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Derivation of collective potential and mass from the generator coordinate method
ANNALS
OF PHYSICS
130,
307-328 (1980)
Derivation of Collective Potential from the Generator Coordinate W.
and Mass Method*
BAUHOFF
Institut fir Theoretische Physik, Universitiit Tiibingen, Tiibingen D-7400, West Germany and 1. Institut fiir Experimentalphysik, Universitiit Hamburg, Hamburg D-2000, West Germany? Received December 18, 1979; revised June 25, 1980
The Hill-Wheeler equation of the generator coordinate method is approximated by a local collective Schradinger equation. General expressions for the potential and the mass parameter are obtained by a symmetrized moment expansion. The validity of the approximation is tested for several examples where the exact solution is known. These include the Gaussian overlap with harmonic and anharmonic interaction, the Lipkin model, and monopole resonances of spherical light nuclei. In all cases, surprisingly close agreement with the exact solution is found. Other possible applications of the formalism are indicated.
1. INTRODUCTION
Quantum mechanical many-particle systems often exhibit a collective behaviour for low-energy excitations. Examples for such a behaviour are the rotational and vibrational states of nuclei, the fission process and the scattering of heavy ions. The examples are taken from nuclear physics since we are mainly interested in applications to this field, but the statement applies also to other areas of physics. These collective phenomena are most economically described by the introduction of collective coordinates. A convenient way to do this is provided by the generator coordinate method (GCM) (for a review see, e.g., [l]). This method does not require the introduction of additional degrees of freedom [2]; it also avoids the difficult problem of constructing collective coordinates in terms of one-particle coordinates [3]. The method is based on a special ansatz for the many-particle wave-function. By inserting this ansatz into the microscopic SchrSdinger equation, an integral equation (the Hill-Wheeler equation) for the superposition amplitude is derived [4]. By a solution of this integral equation one gets information on the colIective behaviour of the system. A drawback of this method is the missing physical interpretation of the integral kernel. In order to make a connection to more phenomenological theories of collective motion [2] it is therefore tempting to approximate the Hill-Wheeler equation for the superposition amplitude by a Schriidinger-like equation with a collective potential * Work supported by the DFG. + Present address. 307 0003-4916/80/140307-22$05.00/O All
Copyright 0 1980 rights of reproduction
by Academic Press, Inc. in any form reserved.
308
W.
BAUHOFF
and a collective mass, which will in general depend on the collective coordinate. Indeed, there are several proposals for such an approximation in the literature [5-g]. In all cases, the small width of the Hamiltonian and wave-function overlap is used for justifying the approximation, which is usually based on some form of the moment expansion of the integral kernel. An extensive review of these approximations is given, e.g., by [I]. It has the merit of providing directly a Schrodinger equation, i.e., a quanta1 description, for the collective motion. So a difficult and ambiguous requantization of a classical equation is avoided which is necessary, e.g., in the BohrMottelson theory [2]. But due to the peculiar normalization condition for the superposition amplitude, it may not be interpreted as a true square-integrable wave-function [lo]. In fact, there are cases where it is a distribution and not a function at all [l 11. So it is not safe to derive an approximate Schrodinger equation for such a possibly singular object. In the next section, we derive therefore a local differential equation as an approximation for a transformed superposition amplitude which has a proper wave-function interpretation. The expansion we propose is a combination of the methods of [7, 81: We use the symmetrized moments of the integral kernels introduced in [7] instead of the usual moments used in [6]. In order to avoid the explicit calculation of the square root of the normalization kernel we employ the method of 181. Our final formulas for the collective potential and mass differ, however, from those of [8]. As discussed in detail at the end of the next section, the formalism of [S] does not lead to a unique value for the mass and is therefore not applicable in concrete calculations (in fact, it has never been used in practice). In actual calculations, the moment expansion has been used only in the special case of a Gaussian form of the overlap kernel [12, 131. Qualitative agreement with known exact results has been obtained. But the assumption of a Gaussian form restricts severely the applicability of the method since in some cases of physical interest it cannot be justified [14]. In the present paper, we try to overcome this difficulty. In order to test the validity of the approximation, we perform a number of tests for different models in the following: We compare the solution of the approximate Schrodinger equation with those of the full Hill-Wheeler equation which are either known analytically (Sections 3-5) or numerically (Section 6). The models we treat explicitly are: Gaussian overlap with harmonic (Section 3) and anharmonic (Section 4) interaction which are the standard test cases for such a study. In Section 5 we investigate the Lipkin model [15] where the influence of the particle number on the quality of the approximation can be studied easily. We compare our results also with those of the moment expansion of [13] where a Gaussian overlap has been assumed. We find a definite improvement in our approach in all cases considered. Giant monopole resonances of spherical Iight nuclei are considered in Section 6. This is probably the most interesting example from a physical point of view. We compare our results with a recent generator coordinate calculation using the Skyrme interaction [16]. As in this case the collective mass is not positive for all values of the coordinate, one may have problems in a variational solution of the collective Schrodinger equation [17, 181. We demonstrate that the expression for the collective mass we obtain does not lead to
COLLECTIVE
POTENTIAL
309
AND MASS FROM GCM
difficulties of this kind. This is in contrast to earlier applications of the moment expansion to this problem where a spectrum was obtained which is not bounded from below [9, 181. Finally, the results are summarized in Section 7. Some technical details not directly related to the main topic of the paper can be found in the appendices. 2. DERIVATION
OF COLLECTIVE
POTENTIAL
AND COLLECTIVE
MASS
The starting point of the GCM is the ansatz for the wave-function
(2.1)
~(s, = jr&r, R)f('k) c/r7
where .Y denotes all the one-particle coordinates and CYis the generator coordinate which we assume to be real. The extension of the ansatz (2.1) to several generator coordinates is straightforward, but will not be considered in this paper. The manyparticle basis functions ~(x, a) are assumed to be given. Usually they are taken as Slater determinants for simplicity. They can be chosen according to one’s physical intuition or determined, e.g., by a constrained Hartree-Fock calculation. Substituting the ansatz (2.1) into the many-particle Schrijdinger equation, formulated as a projection equation [19] (&,A 1H - E/41,\ = 0
leads to the Hill-Wheeler
The Hamiltonian
equation for the superposition
(2.2)
amplitudef(,u)
and overlap kernel are defined by
If the basis function ~(x, IX) are real the integral kernels H(cY, p) and N(cx, /I) are symmetric. Eigenfunctions of Eq. (2.3) satisfy the orthogonality condition:
Since this is not the usual orthogonality condition for a proper quantum mechanical wave-function, f(a) cannot be interpreted in such a manner. So the derivation of a collective Schrodinger equation forf(ol) will in general be a questionable approximation. Therefore it is useful to transform from the superposition amplitudef(n) to an
310
W.
BAUHOFF
object which can be considered as a collective wave-function, by the definition
[IO] (2.7)
From (2.6) we find immediately
the correct orthogonality
condition
for g(a):
gn(4 da=%rzn . sdX4
(2.8)
The dynamical equation for g(a) is deduced from (2.3) (2.9) with the integral kernel K(a, jl) = /N-‘f’(a,
a’) H(d,
B’) N-1f2(j?, ,8) da’ dfi’.
(2.10)
Since the normalization kernel N(ol, /I) is positive definite the square root N1j2 is well defined. Its inverse will also be defined if the Hill-Wheeler equation has no zero-energy solutions. If we exclude this exceptional case, the transformation fromf to g and back is a well-defined operation. The square root and its inverse can be calculated by a spectral decomposition (see Appendix A for such a calculation). But the approximation of (2.9) by a local differential equation does not require the explicit knowledge of the square root as will be demonstrated in the following. So the actual computation of N1f2 will not be needed. The integral kernels N, Hand K will be sharply peaked as a function of 01- /? and slowly varying as a function of o + ,8. This property will become more evident in the examples considered in the next sections. It allows the approximation of the kernels by local differential operators. We will explain this approximation for a general operator A(s /Q h aving the same properties as N, H and K and will specialize to (2.9) below. With the momentum operator (vl = 1)
one readily derives the following equation [7]: j-A@, @s)(/l) d/9 = j-e(i’2’Epu A(a - i/i?, a + 4s) e(i’2’8pa dfif(a).
If we expand the right-hand
s
(2.12)
side in powers of P, we find up to second order
da, k?ls)f(@ 43 = [A,(4
- t V’a2, A,(a)}
+ . ..]f(a).
(2.13)
COLLECTIVE
POTENTIAL
AND
MASS
FROM
311
GCM
where we have introduced the symmetrized moments of the integral kernel
and the notation
v,2, A,(4) = P,242(4 + 2PJ,(a)
P, -t A*(R) P,2.
Since (2.13) is independent of the form of the function5
(2.15)
we have the operator equality (2.16)
A = A, - ${P”, AZ; -; ‘.’ .
