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Zero-point vibrations and the nuclear surface
I
I.E.5
[ [
Nuclear Physics 4 2 (1963) 1 7 7 - - 1 8 2 ; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
Z E R O - P O I N T VIBRATIONS AND T H E NUCLEAR SURFACE G. E. B R O W N *
NORDITA, Copenhagen and GERHARD
J A C O B t)
Institute for Theoretical Physics, University of Copenhagen, Denmark Received 4 O c t o b e r 1962 Abstract: T h e influence o f zero-point vibrations on t h e g r o u n d state o f spherical nuclei is calculated u s i n g the particle-hole f o r m a l i s m to describe the vibrations. W i t h wave f u n c t i o n s obtained f r o m a W o o d s - S a x o n potential, the influence o f t h e first 3-, T = 0 vibration o n the radius a n d the surface thickness o f Ca 40 is estimated a n d the results are discussed.
Calculations of nuclear ground states are generally made within the framework of a Hartree-Fock theory, or some modification of it which takes into account the hard cores in the nucleon-nucleon interaction t). Anderson and Thouless 2) have pointed out that the surface diffuseness should be increased over and above the value obtained in these calculations by the presence of zero-point vibrations in nuclei, and they carried out a semi-classical calculation of this effect for the case of Ca 4°. We shall here describe a quantal calculation of this effect, using a theory which describes the vibrations as particle-hole excitations. Although it would be possible to calculate the effects of at least the odd-parity vibrations accurately, since they can be calculated in a relatively straightforward way 3, 4), we shall do this in a fairly rough approximation, using the schematic model discussed in ref. 3), as this gives a very simple description of the effect. Since we shall use empirical energies to determine the parameters in this model, we shall probably compensate for many neglects. We consider, in what follows, only spherical nuclei (in ordinary, spin and isospin space) and assume that the reader is familiar with the contents of ref. 3). Denoting by x., and y . , , respectively, the forward and backward amplitudes for the perturbed particle-hole state and using the schematic model for the degenerate case, we have 3) xiv = N eOw -E'
Yw = N D w . e+E
(1)
In eq. (1), e is the unperturbed particle-hole energy, E the perturbed one, D., is the * N o w at M a s s a c h u s e t t s Institute o f T e c h n o l o g y , C a m b r i d g e , Mass. )'t O n leave o f absence f r o m Universidade do Rio G r a n d e do Sul, P6rto Alegre, Brasil; n o w at Institute for Theoretical Physics, University o f Heidelberg, G e r m a n y . 177 April 1963
178
q.
E.
B R O W N
A N D
G.
J A C O B
amplitude for absorption of an octupole ?-ray and N is a normalization constant defined in such a way that Y. Ix,,'l 2 - E lYu']" - 1. tl'
il"
We define the operators b~r = a~ar and their hermitian conjugates bit = a~ai, where a~ denotes the creation operator for a particle and a~. that for a hole. These creation and annihilation operators for a particle-hole pair will be treated in the quasiboson approximation (see, e.g., ref. 5)): [bir, b~.] t -- ~J~,'l"
(2)
The annihilation operator for a vibration with total angular momentum J, projection m and isospin T, projection n is am.n
~
/
mn-nmt
= 2.,LYl Oi +xi
--m--n, --m-n,,
ol
),
(3)
1
where we have suppressed the indices J and T, and the index i stands for the pair of indices i, i' (the ~ being, therefore, a double sum ~ , , ) . This simplified notation will be consistently used in what follows. Furthermore, denoting the uncorrelated ground state (bare vacuum) of a spherical nucleus by [), the correlated ground state (physical vacuum) 10) will be, in the random phase approximation 6), of the form
10>
(
1 ~, B~,b~,,,/tb;,~,_nl) xB+ x/(EJ + 1XET+ l) . p - , . , i
1 + (2,/+1)(2T+1)
.a~a
B~,p,~b~'"'*b;'~-"db~"~¢b~"2-"~)+...)
[).
(4)
mlNlmln2
In eq. (4), B, B,p, B,c~a. . . . indicate the amplitudes for the presence of zero, two, four, • . . particle-hole pairs in the ground state due to the virtual vibration J, T; each boson pair b~ 1"at b~mt-mt e t c . . , is coupled to the total angular momentum 0 and to the total isospin 0 of the ground state. The condition that I0) be the ground state of the nucleus to the approximation in which we work is that it contain no real vibration, i.e. A'"I0> ---- 0.
