Correlation of Kinetic Moment of Inertia with Power Formula Index

October 24, 2012 17:3 WSPC/143-IJMPE S0218301312500826

International Journal of Modern Physics EVol. 21, No. 10 (2012) 1250082 (13 pages)c© World Scientific Publishing Company

DOI: 10.1142/S0218301312500826

CORRELATION OF KINETIC MOMENT OF INERTIA WITH

POWER FORMULA INDEX

RAJESH KUMAR

Department of Physics, Noida Institute of Engineering & Technology,

Greater Noida 201306, India

[email protected]

VIKAS KATOCH

Raj Kumar Goel Institute of Technology, Ghaziabad, India

Singhania University, Jhunjhunu, Rajasthan 333515, India

S. SHARMA

Panchwati Institute of Engineering & Technology, Meerut 250005, India

J. B. GUPTA∗

Ramjas College, University of Delhi, Delhi 110007, India

Received 18 July 2012Revised 28 August 2012Accepted 29 August 2012

Published 18 September 2012

The level energies of ground band of even Z, even N nuclei may be reproduced well withgood accuracy by using the power index formula E = aIb. In an earlier study of thedependence of the kinetic moment of inertia (MoI) J(1) on spin I, a possible correlationof the MoI J(1) with power index “b” was suggested. Here we illustrate that the slope ofthe kinetic MoI versus spin I corresponds to the magnitude of the index “b” for severalisotopes in the A = 100–150 region. The validity of the formula is illustrated for lightnuclei in A = 100 region and its use for studying shape phase transition at N = 60.

Keywords: Nuclear structure; kinetic MoI; power index formula.

PACS Number(s): 21.10.Re, 21.60.Ev, 27.60.+j

1. Introduction

In the unified collective model of Bohr–Mottelson,1 the level energies are expressed

in terms of the rotation–vibration of the nuclear core. The rotational motion is

characterized by the moment of inertia (MoI) “J” of the rotating core.

∗Associated.

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mailto:[email protected]


R. Kumar et al.

For a good rotor, the level energies are given as EI = ~2I(I + 1)/2J.

At low spin, the MoI is determined approximately by the expression J(21) =

3/E(21).

In the heavy-ion induced reactions with medium mass nuclei, the nucleus is

excited to high spins and for most nuclei, the MoI varies with spin I. From

the regular γ-ray peak spectra one deduces the kinetic MoI J (1) in terms of the

consecutive Eγ ,

J (1) =(2I − 1)

Eγ(1)

based on the rotation model (RM) formula. For an ideal rotor nucleus, the slope

of J (1) versus spin I would be zero for a K-band. But in general, the average

slope of the MoI curve varies nucleus to nucleus, being greater for a less deformed

nucleus. This signifies the nuclear structure change with N , Z and spin I. Further

information is obtained by calculating the dynamic MoI J (2) = 4/(Eγ2 − Eγ1).

Often, the plot of MoI is given versus ~ω to study back bending phenomenon.

To express the level energy in ground band, a power index formula,

E = aIb (2)

was proposed earlier by Gupta et al.2 It was demonstrated therein that this formula

reproduces the energy spectra of deformed nuclei and of the shape transitional

nuclei in the medium mass region A = 100–200, with reasonable accuracy for low

spins below back bending. Here the index “b” is a measure of the deformation of

the nuclear core, being equal to 1.0 for a spherical nucleus and about ∼ 2.0 for a

deformed rotor. The coefficient “a” plays the role of inverse MoI. For light nuclei

of Xe–Sm nuclei, its validity was verified by Mittal et al.3

Recently, Gupta and Hamilton4 pointed out that the absolute slope of the kinetic

MoI J (1) versus spin I, using the expression (1), represents only the deformation of

the core, not the change in nuclear structure with spin. This is important, especially

for near spherical nuclei, which yield larger slope of J (1) and small value of index

“b”. Instead, it is the change in slope with spin I, which represents the variation

of nuclear structure with spin. The same should be reflected in the change in value

of “b”. In Sec. 2, we illustrate this aspect of nuclear structure by looking at the

correlation of the variation of kinetic MoI J (1) for increasing spin and the value

of “b” in the power index formula for a few nuclei in the A = 100–150 region.

