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Space-like field in a relativistic local density approximation
Volume 156B, n u m b e r 5,6
PHYSICS LETTERS
27 J u n e 1985
S P A C E - L I K E F I E L D IN A R E L A T I V I S T I C L O C A L D E N S I T Y A P P R O X I M A T I O N M. J A M I N O N Institut de Physique BS, Universitb de Libge, Sart Tilman, B-4000 Liege 1, Belgium
and G. DO D A N G Laboratoire de Physique Thborique et Hautes Energies l, Universitb de Paris-Sud, F-91405 Orsay, France Received 9 January 1985; revised manuscript received 18 March 1985
We calculate the local space-like component of the relativistic nucleon-nucleus potential from nuclear matter results via a local density approximation. This component seems too small to remove the discrepancy between calculated and measured magnetic moments of nuclei.
The problem related to the magnetic moment [1,2] prevents the relativistic "scalar and vector potential model" [3] from being completely successful [4]. It may be stated by saying that, in a model consisting only of the scalar and vector potentials U s and U0 with parameters chosen to fit other experimental data, the magnetic moments of single particle or hole states relative to closed shell nuclei deviate from the experimental values. In view of the many virtues of the relativistic formalism in producing better results for elastic proton scatterings [5] as compared to the conventional non-relativistic description, one certainly would like to find a way out of the above difficulty. Up to now, there have been two attempts to tackle the problem. (i) The first of these [6] ascribes the origin of the difficulty to the anomalous component of the magnetic moment operator itself. The idea is attractive but, unfortunately, the seemingly good results obtained are made unreliable by the somewhat arbitrary choice of parameters. (ii) The second one [7] is purely phenomenological. Based on results from ref. [ 1] where it is shown that the magnetic moment depends rather strongly on the space-like component of 1 Laboratoire associ6 au Centre National de la Recherche Scientifique.
0370-2693•85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
the vector potential, a fit is done for a number of nuclei near closed shells. Using a space-like potential of the form Uv(r) = - i U v
df(r)/dr,
(1)
where f(r) is the nuclear mass distribution which is given by a standard Fermi shape, it is found that the fit of experimental data requires rather large values for Uv (Uv ~ 0.5) and furthermore that its sign has to change when one goes from a hole to a particle state relative to a closed shell nucleus. Such features are a priori difficult to understand from any microscopic basis. Therefore it would be interesting to see whether they have any physical meaning or whether they are just the results of an ad hoc fitting procedure. There has been one attempt by Miller [8] to calculate microscopically the space-like potential Uv(r). In the formulation of that work, the potential Uv(r) arises naturally from the exchange potential when one tries to make it local. In accordance with the requirement of time reversal invariance, it is purely imaginary as assumed in eq. (1). Its magnitude, however, is found to be much smaller than the phenomenologieal value (even after a multiplicative factor [9] of two is taken into account). It should also be noticed that the radial dependence of the space-like potential for the differ283
Volume 156B, number 5,6
PHYSICS LETTERS
27 June 1985
Hermiticity, however, is also preserved as is apparent from the way we have written the Dirac equation (3). In the form of eq. (4), the additional term in front of ~t. p restores this property. One can see this in another way by proving directly from eq. (4) that the four-current is conserved, namely
ent single particle levels shows a strange and unexplained behaviour, particularly in the central region. It is the purpose of this work to get further information on the space-like potential within the framework of the local density approximation (LDA). The calculation is done using nuclear matter results obtained in previous works [10]. Let us now sketch the main steps that we follow. Starring from a lagrangian involving the well-known bosons, a Hartree-Fock or Brueckner-Hartree-Fock [11 ] calculation leads to the following Dirac equation for a nucleon in nuclear matter:
In order to reduce eq. (4) to the form that is frequently assumed for the Dirac equation, we divide it by (1 + Uv/m) and reorder terms. The result is the following LDA equation:
{ot.k+3,0[m+G(E,k)]}u(k)=E(k)u(k),
(2a)
{at "p + 70[m + UsLDA(r,E) + 70 uLDA(r, E)
U(E, k) = Us (E) + ~'0U0(E) + Uv(E) Y" k/m,
(2b)
where the momentum-dependence of the various potentials Ui has been transformed into an energy-dependence via a dispersion relation [10]. Notice that these potentials depend on the density through the Fermi momentum kF. According to the spirit of the LDA, this dependence allows one to go from an infinite nuclear medium to a finite nucleus. We choose an LDA which gives, when it is applied to eq. (2), the following equation:
{at .p + 3,0[m + Us(r, E) + ~,0U0(r , E)
+ (1/2m)(Uv(r, E) ~ .p + "f.p Uv(r,E))] } ~bLOh(r,E) = E ffLDA(r, E ) ,
(3)
or equivalently {[1 +Uv(r,E)/m] at .p +-y0[m + Us(r,E) + ?0U0(r,E) -
i't" (V Uv(r,E)/2m)]} $LDA(r,E)
a~,f' - au(~,~'
~,) =
0.
