Axial asymmetry in even rare earth nuclei

1.D.2

I I

Nuclear Physics 29 (1902) 623--634; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permi~ion from the publisher

A X I A L A S Y M M E T R Y IN E V E N R A R E E A R T H N U C L E I ILA DUTT and P A R E S H M U K H E R J E E Saha Institute o/Nuclear Physics, Calcutta, India Received 20 June 1961 The 7-stability of some rare earth nuclei, whose quadrupole deformation/~ is around + 0.28, has been investigated b y calculating the energies of individual nucleons in a c o m p l e t e l y anisotropic harmonic oscillator potential with spin-orbit coupling.

Abstract:

1. I n t r o d u c t i o n

Our understanding of the spectra of even nuclei is becoming more clear and composite due to recent work b y m a n y authors in this field. It has now been realised 1) that the spectra of such simple nuclei as Ca 42are not merely dependent on the motion of two outer neutrons b u t also on the collective vibration of the closed shell core. Although it is not easy to extend such calculations to more complex configurations, it is reasonable to assume that in the neighbourhood of closed shells the excitation spectra can be explained in terms of three factors: (a) the oscillations of the spherical core, (b) motion of the outer nucleons and (c) their coupling to the nuclear surface oscillations. As the number of nucleons is increased the nucleus acquires a permanent spheroidal shape and exhibits the well known rotational excitations of low energies. Over and above these low-lying rotational levels corresponding to the ground state band, there m a y be higher bands which are attributed to the/~ and 7 vibrations of the nucleus. Superposed on these vibrational levels are the rotational levels. The spectra of even nuclei in the deformed region can be satisfactorily accounted for in terms of this simple model, together with finer details as the rotationvibration interaction 2). But the exact level positions of the vibrational states cannot be reproduced since the frequencies of vibration oJp and o~y are markedly dependent on the actual nucleon configuration 3) and as such it is difficult to estimate their magnitudes. The I ( I + 1 ) dependence of the rotational levels indicate an axial symmetry for such nuclei and the slight deviations from theoretical estimates is taken into account b y introducing the rotation vibration interaction. With this point in view the calculation of energies of individual nucleons in a deformed field has been invariably made in an axially symmetric spheroidal field 4). In recent years an alternative model has been proposed b y D a v y d o v et an. 5) to explain the spectra of even deformed nuclei. The model assumes that all the 623

624

ILA D U T T AND PARESH M U K H E R J E E

low-lying levels in such nuclei are generated b y the rotation of an asymmetric ellipsoidal nucleus. The hamiltonian corresponding to the rotational motion is 3

and its eigenfunctions in the frame of principal axes (x = 1, 2, 3) of the ellipsoid are ~KD~K(O,), where O ~ are the elements of the representation of a (2I+l)-dimensional rotation group. Thus for different I, the energy matrix can be constructed and the corresponding eigenvalues can be determined. Since in the limit of hydrodynamic approximation the moment of inertia J r depends both on/5 and 7, the eigenvalues call be expressed as a function of the asymmetry parameter 7. The observed spectra over a wide range of even nuclei are found to be in nice agreement e) with this theory b y suitably choosing the value of 7- However, the 0 + excited states form an exception and cannot be accounted for b y this theory. A still more convincing application of this model is in the variation of transition ratios B(E2, 2' --~ 2)/B(E2, 2 -~ 0) and B(E2, 2' -+ 2)/B(E2, 2' -+ 0) with the energy ratios E(2')/E(2) of the second and first excited 2 + states ~). It is astonishing that this simple model, with assumptions like the constancy of nuclear shape over a wide range of energy, non-existence of vibration even at an energy greater than the energy of a phonon and complete disregard to the intrinsic particle motion, can account for such a large number of data over a wide range of nuclei. Several attempts have been made to prove that the/~ and 7-vibrational model also can explain the above transition rates. The recent work of Tamura 8) and Jean 9) m a y be mentioned in this respect. Thus it has become necessary to investigate whether according to the energy considerations, stable ellipsoidal shapes in nuclei m a y exist or not, which m a y prove the validity of Davydov's model. An attempt to examine the stability of ellipsoidal nuclear shapes has been made b y D a v y d o v a al. lO) for a nucleus consisting of two extra nucleons outside a closed shell, b y extending the method of Bohr. They have shown that in certain cases the minimum energy of a nucleals does correspond to the non-axial shape. Unfortunately this method is not applicable to nuclei with a large number of particles outside the closed shell, as occur in actual cases. Recently Newton 11) has calculated the energies of individual nucleons in an anisotropic harmonic oscillator well for states with principal quantum number N up to 4. But the number of levels he has calculated is not sufficient to build up the actual nuclear configuration, where D a v y d o v and Filippov's model is found to be most satisfactory, namely the mass region around A ~ 160. Thus an estimate of total single particle energy in terms of the a s y m m e t r y parameter 7 is still lacking. The present work is devoted to such an analysis b y extending the treatment of Nilsson to non-axial nuclei.