In Sec. 3, power index formula is used to illustrate the shape phase transition at

N = 60. Conclusions are given in Sec. 4.

2. Results

2.1. Correlation of J(1) with power index “b”

We illustrate this correlation of slope of J (1) with magnitude of index “b” for two

nuclei: 148Ce and 148Nd (N = 90, 88), of different deformation, through the plots

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Correlation of Kinetic MoI with Power Formula Index

0 4 8 12 16 20

20

30

40

50

60

70

148Ce

J(1

) (M

eV

)-1

SPIN (I)

Fig. 1. Plot of MoI J(1) versus spin I for 148Ce.

0 4 8 12 16 20

1.47

1.48

1.49

1.50

1.51

1.52148

Ce

b

SPIN (I)

Fig. 2. Plot of index “b” versus spin I for 148Ce.

of MoI J (1) and of index “b” versus spin I. The Eγ data is taken from Ref. 5, in

which levels up to high spins were assigned for Ba, Ce and Nd isotopes, using high

efficiency detector array and 252Cf as the source. In Figs. 1 and 2, we have plotted

MoI J (1) and index “b” against the level spins of the ground band in 148Ce. The plot

of J (1) versus I for 148Ce shows a 300% rise in value from 20 to 60 MeV−1, up to

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R. Kumar et al.

0 2 4 6 8 10 12 14

10

15

20

25

30

35

40

148Nd

J(1

) (M

eV

)-1

SPIN (I)

Fig. 3. Plot of MoI J(1) versus spin I for 148Nd.

0 2 4 6 8 10 12 14

1.300

1.304

1.308

1.312

1.316

1.320148

Nd

b

SPIN (I)

Fig. 4. Plot of index “b” versus spin I for 148Nd.

I = 20 and the slope is almost constant, except a small rise at I = 12. If the above

hypothesis is valid, then a constant slope of J (1) should correspond to a constant

magnitude of index “b” (see Ref. 4). The same is approximately realized in the plot

of index “b” versus spin I (see Fig. 2). Its magnitude varies less than 3% from 1.52

to 1.47 only. Note the highly magnified scale of “b”. This slight decrease corresponds

to slight move towards vibrational. The decrease in index “b” corresponds to the

increase of MoI.

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4 8 12 16 20

0

1

2

3

4

5

6

152Ce

150Ce

148Ce

diff. J

(1) (

Me

V)-1

Spin (I)

Fig. 5. Differential slope of MoI in Ce.

4 8 12 16 20

-0.01

0.00

0.01

0.02

0.03152

Ce

150

Ce

148

Ce

diff. b

Spin (I)

Fig. 6. Differential of index “b” in Ce.

The same features are reflected in 148Nd (see Figs. 3 and 4). Here J (1) increases

by 350% up to spin I = 12, (for a more vibrational nucleus). Again “b” is almost

constant (within 2%). This clearly illustrates that the constant slope of J (1) implies

a structure, constant with spin. The absolute rise of J (1) with spin I, is due to the

use of rotational model expression (2I − 1) in the numerator of Eq. (1).

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R. Kumar et al.

4 8 12 16 20

0

50

100

150

200

250

152Ce

150Ce

148Ce

Dynam

ic M

oI J

(2) (

Me

V)-1

SPIN (I)

Fig. 7. Plot of dynamic MoI J(2) for Ce.

To further test the above hypothesis, we present the differentials of MoI (= slope)

and of index “b” in Figs. 5 and 6 for 148–152Ce. The slight change (increase) in slope

of MoI with spin (Fig. 5) corresponds to the slight decrease in “b” (Fig. 6). The

slopes of the curves of differentials of J (1) for the three isotopes of Ce are markedly

different. The same is reflected in the slopes of differentials of “b”. For example, the

near constancy at I = 6–20 for 150Ce in Fig. 5 correspond to the near constancy

of differentials of “b” in Fig. 6. The continuous rise of differential of J (1) in 152Ce

(see Fig. 5) corresponds to continuous fall of “b” (see Fig. 6). In 148Ce, the rise of

J (1) at I = 10 corresponds to the fall in differential “b” at the same spin. Thus

a correlation of the slope of J (1) and of index “b” is apparent. In the plot of J (1)

versus ~ω in Ref. 4, the changing slope is well visible. Generally, no quantitative

analysis is done of these structural changes. In the present work, we have expressed

these variations on a numerical basis.