Because of the symmetry of A(cu, /3) the first-order moment A,(a) and all other odd moments will vanish identically. So the Hermiticity of the operator A is preserved term by term in the expansion (2.16). This is in contrast to other moment expansions proposed earlier [6]. Equation (2.16) will now be applied to the transformed Hill-Wheeler equation (2.9) and the definition (2.10). In order to avoid the explicit calculation of the square root we rewrite the expansion for N1j2 in terms of moments of N following [S]: Defining N’ by N = N;i2(1 + N’) Nil”
(2.17)
the expansion (2.16) yields N’
,AJ;;‘P[-i(p2,
=
N,;
+
. ..I
N,-‘I’
(2.18)
and for N1jz we get N-112
By a binomial
=
(1
+
N’)-‘/2
,‘+J;‘P
=
&l/2(1
+
N’)-~‘9.
(2.19)
expansion we get finally using (2.18): (1
+
/7g’)-l/2
=
1 -
$N’+
. ..
=
] -
~&-1/z[+{f’2,
N,}
T
. ..I
N;:?
+
....
(2.20)
It should be noted that Ni’2 is an ordinary square root in the sense of a function and not the square root of an operator. So its calculation is without any problem. The Eqs. (2.19)-(2.20) together with the expansion (2.16) applied to the Hamiltonian overlap H are now inserted into the Hill-Wheeler equation (2.9) and the definition (2.10).
Jf we keep only terms up to second order in P we get [N;l12(Ho - .k{P2, H2}) NilI
+ &N;‘/2{P2,
N2) N;3/2H0 (2.21)
312
W.
BAUHOFF
The terms on the left-hand side can be re-arranged to the form of a Schriidinger equation with a potential and an effective mass: (2.22)
For the effective mass we find (2.23)
and the collective potential is
In (2.24) all moments No, N2 , Ho and H, are functions of 01,and the primes denote differentiation with respect to CL The most important contribution to the potential is the first term Ho/No. All other terms contain second moments and hence are expected to be small. They can be considered as zero-point energy corrections to the potential energy surface [20]. The expression for the potential simplifies drastically if the norm kernel N(ol, ,8) depends only on the difference of its arguments a - ,L?.In this case, all moments of N will be constant as can be seen from the definition (2.14). So all derivatives of No and NZ vanish, and the collective potential is simply 1 H;(a) v(“=$j#+8Y#-&q-4--.
1
N2(4
H,“(4 No2t4
This expression for the collective potential will be used in the next three sections. The expression for the collective mass and potential is formally identical to that given in [S]. A difference lies, however, in the definition of the quantities No , N2 , Ho and H2 which appear in the formulas. In our case, they are the symmetrized moments defined in Eq. (2.14). Instead of this, they are given in [8] by
and similarly for Ho and H2 . The important point is that an equation is given only for the derivative of N2 . So N2 is determined only up to a constant which cannot be fixed. Therefore, the collective mass contains undetermined constants in the formalism of [8). In fact, the formalism has never been applied in an actual calculation, at least to our knowledge.
COLLECTlVE
3. HARMONIC
POTENTIAL
INTERACTION
AND MASS FROM GCM WITH GAUSSIAN
313
OVERLAP
As the first test for the general investigations of the preceding section we treat the harmonic oscillator with a Gaussian overlap. In this case the Hill-Wheeler equation can be solved exactly [4], so we can easily study the validity of our approximations. The model is defined by the overlap kernel qoc, p> = e-(G3)~/A” and the Hamiltonian
(3.1)
kernel
For the extraction of the collective potential and the collective mass we have to calculate the lowest symmetrized moments of these kernels. By using standard integrals for Gaussians we find
(3.3)
Since the overlap depends only on the difference of the arguments we can use Eq. (2.25) for the calculation of the potential. Substituting (3.3) into (2.25) yields V(a) = E, - $Gmw2h2f $~co~cY~.
Similarly
(3.4)
we find from Eq. (2.23) the effective mass M(a) ,= m.
(3.5)
So in this case the collective Schrodinger equation (2.22) is simply the equation of a harmonic oscillator. Its eigenvalues are well known to be E, = E, - ,~~rnw2X2 + (n + Q)h.
(3.6)
This expression for the eigenvalues is identical to the exact solution of the HillWheeler equation [4, 111. This is a definite advantage compared to the moment expansions proposed earlier [6] which yield the eigenvalues only approximately for small X (i.e., a small non-locality of the overlap kernel). Our treatment is correct
314
W. BAUHOFF
even for X2 > 4/mw when the superposition amplitude f (a) is no longer a function but a distribution [ll]. This is due to our transformation to g(a) which is a wellbehaved function also in this case. For the eigenfunctions we find
where H, denote Hermite polynomials. This result also coincides with the exact solution. The belonging &(ol) can be calculated from (3.7) by means of a Gauss transform [I 11.
4. ANHARMONIC INTERACTION WITH GAUSSIAN OVERLAP
If the collective motion is no longer restricted to small deformations, a quadratic expansion for the Hamiltonian is not adequate any more. In this case a more general ansatz instead of (3.2) is in order:
with slowly varying functions W and U, whereas N(oI, /3) is still given by (3.1). This form of the Hamiltonian is of interest since for it the transformed kernel K (see Eq. (2.10)) can be calculated exactly and is in fact a local differential operator. Details of the calculation can be found in Appendix A. If we apply the formalism of Section 2, we find for the moments H,(a) = X7w [W(E) - QhW(a)], f&(a)
(4.2)
= ;x3%-1/2[W(a) - gm.qa)].
This leads to the collective potential V(a) = W(a) - &w(a)
- pPwy01) -
&X4uya)
(4.3)
and the effective mass 1 w4
-
= a hW(a).
An expansion of the exact expression for the potential and the mass in powers of the non-locality range h shows that the approximate quantities (4.3), (4.4) are correct up to terms with X4. This also explains why the exact solutions were obtained for the harmonic interaction in the previous section: In this case the exact Hamiltonian I contains only terms with h4 at most.
COLLECTlVE
POTENTIAL
315
AND MASS FROM GCM
5. THE LIPKIN
MODEL
One of the most popular testing grounds for theories of collective motion is the Lipkin model [lS] since it is exactly solvable for all particle numbers. The model contains two n-fold degenerate levels with unperturbed energies & and -&. The interaction Hamiltonian is given by H T=&Ec oa;?;,a,, + V c
(5.1)
a+ m.a+ m,oam,-oam-o ,
where CJdenotes the two levels i 1 and m, m’ indicate the degeneracies. By the introduction of quasi-spin operators
(5.2)
the Hamiltonian
can be rewritten in the form
H = 4, + V(K+2+
(5.3)
Ks2),
which obviously commutes with K” and hence mixes only the states belonging to a multiplet with a fixed quantum number K. This form of the Hamiltonian is the reason for the possibility of obtaining exact solutions. The unperturbed ground state I 0) has evidently all particles occupying the lower level. A convenient choice of (normalized) generating functions 1OZ)is [13]: \ CY>= coP DL. exp(tan CL. K+) / O>
(5.4)
with 31varying from -an to &T. By the Thouless theorem 1 CX)is the most general Slater determinant which is not orthogonal to I 0). The physical interpretation of the generator coordinate 01is clear from the computation of the particle number in the lower and upper level: (a I C a;t,, a,, / a) = n sin” c(,
m
(5.5) (a 1c uf-, a,-, / a) = n COG a. m With the generating function (5.4) the normalization can be calculated easily:
and Hamiltonian
overlap kernel
N(% p>= cosya - p), H(a, P>= +n{cos(a + /3)cosya - p> + txbs”-“(a
- Ml
+ sin2(a + /3)) - COS~(CY - p)]>,
(5.6)
316
W.
BAUHOFF
where x = V(1 - n)/~ denotes the strength of the two-particle interaction. From (5.6) one recognizes the sharply peaked nature of the overlap and Hamiltonian kernel for large particle numbers. The approximation of the Hill-Wheeler equation belonging to (5.6) by a local Schrodinger equation has been considered before [13], but in the calculation of the moments a Gaussian form of the overlap kernel has been assumed which leads to sizable errors for small particle numbers. The solution of the full HillWheeler equation is identical to the exact (algebraic) solution of the model. This seems to be known to other people [21], but as we do not know of a published proof for it, we present a proof in Appendix B. Introducing the notation
(5.7)
the symmetrized moments of N(cx, /3) and H(u., /3) can be written: (5.8) f&(a) = -&n{Zo(n H,(a) = -&n{Z2(n
- 1) cos 2a + &x[Zo(n - 2)(1 + sin2 2~x) - 1&r)]}, - 1) cos ~LX+ &x[Z,(n - 2)(1 + sin2 201) - I&z)]>.