(5)
Using eqs. (3) and (4) and the commutation relations (2) and bearing in mind that B,p = Bp,, etc., as is obvious from eq. (4), eq. (5) yields the recurrence relations B
y,""
2
-I-
--
~.. Bi, xf"-" = O,
x/(2d + 1X2T+ 1) •
(6)
4
B,py'~'+ 4(2.I+ 1)(2T+ 1) ~ n'tmx;'-" = O, •
°
°
•
•
•
.
.
.
.
.
•
.
•
.
.
°
ZERO-VOl~rrvm~mONS ^r~'D arm NOCta~R StJRr^CE
179
Since we are considering a spherical nucleus (in ordinary, spin and isospin space), y~'" and x~-m-" are independent o f m and n, their values being given by eq. (1). Also, using the schematic model, we can write
B =- Bo,
B,,# = B2D~D #,
B,#va = B4D,,D#DTD,I . . . .
(7)
the D,, etc. having been defined above. Furthermore, defining U ~
--E 4 ( 2 . / + 1)(2 T + 1), ~+~-'~
(8)
one obtains from (6), (7) and (1) the general expression 1
8,. = (-1)"-
un
- 80. 2",~ (Y" o,2)"
(9)
CI
The probabilities of finding, in the correlated ground state, zero, two, four . . . . virtual particle-hole pairs are given by the individual terms of the expression
1 ~. B.a.B.ab#z,.,,_.,,b~/,..,b~.,.,tb~.,t_.d (010) = (I BZ-t- (2,/-t-1X2T+I) -n,'#' \
minim'In' l
+.-.} I>. (I0)
Factor pairing the boson operators and using (7) and (9),one has from (I0)
<010>
Bo2 l+½u2"i- ~I (½u2Y+ ~I (½u,)3+""
.}.
(xl)
In actual cases the values of ½us are of the order of unity, so that only the first few terms in (11) have a significant value. Consequently, although we are dealing with a finite nucleus, we can write to a very good approximation (010) = B~ exp (½u 2) and therefore, since the correlated ground state has to be normalized to unity,
Bo~ -- exp (-½u~).
(12)
From (11) and (12) one obtains for the probability of having 2n particle-hole pairs in the correlated ground state P2. = exp ( - ½ u 2) 1 (½u2).
(13)
and for the average number of particles excited
N = ~ . 2 n P 2 . = u 2. ~=|
(14)
180
G. E. BRO',ArN AND G. JACOB
Finally, using (4), (7), (9) and the fact that the boson creation operators commute, the wave function for the correlated nuclear ground state can be written as [0) = Bo {exp ( - o ~. •, b?n'Oj bf"-n')}l),
(15)
where
v=½
e-E
1
We now apply our results to the special case of Ca 4°. In order to obtain an estimate of the influence of the considered virtual excitations on the radius and on the thickness of the nuclear surface, we take into account only the effect of the first 3 - , T = 0 vibration at E = 3.73 MeV; the unperturbed particle-hole energy we take as e = 14 MeV. With these parameters, we obtain for the average number of particles excited N = 2.35 and for the various probabilities Po = 0.309,
P2 = 0.363,
P4 = 0.213,
P6 = 0.083,
Ps = 0.024,
Plo = 0.006.
In order to calculate the influence of these virtual excitations on the nuclear radius, we used numerical wave functions for Ca 4°, obtained from a Woods-Saxon potential. Specifically, this potential was of the form ! d_f a " L,
(16)
rdr where
f(r) =
1 + e x p {(r-R)/a}
and h/lac is the pion Compton wave length. The following parameters were used: Fco = - 4 9 . 5 MeV, a = 0.48 MeV,
~o = 7.6 MeV, R = 4.45 fm.
