Collective Bohr hamiltonian in the generator coordinate - Biblioteka UMCS

ANNALES

U N I V E R S I T A T I S MARIAE CURIE-SKLODOWSKA LUBLIN —POLONIA

VOL. XL/XLI, 14 SECTIO AAA 1985/1936

Instytut Fizyki UMCS

A. GOŹDŹ

Collective Bohr Hamiltonian in the Generator Coordinate Method

Kolektywny hamiltonian Bohra w metodzie współrzędnej generującej

Коллективный гамильтонианБорав методегенерирующей координаты

Dedicated to Professor Stanisław Szpikowski on occasion

of his 60th birthday

1. INTRODUCTION

The collective Hamiltonian for quadrupole motion originally proposed by A. Bohr in 1952 [1] was successfully used for de­ scription of low lying collective nuclear states [2]. The para­ meters of the Hamiltonian at first were determined phenomenolo­

gically and afterwards on the basis of microscopic theories [2,51* In the estimation of the inertial functions as well as the col­

lective potential function different models have been used. The partially phenomenological pairing-plus-quadrupole model [4-6],

170 A. Góźdż

TDHF approach [7], ATDHP theory of collective motion [8] and a direct application of the Inglis formula [9-111 provide only a few examples of a great variety of methods used to calculate the mass tensor and the potential energy surface. However, to obtain a quantal collective Hamiltonian all the methods require not uni­ que procedure of quantization of a classical collective Hamilto­ nian [121. In addition, e.g. in the popular cranking approach, the potential energy surface [13, 141 is chosen with accuracy to any arbitrary scalar function i.e. the quantal zero-point energy is not taken into account [15, 16].

The criticism of these quasi-quantal methods was a motiva­

tion to look for a more satisfactory approach. The generator co­ ordinate method (GCM) offers a fully quantal theory [17-191 which together with the Gaussian overlap approximation [20] or its ex­ tension [21] allows for microscopical derivation of the collective Hamiltonian. This method allows to derive the collective Hamilto­ nian starting from stationary Schrbdinger equation in a fermionic space. Contrary to the previous approaches no redundant variables are introduced and no quantization procedure of a classical Hamil­

tonian for collective motion is needed. The first attempts in this direction were already made nearly ten years ago. Using an appro­

ximate narrowing kernel approach [221 the rotational kinetic ener­

gy of the Bohr Hamiltonian has been obtained in [231. However, the moments of inertia are derived only for the many-body Hamiltonians and not for the mean fields Hamiltonians. In addition the paper does not offer explicitly any expression for the zero-point cor­

rection to the potential energy. The full Bohr Hamiltonian is also not explicitly derived there because the analytical diago­

nalization of the five-dimensional metric tensor is needed.

It is the aim of the present paper to describe a quantal de­ rivation of the full quadrupole collective Hamiltonian within the GCM and the extended Gaussian overlap approximation [21]. The re­ cent GCM estimates of the mass parameters and potential energy [24-27] are different from those obtained by quantization of the classical collective Hamiltonian. However, to decide which ap­

proximation gives results more closer to the experimental data one needs to solve the full, including the most important degrees of freedom collective Hamiltonian. It will be a topic of our fu­

ture publications. The present paper is only a preliminary step

Collective Bohr Hamiltonian in the Generator ... 171

towards this goal; here we give only the outline of the theo­ retical formalism.

2. THE GENEBATING FUNCTION AND THE METRIC TENSOR The classical Bohr Hamiltonian is dependent on five quadru­

pole complex collective variables a J2 (u=-2,1,...,2) which f

* I

one assumes as generating coordinates. One also assumes that the appropriate normalized generating function I au > describing a quadrupole nuclear motion of many body system satisfies the conditions for the extended Gaussian overlap approximation [21]

z (?) <«> i <W

i .e. the overlap function ^<Хг /is a deformed in the variables Ct .J2) and Gaussian profile (for details see [21, 26 and 27]). After transformation [1] to the shape va­

riables p and у and three Euler angles Л= 52 the generating function can be factorized as

1<2,...,<’> = 1ЙРТ>= R(û)|pT> , (i)'

• Л -ißjüj

where R (Л) = 6 в в is the rotation operator de­

fined in the space fixed frame. The normalized "intrinsic" func­ tion I corresponds to the generating function |<1 > at the moment when the "intrinsic" (rotating) and the laboratory (fixed space) frames coincide. In applications the intrinsic function

j jbf) is usually chosen as a BCS-type function. Following the paper [23] one assumes that the intrinsic function has the reflection (dg-group) symmetry [2, 3] :

-iT3k

£ IpPHpp , k = 1,2,3. (2)

The symmetry allows to obtain the rotational inertia in respect to the principal axes. To apply directly the formulae of Refs.