The integration in (5.7) can be performed analytically, but the results are quite long, so we do not quote them here. IJsing (2.23), (2.25) the mass parameter and the potential are readily obtained from (5.8): 1 1 M(ol)=2En
Mn - 1) m I
- m Zo2(n)
Zo(n - 1) cos .&
+ 1x Zdn- 2) Z,(n)- m Al@- 2) 2
V(R)=+1
Z”(n) Al(n) Lb - 1) +
[l + sin2 2011 , I (5.9)
Z2(4
an - 1)- tun - 1)m) cos2cr ~nYn>
+ lo(n) Zo(n - 2) - z”yn) + z&2 - 2) &(n) - 2&(n) m &‘Yn)
- 2) II*
In Figs. 1 and 2 we show the mass and potential for n = 4 and different strength of the two-particle interaction, x. For comparison, the results for a Gaussian overlap approximation [ 13] are also shown. We find a sizable deviation for the mass parameter,
COLLECTIVE
POTENTIAL
AND
MASS
FROM
317
GCM
‘OI -10' -n/2
0
n/2
FIG. 1 The potential V(a) and the inverse mass parameter h/M(a) of the Lipkin model for n = 4 and x ; 0.6. Both quantities are given in units of E. The dashed lines indicate the results for a Gaussian overlap approximation (131.
0
ni2
FIG. 2. Same as Fig. I for n = 4 and ,y = 1.8.
especially for a strong two-particle interaction. For higher particle numbers the differences become smaller. For x < 1, the mass parameter is no longer positive definite. The negative contribution is, however, much smaller than for a Gaussian overlap assumption.
318
W. BAUHOFF TABLE I The Energies of the Ground and First Excited State of the Lipkin Model for Various Particle Numbers n and Interaction Strengths x n
x
El?
El
4
0.2
-2.013 -2.017 -2.043
- 1.020 - 1.027 -1.033
4
1.0
-2.309 -2.315 - 2.269
-1.414 -1.388 - 1.476
4
5.0
-6.110 -6.043 -5.813
-5.099 -4.943 -5.201
14
0.37
-7.038 -7.040 -7.041
-6.096 -6.108 -6.111
14
0.93
-7.270 -7.275 -7.259
-6.622 -6.621 -6.626
20
0.95
- 10.292 - 10.296 - 10.282
-9.710 -9.710 -9.709
Note. All energies are given in units of E. For all parameter values, the first line gives the exact energy taken from [15] or (211. The second line gives the approximate result of the local Schrijdinger equation (2.22). In the third line, we give the result for the approach of [13].
In the calculation of the eigenvalues, no difficulty arose from this fact. We will come back to this point in the next section and in Appendix C. The eigenvalue spectrum for the Schrbdinger equation with the parameters (5.9) can be obtained quite simply by a Fourier ansatz for the wave-function. In Table I we quote some results for different interaction strengths x. The exact solutions for comparison are taken from [15, and 211. We find a remarkable agreement between exact and approximate solutions already for small particle numbers, where the assumption of a narrow overlap is not necessarily fulfilled (for n = 4, the width of the normalization kernel is 0.36 r which is one-third of the whole range of the variable). The agreement is found for all values of the interaction strength, so our approximation can easily handle the “phase transition” at x = 1. For comparison, we have also listed the results for the approach of [13]. In this paper, only figures were given showing the eigenvalues as functions of x and n. Therefore, we have redone the calculations to obtain numbers for the eigenvalues. As can be seen from the table, the results for the approach of [I 31 differ from ours especially for small n and large x. This could
COLLECTIVE
POTENTIAL TABLE
AND MASS FROM GCM
319
I1
The Complete Spectrum of the Lipkin Model for n = 8 and x = 0.88 @V/r = 1) from the Exact Solution (Left Column), the Approximate Calculation according to Eq. (2.22) (Middle Column) and according to [13] (Right Column) Exact
Eq. (2.22)
Ref. 1131
4.233 3.467 2.403 1.228 0 -1.228
3.646 2.888 1.912 0.941 -0.164 -1.301
3.157 2.590 I .707 0.718 -0.338 -- I .424
-2.403
-2.421
-2.492
-3.467
-3.463
-3.489
-4.233
-4.239
-4.219
have been expected from Figs. 1 and 2. In all cases considered, our formalism leads to closer agreement with the exact results. In Table II we give the complete spectrum obtained for n = 8 and x = 0.88 (n V/c = I ). This shows an increasing discrepancy for the higher eigenvalues, but stiI1 a qualitative agreement is found. Also for the higher eigenvalues our results compare favourable with those of [ 131. Especially the symmetry between positive and negative eigenvalues with is not built in in our approach is approximately reproduced. For larger particle numbers, we expect a closer agreement also for the higher eigenvalues. But this requires such a large number of basis functions in the variational ansatz that we did not perform such a calculation.
6. MONOPOLE
RESONANCE OF SPHERICAL
LIGHT
NUCLEI
As the last and physically most interesting example we treat the isoscalar monopole vibrations of nuclei (breathing mode). For simplicity we consider only spherical, closed-shell nuclei of low mass, i.e., 4He, 160 and 4oCa. The generating functions are Slater determinants of harmonic oscillator functions with the oscillator constant /3 = (m~/yZ)“~ as generator coordinate. For the nucleon-nucleon interaction, the Skyrme interaction [22] is adopted. With this interaction the Hill-Wheeler equation has been solved numerically [16] so that we have again an exact solution to compare with. The normalization kernel is for this choice of the basis functions
320
W.
and the Hamiltonian
BAUHOFF
kernel assumes the form
where we have introduced the abbreviation 6 = ($(f
+ /P)““.
(6.3)
The coefficients t, ,..., t3 are the parameters of the Skyrme interaction. All calculations have been performed with the parametrization S III. The coefficients B,, ,..., B3 , C, D depend on the nucleus considered [16]. For completeness they are given explicitly in Table III. We have accounted for the c.m. motion by subtracting the direct term of the c.m. kinetic energy. For the calculation of the various moments one has to observe that the variables 01, /3 are defined on the interval (0, co). Keeping this in mind, we find for the moments of the norm kernel N,(a) = 401 . n, , (6.4)
iV2(a) = 1601~* n2 with the (constant) integrals
Similarly,
the moments of the Hamiltonian ~,(a)
= 4c&;’
can be written
+ 4a4h$’ + 4cZ’hf) + 4a’ht’
+ 4a3hf’,
~,(a) = 16a5h;’ + 16cfhf’ + 16a*h;’ + 16d’ht’ TABLE
+ 160r5h$’
III
The Various Coefficients appearing in the Definition (6.2) of the Hamiltonian Overlap for the Various Nuclei
BO 4 & B3 C D
4He
I40
1 1 0 1
3114 3514 4 116/9 13514 12
912 2
%a 1945/64 2625164 5512 2060127 117 40
COLLECTIVE
POTENTIAL
AND
MASS
FROM
321
GCM
with the integrals
All the integrals (6.5), (6.7) can be calculated numerically without any difficulty. Since the norm kernel depends not only on the difference of the arguments, the full expression (2.24) has to be used for the calculation of the collective potential. With the explicit form (6.4), (6.6) of the moments the actual calculation is, however, straightforward. The explicit result will be given as function of the variable y = l/a. It has a more direct physical interpretation because it is proportional to the root mean square radius of the nucleus. In terms of this variable, the collective Schrodinger equation reads (6.8)
with the explicit form of the potential 5* +Z
no
1
(6.9) and of the mass parameter -=
h(2) 2 no I (6. IO)
322
W. BAUHOFF 160
r y Ifml
FIG. 3. The collective potential for monopole
vibrations of lo0 (see Eq. (6.9)).
6
FIG. 4. The collective mass parameter for monopole vibrations of .W in units of the nucleon mass (see Eq. (6.10)).
COLLECTIVE
POTENTIAL
AND
MASS
FROM
GCM
323
For 160 the potential is shown in Fig. 3. It has the form expected on physical grounds: For small radii, we have an extremely strong hard-core corresponding to the large compressibility of nuclear matter. It is determined by the three-body part of the Skyrme force. Then we find a minimum which indicates the equilibrium position. For larger radii, the kinetic energy contribution leads to a slightly repulsive part whereas asymptotically the potential vanishes. For 4He and 40Ca the potential has the same shape with a minimum of about -50 MeV at y = 1.35 fm for IHe and of about -420 MeV at y ==- 1.9 fm for 40Ca. The mass parameter for 160 is shown in Fig. 4. The most prominent feature of this graph is the existence of a pole around y = 1.35 fm. For smaller radii, the mass parameter is negative. In the asymptotic region y - co, the mass approaches a constant value which is independent of the nucleon-nucleon interaction (compare (6.10) with the definition of the integrals (6.7)). So the form of the mass parameter is quite different from the cranking mass, which increases monotonically with y [9]. For the other nuclei, the collective mass has a similar form with the pole at y == 1.2 fm for *He and at y == 1.5 fm for 4oCa. Since the collective kinetic energy is no longer positive definite, one can have troubles with a variational solution of the collective SchrGdinger equation [17]. But in our case, the mass parameter is negative only in a region where the potential is strongly repulsive. Therefore we do not get any problems in a variational solution. This can be seen from an investigation of the asymptotic behaviour of the wavefunction at y = 0. From the explicit expressions (6.9), (6.10) we find that it is governed by the equation (6. I I )
with positive coefficients m, and ul. Inserting a simple power ansatz $(?I) :- ~3” yields a quadratic equation for the exponent with the solutions (6.12) Since iz, 1 0 and n, < 0, we have a regular solution and a singular one. The boundary condition at the origin requires the absence of the singular solution. If, however, the potential would be less repulsive for y + 0, the behaviour at the origin were determined by the mass term alone, which does not give a well-behaved solution. An exactly solvable model demonstrating this effect is presented in Appendix C. The actual numerical solution of the collective SchrBdinger equation can thus be done without principal difficulties using a basis of modified Laguerre polynomials. The results are shown in Table IV together with the solutions of the full Hill-Wheeler equation 1161. Also in this case we find a quantitative agreement between exact and approximate solutions. Since the integral kernel is more peaked for the heavier nuclei, one would expect a closer agreement in this case. This is not borne out by the explicit results (see the
324
W.