XXIXLASr~MET~y

625

2. Single-Particle Energy in an Anisotroptc Harmonic Oscillator Well 2.1. C O N S T R U C T I O N O F T H E H ~ T O N I A N

Since the Nilsson potential is found to be most satisfactory for the behaviour of nucleons in a deformed nucleus 22) we start b y modifying this potential so as to include a s y m m e t r y j n shape. The hamiltonian for a single particle in an ellipsoidal harmonic oscillator potential can be expressed as ~2

H --

2M V"+½M(eo®'z"-i-e%'y"+a,,az")-FCl.

s + D l 2.

(I)

The corresponding nuclear surface m a y be represented in terms of (fl, ~,, 0', ~') as

r'(O',

¢') = Ro). [I +flcos ~,Y,.(O',~')-{- ~-~Sl "n~,(Y2,(0,4')-k-Y,_9.(O, 4")}], I

'

'

'

(2) where ,% preserves the volume of the nucleus. It is easy to see that the length of the three principal axes of the surface given by eq. (2) are 5 fl(cos ~,--'V~ sin ~,)], A." = Ro2 [1-- Vi--~

~..=,.,[,-V

°

]

] - ~ fl(cos r+V'-3 sin r) '

(3

A," = Rol [l+ V~ fl cos ~]. Identifying A,,, A ¢ , A,, with the corresponding axes of the anisotropic well given by eq. (1) we can take

[ V~~ fl cos (y -~-~-:~)] , o,,-- o0, [, + 1/-~,, cos ,~+,], oJ,' = ¢oo' 1 qt_

(4)

¢o,a ---- ¢0o~[ 1 - - V ~ fl cos 71, where ~oo is a function of (fl, 7) to be determined from the condition of volume preservation.

ILA D U T T AND PARESH M U K H E R J E E

626

It is evident from (4) t h a t for y = ½ner(n = 0, 1, 2 . . . . ) eq. (1) reduces to the familiar Nilsson h a m i l t o n i a n for a spheroidal nucleus 13). Since the v o l u m e of a nucleus must r e m a i n c o n s t a n t we have

~o,~,

= (~o°p,

where o~o° is the f r e q u e n c y of oscillations in the spherical case. Substituting from eq. (4) it is easy to see t h a t

[

O,o(~, r) = ,~o° 1 -

7=

~

~ / ~ cos

4'

.

(5)

F u r t h e r we introduce dimensionless variables x=~--x,

etc.,

a n d H given b y (1) is t r a n s f o r m e d to H -- -- ½ ]~.~oo (V 2 - r 2) - - X h ~ o 0 R,

(6)

where R is given b y

'I

R = - t5 cos Z ×

'

(7}

E V 1--

]

yr2Y20(O,¢) + ~ fl sin yr2{Y22(O, ¢) +Y2_2(O, ~b)} ~

~

~

cos 37

]'

+21"s+/*l

=,

and the p a r a m e t e r s Z and # are defined as C X --

2?/o)0 o ,

D #

--

Z~a~oO-

\Ve have chosen 7. = 0.05 and t, = 0 to 0.45 as t a k e n b y Nilsson 13) so t h a t in the spherical limit the shell model levels are reproduced. The evaluation of the eigenvalues E(fl, y) corresponding to H given b y eq. (6) can be m a d e in an analogous w a y as in Nilsson's work, with the difference t h a t in this case, since t h e r e is no axis of s y m m e t r y , the projection of t o t a l angular m o m e n t u m on a n y axis, s a y / ' , , is no longer a c o n s t a n t of motion. 2.2. CHOICE

OF

REPRESENTATION

F o r the h a m i l t o n i a n (6) neither of the N, l, l,, j or 7', are good q u a n t u m numbers, b u t to m a k e our calculations simpler we assume t h a t the principal q u a n t u m n u m b e r N is an a p p r o x i m a t e constant of motion. One can thus choose two alternative representations in which the eigenvalue problem of the hamiltonian (6) can be solved.

