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On the Brueckner theory of pairing in semi-infinite nuclear matter beyond the local density approximation
11 May 1995
PHYSICS
ELSEVIER
LETTERS
B
Physics Letters B 350 (1995) 135-140
On the Brueckner theory of pairing in semi-infinite nuclear matter beyond the local density approximation M. Baldo a, U. Lombard0 a, E.E. Saperstein b, M.V. Zverev ’ a INFN, Sezione di Catania, Corso Italia 57, 1-95129 Catania, Italy ’ Kurchatov Institute, 123182 Moscow, Russia c MEPI, I1 5409 Moscow, Russia Received
12 December
1994; revised manuscript
received 13 March 1995
Editor: C. Mahaux
Abstract
The problem of pairing in semi-infinite nuclear matter is investigated within a Bmeckner type approach beyond the local density approximation (LDA). The propagator of two non-interacting particles in the semi-infinite potential well is calculated numerically for the Saxon-Woods potential. The equation for the effective pairing interaction is solved for a bare interaction of the &form with the momentum cutoff which correctly reproduces the NV-scattering data in the low-energy limit. Results show significant deviations from the LDA predictions in the surface region.
Results of the nuclear matter theory are usually applied to finite nuclei within the local density approximation (LDA). This approximation works well in general but it fails in some cases. One of them is the well known problem of calculating the scalar-isoscalar component Fc of the Landau-Migdal interaction amplitude which changes drastically in the surface vicinity. The Pomeranchuk’s stability condition [ 1,2] is violated for this amplitude in the surface region as far as the problem is considered within the LDA. The nuclear pairing problem seems to be another example in which the accuracy of the LDA can be questioned. The reason is the relevance of the nuclear surface for this problem, as several calculations for superfluid nuclei within the self-consistent finite Fermi systems theory have shown [ 3-51. Results of these calculations are in favour of a strong density dependence of the effective interaction with a dominant contribution from the nuclear surface. This strong density de0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)00340-l
pendence of pairing was used recently [ 61 for solving the old problem of the so-called staggering, the oddeven effect in the A-dependence of the r.m.s. charge radii measured by isotopic shifts. On the other hand, recent study of the pairing in nuclear matter within the Brueckner theory [ 71 also support the idea of a major role of the surface in nuclear pairing. Although these calculations indicate the existence of pairing in nuclear matter, the value of the pairing gap A at the saturation density pa turns out to be too small (- 0.2-0.3 MeV) in comparison with the one observed in heavy nuclei (N 1 MeV). The critical density pC at which the pairing in nuclear matter disappears turned out to be just slightly larger than pc, whereas the value of A ( p) grows rapidly as p decreases below po. These results are in agreement with other many-body approaches [ 81 and give evidences in favour of a strong density dependence of A and of an essential role of the surface for the pairing phenomenon in finite nu-
136
M. Boldo et al. /Physics
clei. In such a situation it appears of great interest to check if significant deviations from LDA are present in the vicinity of the nuclear surface. In this letter we present a method of solution of the Brueckner-type theory equations for the semi-infinite nuclear matter beyond the LDA. We applied this method to calculate the effective pairing interaction in the case of a bare interaction of the S-form which reproduces correctly the NN-scattering data in the low-energy limit. The gap equation in the coordinate representation can be written as follows [ 1,2] :
A(rl,rz,~)
x
=
G(r3,r5,;
Letters B 350 (1995) 135-140
and put (G”)AA~ = (G”),JAA,. Then, for the integral over E of the product we get
=
c
‘V(~lrr2,r3,rq;E,~,~‘)
+e’)GS(~4,r6,;
de’ x A(rs,rg9&‘)-dr3dr4dr5dr6, 2k
-8’) (1)
where V is the interaction block which is irreducible in the particle-particle channel, G, is the one-particle Green’s function in the superfluid system whereas G is the one without pairing effects taken into account. The total two-particleenergy E on the right side of this equation should be taken equal to E = 2,~ where p stands for the chemical potential of the system under consideration: ,U = m N -16 MeV for the nuclear matter, and ,X 2~ -8 MeV for finite nuclei. For a while we take the quantity E as a parameter. For brevity we omitted the spin variables in Eq. ( 1) . We consider the singlet (T = 1, S = 0) pairing, therefore V denotes the corresponding component of the interaction block. We shall assume that it is energy independent. Then the gap A is also energy independent and the product of two Green’s functions in Eq. ( 1) can be integrated over e’. We concentrate mainly on the geometrical peculiarities of the semi-infinite system and do not consider the self-consistency aspects, renormalization procedure, etc. Therefore we consider G and GS in Eq. ( 1) as the quasiparticle Green’s functions in the fixed average field. Let us expand now Green’s functions G, GS in Eq. ( 1) into the basis of the functions PA(~) (with energies EA) which diagonalize G. These functions also approximately diagonalize the functions G”. Small non-diagonal terms of ( GS) AA,appear only in the equation for A itself but not in the equation for the effective interaction which will be considered below. Therefore, for the sake of simplicity, we neglect them
A3;*‘(E)qDA(rl)40;(r3)(0A,(r2)(P;,(r4),
AA’ (2)
where A;,,(E)
J
GGS,
=
J
$G”(f
+e)G$(;
-s).