BAUHOFF
TABLE IV The Energies (in MeV) of the Ground and the First Excited State of the Monopole
EO (exact) & (approx) El (exact) El (approx) AE (exact) AE (approx)
-32.91 -33.15 -5.34 -6.45 27.57 26.70
- 140.34 -141.86 - 108.82 -111.82 31.52 30.04
Note. The exact results are taken from [16], whereas the approximate the collective Schradinger equation (6.8).
Modes
-403.27 -403.01 -374.91 -374.23 28.36 28.78 ones are obtained from
results for 4He and IsO). The reason for this is that the wave-functions are also more sharply peaked for heavier nudei [16]. Since the quality of the local approximation depends on the ratio of the range of the kernel and that of the wave-function, the numerical results are intelligible. This shows that it is dangerous to judge on the validity of the local approximation alone from the form of the integral kernel. If we compare our approach with an adiabatic time-dependent Hartree-Fock calculation [23] using the same interaction, we find a difference in the potential and especially in the mass for the collective motion. Consequently, the excitation energies differ by about 5 MeV for la0 and 40Ca. Since the monopole mode has not been found experimentally in these nuclei, the correctness of either procedures cannot be judged from experiment.
7.
CONCLUSIONS
In this paper, we have investigated the approximation of the full Hill-Wheeler equation of the GCM by a local collective SchrGdinger equation. Explicit expressions for the potential and mass parameter were given which are based on a symmetrized moment expansion of the integral kernel. The validity of this approximation has been investigated for several cases where the exact solutions are known, either analytically or numerically. A close correspondence has been obtained in all cases already for a comparatively large width of the integral kernel in which case the approximation is thought to be questionable. Apart from the investigation of monopole modes in spherical nuclei, the examples treated have no great physical relevance. They were chosen for simplicity. Because of the encouraging results, one may now turn to the application to cases of greater physical interest. We mention here only three examples for such an application: The first is the calculation of potential and mass for the fission of heavy nuclei. The generator coordinate to be used there is the distance between the two fragments 124).
COLLECTIVE
POTENTIAL
AND MASS FROM CCM
325
The other example is the rotation-vibration coupling model [25] for deformed nuclei. Here the formalism has to be extended to several generator coordinates which offers no principal difficulties. The variables to be used are the /I and y deformation of the nucleus and the three Euler angles specifying its orientation. The computation of the (multidimensional) potential energy surface would establish a correspondence to shellmodel calculations [2]. Finally, we mention the application to the resonating-group method for scattering of composite particles [19]. The generator coordinate is the distance of the two nuclei. In this case, the norm operator N can have eigenvalues equal to zero. Therefore, the transformation (2.10) can only be performed in the subspace of non-vanishing eigenvalues. The moment expansion will thus lead to the modified orthogonality condition model [26] with a potential which differs from the double-folding potential because of the intercluster antisymmetrization. Another interesting topic is the generalization to time-dependent phenomena. This allows a straightforward transition to the classical limit together with a microscopic basis for the description of dissipation in many-particle systems [27].
APPENDIX
A
ln this appendix, we present the exact calculation of the transformed Hamiltonian (2.10) for a Gaussian overlap (3.1) and an anharmonic interaction (4.1). The first step is the calculation of the inverse square root of the normalization kernel. At first we deal with a general form of the overlap and specialize to a Gaussian form below. Suppose we know the spectral decomposition of the operator N(cY, p) in the form
where fn(n) are eigenfunctions of N(oc, /3) to the eigenvalue qrL Then the inverse square root of N((Y, p) is given by
provided that all eigenvalues ~~ are strictly positive. The determination of the eigenvalues and belonging eigenfunction is particularly easy if the integral kernel depends only on the difference of its arguments 01- /I. The eigenvalue equation
- B>f(P) dP =77f(~) sWa
(A.3)
is then solved by the plane wavesfk(ol) = & as can be seen by a simple transformation of the integration variable. The eigenvalue is yk
=
IN(a)
ecika dn.
(A.4)
326
W.
BAUHOFF
Now we turn to the case of a Gaussian kernel (3.1). For this case the integral (A.4) can be performed easily and we find
The spectral decomposition
(A.]) then reads (A-6)
which is recognized as a well-known formula Gaussian. The inverse square root is
for the Fourier transformation
N-1/2(a, fi) = e-ika
@k’/S
&kS
of a
&,
which does not exist as a function. So care has to be taken in the calculation of K(o1, /3) that the integrations are performed in the correct order. If this is kept in mind the computation is straightforward but tedious, so we quote only the result: 64.8)
The potential
and the inertial parameter are given as double integrals:
By a Taylor expansion of the exponential, we find that the terms up to X4 coincide with the approximate expressions (4.3) (4.4) whereas higher terms are absent there.
APPENDIX
B
In this appendix we demonstrate that for the Lipkin model the solution of the HillWheeler equation with the basis functions (5.4) is equivalent to the exact diagonalization of the Hamiltonian (5.3). Because the Hamiltonian (5.3) commutes with the quasi-spin operator K2, the exact diagonalization can be performed in the space spanned by the states 1i) = K+i IO)
i = O,..., n
(B. 1)
since at most n particles can be moved to the upper level. On the other hand consider
327
COLLECTIVE POTENTIAL AND MASS FROM GCM
the generating states (5.4). If we expand the exponential in a Taylor series, the series will terminate at a finite number of terms because of the algebraic properties of K+ j a) = coP o[ exp (tan &+)I
Oi
(6.2)
n 1
= ~0s’~ c1C 7 tan (Yj i).
i=O ‘.
So the wave-function obtained in the generator coordinate method can be written COP 01tan oif(~~) da: ( i),
lY)=$f$J
(B.3)
i.e., as a superposition of states, which span the space for the exact solution. So the generator coordinate method performs the diagonalization of the Hamiltonian in the same state space as the exact solution and therefore is identical to it. APPENDIX C
In this appendix we demonstrate that a mass parameter, which is negative for some range of the coordinate, does not necessarily lead to a spectrum unbounded from below [17]. This will be done for a square well potential with a discontinuous mass, jumping from fm to -m. The most simple case is defined by the potential V(x)
= 00,
x < a, )
= - vl7 ,
a, < .Y < a, ,
= 0,
a2 c s,
(C.1)
and the mass parameter M(x) = -m, = m,
x < h, b :- x,
(C.2)
if we have b < a, . The boundary condition of a vanishing wave-function at x := a, excludes any influence of the negative mass for x < b. So we find only the usual spectrum of a square-well potential and no anomalous bound states with E < V. occur. The reason for this is the repulsive nature of the potential in the negative mass region. Therefore the simple example is of relevance to the discussion in Section 6.
ACKNOWLEDGMENT I am indebtedto Prof. K. Wildermuthfor usefuldiscussions.
328
W. BAUHOFF REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
14. 15.
16. 17. 18.
19. 20. 21. 22. 23. 24. 25. 26. 27.
C. W. WONG, Whys. Rep. 15 (1975), 283. A. BOHR AND B. R. MOTTEL~~N, “Nuclear Structure,” Vol. II, F. VILLARS, Ann. Rev. Nucl. Sci. 7 (1957), 185. D. L. HILL AND J. A. WHEELER, Phys. Rev. 89 (1953), 1102; J. J. Phys. Rev. 108 (1957), 311. D. M. BRINK AND A. WEIGUNY, Nucl. Phys. A 120 (1968), 58. B. BANERJEE AND D. M. BRINK, Z. Phys. 258 (1973), 46. A. K. KERMAN AND S. E. KOONIN, Phys. Ser. A 10 (1974), 118. H. FL~CARD AND D. VAUTHERIN, Phys. Lett. B 52 (1974), 399. B. GIRAUD AND B. GRAMMATICOS, Nucl. Phys. A 233 (1974), 373; T. FLIESSBACH, Z. Phys. A 272 (1975), 39. L. LATHOUWERS, Ann. Phys. 102 (1976), 347. P. G. REINHARD, Nucl. Phys. A 261 (1976), 291. G. HOLZWARTH, Nucl. Phys. A 207 (1973), 545. F. VILLARS, in “Many-Body Description of Nuclear Structure and
Benjamin, GRIFFIN
New
York,
1965.
AND J. A. WHEELER,
255 (1975), 141.
