Two moments of inertia of bands in the cranking model

Two moments of inertia of bands in the cranking model

Volume 96B, number 3,4 PHYSICS LETTERS 3 November 1980 TWO MOMENTS OF INERTIA OF BANDS IN THE CRANKING MODEL H. SAGAWA The Niels Bohr Institute, U...

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Volume 96B, number 3,4


3 November 1980


H. SAGAWA The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen (~, Denmark and T. DOSSING NORDITA, DK-21 O0 Copenhagen (~, Denmark Received 14 July 1980

We discuss the moments of inertia, associated with the first and second derivatives of the energy as a function of angular momentum of band, by applying the cranking model. A decrease in the second moment of inertia with increasing frequency is connected with the alignment of single particle angular momentum along the rotation axis.

Recent T-spectroscopy experiments have given us new information about rotational bands up to very high angular m o m e n t u m (I ~ 60 h) in rare earth nuclei. Sum crystal and T-multiplicity spectroscopy give information about average slope (the first derivative) of the energies of states, as a function of angular momentum, which are connected by the "f-decay [1,2]. T - T coincidence experiments give information about the second derivative of the energies of states which are connected by the T-decay [2,3]. These quantities are expressed by moments of inertia, c"J:(1)/1 = ( - - ~ ) -1

fi(2)= ( d 2 E i -1 ,

\d/2 ]



As a characteristic feature of these experiments, the moment of inertia fi(1) follows, quite closely, rigid rotation from angular m o m e n t u m 20 h to ~ 6 0 h, whereas the value o f f i (2) shows a 2 0 ~ 5 0 % decrease compared with fi(1) in this spin region. Our aim in this letter is to discuss a microscopic mechanism of this change by using the cranking model. We identify the experimental fir (1) and fi (2) by the first and second derivatives of the bands which appear as the yrast line at some rotational frequency. The calculated quantities are denoted by ~ (1)d, fi(b2)nd . It is not clear to which extent the experiments really measure 9r(bl~d 238

and fi(2a)nd, but we shall argue that, within the cranking model, the bands which appear as the yrast line are representative for the other bands at the same rotational frequency. We consider the cranking hamiltonian h'(ei, 60) = h (ei) - mix


where h(ei) is a single particle hamiltonian and e i are deformation parameters. The pairing interaction is neglected in the present treatment since it is expected to become unimportant in the high-seniority states which are realistic for the very high angular momenta we consider here. The energy of a band, E, is the sum of expectation values of h(ei) over the occupied eigenstates of the cranked hamiltonian, and the angular momentum, I, is given by the sum of the expectation values of Ix" The value fi(22nd can be obtained numerically [4] by diagonalizing h' at some closed spaced values of ¢o (and in principle also el) and evaluating the second derivative. In this letter we follow another approach by deriving an expression that may give more insights into the results. In the cranking model the energy E of a band and the energy E ' in the rotating frame are connected by a Laplace transformation

Volume 96B, number 3,4


E(ei, I) = E'(ei, w) +coI.


The condition for equilibrium deformation lar m o m e n t u m I reads

(j, OE


ei at


\3e i / w=O .



The quantities we want to evaluate are expressed by 11~(1) dE_dE t " b a n d - d/ 01 +

~i (a~) dei _aE -


aI -co,


3 November 1980

e ~ 0.2 in t h e / 3 - 7 plane until very high angular momentum (I ~ 60) [5]. The numerical results are shown in figs. 1 - 3 where we consider only the Inglis moment of inertia. In fig. I we show the single particle contributions to 7ingli s for the yrast configurations at three different frequencies. The main contributions at 09/600 = 0.0 (hco0 = 41/A 1/3 MeV) come from the excitations from [651 3/2] to [642 5/2] in the neutron orbits. The proton excitations [523 7/2] to [514 9/2] have also a large contribution, 15% of the total moment of inertia. Actually the main neutron contributions are

1 / 7 (2) band = d 2 E / d / 2 = dco/d/

166, ,.



Here we used the condition (4). Further we obtain

dI_ aI dw

~i 31dei_

aco i- . aei dw

[e=02 ' ~ I C'~ ~I c~ O~ ~-

32E ' + ~ ] 32E ' d~i aw 2 i 3waei d w " (6)





proton I neut r o n ~ w/uJo =0.0

A A , A ~.gus=115MeV4 I

The equilibrium condition (4) determines the change



Lo un -~ u~

32E' de/ d~iae i ,I aw3e~.

