the pairing force in the quasi-boson approximation

Nuclear Physics A154 (1970) 65--92; ~ ) North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE PAIRING FORCE IN THE QUASI-BOSON APPROXIMATION

OLIVER JOHNS

The Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen O, Denmark

Received 29 July 1968 (Revised 14 May 1970)

Abstract: A modified form of the BCS+RPA method for 0 + states is presented which changes the zero-frequency boson and the effective Hamiltonian. The modified theory is closely related to the Nogami method. Complete BCSWRPA state vectors are calculated and displayed graphically for a model system of two j = ~ levels, illustrating the improvements produced by the modified theory.

Introduction

This paper presents a method of removing several of the difficulties which arise in the application of the quasi-boson (BCS+RPA) method to the calculation of 0 + states in nuclei.

(i) The paper demonstrates (sects. 3 and 4), and illustrates numerically by a graphical display of complete 0 + ground and boson state vectors in a model system (sect. 6), that the ground state correlations typical of the RPA method must be changed if correct state vectors are to be obtained. The proposed new method modifies both the zero-frequency boson and the form of the Hamiltonian.

(ii) The modified zero-frequency boson is finite and non-hermitian, and thus also removes the old problem of infinities in the zero-frequency boson mode (a problem also known as the non-normalizability of the zero-frequency boson).

(iii) The paper demonstrates (subsect. 4.2) the close connection between the BCS + RPA Hamiltonian and the Hamiltonian used in the Nogami method 1- 4). In connection with (i) and (ii) above, a simple modification of the BCS + RPA Hamiltonian is proposed, which completely removes the operators of the zero-frequency mode from it, giving a Hamiltonian which approximates that of the Nogami method.

(iv) The paper presents (sect. 5) a method of correcting for the shift in average particle number which occurs between the ground and 0 + boson states for small pairing force strength.

In sect. 6 we display graphically, in the form of projections onto exact number- conserving eigenstates of the Hamiltonian, the full BCS + RPA ground state and 0 + boson state vectors including the number non-conserving parts. In this way, one is able to visualize in detail important features of the boson states such as the percen- tages of correct and incorrect exact state strength, the spread in particle number, the

65

66 o. JOHNS

effect of correlations on the ground state, etc., which should be of interest to those who use the method. These numerical calculations are made using a pure, constant pairing force in a model single closed-shell nuclear system consisting of two j = levels. This system is complex enough to show the characteristic features and failures of the BCS + RPA method for 0 ÷ states, while being simple enough to allow the full ground and boson state vectors to be displayed graphically in a transparent manner.

The paper also includes an appendix which relates the operators of the zero- frequency boson mode to the generators of gauge transformations of the BCS state.

1. The quasiparticle transformation

In this section and sect. 2, we briefly outline the quasi-boson (BCS + RPA) procedure for calculating 0 + states. Although these first two sections are intended mainly for later reference, some new work is included in sect. 2.

The pairing Hamiltonian 5) with chemical potential 2 is

where

H'= H - 2 N = E ( e l - 2 ) N i - G E x/-~/-~J'4*'41, ./ U

.~* = ~22 ~ (2j + 1)-½(- 1)i- 'c* e* - ] --jm v j - - r a ,

0 )

(2)

E * = = c j , Cim. (3) j jm

After a canonical transformation from particles c* to quasiparticles ~*, the above Hamiltonian can be expressed as the sum of the following six terms:

H'o = ~ f2i(ej-A-½GVf)2Vf -A2/G, J

H2o = E ~/~-~((e~-- 2 -- G V/2D)2 Uj Vj - - A ( U 2 - V2))(A~ .. + Ai), J

H'11 = Z El Ni' 1

H22 _ G E 4 ~ / / - - 2 2 t = .. 4 ~ 2 j ( U i U j + Yi 2 V 2 ) A ~ A j tJ

-- G ~ U, Vi U i Vi(N, N i - ~,j Ni)D, U

H'at = G • U, V~x/-~j(U 2 - Vf)(NiAi+ A*Ni)D, t3

m o ½G X 2 2 , , ' = U i V~ )(A, A i + Aj A,). U

(4)

(5)

(6)

(7)

(8)

(9)

QUASI-BOSON APPROXIMATION

The following definitions have been made:

A* = ~-~1 ~ (2j+l)_~(_l)~_m~. ~j-m,*

N= • Nj= Eot*,,,aj,,,, j jm

~j =j+½,

Ej = [ ( e ~ - 2 - ov iD) ~ + ~]~ ,

= o Z a j u j v j . i

67

(lO)

(11)

(12)

(13)

(14)

The values of the parameters which minimize H~) and make H~o zero are given by the BCS equations

n = E 2vT~j, (15) J

1 = ½G ~ f2~_, (16) • E j

U} = ½[l___(ej-2- GV)D)/Ej], (17) v~ ~

where n is the average particle number. The BCS ground state (vacuum state of quasiparticles) is

[BCS) = l--[ (Vj+ V~c~',,(- 1)J-mc~'_,,)I0). (18) jm> O

The six terms of H ' are all normal ordered. The sum of the six terms is identically equal to H ' only if D = 1, where D is an artificial parameter introduced to facilitate later discussion.

2. The harmonic quasi-boson method

The transformation to quasiparticles in sect. 1 diagonalizes H ' in the space generated by single quasiparticle creation operators acting on the BCS ground state. The residual terms H~z, H3x and H~.o do not act within this space.

The harmonic quasi-boson (BCS + RPA) method for treating these residual parts of H ' rests on two approximations:

(i) The quasi-boson approximation. Since the ground state (0, 4, 8 . . . . quasiparticles) and the low-lying coherent excited states (2, 6 . . . . quasiparticles) will not be mixed by the quasiparticle number operator Ni, and since the amplitudes of quasi-

68 O. JOHNS

particle terms in the ground state are expected to be small, it is reasonable to approximate

[A,,A~]lgnd) = 3,j ( 1 - N j ) g [gnd) ~ 6,;Ignd). (19)

(ii) The harmonic approximation. One can include contributions from some (H~2, H,~o, but not H~a) of the residual terms of H' by approximately diagonalizing the Hamiltonian in the space generated by operators A* and Aj acting on the ground state. Dropping states generated from the ground state by normal ordered operators not in this space, one effectively approximates H' by a system of simple harmonic oscillators.

Throughout this paper, the symbol ~ will denote approximate equality resulting from the use of approximations 1 and/or 2.

Defining the quasi-boson operator B* by

B~ ---- Z (Xkj A * - Ykj A j), (20) J

the condition that B~lgnd) be an approximate eigenstate of H' with excitation energy Wk is

[H', Bk*] ~ WkB~'. (21)

Since H' is hermitian, eq. (21) implies that the eigenoperators of different Wk values commute. Thus,

[Bk, B*] ~ 6kk,, (22)

[Bk, Bk,] ,~ O. (23)

Using the approximations (1) and (2) above, one expresses eq. (21) as a matrix eigenvalue equation for Xkj and Ykj,

( ? M M ~ (Xk~ (X,) (24) -L] \Yk] = Wk Yk '

where

L,, = + v?v:), (25)

M u = - 26ij GU~ VfD + Gx/-~ i x/Z(Uai Vf + U 21:/2). (26)

Some simplification of eqs. (25) and (26) results if one takes D = 0. A consistent justification of this truncation has been given 6) for the case in which the I2j parameters are large compared to unity. A similar simplification can be justified by the Nogami method 1-4). (See also the discussion in subsect. 4.2 of the present paper.) For the remainder of this paper we take D = 0. The D-factor in the first term on the right in eq. (26) has been inserted here for consistency with the other approximations

QUASI-BOSON APPROXIMATION 69

made when D = 0. Note that this term comes from the normal ordering of the commutator with H~o, and was omitted in ref. 6).

