An Investıgatıon Of The Energy Levels And Multipole Mixing Ratio Of Electromagnetic Transitionsin The Even-Even Isotopes

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SAÜ

Fen Bilimleri Enstitüsü Dergisi

2

(1998) 149-153

AN

INVESTIGATION OF THE ENERGY L.EVELS

_{AND MUL}

TIPOLE MIXING

RATIO OF ELECTROMAGNETIC TRANSITIONSIN

_THE

EVEN-EVEN

ISOTOPES

R._AKKAYA, _M.

İL

_HAN

.')'ukaJ�ya

L'niversity, F"acul�v of

.. ·lrts and .\'cıences, J>h.vsics /Jeparnnent-Srık(ıt�va. Turkey Summary- In this work some of the electromagnetic

interactions of even-even Hafnium isotopes in the 150<k:;l90 _{defoıınation region were studied in a} detailed manner.

l n this region� us ing the experimental

8(E2/lv11)

ınultipole ınixing ratios the deformation parameters

�o

and the quadrupole moments q0 and q'2 were calculated. The obtained results are in a good agreement ·with the general systematic of the defoıınation region under consideration.

1-INTRODUCTION

The Theoretical and experirnental investigations of the rare earth elemen ts i

n the 1

50<k:;

190

defornıation region provide the valuable infarınations in explanation of _the characteristic properties of the elements toward the end of the defoıınation region

[2

,3

]

. Since the Hafniwn isotopes

(Z=72)

are the important members of thjs deforınation region they are under the intensive i nvestigations

[ 1-12].

2-_THEORETICAL BACKGRO_UND

As _{it is \Vell} kno\vn� the pairing plus the quadrupole forces have an important role among the nuclear forces[

1,2].

The quadrupole force produces the nuclear deformation while the pairing force tries to retain the spherical _symmetry. The effect of the pairing forces decreases as the added number of valance particles

ıncreases.

In this case the forces, causing to the rotational spectra, appear a dominant effect and nucleus becomes into defornıed state[3]. Dudex suggested the relation of

G=G0+G1(N-Z)

A A

(1)

for calculation of the pairing forces. Using the values of the G0 and _G1 for the protons and neutrons in the Eqn. (

1)

w e obtain the folloving equations�

GP =

ll7.90+0.176(N-Z)

1 A

Gn ==

lı

s. 95+0.078(N-Z)

1

1 A

(2)

Where GP and _Gn represent the pairing forces for the protons and neutrons respectively. As the Eqn.(2) show,

Gp> Gn so that we can write �0(n)>

Po(p).

It must be noted that

Po(P)

and

Po(n)

are the deformation parameters for the protons and neutrons respectively. W e have calculated the defoı ı na tion parameters

Po

from the second reference as a function of the f and

8(E2/Ml)

as given in the Eqn.(3);

Where f is given as follovvs

f = -·---

f3o-flo(P)

�

Po

N

f3o(n)

-

1

A

f3o (p)

(3)

(4)

and it can be put in a useful fornı as in the fallawing Equation from ref.

[14].

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Using the Eqn.(2) values of the GP and _Gn and the Eqn.(5), \Ve reach to the Eqa(6) for calculation of the parameter f

N _GP

f = -ı

A _Gn

(6)

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An lnvestigation of the Energy Levels and Multipele Mixi ng Ratio of Electromagnetic Transitions in the Even-Even Hafnium lsotopes

lt should be ernphasized that the Eqn

(

3) can be used for the calculation of Po the parameters of the isotopes toward the en d seetion of the region of the 150�AS 190. The electric quadrupole moments are the signifıcant factors for the nuclear defomıations. They can be dctcrınincd in terrus of the spectroscopic studies. We

have to renıernber that they can be used as a measure of the shape of the defined charge ditributions and the dimensional antisymmetry of the rotaticnal nuclei. Therefore the q, electric quadrupole, quantities have a

grc�lt significance in invcstigation of the defornıed ııucl�i. The spcctroscopic quadrupole nıoıncııt of

nuclcus is _given by:

q(D

1

3K ..

- 1(1

+ 1)

- (1+1)(21+1') qo ₍₇₎

forınula[ 6]. Here q0 is the intrinsic qudrupole monıent and it is d efi n ed as [sı.

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On the other hand, as the Table 1. shows, the deforınation has an important effect on the excitation energies of the Hf isotopes. Some of the transition energies[7] of these isotopes and their ratios were given in the Table 1. to clarify this effect. And alsa the variation of the excitation energies as a function of the number of neutrons was shown in the Figure _ı.

The folloving conclusion can be obtained from the close ana

l

_ysis of the Table.

ı

for the even-even Hf isotopes.

I.

X9 kc V < E"t < 94 kc V and 290 keV <E . <

298

-g �,

ke V relative lv small ener!,ry values are ıneasurcd experimentally.

II. 901 keV _<E"� � 1277 keV and

"'r

1063 ke _v_·_{<E 4}+ <

13

91 ke V relatively high energy r

values are deteıınined ex.perimentally for they rays.