"7- 3eiaei


c-i m




Since we must have a stable equilibrium, the matrix a2E'/aeiaej is positive definite and we can insert the solution o f eq. (7) with respect to dej/dw into eq. (6):





d/ dw



A'" LIUL~ rp It



~' 24h 3~g~i~= 73 MeV-;


(8) All second partial derivatives in eq. (8) can be evaluated in a second-order perturbation and the result for any two variables a and b among ei and co is


m / m e = 0.03

v o4

i I

3~E'~] aaab v occupied _ ~, 2Re{



colwo:O.O ! ~ L3~ 3z.g,is=59 MeV-I[



e. (9)


I ii

Here, particularly, the first term in eq. (8) is the Inglis and the second term gives a positive contribution to 7(2~)nd caused b y the change o f deformation along the band. For our calculations we use the Nilsson potential with standard parameters. We take the nucleus 166yb as an example, since this nucleus within the framework o f our model has an almost fixed deformation

moment of inertia in the rotating frame

o~ 0


d I ] h ,ll~ I ', ', 1



ep - eh (MeV)

Fig. l. Individual particle-hole contributions for the Inglis moment of inertia 7inglis at three different frequencies in 166yb. We take a fixed quadrupole deformation e = 0.2 and the standard set of parameters in the Nilsson model. The numbers in parenthesis in the upper part of the figure show the asymptotic quantum number in the Nilsson model. 239

Volume 96B, number 3,4


too large in the beginning of the il3/2-shell [6]. The large contributions from the neutron excitations decrease quite rapidly with increasing rotational frequency as is seen in the middle of fig. 1. This tendency is intimately connected with the rapid spin alignment of high-] orbits. For the frequencies we consider here, the states coming from il3/2-orbits with the smaller projections upon the z-axis become almost eigenstates of ix, and the matrix elements of ]x for.the Inglis moment of inertia become small. On the other hand, the proton contributions are rather constant since the proton fermi surface is high among the members of h 11/2orbits, which are more strongly coupled to the deformation, and these contributions start to change only above ca/co 0 = 0.06 (I ~ 40). In fig. 2 the two moments of inertia, cgr(bla)nd and ~'lnglis, are shown as a function of rotational frequency co along several bands. The lower frequency region before the first level crossing is not realistic, since the pairing effect is not taken into account in this calculation. We can see, in the upper part of fig. 2, the value r(1) band is almost constant as the function of rotational frequency because of the band crossings. The rigid 200






body moment of inertia for mass number 166 and deformation e = 0.2 is equal to ~ 7 5 MeV -1. The reason why 9r (1)nd becomes larger than this value by 25% is mainly due to the occupation of highj-orbits from the major shell above, in this case, the proton il3/2-1evels. ~rlngli s decreases rapidly even though the band crossings give local increases. The abnormal increase of ~rlnglis near the frequency 60/600 = 0.035 is caused by the band crossing of two single-particle proton levels [411 1/2] and [404 7/2] with same signatures. We show in fig. 3a the angular m o m e n t u m for each band which corxesponds to each dashed line in fig. 2. The solid lines correspond to the yrast configuration for each rotational frequency. The ratio 1/60 means the value 9:(1) band and the derivative d//d60 give ~r(2) band' The large angular m o m e n t u m is gained by the band crossings of i 13/2 and h9/2 proton-configurations. We draw the angular momentum difference between the excited band and the ground band A / =




co/co 0




Fig. 2. Moments of inertia cJr(bl2nd and ~rlnglis defined by eqs. (5) and (8) as a function of rotational frequency in the cranking model. The dashed line shows the moment of inertia for each band and the solid line corresponds to that of the yrast configuration. The dotted lines show the change of the yrast configuration at band crossing. 240


in the lower part of fig. 3b. This value shows the gain of angular momentum by band crossings. Alternatively [2,3], ~r(2a)nd can be viewed as the moment of inertia of a rotor, which then carries the angular momentum 60cr(2) -'band" In this picture, the rest of the angular momentum ]a = I - co ~(b2a)nd


3 November 1980

(1 1)

corresponds to the aligned angular momentum of particles outside of the core. This value ]a is easily obtained by the difference of 5r(bla)ndto 5rlngli s in fig. 2 in our model, and shown in the upper part of fig. 3b. The difference of ]a and &/in the low frequency region is due to the spin-alignment of [660 1/2] neutron-states. In general, the value A / f o r each band shows a straight line as a function of 60. The main part of ]a attributes to A/, but some part comes from a collective effect of the core which is clearly seen in the increase of the value ]a for each band. This effect is mainly due to the sum of moderate spin-alignments of small ]-orbits. We will discuss, now, a deformed harmonic oscillator model in order to elucidate the picture of the decrease of 9r (b2a)nd.The model consists of the single-particle hamiltonian with prolate deformation at co = 0.0 (601 = 602 4= 6o3) and the cranking term without the spin-dependence:

Volume 96B, number 3,4


[ wa /






60~ = ~(602 + 603) -+ ~(602 - 603)( 1 +

1-C h 9/2

£ = 0.2

3 November 1980

i 13/2


(P --- 26°/(6°2 - 603)).


The total angular momentum is given by

,,,,/~... . " . 4.0

L =

I[/~ I


Z,,~ " -




, ~ - - - - ( 2 ; 3 - 2;2) X/1 + p 2 260

(2;3 -- 2;2)

The moment of inertia 9 (bl~)ndin the fixed deformation case obtained by (602 = dJ2,603 = ~ 3 )


9(1)band = L/60 = C J r i g i d / 1 ~ 2 ZXI 20

1'I' ~ 13/2 I



9(2) band = dL/d60 = 9rigid/(1 + p2)3/2

Fig. 3a. The angular momentum I as a function of rotational frequency. The dashed line corresponds to each configuration and the solid line slJows the yrast configurations for each rotational frequency. For details, see the text. Fig. 3b. Angular momenta ~r and] a defined by eqs. (10) and (11). The lower lines show the angular momentum difference AI and the upper curves are the values/a.