An alternate parameterization of the quasi-boson B* expresses it as the raising operator of an harmonic oscillator. Thus, instead of eq. (20), we may write equivalently

where

Eq. (21) is then equivalent to

with

B* = (2 Wk)- ~r(_ ix/~ Pk + %/~ Qk),

Qk = E qkj(ay + A j), J

-- iPk = ~ pkj(A*-- A/). J

E H', Pk] '~ iVkQk,

[H', Qd ~, -it~e~,

and eqs. (22) and (23) imply that

[Qk, Pk'] ~ iSkk,.

(27)

(2s)

(29)

(30)

(31)

(32)

(33)

It is shown in ref. 6) that, with D = 0, the non-zero eigenenergies of eq. (24) are given by

Wk ___ (y2 +4A2)~r, (34)

where the Yk are solutions of the secular equation

~j y = 0 (35)

Ej[yk~-2(ej-- 2)] "7

The corresponding eigenvectors are then

Xkj Gx/~ (e j -2 Yk) Ok Ykj -- 2Ej T-Wk \-Ej +- Wkk "

(36)

For closed subshell systems, there is always one non-zero eigenenergy Wk of eqs. (34) and (35), which we may denote by k = 1, which vanishes with A at the critical G- value. For certain special cases 7), e.g. a half-filled system of identical equally spaced j-levels, this k = 1 solution is particularly simple since then, for all G-values, Yl = 0 and W1 = 2A. However, all of the theory presented in the present paper applies to the general case and is not restricted to such special cases.

70 o. JOHNS

There is a zero-energy eigensolution denoted by k = 0 which has W o = 0. It is associated with the boson B* and the two hermitian operators Qo and Po. Setting k = 0 in eqs. (28) and (29), Qo and Po are defined by

qoj - 00, (37) Ej

Ax/-~j [rA+s(e,-2).] 0o, (38) P o j - 2E ff \ r2+s2 ]

where V A 3t2j A 2(e j - 2)I2j

r = z~ ~ ' S = Z 3 " (39) • Ej j E i

The arbitrary normalization factor 0o is set equal to one. With this normalization (and the appropriate choice of Ok above) eqs. (22), (23) and (33) are satisfied for all values of k and k', including zero. The two operators Qo and Po obey eqs. (30) and (31) with k = 0, to = 0, and

rA v o - . (40)

r 2 + s 2

This value of vo follows from eqs. (16) and (30) and is independent of the Value of 0o. Note from eqs. (25) and (26) that parameter D cancels from the equation [H', Qo] ~ 0. The zero-frequency root is present even i f D # 0.

The Hamiltonian H ' can be expressed in terms of the boson solutions as

H ' = C'+ ~ WkB*Bk+½VoQ2o, (41) k~0

where C' = H~ - E Wt Y~--½Vo ~, qZj. (42)

k~eO j

The k = 0 zero-energy mode exists because the BCS state, eq. (18), is not an eigenstate of particle number and thus breaks the number conservation symmetry of H' . T o demonstrate the relation o f the k = 0 m o d e to variations in particle number, the identity

A* = UjAj - , Vf.~j-~/-~j U i Vj(1-N~/I2~) (43)

can be differentiated with respect to n, using eqs. (15)-(17), and the result expressed as

aA*~n ~ "~ i[A*, Po]. (44)

Defining the BCS state with average particle number n + ~ by the condition

(Aj+~ ~,4, ~ rBCS( , ,+~)> = O, (45) \ t3n ~/


one obtains to first order in

[BCS(n + ~)) ~ e-'~P°IBCS(n)). (46)

Thus Po is the infinitesimal generator of unitary transformations in average particle number. Also, the identity

- n = ~ {2 Uj Vj x /~(A* + A j) + (U 2 - Vf)Nj} (47) J

implies that .N-n ~ ao. (48)

Thus, in the quasi-boson theory, Qo represents the coordinate and Po the momentum of a zero-frequency vibration in average particle number.

An interpretation of Qo and Po as generators of gauge transformations of the BCS state is given in the appendix at the end of this paper.

3. Ground state correlations: Failure of the usual method

The correlated boson ground state is defined in the usual quasi-boson (BCS + RPA) theory by the equations

Bklgnd) ~ 0, k ~ 0 (49)

Bo[gnd) ~ O, (50)

which have the solution

lgnd) = K e '~ IBCS),

#ij = ( x - 1 y ) i j ,

g = e -¼Tr(#2),

where K is a normalization factor. According to eq. (27), frequency boson operator is infinite since Wo and to are zero while Vo is not. To deal with this infinity without modifying the usual BCS+RPA theory, one can do all calculations (ground state correlations, etc.) with B~ replaced by the finite, normalized boson operator

B~(b) = (2b)-~(- ibPo + Qo). (54)

Then, (½b) ~ is the amplitude of zero-point oscillations in the variable Qo, and by comparison with k = 0 in eq. (27)

b = (to)_ ~, bvo= Wo, (55) \VOI

and lim B~(b) = B~. (56) b~O

(51)

(52) (53)

the normalized zero-

72 o. JOHNS

Eq. (54) is finite for all b # 0. By using eq. (54) in place of B* and taking the b = 0 limit only at the end of the calculations, one can often obtain finite results even though B~ is infinite. For example, one obtains

lim WoB~(b)Bo(b ) = ½Vo Q2, (57) b ~ 0

which appears in eq. (41). Also, one finds that, when ground state correlations are defined by

lim Bo(b)lgnd ) ~ 0, (58) b~O

which reduces to eq. (50) in the b = 0 limit, the infinite (2b) -* factor cancels between X - 1 and Y in eq. (52) giving a finite matrix ft. This fl is then identical to that which would have been obtained had eq. (50) been replaced by

Oolgnd) ~ 0. (59)

We propose in this paper that eq. (59) [or the equivalent eq. (50)] is incorrect in the sense that its use leads to certain failures in the usual BCS + RPA method. First, in the numerical work of sect. 6, comparison of the graph of the BCS state, fig. 2, with that of the correlated ground state defined by eq. (59) and the usual theory, fig. 3, shows that the effect of the correlations is to degrade the state vector by the introduction of significant excited state strength. This is a strong indication that the correlations introduced are incorrect, since the correlations in RPA should represent the effect of the higher order terms H~2 and H~0 and should improve the ground state, not degrade it.

The second failure involves the k = 0 contribution to the equation inverse to eq. (20)

A* = ~ (XkjB~+ YkjBk). (60) k

By use of the b --- 0 limit procedure outlined above, eqs. (54)-(56), this inverse equation can be written in a finite form

A~. = ~, (Xkj B* + Ykj Bk) + (Poj Qo - iqoj Po). (61) k¢:O

However, essentially due to eqs. (59) and (33), the appearance of the operator Po in eq. (61), which is necessary for completeness, leads to infinities when the equation is used. For example, the ground state expectation value of the quasiparticle number operator N 3 calculated using eq. (61) becomes infinite in the b = 0 limit.

