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Centre-of-mass effects in Brueckner calculations of 4He
Nuclear Physics @ North-Holland
A359 (1981) 109-121 Publishing Company
CENTRE-OF-MASS EFFECTS IN BRUECKNER CALCULATIONS OF 4He J. BLANK
and J. HOT\EJSi
Nuclear Centre, Faculty of Mathematics and Physics, Prague, Povltavska’ 1, 180 OOPraha 8-Pelt Tyrolka, Czechoslovakia Received 19 November 1979 (Revised 13 October 1980) Abstract: A general criterion is proposed which states a condition for the accuracy of BruecknerGoldstone perturbation approximations to the expectation value of a given internal operator 0’. The criterion is formulated via the dependence of (0) on a properly chosen parameter. A simple parametrization of the cm. motion is introduced and shown to be suitable for applying the criterion. Simultaneously the parametrization makes it possible to test some methods currently used for making c.m. corrections. Numerical results are obtained for the binding energy and r.m.s. radius of the 4He nucleus using the standard Brueckner-Goldstone theory in an oscillator basis. It is shown that the method based on the Bolsterli-Feenberg-Lipkin approach is incorrect. On the other hand, the method which employs the usual unmodified hamiltonian eliminates c.m. effects satisfactorily and ensures fulfilment of the condition for accuracy of the binding energy provided that first- and second-order diagrams are taken into account when calculating the cm. kinetic energy.
1. Introduction In Brueckner under
of the
theory
consideration,
operators
the calculated
physical
i.e. the expectation
corresponding
characteristics
of the A-body
system
values,
to individual
observables
of the
system,
are
influenced by the c.m. motion. This feature is common to all many-body theories which are based on independent-particle models ‘) (IPM). Dependence of (0) on the c.m. motion persists even if 19 is an internal operator (related to the c.m. coordinate system). This is due to the fact that the theory provides us only with approximations In general, these 4 aPP of $ which are expressed in terms of IPM basis functions. functions contain spurious components, i.e. (Jlappcannot be factorized into c.m. and internal parts and the c.m. dependence does not cancel out in eq. (1). Investigating the c.m. dependence and its best possible elimination is important, in the first place, for comparison of calculated physical quantities with experiment. In many-body perturbation calculations this problem has another, purely theoretical aspect: eliminating the c.m. dependence of a given approximation (O),,, is closely connected with convergence of the Brueckner-Goldstone (BG) expansion for (6). This statement, which follows from the obvious fact that the exact value of (0) is c.m. 109
J. Blank, J. HoiejiLl Centre-of-mass effects
110
independent for any internal operator 0, is the starting We shall formulate them first quite generally. Let p be a parameter use of the perturbation
whose appearance theory
point of our considerations.
in the calculation
is caused merely by the
(e.g. p may be the oscillator
frequency
in a BG
calculation in an oscillator basis); of course, the internal hamiltonian does not depend on p. Then for any given internal operator 6 the exact value (0) also does not depend on p while perturbation approximations (O),,, in general do. Suppose a given (O),,, is known as a function of p. Does it tell us something about its accuracy? Weak dependence of (O),,, on p? is obviously not sufficient, and if the expansion for (0) does not converge uniformly (with respect top) then it is not necessary either: as is shown in fig. 1, (O),,, may be equal (or close) to (0) for some p = p. even if (O),,, depends strongly on p. However, such an (S),,, can be used only if some additional condition implied solely by convergence arguments (e.g. by some sort of selfconsistency requirements) is available for selecting po. If that is not the case, then only that (O),,, can be satisfactory for which a sufficiently large interval 9 = (pl, p2) exists such that the variation of (O),,, over 8; is small. (In the hypothetical situation shown in fig. 1 this is fulfilled by the solid curve only.) In this way we simultaneously arrive at bounds for admissible values of p.
Fig. 1. Illustration of the general considerations of sect. 1: possible dependence of two perturbation approximations to (0) on parameter p. The approximations (a),, (C?),, are represented by the solid and dashed curve, respectively, the exact value by the dotted line. t In view of the qualitative character of these considerations we are speaking about weak and strong dependence in an intuitive sense. No strict specification can be given unless the physical meaning of p is known since one has to consider the variation of (CY).,, over a “sufficiently large” interval of “admissible” values of p.
J. Blank, J. Hoiejcil
How are these abstract effects?
statements
In the next section
related
to the problem
we show that the c.m. motion
way which makes it possible different BG approximations
111
Centre-of-mass effects
of eliminating
the c.m.
can be parametrized
in a
to apply the above approach for testing the accuracy of to the binding energy and other ground-state charac-
teristics of 4He. As a by-product we get an estimate of the applicability of two methods currently used for making c.m. corrections. Numerical results are presented and discussed in sect. 3. Finally, in the last section the results are summarized and some conclusions are drawn.
2. Parametrization
of the c.m. motion
We start with a simple generalization of the well-known trick which consists of modifying the c.m. part of the total hamiltonian while leaving the internal part unchanged I,‘). Let H be the usual=total hamiltonian of an A-nucleon system
Tc.m.= Pf,./2mA
Vjk = Hint + Tc.m.,
i=l
3
and let us introduce
the following
one-parameter
H(a)=Hf$mAa*~*Xz,,,
family of modified =Hi,,+HpE,‘(a).
In particular, H(1) is the Bolsterli-Feenberg-Lipkin hamiltonians standard Brueckner calculations suitable IPM determined by
bation”
hamiltonian
will be specified
H(a) - Ho into one- and two-body H(a)=H,+
=
v ->
(3)
(4) later.
After
decomposing
the “pertur-
parts, we get
C AVjk(a)- t 1 i
vj(a) 9 (5)
2 v(a)
hamiltonians:
(BFL) hamiltonian. For these can be performed starting with a
j=l
This model
(2)
lsj
V(OSC)
)
V(a)=
v-a2v(0sc).
Here V(Osc) (x) = ’2 mw2x2 and the meaning of x is obvious, depending on whether V(Osc)is a part of the one- or two-body operator. Now the reaction matrix must be calculated. Notice that it is the whole two-body part v(a) that enters the equation for t; in this way dependence on (Yis brought into t. When the reaction matrix is found, we can calculate the total ground-state energy E(a) [the minimal eigenvalue of H(a)] and the expectation value of any (internal) operator. For obtaining the binding energy Eb one has to subtract the c.m. energy Ec.m.(a)= (H?$ (a)) from E(cu): Eb E (Hint) = E(a)
-E,.,.(a)
*
(6)
112
J. Blank, J. Hoiejiil
Centre-of-mass effects
In particular, for LY= 0 we get Eb
=
E(O)
-
(~cm.)