These parameters give the I fI, lft and ld! levels at - 1.8 MeV, - 18.2 MeV and - 14.9 MeV, very close to the empirical values for the neutron levels. Of course, the proton levels are less bound, and through our neglect of Coulomb forces we underestimate the effect calculated here, possibly by as much as a factor of two. These wave functions give a root mean square radius of 3.33 fm as compared to the experimental value 7) 3.52 fm. Our theoretical radius is almost large enough when we take into account the fact that the proton structure must be folded in before comparing with experiment. Furthermore, we took the excited particles with their respective probabilities out of the levels of the 2he0 shell in accordance with the statistical weight of each level and located them in the levels of the 3hto shell in accordance with the same
ZERO-POINT
VIBRATIONS
AND
THE NUCLEAR
181
SURFACE
criterium. The new root mean square radius obtained in this way was 3.38 fm, giving an increase of 1.5%. The density distributions for both correlated and uncorrelated nuclei are shown in fig. 1 for the Woods-Saxon potential and, for comparison, also for a harmonic oscillator well with hto = e = 14 MeV. The change in surface thickness through the correlation is not easy to evaluate, the usual definition as the distance between points of 90 and 10 values of the central density being obviously applicable only to two-parameter density distributions. Because the electron scattering will be very insensitive to the bump at the origin in any case, we can get some idea of the change in the surface thickness by taking the density at ~ 2 fm as the central density. In this case, the 90 to 10% distance Q ( r ) in f m -3
SO
H.O.a
~n
~o
30
~o
so
r~fm)
Fig. 1. Density distributions for the ground state in Ca40 with Woods-Saxon (W.S.) and harmonic oscillator (H.O.) potentials: a: . . . . uncorrelated; b: _ _ correlated, with the average number of particles excited equal to 2.35 (see text). The normalization is 5~p(r)r2 dr ~ 40. is 1.76 fm in the uncorrelated case, and 1.84 fm in the correlated one, giving a 4 . 4 ~ increase in surface thickness. In the calculation of Brueckner et al. ~) one has a similar difficulty in defining a surface thickness, since the distribution is not a simple two-parameter one. In either our case or in theirs one obtains already too large a surface thickness in the uncorrelated case by taking literally the distance between 90 and 10 % points. Again, these authors obtain a central bump which will not be sensitive to electron scattering. The only way to settle the question of whether the surface thickness is too small or too large in the uncorrelated distribution is to calculate the electron scattering directly for each distribution. The rms radius is, however, less ambiguous, and comes out 20% too small in the theory of Brueckner et al.; these authors further point out that this discrepancy would be hard to eliminate in their theory.
182
O. E. BROWN AND O. JACOB
Although in the case of a liquid drop 2) the increase in surface diffuseness is of first order, and that in the rms radius only of second order, we see that they are numerically of the same size in our calculation, stemming from the fact that calcium is mainly surface. Anderson and Thouless took into account vibrations of angular m o m e n t u m 2, 3, 4, 5. In fact, a strong 5 - vibration has been observed at 4.4$ MeV in Ca 4°, and would have as large an effect as the 3 - one. In addition, 2 + and 4 + vibrations are probably present, but have not been observed yet. Effects o f even-parity vibrations are difficult to calculate in our formalism, since they lead to particles in unbound states and one does not know how to handle these. It seems that the total increase in rms radius will be several per cent, but will fall short of the 20 ~o required to bring agreement with experiment. These calculations show that particles are to be found a reasonable part of the time in higher shells than those assigned to them by the shell model, the average number of "excited" particles resulting from the 3 - vibration being 2.35. I f these are present, clearly pickup reactions, such as (p, d) and (d, t) will show them. From the nature of the correlations, the pickup of such a particle will leave a hole and a vibration; that is, the final nucleus will be in all of the states which can be reached by coupling the hole and vibration. This will require an average energy higher than that required to reach the ground state o f the final nucleus by about the energy of the vibration. We would like to thank D. J. Thouless and J. A. Evans for valuable advice and criticism. One of us (G. J.) wishes to thank the Ford Foundation for a fellowship and the C a m p a n h a Nacional de Aperfei~oamento de Pessoal de Nivel Superior (CAPES) for a travel grant, which made his stay in the very pleasant and stimulating atmosphere of the Copenhagen Institute possible. Refereuces I) 2) 3) 4) 5) 6)
K. A. Brueckner, A. M. Lockett and M. Rotenberg, Phys. Rev. 121 (1961) 255 P. W. Anderson and D. ft. Thouless, Phys. Lett. I (1962) 155 G. E. Brown, J. A. Evans and D. J. Thouless, Nuclear Physics 24 (1961) 1 Vincent Gillet and Nicole Vinh-Mau, Phys. Lett. 1 (1962) 25 J. HOgaasen-Feldman, Nuclear Physics 28 (1961) 258 D. J. Thouless, Nuclear Physics 22 (1961) 78; D. J. Thouless, The quantum mechanics of malty-body systems (Academic Press, New York, 1961); G. E. Brown, Lectures on Many Body Problems, NORDITA, Copenhagen, 1961 7) Robert Herman and Robert Hofstadter, High-energy electron scattering tables (Stanford University Press, Stanford, 1960)
I.E.5
[ [
Nuclear Physics 4 2 (1963) 1 7 7 - - 1 8 2 ; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
Z E R O - P O I N T VIBRATIONS AND T H E NUCLEAR SURFACE G. E. B R O W N *
NORDITA, Copenhagen and GERHARD
J A C O B t)
Institute for Theoretical Physics, University of Copenhagen, Denmark Received 4 O c t o b e r 1962 Abstract: T h e influence o f zero-point vibrations on t h e g r o u n d state o f spherical nuclei is calculated u s i n g the particle-hole f o r m a l i s m to describe the vibrations. W i t h wave f u n c t i o n s obtained f r o m a W o o d s - S a x o n potential, the influence o f t h e first 3-, T = 0 vibration o n the radius a n d the surface thickness o f Ca 40 is estimated a n d the results are discussed.