The third failure of the usual theory is a violation of the quasi-boson condition, eq. (19). A short calculation in the A*, Aj basis using eq. (51) gives

(gnd[Nflgnd) ~ 2(fl2)jj. (62)


It can be shown that, for systems with a partially filled subshell, the value of (fl2)jj for the partially filledj'-level approaches unity as G becomes small. Thus, in the usual theory, eq. (19) may be badly violated for weakly superconductive spherical nuclei with a partially filled level of low pair degeneracy t2j and for weakly superconductive odd deformed nuclei.

4. Modification of the quasi-boson theory

To remedy the three failures noted in sect. 3, we propose two changes in the quasi- boson theory. First, in subsect. 4.1, the zero-frequency boson of the usual theory Bo is to be replaced everywhere by a finite, non-hermitian zero-frequency boson B*(bo). Secondly, in subsect. 4.2, the Hamiltonian H' is to be replaced by a Hamilto- nian H " which is approximately that used in the Nogami method.

4.1. MODIFICATION OF THE ZERO-FREQUENCY BOSON

This modification consists essentially in not taking the b = 0 limit in eqs. (54)-(56). Instead, b is fixed at some finite value b o to be determined. Thus, By is replaced everywhere by

B*(bo) = (2bo)-~r(-ibo Po +Qo). (63)

In particular, the correlated boson ground state is now to be defined by eq. (49) and eq. (64)

Bo(bo)lgnd ) ~ 0. (64)

These two equations have a solution given by eqs. (51)-(53) but now of course with different values for fl due to the changed k = 0 rows in matrices X and Y.

Note that the isolation of the k = 0 number fluctuation mode from the k ~ 0 modes is not affected by the replacement of B~ by B~(bo). It remains true in the modified theory that the right-hand side of eq. (52) is well defined with (X-1Y)~j = (X-~Y)ji, and that states Qolgnd) and Polgnd) are orthogonal to the states B*lgnd) with k ~ 0.

The above change in the zero-frequency boson treats the first of the three failures of the usual theory noted in sect. 3. Comparison of fig. 4 with figs. 2 and 3 shows that the correlated ground state of fig. 4, defined by the modified theory of this section [eqs. (49) and (64)], is slightly better than the BCS state, indicating that in the modified theory the ground state correlations are smaller and more nearly correct than those in the unmodified theory. Note also that the boson excited states calculated in sect. 6 for several n- and G-values show that the modified theory in each case gives better excited states than the usual theory.

The second failure of sect. 3 is removed by giving b o a finite, non-zero value. Since B*(bo) is then finite and normalized, the inverse equation is then correct as written in eq. (60), and its use will give finite results. Thus, in the modified theory, calculations

74 o. JOHNS

of expectation values, etc., can easily be transformed to the boson basis where they can be performed without explicit use of the matrix ft.

The third failure of sect. 3 is corrected by a proper choice ofbo. With an eye to the quasi-boson condition, eq. (19), we can choose bo to minimize the average total number of quasiparticles in the correlated ground state. From eq. (60), this is

1 <gndlN]gnd) ,~ 7- ~ (boPoi-qo~)2 + ~, Yk 2, (65)

Do j j,k~o

where a comparison of eqs. (63) and (20) has been used to express Yo~ in terms of bo, Pos and qoj. The value of bo which minimizes eq. (65) is

bo = t~po2j ] . (66)

J

This bo is finite for all G-values, and non-zero except at G~, the critical G-value at which A becomes zero, in closed subshell systems. In the small G limit of a system with a partially filled subshell, the above choice of b o makes the number of quasiparticles in the ground state become zero for all j-levels including the partially filled level. Thus, the third failure of sect. 3 is corrected. Note that the number of quasiparticles in the ground state also becomes zero in the limit of large G-values.

The meaning of the parameter b o is clarified by the expressions

(AQo>g,d = (gndlQ2lgnd) ~r ,-~ (½bo) ~, (67)

(-~1~ ~, (68) (AP°)gnd = (gndlp2lgnd)~r ~ \2bo]

for the two conjugate zero-frequency operators Q o ~ N - n and Po "~ - i(d/dn). Eq. (66) guarantees that, in the modified ground state, the ratio (AQo)g,d/(APo)gnd is equal to the corresponding ratio for the bare BCS state. The product (AQo)gnd(APo)g,a is of course one-half, as it must be for a harmonic approximation, while the corresponding product for the BCS state is, except for special cases, greater than one-half.

The motivation for the above choice of bo is that, since bo is not dynamically determined by the modified Hamiltonain of subsect. 4.2, it is reasonable to choose it so as to guarantee the quasi-boson condition, eq. (19). Since this choice minimizes the zero-frequency part of the ground state correlation, it makes the zero-frequency character of [gnd) nearly the same as that of IBCS). In particular, the width of the particle number distribution is approximately the same in the two states.

Fig. 17 shows that, for the two-level system described in sect. 6, the percentage intensity of correct state strength for the ground and one-boson states is rather insensitive to the precise value of bo chosen. When b o is replaced by the variable b, the curves have broad maxima at about 0.8 bo for the ground state and 2.0 bo for the one-boson state. The rms widths of the particle number distribution is also shown.


Note that it increases with b, as predicted by eq. (67). For X = 1.0 (i.e., b = bo), the width is approximately that of the BCS state.

4.2. MODIFICATION OF THE HAMILTONIAN

The second proposed change in the usual quasi-boson theory is to replace H' by the Hamiltonian of the Nogami method

H" = H-21N--22N 2. (69)

It is necessary to make this change since the use of Bo(bo ) in eq. (64) implies that Qolgnd) ~/~ 0. Therefore, due to the Q2 term in H',

H' = C'+ ~ WkB*Bk+½voQ~, (41) k ¢ 0

the modified ground state defined by eqs. (49) and (64) will not be an eigenstate of H'. The form of eq. (41) makes the change to H " very simple. By eq. (40), the factor

½Vo in H' is exactly equal to the value of 22 used by Nogami in his first contribution to the method [see eq. (3.23) of ref. 2)]. This factor is also equal, in the limit of large G-values and in the limit of small G-values in systems with a partially filled subshell, to the 22 used by Goodfellow and Nogami 4) in the later formulation. In these limits, 22 = ¼G.

Since also, by eq. (48), Qo is approximately equal to N - n , it is reasonable to make the approximate identifications

21 = 2-nvo, 22 = ½Vo (70)

and, remembering that H ' = H - 2 N , to write eq. (69) in the following approximate form:

H" ~ H'-½voQ~ +½Vo n2

,~, C" + ~,, WkB~B k , (71) k ¢ 0

where C " = C'+½Vo n2. (72)

Thus, H " has the same k ~ 0 boson spectrum as H' with the same excitation energies Wk, but the operators of the k = 0 zero-frequency mode do not appear in H" . Therefore, in spite of the fact that Qolgnd) ~/~ 0 in the modified theory, the modified ground state defined by eqs. (49) and (64) and the boson states built up from it are eigenstates of H".