(3)
As it is well known, the sum of (2) over (AA’) diverges. This divergence is eliminated in Eq. ( 1) by matrix elements YAA’ of the interaction ‘v. But since AIA; the realistic NN-interaction is short-range the convergence is very slow. Therefore, both in the phenomenological calculations based on the FFS theory [ 3,4] ant1 in the microscopic calculations based on the Brueckner theory [ 71, it is convenient to use a renormalization procedure in Eq. ( 1) by splitting up the sum of Eq. (2) over (AA’) into two parts: AS = A; + A’,
(4)
where the A& term contains only states close to the Fermi level (the “model space”), whereas all the reset is included into A’. In the latter one, the inequality I&A,+ EA, - 2~1 >> A is fulfilled and therefore the dilfference between the Green’s functions GS and G can be neglected (by this reason, we omitted the upperscript “s” in A’). Then we have A;,,(E)
=
1 - nA - nA1 E - E~ - cAr ’
(5)
where nA, np are the quasiparticle occupation numbers (= 0,l) and Eq. ( 1) can be written as follows: A = Ve),ffA;A, where the effective interaction equation: Vefi = V + VA’&.
(6) Vex obeys the following
(7)
Here we shall concentrate on the latter equation and develop the method for solving this equation for semiinfinite nuclear matter within the Brueckner approiach.
137
M. Bald0 et at /Physics Letters B 350 (1995) 135-140
In this case the interaction block V in Eq. (7) is approximated by the bare AN-interaction potential p because all the ladder diagrams are already summed up in this equation. In this paper we define the model space in such a way that the single particle energies appearing in the propagator of Eq. (5) are restricted only to the positive ones. In this case we have no = n,~ = 0. We limit ourselves to the simplest a-interaction:
where (Y,CY’= +, -. This integral is strongly divergent. In the case of a realistic interaction this divergence in Eq. (7) is removed by the the momentum dependence of the free interaction. Here we are forced to introduce a cut-off k,, in the integral “by hand”. In order to connect the cut-off momentum with physical parameters let us consider Elq. (7) in the free space. In a symbolic form, it reads ~=~+~A”~,
vo = A6(rt - r2)S(r2 - t3)8(r3
-
l-4).
(8)
Dealing with the semi-infinite system, it is natural to use a mixed representation, the coordinate one in the x-direction which is perpendicular to the boundary surface plane ( (y, z )-plane, or s-plane) and the momentum one in the s-plane. In this case, the quasiparticle wave functions can be written as 4p~(r) = e’J’l’y,,( X) , where pI stands for the 2-dimensional momentum vectors in the s-plane. It is obvious that in the s-plane the total momentum is conserved, that is we have PI = pi = 0 for the pairing problem. In the potential well which vanishes at large positive x and tends to -Ua at negative X, the complete and orthogonal system of functions for positive energies consists of the set of functions ( yp’, y; ) , where y:(x) 0: e’r” at large positive X, and y; (x) 0: e-“J* at large negative x (q = &T). For negative energies, for each value of the momentum p only one eigenfunction p,, exists which vanishes at large positive X. Let us substitute Eq, (8) into Eq. (7) as l&r and go to the momentum representation in the s-plane. Putting PI = 0, one sees that the effective interaction will be momentum independent in the perpendicular direction. In the x-direction it will depend only on two variables: xt = x2 = x and x3 = x4 = x’. After simple transformations one gets U.*(x,x’;E) +A
J
where A0 stands for the free 2-nucleon propagator which can be readily found from Eq. ( 10) by the substitution yP -+ exp (zp$x) , (and the same for p’). The interaction amplitude Ts is, in fact, the free T-matrix and depends only on the difference (x - x’) . It can be readily obtained from Eq. ( 11) and in the momentum representation has the well-known resonance structure [91: t
dxlA’(x,x,;E)V,ff(x,,x’;E),
(9)
where
(13) Thus, in this case the interaction strength A and the momentum cut-off k,,,, appear only in the combination of JZq.( 13) which defines the parameter ~0. This expression coincides with the low-energy limit of the free NIV T-matrix, thereby KO has the meaning of the opposite dineutron (or diproton) scattering length. It is known to be very small and may be neglected in Eq. ( 12) at energies of the order of E N - 16 MeV we are interested in. Then the Fourier transform of EQ. ( 12) may be readily found to be - x’;E)
=
-+,,(2dmlx
- (pr)2/2m
- k:/m’
t
10)
(14)
(15)
The convergence properties of the difference A’ are much better than those of the propagator A alone. However some slow (logarithmic) divergence remains. Therefore the cut-off parameter k,, (k: 5 A0
x E - p2/2m
- x’l),
where &I( z ) is the modified Bessel function. Let us now express Eq. (9) in terms of the known solution ofEq. (11). Wehave Veti = Ts + 7+(A’ - A’)V,fi.