AXIAL ASYMMETRY"

627

One is the (Nll, s~) scheme in which the wave function can be expressed as

~PN ~ Z a,,~,lNll, s,) •

(8)

~l z sz

The basic vectors INll, s,) are the solutions of the equation --

½h~0 (V2--r ~)IXll, s,> ---- ?~o~0(N+{)INll, s,>.

(9)

The problem of the evaluation of eigenvalues corresponding to H then reduces to the construction of the matrix for R and its diagonalisation. The required eigenvalues rN(fl, 7) of R are given by

rN(l~, y)a,,~,,s, * = ~, a,,,,
(10)

~gz 8z

In the above representation the 12 term is diagonal with eigenvalues l(l+l). The matrix elements of the r2Y~,(O, ¢) and 1 • s terms in R can be evaluated with the use of the following well known relations la):

:
~-[1"

V~

1 sill,, 4--~>

"217-~ =

l,#l'

r,LOOOA'

(,l,

-¢- {l,,

-~

1 V(l~lz)(l_}_lzJf_l).

Thus for a particular value of the principal quantum number N, the matrix for R can be constructed for different values of fl and 7, and the eigenvalues rN(fl, 7) can be calculated. The total single particle energy is then given by

E N = (N+-})/~%(fl, y)--flimo°r~.(fl, 7).

(12)

An alternative representation in which the eigenvalues corresponding to R can be evaluated is the (Nl]i,) scheme. The basic vectors in this representation m a y be expressed as [Nlii,>

= lzESz [1l, ss, i.Z] ,Nllz)[ss,).

In this representation 1 • s and 12 are diagonal. The eigenvalues of l • s in this representation are

= { {i O"+ 1) --l (l + 1) --~}.

6~8

ILA

DUTT

AND

PAR]~SH MUKHERJEB

The matrix elements for Y,a (0, ~) m a y be easily evaluated using the following relation:


,/~

can

be expressed as x4)

For the sake of convenience in numerical calculations we have used here the former representation since in this case there is a smaller number of nonvanishing matrix elements of R than in the latter. 2.3. E F F E C T O F ( N ± 2 ) - M I X I N G

In the above formalism the principal quantum number N has been assumed to be a constant of motion in order to get simplicity in numerical calculations. Actually there will also be mixing of states with N differing b y two. In the limit fl -+ 0, the levels N and N + 2 are separated in energy b y 27k% and the contribution of these coupling terms to energy is therefore assumed to be so small that their neglect is justified. But for large fl some of the levels with N differing b y two come quite close to each other, and the coupling terms connecting the two states become appreciable. Moreover, since we are going to examine a second order effect like the dependence of energy on the a s y m m e t r y parameter ~+,such effects due to the mixing of (N+2)-states must be taken into account, since they m a y give a contribution of comparable order. We have taken into account the mixing of N i n an analogous w a y as in Nilsson's alternative representation is) b y introducing a new deformation parameter ¢t and the volume preserving term (o0(,c, ~,) defined in the following as o~. - - ",o(~, r )

1+

~

{ V {

co, = eaoCac, ~,) 1 +

~

~ cos O,+½e)

ot c o s ( y , - { - ~ )

m

to, = COo(~,~,) 1 --

,

/

}

ot cos • ;

the volume preserving condition requires OJo(~ , ~) = oJo° 1 - - 16-----~

1G~

~

cos

,

(13)

AXIAL ASYMMETRY

629

where our ~ is related to the q u a n t i t y e defined by Nflsson is); e~3

~Z.