Reactions” (C. Bloch, Ed.), Academic Press, New York, 1966. H. J. LIPKIN, N. MESHKOW, AND A. J. GLICK, NucI. Phys. 62 (1965), 188. H. FLOCARD AND D. VAUTHERIN, Phys. Lett. B 55 (1975), 259; Nucl. Phys. A 264 (1976), 197. B. GIRAW AND B. GRAMMATICOS, Ann. Phys. 101 (1976), 670. J. CUGNON, Phys. Lett. B 57 (1975), 320. K. WILDERMUTH AND Y. C. TANG, “A Unified Theory of the Nucleus,” Vieweg, Braunschweig, 1977. P. G. REINHARD, Nucl, Phys. A 252 (1975), 120. J. DA PROVIDENCIA, J. N. URBANO, AND L. S. FERREIRA, Nucl. Phys. A 170 (1971), 129. M. BEINER, H. FLOCARD, N. VAN GIAI, AND P. QUENTIN, Nucl. Phys. A 238 (1975), 29. K. GOEKE AND B. CASTEL, Phys. Rev. C 19 (1979), 201. P. G. REINHARD AND K. GOEKE, Proc. 4th IAEA Symposium on Physics and Chemistry of Fission, Jiilich 1979, to be published. A. FAESSLER, W. GREINER, AND R. K. SHELINE, Nucl. Phys. 80 (1965), 417. S. SAITO, S. OKAI, R. TAMAGAKI, AND M. YASUNO, Progr. Theoret. Phys. 50 (1973), 1561. W. BAIJHOFF AND K. WILDERMUTH, .I. Phys. G. 6 (1980), 1211.
OF PHYSICS
130,
307-328 (1980)
Derivation of Collective Potential from the Generator Coordinate W.
and Mass Method*
BAUHOFF
Institut fir Theoretische Physik, Universitiit Tiibingen, Tiibingen D-7400, West Germany and 1. Institut fiir Experimentalphysik, Universitiit Hamburg, Hamburg D-2000, West Germany? Received December 18, 1979; revised June 25, 1980
The Hill-Wheeler equation of the generator coordinate method is approximated by a local collective Schradinger equation. General expressions for the potential and the mass parameter are obtained by a symmetrized moment expansion. The validity of the approximation is tested for several examples where the exact solution is known. These include the Gaussian overlap with harmonic and anharmonic interaction, the Lipkin model, and monopole resonances of spherical light nuclei. In all cases, surprisingly close agreement with the exact solution is found. Other possible applications of the formalism are indicated.
1. INTRODUCTION
Quantum mechanical many-particle systems often exhibit a collective behaviour for low-energy excitations. Examples for such a behaviour are the rotational and vibrational states of nuclei, the fission process and the scattering of heavy ions. The examples are taken from nuclear physics since we are mainly interested in applications to this field, but the statement applies also to other areas of physics. These collective phenomena are most economically described by the introduction of collective coordinates. A convenient way to do this is provided by the generator coordinate method (GCM) (for a review see, e.g., [l]). This method does not require the introduction of additional degrees of freedom [2]; it also avoids the difficult problem of constructing collective coordinates in terms of one-particle coordinates [3]. The method is based on a special ansatz for the many-particle wave-function. By inserting this ansatz into the microscopic SchrSdinger equation, an integral equation (the Hill-Wheeler equation) for the superposition amplitude is derived [4]. By a solution of this integral equation one gets information on the colIective behaviour of the system. A drawback of this method is the missing physical interpretation of the integral kernel. In order to make a connection to more phenomenological theories of collective motion [2] it is therefore tempting to approximate the Hill-Wheeler equation for the superposition amplitude by a Schriidinger-like equation with a collective potential * Work supported by the DFG. + Present address. 307 0003-4916/80/140307-22$05.00/O All
Copyright 0 1980 rights of reproduction
by Academic Press, Inc. in any form reserved.
308
W.
BAUHOFF
and a collective mass, which will in general depend on the collective coordinate. Indeed, there are several proposals for such an approximation in the literature [5-g]. In all cases, the small width of the Hamiltonian and wave-function overlap is used for justifying the approximation, which is usually based on some form of the moment expansion of the integral kernel. An extensive review of these approximations is given, e.g., by [I]. It has the merit of providing directly a Schrodinger equation, i.e., a quanta1 description, for the collective motion. So a difficult and ambiguous requantization of a classical equation is avoided which is necessary, e.g., in the BohrMottelson theory [2]. But due to the peculiar normalization condition for the superposition amplitude, it may not be interpreted as a true square-integrable wave-function [lo]. In fact, there are cases where it is a distribution and not a function at all [l 11. So it is not safe to derive an approximate Schrodinger equation for such a possibly singular object. In the next section, we derive therefore a local differential equation as an approximation for a transformed superposition amplitude which has a proper wave-function interpretation. The expansion we propose is a combination of the methods of [7, 81: We use the symmetrized moments of the integral kernels introduced in [7] instead of the usual moments used in [6]. In order to avoid the explicit calculation of the square root of the normalization kernel we employ the method of 181. Our final formulas for the collective potential and mass differ, however, from those of [8]. As discussed in detail at the end of the next section, the formalism of [S] does not lead to a unique value for the mass and is therefore not applicable in concrete calculations (in fact, it has never been used in practice). In actual calculations, the moment expansion has been used only in the special case of a Gaussian form of the overlap kernel [12, 131. Qualitative agreement with known exact results has been obtained. But the assumption of a Gaussian form restricts severely the applicability of the method since in some cases of physical interest it cannot be justified [14]. In the present paper, we try to overcome this difficulty. In order to test the validity of the approximation, we perform a number of tests for different models in the following: We compare the solution of the approximate Schrodinger equation with those of the full Hill-Wheeler equation which are either known analytically (Sections 3-5) or numerically (Section 6). The models we treat explicitly are: Gaussian overlap with harmonic (Section 3) and anharmonic (Section 4) interaction which are the standard test cases for such a study. In Section 5 we investigate the Lipkin model [15] where the influence of the particle number on the quality of the approximation can be studied easily. We compare our results also with those of the moment expansion of [13] where a Gaussian overlap has been assumed. We find a definite improvement in our approach in all cases considered. Giant monopole resonances of spherical Iight nuclei are considered in Section 6. This is probably the most interesting example from a physical point of view. We compare our results with a recent generator coordinate calculation using the Skyrme interaction [16]. As in this case the collective mass is not positive for all values of the coordinate, one may have problems in a variational solution of the collective Schrodinger equation [17, 181. We demonstrate that the expression for the collective mass we obtain does not lead to
COLLECTIVE
POTENTIAL
309
AND MASS FROM GCM
difficulties of this kind. This is in contrast to earlier applications of the moment expansion to this problem where a spectrum was obtained which is not bounded from below [9, 181. Finally, the results are summarized in Section 7. Some technical details not directly related to the main topic of the paper can be found in the appendices. 2. DERIVATION
OF COLLECTIVE
POTENTIAL
AND COLLECTIVE
MASS
The starting point of the GCM is the ansatz for the wave-function
(2.1)
~(s, = jr&r, R)f('k) c/r7
where .Y denotes all the one-particle coordinates and CYis the generator coordinate which we assume to be real. The extension of the ansatz (2.1) to several generator coordinates is straightforward, but will not be considered in this paper. The manyparticle basis functions ~(x, a) are assumed to be given. Usually they are taken as Slater determinants for simplicity. They can be chosen according to one’s physical intuition or determined, e.g., by a constrained Hartree-Fock calculation. Substituting the ansatz (2.1) into the many-particle Schrijdinger equation, formulated as a projection equation [19] (&,A 1H - E/41,\ = 0
leads to the Hill-Wheeler
The Hamiltonian
equation for the superposition
(2.2)
amplitudef(,u)
and overlap kernel are defined by
If the basis function ~(x, IX) are real the integral kernels H(cY, p) and N(cx, /I) are symmetric. Eigenfunctions of Eq. (2.3) satisfy the orthogonality condition:
Since this is not the usual orthogonality condition for a proper quantum mechanical wave-function, f(a) cannot be interpreted in such a manner. So the derivation of a collective Schrodinger equation forf(ol) will in general be a questionable approximation. Therefore it is useful to transform from the superposition amplitudef(n) to an
310
W.
BAUHOFF
object which can be considered as a collective wave-function, by the definition
[IO] (2.7)
From (2.6) we find immediately
the correct orthogonality
condition
for g(a):
gn(4 da=%rzn . sdX4
(2.8)
The dynamical equation for g(a) is deduced from (2.3) (2.9) with the integral kernel K(a, jl) = /N-‘f’(a,
a’) H(d,
B’) N-1f2(j?, ,8) da’ dfi’.