Y~tx,Y~:cx,Y,“tx’)Y~~*tx’)
12)
where
p(x
= h&n - n’)
(11)
138
M. Bald0 et al. /Physics
Letters B 350 (1995) 135-140
E =
-16
MeV 1
7
3
‘% z
-0.6
G ti G
-
_~,6~_~~~~~__~~~~~_~_____~~~~___~~-
Fig. 1. The diagonal elements of the propagator A’(x,x, E) for the Saxon-Woods potential (solid line) and of the free propagator A”(x, x, E) (dashed line). The density p(x) of nucleons in that potential is drawn in the upper half of the figure.
k&,) should be used in the integral of Eq. (15) being, in fact, the only parameter of the bare interaction within this approximation. Let us now carry out the integration over kl in Rq. ( 10) explicitly. We have
A'(x,x';E)
=-z cJJ
km
aa’ u
dpdp’
1
x ln kiax/m+ (El + p2/2m+ (p’12/2m /El+p2/2m + (p’>2/2m [ x
y;(x,y;:(x)y;*(x’,y,“:^(x’).
(16)
The same formula is valid for the operator A’, with the plane waves as yP. We calculated the propagator A’ for the sharp boundary and for the Saxon-Woods (SW) potential with E = - 16 MeV. In the case of the sharp potential there are simple analytic solutions for y;. In the case of the SW potential, quasi-analytic solutions in terms of the hypergeometric functions are known [ 91. However, the use of general representations for them is very cumbersome. The direct numerical solution of the Schrodinger equation with the help of the Numerov’s method proves out to be much simpler and takes approximately the same computing time as one for the analytic solutions in the case of the sharp potential. In the numerical realization we have
used the realistic values of Ua = 50 MeV and d = 0.65 fm for the diffuseness parameter. The integration volume of (-L, L) and the step of h, were chosen as Z, = 16 fm and h, = 0.05 fm respectively. Then we have calculated the propagator ( 16) in the fixed grid (xi,XL) wherexi=-L+H,(i-1) withH,=4h,= 0.2 fm. Results for the SW potential are illustrated in Figs. 1, 2. Fig. 1 shows the diagonal elements of the propagator A’( n, X, E) together with those of the free propagator A’. For the negative values of x, the propagator A/(x, x, E) oscillates with a small amplitude. We have also drawn the quasiparticle density p(x) for one type of nucleons. Inside the matter it is a little higher than that of the normal nuclear matter because it corresponds to larger value of the Fermi energy ar = -Ua - ,u = 45 MeV (remind that we have used in calculations the value of p = -8 MeV for sake of closer relevance to the problem of pairing in finite nuclei). It should be noted that the density p(x) is not used in our calculations and is drawn just as an illustration. Fig. 2 shows the profile functions of A’( XI, x2, E) and A’(x~,x~,E) atxi =-8,0,8fm (fortheintervalof 1x1 --x2] < 4 fm). Inside the nuclear matter (xl = -8 fm) , the absolute value of the propagator A’ is significantly smaller than that of the free propagator A’. It vanishes more slowly at increasing values of 1x1 - x2 1 and oscillates. In the free space (xi = 8 fm) , the prop-
M. Bald0 et al. /Physics Letters B 350 (1995) 135-140
139
10-Z
10-a
10-d
10-S
10-B -10
-5
0 Y
5
Fig. 2.