The index “b” has a role similar to the energy ratio R4/2. Herein, one has

numeric measure of the degree of deformation of the nucleus as well as its variation

with spin. By looking at the differential slope of J (1) or at the variation of “b”, one

gets a transparent numeric estimate, which shall provide further clues for nuclear

theory. The dynamic MoI J (2) also gives an indication of the changing structure

(see Fig. 7), which is maximum in 148Ce.

2.2. Comparison with other models

Regan et al.6 proposed a simple expression, called E-GOS=Eγ/I, which helps to

distinguish the vibrational nucleus from the good rotor. The relative rapid fall of

E-GOS versus spin I for 148Ce plot (Fig. 8) than for 150–152Ce exhibits the more

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0 4 8 12 16 20

30

40

50

60

70

80

152Ce

150Ce

148ce

EG

OS

(ke

V)

SPIN I

Fig. 8. Plot of E-GOS versus spin I for Ce.

vibrational status of the former. Recently, the soft rotor formula (SRF)7:

EI =~2I(I + 1)

2J0(1 + σI), (3)

based on the concept of increase of MoI “J” with increasing spin I, due to centrifugal

stretching and coriolis anti-pairing effects, was found fairly successful.7 Applying

SRF Eq. (3) to 148–152Ce, one gets a fairly good reproduction of level energies,

which improves if least square fit is used to deduce σ and J0. Without LS fit, we

get values of σ = 0.0983, 0.0316 and 0.0159 for A = 148–152, respectively. The

respective values for ground state MoI J0 are 15.0, 29.0 and 35.6. Then Eq. (3)

yields values of J (I) increasing with spin I linearly. For example, in 148Ce, the

value of J (I) rises to 40.7 at I = 16, representing a 250% rise. The rise is less for

more deformed N = 92, 94 isotopes.

3. Study of Shape Phase Transition at N = 60 in Zr and Pd

The shape phase transition of nuclei with N , Z is a topic of great interest. Here we

illustrate the application of the power index formula for making these shapes change

transparent. It is well known that at N = 60, there is sharp phase transition, being

maximum for Sr and minimum for Pd, Cd (Ref. 8). Below N = 60 the structure

is vibrator like and at N > 60, the structure corresponds to a rotor. The energy

of 2+1 drops sharply at N = 60 (see Ref. 8). But this drop in energy is very much

dependent on the Z value as well. The sharpest drop is for Sr and it decreases for

Zr, Mo, Ru and Pd successively. This represents the strong effect of the 1νg7/2 sub-

shell closure at N = 58. Here the collectivity builds up quickly with the addition of

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R. Kumar et al.

2 4 6 8 10 12

0.6

0.8

1.0

1.2

1.4

98Pd

100Pd

102Pd

104Pd

106Pd

108Pd

110Pd

b

SPIN

Fig. 9. Plot of index “b” versus spin I for Pd.

valence neutrons, beyond N = 58. The simultaneous effect of the valence protons

is exhibited in the Z-dependent behavior.

The ground band itself exhibits these changes in collectivity. It is of interest

to reproduce these energy variations in a suitable energy formula. The single-term

power index formula [Eq. (2)] has the advantage that only two terms of the spectrum

can yield the values of the two parameters of the formula, which can be used to

predict the energies at higher spins. We illustrate its usefulness for studying the

level structure of 98−110Pd and 100−104Zr around N = 60.

Equation (2) may be solved by taking the ratio for I = 4 and 2. This eliminates

“a”. Then taking log, one gets the index “b”. This is repeated for all available spin

I in the nucleus. Then one finds an average of all “b”. This average “b” is used to

determine the values of “a” for all spin I. Using the average “b” and average “a”,

one recalculates the level energies to test the validity of the power index formula

[Eq. (2)]. A root mean square deviation (RMSD) of the energies from the experiment

determines the goodness of fit of energies to the two empirical parameters “a”

and “b”.