.
(64
The parameter (Yenters each term in the BG expansion for the expectation value of any internal operator (0) (except the zeroth-order term: vacuum expectation value (0>,) via the matrix elements of operators f(a) and/or V(a).On the other hand, the internal part of the exact wave function is a-independent (since Hint is) and the same holds for the exact expectation value of any internal operator. The general statements given in sect. 1 can now be applied for p = (Y.As we also get bounds for admissible values of (Y,we are simultaneously able to test two of the methods for making c.m. corrections that were considered in ref. ‘). The first method (denoted hereafter Ml), which employs the unmodified hamiltonian (2), corresponds to (Y= 0. The second one (M2) is based on the BFL hamiltonian, i.e. cx= 1. The third method mentioned in ref. ‘), which works directly with the internal hamiltonian, cannot be tested by means of the parametrization (3). Instead another family of modified hamiltonians could be considered:
The decomposition
(5) would be then replaced by
or, alternatively,
As we have performed numerical calculations only for the parametrization shall restrict ourselves in the following exclusively to it.
(3), we
3. Results and discussion In this section results of our calculations of the binding energy and r.m.s. radius of the 4He nucleus are presented. The hamiltonian (3) was used with the HamadaJohnston potential v = UHJ[ref. ‘)I for several values of (Ysatisfying 0 S (YC 1. Details of the calculations have been given in our previous papers 4*5)which will be hereafter referred to as Bl and B2. Here we shall recall only some basic features. The standard non-degenerate version of Brueckner theory in a harmonic oscillator (h.o.) basis with a shifted spectrum6-9) is applied. In particular, the IPM hamiltonian reads Ho= t j=l
hi,
h=
E &VP,,
v=o
J. Blank, J. HoiejS/
where
P, projects
eigenvalue Candq:
on the subspace
113
Centre-of-mass effects
of one-body
E, (OS’)= $ hw (2 v + 3). The spectrum
&Y = Elosc) -$UJ
Hilbert
space
of h depends
belonging
to the
on three parameters
(C+S,orl) .
o,
(74
The Brueckner-Hartree-Fock (BHF) conditions for occupied states, which are represented graphically in fig. 2a, are strictly fulfilled. In the case of 4He they reduce
= ---g -1
0
,
fi--
hl
hl
h2
h2
= ii---* (b)
Fig. 2. Graphical
representation
of BHF conditions;
all the &interactions
are on the energy
shell.
for any given w and (Y to one simple relation between C and 7. It is convenient to exclude 77from this relation so that the three free parameters, (Y,w and C, remain. All of them satisfy the general requirements imposed on p in sect. 1 and hence we have investigated the dependence of results not only on (Y but on C and w as well. However, the latter two parameters differ from LYin one important point. Analyzing the BG expansion
for the energy
one can extract
“self-consistent”
values w. and Co
for which contributions of important classes of diagrams vanish or are minimized. In fact the values w. and Co used in our calculations were obtained by requiring the contribution E(‘) of the second-order diagrams (fig. 3b) to be minimal and the contribution Ei3'of diagram 3d to vanish. The former requirement represents a sort of optimal average fulfilment of the BHF conditions shown in figs. 2b, 2c and has been applied by many authors 6’7P9).The latter approximately includes the remaining self-consistency conditions for unoccupied states: (pi Vlp') = 0 [refs. 11,5)]. As is shown in fig. 4, a minimum of lE'2'1 equal to 0.3-0.4 MeV occurs at u = o. almost independently of C and CY,with the corresponding value #iwo lying between 15 and 18 MeV. The self-consistent value of C is practically independent of w. It depends, however, on (Y; in particular Co - 1 for (Y= 0 and Co = 0 for LY= 1 (see table 1). On the other hand, convergence arguments provide no condition for selecting the “best” (~0. Then, according to the general statements of sect. 1, we shall require the cal_culated binding energy and r.m.s. radius to depend weakly on (Y, while the dependence on w and C need not be weak.
J. Blank, J. Hoiej&‘/ Centre-of-mass effects
114
(a)
lb)
f-jJ-0 f---J--*
9 (cl Fig. 3. Diagrams
Id)
of the BG expansion
for the energy
discussed
in the text.
-4
-3
S f 5 -2
-1
0 ;5
20 hw(MeV)
Fig. 4. The second-order
contribution E? versus o for (Y= 0, C = 0 (full line), a = 0, C = C, (dashed line) and a = 1, C = C, = 0 (dash-dotted line).
J. Blank, J. Ho?ejs’i/ Centre-of-mass effects TABLE The self-consistent ho = 15.232 (1
C0
3.1.
1
value C, of parameter hw = 20.735
MeV
115
C versus a and w ho = 24.538
Mev
MeV
0
0.5
0.7
1
0
0.5
0.8
1
0
0.5
0.7
1
0.94
0.71
0.48
0.00
0.95
0.70
0.33
0.00
0.94
0.71
0.47
0.00
BINDING
ENERGY
The formula for binding energy contains two quantities each of which has its own BG expansion [see eq. (6)]. The total energy E(B) is calculated up to the third order, all diagrams being taken into account except those that belong to the three-particle cluster
(fig. 3~): E(a) = 3Ao(l
The first term is the vacuum Ho-
i
+a2)+E(1)+E(2)+E(3)+Ec.
expectation (vj-(y2v~))=
j=l
(8)
value of ;
(~+&7~))
j=l
of fig. 3a, E”‘, r = 2,3, denotes the total [cf. eq. (5)], E(l) is the contribution contribution of rth order diagrams and EC is the vacuum expectation value of the Coulomb interaction. For calculating the c.m. energy we express I&“:.’ in terms of the following one- and two-body operators
where ri, Vi are dimensionless quantities and momenta by rj =~j/b, pi = -ih(l/b)Vj; Hz:.’
that are related b = Jhlo.
to the nucleon coordinates The following then holds:
=$#uJ [a2(R(1)+2R(2))-(D(1)+2D(2))].