One expects that eigenstates of H " should be better than eigenstates of H' , in the following sense. It is proved in ref. 4) that, with a proper choice of the parameters, the Hamiltonian H " has the property that, to second order in/V, the number-projected energy expectation value is

(mlHlm) = (gndlH"lgnd) + 2t m + 22 m 2, (73)

76 o. $OHNS

where m of course is to be set equal to the correct particle number n. In eq. (73), the state vector Im> is the normalized m-particle piece of the correlated ground state and is defined by

Ignd> = ~ Im><mlgnd), (74) m

NIm> = mira>. (75)

[Ref. 4) uses the BCS state rather than the correlated ground state, but the proof is the same in either case.] Although eq. (73) is only taken to second order in N, the essential point of the Nogami method is that, in choosing the variational parameters to minimize <gndlH"lgnd> rather than <gndiH'lgnd>, one is making a better approximation to the minimization of the projected energy <mlHIm>. It follows that the actual number-projected state vector Ira> should be a better eigenstate of H when the Nogami Hamiltonain H " is used. To obtain the full benefits of the Nogami procedure, one must determine the variational parameters self-consistently. In the present paper, we have stayed closer to the usual method and have determined 2 and A by the ordinary BCS equations, eqs. (15)-(17), and 22 approximately by eq. (70). Thus, the modified theory improves the ground and boson state vectors, but does not change the excitation energies W k or the ground state binding energy, eq. (73). Therefore, the failure of the BCS + RPA method at the critical value G¢ in closed subshell systems will remain.

The truncation D = 0 was discussed in sects. 1 and 2 and used in sect. 6. I f we were to write eq. (69) for H " in terms of quasiparticle operators (Hi', H~[,, etc.), and take the limiting value 22 = ¼G, certain terms would cancel, which are similar to the terms removed from H ' (see sect. 1) by taking D = 0. The GV] terms like those in eqs. (5), (13) and (17) would cancel, and in H ~ and H ~ the diagonal terms would be zero. This result offers some justification for the truncation D = 0 even when the pair multiplicities f2~ are small, since taking D = 0 may be viewed as the first step in the replacement of H ' by H " . The second step is the theory of the present section leading to eq. (71) for H".

The parameter bo discussed in subsect. 4.1 is not determined dynamically in the modified BCS+RPA theory. It follows from eq. (71) that H " commutes with both Po and Qo, and hence that [H", B*(bo)] ~ 0 for all bo values. Thus, B~(bo)lgnd> is a valid approximate zero-energy eigenstate of H " for any value of bo. Since bo is not determined dynamically, we are free to choose it as in subsect. 4.1, to match the number distribution of the ground state to that of the BCS state, and hence to opti- mize the quasi-boson condition, eq. (19).

The flatness of the ground correct-state percentage curve in fig. 17 is related to the zero energy character of B~(bo). Since (B~(bo))2/4bo is the infinitesimal generator of changes in the ground state with bo, it follows that the ground state expectation value of H " , and by eq. (73) the expectation values of the full Hamiltonian H in the projected states lm>, should be approximately constant as bo is varied. The ground state


curve shows that, except for extreme b-values at which increasing ground state correlation begins to violate eq.(19), the important states Im) depend only slightly on b, with a broad maximum of correct state strength at approximately the point b = bo which optimizes the quasi-boson approximation, eq. (19). The greater variation for the boson correct-state percentage curve, as well as the location of maxima at points other than b = bo, are probably due to terms in H " which are ignored in the harmonic approximation, and which are more important in the boson states than in the ground state.

5. The shift in average particle number

Figs. I4 and 15 of sect. 6 show a shift of about two in average particle number between the ground and one-boson states. This shift occurs because, as G becomes small, the boson operators B~' change character until at G = 0 they are pure two- particle or two-hole creation operators. In systems with a partially filled subshell, the change in boson character with decreasing G is gradual, and it is possible to correct for the particle number shift.

Denoting the difference in average particle number between the one-bosort state and the ground state by 0~k, we have

~k = (gndlBk(N- n)B~'lgnd) - (gndl(/V - n)lgnd). (76)

Since the (IV- n) factor in eq. (76) is not standing next to the ground state, instead of eq. (48) we must use the longer form eq. (47). The inverse equation, eq. (60), then gives

Ykj)(Vj - vf). (77) J

When G is very small, the values of Ctk from this equation will be nearly -4-_ 2, the sign being the same as the sign of the corresponding solution of eq. (35). When G is large, ~k will be nearly zero (at least for collective roots) due to the spread of non-zero Xkj values over many j-levels and the sign alteration of the (U f - V 2) factor.

We now use eq. (46) to correct the average particle number. Since the difference between the BCS and correlated ground states is small in the modified theory of sect. 4, we expect that the state vector

where

and

T*(ct) lgnd), (78)

T*(~) = K~(1- iOtPo) (79)

Ka = (1+ (80) \ 2bo]

should be approximately equal to the correlated ground state of average particle number (n+ ~). Note that eq. (78) is an eigenstate of H " , eq. (71), but not of the usual

78 O. JOHNS

Hamiltonian H ' , eq. (41). The theory of the present section therefore assumes the modifications proposed in sect. 4. The normalization factor K, is included since eqs. (78)-(80) will be used as an approximation when a is of order unity.

The excited states of the system are now given by bosons B~ acting on the ground state of eq. (78), with a = - a k . Thus,

B ' T * ( - ~k)lgnd) (81)

is an approximate eigenstate of H " with excitation energy W k and approximately the correct average particle number n. As G becomes large and ~k goes to zero, eq. (8 I) automatically goes over to the collective one-boson state B*lgnd). As G becomes small and cq goes to, for example, +2, eq. (81) will approximate the correct excited state which removes one 0 + pair from the partially filled level and puts it in one of the empty j-levels above. For non-collective states or asymmetrical systems, ~k may be non-zero even for large G-values.

Near to subshell closures, an improvement in the state vector given by eq. (81) may be obtained if ~k is replaced by ~;,

t~ k ---- tXkq-2 - 1 ,

where t2x denotes the pair multiplicity of the partially filled level and nx denotes the number of particles in this level in the G = 0 shell-model ground state. The use of ~ produces a small shift in average particle number, and ensures that, in the G = 0 limit when ~k = + 2 (or - 2 ) , the state in eq. (78) will contain no component with t2a (or zero) pairs in the partially filled level. Unless suppressed, this component leads to an incorrect closed-shell ground state component in the one-boson excited states for small G.

The figures and discussion in sect. 6 show that eq. (81), with either ~ = - c q or = - ~;,, gives an excited state which is superior both to that of the usual theory,

and to the simple one-boson state of the modified theory of sect. 4. It is suggested that a good representation of the low-lying excited 0 + states of

deformed even nuclei may be obtained if the average particle number n is set equal to an odd number of nucleons and the theory of the present section is applied. (Note that ~;, = ~k in this case.) Even though n is odd, the state vector of eq. (81) will be spread over correct excited states of neighboring even systems. The failure of the BCS + RPA method for weak pairing forces in deformed even nuclear systems may thus be avoided.

