The Extended gaussian overlap for the interacting boson model hamiltonian - Biblioteka UMCS

UNIVERSITATIS MARIAE C U R I E - S К Ł O D O W S К A LUBLIN —POLONIA

VOL. XL/XLI. 42 SECTIO AAA 1985/1986

Instytut Fizyki UMCS

К. ZAJĄC, A. GÓŹDŹ

The Extended Gaussian Overlap for the Interacting Boson Model Hamiltonian

Uogólnioneprzybliżenie gaussowskie dla hamiltonianu oddziałujących bozonów

Обобщенное приближение гауссовского перекрытия для гамильтонианамодели взаимодействующих бозонов

Dedicated to Professor Stanisław Szpikowski on occasion

of his 60th birthday 1. INTRODUCTION

In the present paper we propose an unitary approximate trans­

formation of the IBU hamiltonian to the Bohr-Mottelson model (BMM) collective space. The Interacting Boson Model [1] has been success­ ful in providing a systematic interpretation of collective excita­

tions over a wide range of nuclei but conceptually it is quite dif­

ferent from geometrical models based on BKM. A few various attempts [^2-5] » therefore, have been made to find relations between both ap­

proaches. However, in all cases either a restricted boson hamilto­ nian has been exactly transformed, e.g. Г2], to the BMil space or

528 К. Zając, A. Góźdź

the Hollstein-Primakoff [6] expansion has been finally used. The last case leads to an infinite order differential form which, in practice, must be restricted up to the second order terras. In addition, it is not quite clear if the Hollstein-Primakoff series is appropriately well convergent to apply such a procedure.

Taking into account these results we propose the approximate transformation of the IBM hamiltonian

H - £s5+S + faEd+dm + ÿs+3+55

+ [(d+d+)L(daiL]°

Z L-

4 5+S(d+d)° 4 ^{(d+d+)°5S + 5+5+(сГ0О°

+ ^C[(d+d+)ZC^5)z]° + [Gd+Ś+)Ł(d d )z]°}

where d*(s +) stands for quadrupole (monopole) boson creation operator and dm = (-)md_m, to the M collective space making u se of the Generator Coordinate Method with Gaussian Overlap Approximation (GCM + G0A)£6, 7] . As it will be shown later, the hamiltonian of noninteracting bosons is transformed exactly in this method and the interaction modifies only the BMM mass para­ meters and the collective potential energy.

2. TRANSFORMATION TO THE ВШ. COLLECTIVE SPACE

Mithin the Generator Coordinate Method the collective sub­

space is defined uniquely by a choice of the generating function jq>. In our case, to obtain the six-dimensional harmonic oscil­

lator space (natural for the monopole + quadrupole collective models ) we assume | q> in the form of rotating coherent state parametrized by Euler angles gj = (ajf, ^gj2, gJj), shape para­ meters and У and the collective variable ot corresponding to the monopole degree of freedom:

(2)

where

|oCjb^> = exp [- 2^- ] x

*expt^ó+ +~cortdź + I sin* (d£+с£)]|0> <3/

R(W) is the rotation operator in the laboratory frame. The ge­

nerating function (2 ) fulfils all conditions for the extended Gaussian Overlap Approximation Q7» 8]. It can be proved easy re­

presenting I q> in cartesian coordinates:

Q,oo = Jp

(4)

++]

The internal generating function (3) is invariant under the transformations of the standard octahedral group a.?:

A

e'^ -|^«> ,<-<2,3 (5)

and formulas derived in the paper [9], contained in this volume, could be applied in our case. For the boson hamiltonian (1) and the generating function (2) we obtained the metric tensor

the inverse mass tensor , the moment of inertia

(k=1,2,3) and the collective potential energy in a form of simple analytical expressions.

The metric tensor forms 6x6 matrix with non-zero components:

— b|/>LC<Aj) 51И 3 (6)

where

óinCGJ^óinCu^) GoS(cjj')

[bfcL CaJ)J = ^5in(cu-3) costal^) 0

0 • 0 <L

530 K. Zając, A. Góźdź

and

The square root of its determinant, which is known expression for the weight in the definition of the scalar product in the collec­ tive quadrupole space, is given by

D = 1|deł

[<J

]Ï

the kinetic vibrational energy, the rotational energy and the collective potential:

q<

=_

L £

2D AO.q

-b 4- Ä (9)

K=1 '

+ V(oCjbs)

where q^ = oL , q^ =j%, q$ =X and L£ denotes the angular momentum operator in the rotating frame.

The inverse mass tensor has six independent components:

= _ ^|łlc«3K

= ‘10)

435 >

(M-^= £d +

(IT1)** = +■

The inverse moment of inertia 3 is a simple function of the collective variables aL , p> and / :

<11>

where 3 135 '

XB> = (12)

represents Bohr’s inertia. For fixed oć = oCQ and small Jb de­

formations one can expand (11) in the Taylor series and obtain

В Cd 4- —■ ßcosU-(13)

where

&з” ÎÆ" °00

This case corresponds to the expansion of the classical kinetic energy containing third order tenus in the collective velocity [ю]:

(14)

532 К. Zając, A. Góźdź

The parameter B^ is'uniquely determined by the interaction strength between s and d bosons.