Further it is easy to see that ~ is related to the former p by

::, v: <,_ v!,)'+= <,+

,)' J

+'

As has been shown b y Nilsson 13), making a transformation to the dimensionless co-ordinates, the hamiltonian H is diagonal in the alternative representation (N~ l, Atsz), the matrix elements for a particular ,t being the same as for the corresponding/~ except for a small perturbation term Hvert. The explicit calculation of total energies with this term is rather tedious and has not been worked out, but it is expected that the dependence of the term H~ert on the parameters /~ and 7 is very small. Hence in our considerations, since we are interested here only in the variation of total energies of nuclei with 7, we shall neglect this term. The total single-particle energy is then expressed as

EN =

(16)

2.4. N U M E R I C A L CALCULATION AND RESULTS

We have investigated the variation of total energy for a number of nuclei lying in the region A m 150--180 for various values of y. The nuclei in this region are known to possess strong deformations and the deformation parameter fl for all these nuclei lie in the range 12) fl m 0.26 and 0.29. Hence in our calculations we have taken ,t -----0. 30 which corresponds to fl ~ 0.28 (cf. eq. (15)). As described above the single particle eigenvalues in a non-axial nucleus have been evaluated b y constructing the matrix of R in the (NU~s~) scheme using eq. (11). We have performed the calculations for the four values y = 0, i ~ , i 2y~ T-f ~ , i~2:~" Since for N ~ 5 there are 21 sets of [Nl~,s,) in the wave function (8) the corresponding matrix R is of order 21 × 21, which is the highest order matrix in the present calculations. The eigenvalues corresponding to higher N states have not been evaluated since t h e y give still higher orders of matrices which are difficult to handle. Moreover for the nuclei in which we are interested there are only two or three levels having N ----- 6 which m a y be filled up. The matrices R which are of higher order t h a n 3, i.e. those for N = 2, 3, 4, 5 have been diagonalised with the help of the U R A L computer at the Indian Statistical Institute, Calcutta. Using the eigenvalues of R the single particle energies are

ILA DUTT AND PARESH MUKHERJEE

630

then calculated from (16). In Table 1 we have presented these single particle energies for states with N up to 5 in units of X~o~0°. These are correct up to three decimal places. For a particular value of N the single particle energies in the above table for each value of 7 have been arranged according to increasing energy. This ordering TABLE 1 Single particle energies in an elli )soidal nucleus

~ O.

30.2910

30.2862

30.2736

30.2607

46.3358 51.4184 53.7007

46.4545 51.3431 53.6334

46.7997 51.126I 53.4422

47.3452 50.7956 53.1626

N=2

62.3892 67.3124 69.8917 72.5458 74.9559 76.9790

62.6317 67.3593 69.8890 71.981i 75.3033 76.8424

63.3361 67.4967 69.6439 71.1020 75.8016 76.4502

64.4404 67.7214 68.5815 70.9973 75.8563 76.0528

N=3

75.7404 79.4783 83.0248 84.1434 88.5159 89.3910 89.4732 94.6498 96.0028 96.8104

76.0728 79.6310 83.0844 83.9850 88.7665 88.8078 89.5063 94.6651 94.8289 97.7381

77.0135 80.0854 82.7809 84.2330 87.2931 89.4630 89.5295 92.7292 96.0855 98.4948

78.3623 80.7694 82.2509 85.1784 85.4686 89.5976 90.1805 90.6863 95.0764 98.7505

N=4

88.9098 91.6943 95.6438 96.6406 100.4033 101.5103 102.0916 105.8007 107.2529 108.0099 109.8284 113.6476 114.3978 117.7118 117.9624

89.2706 91.8945 95.4750 97.0371 100.0047 101.8555 102.2632 105.1338 107.1643 107.4590 110.5642 112.8309 113.4373 118.3551 118.3761

90.2871 92.6054 95.3127 98.1612 99.0973 102.7215 102.7553 103.3218 i05.8464 107.9618 111.1520 111.3811 111.7437 118.8625 119.3384

91.5173 93.5093 95.6286 98.1858 99.7835 100.8896 103.4051 103.9410 104.7018 108.4521 108.9905 109.4995 112.5715 118.8762 119.8870

~=

1

631

AXIAL ASYMMETRY TABLE I (continued)