6. The boson states in a numerical example

The usual quasi-boson method (sects. 1, 2 and 3) and the modified form of it (sects. 4 and 5) are applied to the system of twoj = ~ levels separated by energy ( e 2 - e l ) a n d containing either eight particles or ten particles (closed subshell). We use a constant,


pure pairing force and take D = 0 as discussed irt sects. 1, 2 and 4.2. This model single closed-sheU nuclear system has been used in other studies 6, s, 9).

The excitation energy of the single non-zero energy boson mode (which is the same in the usual and modified theories) is compared with the exact excitation energy in fig. 1, as a function of G' = G/Gc. As seen in the figure, G= is the critical value of the pairing force strength at which the BCS + RPA method fails in the closed subshell system, and the lowest boson excitation energy becomes zero. For the system studied here, Gc = 0 .1(e2-e l ) .

2.0 T T T ~

e z l j 2 = 9/2 e l - - J l = 9/2 / Two-level system / /

: TM -

o

/ /

[ Twt;boson ~ /

o 0 1.0 2.0 3.0

G'= [(111 +~2)G]/(e2-el)

Fig. 1. The BCS+RPA (dashed lines) and exact (solid lines) excitation energies are compared. For n = 10 and G" =< 1.0, the pairing potential A is zero.

The ground state binding energy given by eqs. (4) and (42), which is equal to that given by eqs. (70), (72) and (73), is in fairly good agreement with the exact value. For n = 8, the error is less than 3 % for all G' values. For n = 10, the error is less than 2 % at G' = 1.0, rises to a peak of 7 ~ at G' = 1.6, and is less than 3 % for G' = 3.0. In each case, the model underestimates the binding energy.

Figs. 2 to 16 are contour graphs of state vectors of the usualand modified BCS + RPA theory. In these figures, the number at the intersection of the zth row (z = ground, 2nd, 3rd . . . . ) and the mth column (m = 0, 2 . . . . . 20) is the intensity (z, mlstate) 2, where Iz, m) is the exact, number-conserving, normalized zth eigenvector of the m-particle system and where Istate) is the BCS+ RPA state being graphed (e.g., in fig. 2 it is the BCS state). The states Iz, m) are obtained by an exact, number-conserving diagonalization of H in the m-particle system. Each intensity in the figures is followed by the sign of the corresponding (real) amplitude. Contour lines are drawn at 10 ~ ,

80 0. JOHNS

0

0

0

W -: II

b d

:

+ t 9 -

+

i

+ k ‘?

+

z - + 2 __. : 5 - +

SS(D+S +O OJW-h~!JO!U~S

0

ij

0

W -’

_;

0 :

r : G

L

E~(D+S +o oJaz-h(!JO!Uag

4t h

---,-

03

+ 3r

d --

" O o ,

2nd

~ gr

ound

-*

n :1

0, G

':I.

6

0 0

0

0 0

0 .0

001+

0

.000

1+

0 0 0

.000

2+

[ r ..

......

......

......

......

...., •

i] .0

00,+

.o

oz+

.0,8

+ .0

80.

1.2z

4+

.350

÷ .2

24+1

.0

80+

.0,8

+ .0

02+

.ooo

t.

• .

..

..

..

..

..

..

..

..

..

..

..

..

..

.

I I

I I

I I

I I

t I

I 0

2 4

6 8

I0

12

14

16

18

20

Por

ticl

e-nu

mbe

r (m

)

.000

2+

0 .0

002-

.0

006-

0

.000

6-

.000

2-

Fig

. 4.

C

orre

late

d g

rou

nd

sta

te ]

gnd>

in

the

mod

ifie

d B

CS

+R

PA

th

eory

of

sect

. 4.

(n =

10

, G

' =

1.6,

Tr(

fl 2

) =

0.01

5, b

o =

12.2

, lg

na(t

otal

) =

99.8

~,

lgad

(m

=

10)

= 10

0.0

%.)

4th

--,-

b') +

3rd

-'~

O o e~

N , 2

nd--

,'-

>,

.:..

o ,~ g

r0un

d

n=lO

, G

'= 1

.6

.000

2-

.000

4+

.000

2-

.004

+ .0

04+

.001

+ .0

01+

.001

+ .0

04+

.004

+

.039

+

.00

1-1

.01

9-

.01

4-I

.0

07

-.0

05

-.0

07

-1.0

14

-.0

19

- [

.001

-

it" .

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

~

l i

083+

.0

83+

.09

1+

.0

94+

~ .0

94+

.091

+ .0

83+

.083

+ I

.039

+ J

I I

I I

I I

I I

I I

I 0

2 4

6 8

10

12

14

16

18

20

Po

rtic

le-n

um

ber

(m

)

Fig

. 5.

H

ypot

heti

cal

gro

un

d s

tate

wit

h P

olgn

d> ~

0.

Th

is s

tate

is

an a

pp

rox

imat

e ze

ro e

igen

stat

e o

f ga

uge

phas

e ~.

c >.

T o z > o o z 00

2 4t

h’

z zJ

z 3r

d -

2 u N I

Pnd-

2 ._

: ._ C ;

grou

nd-

n s

IO,

G’=

4.0

.OO

Ol-

.000

5+

.007

+ .0

005+

.o

oo

I-

.000

2-

.ooo

+[~~

.0

004-

.0

002-

.000

2-

.oo

I -

.ooe

+ [p

j---~

~~

~;~

~~

jj .0

06+

.o

oi-

.000

2-

0 0

.ooo

s+

.010

+

.OO

Ol-

[.o

ie- ]

.OO

Ol-

.0

10+

.0

009+

0

0

I 1

I I

I I

I I

I I

I I

0 2

4 6

8 IO

12

14

16

18

20

Parti

cle-

num

ber

(m)

w

Fig.

6.

One

-bos

on

stat

e B

I*[g

nd)

in

the

usua

l, un

mod

ifie

d B

CSf

RPA

th

eory

. (n

=

10,

G’

=

4.0,

I/(

ez-e

l)

=

0.5,

A/G =4.84, TrG2) = 1.0,

Zr,,,

,(tot

al)

= 88

.5

%,

Zr&m

=

10)

= 86

.7

%.)

a I

B

n =

IO,

G’=

4.

0

E 4t

h-

0 0

.000

3+

0 0

s lTi

:, 3r

d-

.ooo

I t

.0

008+

.0

04+

.007

+ .0

04+

.000

6 t

.ooo

I t

0 i

,~__

_-__

____

_-__

--__

----

-__-

,

I 2n

d*

2 .0

05t

.025

+ (*loo+ .21e+ .279+

.21e

t .1

00+

I .0

25+

.005

+ ._

L_

____

____

____

____

_-__

-__?

‘d

._

: v)

grou

nd-

.000

5t

.000

2+

.003

t .0

006+

.0

006-

.0

02-

.000

6-

.000

6+

.003

t

.000

2+

.000

5+

I

I

I

I

I

I

I

I

I

I

I

0

2 4

6,

8 IO

12

I4

16

18

20

Parti

cle-

num

ber

(m)

Fig.

7.

One

-bos

on

stat

e B

,‘\g

nd>

in

the

mod

ifie

d B

CS+

RPA

th

eory

of

se

ct.

4.

(n =

10

, G

’ =

4.

