Microscopic foundation for the particle-vibrator model

Volume 92B, number 3,4

PHYSICS LETTERS

19 May 1980

MICROSCOPIC FOUNDATION FOR THE PARTICLE-VIBRATOR MODEL E.R. MARSHALEK Department o f Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA Received 13 February 1980

The tools for deriving microscopically the particle-vibrator model are presented in the form of a new fermion-boson expansion in which valence particles or holes added to a closed shell are treated essentially as fermion degrees of freedom, while particle-hole excitations of the closed shell are treated as bosons. The Pauli principle is completely fulfilled without redundancy. Both the nonunitary (Dyson) and unitary representations are given.

The particle-vibrator model, in which several particles or holes are coupled to a closed-shell vibrating core, has been a venerable part of nuclear physics, dating from the seminal paper of Bohr [1]. Although various phenomenological versions have evolved, the latest being the cluster-vibration model [2], no satisfactory microscopic derivation has yet been given, fully taking into account the Pauli principal. The main purpose of this note is to provide the tools for such a derivation in the framework o f boson expansion theory. The details must await a longer publication. The boson expansion method is based on the remarkable fact that fermion dynamics can be simulated perfectly within a subspace o f a suitable boson Hilbert space [3], called the physical subspace, whose orthogonal complement, being unrelated to fermions, is called the unphysical subspace. In practice, the entire fermion space is mapped injectively into the physical subspace of the so-called ideal space, which for even particle numbers is a boson space with one boson representing each fermion pair excitation. Systems o f odd particle number are normally described in terms o f the coupling of an ideal o d d f e r m i o n to the bosons of the even system. Thus, all fermion pairs are bosonized, and the formalism is designed to exclude more than one ideal fermion to prevent redundancy [4,5]. To obtain the particle-vibrator description it is necessary n o t to bosonize all pairs, but to allow some to retain their fermion identities when several valence particles or holes are a d d e d to a closed shell. Then, one

should pair only particle-hole excitations into bosons, while treating the excess particles or holes essentially as fermions. It shall be shown that the formalism o f refs. [4.5] can be generalized in a nontrivial way to cover tbis situation * 1. The advantage is that an openshell model calculation can then be performed, taking into account collective core excitations, either indirectly, through a core-polarization contribution to the effective interaction, or directly, in the spirit o f the cluster-vibration model [2]. It should be emphasized that the formalism of refs. [4,5] could also be applied to the same nuclei, but pairs of valence particles or holes would be then described as pair-transfer bosons (pairing vibrations) leading to different approximations. Of course, experiment must ultimately determine which of the two alternative descriptions is better. For brevity, tensor notation is used throughout as in ref. [5], with the usual summation convention. Particle states are denoted by lower case Latin and holes by Greek indices. Capital Latin indices cover both possibilities. In the fermion space, annihilation and creation operators for particles are denoted by a i and Oti a~ and for holes by otu and a ~ = -- a t / d ' obeying the usual anticommutation rules

(~A,~8)-(~A,~8)--0,

(~A,~B)=sA.

(1)

An orthonormal basis for the fermion space is provid:~a Another kind generalizatiola restricted to an SU2 algebra has been carried out by Suzuki and Matsuyanagi [6]. 245

Volume 92B, number 3,4

PHYSICS LETTERS

ed by the vacuum 10>E (the closed shell), satisfying OLA}0>F = 0, together with the antisymmetric states NB {il/al ... iNBI~NB> F = H c~imo~UmlO>F , m=l [Jl ... ]np il/al "-

(2a)

iNB/aNB>F

np = [I

n=l

(2b)

o~JnlilUl ... iNBUNB> F ,

{Pl "" Vnh il/'Zl "" YNB/INB>F nh (2c)