Let us denote by (6)~ the sum of contributions of all diagrams up to Nth order in the BG expansion for (6). We shall consider three different approximations for the binding energy: (EdN, N = 0, 1,2, which are obtained by substituting (HF;‘), into eq. (6), E(a) being always taken according to eq. (8)‘. Our results can be summarized as follows: (i) Conclusions which were drawn in B2 where the total hamiltonian (2) was used remain valid also for the modified hamiltonian (3), at least for 0 s (Ys 1. These + Contributions our recent paper
of the diagrams lo).
considered
were calculated
using explicit formulae
which are given in
116
J. Blank, J. HofejG/
Centre-of-mass
TABLE
effects
2
Individual terms of expansion (8) and the contribution E \” of diagram 3d in MeV; AE - E”‘+ a = 0 hw = 15.232 MeV C E(l) E’z’ Ei3’ E’3’
AE
-0.104 -47.16 -0.289 -3.094 -4.908 -52.35
0.388 -48.48 -0.317 -1.747 -3.686 -52.48
E(l)
E’z’ Ei3’ E’3’
AE
Ec3’.
Q = 0 hw = 24.538 MeV 0.999
-50.25 -0.381 0.200 -1.927 -52.56
0.000 -59.46 -3.723 -6.309 -10.84 -73.82
(Y= 1 ho = 15.232 MeV C
E”‘+
0.501 -62.27 -3.348 -3.094 -7.720 -73.34
1.001 -65.18 -2.970 0.404 -4.376 -72.53
a = 1 ho = 24.538 MeV
-0.042
0.457
0.936
0.062
0.545
1.006
-79.18 -0.337 -0.114 -2.499 -82.01
-80.38 -0.429 1.299 -1.226 -82.04
-81.59 -0.540 2.793 0.127 -82.00
-111.5 -1.920 0.390 -4.682 -118.1
-114.0 -1.652 3.520 -1.782 -117.4
-116.5 -1.408 6.724 1.154 -116.7
conclusions concern various aspects of calculating the total energy using eq. (8) and are illustrated in table 2. Firstly, third-order diagrams, especially that shown in fig. 3d, must not be neglected when calculating E(U). If these diagrams are not taken into account, the binding energy depends strongly on C and no condition for selecting CO follows from convergence arguments. Secondly, the BHF conditions (figs. 2b, 2c) are satisfied almost exactly for the self-consistent value w = w. (see also fig. 4). The importance of choosing o = 00 is manifested by the fact that the weakest dependence of the binding energy on C is attained in this case [the same statement holds for a-dependence - see below (ii)]. (ii) Testing the method Ml ((Y= 0) shows that only the approximation (E& satisfies our accuracy condition (independence on a), whereas the currently used zeroth-order approximation 6*7’9’12) is not accurate (see table 3 and fig. 5). The reason is clear from table 4: the first- and second-order diagrams cannot be neglected when calculating ‘J-J?:.‘). Their contributions lower this quantity to about 75% of (HF$)o =$hw(l +(u*) and Eb is lowered by more than 10% of its experimental value. On the other hand, the fact that (E& varies slowly over 0 G LY< 0.5, and that there is a rapid decrease of the differences (N?$))N+l -(&‘,$), with increasing N suggest that diagrams of higher orders, N L 3, need not be considered. Notice that the weakest dependence of (E& on (Y is attained for hw = 15.232 MeV which is quite close to the self-consistent value koo. The variation of (E& over the intervals OGCU
J. Blank, J. Hoiejiil
117
Centre-of-mass effects
TABLE 3 Binding energy in MeV versus (I for C = 0 and for the self-consistent value Co ho = 15.232 MeV Ec= 0.6961 MeV c=o a
0.5
0 -17.43 -15.67 -14.81
U5Jo (ErA (Eb)z
c = c, 0.7
-15.98 -15.03 -14.65
0
1
-14.81 -14.39 -14.23
-17.58 ‘-15.64 -14.92
-12.78 -12.78 -12.74
hw=20.735
MeV Ec=O.8122
0.5
0.7
-16.08 -15.09 -14.77
-14.87 -14.45 -14.31
(Eb)o (Et,)1 (Eb)z
c=c,
0
0.5
0.8
-17.77 -14.76 -13.36
-15.25 -13.57 -12.94
-12.11 -11.68 -11.51
0
1
0.5
-17.20 -13.95 -12.69
-9.83 -9.83 -9.73
ho=24.538
0.5
0
(EtJo (Et& (E&
-17.93 -13.88 -12.22
-14.41 -12.07 -11.39
0.8
-14.84 -13.06 -12.50
MeV Ec30.8836
-11.96 -11.53 -11.35
1 -9.83 -9.83 -9.73
MeV
c=c,
c=o Q
-12.78 -12.78 -12.74
MeV
c=o a
1
0.7
0
1
-11.57 -10.41 -10.19
0.5
-16.58 -12.22 -10.73
-6.82 -6.82 -6.78
0.7
-13.37 -10.92 -10.31
-10.92 -9.74 -9.53
1 -6.82 -6.82 -6.78
TABLE 4 Different approximations of (HFE.‘) (in MeV) versus a and C (hw = 15.232 MeV) a=0 (HFA. ). = 11.424 MeV -0.104
C (H?:. (H%.
)I )z
9.664 8.803
(Hfi.),
0.383
0.999
-0.020
9.587 8.777
9.475 8.760
13.329 12.938
(Y= 0.5 = 14.280 MeV
(H?;.),
0.476
0.977
-0.042
13.293 12.943
13.250 12.957
22.848 22.811
(Y=l = 22.848 MeV 0.457
0.936
22.848 22.813
22.848 22.816
and C considered if (Y- 1. We conclude that the method M2 does not satisfactorily eliminate c.m. effects in spite of (H?Z.‘)o being very probably a good approximation however, one to J%,. if LYis close to unity+. This conclusion may seem paradoxical; must not forget that it is the convergence of BG expansions for expectation values of + Let USrecall that for a = 1 (Hz:’
)I = (HP:’
)O holds exactly owing to properties of the h.o. basis.
118
J. Blank, J. Hoiejii / Centre-of-mass effects
I
hw- 20.735 MeV
41
I
0
0.5
m
F’ig. 5. Binding energy versus (Y for C = CO. The full, dash-dotted and dashed curves refer to the (E&, (E& and (&JO approximations, respectively. The curves (I?& and (E,Jl are given for ho = 15.232 MeV only.
internal operators, elimination
in particular
the
internal
of c.m. effects. Our results indicate
Eb = (Hint) is bud if a + 1. This can be explained
hamiltonian,
that
assures
correct
that convergence of the expansion for
by examining how individual terms on (Y as (Y+ 1. In this region one has (H:~‘)s = (H?E?), = (8) then yields (E&=(E&=(E~,)l=$ko(l+cr*)+E(~)+ E(*)+Ec3)+Ec. Now, for the self-consistent values C,,, wo, the terms E’*’ and Ec3’ ~a s 1; e.g. for C = Co and fro = 15.232 MeV become small and vary slowly over 0 we found that E’*‘+Ec3) lies between -3 and -2 MeV (see table 2). On the other hand, EC’)rapidly decreases because of the term proportional to (Y*in v(a) [eq. (5)]: for (Y= 0.5, 0.7, 1 it assumes values -57.17, -64.05, -79.25 MeV (C = Co, Rw = 15.232 MeV). Thus (Et,)* equals the difference of two large positive quantities each of which increases as (Y*and this essentially worsens the accuracy of this approximationif (~+l.
in eq. (8) depend arL(l+n*). Eq.