The diagonal elements of the propagator A’(xl, x2. E) for the Saxon-Woods A”(xl, x2. E) (dashed line) versus x2 at the fixed values of XI = -8, 0, 8 fm.
agators A’ and A0 practically coincide. Nearby the boundary vicinity, an intermediate behavior is present and the function of A’( XI, x2, E) is asymmetrical. After calculating the propagators A’ and A0 we can solve the integral Eq. ( 15). We change the integration over dxl by summation over the grid values Xi. Then the integral equation is reduced to a set of linear equations which is solved by usual methods. We analysed also the problem in the local approximation. In the present approach, we fix the form of the potential well U(x) which is not related directly to the density distribution p(x). Therefore, instead of the usual LDA, it is natural to use a version of the local approximation with the effective interaction V$ (X - x/2, X + x/2) defined as a set of the functions Vi,+!“.(x) , calculated for the “nuclear matter” with the constant potential UO which is equal to U(X) . The propagator A”.,.( UO) can be readily calculated with the help of Eq. (16), just in the same manner as the ro agator Ao, by substituting the quantity q = F=-p /2m - Uo for p (and the same for p’). The obtained propagator can be substituted into Eq. (15) which yields the effective interaction vg
y(X)
10
[fml
=
potential
(solid
line)
and the free propagator
V,, (X - x/2, x + x/2)&. s
(17)
and the similar quantity I’“” . The results are displayed in Fig. 3 together with corresponding value r” for the bare T-matrix of IQ. (14). The latter one is, of course, X-independent. We see that the effective interaction changes in the surface region very smoothly. Its strength inside the matter (N -1.1 in the Landau normalization) differs significantly from the freevalue (N -2.8). The effective interaction in the free space tends to the free limit very slowly, much slower than that calculated within the local approximation. Similar results, with a greater deviations from the local approximation predictions, were obtained also for the case of the potential well with the sharp boundary. This result should be qualitatively valid also in the case of the realistic interaction. Such calculations with the separable representation of the Paris potential are now in progress. It should be noticed that the local density approximation appears actually reliable down to very low density. However, deviations are present, and for all phenomena involving the tail of the nuclear density they should be relevant. Indeed, in the pairing problem the states nearby the Fermi energy are involved, which often have slowly falling tails. On the
140
hi. Bald0 ef aI./Physics
-10
-5
Letters B 350 (I995) 135-140
0
5 x
10
15
[fml
Fig. 3. The interaction strength y(x) (solid line), the strength p(x) of the bare interaction (dotted line) am compared with the local approximation result yJ” (x) (dashed line) for the Saxon-Woods potential. The potential f.!(x) is given in the lower half of the figure in arbitrary units.
other hand, for other phenomenological data or phenomena, the deviations may not play any role, e.g. for the total correlation energy, and in every specific case a new analysis is necessary. The research described in this publication was made possible in part by Grant No. ML8000 from the International Science Foundation. The authors wish to acknowledge I? Schuck, S. Fayans, V. Khodel and S. Tolokonnikov for many fruitful discussions. Two of us (ES. and M.Z.) thank INFN, Sezione di Catania, and Catania University for hospitality during their stay in Catania. References [ I] A.A. Abrikosov, L.P Gorkov and LE. Dzyaloshinsky, Methods of quantum field theory in statistical physics (Prentice-Hall, New York, 1964).
I21 A.B. Migdal, Theory of finite Fermi systems and applications to atomic nuclei (Wiley, New York, 1967). [3] E.E. Saperstein and M.V. Zverev, Sov. J. Nucl. Phys. 39 (1984) 1390. [4] E.E. Saperstein and M.V. Zverev, Sov. J. Nucl. Phys. 42 (1985) 1082. [51 S.A. Fayans, A.V. Smirnov and S.V. Tolokonnikov, Sov. J. Nucl. Phys. 48 (1988) 1661. [6] S.A. Fayans, S.V. Tolokonnikov, E.L. Trykov and D. Zawischa, Phys. L&t. B 338 (1994) 1. 171 M. Baldo, J. Cugnon, A. Lejeune and U. Lombardo, Nucl. Phys. A 515 (1990) 409. [S] T.L. Aisworth, J. Wambach and D. Pines, Phys. Len. B 222 (1989) 173; J.M.C. Chen, J.W. Clark, E. Rrotscheck and R.A. Smith, Nucl. Phys. A 451 (1986) 509. [9] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non Relativistic Theory (Addison-Wesley, Reading, Mass., 1958).