(21, 26, 27J one requires that the matrix elements ?

к rs 4. 5

(q = k=1,2,3, q^ =p » q3 = ]• ) have to be equal zero. It is alyways fulfilled by the operators ~— due to do-symmetry (2 ).

172 A. Góźdź

On the other hand, from the normalization property<p-j-= 1, one can easily obtained that Re<2pj'|Aj |ßp[> = О and the ap- priate choice of (p>y )-dependent phaseł factor in the generating function (1) allows always to satisfy the required condition (see e.g.[27] ).

Following [21, 27] the metric tensor in the collective space can be calculated from the formula

Î/W = Re ^9 I д^Г I <0 , (5)

where q= (q*) = ( Л 1, Ä2,5?3,p ) and Д- JL act on bra and ket, respectively. Owing to d2-symmetry the metric tensor has a reduced form:

0

‘hß Sn

(4;

Using the relation (it can be obtained of the rotation operator)

by direct differentiation

R

*

(a) ^{a g (Д)} _(?)

where the matrix

( иг)) -

- sin

22

£3

sin &3

Sin cos

COS &3 ⁰ ₍₆₎

0 1

0

one can obtain the expression for the "rotational" part of the metric tensor

Collective Bohr Hamiltonian in the Generator 173

bklifl)bvt(Ä)<?v> П)

In (7 ) and further the notation <( A ) = < A | У is used. One can easily check that the corresponding contravariant components can be expressed in a similar way:

(b”(®)lk(b"(«)lkl<5t>'' (6)

The "vibrational" part of the metric tensor is dependent on ex­ plicite definition of the generating function and in general can be written as (3) (}i»b = 4»5):

<

„-<P

I

Note that the vibrational components (9) do not depend on Euler angles. To derive expressions for the mass tensor and the poten­

tial energy one needs to calculate a part of Christoffel coeffi­ cients connected with the full metric tensor (g^)» One can easily

prove that

<ггГмгГ^гм'° <1°>

and in practice only the coefficients related to the vibrational tensor (9) have to be evaluated i.e.

3. THE MASS PARAMETERS AND THE ZERO-POINT ENERGY

Denote by H the many-body (or j3 , у dependent mean field) Hamiltonian of the nucleus at the moment when the laboratoxy and intrinsic frames coincide i.e. for £2=0. After rotation the Ha­ miltonian is equal

174 A. Góźdź

H' = R (£) НЙ*(Л). (12)

Obviously, if HA is a true many-body Hamiltonian then it is in­

variant under space rotation and H' = H, For the mean field Hamil­ tonians we assume only that H is invariant under dg-group, i.e.

e’lir^k He = i-i ; к =1,2,3. (13)

It allows to avoid the asymmetry terms linear terms) [27] in the collective Hamiltonian which are not observed in experiment. The covariant components of the inverse mass tensor can be calculated from the formula [21, 26, 2?]:

r e {ч ' < * ■*' 4^ > l *

1 zn I э эИz I \ _ JL -^1

2 <CH 3qv 2 J ’

where the linked matrix element is, as usually defined [18]

нâq

* H

9-^41 ** 1

15)

<q

< a|Q'> /a.a»q U =1,2,...,5) ₍₁₆₎

and ---- -r denotes a covariant derivative e.g.

△ ho _ Shy X _? ⁱ

(17)

Using (10), d2-symmetry property (2) and (13) one can show that the mass tensor has also a reduced form (4)

Collective Bohr Hamiltonian in the Generator ... 175

0 1171 ]pp (177

( 177 ) JP ( rn’1) y|

(18)

After straightforward but lengthy contravariant inverse mass tensor

calculation.one can obtain the

<3’>]

(19)

and for u, 9 = 4,5

£ £ cff М<рт| ^- tH^|pT>L

jL / a r I Э Э M i . i Ahj- 1 (2° ) 2 <Pï' acf " 2 ûq€

j '

Similarly to the vibrational part (9) of the metric tensor, the vibration part (20) of the inverse mass tensor does not depend on Buier angles.