= I-I oLUnlill~l ... iNB)JNB> E ,

n=l

including the subspaces with equal numbers N B of particles and holes, and those with an excess np of particles or n h of holes. In the ideal space, the fundamental operators include the boson annihilation and creation operators B i , and B i" -~ B ~ , respectively, obeying the commutation rules [Biu, ~j,] = [B;", B j~] = 0 ,

where ~p(5~h) are permutations of particle (hole) indices. These vectors, which are mutually orthogonal but not normalized, constitute a D y s o n representation. Since the degrees of freedom of particle-hole pairs are completely preempted by the bosons in the ideal space, ideal particle-hole pairs a i aU should be eliminated to prevent redundancy. The way to accomplish this, if the aA and a A are chosen to obey the usual anticommutation rules analogous to (1), is to simply banish such pairs from the physical subspace, as in eq. (4), although they must then appear in the unphysical subspace. Another method, the one actually used here, is to modify the algebra of the ideal particles and holes so as to incorporate consistently the requirement aia u = 0, thereby abolishing such pairs in the whole ideal space. The required algebra is found to be: £ai, a j } = (ai, a / } = 0 , (au,av} = (aU,av} =O,

NB ~ p I-I B i m # m [ o ) , m=l (4a)

If1 "" fnp il/al -'- iNB/aNB>D = (NB f) -1

np X ~ (-1)70 I-I aYnlil/al ... iNB/aNB> D ,(4b) ~p ~P n=l Iv 1 ... vnh il/at ... iNBPNB> D = ( N B [ ) - I nh X ~ ( - 1 ) 5~h 5~h [-I aVnlil/al ...iNB/aNB>D , 9h n=1 (4c) 246

,

{aU, a v ) = S U v E p , EbaU = a,aE h = 0 ,

Eh ai = ai Eh = a i ,

Eh ai = ai E h = a i ,

EpaU = a U E p = aU ,

Epa, = auE p = a, ,

(5a)

(5b)

[B;,, BYV] - 6 ; j"G - ~ '(3)

and the annihilation and creation +operators for ideal particles and holes, a A and a A = a j4, which commute with all boson operators and obey an algebra discussed below. The physical subspace is defined as that generated by the boson-fermion vacuum 10>, satisfying BiulO> = aA[O> = 0, together with the following antisymmetric vectors which are in correspondence with

{/i/at .,. iNB/aNB> D = ~ ( - - 1 ) ~ P ~p

[ai, a j } = @ E h

Epa i = ale p = O,

E2=Ep,

eq. (2):

19 May 1980

E2 =E h .

(5c)

Eqs. (Sb) immediately imply aiau = auai = aiaU = aUai = aiaU = auai = O .

(6)

The operator 1 - Ep is the projector to the subspace containing at least one ideal particle and 1 - E h is the projector to the subspace containing at least one ideal hole. If desired, the projectors may be represented as Ep = 1 - ai(n.p + 1 ) - l a i , E h = 1 - a U ( h h + 1 ) - l a , , where hp = a~ai and h h = aUa u are number operators for ideal particles and holes, respectively. I f P 0 is the projector to the pure boson subspace (4a), then P0 = Ep + E h - 1. When operating on states containing no ideal holes (particles), the ai, a i ( a , , a ~) behave as ordinary fermions, but otherwise they give zero. Although not necessary, a realization of the algebra (5) can be given in terms of ordinary particles and holes, call them fill, ~A, having anticommutators like (1). Let a i - Eh"d i = ffiEh, a N -=Ep'a~ = "ff~Ep, with the projectors now given by Ep = 1 - a'i(np + 1 ) - l a i , E h = 1 - d u ( ~ h + 1 ) - l a ~ , where Bp = Yi'~i, nh = Yu h'u' Then these a A and a A =a~l satisfy the algebra (5), and the physical states (4) are

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equivalent to those obtained by replacing a A -> dA. Following Okubo, let V be a linear injective mapping from the fermion states (2) to the corresponding ideal states (4): ] }D = V] )F. Since the latter are unnormalized, Vis not isometric, but satisfies V -1 V = IF, V V - I = p , P V = V, where v - l = / : Vt; I F isthe identity in the fermion space, and P is the projector to the physical subspace. For any fermion operator F, F D = V F V -1 is called its Dyson image, and F D P = P F D = F D. Since V is nonisometric, in general ( F ? ) D ve (FD)?. Of special interest are the images of single fermion and number-conserving bifermion operators ,2, expressive in the form:

(ai~/)D = A } P = PA} , (auav) D = A uvP = PAUv , (aia`u)D = pRi`u , (a`u)D =PA`U ,

(a`uOq)D= Ri`up ,

(%')D = A i P ,

(a`u)D =A`uP"

(7)

Aug = Bi`uBiv + a`uau ,

Ri`u = Bi`u _ BJ`UA~ _ BiVa`uau ,

A i = a i + Bi`ua`u(ft h +

1)-1

(AB,Ac}P

AiA/p = A}P,

A"AvP=A~P,

A u A i P = Ri`up ,

A i A ,up = Ri`up.

= 5BP,

(9)

Incidentally, hp and ti h may be replaced by h = h h + hp in eqs. (8) because of eqs. (6). Setting hp = h h = 0 retrieves the forms of ref. [5]. As is easily checked, the physical subspace (4) can be constructed from the operators (8) as follows: lilp t ... iNBNNB).D NB NB = I-I Rim`um[o)= I-I A i m ` u m [ o ) , m=l m=l

(lOa)

1]1 "" ]np i l P l '" iNBPNB)D np = FI

n=l

A /nl ilP t ... iNBUNB> D ,

(10b)

Iv I .,. Pnh i l P l ... iNBPNB) D nh

(10c)

Ri u = Biu ,

ai Bi`u ,

,

A`U = a u - Bi`u ai(np + I) -1 ,

= { A B , A C } p = O,

= I-I Avnlilt.tl ... iNBPNB) D . n=l

A i = (a i - a / A } ) ( h p + 1) -1 + a`uB i`u , A`u = (a`u - a~A~)(hh + 1) -1

(AB,Ac}P

(oti)D = p A i ,

Unlike the Dyson images, which are unique and defined only in the physical subspace, the corresponding operators A}, AUv, R i`u, R i `u , A i, A`u, A i and A`u are nonunique, and can be chosen as extensions of the Dyson operators defined on a dense subset of the ideal space. These extensions may be written as follows: Afi = Bi`UB/`U + aiaf ,

19 May 1980

(8)

since, as is readily checked, the vacuum conditions A}I0) =AvU[0> = Ri`ul0) = A i l 0 ) = A`u[0) = 0 are preserved, and also the commutator algebra of the bifermion operators amongst themselves and with the single fermion operators. It can be shown, along the lines of a similar derivation of Okubo, that all other Pauliprinciple constraints are satisfied inside the physical subspace; for example:

Although the nonunitary Dyson representation may be useful, a unitary representation is usually more convenient. Following Okubo, for a fermion operator F, the unitary representation F U is obtained from the Dyson representation through the similarity transformation F U = S F D S - 1 , where S normalizes the physical states (4) or (10), and satisfies S P = PS. Moreover, since S V is isometric, the condition (F~)u = ( F u ) t is now satisfied. As iseasily checked, S may be chosen in the form S = f ( N B , h), where f ( x , y ) = [P(y + 1)/ F(x + y + 1)] 1/2 F being the gamma function, 3: B =-Bi`uBiu, the boson number operator, and h = lip +/1hCorresponding to eqs. (7) and (8) the following unitary representation is obtained: (~i0~])U : (0~i0l])D ,

(0t`u~u)v : (0~`U0~v)D ,

(1 la)

(O~u~')U '= (/VB + • + 1)l/2Bi`u e = (JVB + tl + 1) -1/2 ,2 The images of all other operators can be constructed from these. For example, the two.particle and two-hole transfer i operators are given by (oJal)D =Pg [A i, AJ] and ( a ` u ~ D = P ~ [A`U,AV], respectively.

