SOME DEFORMATION PROPERTIES OF THE EVEN EVEN YTTERBIUM, HAFNIUM AND TUNGSTEN NUCLEI

1st ReadingAugust 17, 2012 10:34 WSPC/143-IJMPE S0218301312500772

International Journal of Modern Physics E1

Vol. 21, No. 0 (2012) 1250077 (38 pages)c© World Scientific Publishing Company3

DOI: 10.1142/S0218301312500772

SOME DEFORMATION PROPERTIES OF THE EVEN EVEN5

YTTERBIUM, HAFNIUM AND TUNGSTEN NUCLEI

S. B. DOMA∗ and H. S. EL-GENDY†7

Faculty of Science, Alexandria University, Moharram Bay, Alexandria, Egypt∗[email protected]

†[email protected]

Received 9 May 201211

Revised 1 August 2012Accepted 6 August 201213

The deformation structure of the even–even ytterbium, hafnium and tungsten nuclei isinvestigated in framework of the collective model, the single-particle Schrodinger fluid15

model and the cranked Nilsson model. Accordingly, we have calculated the rotational andvibrational energies, the nuclear moments of inertia, the total ground-state energy, the17

quadrupole moment, the LD energy, the Strutinsky inertia, the LD inertia, the volumeconservation factor ω0/ω

0 , the smoothed energy, the Bardeen, Cooper and Schrieffer19

(BCS) energy and the G-value of the ytterbium: 170Yb, 172Yb and 174Yb, hafnium:176Hf, 178Hf and 180Hf and tungsten: 182W, 184W and 186W nuclei as functions of the21

deformation parameters βγ, which are assumed to vary in the ranges (−0.50 ≤ β ≤ 0.50)and (0 ≤ γ ≤ 60). Also, two polynomials in β are obtained to produce results in good23

agreement with the corresponding results for the total ground-state energy and thequadrupole moment of the mentioned nine nuclei.25

Keywords: Heavy nuclei; even–even nuclei; collective model; cranked Nilsson model;single-particle Schrodinger fluid; total energy; quadrupole moment; moment of inertia.27

PACS Number(s): 21.60.Ev, 27.70.+q

1. Introduction29

It is well-known that the shell-and independent-particle models explain many

nuclear properties, but fail to account large nuclear quadrupole moments and31

spheroidal shapes, which many nuclei possess. It is also, clear that such effects

cannot be obtained from any model, which considers the pair-wise filling of the33

individual orbits of spherical potential to be a good approximation to nuclear struc-

ture. Such large effects can only arise from coordinate motion of many nucleons.35

We may characterize such motion by assuming that the particle motion and surface

motion are couples.37

In the shell model, there is a core made up of paired nucleons. This core may be

spherically symmetric in which case it gives rise to the spherically symmetric of the39

1250077-1

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S. B. Doma & H. S. El-Gendy

independent particle model or axially symmetric, as in the Nilsson model1 which1

also be referred to as the deformed independent particle model. The deformed shell

model is a mean field approach that is more illustrative but gives a less accurate3

agreement with data.

Because the surface is distorted at some moment the potential felt by the par-5

ticles is not spherically symmetric, the particles will move in orbits appropriate to

an anisotropic shell-model potential.1,27

To express the particle- surface coupling mathematically, it is necessary to in-

troduce some collective variables to describe the cooperative modes of motion. The9

simpler model has sometimes been called the collective model and the distorted shell

model the unified model.3,4 Both represent collective effects, although in different11

ways.

The quantum fluid5 is considered to be completely transparent internally with13

respect to motion of the constituent particles, and to receive disturbances solely by

way of surface deformations. Its near incompressibility comes about, not by particle15

to particle push, as in an ordinary liquid, but by more subtle means. It is capable

of collective oscillations, but it is the wall which organizes these disturbances, not17

nucleon to nucleon interactions. Oscillations experience a damping, but the mecha-

nism of the damping is unlike that encountered in ordinary liquids. The rotational19

properties of the quantum fluid are quite different from those of ordinary fluids.

The absolute values of the rotational energies or equivalently the moments of21

inertia require knowledge of the fine details of the intrinsic nuclear structure. Con-

sequently, the investigation of the nuclear moments of inertia is a sensitive check23

for the validity of the nuclear structure theories.6–8

Moreover, the study of the velocity fields for the rotational motion led to the25

formulation of the so-called the Schrodinger fluid.9,10 Since the Schrodinger fluid

theory is at present an independent particle model, the cranking model approxi-27

mation for the velocity fields and the moments of inertia play the dominant role in

this theory.29

The pure Nilsson model cannot be used neither for the calculations of the to-

tal energies nor for the calculations of the shape of the energy surfaces at large31

deformation. So, Strutinsky11 had suggested a renormalization procedure, shell-

correction method, and it became possible to calculate realistic potential energy33

surfaces. Therefore, the oscillator part of the deformation energy has been calcu-

lated within this model and replaced the smooth part by the liquid drop energy at35

the same deformation.2 The systematic solutions for an axially symmetric deformed

nucleus with β- and γ-vibrations, the rotation–vibration model, are later obtained37

by Faessler and Greiner.12–18

The cranked Nilsson Strutinsky (CNS) model19–21 is a theoretical approach39

that provides us with good physics interpretation of the different properties of

deformed nuclei and at the same time allows us to carry out systematic and accurate41

calculations of the different properties of the deformed even–even nuclei.

1250077-2


Deformation Properties of the Even–Even Yb, Hf and W Nuclei

Ya-Wei and Jian-Yu studied doubly magic properties in super heavy nuclei.221

Furthermore, Rajasekaran and Kanthimathi discussed super deformed bands in the

mass regions A ≈ 150 and A ≈ 60 by using the statistical theory and the config-3

uration dependent CNS calculations.23 Moreover, Vijayakumari and Ramasubra-

manian24 studied the structural changes in the Ti nuclei at values of the angular5

momentum which are relatively low and below the fission limit. The main interest

in these studies was concentrated to finite temperature CNS calculations with tun-7

ing to fixed spin. The studies showed that all the considered Ti isotopes are good

candidates for detecting Jacobi shapes. Also, Anu Radha, Ramasubramanian and9

Samuel studied the role of quadrupole deformation in proton emitting nuclei in the

medium mass region.2511

Furthermore, we have recently applied the CNS model, the single-particle

Schrodinger fluid model and the nuclear superfluidity model to calculate the electric13

quadrupole moments and the moments of inertia of the even–even p- and sd-shell

nuclei26 and the obtained results are in good agreement with the available experi-15

mental data.