The collective potential is given by the expectation value of H (1) and so-called zero-point energy £0:

V(oipy') = j H| ot[i - £o

~ i ( _ AAl ") J2

2 Z k 2 2 4

I- Ц"° _/4 i ( i --2. ** _ Co _ Cz _ > pż

' à 1 2 20 AO Z AO ‘f i A , Co , ZCz \ V* i ( Vo , U-z, \ ,Z n 2.

V2 , «3 05)

2V351 <=&>?>*>

for pure quadrupole case ( oć = oLQ) the potential (15) relates to the potential analysed in the chapter 4 of ßo]:

V’cpx) - irzjÿ-_ +■ fï.f,’ <«>

with coefficients:

Л- 4- 4-

= —Ł- л Z-TT000

W _ 5 ( Co 2.C-Z • 48Cq \ c4 -

sly +— +-эг).

From this analysis one can conclude the condition for phyai rai be­

haviour of the potential (16) while > oo • the coefficient T?4 must be positive, i.e. + £^2 + 0. In the opposite

case it leads to nuclear instability. The further conclusion is that the s-d boson interaction is responsible for existence of a static nonzero ß deformation. It supports the IBM approximation

where the strength of s-d boson interaction classifies nuclei as transitional or rotational ones; for = 0 we get the "vi­ brational" limit SU (6 ) D SU (5)® U (1 ).

Por the case of vanishing boson interaction the hamiltonian (9) reduces to the six-dimensional harmonic oscillator (with

€. = £ . = 1 it corresponds to the total boson number operator a û d

К) which can be written in the form:

+ jÿ--s)

where the second term is. just the simplest version of the BM col­

lective hamiltonian.

One can easily check that GCM + GOA transformation to the BKM collective space reproduces the exact structure of one boson ope­

rators but the interaction is mapped only approximately and, in general,the hamiltonian (9) does not commute with the boson number operator. It makes some difficulties in comparison between IBU and ВШ calculations because all fits for the boson model parameters are done only for an effective hamiltonian in which a renormaliza­

tion of the interaction parameters has remarkable influence on form of the masses and the potential in (9).

3. APPLICATIONS AND DISCUSSION.

In general the IBH effective hamiltonian, which one compares with experimental data, is usually obtained from (1) by subtract­ ing the operator dependent only on the total number of bosons:

HO(N)= <£s- -t-

N

Such a renormalization allows to diminish the number of free para­

meters by two: £s - the single energy of s-bosons and uQ - the

534 К. ZajĄC, A. Góźdź

interaction strenght between then. Note that for a fixed nucleus the operator (18) can be replaced, by a constant number. However, as we mentioned above, the collective masses and potential are strongly dependent on this renormalization, i.e. on values of

£s and u0< To overcome this difficulty in general One needs to project the hamiltonian (9) into subspaces with definite number of bosons N:

-

Ä

<«>

where

N

and N is given by (17) with £g = Ед = 1. Because this method is rather cumbersome in practice we use only the first order Kam­ iah approximate projection [6J of Ho (N ) in order to subtract from hamiltonian (9) a bulk ground state energy connected with the va­ lence nucleon pairs:

Ttqf = i - (N - N) (21)

where

> = <äjbxlHo(N)(N-<N>)kßY>

° (22)

with

<N> = Oj^lNjctpo

According to the IBM interpretation fl, 11, 12 J the vibrational isotope 1^°Gd does not contain any d-boson admixture in its ground state (or this admixture can be neglected). Then

К = e5+

and. the single s-boson energy £ and uQ can he estimated with­

in the generalized seniority scheme £б, 13]:

£$ Ä G ^lQj = 0.33 KeV

= -0.166 ЫеУ (24)

where iOj= j+1/2 = 5 corresponds to 4 valence neutrons on 1 h$y 2 le"el (we assume Z = 64 as magic number) and G ~r KeV denotes the usual pairing strength. In the "vibrational" limit of IBM the boson renormalized hamiltonian for 1^°Gd has the following form:

He£f = +■ 4- JZ C^VZL+l'E(d+'d+) (dcF) J (25)

where [12] t = 0.63 MeV, = -O.O7O5 MeV, Ch, = 0.0095 KeV, C4 = -O.O44 MeV. The corresponding collective hamiltonian is de­

termined by the collective masses:

( _ io p2.

(26)

the effective moment ox’ inertia

X - X7 Се - +

(27)

+ v (C 4 - П

and the X -independent potential (Big. 1 ) with the minimum at cZeq = t 8, jï>eq = 0. The potential is plotted at is equilibrium point in o£ . The collective masses Мэд, and Ц'^ and moment of inertia J' (k = 1 or 2) ate plotted in Fig. 2 for fixed

oL = oć and cq Ï = 0.

536 К. ZajĄC, A. Góźdi

Because the potential nearly

mass M эдз, is nearly constant parabolic one can estimate the

and the■collective excitation energy of the first jd -vibrational state -

eV (experimental value E^+P =

it is given by harp» 0.69 0.633 MeV).