Table. l.Some of the Transition Energy Values of the _Hf lsotopes Energies(in k e V)

E ı+ E4. E.,. E4+

Isotopes g g "'r r l74Hf 92 298 901 1063 l76Hf ₈₉ ₂₉₀ ₁₂₂₇ ₁₃₉₁ ı:8HI

94

- 1277 -180Hf

₉₄

_- ₁₂₀₁

-3- INVESTIGATION OF _THE ROTATIONAL

ENERGY LEVELS OF THE _Hf ISOTOPES

The deforınation parameters

�o

, the pairing forces of

the neutrons and protons Gn, Gp, the f parameters, the intrinsic quadmpole moments q0 and the quadrupole

nıoınents

q (2 +)

were calculated using the experirnental

energy values and 8(E2/M 1) ınixing ratios. For the calculated values see the Figures(l trough 6).

150

E4+ 1 E2+ g g

E2+ 1 E2+

r ı E4• 1 E2+ _r _&

3.24 9.79 11.55

'

3.26 13.79 15.63

- 13.05

-- 12.78

-(The variation·of the energies _{E_,+} and _E + was dra\vn

"r 4r

in the Figure.2 as a function of the neutron number).

III) 3.24� _{E4T 1}E2+ �3.26, 9.79 � _E')+

1

E _{• :Ç} 13.79

K g �r 2g

and 1.55 _{< E4• 1 E2+} s15.63 ratios can _be

r g

calculated by dividing the corresponding

energy values. The first ratio has a relativelv _•

lovv value with respect to the two last ratio

values.(see _{the F igure.3). Since an ideal} _y

instability is a b out the 2 � 5 [ 1] and E 4 _.. 1 E.,+ g -g

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•

R.Akkaya, M .ll han

so me w hat greather than 3, the re is a y instability

in this region. . ' 336 _.---� 288·_---· 240 192 144 96 48 -·- -·- -· 0 +---�---�---+---� 102 104 106 108

Figure.

1.

The variation of the ground state E..,+ and E 1 � energies depending on the neutron number N for the even-"'g -+g

even _Hf _{Isotopes,(dashed line represents E.,+ the and solid line corresponds to the} _E4+

₎

-g _g 1700 _�---, 1360 1020 _, ,. .... 680 340 , ··- - - - --- - - - -· 0 +---�---�---�---� 102 104 106 108

Figure.2. Neutron nun1ber dependence of the ganıma band energies

E2+

and E4+ for the even-even Hfisotops (Dashed

r r

: • t . '

line corresponds to the E2+ and solid line represents the E 4+

)

r _r 18 _T---� ıs 12 9 • 6 o

+---+---��---�---�

102 ₁₀₄ ₁₀₆ ₁₀₈

Figure 3. Behavior of the (E2; 1 E2;) , (E

-ı;

1

E2;)

and (E

4;

1

E2;)

ratios depeneling on the neutron number N for

the even-even _Hfisotops

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An

lnvestigation of the Energy Levels and Multipole M ix ing Ratio of Electromagnetic Transitions in the Even-Even Hafnium lsotopes

�ll!Sl ...-- --.-... o.ı>W

/

·-· 0,1336

1

a.ım • O.lll .._..,.__..,_...,_ .... 102 lo-& 106 lOI Gp 0,80.5 ....---,

0.77.,

• 0,735

'

."' 0,7 � 0,665 0,63 .,___.,._...,_....,._ ... 102 104 106 108 Gn 0,28 ,.---�---· 0,21 � _._-- _• . --0.14 0,07 o ..._ ... ...,._...,__, 102 104 106 108

f

Figure 4.

The

v

ar

ia

t

i

o

n of

the parameters

GP , Gn an

d

f depending on

t

he

neutron number N

• • •

ô(E2/rv1 ı) ınixing

rat

i

o \·alucs

of ınultipolc

ııüxtures for

tr

a

n

s

iti

o

n

s

fron1

the

ex

c

ite

d

higher

to

the lower levels

in

the

Hf i sotapes were given in

the

Tab le 2.

Tab le 3.

The

Coınputcd value

s for the

so m

e im portant

parameters for the even-ev en Hf Isotopes. • Calculated

values in this work

Tab le

2. The Experimental 8(E21M 1) Müring

Ratios for

the eve

n-ev

e

n Hf

Isotopes.

T

r

a

n

sit

i

o

ns

Iso to pes

l74Hf 176Hf l"i�Hf 1 . l�OHf ") . ,., . � -s

'

... r

( )(7)

- ll·-� 1 �4(7)

(

Ü 41 . �H�

)��()

-t) tl"' O. 7(2)<8) 4. _r _., 4. _&_.

-(2.5:��

₎₍₇₎

2.0. 7°) -6; - 6. _' -0. 9(2)(?)

-The calculated defonnation parameters as a function

of

the rnultipole

nıixing ratios, the pairing forces for the

pro

tons

and neutrons�

and tJıe

f parameters for the even

even

Hf

isotopes

vv

ere

presented

in the Tab

le 3.