+ "yOuLDA(r, E)] ) I~LDA(r, E ) = E ~oLDA(r,E ) ,
(4)
where the gradient operator acts only on Uv(r, E). This LDA has the advantage of preserving hermiticity while yielding a space-like field given by the last term between brackets in eq. (4). Indeed, if this term appeared alone, as usually assumed in a phenomenological approach, the requirement of hermiticity or of time reversal invariance would force it to be either real or imaginary. Assuming for the moment that Uv(r, E) is real, then one gets a purely imaginary spacelike potential which is therefore time reversal invariant. 284
(6)
where UsLDA and ULDA are of the same forms as given previously [ 12] and where the space-like potential is given by UvLDA(r,E) = -(i/2m)[1 + Uv(r, E)/m]-I X d Uv(r, E)/dr.
(7)
The study of this potential can now be made by taking over the results from nuclear matter calculations. As far as the calculation of magnetic moments is concerned, one can neglect in a first approximation its energy-dependence since it is known that the k-dependence of Uv(k) is weak for k < kF [1 I]. Furthermore, we also learn from nuclear matter calculations [11] that Uv is real and depends almost linearly on the density. In the spirit of the LDA, we therefore write Uv(r) = Uvf(r),
= E ~LDA(r, E ) ,
(5)
(8)
where f(r) is proportional to the nuclear density and may be given by the same form as in eq. (1). One thus recovers the main assumptions of ref. [7] namely that the space-like potential is purely imaginary and that its radial dependence is that of the derivative of the mass distribution. Indeed this potential can be written as uLDA(r) = --i(Uv/2m) df(r)/dr,
(9)
since the denominator in eq. (7) is practically equal to unity.
Volume 156B, number 5,6
of several Lorentz components of UF(E, k). Moreover, one finds that UFM(E,k) contains an additional tensor component. Using these properties and adding the direct (Hartree) contribution which is the same in both models, Miller's potential satisfies:
0
-2
-4
3
-6
\,,
/
\
/
\
/,,
{it. k+ 70[m + UsM(E)+ "r°uM(E) +(¥.