(2.10)
Since the normalization kernel N(ol, /I) is positive definite the square root N1j2 is well defined. Its inverse will also be defined if the Hill-Wheeler equation has no zero-energy solutions. If we exclude this exceptional case, the transformation fromf to g and back is a well-defined operation. The square root and its inverse can be calculated by a spectral decomposition (see Appendix A for such a calculation). But the approximation of (2.9) by a local differential equation does not require the explicit knowledge of the square root as will be demonstrated in the following. So the actual computation of N1f2 will not be needed. The integral kernels N, Hand K will be sharply peaked as a function of 01- /? and slowly varying as a function of o + ,8. This property will become more evident in the examples considered in the next sections. It allows the approximation of the kernels by local differential operators. We will explain this approximation for a general operator A(s /Q h aving the same properties as N, H and K and will specialize to (2.9) below. With the momentum operator (vl = 1)
one readily derives the following equation [7]: j-A@, @s)(/l) d/9 = j-e(i’2’Epu A(a - i/i?, a + 4s) e(i’2’8pa dfif(a).
If we expand the right-hand
s
(2.12)
side in powers of P, we find up to second order
da, k?ls)f(@ 43 = [A,(4
- t V’a2, A,(a)}
+ . ..]f(a).
(2.13)
COLLECTIVE
POTENTIAL
AND
MASS
FROM
311
GCM
where we have introduced the symmetrized moments of the integral kernel
and the notation
v,2, A,(4) = P,242(4 + 2PJ,(a)
P, -t A*(R) P,2.
Since (2.13) is independent of the form of the function5
(2.15)
we have the operator equality (2.16)
A = A, - ${P”, AZ; -; ‘.’ .
Because of the symmetry of A(cu, /3) the first-order moment A,(a) and all other odd moments will vanish identically. So the Hermiticity of the operator A is preserved term by term in the expansion (2.16). This is in contrast to other moment expansions proposed earlier [6]. Equation (2.16) will now be applied to the transformed Hill-Wheeler equation (2.9) and the definition (2.10). In order to avoid the explicit calculation of the square root we rewrite the expansion for N1j2 in terms of moments of N following [S]: Defining N’ by N = N;i2(1 + N’) Nil”
(2.17)
the expansion (2.16) yields N’
,AJ;;‘P[-i(p2,
=
N,;
+
. ..I
N,-‘I’
(2.18)
and for N1jz we get N-112
By a binomial
=
(1
+
N’)-‘/2
,‘+J;‘P
=
&l/2(1
+
N’)-~‘9.
(2.19)
expansion we get finally using (2.18): (1
+
/7g’)-l/2
=
1 -
$N’+
. ..
=
] -
~&-1/z[+{f’2,
N,}
T
. ..I
N;:?
+
....
(2.20)
It should be noted that Ni’2 is an ordinary square root in the sense of a function and not the square root of an operator. So its calculation is without any problem. The Eqs. (2.19)-(2.20) together with the expansion (2.16) applied to the Hamiltonian overlap H are now inserted into the Hill-Wheeler equation (2.9) and the definition (2.10).
Jf we keep only terms up to second order in P we get [N;l12(Ho - .k{P2, H2}) NilI
+ &N;‘/2{P2,
N2) N;3/2H0 (2.21)
312
W.
BAUHOFF
The terms on the left-hand side can be re-arranged to the form of a Schriidinger equation with a potential and an effective mass: (2.22)
For the effective mass we find (2.23)
and the collective potential is
In (2.24) all moments No, N2 , Ho and H, are functions of 01,and the primes denote differentiation with respect to CL The most important contribution to the potential is the first term Ho/No. All other terms contain second moments and hence are expected to be small. They can be considered as zero-point energy corrections to the potential energy surface [20]. The expression for the potential simplifies drastically if the norm kernel N(ol, ,8) depends only on the difference of its arguments a - ,L?.In this case, all moments of N will be constant as can be seen from the definition (2.14). So all derivatives of No and NZ vanish, and the collective potential is simply 1 H;(a) v(“=$j#+8Y#-&q-4--.
1
N2(4
H,“(4 No2t4
This expression for the collective potential will be used in the next three sections. The expression for the collective mass and potential is formally identical to that given in [S]. A difference lies, however, in the definition of the quantities No , N2 , Ho and H2 which appear in the formulas. In our case, they are the symmetrized moments defined in Eq. (2.14). Instead of this, they are given in [8] by
and similarly for Ho and H2 . The important point is that an equation is given only for the derivative of N2 . So N2 is determined only up to a constant which cannot be fixed. Therefore, the collective mass contains undetermined constants in the formalism of [8). In fact, the formalism has never been applied in an actual calculation, at least to our knowledge.
COLLECTlVE
3. HARMONIC
POTENTIAL
INTERACTION
AND MASS FROM GCM WITH GAUSSIAN
313
OVERLAP
As the first test for the general investigations of the preceding section we treat the harmonic oscillator with a Gaussian overlap. In this case the Hill-Wheeler equation can be solved exactly [4], so we can easily study the validity of our approximations. The model is defined by the overlap kernel qoc, p> = e-(G3)~/A” and the Hamiltonian
(3.1)
kernel
For the extraction of the collective potential and the collective mass we have to calculate the lowest symmetrized moments of these kernels. By using standard integrals for Gaussians we find
(3.3)
Since the overlap depends only on the difference of the arguments we can use Eq. (2.25) for the calculation of the potential. Substituting (3.3) into (2.25) yields V(a) = E, - $Gmw2h2f $~co~cY~.
Similarly
(3.4)
we find from Eq. (2.23) the effective mass M(a) ,= m.
(3.5)
So in this case the collective Schrodinger equation (2.22) is simply the equation of a harmonic oscillator. Its eigenvalues are well known to be E, = E, - ,~~rnw2X2 + (n + Q)h.
(3.6)
This expression for the eigenvalues is identical to the exact solution of the HillWheeler equation [4, 111. This is a definite advantage compared to the moment expansions proposed earlier [6] which yield the eigenvalues only approximately for small X (i.e., a small non-locality of the overlap kernel). Our treatment is correct
314
W. BAUHOFF
even for X2 > 4/mw when the superposition amplitude f (a) is no longer a function but a distribution [ll]. This is due to our transformation to g(a) which is a wellbehaved function also in this case. For the eigenfunctions we find
where H, denote Hermite polynomials. This result also coincides with the exact solution. The belonging &(ol) can be calculated from (3.7) by means of a Gauss transform [I 11.
4. ANHARMONIC INTERACTION WITH GAUSSIAN OVERLAP
If the collective motion is no longer restricted to small deformations, a quadratic expansion for the Hamiltonian is not adequate any more. In this case a more general ansatz instead of (3.2) is in order:
with slowly varying functions W and U, whereas N(oI, /3) is still given by (3.1). This form of the Hamiltonian is of interest since for it the transformed kernel K (see Eq. (2.10)) can be calculated exactly and is in fact a local differential operator. Details of the calculation can be found in Appendix A. If we apply the formalism of Section 2, we find for the moments H,(a) = X7w [W(E) - QhW(a)], f&(a)
(4.2)
= ;x3%-1/2[W(a) - gm.qa)].
This leads to the collective potential V(a) = W(a) - &w(a)
- pPwy01) -
&X4uya)
(4.3)
and the effective mass 1 w4
-
= a hW(a).
An expansion of the exact expression for the potential and the mass in powers of the non-locality range h shows that the approximate quantities (4.3), (4.4) are correct up to terms with X4. This also explains why the exact solutions were obtained for the harmonic interaction in the previous section: In this case the exact Hamiltonian I contains only terms with h4 at most.
COLLECTlVE
POTENTIAL
315
AND MASS FROM GCM
5. THE LIPKIN
MODEL
One of the most popular testing grounds for theories of collective motion is the Lipkin model [lS] since it is exactly solvable for all particle numbers. The model contains two n-fold degenerate levels with unperturbed energies & and -&. The interaction Hamiltonian is given by H T=&Ec oa;?;,a,, + V c
(5.1)
a+ m.a+ m,oam,-oam-o ,
where CJdenotes the two levels i 1 and m, m’ indicate the degeneracies. By the introduction of quasi-spin operators
(5.2)
the Hamiltonian
can be rewritten in the form
H = 4, + V(K+2+
(5.3)
Ks2),
which obviously commutes with K” and hence mixes only the states belonging to a multiplet with a fixed quantum number K. This form of the Hamiltonian is the reason for the possibility of obtaining exact solutions. The unperturbed ground state I 0) has evidently all particles occupying the lower level. A convenient choice of (normalized) generating functions 1OZ)is [13]: \ CY>= coP DL. exp(tan CL. K+) / O>
(5.4)
with 31varying from -an to &T. By the Thouless theorem 1 CX)is the most general Slater determinant which is not orthogonal to I 0). The physical interpretation of the generator coordinate 01is clear from the computation of the particle number in the lower and upper level: (a I C a;t,, a,, / a) = n sin” c(,
m
(5.5) (a 1c uf-, a,-, / a) = n COG a. m With the generating function (5.4) the normalization can be calculated easily:
and Hamiltonian
overlap kernel
N(% p>= cosya - p), H(a, P>= +n{cos(a + /3)cosya - p> + txbs”-“(a
- Ml
+ sin2(a + /3)) - COS~(CY - p)]>,
(5.6)
316
W.