These are also plotted (dot lines) the collective potential (Fig. 1) and M (Fig. 2) obtained for the effective hamilto­ nian proposed by Casten 01]:

Hgxf. =* x.Q*Q,

> ¥V\

(28)

where Q2u ~ s+du + dus + X./'tfs’ (d+d)^ and the effective para­

meters are: £. = 0.45 MeV, at = 0.019 MeV, X = -°«9 MeV. We did not plot the mass M V« and moment of inertia for Castens’ case because they are very similar to "vibrational” limit calculations.

In both cases the mass parameters and moments of inertia de­

pend smoothly on deformation parameters. They satisfy the appro­

priate (for the generalized BMM ) boundary conditions [14J and re­

produce well the first excited state in the isotope 1^°Gd. The magnitudes of the boson mass parameters are much smaller than cor­

responding masses in the cranking model but also the collective

Fig. 2

potential is much softer. For Gd150 the moment of inertia J is very small - it is consistent with interpretation of this nucleus as nearly pure, not deformed, vibrator. However, to such a quality of this approximation for the interacting bosons hamiltonian, in general, one needs to fit the parameters of (1 ) for nuclei through­ out the whole periodic table elements with additional conditions implied by the BM model (i.e. positive defined mass tensor and non- -negative moment of inertia) and calculate the appropriate masses and potential. It is a purpose of a future publication.

This work was partially supported by contract CPBP 01.09

538 К. Zając, A. Góidi

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STRESZCZENIE

W pracy zastosowano metodę współrzędnej generującej wraz z przybliżeniem gaussowskim do wyprowadzenia kolektywnego hamil­ tonianu w formie operatora różniczkowego drugiego rzędu odpowia­

dającego hamiltonianowi modelu oddziałujących bozonów.

РЕЗЮМЕ

В работе применен метод генерирующей координаты вместе с гауссовским приближением для выведения коллективного гамиль­

тониана в форме дифференциального оператора второй степени со­

ответствующего гамильтониану модели взаимодействующих бозонов.

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11. J. Wawryszczuk, I. Gromowa, W. Żuk, R. Jon-Macha j, E. Kru­

pa, G. Lizuriej, T. M u m i n o w, W. Tańska-Krupa, I. Choł- b a e w: Zaburzone korelacje kątowe promieniowania gamma emitowa­

nego przy rozpadzie jąder 155’ 15J’ 151- 149Tb zaimplantowanych do folii Fe i Ni.

Perturbed Gamma-Gamma Angular Correlations in the Decay of 155' Ш, петь Nuclei Implanted into Fe and Ni Foils.

12. I. Bryłowska, P. Mazurek, K. Paprocki, M. Subotowicz, H. Sci- b i o r: Własności elementów fotowoltaicznych, otrzymanych metodą im- plantacji jonów na GaAs.

Properties of the Photovoltaic GaAs-cells Produced by Implantation Method.

13 I. Bryłowska, К. Paprocki, M. Subotowicz, H. Scibior: Krze­

mowe ogniwa słoneczne otrzymywane metodą implantacji jonów.

Silicon Photovoltaic Cells Produced by Implantation Method.

14. S. Szpikowski: Mikroskopowa interpretacja bozonów „s” w formalizmie izospinu.

Microscopical Interpretation of s-bosons in the Isospin Formalism.

15. В. Adamczyk, L. Michalak: Badanie rozkładu natężenia wiązki atomo­

wej w źródle jonów spektrometru mas przy pomocy modelu optycznego.

Investigation of a Distribution of Intensity of Atom Beam in the Ion Source of Mass Spectrometer by Using an Optical Model.

16. A. Modrzejewski: In Memory of Professor Włodzimierz Żuk.

Pamięci Profesora Włodzimierza Żuka.

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Energy Distribution of Helium Ions in Glow Discharge Source.

2 H. Murlak-Stachura, D. Stachórska, Z. Wroński: Ekstrakcja jo­

nów z plazmy wyładowania jarzeniowego.

Ion Extraction from the Glow Discharge Plasma.

3. J. Sielanko, W. Szyszko: Simple Monte Carlo Computer Procedure for the Depth Parameters Determination of Implanted Ions in Amorphous Targets.

Prosta metoda Monte-Carlo określania parametrów rozkładu implan- towanych jonów w amorficznych tarczach.

4. L. Gładyszewski: O pomiarach parametrów funkcji statystycznych sto­

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Ion Thermoemission Properties of Oxidized Tungsten.

6. L. Gładyszewski, R. Kazański: Analogowy analizator widma małej częstości.

Low Frequency Analyser of Noises Spectra.

V. W. О к u 1 s к i, M. Z a ł u ż n y, M. P i ł a t: O rezonansie cyklotronowym w cien­

kich warstwach PbS i Bi.

On the Cyclotron Resonance in Thin Films of PbS and Bi.

8. M. Kulik, J. Żuk: Ellipsometric Studies of SiO2 Films on Si Substrates.

Badania elipsometryczne warstw SiO2 na podkładach Si.

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