/,,
\',
/ \
k/m) uM(E)
--
/ \
27 June 1985
PHYSICS LETTERS
/ v
-
I
2
-
-
LDA ls1/2
I
r {fm}
4
Fig. 1. Local space-likefield calculated via a local density approximation applied to eq. (2) (full curve) and to eq. (12) (short-dashed curve). The long-dashed curve is the result of ref. [8] for the ls 1/2 shell. To have an estimate of its magnitude, let us use nuclear matter results [10,11]. One has Uv = 68.5 MeV. For 4°Ca, the parameters of the Fermi shape f(r) are a = 0.54 fm andR = 3.58 fm. The equivalent value of Uv in eq. (1) is Uv/2m ~ 0.04. This is smaller than the phenomenological value by one order of magnitude. In fig. 1, we show (full curve) the radial dependence of uLDA(r) using the above parameters. At this point, it would be tempting to compare our results with those of Miller [8,9]. This however would not be very meaningful. Indeed, the Dirac potential of Miller cannot be identified with a potential obtained via an LDA applied to eqs. (2). This can be understood by studying the exchange (Fock) contribution to the Dirac potentials in nuclear matter. It has been shown [10] that the Fock components UF(E, k) of U(E, k) [see eq. (2b)] and uM(E, k) of Miller's potential are related by the following equation:
+ ~/0(~. k/m)
Vtra(E)] } u(k)
= ~(k) u (g),
where the potentials uM(E) (i = s, 0, v, t) are complicated functions of the components Us, U0 and Uv, as well as of the energy E. It is then evident that it is the result of an LDA applied to eq. (12) and not to eq. (2), that should be compared with Miller's ones. Following the same procedure as in ref. [13] in order to eliminate nonlocal terms, the resulting space-like potential UM LDA(r, E) is given by U y LDA(r, E) = -- ~i
X~{[1 + U~(r,E)lm]2- [U~(r,E)lml2}, (13) with
UvM(r,E)=I(uv(r,E) + E - U0(r,E ) m uF(r,E)[1 + Uv(r,E)/m]) ,
E- 00(r,~
UM(E, k) us(k) = UF(E, k) Us(k) .
(1 l)
The sum in eq. (10) is a 4 × 4 matrix which can be expanded in the 16 linearly independent matrices formed from products of the four Dirac matrices. Each Lorentz component of U~I(E, k) can then be expressed in terms
(14)
__ mUF(r,E) UM(r,E) = 1 ( [ 1 + Uv(r,E)/m] if-_ -ffI~o(r:~-,) _ m + Us(r,E) Uv(r,E))"
where us(k) is the self-consistent positive energy Dirac spinor of spin S. Both potentials are strictly equivalent in the sense that they yield the same physical observables since
(12)
(15)
In eqs. (14) and (15), the superscript " F " refers to the Fock contribution to the Lorentz potentials Us and 00. Assuming the same shape f(r) for all the components Ui(r, E) (i = s, 0, v) and taking [10] U H = 240 MeV, 0 F = 90 MeV (U n = - 3 3 0 MeV, 0 F = - 7 0 MeV) for the depth of the direct and exchange parts of U0 (Us), we get the short-dashed curve of fig. 1 for the space-like potential UM LDA. Fig. 1 also shows Miller's potential for the Is 1/2 shell (long-dashed curve). Aside from the fluctuation of the latter near
285
Volume 156B, number 5,6
PHYSICS LETTERS
the origin, the two potentials agree nicely in the region of interest. In conclusion, the space-like component turns out to be small in Miller's calculations as well as in an LDA applied to eq. (2) or to eq. (12). Therefore, the main origin of the difficulty with the magnetic moments can probably not be ascribed to this space-like field.
References [1] L.D. Miller, Ann. Phys. 91 (1975) 40. [2] M. Bawin, C.A. Hugues and G.L. Strobel, Phys. Rev. C28 (1983) 456.
286
27 June 1985
[3] J.D. Walecka, Ann. Phys. 83 (1974) 491. [4] J.M. Eisenberg, NucL Phys. A355 (1981) 269. [5] L.G. Arnold, B.C. Clark, R.C. Mercer and P. Schwandt, Phys. Rev. C23 (1981) 1949. [6] J.V. Noble, Phys. Rev. C20 (1979) 1188. [7] A. Bouyssy, S. Marcos and J.F. Mathiot, Nuel. Phys. A415 (1984) 497. [8] L.D. Miller, Phys. Rev. C9 (1974) 537. [9] L.D. Miller, Phys. Rev. C12 (1975) 710. [10] M. Jaminon, C. Mahaux and P. Rochus, Nucl. Phys. A365 (1981) 371; M. Jaminon, Ph.D. Thesis (Liege, 1982), unpublished. [11] M.R. Anastasio, L.S. Celenza, W.S. Pong and C.M. Shakin, Phys. Rep. 100 (1983) 327. [12] M. Jaminorb NucL Phys. A402 (1983) 366. [ 13] M. Jaminon, Lett. Nuovo Cimento 36 (1983) 481.