0, T

r@12

) =

0.00

03,

b,

=

18.7

, aI

=

0,

Z&

tota

l)

=

97.1

“/

Zpl

,Jm

=

10

) =

96

.4 Y

.1

n =

IO,

G’=

2.

0

41h-

2nd’

.0

006-

.0

03-

.003

- .0

006-

I I

I ,

I z

grou

nd-

0 .0

007-

.o

o It

.044

t .0

03t

! 07

6-

! .0

03t

.044

+ .0

01+

.000

7-

0 m

LL

____

J

0 2

4 6

8 IO

I2

I4

I6

I6

20

Parti

cle-

num

ber

(ml

Fig.

8.

One

-bos

on

stat

e B

,*lg

nd>

in t

he

unm

odif

ied

BC

S+R

PA

theo

ry.

(n =

10

, G

’=

2.0,

&(e

,-e,)

=

0.5,

A/G

= 4

.33,

TI

Zl,,

,,(to

tal)

=

66

.7 %

, If

&n

=

10)

=

59.8

%.)

n =

IO,

G’=

2.

0

4th~

-

3rd-

0

2nd-

.0

03t

.010

+

.ooo

I +

.0002 t

,000

I t

~~~~

~,

.,:.

.oo3

+

I r

, 1

r 1

\

f gr

ound

-

I

.000

7+

.004

+ [.0

15+

J.01

0t

.000

6-

[ .O

ZO-

J .0

006-

.O

iOt[

.Ol5

+ J

.004

+ .0

007+

!) =

1.

005,

I I

I I

I I

I I

I I

I I

0 2

4 6

8 IO

12

14

I6

I8

20

Parti

cle-

num

ber

(m)

Fig.

9.

One

-bos

on

stat

e B

,*lg

nd)

in

the

mod

ifie

d B

CS+

RPA

th

eory

of

se

ct.

4.

(II

=

10,

G’

=

2.0,

Tr@

) =

0.

005,

b,

=

14.9

, al

=

0,

Z

,,,,(

tota

l)

=

89.1

%,

Z,,,

,&n

=

10)

=

88.1

%.)

E

1 n=

IO

. G

’=

1.2

lz

3rd

-

0 ; 2n

d-

F ._ L .o g

grou

nd’

v)

I I

I I

I I

I I

I I

I

I

0

2 4

6 6

IO

12

14

16

16

20

Parti

cle-

num

ber

(ml

Fig.

10.

One

-bos

on

stat

e B

,*lg

nd)

in t

he

unm

odif

ied

BC

S+R

PA

theo

ry.

(n =

10

, G

’ =

1.

2, Q

(e,-

e,)

=

0.5,

A/G =

2.76

,Tr(

,V)

= 1

.083

, &

&to

tal)

=

32

.6 %

, Ip

,&rn

=

10

) =

46

.3 %

.)

P =t

1 n-

IO

, G

’=

1.2

z 4t

h-

tj ;j 6 3r

d-

0 .0

003+

0 E

2nd

-c

.000

6+

.001

+ .0

00t

,”

._

ml

.008

+ .0

01

t .0

006+

z J

c ‘E

gr

ound

’ *:

.000

2t

.003

t

.020

+ ~.

055+

--.~9

7+J

LJ .1

38-

L---_

_----

, .0

97+

.055

+ 1

.020

+

,003

t

.000

2 t

\-----

--_---

__

____

__

-_-_

-_,

I I

I I

I I

I I

I I

I

I

I

1

0

2 4

6 8

IO

12

14

16

I8

20

Parti

cle-

num

ber

(ml

Fig.

11

. O

ne-b

oson

st

ate

B,*

lgnd

) in

the

m

odif

ied

BC

SSR

PA

theo

ry

of s

ect.

4. (

n =

10

, G

’ =

1.

2, T

r(B

Z)

=

0.08

3, b

, =

6.

11,

OL

, =0,

I,

,,,(t

otal

) =

50

.7 %

. &

,(m

=

10

) =

75

.0 %

.)

4 t h

..-...

¢n

o o3

3r

d

o N

2nd

''~

i ";

gro

un

d~

g,

n=

8,

G'=

2

.0

.002

+

J .0

14+

~

.000

3+

i r .

...

.ooo

-

o .o

7+

i.o.+

i[.

0o,+

I

I _

__

_J

....

..

. ,

/

I!-

- .0

005-

0

.076

+

1.3

36

+

.257

+

.035

+

....

..

-~

f ...

....

..

I i

0oo

oo

,.

o o.

I I

I I

I I

I 0

2 4

6 8

I0

12

Pa

rtic

le-n

um

be

r (m

)

.002

- .0

001-

.00

5-

.00

2-

0

.002

+ .0

05

- .0

001-

0

I I

I I

14

16

18

20

Fig

. 12

. O

ne-b

oson

sta

te B

l*lg

nd)

in t

he u

nmod

ifie

d B

CS

+R

PA

th

eory

. (n

=

8, G

' =

2.0,

2/(

e2--

el)

= 0.

236,

A/G

=

4.

27,

Tr~

2)

= 1.

003,

Ir

,,t(

tota

l)

= 7

1.1

%,

Im~

t(m

=

8) =

78

.3 %

.)

4 t h

""

o

3rd

+ 0 o NO

2 ri

d"*"

o °_

~,

grou

nd'-

"

n=8,

G'

' 2.

0 ,0

001+

.0

002+

0

t~

t

.000

1+

.001

+

.005

4+

1.02

0+ I

J t

l r{

.011

+

.04

8+

I

.16

4+

.3

01

+

.27

2+

1

.09

8*

,

.01

5"

k. ..

....

....

....

....

...

/ /

1

,002

+

.002

+

.006

+

0 ,0

09

- .0

07

- .0

08 +

/

.020

+

I I

I I

I I

I I

0 2

4"

6 8

I0

12

14

Pa

rtic

le-n

um

be

r (m

)

.000

9+

0 0

.001

+

.000

4+

.009

+

.001

+

.000

1 +

I I

I 16

18

20

Fig

. 13

. O

ne-

bo

son

st

ate

Bl*

lgn

d>

in

th

e m

od

ifie

d B

CS

+R

PA

th

eory

o

f se

ct.

4.

(n =

8,

G'

= 2.

0, T

r(fl

2)

= 0

.00

8,

b o

= 14

.8,

~1 =

0

.69

6,

Ir,,

,t(t

ota

l)

= 9

10

~,.

It

,rst

(m

= 8

)=

95

.7 %

.)

C

;>

rAo

O

~o

4 t h

""

g w

CO

3

rd--

~

4-

o o ~ 2

rid

-"

.:-

o "~

gro

un

d"

¢D

n=

8,

G'=

0.8

.000

5-

.000

9+

.000

1+

.000

4+

0 0

0 .0

08

- .0

00

2-

I .0

26+

I .0

10+

I I

.o~-

.o

,o-I

I.~8

,+

-

- .

..

..

..

..

.

. __:

!l o,

.+1

ooo8

oo

. '.;

,;;:

l]

.o,o

+,,'

t ...

....

"J

I I

I I

I 4

6 8

I0

12

Po

rtic

le-n

um

be

r (r

n)

.000

5-

0 0

.eO

08-

.000

8+

.003

- .0

001-

0

0

I I

I I

I I

0 2

14.

16

18

20

Fig

. 14

. O

ne-

bo

son

sta

te B

l*lg

nd

) in

the

un

mo

dif

ied

B

CS

+R

PA

th

eory

. (n

=

8, G

" =

0.8,

2/(

e2--

ez)

= 0

.14

8,/

I/G

=

2.66

, T

r(~

2)

= 1.

003,

(/

52)1

1 =

0.92

, If

,st(

tota

l)

= 8

6.1

%,/

flrs

t(m

=

8) =

97

.7 ~

o.)

4th

-"

g o

3rd

"-"

+

o " d

N

2n

--,-

o '7-

gro

un

d""

g~

n =

8,

G'=

0.8

.000

1+

.000

2+

0 "'

.000

1+

.000

6+

.002

+ .0

02+

.0

001+

0

li' ....

......

......

Loo.

.000

4+

.007

+ .0

53+

]-20

9+

.408

+

0 0

0

It I

t .....

-i,

] '

' ,

Iii

o o

ooo2

+ o

ooo.

oo

, IL

~J]

oo2+

oo

o,

o o

I I

I I

I I

I I

I I

I 0

2 4

6 8

10

12

14

16

18

20

Pa

rtic

le-n

um

be

r (m

)

Fig

. 15

. O

ne-

bo

son

sta

te B

l*[g

nd

) in

the

mod

ifie

d B

CS

+R

PA

th

eory

of

sect

. 4.

(n

= 8,

G'

= 0.

8, T

r(~

2)

= 0.

071,

b o

= 6.

84,

0¢1

= 1.

94,

Itl,

t(to

tal)

=

66.9

~o,

Irl

,t(m

=

8) =

98

.9 ~

o.)

O0

Os 9 ¢ z

4 t h

"*

o

3rd

-"

2 nd

---~

i

grou

nd .-

=

n=

8,

G'=

0.8

.00

04

+

.0003+

0

.000

8+

.00

5+

.0

08

+

.004

+

0 0

0

/ ....

....

....

....

....

...

]~

.004

+ _f

.055

+ 1.

245+

.4

37+

.226

+__

0 .0

001-

0

0 I' .

.....

'-- ..

......

......

. ~;

0 .0

00

1-

.00

04

- .0

03

- .0

02

- .0

03+

.0

009+

.0

05

- 0

0 0

I I

I I

I I

I I

I I

I 0

2 4

6 8

I0

12

14

16

18

20

Po

rtic

le-n

um

be

r (m

)

Fig

. 16

. T

wo-

oper

ator

st

ate

Bl*

T*

(--c

t't)