As

the Tab

le 3.

show

s

the pairing force of the protons

increases regularly,

from

0.133 to

O. 135

as the neutron

nunıber increases. In a contradiction� the pairing force of

neutrons r

egul

e

rly

decreases� from 0.078 to

0.070�

as the

neutron nuınber increases. Similarly� there

is

a reguler

incren1ent in the f parameters of

·

the

even-even

Hf

isotopes. And there is a good agreement bet\veen the

values calculated in this study and Kun1ar and Löbner

values for

1 74Hf

isotope[3,9,13] and the obtained values

for the defoıınation p a

nimete

r

s

�0.

152

Iso to

pes Gp Gn 174Hf 0.13

3

0.078 176Hf _0.134 _0.075 178Hf _0.134 _0.072 l�OHf _0.135 _0.070 f

_Bo·

0.179 0.270 O. 197 0.120 0.215 0.020 0.234 0.020

Bo

ref.l3

]

0.250 0.250 0.240 0.230

The

values of

the

parameter

�o are (0.02 <

�o�

0.27) in

accordance

vv ith systematic in the defoı ınation region.

The results for the

quadrupole

ın

om

en

t

s

q;

and intrinsic

quadrupole

monıents q0 for the even-even Hf isotopes

-vvil

l be as

given

in

the Table 4.

' ₊

Tab

le 4.

calculated

q0

a

n

d q2

values

of

the even-even Hf

isotopes

Iso to pes

l74Hf l76Hf }78Hf ı

soru-qo

e.barn

(this vvork)

7.19 3.01 0.48 0.45

qo

e.

bam [

13] 6.95 7.18 6.96 6.84

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•

R.Akkaya,

M.llhan

.ı.RESUL TS AND DISCUSSIONS

In the present work, variations of the energy values of the excited states, multipole mixing ratios 8(E2/M 1) , the pairing forces of protons GP ,the pairing force of neutrons Gn, the f pararneters . the defomıation paraıneters

�n

and the quadrupole moments have been subjected in all details for the even-even Hf isotopes found in the end seetion of the 150 < A < 190 deformed region. lt has been observed that the pa i ring for ce of the protons Gr increases as the neutron nun1ber N increascs \\hi k the pairing forcc Gıı of _the ncutrons dccrcascs. Th�

paranıeter r has the siınilar character \Vitlı the Gp . The deforınation paran1etcr

�o

has a second order dependence on the negative of the parameter f and it decrcases according to the Eq.

(J).

The details of the behaviours of the parameters , mentioned above.can be seen froın the Tab le 3. The values of the parameter

�o

for 1-4Hf isotepe is in a good agreement \vith the calculations given by Kumar and Löbner[3, 9, 1 3]. The Kuınar's values are -0.5 $

�o

< 0.5 for the deformation region of 150 < A

�

190.

As the Tab le 4. shows the quadrupole mornents of the Hf

iso to pes are good indicator of the d efor nıation amounts of the nuclei and they rapidly decrease as the neutron numbers increase. These results are also in a good agreement with the Kumar's work[9] .

REFERENCES

[

1] A. Ari ma, F. Iachello, Interactıng Bo son Model of

Collective Nuclear States, Annalen of Physics III,20

1-238(1978)

[2}

W.Greiner. Nuclear Phys.,80,417-433(1966)

[3]

K. Kumar M.Baranger. Nuclear Physics

A.l22,273-32-t( ı 966)

[4]

J.Dudex, A. Macjehofer, J.Skalski,J.Phys. G,Nucl. Phys. 6.-t47-45-l(l980)

[5]

P.Taras�J.Keinonen�Nuclear Physics.A,390.,287-312

( 1982)

[6] J.B.Gupta, K.Kumar,J.H.Hamılton, Phys. Reviev C,Vol.l6,No: 1(1977)

[7] Rev. Mod.Phys Vol.54:-No:1,(1982)

l8]

M.SakaL Atomic Data Ta b 1 e. 3 ı . N o : 3 � 3 9 9 --+ 3 2

(

1 9

8-+)

and Nuclear Data

[9] K. Kumar J.B.Gupta,J.Phys. 10,525-537(1 984)

[ 10] K. Kumar Collective Hamiltonian Derived From the Pairing-Plus-Quadrupole Model (PPQ) Nucl.Phys. A 321,189-232 (1974)

[ l l] M. Çiftçi,Turkish Journal of Physics, Vol.l5,No:4, 496(1991)

[12] R.K.Sheline,D.G.Burke, M.M.Minor and P.C.Soo4, Physical Review C,Vol.48, 911(1993)

ll3] K.E.Löbner,M. Vetter and V.Hönig, Nuclear lntrinsic Quadrupole Moments and Defonnation Parameters, N uel. Data Tables A 7, 514-51 5(1970)

[

14

]

R.Akkaya. Multi po le Admixtures of Electroınagnetic Transitions in Some Even-even Defoııned Nuclei, Phd., Thesis, I.T.

Ü

Sciences institute, ( 1987)

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