PHYSICS LETTERS
27 J u n e 1985
S P A C E - L I K E F I E L D IN A R E L A T I V I S T I C L O C A L D E N S I T Y A P P R O X I M A T I O N M. J A M I N O N Institut de Physique BS, Universitb de Libge, Sart Tilman, B-4000 Liege 1, Belgium
and G. DO D A N G Laboratoire de Physique Thborique et Hautes Energies l, Universitb de Paris-Sud, F-91405 Orsay, France Received 9 January 1985; revised manuscript received 18 March 1985
We calculate the local space-like component of the relativistic nucleon-nucleus potential from nuclear matter results via a local density approximation. This component seems too small to remove the discrepancy between calculated and measured magnetic moments of nuclei.
The problem related to the magnetic moment [1,2] prevents the relativistic "scalar and vector potential model" [3] from being completely successful [4]. It may be stated by saying that, in a model consisting only of the scalar and vector potentials U s and U0 with parameters chosen to fit other experimental data, the magnetic moments of single particle or hole states relative to closed shell nuclei deviate from the experimental values. In view of the many virtues of the relativistic formalism in producing better results for elastic proton scatterings [5] as compared to the conventional non-relativistic description, one certainly would like to find a way out of the above difficulty. Up to now, there have been two attempts to tackle the problem. (i) The first of these [6] ascribes the origin of the difficulty to the anomalous component of the magnetic moment operator itself. The idea is attractive but, unfortunately, the seemingly good results obtained are made unreliable by the somewhat arbitrary choice of parameters. (ii) The second one [7] is purely phenomenological. Based on results from ref. [ 1] where it is shown that the magnetic moment depends rather strongly on the space-like component of 1 Laboratoire associ6 au Centre National de la Recherche Scientifique.
0370-2693•85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
the vector potential, a fit is done for a number of nuclei near closed shells. Using a space-like potential of the form Uv(r) = - i U v
df(r)/dr,
(1)
where f(r) is the nuclear mass distribution which is given by a standard Fermi shape, it is found that the fit of experimental data requires rather large values for Uv (Uv ~ 0.5) and furthermore that its sign has to change when one goes from a hole to a particle state relative to a closed shell nucleus. Such features are a priori difficult to understand from any microscopic basis. Therefore it would be interesting to see whether they have any physical meaning or whether they are just the results of an ad hoc fitting procedure. There has been one attempt by Miller [8] to calculate microscopically the space-like potential Uv(r). In the formulation of that work, the potential Uv(r) arises naturally from the exchange potential when one tries to make it local. In accordance with the requirement of time reversal invariance, it is purely imaginary as assumed in eq. (1). Its magnitude, however, is found to be much smaller than the phenomenologieal value (even after a multiplicative factor [9] of two is taken into account). It should also be noticed that the radial dependence of the space-like potential for the differ283
Volume 156B, number 5,6
PHYSICS LETTERS
27 June 1985
Hermiticity, however, is also preserved as is apparent from the way we have written the Dirac equation (3). In the form of eq. (4), the additional term in front of ~t. p restores this property. One can see this in another way by proving directly from eq. (4) that the four-current is conserved, namely
ent single particle levels shows a strange and unexplained behaviour, particularly in the central region. It is the purpose of this work to get further information on the space-like potential within the framework of the local density approximation (LDA). The calculation is done using nuclear matter results obtained in previous works [10]. Let us now sketch the main steps that we follow. Starring from a lagrangian involving the well-known bosons, a Hartree-Fock or Brueckner-Hartree-Fock [11 ] calculation leads to the following Dirac equation for a nucleon in nuclear matter:
In order to reduce eq. (4) to the form that is frequently assumed for the Dirac equation, we divide it by (1 + Uv/m) and reorder terms. The result is the following LDA equation:
{ot.k+3,0[m+G(E,k)]}u(k)=E(k)u(k),
(2a)
{at "p + 70[m + UsLDA(r,E) + 70 uLDA(r, E)
U(E, k) = Us (E) + ~'0U0(E) + Uv(E) Y" k/m,
(2b)
where the momentum-dependence of the various potentials Ui has been transformed into an energy-dependence via a dispersion relation [10]. Notice that these potentials depend on the density through the Fermi momentum kF. According to the spirit of the LDA, this dependence allows one to go from an infinite nuclear medium to a finite nucleus. We choose an LDA which gives, when it is applied to eq. (2), the following equation:
{at .p + 3,0[m + Us(r, E) + ~,0U0(r , E)
+ (1/2m)(Uv(r, E) ~ .p + "f.p Uv(r,E))] } ~bLOh(r,E) = E ffLDA(r, E ) ,
(3)
or equivalently {[1 +Uv(r,E)/m] at .p +-y0[m + Us(r,E) + ?0U0(r,E) -
i't" (V Uv(r,E)/2m)]} $LDA(r,E)
a~,f' - au(~,~'
~,) =
0.