Calculations of nuclear ground states are generally made within the framework of a Hartree-Fock theory, or some modification of it which takes into account the hard cores in the nucleon-nucleon interaction t). Anderson and Thouless 2) have pointed out that the surface diffuseness should be increased over and above the value obtained in these calculations by the presence of zero-point vibrations in nuclei, and they carried out a semi-classical calculation of this effect for the case of Ca 4°. We shall here describe a quantal calculation of this effect, using a theory which describes the vibrations as particle-hole excitations. Although it would be possible to calculate the effects of at least the odd-parity vibrations accurately, since they can be calculated in a relatively straightforward way 3, 4), we shall do this in a fairly rough approximation, using the schematic model discussed in ref. 3), as this gives a very simple description of the effect. Since we shall use empirical energies to determine the parameters in this model, we shall probably compensate for many neglects. We consider, in what follows, only spherical nuclei (in ordinary, spin and isospin space) and assume that the reader is familiar with the contents of ref. 3). Denoting by x., and y . , , respectively, the forward and backward amplitudes for the perturbed particle-hole state and using the schematic model for the degenerate case, we have 3) xiv = N eOw -E'
Yw = N D w . e+E
(1)
In eq. (1), e is the unperturbed particle-hole energy, E the perturbed one, D., is the * N o w at M a s s a c h u s e t t s Institute o f T e c h n o l o g y , C a m b r i d g e , Mass. )'t O n leave o f absence f r o m Universidade do Rio G r a n d e do Sul, P6rto Alegre, Brasil; n o w at Institute for Theoretical Physics, University o f Heidelberg, G e r m a n y . 177 April 1963
178
q.
E.
B R O W N
A N D
G.
J A C O B
amplitude for absorption of an octupole ?-ray and N is a normalization constant defined in such a way that Y. Ix,,'l 2 - E lYu']" - 1. tl'
il"
We define the operators b~r = a~ar and their hermitian conjugates bit = a~ai, where a~ denotes the creation operator for a particle and a~. that for a hole. These creation and annihilation operators for a particle-hole pair will be treated in the quasiboson approximation (see, e.g., ref. 5)): [bir, b~.] t -- ~J~,'l"
(2)
The annihilation operator for a vibration with total angular momentum J, projection m and isospin T, projection n is am.n
~
/
mn-nmt
= 2.,LYl Oi +xi
--m--n, --m-n,,
ol
),
(3)
1
where we have suppressed the indices J and T, and the index i stands for the pair of indices i, i' (the ~ being, therefore, a double sum ~ , , ) . This simplified notation will be consistently used in what follows. Furthermore, denoting the uncorrelated ground state (bare vacuum) of a spherical nucleus by [), the correlated ground state (physical vacuum) 10) will be, in the random phase approximation 6), of the form
10>
(
1 ~, B~,b~,,,/tb;,~,_nl) xB+ x/(EJ + 1XET+ l) . p - , . , i
1 + (2,/+1)(2T+1)
.a~a
B~,p,~b~'"'*b;'~-"db~"~¢b~"2-"~)+...)
[).
(4)
mlNlmln2
In eq. (4), B, B,p, B,c~a. . . . indicate the amplitudes for the presence of zero, two, four, • . . particle-hole pairs in the ground state due to the virtual vibration J, T; each boson pair b~ 1"at b~mt-mt e t c . . , is coupled to the total angular momentum 0 and to the total isospin 0 of the ground state. The condition that I0) be the ground state of the nucleus to the approximation in which we work is that it contain no real vibration, i.e. A'"I0> ---- 0.
