Moment of Inertia Calculations for Some Even-Even Rare Earth Nuclei

Moment o f inertia calculations for some even-even rare earth nuclei*

D. S. Ra|

Departmeni o f Physics, OovemmeiU Post GradvuUe College, Bilaspur, M'.P.

{Received 15 Ma0 1973)

A comparative study of various formulations, based on three different models, viz., Cranking model, Centrffugal stretching model and Governor model, for the angular momentum dependent moment of inertia has been made. Two other formulations based on these models and their comparative merits have also been discussed.

On the basis o f the non-rigid rotator model a new formula has been proposed for the variation of moment of inertia with angular momentum. Rare-earth even-even nuclei have been considered throughout the treatment.

1. Intbodttotion

The rotational energies o f deformed nuclei are given on the basis of the hydro- dynamical model (Bohr & Mottelson 1953) by

Ei = A I ( I + l ) ^ B P { I + l ) ^ ... (1) where B is rotation-vibration coupling constant and can be determined from the energies o f excited 2*^ and 4“^^ states. But as is evident from consequent studies (Kane et al 1953, Emery et al 1963), this model does not give values in agreement with experiment for all spin states.

As the vibrations o f a rotating nucleus due to nuclear shape elasticity (centrifugal stretching) are o f two types, viz., /ff-vibrations and y-vibrations, Davydov & Ghaban (1960) considered the coupling between rotation and /?-vibration for explaining the dependence of moment of inertia on the degree of rotation. This leads to an increase in the moment o f inertia. Contrary to the assumption that /?-vibrations are balanced by harmonic restoring forces, some authors (Davydov & Filippov 1958, Davydov & Rostovsky 1959) pro­

posed a theory with adiabatic approximation where the rotation of the nucleus was assumed as taking place without the change o f intrinsic state. But in even- even deformed nuclei considered in this work, the adiabatic condition is violated (Davydov I960). On the basis o f the Cranking model, moment of inertia may

* This work was undertaken at the Government Engineering College, Bilaspur (M.P.) 677

678

D. S. R ai

bo calculated from the dependence of the total energy of the sygJtem on the rate of rotation of average nuclear field in which the nucleons move. Incorporating higher order perturbation terms, Harris (1964a, 1964b) gave the formulation,

“ 2In (1 .24a;®+.. .)> (2⁾

where x ==

and

T 3

r o v < 01 > < wl J * lw > < w ] Ja,IO >

■ “ ~ {E n -E ^ (Em -E^) (E p -E ^)

mn p

-io S m

<^01

{Efn'-EQ)^

for rotational energies.

Sood (1967) used the theory of molecular spectra to obtain an estimate of the higher order corrections to energy levels of a non-rigid rotator. Ho arrivf^l at a two-parameter description of rotational energies by an effective summation of the geometric series guided by the relative magnitude of the successive torniK, thereby presenting a simple expression for the angular momentum-dependent moment of inertia. On tlie basis of the classical hydrodynamical model a two- parameter formula for rotational energies has also been suggested by Moszowski (1966). Again Sood (1968) suggested a formulation (within hydrodynamical model) for the energy o f a classical rotator which includes the effect of centrifugal forces.

Recently a new theory of rotational states called the Governor model has been proposed by defining the portion of the nucleus taking part in rotation as the mass lying outside a rotationally invariant core (Gupta & Trainor 1968, Gupta & Trainor 1969, Gupta 1967, Gupta 1969). In this theory the nucleus is considered as an axially symmetric non-rigid rotator and the stretching effect>s are accounted for in terms of harmonic restoring forces. Krutov (1968), on the- other hand, has defined the collective rotational motion as change o f mass donsity distribution in time such that any motion not changing the mass density din- tribution also does not contribute to the energy of collective motion. Expression for the effective moment of inertia has also been obtained cm this basis (Krutov 1968, Krutov el al 1969).

In the present work, various formulations fo r the angular momentum dependent moment of inertia, based on the above models have been com paratively studied. The wc»^k o f Wold (1969) has been disoussed in some detail.

A formula different from onea used by various authors lias been proposed and its performance discussed in the present paper.