+60 x )

60t 1. (12)

This hamiltonian can be solved in an analytic way and the detailed results are shown elsewhere [7,8]. We discuss the moment o f inertia problem using this model without the A N = 2 term since the A N = 2 contribution is of the order o f (6o 2 - 603)2/(602 + 603)2. The cranking hamiltonian ( l 2) can be diagonalized b y a linear transformation, the result being h c = ( C ~ C 1 + ~) h601 +





h e = p 2 / Z m + ( m / Z ) ( 6 0 2 X 2 + w 22X 2


where 9rigid = 2(2 3 -- 2;2)/(c52 -- ~5,3) ~ 2;2/~2 + 2;3/ c53 if one takes the self-consistent candition d) 1E 1 = d~2Y,2 = c~3N 3 at the rotational frequency 60 = 0.0. The moment of inertia c~(2) is also obtained by "band using eq. (14),

1"~ i 13/2

: /

'n: h 9/2




(b} 4.0


,~/4602 + (6O2 -- 603)2 A

(a) 0


(1 6)

Eqs. (15) and (16) show that b o t h 9(bla)ndand 9(b2a)nd change as a function of 60, but the value 9(2~) d is rather rapidly decreasing. When one considers a situation Co2 - cb 3 = 0.2 w 0 and ca = 0.05 600 for a nucleus A = 160 (this is a typical deformation in the rare-earth nuclei and ca corresponds to L ~ 30 h), the value 9(2a)nd becomes 70% of 9rigid , but "bandqr(1)is still 90% o f the value Carrigid. As an additional condition to the hamiltonian (12), some authors take that o f the isotropic velocity field in the nucleus for each rotational frequency [7,8]. This condition is equivalent to the self-consistent condition at co = 0 and gives the following condition between the deformation and the rotation: (602 - 603) 2 = ( ~ 2 - ~ 3 ) 2 - 4602 ,


which means the deformation changes gradually from the axial-symmetric prolate to triaxial and eventually to oblate deformation. When we take this condition for the moment of inertia problem, the value 9 (1~)a will be constant: 241

Volume 96B, number 3,4


~r(22nd = 2(N 3 - N2)/'~/4602 + (6o2 - 603) 2 = 2 ( ~ 3 - ]~2)/(C02 - cb3) = 7rigid


and CJ(b2a)nais also constant and equal to ~r (bl)na under the condition (17). So for the harmonic oscillator model the lnglis moment of inertia decreases with increasing rotational frequency, but the second term of eq. (8) completely compensates for this decrease. We should stress the difference between the harmonic oscillator model and other potentials. First, the harmonic oscillator model with the condition (17) gives a definite path from prolate to oblate in the/3-3' plane. On the contrary, the Nilsson potential shows many different paths [e.g. 5] depending on mass number. Secondly, the harmonic oscillator model does not have band crossings, whereas in the present calculation, band crossings play a very important role to gain angular momentum of the nucleus. However, it is not clear how a consistent treatment of deformation by evaluating the second term in eq. (8) may change our result. We have restricted our attention to the few bands which are involved in yrast configurations at some rotational frequency. Experimentally, it is quite difficult to specify where the ?,-transitions come from the very high-spin region. However, our result in connection


3 November 1980

with fig. (1) indicates that the higher bands above the yrast configuration will show the same behaviour of cjr(2) band since the spin-alignment, being responsible for the decrease of cr(2) J band, also occurs in the excited bands. We gratefully acknowledge discussions with S. Bj~brnholm, J.D. Garrett and G. Leander. Especially, we thank B. Mottelson for critical comments to the manuscript and K. Neerg~rd for help in deriving the relations from (3) to (9). An acknowledgement for financial support from the Danish National Science Research Council is also expressed herewith.

References [ 1] D.L. Hills et al., Nucl. Phys. A325 (1979) 216. [2] J.D. Garrett and B. Herskind, Talks at the Study Weekend on Nuclei far from stability (Deresbury, Sept. 1979). [3] M.A. Deleplanque et al., Phys. Rev. Lett., to be published. [4] G. Leander, preprint (Lund, October 1979). [5] G. Anderson et al., Nucl. Phys. A268 (1976) 205. [6] R. Bengtson, I. Hamamoto and R.H. Ibarra, Phys. Scripta 17 (1978) 583. [7] A. Bohr and B.R. Mottelson, Nuclear structure, Vol. 2 (W.A. Benjamin, 1975). [8] G. Ripka, J.P. Blaizot and N. Kossis, Heavy-ion, high-spin states and nuclear structure, Vol. 1 (IAEA, Vienna, 1975) 445.