The values of b(I) versus spin I are plotted for each of the Pd and Zr isotopes in

Figs. 9 and 10, respectively. In Pd, the N = 62–64 isotopes have larger “b”, steady

with spin I. At N = 58, 60 fewer data are available,9 and not much different from

those at N = 62, 64. At N = 56 there is some drop in “b”, which drops further at

N = 54. At N = 52, an almost spherical nucleus, not only “b” is small (less than

1.0), it varies with spin sharply at I = 6–10. Average value of “b” = 1.0 corresponds

to a spherical vibrator. The maximum value of “b” in Pd is ≤ 1.4, which is far below

the value of 2 for a good rotor. Thus, in this region of A = 100, even at mid shell

a full deformation is not reached in Pd.

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2 4 6 8 10 12

1.12

1.28

1.44

1.60

1.76 104Zr

102Zr

100Zr b

SPIN

Fig. 10. Plot of index “b” versus spin I for Zr.

Table 1. Average “b”, “a” (keV) and RMSD (keV) of energy values in Pd and Zr.

N = 52 54 56 58 60 62 64

Pd

Avg. b 0.861 1.090 1.217 1.264 1.269 1.273 1.313

Avg. a 480.5 313.0 240.2 231.8 212.4 179.4 150.8

RMSD 178 32.8 48.2 12.3 7.3 34 38.61

Zr

Avg. b 1.481 1.686 1.732

Avg. a 77.2 47.4 42.2

RMSD 93 24 41

As seen in Fig. 10 for Zr, “b” is highest at N = 64 and is ∼1.73, maximum being

≤ 1.9 for good rotors. Also, it is nearly constant with spin for N = 62, 64 within 3%.

At N = 60 in 100Zr, average “b” drops to 1.48 and there is significant variation (8%)

of “b” with spin. The increasing “b” reflects the increase in deformation with spin.

That is, it is a “β” soft nucleus. This is expected at the shape transition point. The

energy ratio R4, is only 2.66 which increases to 3.15 in 102Zr and to 3.22 in 104Zr.

Hamilton et al.10 explained this sharp phase transition in terms of the reinforcement

of Z = 38, 40 sub-shell gap and the N = 60 neutron deformed sub-shell gap.

From the life time measurement of the 2+1 state in 104Zr, the extracted deformation

parameter β ∼ 0.47 (see Ref. 11) indicates that 104Zr has large deformation. At

N = 58, there is a sharp drop in the value of R4 and the rise in E(21). For N ≤ 58,

only few levels in ground band are available. So no plot is shown for 94−98Zr.

In Table 1, the values of average “b” and “a” are listed for Pd and Zr isotopes.

The increase of “b” with N signifies increasing deformation in Pd, Zr. On the other

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R. Kumar et al.

Sr Zr Mo Ru Pd Cd

1.2

1.6

2.0

2.4

2.8

3.2

N=64N=62

N=60

N=58N=56

N=54

R4

Fig. 11. Plot of R4 versus Z.

Sr Zr Mo Ru Pd Cd0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

N=64N=62

N=60

N=58

N=56

N=54

Avg. 'b

'

Fig. 12. Plot of average “b” versus Z. For R4 < 2.0, “b”< 1.0.

hand, average “a”, which corresponds to inverse MoI, decreases with N . An RMSD

of about 30 keV (∼ 2% deviation) is obtained in the use of the power index formula

in these not so well deformed nuclei. For R4 < 2.0, the value of index “b” falls below

1.0 and value of coefficient “a” is irregular. This sets a limit on the use of Eq. (2).

In Figs. 11 and 12, we illustrate the role of atomic number Z in producing the

deformation of a nucleus for N = 54–64 in a plot of R4 (= E4/E2) versus Z. The

shape phase transition is sharp at Z = 40 for N = 54–58, but smoothens out for

N ≥ 60. Same is exhibited in the plot of index “b” versus Z (see Fig. 12). This

indicates “b” is as good a measure of the core deformation as the ratio R4.

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38 40 42 44 46 480.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

CdPdRuMoZrSr

N=64

N=62

N=60

N=58

N=58

N=54

N=56

B(E

2)

Z

54

56

58

60

62

64

Fig. 13. Plot of B(E2, 0g → 2g) versus Z.