2. Co m p a r a t i v e St u d y f o r Ro t a t io n a l En e r o ie s

Proposing that higher order terms in the perturbation expansion of Cranking model formulae are important, Harris (196A) suggestoti the oxx)rcasion (2) for rotational energy. This expression is found to reproduce the experimentally observed rotational spectra satisfactorily. |But since different effectives values of moment of inertia inertia enter in calci|lations of rotational energies and angular momentum, it seems as if a uniqu^ moment of inertia does not exist in the usual sense when terms o f higher ordoj in w are considered in the ex­

pressions

E l = Ja»*(Io+3Co>2), [/(/■ f 1)11 = 6>(Io+2Cto2),

(3a) (3b) wiioro o) is the angular velwhty. This work therefore sheds no light on the angular momentum dependent moment o f inertia througli experimental rotational energies are reproduced within an error o f ±0*3% .

Sood (1967) has derived the relation

E / - A 7 ( / + l ) 1 + -

B (4)

l + A " 7(7+1)

assuming that the terms within the square brackets in the following equation form an infinite geometric series.

AV = A 7(7+ 1) I 1 - | - / ( / + ! ) [ • ]}■ ••• (5) Tlio value o f N was found to be equal to (2-85--0 05/). The above expression reproduces experimental spectra within 1% for rare-earth even-o,ven nuclei.

However, satisfacaory results are not obtained for highly neutron deficient nuclei, although the agreement for nuclei at both ends o f the deformation spectra is faii\

An expression has also been presented (Sood 1967) for the angular momentum dependent moment o f inertia

(6 )

680

D. S. R ai

Here N has to be given different values for different groups o f deformed nulcei if the pattern o f agreeiment of rotational energies with the experimental values is to be followed. The other formulation by the same author (Sood 1968) also gives better and simpler description o f observed rotational energies than other models e.g., B -M model and rotation-vibration model.

Assuming that no y-vibrationa are present, Gupta & Trainer (1968, 1969) consider that the nuclear moment o f inertia is modified duo to centrifugal stretch­

ing and the yff-vibrations are balanced by harmonic restoring forces. They have presented for the rotational energies the expression

El A*

= T 7 ( /- f l)[l-a;+3aT*-13a:3+68a^ •••],

•*^0 (7)

where

X ■■ B ft*

j / ( / + ! ) and = A,

which contains parameters of both nuclear softness and stretching {cr),

[(Tj = ar(l--3ar+15a:2~-91a?3+612a:4+...)]. ... (8) Compared to the pure rotation, the two parameters and the throe parameters descriptions, this formulation gives certainly better results when compared to the experimentally observed energies {Gupta 1967). However, the agreement is found to be dependent on the value of the parameter BfA. Also as the series (7) is asymptotic in nature, the rotational spectra given by it breaks down for large values o f BjA and for high spin states. Compared to the other models (Harris 1965, Stephens et al 1966) this model reproduces the experimental rota­

tional energies bettor upto tho spin state 10+. But after this state the agree ment worsens and the model o f Stephens et al (1965) fits better.

As an extension of the above model an expression for the angular momentum dependent moment of inertia has also been derived (Gupta 1969b)

... (9) 3 ^ < J ^ 9 ’

where S = S|₄^/₅|₇r ^ = deformation parameter )and ₈j is relative atretohing parameter. Later in this work, this expression and the results obtained with it will be discussed in detail.

3. D jm iv A T io K OS' Ex p r e s s io n p o b An o u l a b MoMENTtmt De p e n d e n t

Mo m e n t o p In e r t i a

This is based on the non-rigid rotator mc^el (Gupta 1969). Assuming that it {weserves axial symmetry and it has prolate deformation, the nuoleua is con­

sidered to bo divided by a plane conl|fiming the axis o f rotation and perpendicular to the axis o f symmetry, into two parts. Proceeding on the basis o f these con­

siderations we get the total energy of rotation i.c., the sum of kinetic and potential energies

(10⁾

where = distance between the centres o f liiass of the two parts in the stretched condition o f the nucleus, and — distanc^ between the centres of mass o f the two parts in the equilibrium condition of Sthe nucleus. Then

(

11

where cr (relative stretching) = and A = 2 Io-

Putting the values o f K and Iq (ground state moment o f inertia) from B ^

A ~ ’

and

I„ = mr^

in expression (11) and adjusting terms,

^ / ( / - l - l ) . . I B

Er 1 7 7 ^ ( f ^ ) ' / * ( i + ! ) * ( ! - 3 * + 16a;*-91*»+...)*,

(12a)

(12b)

(13)

where a: = § / ( / + l ) .