In the present paper, we have applied the collective model to calculate the17

rotational and vibrational energies of the even–even ytterbium: 170Yb, 172Yb and174Yb, hafnium: 176Hf, 178Hf and 180Hf and tungsten: 182W, 184W and 186W nuclei.19

Moreover, we have applied the single-particle Schrodinger fluid model to calculate

the nuclear moment of inertia of the nine mentioned nuclei by using the rigid-body21

model and the cranking model. Furthermore, we have applied the CNS model to

calculate the LD energy, the Strutinsky inertia, the LD inertia, the volume conser-23

vation factor ω0/ω

0 , the smoothed energy, Bardeen, Cooper and Schrieffer (BCS)

energy, the G-value, the total ground-state energy and the quadrupole moment of25

the nine mentioned nuclei as functions of the deformation parameters βγ, which

are assumed to vary in the ranges (−0.5 ≤ β ≤ 0.5) and (0 ≤ γ ≤ 60). Also, two27

polynomials in β are fitted to obtain formulas, which produce results in good agree-

ment with the corresponding CNS- calculations for the total ground-state energy29

and the quadrupole moment of the mentioned nine nuclei.

2. Triaxial Deformed Cranked Nilsson Strutinsky Method31

In the triaxial deformed CNS method, the nucleons are assumed to move in a CNS

potential25 with the deformation being described by the deformation parameters β33

and γ. The cranking is performed around one of the principal axes: the z-axis, and

the cranking frequency is given by ω. In these calculations the triaxial CNS model35

is used in the rotating frame. The model provides a microscopic description of the

influence of rotation on single-particle motion. The rotation is treated classically37

and the nucleons are considered as independent particles moving in an average

rotating potential. The basic developments leading to the modified single-particle39

oscillator potential are described in Refs. 1, 19, 27, while cranking was introduced

1250077-3



in Refs. 20, 28. The single-particle Hamiltonian used here is in the form29:1

Hω = H − ωjx = Hho(β, γ) + V ′ − ωjx , (1)

where Hho(β, γ) is the anisotropic harmonic oscillator Hamiltonian, given by:3

Hho(β, γ) =P 2

2m+

1

2m

3∑

i=1

ω2i x

2i . (2)

The frequencies ωx, ωy and ωz are expressed in the quadrupole deformation pa-5

rameters, in the usual way, such that the signs are chosen according to the Lund

convention,30,31 so that7

ωj = ω0(β, γ)

[

1−2

3β cos

(

γ +2πvj3

)]

, j ∈ x, y, z (3)

with vx = 1, vy = −1 and vz = 0. The calculations are carried out in the stretched9

coordinate system: ξ = x√

Mωx/~ etc., and the higher multipoles in the potential

are also defined in these coordinates; i.e., the spherical harmonics Yλµ are functions11

of the angles θt and φt where the index t refers to the stretched system. The term

V ′, which is also defined in the stretched coordinates, is introduced to reproduce13

the level ordering as observed in nuclei, and is given by

V ′ = −κ(N)~ω

02lt · s+ µ(N)(ℓt − ℓN) . (4)15

In the above equation the parameters κ and µ might either be given the same values

for each shell or, alternatively, as indicated in (4), they can be made dependent on17

the main oscillator quantum number N = Nt.

The diagonalization of the Hamiltonian (1) gives the eigenvalues eωi and the19

eigenvectors χωi . Furthermore, the single-particle energies in the laboratory system

and the single-particle spin contributions mi are obtained as21

ei = 〈χωi |H

|χωi 〉, mi = 〈χω

i |jx|χωi 〉 , (5)

where H is the static single-particle Hamiltonian. [See (1)].23

The total energy is then obtained as

E =∑

i

ei + Ec =∑

i

eωi + ~ω∑

i

mi + Ec , (6)25

with the total spin given by

I =∑

mi . (7)27

These sums should be carried out over the occupied states where the occupation is

determined from the order of the quantities eωi . Ec in Eq. (6) is the nuclear Coulomb29

energy which depends on deformation.

The nuclear Coulomb energy should, in principle, be treated as a residual force31

for the particles moving in the single-particle potential (1). The most accurate pro-

cedure is, however, very cumbersome, and one, therefore, determines the Coulomb33

energy of a homogeneous proton distribution with an ellipsoidal shape. The exact

1250077-4



expression for the Coulomb energy of an ellipsoid Ec in units of Coulomb energy of1

a sphere E(0)c was derived by Pal,32 Gob et al.33 and Leander.34

To overcome the difficulties encountered in the evaluation of the total energy3

for large deformation through the summation of the single-particle energies, the

Strutinsky shell correction method is adapted to I 6= 0 cases by suitably tuning35,365

the angular velocities to yield fixed spins.

3. The Single-Particle Schrodinger Fluid7

According to the semi classical approach of dealing the motion of the nucleon inside

the nucleus, we assume that each nucleon in the nucleus is moving in a single-9

particle potential V (r, α(t)), which is deforming with time t, through its parametric

dependence on a classical shape variable α(t). Thus the Hamiltonian for the present11

problem is given by9,10

H(r,p;α(t)) =p2

2m+ V (r, α(t)) . (8)13

The single-particle wave function Ψ(r, α(t)t), which describes the motion of a nu-

cleon, satisfies the time-dependent Schrodinger equation15

H(r,p; a(t))Ψ(r;α(t), t) = i~∂

∂tΨ(r;α(t), t) . (9)

We use polar form of the wave function and isolate the explicit time dependence in17

Ψ(r;α(t), t) by an energy phase factor, i.e., we write9

Ψ(r;α(t), t) = Ψ(r;α(t)) exp

−i

~

∫ t

0

(α(t′))dt′

, (10)19

where ε(α(t)) is the intrinsic energy of the nucleon that depends on time through

α. Then we write the complex wave function Ψ(r, α(t)) in polar form21

Ψ(r, α(t)) = Φ(r, α(t)) exp

−iM

~S(r, α(t))

, (11)

where Φ(r, α(t)) and S(r, α(t)) are assumed to be real functions of r and α.23

Separating the real and imaginary parts in (9) we obtain a pair of coupled equations

for Φ and S925

1

2Φ(∇2S) + (∇Φ) • (∇S) =

∂Φ

∂t(12)

and27[

H −M

(

∂S

∂t−

1

2∇S • ∇S

)]

Φ = ǫΦ . (13)

We may call Eq. (13) modified Schrodinger equation because it differs from the29

usual time-independent Schrodinger equation HΦ = ǫΦ by an added term which

we refer to as the “dynamical modification potential”31

Vdyn = −M

[

∂S

∂t−

1

2(∇S) • (∇S)

]

. (14)

1250077-5



Multiplying (12) by 2Φ and noting that the probability density of the single-1

particle, ρ, is just the square of the amplitude, |Φ|2, we get the equation of continuity

familiar from classical fluid dynamics3

∇ • (ρv) = −∂ρ

∂t, (15)

where the velocity field v is identified as5

v = −∇S . (16)

Because of the continuity Eq. (15) we can interpret the changing in the proba-7

bility distribution of the single-particle as a fluid of density ρ = |Φ|2 whose motion

is described by the velocity potential v of Eq. (16). We shall refer to this fluid as9

the single-particle Schrodinger fluid. The velocity field can be expressed as

v =i~

2M |ψ|2[ψ∇ψ∗ − ψ∗∇ψ] . (17)

11

On the other hand, the current of the single-particle state |ψ〉 is defined as

J =i~

2M[ψ∇ψ∗ − ψ∗∇ψ] . (18)13

It follows by comparing the Eqs. (17) and (18) that

J = ρv , (19)15

which is exactly the relationship between current and velocity in classical fluid

dynamics.17

For a nonviscid fluid which admits pressure P , the equation of motion is the

Euler’s equation (the Navier–Stokes) equation919

∂v

∂t+ (v • ∇)v = −

∇P

ρ. (20)