BAUHOFF
where x = V(1 - n)/~ denotes the strength of the two-particle interaction. From (5.6) one recognizes the sharply peaked nature of the overlap and Hamiltonian kernel for large particle numbers. The approximation of the Hill-Wheeler equation belonging to (5.6) by a local Schrodinger equation has been considered before [13], but in the calculation of the moments a Gaussian form of the overlap kernel has been assumed which leads to sizable errors for small particle numbers. The solution of the full HillWheeler equation is identical to the exact (algebraic) solution of the model. This seems to be known to other people [21], but as we do not know of a published proof for it, we present a proof in Appendix B. Introducing the notation
(5.7)
the symmetrized moments of N(cx, /3) and H(u., /3) can be written: (5.8) f&(a) = -&n{Zo(n H,(a) = -&n{Z2(n
- 1) cos 2a + &x[Zo(n - 2)(1 + sin2 2~x) - 1&r)]}, - 1) cos ~LX+ &x[Z,(n - 2)(1 + sin2 201) - I&z)]>.
The integration in (5.7) can be performed analytically, but the results are quite long, so we do not quote them here. IJsing (2.23), (2.25) the mass parameter and the potential are readily obtained from (5.8): 1 1 M(ol)=2En
Mn - 1) m I
- m Zo2(n)
Zo(n - 1) cos .&
+ 1x Zdn- 2) Z,(n)- m Al@- 2) 2
V(R)=+1
Z”(n) Al(n) Lb - 1) +
[l + sin2 2011 , I (5.9)
Z2(4
an - 1)- tun - 1)m) cos2cr ~nYn>
+ lo(n) Zo(n - 2) - z”yn) + z&2 - 2) &(n) - 2&(n) m &‘Yn)
- 2) II*
In Figs. 1 and 2 we show the mass and potential for n = 4 and different strength of the two-particle interaction, x. For comparison, the results for a Gaussian overlap approximation [ 13] are also shown. We find a sizable deviation for the mass parameter,
COLLECTIVE
POTENTIAL
AND
MASS
FROM
317
GCM
‘OI -10' -n/2
0
n/2
FIG. 1 The potential V(a) and the inverse mass parameter h/M(a) of the Lipkin model for n = 4 and x ; 0.6. Both quantities are given in units of E. The dashed lines indicate the results for a Gaussian overlap approximation (131.
0
ni2
FIG. 2. Same as Fig. I for n = 4 and ,y = 1.8.
especially for a strong two-particle interaction. For higher particle numbers the differences become smaller. For x < 1, the mass parameter is no longer positive definite. The negative contribution is, however, much smaller than for a Gaussian overlap assumption.
318
W. BAUHOFF TABLE I The Energies of the Ground and First Excited State of the Lipkin Model for Various Particle Numbers n and Interaction Strengths x n
x
El?
El
4
0.2
-2.013 -2.017 -2.043
- 1.020 - 1.027 -1.033
4
1.0
-2.309 -2.315 - 2.269
-1.414 -1.388 - 1.476
4
5.0
-6.110 -6.043 -5.813
-5.099 -4.943 -5.201
14
0.37
-7.038 -7.040 -7.041
-6.096 -6.108 -6.111
14
0.93
-7.270 -7.275 -7.259
-6.622 -6.621 -6.626
20
0.95
- 10.292 - 10.296 - 10.282
-9.710 -9.710 -9.709
Note. All energies are given in units of E. For all parameter values, the first line gives the exact energy taken from [15] or (211. The second line gives the approximate result of the local Schrijdinger equation (2.22). In the third line, we give the result for the approach of [13].
In the calculation of the eigenvalues, no difficulty arose from this fact. We will come back to this point in the next section and in Appendix C. The eigenvalue spectrum for the Schrbdinger equation with the parameters (5.9) can be obtained quite simply by a Fourier ansatz for the wave-function. In Table I we quote some results for different interaction strengths x. The exact solutions for comparison are taken from [15, and 211. We find a remarkable agreement between exact and approximate solutions already for small particle numbers, where the assumption of a narrow overlap is not necessarily fulfilled (for n = 4, the width of the normalization kernel is 0.36 r which is one-third of the whole range of the variable). The agreement is found for all values of the interaction strength, so our approximation can easily handle the “phase transition” at x = 1. For comparison, we have also listed the results for the approach of [13]. In this paper, only figures were given showing the eigenvalues as functions of x and n. Therefore, we have redone the calculations to obtain numbers for the eigenvalues. As can be seen from the table, the results for the approach of [I 31 differ from ours especially for small n and large x. This could
COLLECTIVE
POTENTIAL TABLE
AND MASS FROM GCM
319
I1
The Complete Spectrum of the Lipkin Model for n = 8 and x = 0.88 @V/r = 1) from the Exact Solution (Left Column), the Approximate Calculation according to Eq. (2.22) (Middle Column) and according to [13] (Right Column) Exact
Eq. (2.22)
Ref. 1131
4.233 3.467 2.403 1.228 0 -1.228
3.646 2.888 1.912 0.941 -0.164 -1.301
3.157 2.590 I .707 0.718 -0.338 -- I .424
-2.403
-2.421
-2.492
-3.467
-3.463
-3.489
-4.233
-4.239
-4.219
have been expected from Figs. 1 and 2. In all cases considered, our formalism leads to closer agreement with the exact results. In Table II we give the complete spectrum obtained for n = 8 and x = 0.88 (n V/c = I ). This shows an increasing discrepancy for the higher eigenvalues, but stiI1 a qualitative agreement is found. Also for the higher eigenvalues our results compare favourable with those of [ 131. Especially the symmetry between positive and negative eigenvalues with is not built in in our approach is approximately reproduced. For larger particle numbers, we expect a closer agreement also for the higher eigenvalues. But this requires such a large number of basis functions in the variational ansatz that we did not perform such a calculation.
6. MONOPOLE
RESONANCE OF SPHERICAL
LIGHT
NUCLEI
As the last and physically most interesting example we treat the isoscalar monopole vibrations of nuclei (breathing mode). For simplicity we consider only spherical, closed-shell nuclei of low mass, i.e., 4He, 160 and 4oCa. The generating functions are Slater determinants of harmonic oscillator functions with the oscillator constant /3 = (m~/yZ)“~ as generator coordinate. For the nucleon-nucleon interaction, the Skyrme interaction [22] is adopted. With this interaction the Hill-Wheeler equation has been solved numerically [16] so that we have again an exact solution to compare with. The normalization kernel is for this choice of the basis functions
320
W.
and the Hamiltonian
BAUHOFF
kernel assumes the form
where we have introduced the abbreviation 6 = ($(f
+ /P)““.
(6.3)
The coefficients t, ,..., t3 are the parameters of the Skyrme interaction. All calculations have been performed with the parametrization S III. The coefficients B,, ,..., B3 , C, D depend on the nucleus considered [16]. For completeness they are given explicitly in Table III. We have accounted for the c.m. motion by subtracting the direct term of the c.m. kinetic energy. For the calculation of the various moments one has to observe that the variables 01, /3 are defined on the interval (0, co). Keeping this in mind, we find for the moments of the norm kernel N,(a) = 401 . n, , (6.4)
iV2(a) = 1601~* n2 with the (constant) integrals
Similarly,
the moments of the Hamiltonian ~,(a)
= 4c&;’
can be written
+ 4a4h$’ + 4cZ’hf) + 4a’ht’
+ 4a3hf’,
~,(a) = 16a5h;’ + 16cfhf’ + 16a*h;’ + 16d’ht’ TABLE
+ 160r5h$’
III
The Various Coefficients appearing in the Definition (6.2) of the Hamiltonian Overlap for the Various Nuclei
BO 4 & B3 C D
4He
I40
1 1 0 1
3114 3514 4 116/9 13514 12
912 2
%a 1945/64 2625164 5512 2060127 117 40
COLLECTIVE
POTENTIAL
AND
MASS
FROM
321
GCM
with the integrals
All the integrals (6.5), (6.7) can be calculated numerically without any difficulty. Since the norm kernel depends not only on the difference of the arguments, the full expression (2.24) has to be used for the calculation of the collective potential. With the explicit form (6.4), (6.6) of the moments the actual calculation is, however, straightforward. The explicit result will be given as function of the variable y = l/a. It has a more direct physical interpretation because it is proportional to the root mean square radius of the nucleus. In terms of this variable, the collective Schrodinger equation reads (6.8)
with the explicit form of the potential 5* +Z
no
1
(6.9) and of the mass parameter -=
h(2) 2 no I (6. IO)
322
W. BAUHOFF 160
r y Ifml
FIG. 3. The collective potential for monopole
vibrations of lo0 (see Eq. (6.9)).
6
FIG. 4. The collective mass parameter for monopole vibrations of .W in units of the nucleon mass (see Eq. (6.10)).