J. Blank, J. Hoiejill
Centre-of-mass
We have shown that the (E& approximation
119
effects
satisfies the condition
(Y+ 0. This fact by itself does not guarantee
that (E&
for accuracy
is sufficiently
if
close to the
binding energy, i.e. to the minimal eigenvalue of Hi,,. However, of various known approaches which permit the inclusion of a greater part of the BG expansion of the energy [calculating the total contribution A3 of the three-particle cluster 16) with the help of the Bethe-Faddeev equations, renormalization of BHF conditions via occupation probabilities ‘l), summation of repeated V-insertions into particle lines I’) only the first one yields a non-negligible contribution to Eb [As = -2 MeV for soft-core potentials ‘“)I. Now reaction matrix elements are essentially quadratic functions of (Y[see eq. (5)], and thus we expect a weak a-dependence of higher-order corrections for small (Y (0 6 (Ys i)‘. A s A 3 contains diagrams of at least third order, our conclusions concerning the a-dependence for 0 s cx s i will not change by adding A3 to (E&. Hence the (E& approximation yields a reliable value of the binding energy. We shall not discuss here the reasons for the existing discrepancy with the experimental value (see B2).
3.2. RMS
RADIUS
The nuclear
r.m.s. radius
rch is given by 13) r:,, =($)+r;+&
where r,, (r,) is the proton as follows++ [cf. eq. (9)]
Thus rch is calculated dered:
similarly bch)N
In contrast
(neutron)
to the energy,
=
r.m.s. radius
to E,.,.. J(~2>~
+
Again ri
+
6
14) and the operator
three ,
rc2 is expressed
approximations
will be consi-
N = 0, 1,2.
these differ little from each other over the whole interval
0 s (Y< 1 (see fig. 6). As (T&)Odoes not depend on (Y*, the other two approximations depend on cx weakly, and testing their accuracy by means of our procedure is not too significant. Nevertheless, insufficient accuracy of method M2 is manifested at least for ho = 15.232 MeV where
both (rch)l and (r&)2 begin to increase
as (Y+ 1.
Recently, Jadid and Mavromatis is) calculated the r.m.s. radius of 4He, 160 and 40Ca using the Goldstone expansion (instead of the BG one) and considered the corresponding Goldstone diagrams up to the third order. Their results are qualitatively similar to ours: the higher-order contributions represent only a small correction to (T.-h)@ ’ This is confirmed numerically for the second- and third-order corrections E”’ and E?‘. ‘+ As we have neglected the Coulomb interaction when calculating the reaction matrix, common value of both charge and mass nuclear radii. ’ Unlike
Eb, r=,, depends
on LYonly “implicitly”
via matrix
elements
of
t and V(a).
rch is the
120
J. Blank, J. HoiejE/
Centre-of-mass effects
1.a
5 t .-._._ _,_,_ - -.-
---
c
---
‘I d
1.5
N
j
>
--
---_.____,__.----
2
-------$
i
_J I,
1.6 .----------
3 If=
------_----
0.5
1
a Fig. 6. R.m.s. radius versus cx for C = Co. The full, dash-dotted and dashed and (r& approximations, respectively.
curves refer to the (r&
(r,_&
Another important ground-state characteristic of 4He is the form factor. Unfortunately, this is expressed via the expectation value of an A-body operator F(q) which is very difficult to calculate even for A = 4. The usual factorization, 0%))
- (K(4))l(~C.,.(q))
3
by means of operators FL(q) = C;f=, exp (iqxk) and F,.,.(q) = exp (iqX,.,.) can hardly assure correct elimination of c.m. effects. This is suggested e.g. by the fact that the zero of (F(q)X which determines the position of the dip, is identical with that of (F,(q)) where no c.m. corrections are included. Therefore it would be of little use applying our criterion of accuracy in this case. 4. Conclusions Results of our calculations of the binding energy and r.m.s. radius of 4He imply that two of the methods currently used (Ml, M2) for making c.m. corrections are incorrect; in both cases the binding energy depends strongly on the parameter (Y which characterizes the c.m. motion. In the method M2 which uses the BolsterliFeenberg-Lipkin hamiltonian (corresponding to (Y= 1) all the approximations considered depend strongly on cy.This is caused by the binding energy being equal to the difference of two positive quantities each of which depends essentially quadra-
J. Blank, J. Hoiejiij
Centre-of-mass effects
121
tically on (Y.It is questionable whether the strong a-dependence can be removed by including further diagrams; the BG expansion for the energy may be divergent due to the strong oscillator term in the two-body potential U(CX)[see eq. (5)]. Anyway our results show that in practical, numerically manageable calculations this method does not give reliable results. The method, which corresponds to (Y= 0, gives bad results (strong a-dependence) if the usual zeroth-order approximation for the c.m. energy is used (method Ml). On the other hand, if first- and second-order diagrams in the BG expansion for (H%?) are included, then the binding energy becomes practically constant over the interval 0 c CY
for stimulating discussions and permanent
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)
K.T.R. Davies and R.L. Becker, Nucl. Phys. Al76 (1971) 1 M. Bolsterli and E. Feenberg, Phys. Rev. lOl(1956) 1349; H.J. Lipkin, Phys. Rev. 109 (1958) 2071 T. Hamada and I.D. Johnston, Nucl. Phys. 34 (1962) 382 J. Blank, Czech. J. Phys. B29 (1979) 597 J. Blank, Czech, J. Phys. B29 (1979) 718 R.L. Becker, A.D. MacKellar and B.M. Morris, Phys. Rev. 174 (1968) 1264 J.W. Woodring, A.D. MacKeIIar and R.K. Tripathi, Nucl. Phys. A228 (1974) 345 I. Ulehla and N.T. Nguyen, Nucl. Phys. A260 (1976) 253 M. Gmitro and M. Sotona, Phys. Rev. C2 (1970) 356; M. Sotona, Czech. J. Phys. B27 (1977) 386 J. Blank and J. HoiejSi, Acta Univ. Carol.-Math. et Phys. 20 (1979) 27 B.H. Brandow, Phys. Rev. 152 (1966) 863 A. Kallio and B.D. Day, Nucl. Phys. Al24 (1969) 177 V.C. Aguillera-Navarro, M. Moshinsky and W.W. Yeh, Ann. of Phys. 51 (1969) 312 T. Janssens, R. Hofstadter, E.B. Hughes and M.R. Yearian, Phys. Rev. 142 (1966) 922 M.A. Jadid and H.A. Mavromatis, Nucl. Phys. A317 (1979) 399 R. Rajaraman and H.A. Bethe, Rev. Mod. Phys. 39 (1967) 745 H.S. Kiihler, Nucl. Phys. A98 (1967) 569 J.G. Zabolitzky, Nucl. Phys. A228 (1974) 285