PHYSICS
ELSEVIER
LETTERS
B
Physics Letters B 350 (1995) 135-140
On the Brueckner theory of pairing in semi-infinite nuclear matter beyond the local density approximation M. Baldo a, U. Lombard0 a, E.E. Saperstein b, M.V. Zverev ’ a INFN, Sezione di Catania, Corso Italia 57, 1-95129 Catania, Italy ’ Kurchatov Institute, 123182 Moscow, Russia c MEPI, I1 5409 Moscow, Russia Received
12 December
1994; revised manuscript
received 13 March 1995
Editor: C. Mahaux
Abstract
The problem of pairing in semi-infinite nuclear matter is investigated within a Bmeckner type approach beyond the local density approximation (LDA). The propagator of two non-interacting particles in the semi-infinite potential well is calculated numerically for the Saxon-Woods potential. The equation for the effective pairing interaction is solved for a bare interaction of the &form with the momentum cutoff which correctly reproduces the NV-scattering data in the low-energy limit. Results show significant deviations from the LDA predictions in the surface region.
Results of the nuclear matter theory are usually applied to finite nuclei within the local density approximation (LDA). This approximation works well in general but it fails in some cases. One of them is the well known problem of calculating the scalar-isoscalar component Fc of the Landau-Migdal interaction amplitude which changes drastically in the surface vicinity. The Pomeranchuk’s stability condition [ 1,2] is violated for this amplitude in the surface region as far as the problem is considered within the LDA. The nuclear pairing problem seems to be another example in which the accuracy of the LDA can be questioned. The reason is the relevance of the nuclear surface for this problem, as several calculations for superfluid nuclei within the self-consistent finite Fermi systems theory have shown [ 3-51. Results of these calculations are in favour of a strong density dependence of the effective interaction with a dominant contribution from the nuclear surface. This strong density de0370-2693/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO370-2693(95)00340-l
pendence of pairing was used recently [ 61 for solving the old problem of the so-called staggering, the oddeven effect in the A-dependence of the r.m.s. charge radii measured by isotopic shifts. On the other hand, recent study of the pairing in nuclear matter within the Brueckner theory [ 71 also support the idea of a major role of the surface in nuclear pairing. Although these calculations indicate the existence of pairing in nuclear matter, the value of the pairing gap A at the saturation density pa turns out to be too small (- 0.2-0.3 MeV) in comparison with the one observed in heavy nuclei (N 1 MeV). The critical density pC at which the pairing in nuclear matter disappears turned out to be just slightly larger than pc, whereas the value of A ( p) grows rapidly as p decreases below po. These results are in agreement with other many-body approaches [ 81 and give evidences in favour of a strong density dependence of A and of an essential role of the surface for the pairing phenomenon in finite nu-
136
M. Boldo et al. /Physics
clei. In such a situation it appears of great interest to check if significant deviations from LDA are present in the vicinity of the nuclear surface. In this letter we present a method of solution of the Brueckner-type theory equations for the semi-infinite nuclear matter beyond the LDA. We applied this method to calculate the effective pairing interaction in the case of a bare interaction of the S-form which reproduces correctly the NN-scattering data in the low-energy limit. The gap equation in the coordinate representation can be written as follows [ 1,2] :
A(rl,rz,~)
x
=
G(r3,r5,;
Letters B 350 (1995) 135-140
and put (G”)AA~ = (G”),JAA,. Then, for the integral over E of the product we get
=
c
‘V(~lrr2,r3,rq;E,~,~‘)
+e’)GS(~4,r6,;
de’ x A(rs,rg9&‘)-dr3dr4dr5dr6, 2k
-8’) (1)
where V is the interaction block which is irreducible in the particle-particle channel, G, is the one-particle Green’s function in the superfluid system whereas G is the one without pairing effects taken into account. The total two-particleenergy E on the right side of this equation should be taken equal to E = 2,~ where p stands for the chemical potential of the system under consideration: ,U = m N -16 MeV for the nuclear matter, and ,X 2~ -8 MeV for finite nuclei. For a while we take the quantity E as a parameter. For brevity we omitted the spin variables in Eq. ( 1) . We consider the singlet (T = 1, S = 0) pairing, therefore V denotes the corresponding component of the interaction block. We shall assume that it is energy independent. Then the gap A is also energy independent and the product of two Green’s functions in Eq. ( 1) can be integrated over e’. We concentrate mainly on the geometrical peculiarities of the semi-infinite system and do not consider the self-consistency aspects, renormalization procedure, etc. Therefore we consider G and GS in Eq. ( 1) as the quasiparticle Green’s functions in the fixed average field. Let us expand now Green’s functions G, GS in Eq. ( 1) into the basis of the functions PA(~) (with energies EA) which diagonalize G. These functions also approximately diagonalize the functions G”. Small non-diagonal terms of ( GS) AA,appear only in the equation for A itself but not in the equation for the effective interaction which will be considered below. Therefore, for the sake of simplicity, we neglect them
A3;*‘(E)qDA(rl)40;(r3)(0A,(r2)(P;,(r4),
AA’ (2)
where A;,,(E)
J
GGS,
=
J
$G”(f
+e)G$(;
-s).