lgn

d)

in t

he m

odif

ied

BC

S+

RP

A

theo

ry o

f se

ct.

5.

(n =

8,

G'

= 0.

8, T

r(fl

2)

= 0.

071,

b o

= 6.

84,

ct'x

=

3.14

, It

,st(

tota

l)

= 96

.7 %

, lr

irst

(m

= 8)

=

97.7

%.)

C

i w 0 M

Z

88 o. JOHNS

5 ~ (dashed line) and 1 Y/oo intensities. Intensities less than 5 x 10- 5 are written as zero in the figures. In the caption to each figure, we give/state(total), the total intensity (summed over all m) of the overlap of the BCS + RPA state with the desired exact states. This is a figure of merit which indicates the validity of calculations performed in the non-number-projected quasiparticle or boson basis. Also in the captions we give Istate(m = n), the overlap intensity with the desired exact state after number projection of the BCS + RPA state. (The number-projected state is the renormalized mth column, where m -- n, the correct particle number.)

Figs. 2, 3 and 4 show ground states calculated by three methods. The results are typical of all n and G' values studied. The correlated ground state in the usual BCS + RPA theory, fig. 3, is seen to be inferior to the simple BCS state, fig. 2. The number projection from fig. 3 is particularly bad, giving only a 87 ~o overlap intensity with the exact m = 10 ground state. The correlated ground state in the modified theory, fig. 4, is slightly superior to the BCS state (see the discussion in sects. 3 and 4).

Fig. 5 is not proposed as a valid state vector. It is a hypothetical ground state calculated in the same way as fig. 3, but with eq. (59), Qolgnd) ~ 0, replaced by Polgnd> ~ 0. Thus figs. 3 and 5 are approximate zero eigenstates of the conjugate variables Qo and Po, respectively (or of I and 4, respectively, see appendix).

Figs. 6 to 16 show BCS+ RPA predictions for the lowest excited state. Figs. 6, 8 and 10 for the usual theory and figs. 7, 9 and 11 for the modified theory show the progressive failure of both methods at a closed subshell (n = 10) as G' is reduced toward the critical value G' = 1.0. However, the state vectors of the modified theory are in each case superior to those of the usual theory, and the modified theory gives excellent results at the largest G' value. For comparison, we note that a typical shell-model study lo) using the BCS theory had G' = 3.4, 2.5 and 1.3 for 6°Ni, z4°Ce and 114Sn, respectively. Studies of deformed nuclei, e.g. ref. t l) , use G' values of about 1.3, but the great difference between such systems and the present one makes comparison difficult.

Figs. 12 and 14 for the usual theory and figs. 13, 15 and 16 for the modified theory show boson states when the lower j-level is partially filled (n = 8). Fig. 13 is a one- boson state of the modified theory. Since 0q = 0.7 in this figure, inclusion of T * ( - 0q) would give a better picture (see sect. 5). However, the state shown is superior to that of the usual theory, fig. 12. Figs. 15 and 16 show the application of the theory of sect. 5 for a weak pairing force, G' = 0.8. Fig. 15 illustrates that, even in the modified theory of sect. 4, the simple one-boson state is not a sufficient representation of the first excited state of the system. The two-operator state of eq. (81), with 0q given by eq. (77), is superior both to fig. 15 and to the usual theory, fig. 14. This state is not shown in the figures, but it is essentially the same as fig. 16 with 0.047+ in place of 0.0009 + in the m = 12 ground state entry. Fig. 16 shows the state which results when 0~ is used in place of ~1 in eq. (81). There is a striking improvement over the unmodified state vector of fig. 14.

Fig. 17, which is discussed in subsect. 4.1, gives the percentage of the correct state

Q U A S I - B O S O N A P P R O X I M A T I O N 89

strength and the rms width of the distribution in particle number, when the parameter bo in the modified theory is replaced by variable b. The left-hand zero corresponds to B~(b = O) oc Qo and to fig. 3; the right-hand zero corresponds to B'~(b = oo) oz Po and to fig. 5; the point X = 1.0 (i.e., b = bo) corresponds to B~(bo) and to fig. 4.

The figures shown in this section were selected as the minimum number required to illustrate the effect of the modifications proposed in sects. 4 and 5. Other figures and the theory in a somewhat different notation may be found in ref. 12).

100 w

bO so

U W eo 0£ 0~ E] U 7o

Z W U 6o 0£ w 0_

50

GROUND ~'---.~, STATE

.°

STATE ......................

0-0 0.2 0-4 O.S 0.8 i -O 0-8 O-S 0.4 0.2

X $/X

I---4 C3

0 -q

iO -D

a P F q

6

---4 7O I---I

4 m

--4

2 0 . 0

Fig. 17. For the ground and one boson states of the modified BCS+RPA theory, the solid lines show total intensity of correct states( exact ground and first excited states, respectively) when b o is replaced by various b-values (n = 10, G' = 1.6, curves plotted as functions of X = b/bo). The dotted line

shows the rms width of the particle number distribution in the correlated ground state.

7. Conclusion

The numerical results of sect. 6 indicate that the modified BCS + R P A theory proposed in this paper results in more nearly correct ground and boson states. More- over, computations of expectation values and transition probabilities are straight- forward and free of spurious infinities. Due to the finite, well-defined character of the zero-frequency boson B'~(bo), the inverse equation, eq. (60), can be used to compute physical operators by expressing them directly in the boson basis. Also, spectroscopic factors for two-nucleon transfer reactions can be calculated, using the theory of sect. 5 to write the ground and boson states of the n + 2 particle system in terms of those of the n-particle system.

The figures in sect. 6 show that projection of correct particle number 13) from states of the usual theory would give rather poor state vectors, while much better results would be obtained by projection from states of the modified theory. Also, in ref. 14), ground state expectation values of Nj (occupation numbers) agreeing well with exact