A359 (1981) 109-121 Publishing Company
CENTRE-OF-MASS EFFECTS IN BRUECKNER CALCULATIONS OF 4He J. BLANK
and J. HOT\EJSi
Nuclear Centre, Faculty of Mathematics and Physics, Prague, Povltavska’ 1, 180 OOPraha 8-Pelt Tyrolka, Czechoslovakia Received 19 November 1979 (Revised 13 October 1980) Abstract: A general criterion is proposed which states a condition for the accuracy of BruecknerGoldstone perturbation approximations to the expectation value of a given internal operator 0’. The criterion is formulated via the dependence of (0) on a properly chosen parameter. A simple parametrization of the cm. motion is introduced and shown to be suitable for applying the criterion. Simultaneously the parametrization makes it possible to test some methods currently used for making c.m. corrections. Numerical results are obtained for the binding energy and r.m.s. radius of the 4He nucleus using the standard Brueckner-Goldstone theory in an oscillator basis. It is shown that the method based on the Bolsterli-Feenberg-Lipkin approach is incorrect. On the other hand, the method which employs the usual unmodified hamiltonian eliminates c.m. effects satisfactorily and ensures fulfilment of the condition for accuracy of the binding energy provided that first- and second-order diagrams are taken into account when calculating the cm. kinetic energy.
1. Introduction In Brueckner under
of the
theory
consideration,
operators
the calculated
physical
i.e. the expectation
corresponding
characteristics
of the A-body
system
values,
to individual
observables
of the
system,
are
influenced by the c.m. motion. This feature is common to all many-body theories which are based on independent-particle models ‘) (IPM). Dependence of (0) on the c.m. motion persists even if 19 is an internal operator (related to the c.m. coordinate system). This is due to the fact that the theory provides us only with approximations In general, these 4 aPP of $ which are expressed in terms of IPM basis functions. functions contain spurious components, i.e. (Jlappcannot be factorized into c.m. and internal parts and the c.m. dependence does not cancel out in eq. (1). Investigating the c.m. dependence and its best possible elimination is important, in the first place, for comparison of calculated physical quantities with experiment. In many-body perturbation calculations this problem has another, purely theoretical aspect: eliminating the c.m. dependence of a given approximation (O),,, is closely connected with convergence of the Brueckner-Goldstone (BG) expansion for (6). This statement, which follows from the obvious fact that the exact value of (0) is c.m. 109
J. Blank, J. HoiejiLl Centre-of-mass effects
110
independent for any internal operator 0, is the starting We shall formulate them first quite generally. Let p be a parameter use of the perturbation
whose appearance theory
point of our considerations.
in the calculation
is caused merely by the
(e.g. p may be the oscillator
frequency
in a BG
calculation in an oscillator basis); of course, the internal hamiltonian does not depend on p. Then for any given internal operator 6 the exact value (0) also does not depend on p while perturbation approximations (O),,, in general do. Suppose a given (O),,, is known as a function of p. Does it tell us something about its accuracy? Weak dependence of (O),,, on p? is obviously not sufficient, and if the expansion for (0) does not converge uniformly (with respect top) then it is not necessary either: as is shown in fig. 1, (O),,, may be equal (or close) to (0) for some p = p. even if (O),,, depends strongly on p. However, such an (S),,, can be used only if some additional condition implied solely by convergence arguments (e.g. by some sort of selfconsistency requirements) is available for selecting po. If that is not the case, then only that (O),,, can be satisfactory for which a sufficiently large interval 9 = (pl, p2) exists such that the variation of (O),,, over 8; is small. (In the hypothetical situation shown in fig. 1 this is fulfilled by the solid curve only.) In this way we simultaneously arrive at bounds for admissible values of p.
Fig. 1. Illustration of the general considerations of sect. 1: possible dependence of two perturbation approximations to (0) on parameter p. The approximations (a),, (C?),, are represented by the solid and dashed curve, respectively, the exact value by the dotted line. t In view of the qualitative character of these considerations we are speaking about weak and strong dependence in an intuitive sense. No strict specification can be given unless the physical meaning of p is known since one has to consider the variation of (CY).,, over a “sufficiently large” interval of “admissible” values of p.
J. Blank, J. Hoiejcil
How are these abstract effects?
statements
In the next section
related
to the problem
we show that the c.m. motion
way which makes it possible different BG approximations
111
Centre-of-mass effects
of eliminating
the c.m.
can be parametrized
in a
to apply the above approach for testing the accuracy of to the binding energy and other ground-state charac-
teristics of 4He. As a by-product we get an estimate of the applicability of two methods currently used for making c.m. corrections. Numerical results are presented and discussed in sect. 3. Finally, in the last section the results are summarized and some conclusions are drawn.
2. Parametrization
of the c.m. motion
We start with a simple generalization of the well-known trick which consists of modifying the c.m. part of the total hamiltonian while leaving the internal part unchanged I,‘). Let H be the usual=total hamiltonian of an A-nucleon system
Tc.m.= Pf,./2mA
Vjk = Hint + Tc.m.,
i=l
3
and let us introduce
the following
one-parameter
H(a)=Hf$mAa*~*Xz,,,
family of modified =Hi,,+HpE,‘(a).
In particular, H(1) is the Bolsterli-Feenberg-Lipkin hamiltonians standard Brueckner calculations suitable IPM determined by
bation”
hamiltonian
will be specified
H(a) - Ho into one- and two-body H(a)=H,+
=
v ->
(3)
(4) later.