(3)
As it is well known, the sum of (2) over (AA’) diverges. This divergence is eliminated in Eq. ( 1) by matrix elements YAA’ of the interaction ‘v. But since AIA; the realistic NN-interaction is short-range the convergence is very slow. Therefore, both in the phenomenological calculations based on the FFS theory [ 3,4] ant1 in the microscopic calculations based on the Brueckner theory [ 71, it is convenient to use a renormalization procedure in Eq. ( 1) by splitting up the sum of Eq. (2) over (AA’) into two parts: AS = A; + A’,
(4)
where the A& term contains only states close to the Fermi level (the “model space”), whereas all the reset is included into A’. In the latter one, the inequality I&A,+ EA, - 2~1 >> A is fulfilled and therefore the dilfference between the Green’s functions GS and G can be neglected (by this reason, we omitted the upperscript “s” in A’). Then we have A;,,(E)
=
1 - nA - nA1 E - E~ - cAr ’
(5)
where nA, np are the quasiparticle occupation numbers (= 0,l) and Eq. ( 1) can be written as follows: A = Ve),ffA;A, where the effective interaction equation: Vefi = V + VA’&.
(6) Vex obeys the following
(7)
Here we shall concentrate on the latter equation and develop the method for solving this equation for semiinfinite nuclear matter within the Brueckner approiach.
137
M. Bald0 et at /Physics Letters B 350 (1995) 135-140
In this case the interaction block V in Eq. (7) is approximated by the bare AN-interaction potential p because all the ladder diagrams are already summed up in this equation. In this paper we define the model space in such a way that the single particle energies appearing in the propagator of Eq. (5) are restricted only to the positive ones. In this case we have no = n,~ = 0. We limit ourselves to the simplest a-interaction:
where (Y,CY’= +, -. This integral is strongly divergent. In the case of a realistic interaction this divergence in Eq. (7) is removed by the the momentum dependence of the free interaction. Here we are forced to introduce a cut-off k,, in the integral “by hand”. In order to connect the cut-off momentum with physical parameters let us consider Elq. (7) in the free space. In a symbolic form, it reads ~=~+~A”~,
vo = A6(rt - r2)S(r2 - t3)8(r3
-
l-4).
(8)
Dealing with the semi-infinite system, it is natural to use a mixed representation, the coordinate one in the x-direction which is perpendicular to the boundary surface plane ( (y, z )-plane, or s-plane) and the momentum one in the s-plane. In this case, the quasiparticle wave functions can be written as 4p~(r) = e’J’l’y,,( X) , where pI stands for the 2-dimensional momentum vectors in the s-plane. It is obvious that in the s-plane the total momentum is conserved, that is we have PI = pi = 0 for the pairing problem. In the potential well which vanishes at large positive x and tends to -Ua at negative X, the complete and orthogonal system of functions for positive energies consists of the set of functions ( yp’, y; ) , where y:(x) 0: e’r” at large positive X, and y; (x) 0: e-“J* at large negative x (q = &T). For negative energies, for each value of the momentum p only one eigenfunction p,, exists which vanishes at large positive X. Let us substitute Eq, (8) into Eq. (7) as l&r and go to the momentum representation in the s-plane. Putting PI = 0, one sees that the effective interaction will be momentum independent in the perpendicular direction. In the x-direction it will depend only on two variables: xt = x2 = x and x3 = x4 = x’. After simple transformations one gets U.*(x,x’;E) +A
J
where A0 stands for the free 2-nucleon propagator which can be readily found from Eq. ( 10) by the substitution yP -+ exp (zp$x) , (and the same for p’). The interaction amplitude Ts is, in fact, the free T-matrix and depends only on the difference (x - x’) . It can be readily obtained from Eq. ( 11) and in the momentum representation has the well-known resonance structure [91: t
dxlA’(x,x,;E)V,ff(x,,x’;E),
(9)
where
(13) Thus, in this case the interaction strength A and the momentum cut-off k,,,, appear only in the combination of JZq.( 13) which defines the parameter ~0. This expression coincides with the low-energy limit of the free NIV T-matrix, thereby KO has the meaning of the opposite dineutron (or diproton) scattering length. It is known to be very small and may be neglected in Eq. ( 12) at energies of the order of E N - 16 MeV we are interested in. Then the Fourier transform of EQ. ( 12) may be readily found to be - x’;E)
=
-+,,(2dmlx
- (pr)2/2m
- k:/m’
t
10)
(14)
(15)
The convergence properties of the difference A’ are much better than those of the propagator A alone. However some slow (logarithmic) divergence remains. Therefore the cut-off parameter k,, (k: 5 A0
x E - p2/2m
- x’l),
where &I( z ) is the modified Bessel function. Let us now express Eq. (9) in terms of the known solution ofEq. (11). Wehave Veti = Ts + 7+(A’ - A’)V,fi.