54 56 58 60 62 64

0.0

0.4

0.8

1.2

1.6

Cd

Pd

Ru

Mo

Zr

Sr

B(E

2)

N

Sr

Zr

Mo

Ru

Pd

Cd

Fig. 14. Plot of B(E2, 0g → 2g) versus N .

The variation of nuclear structure is also reflected in the E2 transition strength.

The plot of B(E2, 0g → 2g) versus Z (Fig. 13) exhibits the large variation with

N at Z = 38, 40. The variation with N is increasingly less for higher Z. This also

corresponds to the variation of R4 and index “b” (Figs. 11 and 12). It is instructive

to see the variation of B(E2, 0g → 2g) versus N (Fig. 14). The variation with N is

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54 56 58 60 62 64

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Cd

Pd

Ru

Mo

Zr

Sr

Avg

. 'b

'

N

Sr

Zr

Mo

Ru

Pd

Cd

Fig. 15. Plot of average “b” versus N .

almost zero for Cd (with two proton holes) and is maximum for Zr, Sr, especially

at N = 58–60. Again there is correspondence of variation of index “b” with N

(Fig. 15) and of B(E2).

4. Discussion

The original suggestion of Power index formula in Ref. 2, represented an alternative

simple expression for relating E(I) to spin I, as compared to several other formulae

in literature. In Ref. 3, the validity of the index formula was tested for light nuclei

and a comparison with other formulae was presented in terms of RMSD. However,

in Refs. 4 and 12, a very basic new concept was put forth to consider the role of

rotor model in shape transitional nuclei. This was illustrated by a comparison of

variation of kinetic MoI J (1) with spin, which apparently gives an impression of

large increase in J (1) with spin I in shape transitional nuclei, but was not realistic.

The use of power index formula yields a more realistic relation of E(I) with spin I.

Here one does not get a rapid change with spin I. Since this involves a significant

conceptual alternative, in the present work, the correspondence of the change in

the slope of the kinetic MoI J (1) and the power formula index “b” is made more

transparent. We have shown that the structural variation with spin is better seen in

a differential plot of MoI or index “b”. This also corresponds to the role of dynamic

MoI, which however, is not interpreted numerically. These deductions yield a more

quantitative picture of the changes in nuclear structure with spin I.

Further, we have illustrated the use of power index formula of level energies for

the study of shape phase transition at N = 60. The much sharper shape transition

in Zr, Sr, and slower for Pd are exhibited. There being only two free parameters,

one can study the nuclei with fewer levels as well. Further the constancy of “b” with

spin in nuclei not under going shape change with spin is also tested.

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References

1. A. Bohr and B. R. Mottelson, Nuclear Structure (W. A. Benjamin, 1975).2. J. B. Gupta, A. K. Kavathekar and R. Sharma, Phys. Scripta 51 (1995) 316.3. H. M. Mittal, V. Devi and J. B. Gupta, Phys. Scripta 81 (2010) 01-5202.4. J. B. Gupta and J. H. Hamilton, Phys. Rev. C 83 (2011) 064312.5. S. J. Zhu et al., J. Phys. G 21 (1995) L75.6. P. H. Regan et al., Phys. Rev. Lett. 90 (2003) 152502.7. P. von Brentano, N. V. Zamfir, R. F. Casten, W. G. Rellergert and E. A. McCutchen,

Phys. Rev. C 69 (2004) 044314.8. R. F. Casten, Nuclear Structure from a Simple Perspective (Oxford University Press,

1990).9. Brookhaven National Laboratory (USA), Chart of Nuclides of National Nuclear Data

Center, www.nndc.bnl.gov.nsdf.10. J. H. Hamilton et al., Prog. Part. Nucl. Phys. 35 (1965) 635.11. J. K. Hwang et al., Phys. Rev. C 73 (2006) 044316.12. J. B. Gupta, Third International Conference on Fission an Properties of Neutron-

Rich Nuclei, eds. J. H. Hamilton, A. V. Ramayya and H. K. Carter (World Scientific,Singapore, 2003), p. 320.

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