After
decomposing
the “pertur-
parts, we get
C AVjk(a)- t 1 i
vj(a) 9 (5)
2 v(a)
hamiltonians:
(BFL) hamiltonian. For these can be performed starting with a
j=l
This model
(2)
lsj
V(OSC)
)
V(a)=
v-a2v(0sc).
Here V(Osc) (x) = ’2 mw2x2 and the meaning of x is obvious, depending on whether V(Osc)is a part of the one- or two-body operator. Now the reaction matrix must be calculated. Notice that it is the whole two-body part v(a) that enters the equation for t; in this way dependence on (Yis brought into t. When the reaction matrix is found, we can calculate the total ground-state energy E(a) [the minimal eigenvalue of H(a)] and the expectation value of any (internal) operator. For obtaining the binding energy Eb one has to subtract the c.m. energy Ec.m.(a)= (H?$ (a)) from E(cu): Eb E (Hint) = E(a)
-E,.,.(a)
*
(6)
112
J. Blank, J. Hoiejiil
Centre-of-mass effects
In particular, for LY= 0 we get Eb
=
E(O)
-
(~cm.)
.
(64
The parameter (Yenters each term in the BG expansion for the expectation value of any internal operator (0) (except the zeroth-order term: vacuum expectation value (0>,) via the matrix elements of operators f(a) and/or V(a).On the other hand, the internal part of the exact wave function is a-independent (since Hint is) and the same holds for the exact expectation value of any internal operator. The general statements given in sect. 1 can now be applied for p = (Y.As we also get bounds for admissible values of (Y,we are simultaneously able to test two of the methods for making c.m. corrections that were considered in ref. ‘). The first method (denoted hereafter Ml), which employs the unmodified hamiltonian (2), corresponds to (Y= 0. The second one (M2) is based on the BFL hamiltonian, i.e. cx= 1. The third method mentioned in ref. ‘), which works directly with the internal hamiltonian, cannot be tested by means of the parametrization (3). Instead another family of modified hamiltonians could be considered:
The decomposition
(5) would be then replaced by
or, alternatively,
As we have performed numerical calculations only for the parametrization shall restrict ourselves in the following exclusively to it.
(3), we
3. Results and discussion In this section results of our calculations of the binding energy and r.m.s. radius of the 4He nucleus are presented. The hamiltonian (3) was used with the HamadaJohnston potential v = UHJ[ref. ‘)I for several values of (Ysatisfying 0 S (YC 1. Details of the calculations have been given in our previous papers 4*5)which will be hereafter referred to as Bl and B2. Here we shall recall only some basic features. The standard non-degenerate version of Brueckner theory in a harmonic oscillator (h.o.) basis with a shifted spectrum6-9) is applied. In particular, the IPM hamiltonian reads Ho= t j=l
hi,
h=
E &VP,,
v=o
J. Blank, J. HoiejS/
where
P, projects
eigenvalue Candq:
on the subspace
113
Centre-of-mass effects
of one-body
E, (OS’)= $ hw (2 v + 3). The spectrum
&Y = Elosc) -$UJ
Hilbert
space
of h depends
belonging
to the
on three parameters
(C+S,orl) .
o,
(74
The Brueckner-Hartree-Fock (BHF) conditions for occupied states, which are represented graphically in fig. 2a, are strictly fulfilled. In the case of 4He they reduce
= ---g -1
0
,
fi--
hl
hl
h2
h2
= ii---* (b)
Fig. 2. Graphical
representation
of BHF conditions;
all the &interactions
are on the energy
shell.
for any given w and (Y to one simple relation between C and 7. It is convenient to exclude 77from this relation so that the three free parameters, (Y,w and C, remain. All of them satisfy the general requirements imposed on p in sect. 1 and hence we have investigated the dependence of results not only on (Y but on C and w as well. However, the latter two parameters differ from LYin one important point. Analyzing the BG expansion
for the energy
one can extract
“self-consistent”
values w. and Co
for which contributions of important classes of diagrams vanish or are minimized. In fact the values w. and Co used in our calculations were obtained by requiring the contribution E(‘) of the second-order diagrams (fig. 3b) to be minimal and the contribution Ei3'of diagram 3d to vanish. The former requirement represents a sort of optimal average fulfilment of the BHF conditions shown in figs. 2b, 2c and has been applied by many authors 6’7P9).The latter approximately includes the remaining self-consistency conditions for unoccupied states: (pi Vlp') = 0 [refs. 11,5)]. As is shown in fig. 4, a minimum of lE'2'1 equal to 0.3-0.4 MeV occurs at u = o. almost independently of C and CY,with the corresponding value #iwo lying between 15 and 18 MeV. The self-consistent value of C is practically independent of w. It depends, however, on (Y; in particular Co - 1 for (Y= 0 and Co = 0 for LY= 1 (see table 1). On the other hand, convergence arguments provide no condition for selecting the “best” (~0. Then, according to the general statements of sect. 1, we shall require the cal_culated binding energy and r.m.s. radius to depend weakly on (Y, while the dependence on w and C need not be weak.
J. Blank, J. Hoiej&‘/ Centre-of-mass effects
114
(a)
lb)
f-jJ-0 f---J--*
9 (cl Fig. 3. Diagrams
Id)
of the BG expansion
for the energy
discussed
in the text.
-4
-3
S f 5 -2
-1
0 ;5
20 hw(MeV)
Fig. 4. The second-order
contribution E? versus o for (Y= 0, C = 0 (full line), a = 0, C = C, (dashed line) and a = 1, C = C, = 0 (dash-dotted line).
J. Blank, J. Ho?ejs’i/ Centre-of-mass effects TABLE The self-consistent ho = 15.232 (1
C0
3.1.
1
value C, of parameter hw = 20.735
MeV
115
C versus a and w ho = 24.538
Mev
MeV
0
0.5
0.7
1
0
0.5
0.8
1
0
0.5
0.7
1
0.94
0.71
0.48
0.00
0.95
0.70
0.33
0.00
0.94
0.71
0.47
0.00
BINDING
ENERGY
The formula for binding energy contains two quantities each of which has its own BG expansion [see eq. (6)]. The total energy E(B) is calculated up to the third order, all diagrams being taken into account except those that belong to the three-particle cluster
(fig. 3~): E(a) = 3Ao(l
The first term is the vacuum Ho-
i
+a2)+E(1)+E(2)+E(3)+Ec.
expectation (vj-(y2v~))=
j=l
(8)
value of ;
(~+&7~))
j=l
of fig. 3a, E”‘, r = 2,3, denotes the total [cf. eq. (5)], E(l) is the contribution contribution of rth order diagrams and EC is the vacuum expectation value of the Coulomb interaction. For calculating the c.m. energy we express I&“:.’ in terms of the following one- and two-body operators
where ri, Vi are dimensionless quantities and momenta by rj =~j/b, pi = -ih(l/b)Vj; Hz:.’
that are related b = Jhlo.
to the nucleon coordinates The following then holds:
=$#uJ [a2(R(1)+2R(2))-(D(1)+2D(2))].