For ideal fluids, the gradient of the pressure P is related to the enthalpy per unit21

mass as

∇P

ρ= ∇ . (21)

23

Then Euler’s equation (20), can be rewritten in the form:

∂v

∂t+ (v • ∇)v = −∇ . (22)25

A first integral of Eq. (22) can be obtained as

∂S

∂t−

1

2(∇S)2 = . (23)27

If we assume that Euler’s equation in the form (22) holds for the single-particle

Schrodinger fluid, then the modified Schrodinger equation takes the form:29

(H −m)Φk = ǫkΦk , (24)

1250077-6



where is now the “enthalpy” of the single-particle Schrodinger fluid.1

Hence, we have a set of fluid dynamical equations completely analogous to those

which describe a classical fluid. This set consists of the continuity Eq. (15), Euler’s3

equation (22), and the equation of state (24). By derivation, their content is pre-

cisely that of the original time-dependent Schrodinger equation.5

The collective kinetic energy T for the entire nucleus, which is the sum of the

single-particle contributions, reduces to the form:7

T =

(

m

2

)∫

ρTvT · (Ω ∩ r)dτ , (25)

where ρT is the total density distribution of the nucleus, which is the sum of the9

single-particle density distributions over all occupied states

ρT =∑

i=occ

ρi (26)11

and vT is the total velocity field

vT =1

ρT

∑

i=occ

ρivi . (27)13

The average potential field is assumed to be in the form of a harmonic oscillator

potential. The intrinsic energy of the single-particle state is, then15

Enxnynz= ~ωx(nx + ny + 1) + ~ωz(nz + 1) . (28)

In terms of the frequencies ωx, ωy and ωz we introduce one single parameter of

deformation δ given by37

ω2z = ω2

0

(

1−4

3δ

)

, (29)

ω2x = ω2

y = ω20

(

1 +2

3δ

)

. (30)

Accordingly, the condition of constant volume of the nucleus is guaranteed. The17

parameter δ is related to the well-known deformation parameter β by

δ =3

2

√

5

4πβ . (31)

19

Using the equation arising from the first-order perturbation of the wave function

we can calculate the first-order time-dependent perturbation correction to the wave21

function explicitly as function of the numbers of quanta of excitations corresponding

to the Cartesian coordinates and the quantity σ, defined by923

σ =ωy − ωz

ωy + ωz

, (32)

which is a measure of the deformation of the potential.25

1250077-7



We now examine the cranking moment of inertia in terms of the velocity fields.1

Bohr and Mottelson3,4 show that for harmonic oscillator case at the equilibrium

deformation, where3

d

dδ

∑

i=1

(Enxnynz)i = 0 , (33)

the cranking moment of inertia is identically equal to the rigid moment of inertia:5

ℑcr = ℑrig =∑

i=1

m〈y2i + z2i 〉 . (34)

Hence, this result asserts the equality of the collective kinetic energy of the7

Schrodinger fluid and that of rigidly rotating classical fluid

m

2

∫

ρTvT • (Ω× r)dτ =1

2ℑrigΩ

2 =m

2

∫

ρT (Ω× r)2dτ , (35)9

at the equilibrium deformation. We emphasize that Eqs. (34) and (35) hold for any

number of nucleons occupying any set of single-particle harmonic oscillator states11

at the deformation defined by equilibrium condition (33). In particular, it holds for

a one particle state. For this case, Eq. (35) becomes13

m

2

∫

ρivi • (Ω× r)dτ =m

2

∫

ρi(Ω× r)2dτ , (36)

at the equilibrium deformation of the single-particle state

|i〉 ≡ |nxnynz〉 .

Equation (36) is a remarkable identity. The scalar product of vi and (Ω× r) which15

occurs on the left-hand side is replaced on the right-hand side, by the absolute

square of (Ω× r).17

We note that the cranking moment of inertia ℑcr and the rigid moment of inertia

ℑrig are equal only when the harmonic oscillator is at the equilibrium deformation.19

At other deformations, they can, and do, deviate substantially from one another.10

The following expressions for the cranking moment of inertia and the rigid-body

moment of inertia ℑrig are then obtained10:

ℑcr =E

ω20

(

1

6 + 2σ

)(

1 + σ

1− σ

)1

3

[

σ2(1 + q) +1

σ(1− q)

]

, (37)

ℑrig =E

ω20

(

1

6 + 2σ

)(

1 + σ

1− σ

)1

3

[(1 + q) + σ(1 − q)] , (38)

where E is the total single-particle energy, given by (28) and q is the ratio of the21

summed single-particle quanta in the y- and z-directions

q =

∑

occ

(ny + 1)

∑

occ

(nz + 1). (39)

23

q is known as the anisotropy of the configuration.

1250077-8



4. The New Rotational and Vibrational Formulas1

By analyzing the well-known experimental rotational energy levels of the even–

even deformed nuclei in the high mass region we have derived a new formula for3

the rotational energy levels, that depends upon the total spin momentum I and the

nuclear moment of inertia ℑ in the following simple form:5

E(I) =AI(I + 1)

[

1 +DI(I + 1)

1− CI(I + 1)

] . (40)

Here, A is the reciprocal-moment of inertia of the nucleus, A = ~2

2I . The value of7

A has been determined for all the considered isotopes by using the concept of the

single-particle Schrodinger fluid.109

Accordingly, our formula contains two parameters beside the nuclear moment of

inertia. In our fitting we determined C and D by inserting two values of the experi-11

mental rotational energies for middle values of I. In our calculations we considered

I= 10 and I= 12 to determine C and D.13

When D = C formula (40) gives the following simple relation

E(I) = AI(I + 1)−BI2(I + 1)2 , (41)

where B = AC This special case coincides with the AB-formula.38

Accordingly, our new formula modifies the AB-formula by the correction factor:

1

1 + (D − C)[I(I + 1)].