COLLECTIVE
POTENTIAL
AND
MASS
FROM
GCM
323
For 160 the potential is shown in Fig. 3. It has the form expected on physical grounds: For small radii, we have an extremely strong hard-core corresponding to the large compressibility of nuclear matter. It is determined by the three-body part of the Skyrme force. Then we find a minimum which indicates the equilibrium position. For larger radii, the kinetic energy contribution leads to a slightly repulsive part whereas asymptotically the potential vanishes. For 4He and 40Ca the potential has the same shape with a minimum of about -50 MeV at y = 1.35 fm for IHe and of about -420 MeV at y ==- 1.9 fm for 40Ca. The mass parameter for 160 is shown in Fig. 4. The most prominent feature of this graph is the existence of a pole around y = 1.35 fm. For smaller radii, the mass parameter is negative. In the asymptotic region y - co, the mass approaches a constant value which is independent of the nucleon-nucleon interaction (compare (6.10) with the definition of the integrals (6.7)). So the form of the mass parameter is quite different from the cranking mass, which increases monotonically with y [9]. For the other nuclei, the collective mass has a similar form with the pole at y == 1.2 fm for *He and at y == 1.5 fm for 4oCa. Since the collective kinetic energy is no longer positive definite, one can have troubles with a variational solution of the collective SchrGdinger equation [17]. But in our case, the mass parameter is negative only in a region where the potential is strongly repulsive. Therefore we do not get any problems in a variational solution. This can be seen from an investigation of the asymptotic behaviour of the wavefunction at y = 0. From the explicit expressions (6.9), (6.10) we find that it is governed by the equation (6. I I )
with positive coefficients m, and ul. Inserting a simple power ansatz $(?I) :- ~3” yields a quadratic equation for the exponent with the solutions (6.12) Since iz, 1 0 and n, < 0, we have a regular solution and a singular one. The boundary condition at the origin requires the absence of the singular solution. If, however, the potential would be less repulsive for y + 0, the behaviour at the origin were determined by the mass term alone, which does not give a well-behaved solution. An exactly solvable model demonstrating this effect is presented in Appendix C. The actual numerical solution of the collective SchrBdinger equation can thus be done without principal difficulties using a basis of modified Laguerre polynomials. The results are shown in Table IV together with the solutions of the full Hill-Wheeler equation 1161. Also in this case we find a quantitative agreement between exact and approximate solutions. Since the integral kernel is more peaked for the heavier nuclei, one would expect a closer agreement in this case. This is not borne out by the explicit results (see the
324
W.
BAUHOFF
TABLE IV The Energies (in MeV) of the Ground and the First Excited State of the Monopole
EO (exact) & (approx) El (exact) El (approx) AE (exact) AE (approx)
-32.91 -33.15 -5.34 -6.45 27.57 26.70
- 140.34 -141.86 - 108.82 -111.82 31.52 30.04
Note. The exact results are taken from [16], whereas the approximate the collective Schradinger equation (6.8).
Modes
-403.27 -403.01 -374.91 -374.23 28.36 28.78 ones are obtained from
results for 4He and IsO). The reason for this is that the wave-functions are also more sharply peaked for heavier nudei [16]. Since the quality of the local approximation depends on the ratio of the range of the kernel and that of the wave-function, the numerical results are intelligible. This shows that it is dangerous to judge on the validity of the local approximation alone from the form of the integral kernel. If we compare our approach with an adiabatic time-dependent Hartree-Fock calculation [23] using the same interaction, we find a difference in the potential and especially in the mass for the collective motion. Consequently, the excitation energies differ by about 5 MeV for la0 and 40Ca. Since the monopole mode has not been found experimentally in these nuclei, the correctness of either procedures cannot be judged from experiment.
7.
CONCLUSIONS
In this paper, we have investigated the approximation of the full Hill-Wheeler equation of the GCM by a local collective SchrGdinger equation. Explicit expressions for the potential and mass parameter were given which are based on a symmetrized moment expansion of the integral kernel. The validity of this approximation has been investigated for several cases where the exact solutions are known, either analytically or numerically. A close correspondence has been obtained in all cases already for a comparatively large width of the integral kernel in which case the approximation is thought to be questionable. Apart from the investigation of monopole modes in spherical nuclei, the examples treated have no great physical relevance. They were chosen for simplicity. Because of the encouraging results, one may now turn to the application to cases of greater physical interest. We mention here only three examples for such an application: The first is the calculation of potential and mass for the fission of heavy nuclei. The generator coordinate to be used there is the distance between the two fragments 124).
COLLECTIVE
POTENTIAL
AND MASS FROM CCM
325
The other example is the rotation-vibration coupling model [25] for deformed nuclei. Here the formalism has to be extended to several generator coordinates which offers no principal difficulties. The variables to be used are the /I and y deformation of the nucleus and the three Euler angles specifying its orientation. The computation of the (multidimensional) potential energy surface would establish a correspondence to shellmodel calculations [2]. Finally, we mention the application to the resonating-group method for scattering of composite particles [19]. The generator coordinate is the distance of the two nuclei. In this case, the norm operator N can have eigenvalues equal to zero. Therefore, the transformation (2.10) can only be performed in the subspace of non-vanishing eigenvalues. The moment expansion will thus lead to the modified orthogonality condition model [26] with a potential which differs from the double-folding potential because of the intercluster antisymmetrization. Another interesting topic is the generalization to time-dependent phenomena. This allows a straightforward transition to the classical limit together with a microscopic basis for the description of dissipation in many-particle systems [27].
APPENDIX
A
ln this appendix, we present the exact calculation of the transformed Hamiltonian (2.10) for a Gaussian overlap (3.1) and an anharmonic interaction (4.1). The first step is the calculation of the inverse square root of the normalization kernel. At first we deal with a general form of the overlap and specialize to a Gaussian form below. Suppose we know the spectral decomposition of the operator N(cY, p) in the form
where fn(n) are eigenfunctions of N(oc, /3) to the eigenvalue qrL Then the inverse square root of N((Y, p) is given by
provided that all eigenvalues ~~ are strictly positive. The determination of the eigenvalues and belonging eigenfunction is particularly easy if the integral kernel depends only on the difference of its arguments 01- /I. The eigenvalue equation
- B>f(P) dP =77f(~) sWa
(A.3)
is then solved by the plane wavesfk(ol) = & as can be seen by a simple transformation of the integration variable. The eigenvalue is yk
=
IN(a)
ecika dn.
(A.4)
326
W.
BAUHOFF
Now we turn to the case of a Gaussian kernel (3.1). For this case the integral (A.4) can be performed easily and we find
The spectral decomposition
(A.]) then reads (A-6)
which is recognized as a well-known formula Gaussian. The inverse square root is
for the Fourier transformation
N-1/2(a, fi) = e-ika
@k’/S
&kS
of a
&,
which does not exist as a function. So care has to be taken in the calculation of K(o1, /3) that the integrations are performed in the correct order. If this is kept in mind the computation is straightforward but tedious, so we quote only the result: 64.8)
The potential
and the inertial parameter are given as double integrals:
By a Taylor expansion of the exponential, we find that the terms up to X4 coincide with the approximate expressions (4.3) (4.4) whereas higher terms are absent there.
APPENDIX
B
In this appendix we demonstrate that for the Lipkin model the solution of the HillWheeler equation with the basis functions (5.4) is equivalent to the exact diagonalization of the Hamiltonian (5.3). Because the Hamiltonian (5.3) commutes with the quasi-spin operator K2, the exact diagonalization can be performed in the space spanned by the states 1i) = K+i IO)
i = O,..., n
(B. 1)
since at most n particles can be moved to the upper level. On the other hand consider
327
COLLECTIVE POTENTIAL AND MASS FROM GCM
the generating states (5.4). If we expand the exponential in a Taylor series, the series will terminate at a finite number of terms because of the algebraic properties of K+ j a) = coP o[ exp (tan &+)I
Oi
(6.2)
n 1
= ~0s’~ c1C 7 tan (Yj i).
i=O ‘.
So the wave-function obtained in the generator coordinate method can be written COP 01tan oif(~~) da: ( i),
lY)=$f$J
(B.3)
i.e., as a superposition of states, which span the space for the exact solution. So the generator coordinate method performs the diagonalization of the Hamiltonian in the same state space as the exact solution and therefore is identical to it. APPENDIX C
In this appendix we demonstrate that a mass parameter, which is negative for some range of the coordinate, does not necessarily lead to a spectrum unbounded from below [17]. This will be done for a square well potential with a discontinuous mass, jumping from fm to -m. The most simple case is defined by the potential V(x)
= 00,
x < a, )
= - vl7 ,
a, < .Y < a, ,
= 0,
a2 c s,
(C.1)
and the mass parameter M(x) = -m, = m,
x < h, b :- x,
(C.2)
if we have b < a, . The boundary condition of a vanishing wave-function at x := a, excludes any influence of the negative mass for x < b. So we find only the usual spectrum of a square-well potential and no anomalous bound states with E < V. occur. The reason for this is the repulsive nature of the potential in the negative mass region. Therefore the simple example is of relevance to the discussion in Section 6.
ACKNOWLEDGMENT I am indebtedto Prof. K. Wildermuthfor usefuldiscussions.
328
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