Let us denote by (6)~ the sum of contributions of all diagrams up to Nth order in the BG expansion for (6). We shall consider three different approximations for the binding energy: (EdN, N = 0, 1,2, which are obtained by substituting (HF;‘), into eq. (6), E(a) being always taken according to eq. (8)‘. Our results can be summarized as follows: (i) Conclusions which were drawn in B2 where the total hamiltonian (2) was used remain valid also for the modified hamiltonian (3), at least for 0 s (Ys 1. These + Contributions our recent paper
of the diagrams lo).
considered
were calculated
using explicit formulae
which are given in
116
J. Blank, J. HofejG/
Centre-of-mass
TABLE
effects
2
Individual terms of expansion (8) and the contribution E \” of diagram 3d in MeV; AE - E”‘+ a = 0 hw = 15.232 MeV C E(l) E’z’ Ei3’ E’3’
AE
-0.104 -47.16 -0.289 -3.094 -4.908 -52.35
0.388 -48.48 -0.317 -1.747 -3.686 -52.48
E(l)
E’z’ Ei3’ E’3’
AE
Ec3’.
Q = 0 hw = 24.538 MeV 0.999
-50.25 -0.381 0.200 -1.927 -52.56
0.000 -59.46 -3.723 -6.309 -10.84 -73.82
(Y= 1 ho = 15.232 MeV C
E”‘+
0.501 -62.27 -3.348 -3.094 -7.720 -73.34
1.001 -65.18 -2.970 0.404 -4.376 -72.53
a = 1 ho = 24.538 MeV
-0.042
0.457
0.936
0.062
0.545
1.006
-79.18 -0.337 -0.114 -2.499 -82.01
-80.38 -0.429 1.299 -1.226 -82.04
-81.59 -0.540 2.793 0.127 -82.00
-111.5 -1.920 0.390 -4.682 -118.1
-114.0 -1.652 3.520 -1.782 -117.4
-116.5 -1.408 6.724 1.154 -116.7
conclusions concern various aspects of calculating the total energy using eq. (8) and are illustrated in table 2. Firstly, third-order diagrams, especially that shown in fig. 3d, must not be neglected when calculating E(U). If these diagrams are not taken into account, the binding energy depends strongly on C and no condition for selecting CO follows from convergence arguments. Secondly, the BHF conditions (figs. 2b, 2c) are satisfied almost exactly for the self-consistent value w = w. (see also fig. 4). The importance of choosing o = 00 is manifested by the fact that the weakest dependence of the binding energy on C is attained in this case [the same statement holds for a-dependence - see below (ii)]. (ii) Testing the method Ml ((Y= 0) shows that only the approximation (E& satisfies our accuracy condition (independence on a), whereas the currently used zeroth-order approximation 6*7’9’12) is not accurate (see table 3 and fig. 5). The reason is clear from table 4: the first- and second-order diagrams cannot be neglected when calculating ‘J-J?:.‘). Their contributions lower this quantity to about 75% of (HF$)o =$hw(l +(u*) and Eb is lowered by more than 10% of its experimental value. On the other hand, the fact that (E& varies slowly over 0 G LY< 0.5, and that there is a rapid decrease of the differences (N?$))N+l -(&‘,$), with increasing N suggest that diagrams of higher orders, N L 3, need not be considered. Notice that the weakest dependence of (E& on (Y is attained for hw = 15.232 MeV which is quite close to the self-consistent value koo. The variation of (E& over the intervals OGCU
J. Blank, J. Hoiejiil
117
Centre-of-mass effects
TABLE 3 Binding energy in MeV versus (I for C = 0 and for the self-consistent value Co ho = 15.232 MeV Ec= 0.6961 MeV c=o a
0.5
0 -17.43 -15.67 -14.81
U5Jo (ErA (Eb)z
c = c, 0.7
-15.98 -15.03 -14.65
0
1
-14.81 -14.39 -14.23
-17.58 ‘-15.64 -14.92
-12.78 -12.78 -12.74
hw=20.735
MeV Ec=O.8122
0.5
0.7
-16.08 -15.09 -14.77
-14.87 -14.45 -14.31
(Eb)o (Et,)1 (Eb)z
c=c,
0
0.5
0.8
-17.77 -14.76 -13.36
-15.25 -13.57 -12.94
-12.11 -11.68 -11.51
0
1
0.5
-17.20 -13.95 -12.69
-9.83 -9.83 -9.73
ho=24.538
0.5
0
(EtJo (Et& (E&
-17.93 -13.88 -12.22
-14.41 -12.07 -11.39
0.8
-14.84 -13.06 -12.50
MeV Ec30.8836
-11.96 -11.53 -11.35
1 -9.83 -9.83 -9.73
MeV
c=c,
c=o Q
-12.78 -12.78 -12.74
MeV
c=o a
1
0.7
0
1
-11.57 -10.41 -10.19
0.5
-16.58 -12.22 -10.73
-6.82 -6.82 -6.78
0.7
-13.37 -10.92 -10.31
-10.92 -9.74 -9.53
1 -6.82 -6.82 -6.78
TABLE 4 Different approximations of (HFE.‘) (in MeV) versus a and C (hw = 15.232 MeV) a=0 (HFA. ). = 11.424 MeV -0.104
C (H?:. (H%.
)I )z
9.664 8.803
(Hfi.),
0.383
0.999
-0.020
9.587 8.777
9.475 8.760
13.329 12.938
(Y= 0.5 = 14.280 MeV
(H?;.),
0.476
0.977
-0.042
13.293 12.943
13.250 12.957
22.848 22.811
(Y=l = 22.848 MeV 0.457
0.936
22.848 22.813
22.848 22.816
and C considered if (Y- 1. We conclude that the method M2 does not satisfactorily eliminate c.m. effects in spite of (H?Z.‘)o being very probably a good approximation however, one to J%,. if LYis close to unity+. This conclusion may seem paradoxical; must not forget that it is the convergence of BG expansions for expectation values of + Let USrecall that for a = 1 (Hz:’
)I = (HP:’
)O holds exactly owing to properties of the h.o. basis.