From eq. (13) we can get

IrrJL — A I ( I + l ) (14)

which is the propoi^ed expression for angular momentum dependent moment of inertia.

4. Eestots and Discussion

Table 1 gives the values o f the angular momentum dependent moment of inertia, expressed as the ratio Ij/Iq, for some rare earth even-oven nuclei. Sood’s expression (6), tho proposed expression (14) and the exprossiori of Wold (1969) have been used upto the spin state 10“^. The latter expression will be discussed

682

D . S, R ai

later in this work. The values o f the parameters A and BjA have been taken from Gupta (1969a).

Table 1. Angular momentum dependence o f moment of inertia for some rare earth nuclei

Nucleus BjA % h llo

1^ 2 + 4'^ 6+ 8+ 10^

Yh^70 0.078 Sood 1.006 1.016 1.031 1.060 1.077

Author 1.008 1.029 1.064 1.103 1.143

Wold 1.148 1.261 1.395 1.464 1.543

Hfm 0.147 Sood 1.009 1.028 1.066 1.088 1.134

Auahor 1.017 1.065 1.106 1.151 1.177

Wold 1.183 1.321 1.487 1.669 1.669

W17B 0.102 Sood 1.006 1.019 1.040 1.066 1.097

Author 1.011 1.039 1.089 1.162 1.225

Wold 1.162 1.283 1.428 1.493 1.690

Gdise 0.235 Sood 1.013 1.043 1.086 1.129 1.191

Author 1.027 1.085 1.166 1.172 1.195

Wold 1.220 1.385 1.684 1.670 1.801

0.346 Sgod 1.020 1.062 1.120 1.171 1,251

Author 1.040 1.087 1.173 1.191 1.277

Wold 1.269 1.465 1.689 1.792 1.946

Wold (1969) has tried to explain the deviation from the / ( / ^ - l ) law in terms of the centrifugal stretching model of Sood (1967). Assuming the effective resort­

ing force to bo equal to mato^ (where a is a function o f and r^), the expression E 2 1l r i i + i ) \

1 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1+/?*/(/+ l)+2y?{/(/+l)}*

for rotational onorgiea been derived. Here is a positive parameter dolor mined from the two lowest energy levels for each nucleus. Two expressions

Ti and

(

10

(17) have also been provided for the angular m om entum variable m om ent o f inertie- In expression (16) n in such that 0 ^ n ^ 1. Now, if the experimental data fo r rotational energies is to be reproduced satisfactorily n should have the value .0*8 (W pld 1969).. If, how ever, » « 1*0, the curves o f W old (1969) showing

oq. (16) gives smaller and thus better values of T//T„ for 7? — 0*8 than oq.

(17). Table 1 shows these values calculated from the values of parameter /i given in Wold (1969).

From table 1 it is clear that W old’s formulation for the angular momen­

tum dependent moment of inertia fails for all spin values as it gives values very different from ours as well as from Sood’H values. Values of !// ! „ calculated on the basis o f the proposed relation are syslematically greater than those of Sood wliich points to the somewhat better accuraoy of Sood’s formulation. As explain- cd in detail in another work (Gupta 1969a) the parameter x =- ^ / ( /B 4-I) plays an important part in the summation of the sedes occurring in expression (13). For values o f x less than 0*1 the series converges rapidly. For values greater than tlus, the summation can only be performed by taking the last terms of the series as th(^ average of the two previous successiye terms. The five nuckn for which tlui values of I //I0 have been calculated, have been chosen with a view to compare the calculated values o f I //I0 for nuclei having the minimum as well as the maxi­

mum value of BfA, It should be remembered that this factor, as it constitutes Urn weight factor x, also has a significant effect on the values of I //I0 parti­

cularly when its value is high.

It may also be remarked that the formulation proposed by us has an added advantage in that it is simpler and more systematic in appearance and compares favourably to that of Gupta (1969b).

Acknowledgment

The author is grateful to Professor 0 . N. Kaul for encouragement and guidance. The author also wishes to thank the Principal, R. A. Doshpando of Government Engineering College, Bilaspur, M.P., for providing necessary facilities.

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