The zero-point energy can be obtained from [21, 26, 27]

V? ^l4>] (21)

i »

and due to dg-symmetry property reduces to a sum of the rotational о ‘ and vibrational S corrections to the standard col­ lective potential energy <яру|н'|йр»|> = :

c ctroi) (vib)

% = -^o (22)

176 A. Góźdź

In a similar way as for (19), making use of the formulae

<$РТ| эа;Н'Д;|ярг\-- £ b^b^WÓtHO^L

(23) and

(24) one can obtain the rotational zero-point correction:

C 4 è < 3 ‘ > [< >

4 < A « Л > ]

where, as it will be shown in next paragraph, 4 represents the rotational inertia:

4 < & .O

,«]])

(26)

Note that, the moments of inertia (ßf ) resembles the Peiers- Toccoz and Une et al. [23] moments of inertia. The vibrational zero-point energy can be written as

e W 4è <4е{<Рг1^Н^1рт > и .

*<рт!з^ 3^lPï>] ; vP л4 5=г

4. THE COLLECTIVE HAMILTONIAN

The collective Hamiltonian derived in the GCM with extended gaussian overlap approximation [21] is veiy similar in form to the collective Hamiltonian obtained after the Pauli-Podoisky quantiza­ tion procedure [3, 12]

Collective Bohr Hamiltonian in the Generator ... 177

<28) where

<rdeHfyJ ; tu,v = t5

and the potential energy is

V (4) = <ql H'l4> - £«(4) (29) The main difference between (28) and the traditional collective Hamiltonian is that the metric tensor is different from the mass tensor and the potential is corrected by the zeio-point energy.

In the case of the Bohr Hamiltonian (28) the rotational energy can be easily written in the standard fono [23] :

э-;1(ртК1к)\ о»)

where the inertia parameters are explicitly given by (26) and

3 д

i'k=-i£(b <»)цж <”)

are the angular momentum operators in the rotating (intrinsic) frame expressed in terms of Euler angles (see Chpt. 5 of [3] )•

In derivation of (30) the following relation was used

è [ fa ■ ’Пуг 0?)

The vibration kinetic energy does not depend on Euler angles and is given by the operator

£■ _ 1 г a „z .Л a a _npr a

’vib 2D 19(3 D^m ' 3J3 + ap D(m ) ay

(55)

178 ^A* Góźdź

where D = J l<|0p<jTr'(9pr)2|<34 < Эг )< Э j>

The potential energy (29), as it is expected, is a function of only shape parameters:

A . (rot) (vib)

V=<PT |H|pr> -€o (PT)-SO (pj), (34) where £ and 6 can be calculated from (25) and (27)« To complete formulae, one can quote the volume element

dT=^det(^)r dpdT dÆ,0Ягdfi3 HI(]ßJh-(<jр/1 '•

• <>< 3’ >(Э, > • dpdæ sin Лг dĄ d£2 dP3 (35) which ensures the Bohr Hamiltonian

(56) to be hermitian. It is also important to note that the Hamiltonian

(56) is invariant under space rotations even when the Hamiltonian H’, (12), is not invariant.

This way we derived the full quantal Bohr Hamiltonian with unique expressions for the mass tensor and potential energy for

both a many-body and effective nuclear Hamiltonians. The deriva­ tion is independent on somewhat artificial quantisation of a class­ ical collective Hamiltonian as it was in the cranking or similar approaches. The collective Hamiltonian (56) has required symmetry properties and, on the other hand, allows for a very flexible choice of the collective subspace in the full fermion space. This fully quantal derivation of the Bohr Hamiltonian is limited only hy rather general conditions under which the extended Gaussian overlap approximation can be used.

The author is greatly indebted to Professor K. Pomorski for suggesting this problem and for stimulating discussion. The author also would like to thank Professor M. Brack and B. Werner for warm hospitality at the University of Regensburg.

This work has been partially supported by contract CPBP 01.06.

Collective Bohr Hamiltonian in the Generator 179

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A. GóźdA 180

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STRESZCZENIE

'.7 pracy zastosowano metodę współrzędnej generującej f+ ogólnione przybliżenie gaussowskiej do otrzymania kwantowe­

go hamiltonianu opisującego kwadrupolowe ruchy kolektywne ją­ dra atomowego. Jako zmiennych kolektywnych użyto trzech kątów .Lilera oraz dwóch standardowych parametrów deformacji (bi . Uzyskano vj pełni mikroskopowe wyrażenia na tensor masowy i po­ tencjał kolektywny.

РЕЗЮМЕ

В работе применяется генерирующей координаты (+ обоб­

щенное гауссово приблежение) для получения квантового гамиль­ тониана, описывающего квадрупольные коллективные движения атом­ ного ядра. Как коллективные переменные применялись три углы Эйлера и два стандартные параметры деформации Jb и Т. Получены полностью микроскопические выражения на массовый тензор и кол­ лективный потенциал.