118
J. Blank, J. Hoiejii / Centre-of-mass effects
I
hw- 20.735 MeV
41
I
0
0.5
m
F’ig. 5. Binding energy versus (Y for C = CO. The full, dash-dotted and dashed curves refer to the (E&, (E& and (&JO approximations, respectively. The curves (I?& and (E,Jl are given for ho = 15.232 MeV only.
internal operators, elimination
in particular
the
internal
of c.m. effects. Our results indicate
Eb = (Hint) is bud if a + 1. This can be explained
hamiltonian,
that
assures
correct
that convergence of the expansion for
by examining how individual terms on (Y as (Y+ 1. In this region one has (H:~‘)s = (H?E?), = (8) then yields (E&=(E&=(E~,)l=$ko(l+cr*)+E(~)+ E(*)+Ec3)+Ec. Now, for the self-consistent values C,,, wo, the terms E’*’ and Ec3’ ~a s 1; e.g. for C = Co and fro = 15.232 MeV become small and vary slowly over 0 we found that E’*‘+Ec3) lies between -3 and -2 MeV (see table 2). On the other hand, EC’)rapidly decreases because of the term proportional to (Y*in v(a) [eq. (5)]: for (Y= 0.5, 0.7, 1 it assumes values -57.17, -64.05, -79.25 MeV (C = Co, Rw = 15.232 MeV). Thus (Et,)* equals the difference of two large positive quantities each of which increases as (Y*and this essentially worsens the accuracy of this approximationif (~+l.
in eq. (8) depend arL(l+n*). Eq.
J. Blank, J. Hoiejill
Centre-of-mass
We have shown that the (E& approximation
119
effects
satisfies the condition
(Y+ 0. This fact by itself does not guarantee
that (E&
for accuracy
is sufficiently
if
close to the
binding energy, i.e. to the minimal eigenvalue of Hi,,. However, of various known approaches which permit the inclusion of a greater part of the BG expansion of the energy [calculating the total contribution A3 of the three-particle cluster 16) with the help of the Bethe-Faddeev equations, renormalization of BHF conditions via occupation probabilities ‘l), summation of repeated V-insertions into particle lines I’) only the first one yields a non-negligible contribution to Eb [As = -2 MeV for soft-core potentials ‘“)I. Now reaction matrix elements are essentially quadratic functions of (Y[see eq. (5)], and thus we expect a weak a-dependence of higher-order corrections for small (Y (0 6 (Ys i)‘. A s A 3 contains diagrams of at least third order, our conclusions concerning the a-dependence for 0 s cx s i will not change by adding A3 to (E&. Hence the (E& approximation yields a reliable value of the binding energy. We shall not discuss here the reasons for the existing discrepancy with the experimental value (see B2).
3.2. RMS
RADIUS
The nuclear
r.m.s. radius
rch is given by 13) r:,, =($)+r;+&
where r,, (r,) is the proton as follows++ [cf. eq. (9)]
Thus rch is calculated dered:
similarly bch)N
In contrast
(neutron)
to the energy,
=
r.m.s. radius
to E,.,.. J(~2>~
+
Again ri
+
6
14) and the operator
three ,
rc2 is expressed
approximations
will be consi-
N = 0, 1,2.
these differ little from each other over the whole interval
0 s (Y< 1 (see fig. 6). As (T&)Odoes not depend on (Y*, the other two approximations depend on cx weakly, and testing their accuracy by means of our procedure is not too significant. Nevertheless, insufficient accuracy of method M2 is manifested at least for ho = 15.232 MeV where
both (rch)l and (r&)2 begin to increase
as (Y+ 1.
Recently, Jadid and Mavromatis is) calculated the r.m.s. radius of 4He, 160 and 40Ca using the Goldstone expansion (instead of the BG one) and considered the corresponding Goldstone diagrams up to the third order. Their results are qualitatively similar to ours: the higher-order contributions represent only a small correction to (T.-h)@ ’ This is confirmed numerically for the second- and third-order corrections E”’ and E?‘. ‘+ As we have neglected the Coulomb interaction when calculating the reaction matrix, common value of both charge and mass nuclear radii. ’ Unlike
Eb, r=,, depends
on LYonly “implicitly”
via matrix
elements
of
t and V(a).
rch is the
120
J. Blank, J. HoiejE/
Centre-of-mass effects
1.a
5 t .-._._ _,_,_ - -.-
---
c
---
‘I d
1.5
N
j
>
--
---_.____,__.----
2
-------$
i
_J I,
1.6 .----------
3 If=
------_----
0.5
1
a Fig. 6. R.m.s. radius versus cx for C = Co. The full, dash-dotted and dashed and (r& approximations, respectively.
curves refer to the (r&
(r,_&
Another important ground-state characteristic of 4He is the form factor. Unfortunately, this is expressed via the expectation value of an A-body operator F(q) which is very difficult to calculate even for A = 4. The usual factorization, 0%))
- (K(4))l(~C.,.(q))
3
by means of operators FL(q) = C;f=, exp (iqxk) and F,.,.(q) = exp (iqX,.,.) can hardly assure correct elimination of c.m. effects. This is suggested e.g. by the fact that the zero of (F(q)X which determines the position of the dip, is identical with that of (F,(q)) where no c.m. corrections are included. Therefore it would be of little use applying our criterion of accuracy in this case. 4. Conclusions Results of our calculations of the binding energy and r.m.s. radius of 4He imply that two of the methods currently used (Ml, M2) for making c.m. corrections are incorrect; in both cases the binding energy depends strongly on the parameter (Y which characterizes the c.m. motion. In the method M2 which uses the BolsterliFeenberg-Lipkin hamiltonian (corresponding to (Y= 1) all the approximations considered depend strongly on cy.This is caused by the binding energy being equal to the difference of two positive quantities each of which depends essentially quadra-
J. Blank, J. Hoiejiij
Centre-of-mass effects
121
tically on (Y.It is questionable whether the strong a-dependence can be removed by including further diagrams; the BG expansion for the energy may be divergent due to the strong oscillator term in the two-body potential U(CX)[see eq. (5)]. Anyway our results show that in practical, numerically manageable calculations this method does not give reliable results. The method, which corresponds to (Y= 0, gives bad results (strong a-dependence) if the usual zeroth-order approximation for the c.m. energy is used (method Ml). On the other hand, if first- and second-order diagrams in the BG expansion for (H%?) are included, then the binding energy becomes practically constant over the interval 0 c CY
for stimulating discussions and permanent
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