Furthermore, by analyzing the well-known experimental β-band energy levels

of the even–even deformed nuclei in the high mass region we have derived a new

formula for the β-band energy levels, that depends upon the total spin momentum

I, the head band of β and the head band of γ and the nuclear moment of inertia J

in the following simple form:

Eβ−band = −~2

J

1

Eγ

I2(I+1)2+~2

2JI(I+1)+Eβ . (42)

Also, by analyzing the well-known experimental γ-band energy levels of even–15

even deformed nuclei in the high mass region we have derived a new formula for

the γ-band energy levels, that depends upon the total spin momentum I, the head17

band of γ, the number of neutrons N and the nuclear moment of inertia J in the

following simple form.19

Eγ−band =~2

2J

1

NI2(I + 1)2 +

1

2

~2

2JI(I + 1) + Eγ . (43)

5. Results and Conclusion21

We have calculated the reciprocal moments of inertia according to the cranking

model and the rigid-body model of the single-particle Schrodinger fluid for the23

1250077-9



Table 1. Reciprocal moments of inertia by using

Schrodinger fluid for the even–even deformed iso-topes: 170

70 Yb, 17270 Yb, 174

70 Yb, 17672 Hf, 178

72 Hf, 18072 Hf,

18274 W, 184

74 W and 18674 W.

Nucleus β~2

2Jcr

~2

2Jrig

~2

2Jexp

17070 Yb −0.25 14.12 3.62 14.1

0.26 14.09 3.317270 Yb −0.31 13.26 3.67 13.16

0.33 13.01 3.2917470 Yb −0.30 12.59 3.61 12.76

0.32 12.54 3.2317672 Hf −0.23 14.55 3.39 14.4

0.23 14.59 3.1317872 Hf −0.47 15.39 3.40 NA

0.31 15.13 2.9218072 Hf −0.43 15.7 3.31 15.6

0.29 15.2 2.8818274 W −0.43 16.75 3.2 16.78

0.29 16.6 2.818474 W −0.5 18.15 3.3 18.5

0.31 18.59 2.7718674 W −0.5 17.56 3.25 20.3

0.33 20.21 2.74

even–even deformed isotopes; ytterbium: 170Yb, 172Yb and 174Yb, hafnium: 176Hf,1

178Hf and 180Hf and tungsten: 182W, 184W and 186W nuclei as functions of the

deformation parameter β, which is allowed to vary in the range from −0.50 to 0.503

with a step equals 0.01.

In Table 1 we present the best values of the reciprocal moments of inertia by5

using Schrodinger fluid for the even–even deformed isotopes: 17070 Yb, 172

70 Yb, 17470 Yb,

17672 Hf, 17872 Hf, 18072 Hf, 18274 W, 18474 W and 186

74 W. The values of the deformation parameter7

β are also given in this table. The corresponding experimental values are also given

in the last column.39–479

It is seen from Table 1 that the calculated values of the moments of inertia

by using the cranking model are in excellent agreement with the corresponding11

experimental values. It is also seen that there are two possible values of the defor-

mation parameter for each nucleus which produce the best agreement, one of which13

is positive and the other is negative. As expected, the rigid-body values of the re-

ciprocal moments of inertia fall within the range (20%–30%) of the corresponding15

experimental values.

In the numerical calculations of the rotational energies of the even–even de-17

formed isotopes: 17070 Yb, 172

70 Yb, 17470 Yb, 176

72 Hf 17872 Hf, 180

72 Hf, 18274 W,18474 Wand18674 W, we

have used our new formula, Eq. (40). Furthermore, we have also calculated the19

rotational energies by using the AB-formula,38 the Wrake–Khadikikar formula,48

Harris-formula,49 the variable moment of inertia-formula50 and the ab-formula.5121

1250077-10


Deformation Properties of the Even–Even Yb, Hf and W NucleiTable

2.

Rotationalen

ergiesofthenineeven

–even

deform

edisotopes;Ytterbium:170

70Yb,172

70Yband

174

70Yb,Hafnium:176

72Hf

178

72Hfand

180

72Hf

andtungsten

:182

74W

,184

74W

and

186

74W

asfunctionsofthetotalsp

inIbyusingtheab-form

ula

50andthenew

form

ula,Eq.(40).

Theex

perim

ental

values

are

taken

from

Refs.

39–47.

E(I)in

KeV

Nucleu

sCase

I=

2I=

4I=

6I=

8I=

10

I=

12

I=

14

I=

16

I=

18

I=

20

170

70Yb

172

70Yb

174

70Yb

176

72Hf

178

72Hf

180

72Hf

182

74W

184

74W

186

74W

ab

New

Exp.

ab

New

Exp.

ab

New

Exp.

ab

New

Exp.

ab

New

Exp.

ab

New

Exp.

ab

New

Exp.

ab

New

Exp.

ab

New

Exp.

84.25

83.75

84.25

78.75

78.61

78.75

76.47

77.07

76.47

88.35

87.97

88.35

93.18

92.75

93.18

93.32

93.10

93.32

100.1

99.46

100.1

111.2

109.9

111.2

122.6

119.1

122.6

277.59

276.64

277.44

260.36

260.05

260.29

253.16

254.54

253.12

290.42

289.53

290.18

306.81

305.90

306.62

308.65

308.24

308.58

329.63

328.38

329.43

364.51

362.25

364.07

397.86

391.78

396.55

572.84

572.80

573.41

539.97

539.91

540.0

526.12

527.51

526.03

597.27

596.48

596.82

632.51

631.75

632.18

640.51

640.70

640.86

679.67

679.8

680.5

746.67

747.46

748.32

803.80

806.85

809.25

960.15

963.53

963.53

910.68

912.16

912.16

889.59

889.93

889.93

997.01

997.74

997.74

1058.93

1058.56

1058.56

1081.06

1083.94

1083.94

1138.08

1144.4

1144.4

1240.60

1252.2

1252.2

1315.11

1349.2

1349.2

1428.45

1437.97

1437.97

1364.29

1370.11

1370.11

1336.64

1336

1336

1476.69

1481.07

1481.07

1573.37

1571

1571.0

1620.92

1630.40

1630.4

1691.32

1711.9

1711.9

1828.38

1860.8

1860.8

1908.23

2002.4

2002.4

1966.57

1983.80

1983.8

1892.04

1907.21

1907.21

1859.63

1861

1861

2023.74

2034.67

2034.67

2163.20

2150.7

2150.7

2250.03

2272.4

2272.4

2325.85

2372.5

2372.5

2493.29

2557

2557.0

2563.82

2750.9

2750.9

2564.07

2588.01

2580.9

2485.23

2517.67

2518.4

2450.79

2461.61

2457

2626.81

2648.25

2646.6

2816.71

2776.9

2777.6

2958.29

3002.8

...

3029.13

3117.7

3112.6

3220.86

3325.3

3319.9

3266.92

3581.44

3562.8

3211.63

3237.57

...

3135.62

3196.78

3198.1

3102.53

3135.82

3117

3276.15

3313.62

...

3523.59

3427.14

3436.2

3736.05

3815.47

...

3790.08

3940.95

...

3999.11

4151.9

4116.9

4006.37

4483.7

...

3901.24

3920.04

...

3835.69

3941.02

...

3807.76

3882.66

3836

3963.63

4024.76

...

4275.04

4077.92

...

4574.46

4705.4

...

4599.23

4837.3

...

4818.35

5025.5

...

4773.86

5450.4

...

4626.13

4623.96

...

4578.72

4747.95

...

4559.98

4701.8

4610

4682.58

4777.6

...

5063.69

4705.35

...

5465.56

5669.02

...

5448.64

5803.6

...

5670.87

5937.06

...

5563.25

6476.8

...

1250077-11



Table 3. β-band energies of the nine isotopes 17070 Yb, 172

70 Yb, 17470 Yb, 176

72 Hf 17872 Hf, 180

72 Hf, 18274 W,

18474 W and 186

74 W as functions of the total spin I by using the new formula, equation (42). Theexperimental values are taken from Refs. 39–47.