+ "yOuLDA(r, E)] ) I~LDA(r, E ) = E ~oLDA(r,E ) ,
(4)
where the gradient operator acts only on Uv(r, E). This LDA has the advantage of preserving hermiticity while yielding a space-like field given by the last term between brackets in eq. (4). Indeed, if this term appeared alone, as usually assumed in a phenomenological approach, the requirement of hermiticity or of time reversal invariance would force it to be either real or imaginary. Assuming for the moment that Uv(r, E) is real, then one gets a purely imaginary spacelike potential which is therefore time reversal invariant. 284
(6)
where UsLDA and ULDA are of the same forms as given previously [ 12] and where the space-like potential is given by UvLDA(r,E) = -(i/2m)[1 + Uv(r, E)/m]-I X d Uv(r, E)/dr.
(7)
The study of this potential can now be made by taking over the results from nuclear matter calculations. As far as the calculation of magnetic moments is concerned, one can neglect in a first approximation its energy-dependence since it is known that the k-dependence of Uv(k) is weak for k < kF [1 I]. Furthermore, we also learn from nuclear matter calculations [11] that Uv is real and depends almost linearly on the density. In the spirit of the LDA, we therefore write Uv(r) = Uvf(r),
= E ~LDA(r, E ) ,
(5)
(8)
where f(r) is proportional to the nuclear density and may be given by the same form as in eq. (1). One thus recovers the main assumptions of ref. [7] namely that the space-like potential is purely imaginary and that its radial dependence is that of the derivative of the mass distribution. Indeed this potential can be written as uLDA(r) = --i(Uv/2m) df(r)/dr,
(9)
since the denominator in eq. (7) is practically equal to unity.
Volume 156B, number 5,6
of several Lorentz components of UF(E, k). Moreover, one finds that UFM(E,k) contains an additional tensor component. Using these properties and adding the direct (Hartree) contribution which is the same in both models, Miller's potential satisfies:
0
-2
-4
3
-6
\,,
/
\
/
\
/,,
{it. k+ 70[m + UsM(E)+ "r°uM(E) +(¥.