9 0 O. JOHNS

values in a model deformed nuclear system, are calculated by a method which uses approximate number projection from a state vector similar to the modified ground state defined in sect. 4 of the present paper.

I wish to thank Professor J. O. Rasmussen for suggesting this project and for making many substantive and stimulating contributions to it. His encouragement and support are gratefully acknowledged. I thank the Nuclear Research Institute of the Czechoslovak Academy of Sciences and the T~inec Steel Corporation Scientific Foundation for their hospitality and financial support during a brief visit. I wish to thank Professor A. Bohr for the generous hospitality of the Niels Bohr Institute, and the Danish Ministry of Education for fellowship assistance. The work was supported in part by USAEC contract AT 11-1 (34).

Appendix

When an electromagnetic field coupled to a many-particle system is gauge transformed, each m-particle state vector must be multiplied by a phase factor exp [ - i(m- n)rp ], where the arbitrariness of overall phase has been used to define phases relative to the average particle number n. (For an extensive discussion of gauge transformations of nuclear states, see ref. 15).) From the number-projected form of eq. (18), it follows that the unitary transformation of the BCS state, when the gauge phase changes by an amount rp, is

U,(rp)IBCS> = y" e-'('-")~lm><mlBCS> rtl

= e-~c~-")*IBCS). (82)

It follows from eq. (48) that, in the quasi-boson theory, the operator Qo is the infinitesimal generator of gauge transformations and the conjugate operator P0 represents the phase - 9 -

Gauge transformation of a number non-conserving (pair-field deformed) BCS state by a phase q~ is analogous to a rotation of a shape-deformed nucleus (but of course with only the one degree of freedom). The angular momentum I is the analog of Q0 and the rotation angle • is the analog of the phase q~ (which is represented by operator -P0). The analogy is summarized in table 1. Note the reversal of roles of coordinate and momentum between the first and third columns of the table. The notation Ux(y) stands for the unitary transformation generated by changing the expectation value of the the variable x by an amount y.

The Nogami method can be viewed as the analog of the Peierls 16) method of calculating the moment of inertia of shape-deformed nuclei. In the Nogami method, one approximately projects an eigenstate of particle number (analog of angular momentum) from the correlated ground state vector; and then one chooses the ground state to miminize the projected energy, eq. (73), subject to the constraint that (gndlNIgnd) = n (analog of cranking). The moment of inertia is then taken from


the coefficient o f the m 2 term [analog of I ( I + 1)] in the projected energy. I t is j = (222)- 1. An operator formulat ion o f the Peierls method is given in ref. 17), and the analogy with the pairing problem is exploited in ref. 4). In the usual, unmodified B C S + R P A theory, the ½ v o Q ~ term in H ' is the analog o f a rigid ro ta tor with J = Vo i. By eq. (70) this expression for J is the same as the above expression in

terms o f 22.

TABLE 1

Quasi-boson Many-body Analogy with spatial rotation

generator of gauge N--n angular momentum: I transformations: Qo

operator representing - - i ( d / d n ) angle of spatial rotation: gauge phase: Po = --~ - -~

vff I = (22z) - I vo 1 = (22~) -1 effective moment of inertia: J

the k = 0 term in H': ½Vo(N- -n ) 2 Hamiltonian of a rigid ½voQ 2 rotator: I 2 / 2 j

unitary operator which U~(~) = e-i~(/~-n) unitary rotation operator: gauge transforms the U~(~) = e - i ~ l BCS state: U~(~0) = e-i~Qo

unitary operator which U(I~ - n) (~) = e - in( - i(d/dn)) unitary operator which changes average particle changes expectation number of BCS state: value of I: UQo(~ ) = e- i~Po UI(~) = e ia~

In certain special cases, e.g. a half-full system of identical equally-spaced levels, it follows f rom eqs. (39) and (40) that s = 0 and the momen t o f inertia takes the simple fo rm J = A - l r . However, the quasi-boson method makes the k = 0 gauge mode or thogonal to the k ~ 0 modes in general, and not just in the special cases ment ioned above. Thus, the identification J = Vo 1 is valid even when s ~ 0.

Fig. 5 o f sect. 6 shows the analog of an approximate zero eigenstate o f the angle

(eigenstate o f Po). Note that, just as in the ordinary m o m e n t u m representation o f a spatial delta-function, all momen ta (all particle number projections) have about equal amplitude. Similarly, fig. 3 shows the analog o f an approximate eigenstate of angular m o m e n t u m I (eigenstate o f Qo)-

References

1) S. G. Nilsson, Nucl. Phys. 55 (1964) 97 2) Y. Nogami, Phys. Rev. 134 (1964) B313 3) Y. Nogami and I. J. Zucker, Nucl. Phys. 60 (1964) 203 4) J. F. Goodfellow and Y. Nogami, Can. J. Phys. 44 (1966) 1321 5) A. M. Lane, Nuclear theory (Benjamin, New York, 1964)

92 o. JOHNS

6) J. H~gaasen-Feldman, Nucl. Phys. 7.8 (1961) 258 7) D. R. B6s and R. A. Broglia, Nucl. Phys. 80 (1966) 289 8) M. Rho and J. O. Rasmussen, Phys. Rev. 135 (1964) B1295 9) H. J. Mang, J. O. Rasmussen and M. Rho, Phys. Rev. 141 (1966) 941

10) A. Plastino, R. Arvieu and S. A. Moszkowski, Phys. Rev. 145 (1966) 837 11) V. G. Soloviev, Atomic energy review (IAEA, Vienna, 1965) vol. 3, no. 2, p. 117 12) O. Johns, Pairing in nuclei, Ph.D. Dissertation, Physics (University of California, Berkeley, 1968) 13) H. J. Mang, J. K. Poggenburg and J. O. Rasmussen, Nucl. Phys. 64 (1965) 353 14) J. Bang and J. Krumlinde, Nucl. Phys. A141 (1970) 18 15) A. Bohr and B. Mottelson, Nuclear structure (to be published) vol. 3, chap. 8 16) R. E. Peierls and J. Thouless, Nucl. Phys. 38 (1962) 154 17) C.-Y. Hu, Nucl. Phys. 66 (1965) 449

the pairing force in the quasi-boson approximation

Documents