N=5

102.3209 104.5030 107.8741 111.0210 112.0926 114.9865 115.6725 116.9805 119.9961 120.9555 122.4281 123.0704 125.7584 126.7786 130.3834 130.7261 132.112~ 133.0754 138.0402 138.5229 140.1825

102.7025 104.8013 107.9259 111.5354 111.8848 115.4088 115.9659 116.5084 120.2952 120.3462 121.6955 123.8694 125.1734 126.0517 131.0558 131.1186 131.3657 132.0542 136.9942 138.5799 141.7117

103.6622 ]05.5944 108.2997 111.4932 112.9506 115.2741 116.5474 116.7680 119.5538 119.6761 121.0517 123.9985 124.5712 125.3778 129.3399 129.4155 132.0889 132.3174 135.2054 139.4322 143.2798

]04.6960 106.5413 109.3512 111.4129 113.8~2 114.9241 116.8864 117.9061 118.0904 119.3298 121.9919 123.0747 123.1163 125.9073 126.8841 127.3155 132.7492 133.3790 133.4577 139.8151 144.0560

of levels for different N differs from t h a t of Newton's Xl) work where the eigenvalues in a given row h a v e been a r r a n g e d in such a w a y t h a t the scalar p r o d u c t of the sets of adjacent vectors in a row is m a x i m u m . T h e calculations for single particle energies h a v e been p e r f o r m e d for 56 levels which can a c c o m m o d a t e 112 particles of each kind. B u t since we h a v e assumed the d e f o r m a t i o n p a r a m e t e r ~ to be quite large some of the states with N ~ 6 m a y a p p e a r low in e n e r g y n e a r the N = 5 states. The neglect of these states m a y i n t r o d u c e an error in the t o t a l e n e r g y calculations of nuclei which h a v e n e u t r o n or p r o t o n n u m b e r g r e a t e r t h a n 100. H e n c e in the following treatm e n t we h a v e limited our discussion to those nuclei which h a v e n e u t r o n n u m b e r less t h a n 100, We h a v e calculated the t o t a l energies for a n u m b e r of nuclei, n a m e l y Sm xS*, Gd x54, Gd 156, Gd x~s, D y xs°, D y xs2, E r xs~, E r x68, Yb ls8 and Yb 17° for various values of 7, b y s u m m i n g the single particle energies a n d assuming t h a t the levels of lower e n e r g y are filled u p first. I t can be seen from Nilsson's level scheme 13) t h a t in the limit of 7 = 0, there will be o n l y two or three levels with N = 6 in the above m e n t i o n e d nuclei. In order to investigate the a p p r o x i m a t i o n involved in our calculations due to the neglect of N = 6 states, we h a v e shown in fig. 1 the t o t a l e n e r g y curves for Sm m in two cases, one neglecting, and a n o t h e r t a k i n g into account the N = 6 states in the limit of spheroidal deformations. It is obvious t h a t the change in equilibrium deformation is not v e r y appreciable.

632

ILA D U T T AND PAI~.SH M U K H E R J E E

Hence we have assumed t h a t the neglect of the few higher N states will not be important in the region of nuclei t h a t we are going to consider. The calculated values of total single particle energies for each of the above mentioned nuclei are shown in fig. 2 for various values of 7. It can be seen t h a t for almost all the nuclei considered here the total energy increases as 7 is increased but the variation of energy over the range 0 ~ 7 ~ 1 - ~ is very small. M

t

b%E % " %% %

o&

~/Y.d

J o

~o.1

o.1

--

0.3

Fig. 1. T h e t o t a l e n e r g y curve E~8) in S m ~" as a f u n c t i o n o f / L The full d r a w n curve is for s t a t e s w i t h N u p to 5 and t h e dotted curve takes into a c c o u n t t h e states w i t h N u p to 6.