Nucleus 0+ 2+ 4+ 6+ Eβ Eγ

17070 Yb Exp. 1069.35 1138.57 1293.5 1521.26 1069.35 1119.72

Cal. 1069.35 1138.20 1293.06 1520.017270 Yb Exp. 1042.91 1117.87 1286.54 1537.5 1042.91 1439.56

Cal. 1042.91 1121.21 1298.80 1563.3917470 Yb Exp. 1487.12 1561.02 1715.45 . . . 1487.12 1608.45

Cal. 1487.12 1563.11 1735.97 1995.0517672 Hf Exp. 1149.94 1226.63 1390.19 1628.55 1149.94 1317.71

Cal. 1149.94 1220.10 1378.78 1613.9517872 Hf Exp. 1199.39 1276.69 1450.36 1731.06 1199.39 1146.91

Cal. 1199.39 1281.68 1466.9 1738.8718072 Hf Exp. 1101.9 1183.32 1369.48 . . . 1101.9 1168.06

Cal. 1101.9 1182.66 1364.59 1632.0218274 W Exp. 1135.82 1257.41 1510.22 . . . 1135.82 1187.84

Cal. 1135.82 1235.48 1460.1 1790.718474 W Exp. 1002.49 1121.44 1360.38 . . . 1002.49 867.91

Cal. 1002.49 1112.54 1357.34 1708.0818674 W Exp. 883.59 1030.23 1298.93 . . . 881.74 696.41

Cal. 883.59 1005.06 1271.93 1644.43

Table 4. γ-band energies of the nine isotopes 17070 Yb, 172

70 Yb, 17470 Yb, 176

72 Hf 17872 Hf, 180

72 Hf,18274 W,18474 W and 186

74 W as functions of the total spin I by using the new formula, equation(43). The experimental values are taken from Refs. 39–47.

Nucleus 2+ 3+ 4+ 5+ Eβ Eγ

17070 Yb Exp. 1145.72 1225.36 1329.8 1479.8 1069.35 1119.72

Cal. 1163.4 1216.44 1301.7 1431.717270 Yb Exp. 1465.88 1549.15 1657.79 1778.86 1042.91 1439.56

Cal. 1483.69 1537.09 1622.77 1753.0817470 Yb Exp. 1633.97 1709.42 1805.40 1926 1487.12 1608.45

Cal. 1651.15 1702.68 1785.13 1910.2717672 Hf Exp. 1341.31 1445.80 1540.3 1727.8 1149.94 1317.71

Cal. 1365.9 1424.05 1517.1 1658.3217872 Hf Exp. 1174.63 1268.54 1384.46 1533.15 1199.39 1146.91

Cal. 1193.19 1248.9 1337.8 1472.518072 Hf Exp. 1199.66 1291.04 1409.25 1556.81 1101.9 1168.06

Cal. 1220.72 1283.9 1384.57 1536.7218274 W Exp. 1221.40 1331.12 1442.84 1623.51 1135.82 1187.84

Cal. 1243.7 1310.89 1417.78 1579.3718474 W Exp. 903.31 1005.97 1133.85 1294.94 1002.49 867.91

Cal. 926.8 997.28 1109.27 1278.2218674 W Exp. 737.61 861.77 1006.45 1195 881.74 696.41

Cal. 764.95 846.98 977.31 1173.60

Among the previous five formulas the results obtained by using the ab-formula are1

to some extent better than those of the other four formulas. Accordingly, we present

only in Table 2 the calculated values of the rotational energies of the mentioned3

nine isotopes, for even values of the total angular momentum I in the interval from

1250077-12



-0.4 -0.2 0.2 0.4-2

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

1

2

3

4

5

6

7

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4-2

2

4

6

8

Fig. 1. β-dependence of the total ground state energy of 17070 Yb at various γ-values from 0 to

60 respectively.

-0.4 -0.2 0.2 0.4-2

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

1

2

3

4

5

6

7

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4-2

2

4

6

8

Fig. 2. Same as Figure-1 for 17270 Yb.

-0.4 -0.2 0.2 0.4-2

2

4

6

8

10

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

1

2

3

4

5

6

7

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4-2

2

4

6

8

10

Fig. 3. Same as Figure-1 for 17470 Yb.

2 to 20, by using the ab-formula and the new formula together with the available1

experimental values. The experimental values are taken from Refs. 39–47.

1250077-13



-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

1

2

3

4

5

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8

Fig. 4. Same as Fig. 1 for 17672 Hf.

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

1

2

3

4

5

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8


-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

1

2

3

4

5

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

8


It is seen from Table 2 that the calculated values of the rotational energies1

of the nine isotopes by using the new formula are in better agreement with the

corresponding experimental ones than those obtained by using the ab-formula.3

1250077-14



-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4-1

1

2

3

4

5

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

8

Fig. 7. Same as Fig. 1 for 18274 W.

-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

8


-0.4 -0.2 0.2 0.4

2

4

6

8

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

-0.4 -0.2 0.2 0.4

2

4

6

8


In Table 3 we present the βband energies of the nine isotopes 17070 Yb, 172

70 Yb,1

17470 Yb, 176

72 Hf 17872 Hf, 180

72 Hf, 18274 W,18474 W and 186

74 W as functions of the total spin

I by using the new formula, Eq. (43). The experimental values are taken from3

Refs. 39–47.

1250077-15



-0.4 -0.2 0.2 0.4

-6

-4

-2

2

4

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1

-0.4 -0.2 0.2 0.4

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

4

Fig. 10. β-dependence of the quadrupole moment of 17070 Yb at various γ-values from 0 to 60

respectively.

-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1

-0.4 -0.2 0.2 0.4

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4

Fig. 11. Same as Fig. 10 for 17270 Yb.

-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1

-0.4 -0.2 0.2 0.4

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4

Fig. 12. Same as Fig. 10 for 17470 Yb.

In Table 4 we present the γ-band energies of the nine isotopes 17070 Yb, 172

70 Yb,1

17470 Yb, 176

72 Hf 17872 Hf, 180

72 Hf, 18274 W,18474 W and 186

74 W as functions of the total spin

I by using the new formula, Eq. (43). The experimental values are taken from3

Refs. 39–47.

1250077-16



-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1

-0.4 -0.2 0.2 0.4

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4


-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1

-0.4 -0.2 0.2 0.4

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4


-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1

-0.4 -0.2 0.2 0.4

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4


It is seen from Tables 3 and 4 that the calculated values of the β-band and γ-1

band energies of the nine isotopes by using the new formulas are in good agreement

with the corresponding experimental values.3

From the obtained results we have seen that a new three-parameter formula

for the rotational band of a well-deformed nucleus is suggested on the basis of the5

1250077-17



-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1

-0.4 -0.2 0.2 0.4

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4


-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1

-0.4 -0.2 0.2 0.4

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4


-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4

-0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1

-0.4 -0.2 0.2 0.4

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

-0.4 -0.2 0.2 0.4

-5

-4

-3

-2

-1 -0.4 -0.2 0.2 0.4

-6

-4

-2

2

-0.4 -0.2 0.2 0.4

-8

-6

-4

-2

2

4


phenomenological Bohr Hamiltonian. In the derivation of this formula a small axial1

asymmetry and vibrational effects (including anharmonicity) have been taken into

account. The agreement between the obtained results by using this formula and the3

1250077-18



observed ground bands is astonishingly excellent, which perhaps implies that the1

formalism described here may have some truth.