Y~tx,Y~:cx,Y,“tx’)Y~~*tx’)
12)
where
p(x
= h&n - n’)
(11)
138
M. Bald0 et al. /Physics
Letters B 350 (1995) 135-140
E =
-16
MeV 1
7
3
‘% z
-0.6
G ti G
-
_~,6~_~~~~~__~~~~~_~_____~~~~___~~-
Fig. 1. The diagonal elements of the propagator A’(x,x, E) for the Saxon-Woods potential (solid line) and of the free propagator A”(x, x, E) (dashed line). The density p(x) of nucleons in that potential is drawn in the upper half of the figure.
k&,) should be used in the integral of Eq. (15) being, in fact, the only parameter of the bare interaction within this approximation. Let us now carry out the integration over kl in Rq. ( 10) explicitly. We have
A'(x,x';E)
=-z cJJ
km
aa’ u
dpdp’
1
x ln kiax/m+ (El + p2/2m+ (p’12/2m /El+p2/2m + (p’>2/2m [ x
y;(x,y;:(x)y;*(x’,y,“:^(x’).
(16)
The same formula is valid for the operator A’, with the plane waves as yP. We calculated the propagator A’ for the sharp boundary and for the Saxon-Woods (SW) potential with E = - 16 MeV. In the case of the sharp potential there are simple analytic solutions for y;. In the case of the SW potential, quasi-analytic solutions in terms of the hypergeometric functions are known [ 91. However, the use of general representations for them is very cumbersome. The direct numerical solution of the Schrodinger equation with the help of the Numerov’s method proves out to be much simpler and takes approximately the same computing time as one for the analytic solutions in the case of the sharp potential. In the numerical realization we have
used the realistic values of Ua = 50 MeV and d = 0.65 fm for the diffuseness parameter. The integration volume of (-L, L) and the step of h, were chosen as Z, = 16 fm and h, = 0.05 fm respectively. Then we have calculated the propagator ( 16) in the fixed grid (xi,XL) wherexi=-L+H,(i-1) withH,=4h,= 0.2 fm. Results for the SW potential are illustrated in Figs. 1, 2. Fig. 1 shows the diagonal elements of the propagator A’( n, X, E) together with those of the free propagator A’. For the negative values of x, the propagator A/(x, x, E) oscillates with a small amplitude. We have also drawn the quasiparticle density p(x) for one type of nucleons. Inside the matter it is a little higher than that of the normal nuclear matter because it corresponds to larger value of the Fermi energy ar = -Ua - ,u = 45 MeV (remind that we have used in calculations the value of p = -8 MeV for sake of closer relevance to the problem of pairing in finite nuclei). It should be noted that the density p(x) is not used in our calculations and is drawn just as an illustration. Fig. 2 shows the profile functions of A’( XI, x2, E) and A’(x~,x~,E) atxi =-8,0,8fm (fortheintervalof 1x1 --x2] < 4 fm). Inside the nuclear matter (xl = -8 fm) , the absolute value of the propagator A’ is significantly smaller than that of the free propagator A’. It vanishes more slowly at increasing values of 1x1 - x2 1 and oscillates. In the free space (xi = 8 fm) , the prop-
M. Bald0 et al. /Physics Letters B 350 (1995) 135-140
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10-Z
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10-d
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Fig. 2.