In fact, in a few cases, namely, in Gd 1~, D y xea and Er xee, the total energy at ~ is slightly less than at 7 = 0, showing t h a t an asymmetric shape with 7 ~ 15° is more favoured than the spheroidal shape. In the cases of Gd 1~, D y le°, Er 16s and Yb les, though the minimum occurs at 7 ---- 0°, the change in total energy over the range 0 ~ 7 ~ 15° is so small t h a t the spheroidal nuclear shape is not well defined and the nuclei are soft to ellipsoidal deformations with 0 ° ~ 7 15°. In table 2 we present the magnitudes of the a s y m m e t r y parameter 7 TABL~ 2 C o m p a r i s o n of the calculated values of the a s y m m e t r y p a r a m e t e r ~ w i t h those resulting I r o m t h e D a v y d o v model (the energies are in keV) Nucleus Stair|

C~lSe CTdlSs DyXSO DyX" E r t S s b)

ErX.S X~Fbllt8 y~170

E(2) ") 122 123 89 79.5 86.6 80.7 80 87 84.2

E ( 2 ' ) ")

E(4)')

1086 998 1154 261.9 966 787.4 822 987 277.7

a) The energies are t~t~en f r o m Nuclear D a t a Cards b) Grigoriev et ~d., N u c l e a r Physics 19 (1960) 248

[7(DavTdov) 13.2 ° 13.8 ° 11.0 ° 13.6 ° 11.8 ° 12.9 ° 12.3 ° 11.7 ° 13.2 °

7, (cal) 0o

0°---15 ° 15 o 0o 0°--15 ° 15 ° 15 ° 0°--15o 0°--15 ° 0o

633

J~JM., AS~MMCntY 711.3

687.,

E

I

G~.

xA2

I

!

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Fig. 2. The v a r ~ t i o n of total eners~ E(0(, 7) in various nuclei with the asymmetry parameter 7"

634

ILA

DUTT

AND

PARESH

MUKHERJEE

as obtained from the ratio EC2"i/E(2 ) in a nucleus according to Davydov's model along with the possible equilibrium value of 7 from our total energy calculations. In the two cases in which the second 2 ' + state is not well established, 7 has been calculated from the ratio E(4)/E(2) given in the table. It is interesting to note that in all the nuclei considered here the magnitude of the asymmetry parameter 7 required by Davydov's model also lies between 0° ( 7 ( 15°" As has been mentioned earlier, the present calculations are limited to those nuclei which have a stable deformation around ~ ~ 0.28. The results of our calculation reveal that it is not possible to establish any definite conclusion on the possibility of the existence of stable ellipsoidal nuclear shapes. But it is evident that at least for the nuclei that we have considered, the total single particle energies calculated from first principles do not change appreciably up to 7 = 15°- This only suggests that these nuclei are soft towards ellipsoidal deformations. However, due to the absence of any sharp minima in the E (~, 7) versus 7 curves, it is hard to realise how such nuclei can retain a stable non-axial shape throughout the whole excitation range. It seems quite unlikely that the inclusion of two or three N = 6 levels in the above calculations should change the general behaviour of E(x, 7) curves. We t h a n k Prof. A. K. Saha for his interest in the problem. References 1) 2) 3) 4)

5) 6) 7) 8) 9) 10) 11) 12) 13) 14)

B. J. Raz, Phys. Rev. 114 (1959) 1116 A. K. Kerman, Mat. Fys. Medd. Dan. Vid. Selsk. 30, No. 15 (1956) S. A. Moszkowski, Handbuch der Physik Vol. 39 (1957) 411 S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955); K. Gottfried, Phys. Rev. 103 (1956) 1017; S. A. Moszkowski, Phys. Rev. 99 (1955) 803 A. Davydov and G. Filippov, Nuclear Physics 8 (1958) 237 E. P. Grigoriev, and M. P. Avotina, Nuclear Physics 19 (1960) 248 D. M. Van Patter, Nuclear Physics 14 (1959) 42 T. Tamura and L. Komai, Phys. Rev. Let. 3 (1959) 344; T. Tamura and T. Udagawa, Nuclear Physics 16 (1959) 460 M. Jean, Nuclear Physics 21 (1960) 142 A. Davydov and G. Filippov, Nuclear Physics 10 (1959) 654 T. D. Newton, Can. J. Phys. 38 (1960) 700; Chalk River Report CRT-886 (1960) B. Mottelson and S. G. Nilsson, Mat. Fys. Skr. Dan. Vid. Selsk. 1, No. 8 (1959) S. G. Nilsson, Mat. Fys. Medd. Dan. Vid. Selsk. 29, No. 16 (1955) K. \V. Ford and C. Levinson, Phys. Rev. I 0 0 (1955) 1