In framework of the Nilsson cranked model, we have calculated the total energy3

and the quadrupole moment of the nine mentioned deformed nuclei, for values of

the deformation parameters in the ranges: −0.5 ≤ β ≤ 0.5, 0 ≤ γ ≤ 60, with5

steps of ∆β = 0.1 and ∆γ = 10. The obtained results are given in Figs. 1 to 9 for

the β-dependence of the total ground-state energy of the nine nuclei and in Figs. 107

to 18 for the β-dependence of the quadrupole moments of the nine nuclei.

After analyzing the obtained results for each nucleus, we have carried out a9

β-dependence fit to the total ground-state energy of each nucleus, in the form of a

polynomial of degree 8 in β11

ET =

8∑

i=0

aiβi , (44)

where ai are parameters, which have been obtained from the corresponding fit to13

our calculated values. Furthermore, the best fit for the values of the calculated

quadrupole moments results in a polynomial in β of degree 6 in the following form:15

Q =6

∑

i=1

biβi . (45)

The parameters ai are given in Table 5, while the parameters bi are given in Table 6.17

The fitted values for the total energy and the quadrupole moment of the nine

nuclei are given in solid curves on Figs. 1–18.19

From the figures of the total energy and the quadrupole moment we find that for

γ = 0 and γ = 60 the values of the total energy (quadrupole moment) are equal21

but with different signs of β. Not only this symmetry occurs but also for γ = 10

and γ = 50, γ = 20 and γ = 40. On the other hand, for γ = 30 the values are23

symmetric around β = 0.

In Table 7, we present the calculated values of the LD Energy, the Strutinsky25

inertia, the LD inertia, the volume conservation factor ω0/ω

0 , the smoothed energy,

the BCS energy and the G-value of the nine isotopes 17070 Yb, 172

70 Yb, 17470 Yb, 176

72 Hf27

17872 Hf, 18072 Hf, 18274 W, 184

74 W and 18674 W for values of the deformation parameter β, and

the nonaxiality parameter γ, which produce good agreement with the corresponding29

findings.

On the other hand, we had seen in Figs. 19–27 how the triaxiality of the nine31

isotopes 17070 Yb, 172

70 Yb, 17470 Yb, 176

72 Hf 17872 Hf, 180

72 Hf, 18274 W, 184

74 W and 18674 W changes

with spin. We fix the quadrupole deformations ε, ε4 and vary γ between −10 and33

90 in steps of 10. The important symmetry occurs at γ = 60. At I = 0 the energy

is also symmetric around γ = 0 but this symmetry is violated at I > 20, since35

positive values of γ corresponds to rotation around the smallest axis and negative

values of γ corresponds to rotation around the intermediate axis. Accordingly, at37

higher spin values the liquid-drop model may be taken to indicate a stably oblate

spin region which is realized, in existing nuclei, only in special regions of mass. For39

1250077-19



Table

5.

Totalen

ergyparameters,

therelation(44),

fortheninenuclei.

Nucleu

sγ

a0

a1

a2

a3

a4

a5

a6

a7

a8

170

70Yb

05.20

−0.73

−195.18

−201.41

2307.03

1988.6

−9322.26

−4726.9

13857.6

10

5.19

−0.514

−192.11

−183.03

2260.36

1836.3

−9010.53

−4486.5

12983

20

5.17

0.066

−188.92

−121.70

2183.88

1285.0

−8613.01

−3298.9

12175

30

5.15

0.0

−185.06

0.0

2084.12

0.0

−8030.66

0.0

11226

40

5.17

−0.066

−188.92

121.70

2183.88

−1285.0

−8613.01

3298.9

12175

50

5.19

0.514

−192.11

183.03

2260.36

−1836.3

−9010.53

4486.46

12983

60

5.20

0.73

−195.18

201.41

2307.03

−1988.6

−9322.26

4726.89

13857.6

172

70Yb

05.35

−0.74

−209.16

−178.65

2483.81

1713.7

−10059

−4025.4

14826.1

10

5.36

−0.82

−206.42

−162.06

2467.09

1640.5

−10088.3

−4016.7

14845.7

20

5.34

−0.117

−198.01

−105.92

2268.18

1100.3

−8649.63

−2783.0

11519.7

30

5.34

0.0

−189.91

0.0

2042.73

0.0

−7063.17

0.0

8223.68

40

5.34

0.117

−198.01

105.92

2268.18

−1100.3

−8649.63

2783.0

11519.7

50

5.36

0.82

−206.42

162.06

2467.09

−1640.5

−10088.3

4016.7

14845.7

60

5.35

0.74

−209.16

178.65

2483.81

−1713.7

−10059

4025.44

14826.1

174

70Yb

05.49

−2.127

−224.93

−115.47

2731.99

1138.4

−11328.3

−2677.4

16886

10

5.48

−1.61

−221.58

−116.83

2715.75

1204.9

−11381.7

−2942.9

16957.8

20

5.43

−1.11

−207.32

−68.204

2410.01

747.9

−9250.18

−1898.3

12261.1

30

5.43

0.0

−199.55

0.0

2208.83

0.0

−7793.62

0.0

9014.72

40

5.43

1.11

−207.32

68.204

2410.01

−747.9

−9250.18

1898.3

12261.1

50

5.48

1.61

−221.58

116.83

2715.75

−1204.9

−11381.7

2942.9

16957.8

60

5.49

2.127

−224.93

115.47

2731.99

−1138.4

−11328.3

2677.4

16886

1250077-20



Table

5.

(Continued

).