(5)
Using eqs. (3) and (4) and the commutation relations (2) and bearing in mind that B,p = Bp,, etc., as is obvious from eq. (4), eq. (5) yields the recurrence relations B
y,""
2
-I-
--
~.. Bi, xf"-" = O,
x/(2d + 1X2T+ 1) •
(6)
4
B,py'~'+ 4(2.I+ 1)(2T+ 1) ~ n'tmx;'-" = O, •
°
°
•
•
•
.
.
.
.
.
•
.
•
.
.
°
ZERO-VOl~rrvm~mONS ^r~'D arm NOCta~R StJRr^CE
179
Since we are considering a spherical nucleus (in ordinary, spin and isospin space), y~'" and x~-m-" are independent o f m and n, their values being given by eq. (1). Also, using the schematic model, we can write
B =- Bo,
B,,# = B2D~D #,
B,#va = B4D,,D#DTD,I . . . .
(7)
the D,, etc. having been defined above. Furthermore, defining U ~
--E 4 ( 2 . / + 1)(2 T + 1), ~+~-'~
(8)
one obtains from (6), (7) and (1) the general expression 1
8,. = (-1)"-
un
- 80. 2",~ (Y" o,2)"
(9)
CI
The probabilities of finding, in the correlated ground state, zero, two, four . . . . virtual particle-hole pairs are given by the individual terms of the expression
1 ~. B.a.B.ab#z,.,,_.,,b~/,..,b~.,.,tb~.,t_.d (010) = (I BZ-t- (2,/-t-1X2T+I) -n,'#' \
minim'In' l
+.-.} I>. (I0)
Factor pairing the boson operators and using (7) and (9),one has from (I0)
<010>
Bo2 l+½u2"i- ~I (½u2Y+ ~I (½u,)3+""
.}.
(xl)
In actual cases the values of ½us are of the order of unity, so that only the first few terms in (11) have a significant value. Consequently, although we are dealing with a finite nucleus, we can write to a very good approximation (010) = B~ exp (½u 2) and therefore, since the correlated ground state has to be normalized to unity,
Bo~ -- exp (-½u~).
(12)
From (11) and (12) one obtains for the probability of having 2n particle-hole pairs in the correlated ground state P2. = exp ( - ½ u 2) 1 (½u2).
(13)
and for the average number of particles excited
N = ~ . 2 n P 2 . = u 2. ~=|
(14)
180
G. E. BRO',ArN AND G. JACOB
Finally, using (4), (7), (9) and the fact that the boson creation operators commute, the wave function for the correlated nuclear ground state can be written as [0) = Bo {exp ( - o ~. •, b?n'Oj bf"-n')}l),
(15)
where
v=½
e-E
1
We now apply our results to the special case of Ca 4°. In order to obtain an estimate of the influence of the considered virtual excitations on the radius and on the thickness of the nuclear surface, we take into account only the effect of the first 3 - , T = 0 vibration at E = 3.73 MeV; the unperturbed particle-hole energy we take as e = 14 MeV. With these parameters, we obtain for the average number of particles excited N = 2.35 and for the various probabilities Po = 0.309,
P2 = 0.363,
P4 = 0.213,
P6 = 0.083,
Ps = 0.024,
Plo = 0.006.
In order to calculate the influence of these virtual excitations on the nuclear radius, we used numerical wave functions for Ca 4°, obtained from a Woods-Saxon potential. Specifically, this potential was of the form ! d_f a " L,
(16)
rdr where
f(r) =
1 + e x p {(r-R)/a}
and h/lac is the pion Compton wave length. The following parameters were used: Fco = - 4 9 . 5 MeV, a = 0.48 MeV,
~o = 7.6 MeV, R = 4.45 fm.
