cornmumcat~on G Rakavy, Nuclear Physics 4 (1957) 289 L. S. Ktssllnger and R. A. Sorensen, Mat. Fys. Medd. Dan. Vld. Selsk. 32, (1960) No. 9; referred to as KS S G Ndsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, (1955) No. 16 T. Tamura and T. Udagawa, to be published K. Sawada, Phys. Rev. 106 (1957) 372 R. Arvleu and M. Veneroni, Compt. rend. 250 (1960) 992, 2155; T. Marumon, Prog. Theor. Phys. 24 (1960) 331; M. Baranger, Phys. Rev. 120 (1960) 957 A Bohr, Mat. Fys. Medd. Dan. Vld. Selsk. 26, (1952) No. 14 F. K. McGowan and P. H. Stelson, Phys. Rev. (to be pubhshed)