Nucleu

sγ

a0

a1

a2

a3

a4

a5

a6

a7

a8

176

72Hf

04.62

−3.004

−213.22

−63.99

2813.33

707.07

−12731.8

−1735.5

20561.8

10

4.62

−2.518

−211.21

−65.04

2807.92

777.78

−12753.6

−2013.9

20388.2

20

4.56

−1.348

−201.36

−43.88

2622.98

553.84

−11472.7

−1496.3

17408.1

30

4.56

0.0

−193.20

0.0

2411.22

0.0

−9910.56

0.0

13904.6

40

4.56

1.348

−201.36

43.88

2622.98

−553.84

−11472.7

1496.3

17408.1

50

4.62

2.518

−211.21

65.04

2807.92

−777.78

−12753.6

2013.9

20388.2

60

4.62

3.004

−213.22

63.99

2813.33

−707.07

−12731.8

1735.5

20561.8

178

72Hf

04.42

−3.668

−223.53

−16.63

3050.18

256.58

−14037.6

−676.35

22701.3

10

4.41

−3.171

−219.93

−22.81

3012.27

361.07

−13897.4

−965.80

22322.7

20

4.35

−1.901

−208.42

−12.12

2824.02

213.15

−12671.1

−578.90

19561.9

30

4.34

0.0

−205.58

0.0

2797.65

0.0

−12494.2

0.0

18917.1

40

4.35

1.901

−208.42

12.12

2824.02

−213.15

−12671.1

578.90

19561.9

50

4.41

3.171

−219.93

22.81

3012.27

−361.07

−13897.4

965.80

22322.7

60

4.42

3.668

−223.53

16.63

3050.18

−256.58

−14037.6

676.35

22701.3

180

72Hf

03.94

−4.164

−217.77

36.95

3118.05

−266.99

−14821.2

557.31

24400.8

10

3.94

−3.77

−216.53

29.57

3149.21

−183.95

−15242.3

423.67

25360.3

20

3.94

−2.625

−213.91

30.31

3166.49

−214.50

−15503.2

522.88

25879.8

30

3.91

0.0

−209.72

0.0

3126.52

0.0

−15203.5

0.0

24932.1

40

3.94

2.625

−213.91

−30.31

3166.49

214.50

−15503.2

−522.88

25879.8

50

3.94

3.77

−216.53

−29.57

3149.21

183.95

−15242.3

−423.67

25360.3

60

3.94

4.164

−217.77

−36.95

3118.05

267

−14821.2

−557.31

24400.8

1250077-21



Table

5.

(Continued

).

Nucleu

sγ

a0

a1

a2

a3

a4

a5

a6

a7

a8

182

74W

02.72

−4.43

−192.87

64.78

3007.68

−485.05

−15182.8

983.9

26167

10

2.74

−3.86

−194.04

49.43

3067.67

−330.52

−15656.9

688.03

27001.1

20

2.72

−2.57

−191.65

33.69

3088.33

−215.69

−15864.7

470.94

27270

30

2.72

0.0

−189.96

0.0

3103.5

0.0

−15930.5

0.0

27085.9

40

2.72

2.57

−191.65

−33.69

3088.33

215.69

−15864.7

−470.94

27270

50

2.74

3.86

−194.04

−49.43

3067.67

330.52

−15656.9

−688.03

27001.1

60

2.72

4.43

−192.87

−64.78

3007.68

485.05

−15182.8

−983.9

26167

184

74W

02.13

−5.12

−179.58

117.37

3073.71

−987.70

−16501.2

2175

29598.7

10

2.16

−4.539

−186.05

104.79

3202.21

−854.90

−17287.6

1960.78

30986.3

20

2.19

−3.272

−193.74

84.72

3430.07

−732.97

−18917.1

1787.46

34292.8

30

2.24

0.0

−200.06

0.0

3581.73

0.0

−19724.8

0.0

35442.8

40

2.19

3.272

−193.74

−84.72

3430.07

732.97

−18917.1

−1787.5

34292.8

50

2.16

4.539

−186.05

−104.79

3202.21

854.90

−17287.6

−1960.8

30986.3

60

2.13

5.12

−179.58

−117.37

3073.71

987.70

−16501.2

−2175

29598.7

186

74W

01.34

−5.46

−164.95

172.57

3209.34

−1561.1

−18611.5

3644.37

35120.4

10

1.37

−4.871

−172.75

159.76

3341.87

−1442.7

−19378.4

3464.05

36453.1

20

1.44

−3.47

−187.49

126.07

3659.19

−1171.1

−21381.5

2902.08

40297.4

30

1.49

0.0

−198.3

0.0

3903.45

0.0

−22791.5

0.0

42731.8

40

1.44

3.47

−187.49

−126.07

3659.19

1171.12

−21381.5

−2902.1

40297.4

50

1.37

4.871

−172.75

−159.76

3341.87

1442.69

−19378.4

−3464.1

36453.1

60

1.34

5.46

−164.95

−172.57

3209.34

1561.07

−18611.5

−3644.4

35120.4

1250077-22



Table 6. Quadrupole moment parameters, the relation (45), for the nine nuclei.

nucleus γ a1 a2 a3 a4 a5 a6

17070 Yb 0 −14.42 −3.271 34.025 −40.981 −108.81 130.74

10 −10.81 −4.956 32.518 −57.923 −88.462 171.8120 −4.785 −7.666 5.6928 −26.540 −7.2115 37.98830 0.0 −16.470 0.0 34.311 0.0 −76.37640 4.785 −7.666 −5.6928 −26.540 7.2115 37.98850 10.81 −4.956 −32.518 −57.923 88.462 171.8160 14.42 −3.271 −34.025 −40.981 108.81 130.74

17270 Yb 0 −14.52 −3.313 34.326 −40.889 −109.94 130.41

10 −10.90 −4.9157 32.882 −59.316 −89.583 175.7620 −4.819 −7.7415 5.6694 −26.507 −7.2115 37.76230 0.0 −8.621 0.0 −58.874 0.0 170.6740 4.819 −7.7415 −5.6694 −26.507 7.2115 37.76250 10.90 −4.9157 −32.882 −59.316 89.583 175.7660 14.524 −3.313 −34.33 −40.889 109.94 130.41

17470 Yb 0 −14.66 −3.3235 35.055 −41.692 −112.18 133.47

10 −10.96 −4.948 32.874 −59.853 −89.423 177.4920 −4.870 −7.787 5.989 −26.55 −8.013 37.37630 0.0 −8.627 0.0 −59.142 0.0 170.5540 4.870 −7.787 −5.989 −26.55 8.013 37.37650 10.96 −4.948 −32.87 −59.853 89.423 177.4960 14.66 −3.3235 −35.055 −41.692 112.18 133.47

17672 Hf 0 −14.29 −6.433 13.279 −20.804 −45.994 84.641

10 −9.962 −15.167 9.1689 31.878 −12.981 −28.72720 −3.586 −5.791 −15.852 −65.862 47.115 181.9130 0.0 5.1117 0.0 −207.60 0.0 538.99

40 3.586 −5.791 15.852 −65.862 −47.115 181.9150 9.962 −15.167 −9.1689 31.878 12.981 −28.72760 14.29 −6.433 −13.279 −20.804 45.994 84.641

17872 Hf 0 −14.36 −6.599 13.031 −19.954 −45.353 83.263

10 −10.02 −15.160 8.7974 31.115 −11.378 −26.7320 −3.73 −5.3443 −14.757 −71.96 44.551 198.2230 0.0 4.975 0.0 −206.45 0.0 535.3040 3.73 −5.3443 14.757 −71.96 −44.551 198.2250 10.02 −15.160 −8.7974 31.115 11.378 −26.7360 14.36 −6.599 −13.031 −19.955 45.353 83.263

18072 Hf 0 −14.46 −6.621 13.27 −20.503 −46.154 85.17

10 −10.12 −15.35 9.361 32.472 −12.981 −30.40320 −3.624 −5.919 −16.26 −66.54 48.24 184.4230 0.0 4.988 0.0 −207.98 0.0 539.2940 3.624 −5.919 16.26 −66.54 −48.24 184.4250 10.12 −15.35 −9.361 32.472 12.981 −30.40360 14.46 −6.621 −13.27 −20.503 46.154 85.17