These parameters give the I fI, lft and ld! levels at - 1.8 MeV, - 18.2 MeV and - 14.9 MeV, very close to the empirical values for the neutron levels. Of course, the proton levels are less bound, and through our neglect of Coulomb forces we underestimate the effect calculated here, possibly by as much as a factor of two. These wave functions give a root mean square radius of 3.33 fm as compared to the experimental value 7) 3.52 fm. Our theoretical radius is almost large enough when we take into account the fact that the proton structure must be folded in before comparing with experiment. Furthermore, we took the excited particles with their respective probabilities out of the levels of the 2he0 shell in accordance with the statistical weight of each level and located them in the levels of the 3hto shell in accordance with the same
ZERO-POINT
VIBRATIONS
AND
THE NUCLEAR
181
SURFACE
criterium. The new root mean square radius obtained in this way was 3.38 fm, giving an increase of 1.5%. The density distributions for both correlated and uncorrelated nuclei are shown in fig. 1 for the Woods-Saxon potential and, for comparison, also for a harmonic oscillator well with hto = e = 14 MeV. The change in surface thickness through the correlation is not easy to evaluate, the usual definition as the distance between points of 90 and 10 values of the central density being obviously applicable only to two-parameter density distributions. Because the electron scattering will be very insensitive to the bump at the origin in any case, we can get some idea of the change in the surface thickness by taking the density at ~ 2 fm as the central density. In this case, the 90 to 10% distance Q ( r ) in f m -3
SO
H.O.a
~n
~o
30
~o
so
r~fm)
Fig. 1. Density distributions for the ground state in Ca40 with Woods-Saxon (W.S.) and harmonic oscillator (H.O.) potentials: a: . . . . uncorrelated; b: _ _ correlated, with the average number of particles excited equal to 2.35 (see text). The normalization is 5~p(r)r2 dr ~ 40. is 1.76 fm in the uncorrelated case, and 1.84 fm in the correlated one, giving a 4 . 4 ~ increase in surface thickness. In the calculation of Brueckner et al. ~) one has a similar difficulty in defining a surface thickness, since the distribution is not a simple two-parameter one. In either our case or in theirs one obtains already too large a surface thickness in the uncorrelated case by taking literally the distance between 90 and 10 % points. Again, these authors obtain a central bump which will not be sensitive to electron scattering. The only way to settle the question of whether the surface thickness is too small or too large in the uncorrelated distribution is to calculate the electron scattering directly for each distribution. The rms radius is, however, less ambiguous, and comes out 20% too small in the theory of Brueckner et al.; these authors further point out that this discrepancy would be hard to eliminate in their theory.
182
O. E. BROWN AND O. JACOB
Although in the case of a liquid drop 2) the increase in surface diffuseness is of first order, and that in the rms radius only of second order, we see that they are numerically of the same size in our calculation, stemming from the fact that calcium is mainly surface. Anderson and Thouless took into account vibrations of angular m o m e n t u m 2, 3, 4, 5. In fact, a strong 5 - vibration has been observed at 4.4$ MeV in Ca 4°, and would have as large an effect as the 3 - one. In addition, 2 + and 4 + vibrations are probably present, but have not been observed yet. Effects o f even-parity vibrations are difficult to calculate in our formalism, since they lead to particles in unbound states and one does not know how to handle these. It seems that the total increase in rms radius will be several per cent, but will fall short of the 20 ~o required to bring agreement with experiment. These calculations show that particles are to be found a reasonable part of the time in higher shells than those assigned to them by the shell model, the average number of "excited" particles resulting from the 3 - vibration being 2.35. I f these are present, clearly pickup reactions, such as (p, d) and (d, t) will show them. From the nature of the correlations, the pickup of such a particle will leave a hole and a vibration; that is, the final nucleus will be in all of the states which can be reached by coupling the hole and vibration. This will require an average energy higher than that required to reach the ground state o f the final nucleus by about the energy of the vibration. We would like to thank D. J. Thouless and J. A. Evans for valuable advice and criticism. One of us (G. J.) wishes to thank the Ford Foundation for a fellowship and the C a m p a n h a Nacional de Aperfei~oamento de Pessoal de Nivel Superior (CAPES) for a travel grant, which made his stay in the very pleasant and stimulating atmosphere of the Copenhagen Institute possible. Refereuces I) 2) 3) 4) 5) 6)
K. A. Brueckner, A. M. Lockett and M. Rotenberg, Phys. Rev. 121 (1961) 255 P. W. Anderson and D. ft. Thouless, Phys. Lett. I (1962) 155 G. E. Brown, J. A. Evans and D. J. Thouless, Nuclear Physics 24 (1961) 1 Vincent Gillet and Nicole Vinh-Mau, Phys. Lett. 1 (1962) 25 J. HOgaasen-Feldman, Nuclear Physics 28 (1961) 258 D. J. Thouless, Nuclear Physics 22 (1961) 78; D. J. Thouless, The quantum mechanics of malty-body systems (Academic Press, New York, 1961); G. E. Brown, Lectures on Many Body Problems, NORDITA, Copenhagen, 1961 7) Robert Herman and Robert Hofstadter, High-energy electron scattering tables (Stanford University Press, Stanford, 1960)