/,,
\',
/ \
k/m) uM(E)
--
/ \
27 June 1985
PHYSICS LETTERS
/ v
-
I
2
-
-
LDA ls1/2
I
r {fm}
4
Fig. 1. Local space-likefield calculated via a local density approximation applied to eq. (2) (full curve) and to eq. (12) (short-dashed curve). The long-dashed curve is the result of ref. [8] for the ls 1/2 shell. To have an estimate of its magnitude, let us use nuclear matter results [10,11]. One has Uv = 68.5 MeV. For 4°Ca, the parameters of the Fermi shape f(r) are a = 0.54 fm andR = 3.58 fm. The equivalent value of Uv in eq. (1) is Uv/2m ~ 0.04. This is smaller than the phenomenological value by one order of magnitude. In fig. 1, we show (full curve) the radial dependence of uLDA(r) using the above parameters. At this point, it would be tempting to compare our results with those of Miller [8,9]. This however would not be very meaningful. Indeed, the Dirac potential of Miller cannot be identified with a potential obtained via an LDA applied to eqs. (2). This can be understood by studying the exchange (Fock) contribution to the Dirac potentials in nuclear matter. It has been shown [10] that the Fock components UF(E, k) of U(E, k) [see eq. (2b)] and uM(E, k) of Miller's potential are related by the following equation:
+ ~/0(~. k/m)
Vtra(E)] } u(k)
= ~(k) u (g),
where the potentials uM(E) (i = s, 0, v, t) are complicated functions of the components Us, U0 and Uv, as well as of the energy E. It is then evident that it is the result of an LDA applied to eq. (12) and not to eq. (2), that should be compared with Miller's ones. Following the same procedure as in ref. [13] in order to eliminate nonlocal terms, the resulting space-like potential UM LDA(r, E) is given by U y LDA(r, E) = -- ~i
X~{[1 + U~(r,E)lm]2- [U~(r,E)lml2}, (13) with
UvM(r,E)=I(uv(r,E) + E - U0(r,E ) m uF(r,E)[1 + Uv(r,E)/m]) ,
E- 00(r,~
UM(E, k) us(k) = UF(E, k) Us(k) .
(1 l)
The sum in eq. (10) is a 4 × 4 matrix which can be expanded in the 16 linearly independent matrices formed from products of the four Dirac matrices. Each Lorentz component of U~I(E, k) can then be expressed in terms
(14)
__ mUF(r,E) UM(r,E) = 1 ( [ 1 + Uv(r,E)/m] if-_ -ffI~o(r:~-,) _ m + Us(r,E) Uv(r,E))"
where us(k) is the self-consistent positive energy Dirac spinor of spin S. Both potentials are strictly equivalent in the sense that they yield the same physical observables since
(12)
(15)
In eqs. (14) and (15), the superscript " F " refers to the Fock contribution to the Lorentz potentials Us and 00. Assuming the same shape f(r) for all the components Ui(r, E) (i = s, 0, v) and taking [10] U H = 240 MeV, 0 F = 90 MeV (U n = - 3 3 0 MeV, 0 F = - 7 0 MeV) for the depth of the direct and exchange parts of U0 (Us), we get the short-dashed curve of fig. 1 for the space-like potential UM LDA. Fig. 1 also shows Miller's potential for the Is 1/2 shell (long-dashed curve). Aside from the fluctuation of the latter near
285
Volume 156B, number 5,6
PHYSICS LETTERS
the origin, the two potentials agree nicely in the region of interest. In conclusion, the space-like component turns out to be small in Miller's calculations as well as in an LDA applied to eq. (2) or to eq. (12). Therefore, the main origin of the difficulty with the magnetic moments can probably not be ascribed to this space-like field.
References [1] L.D. Miller, Ann. Phys. 91 (1975) 40. [2] M. Bawin, C.A. Hugues and G.L. Strobel, Phys. Rev. C28 (1983) 456.
286
27 June 1985
[3] J.D. Walecka, Ann. Phys. 83 (1974) 491. [4] J.M. Eisenberg, NucL Phys. A355 (1981) 269. [5] L.G. Arnold, B.C. Clark, R.C. Mercer and P. Schwandt, Phys. Rev. C23 (1981) 1949. [6] J.V. Noble, Phys. Rev. C20 (1979) 1188. [7] A. Bouyssy, S. Marcos and J.F. Mathiot, Nuel. Phys. A415 (1984) 497. [8] L.D. Miller, Phys. Rev. C9 (1974) 537. [9] L.D. Miller, Phys. Rev. C12 (1975) 710. [10] M. Jaminon, C. Mahaux and P. Rochus, Nucl. Phys. A365 (1981) 371; M. Jaminon, Ph.D. Thesis (Liege, 1982), unpublished. [11] M.R. Anastasio, L.S. Celenza, W.S. Pong and C.M. Shakin, Phys. Rep. 100 (1983) 327. [12] M. Jaminorb NucL Phys. A402 (1983) 366. [ 13] M. Jaminon, Lett. Nuovo Cimento 36 (1983) 481.