The diagonal elements of the propagator A’(xl, x2. E) for the Saxon-Woods A”(xl, x2. E) (dashed line) versus x2 at the fixed values of XI = -8, 0, 8 fm.
agators A’ and A0 practically coincide. Nearby the boundary vicinity, an intermediate behavior is present and the function of A’( XI, x2, E) is asymmetrical. After calculating the propagators A’ and A0 we can solve the integral Eq. ( 15). We change the integration over dxl by summation over the grid values Xi. Then the integral equation is reduced to a set of linear equations which is solved by usual methods. We analysed also the problem in the local approximation. In the present approach, we fix the form of the potential well U(x) which is not related directly to the density distribution p(x). Therefore, instead of the usual LDA, it is natural to use a version of the local approximation with the effective interaction V$ (X - x/2, X + x/2) defined as a set of the functions Vi,+!“.(x) , calculated for the “nuclear matter” with the constant potential UO which is equal to U(X) . The propagator A”.,.( UO) can be readily calculated with the help of Eq. (16), just in the same manner as the ro agator Ao, by substituting the quantity q = F=-p /2m - Uo for p (and the same for p’). The obtained propagator can be substituted into Eq. (15) which yields the effective interaction vg
y(X)
10
[fml
=
potential
(solid
line)
and the free propagator
V,, (X - x/2, x + x/2)&. s
(17)
and the similar quantity I’“” . The results are displayed in Fig. 3 together with corresponding value r” for the bare T-matrix of IQ. (14). The latter one is, of course, X-independent. We see that the effective interaction changes in the surface region very smoothly. Its strength inside the matter (N -1.1 in the Landau normalization) differs significantly from the freevalue (N -2.8). The effective interaction in the free space tends to the free limit very slowly, much slower than that calculated within the local approximation. Similar results, with a greater deviations from the local approximation predictions, were obtained also for the case of the potential well with the sharp boundary. This result should be qualitatively valid also in the case of the realistic interaction. Such calculations with the separable representation of the Paris potential are now in progress. It should be noticed that the local density approximation appears actually reliable down to very low density. However, deviations are present, and for all phenomena involving the tail of the nuclear density they should be relevant. Indeed, in the pairing problem the states nearby the Fermi energy are involved, which often have slowly falling tails. On the
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Fig. 3. The interaction strength y(x) (solid line), the strength p(x) of the bare interaction (dotted line) am compared with the local approximation result yJ” (x) (dashed line) for the Saxon-Woods potential. The potential f.!(x) is given in the lower half of the figure in arbitrary units.
other hand, for other phenomenological data or phenomena, the deviations may not play any role, e.g. for the total correlation energy, and in every specific case a new analysis is necessary. The research described in this publication was made possible in part by Grant No. ML8000 from the International Science Foundation. The authors wish to acknowledge I? Schuck, S. Fayans, V. Khodel and S. Tolokonnikov for many fruitful discussions. Two of us (ES. and M.Z.) thank INFN, Sezione di Catania, and Catania University for hospitality during their stay in Catania. References [ I] A.A. Abrikosov, L.P Gorkov and LE. Dzyaloshinsky, Methods of quantum field theory in statistical physics (Prentice-Hall, New York, 1964).
I21 A.B. Migdal, Theory of finite Fermi systems and applications to atomic nuclei (Wiley, New York, 1967). [3] E.E. Saperstein and M.V. Zverev, Sov. J. Nucl. Phys. 39 (1984) 1390. [4] E.E. Saperstein and M.V. Zverev, Sov. J. Nucl. Phys. 42 (1985) 1082. [51 S.A. Fayans, A.V. Smirnov and S.V. Tolokonnikov, Sov. J. Nucl. Phys. 48 (1988) 1661. [6] S.A. Fayans, S.V. Tolokonnikov, E.L. Trykov and D. Zawischa, Phys. L&t. B 338 (1994) 1. 171 M. Baldo, J. Cugnon, A. Lejeune and U. Lombardo, Nucl. Phys. A 515 (1990) 409. [S] T.L. Aisworth, J. Wambach and D. Pines, Phys. Len. B 222 (1989) 173; J.M.C. Chen, J.W. Clark, E. Rrotscheck and R.A. Smith, Nucl. Phys. A 451 (1986) 509. [9] L.D. Landau and E.M. Lifshitz, Quantum Mechanics: Non Relativistic Theory (Addison-Wesley, Reading, Mass., 1958).