18274 W 0 −13.36 −3.193 −1.254 −45.94 −9.295 125.32

10 −8.338 −9.132 −12.57 −29.61 36.38 89.31320 −4.179 −0.785 −9.978 −125.88 27.885 327.2830 0.0 −7.680 0.0 12.497 0.0 −161.3040 4.179 −0.785 9.978 −125.88 −27.885 327.2850 8.338 −9.132 12.57 −29.61 −36.38 89.31360 13.36 −3.193 1.254 −45.94 9.295 125.32

1250077-23



Table 6. (Continued ).

nucleus γ a1 a2 a3 a4 a5 a6

18474 W 0 −13.45 −3.1865 −1.3403 −46.701 −8.974 127.31

10 −8.429 −9.2491 −12.206 −29.62 35.256 90.25520 −4.206 −0.6505 −10.189 −128.20 28.846 333.0930 0.0 −7.6296 0.0 11.80 0.0 −161.2540 4.206 −0.6505 10.189 −128.20 −28.846 333.09150 8.429 −9.2491 12.206 −29.62 −35.256 90.25560 13.452 −3.1865 1.3403 −46.701 8.974 127.31

18674 W 0 −13.54 −3.2546 −1.2952 −46.13 −9.295 125.47

10 −8.448 −9.4597 −12.652 −27.377 36.38 83.38320 −4.211 −0.7006 −10.452 −128.64 29.33 334.4030 0.0 −8.0042 0.0 15.2651 0.0 −170.4140 4.211 −0.7006 10.452 −128.64 −29.33 334.4050 8.448 −9.4597 12.652 −27.377 −36.38 83.38360 13.543 −3.2546 1.2952 −46.13 9.295 125.47

Table 7. The L. D. Energy, the Strutinsky inertia, the L. D. inertia, the volume conservationfactor ω0/ω

0 , the smoothed energy, the BCS energy and the G-value of the nine isotopes 17070 Yb,

17270 Yb, 174

70 Yb, 17672 Hf 178

72 Hf, 18072 Hf, 182

74 W, 18474 W and 186

74 W.

LD Strutinsky L.D. Smoothed BCSγ energy inertia inertia energy energy G-value

Nucleus β degrees MeV 1/MeV 1/MeV ω0/ω0 MeV MeV MeV

17070 Yb 0.299 50 3.455 110.20 87.70 1.0096 3950.9 −1.705 0.105

17270 Yb 0.304 50 3.604 112.69 89.71 1.0099 4052.6 −1.380 0.104

17470 Yb 0.301 50 3.556 114.32 91.28 1.0097 4154.4 −0.747 0.102

17672 Hf 0.284 20 2.920 114.00 91.05 1.0094 4120.97 −0.927 0.101

17872 Hf 0.280 20 2.864 115.46 92.52 1.0091 4222.57 −0.275 0.10

18072 Hf 0.275 20 2.784 117.25 93.93 1.0088 4324.4 −1.128 0.098

18272 W 0.265 20 2.478 118.78 95.01 1.0082 4290.5 −1.091 0.098

18472 W 0.256 20 2.333 120.08 96.16 1.0076 4392.1 −1.436 0.096

18672 W 0.241 20 2.083 120.61 96.92 1.0067 4493.9 −1.127 0.095

still higher spins there is again a trend towards triaxiality. The termination of the1

yrast band for heavy elements is usually decided by the instability to fission. This

instability is caused by the resistance to deformation, provided that the increase3

in surface energy can be eliminated by the decrease in rotational energy, which in

turn is due to the increase in the value of the moment of inertia with deformation.5

In Figs. 28–36 we present the variation of the quadrupole moment of the nine

mentioned nuclei as functions of the total spin for the specific values of the defor-7

mation parameter β and the nonaxiality parameter γ.

1250077-24



Fig. 19. Total energy at different spins for the nucleus 17070 Yb drawn as function of γ for ε = 0.299,

ε4 = 0. For I = 0, the two curves give the energy with and without pairing included. For I > 0,no pairing is included in the calculations.

1250077-25



-2

1

4

7

10

13

16

19

22

25

28

-2

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

-10 0 10 20 30 40 50 60 70 80 90

+Epair

Etot

I=60

I=50

I=40

I=20

I=10

I=0

I=0

I=30



1250077-26



-3

0

3

6

9

12

15

18

21

24

27

-3

0

3

6

9

12

15

18

21

24

27

-10 0 10 20 30 40 50 60 70 80 90

+Epair

Etot

I=60

I=50

I=40

I=30

I=20

I=10

I=0

I=0



1250077-27



-3

0

3

6

9

12

15

18

21

24

27

-3

0

3

6

9

12

15

18

21

24

27

-10 0 10 20 30 40 50 60 70 80 90

Etot

+Epair

I=60

I=50

I=30

I=20

I=10

I=0

I=0

I=40

Fig. 22. Total energy at different spins for the nucleus 17672 Hf drawn as function of γ for ε = 0.299,


1250077-28



-3

0

3

6

9

12

15

18

21

24

27

-3

0

3

6

9

12

15

18

21

24

27

-10 0 10 20 30 40 50 60 70 80 90

Etot

+Epair

I=60

I=50

I=40

I=30

I=20

I=10

I=0

I=0



1250077-29



-2

1

4

7

10

13

16

19

22

25

-2

1

4

7

10

13

16

19

22

25

-10 0 10 20 30 40 50 60 70 80 90

Etot

+Epair

I=60

I=50

I=40

I=30

I=20

I=10

I=0

I=0



1250077-30



-2

1

4

7

10

13

16

19

22

25

-2

1

4

7

10

13

16

19

22

25

-10 0 10 20 30 40 50 60 70 80 90

+Epair

Etot

I=60

I=50

I=40

I=30

I=20

I=10

I=0

I=0

Fig. 25. Total energy at different spins for the nucleus 18274 W drawn as function of γ for ε = 0.299,


1250077-31



-2

1

4

7

10

13

16

19

22

25

-2

1

4

7

10

13

16

19

22

25

-10 0 10 20 30 40 50 60 70 80 90

+Epair

Etot

I=60

I=50

I=40

I=30

I=20

I=10

I=0

I=0



1250077-32



-2

3

8

13

18

23

-2

3

8

13

18

23

-10 0 10 20 30 40 50 60 70 80 90

+Epair

Etot

I=60

I=50

I=40

I=30

I=20

I=10

I=0

I=0



1250077-33



Fig. 28. Quadrupole moment of the nucleus 17070 Yb drawn as function of spin for ε = 0.299, ε4 = 0

and γ = 50.

1.51.61.71.81.92

2.12.22.32.42.5

0 10 20 30 40 50 60 70 80 90 100

Q2

I

Q2(Exp)=2.16


and γ = 50.

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

0 10 20 30 40 50 60 70 80 90 100

Q2

I

Q2(exp)=2.12


and γ = 50.

1250077-34



Fig. 31. Quadrupole moment of the nucleus 17672 Hf drawn as function of spin for ε = 0.284, ε4 = 0

and γ = 20.


and γ = 20.


and γ = 20.

1250077-35



Fig. 34. Quadrupole moment of the nucleus 18274 W drawn as function of spin for ε = 0.265, ε4 = 0

and γ = 20.


and γ = 20.


and γ = 20.

1250077-36



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