X (Bi`u - g f v B i v B i `u - Bj`ua]a i - BivbUb`u)P,

(0~')U = [(]VB +/}p + 1)l/2g//+ [l`uBi`u]P = [(/VB + n p + 1)-l/2(ai - Aia]) + a`uBi`ulP, 247

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Volume 92B, number 3,4

(au)u = [(NB + nh + l)l/2ti, - 8iBiu] e

= [(NB + '~h + 1)-1/2(a. _ ajar)

- a;B~.]e,

where (l lb)

~ the abbreviations: a_ i -- (nA + 1)- 1 /"a i, a- = (n h + 1)-1/2a,, ~z = al(np + 1)-1/2,~2, -aU(n h + 1) -I/2. Actually, (11) remains correct if the replacements hp, h h ~ h = hp + h h are made everywhere, because of (6). The remaining operators are obtained by hermitian conjugation; thus, (~/a")u = (~,~/)~ and (a A)U = (aA)4. Eqs. (1 lb) are still not in the most convenient form for perturbative applications. Such a form is obtained by extending to the present problem the identities derived in the appendix of ref. [5], with the aid of which one may derive the relations with

[f(A)I}P = (1 +JVB + h p ) - I X {f [-(NB + hp)](~} - A})

+ f(1)[A} + 8}(9~ + ~p)l}e, f ( _ h p ) a i = [f(~)]{i a] ,

(12)

plus similar equations obtained by letting i -+/~, j ~ v, hp -* h h. Here, f(Z) is an arbitrary holomorphic function f(Z) = ~m=.oCmZm., withZ replaced by tensor operators (Am)} or 05m)}, defined recursively by (A0)~ = 6~,

~-aiai,

(Am+l)~ =(A m ki)~Ak ' (~30)~=6},

(/sm+l)~=(,bm)k/3~c, (13)

and similarly with Latin letters replaced by Greek. With the aid of (12), eqs. (1 lb) may be rewritten as follows: (oluoq.)U

=

(S/B~. + SSBiv - D/B/u)P,

(~')U = (s/rkak + aV rff Bi.) P , (au) U = (Sur v vxa?. - aYrjBiu)p,

(14)

S[ = [(1-A)1/2]~I,

S ,v = [ ( l _ A ) 1/2]u, v

D] = [(1 - AB)I/2]~,

(AB) / =- B/U Biu ,

r."j [(I - b)-1/21~:,

rvu = [(1 - ~ ) - l / 2 ] v u .

=

(15)

In practice, the Taylor expansion of the squareroot operators (15) determines the successive orders of a perturbation expansion. The nonconvergence of these formal expansions can be averted by working with S(Z) = (1 - ZA) 1/2 and r(Z) = (1 - Zp) -1/2 and letting Z -+ 1 after applying the operators [4]. In actual practice, this nicety can be ignored since one works with operators coupled to good angular momentum. The Racah coefficients then automatically introduce small parameters of order (2/" + 1) -1/2 [4]. More important, the introduction of random-phase approximation (RPA) normal modes, obtained in lowest order, multiplies the small parameters by RPA zero-point amplitudes, the smallness of which ultimately controls the convergence. Finally, it should be pointed out that the relation between the forms (14) and (7) is similar to that between the Holstein-Primakoff and Dyson representations of angular-momentum operators [3]. The author is grateful for the hospitality of the Niels Bohr Institute in the summer of 1979 when this work was initiated, and for the support of the National Science Foundation.

References [1] A. Bohr, Dan. Mat. Fys. Medal, 26, no. 14 (1952). [2] V. Paar, in: Problems of vibrational nuclei, Proc. Topical Conf. on Problems of vibrational nuclei, Zagreb, Croatia, Yugoslavia, 1974 (North-Holland, Amsterdam, 1975) p. 15. [3] P. Garbaczewski, Physics Reports 36, No. 2 (1978) 65. [4] E.R. Marshalek, Nucl. Physics, A224, 221 (1974) 245. [5] S. Okubo, Phys. Rev. C10 (1974) 2048. [6] T. Suzuki and K. Matsuyanagi, Prog. Theor. Phys. 56

(1976) 1156.

248

19 May 1980