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Level densities for random one- or two-body potentials
I lsC I
LEVEL
Nuclear Physics Al84 (1972) 507-532;
@ North-Holland
Publishing
Co.,
Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
DENSITIES
FOR RANDOM
ONE-
OR TWO-BODY
POTENTIALS
ANNIE GERVOIS Service de Physique Centre d’Etudes
Thbrique,
NuclLaires
de Saclay,
BP. no. 2 - 91 - Gif-sur-Yvette
Received 30 November
1971
Abstract: The level density for a given number of particles in one shell is calculated from the grand partition function for random one- or two-body interactions. We find a Gaussian distribution, in agreementwithrecent numericalresults,except for low particle (or hole) density i.e. near closed shells, where the semi-circle law is obtained.
1. Introduction Shell-model calculations often lead to huge diagonalizations in a configuration space of large dimension. It is sometimes sufficient to simulate a part or the whole nuclear interaction by a random Hermitian matrix V. It is then assumed that the behaviour of the set of energy levels is well described by average functions such as level densities, level spacing distributions, etc. In the statistical model proposed by Wigner the matrix elements of V in the configuration space with N particles are taken as random independent variables. Then, the level density p(N, E) associated with V follows the semi-circle law ls2):
P(N,
E) = -&
44x3
when 1~1 < &%?,
= 0
otherwise,
(1.1)
where JV’ is the dimension of the space and o the dispersion of one-matrix element. Numerical calculations with real one- or two-body matrix elements 3, 4), however, show that the level density looks rather like a Gaussian p(N, E) cc --L
exp ( -E2/202), J 2l7ca
(1.2)
of dispersion c (CJdepending on N). The difference between results (1.1) and (1.2) seems to indicate that the most important features of the potential have not been described in the statistical model: first, the nuclear interaction is a one- and two-body force, and secondly, we deal with fermions. There are two ways of taking this into account: 507
508
ANNIE
GERVOIS
(a) Either we impose more restrictive symmetry conditions on the matrix elements of the N-particle configuration space. A model was proposed by Dubner “). He found a Gaussian for the level density but, because of the linear dependence of the eigenvalues in the parameters he had no level repulsion. (b) Either we take the one- or two-body matrix elements as random independent variables. Numerical tests were performed in the p-f and s-d shells 6,‘). The level density is a Gaussian in nearly all cases. In this paper, we choose the latter point of view. We examine under which conditions the level density is a Gaussian and we study the transition to other forms, for example to the semi-circle law. In sect. 2, we derive from the grand partition function Z(a, /3) the characteristic function 4p(N, /3) associated with the level density p(N, 15). Each term of the expansion of In Z(cr, p) in powers of the inverse temperature j? is a sum of terms associated with connected diagrams; their contribution is calcufated using the generalization of Wick’s theorem for statistical mechanics. When the summation of the dominant diagrams is possible, we get (p(N, j?) by integration over the variable a. Practically, in many cases, we must restrict ourselves to the calculation of the lower-order diagrams and therefore of the first reduced cumulants of the distribution. We assume that their knowledge is sufficient to describe the behaviour of&N, E), mainly to decide if it looks rather like a Gaussian or a semi-circle. In sects. 3 and 4, we discuss the case of one- and two-body potentials. The level density is in general a Gaussian especially for mid-shell nuclei. We recover the semi-circle law in some cases of the low particle density limit, i.e. near closed shells. The definition of the low particle density will depend on the type of nuclear interaction, and from the results obtained for one- and two-body potentials, we can see in particular that for more complicated forces (three-body or four-body potentials, etc.) and smafl shells the level density is no longer a Gaussian but rather a semi-circle. 2. General formalism The level density p(N, E) at the energy E for a system of N particles is the Fourier transform of the grand partition function Z(a, j3):
P(K E) = &
da d/? Z(a, P)e-OLNSBE,
(2.la)
where the integrations run along parallels to the imaginary axis. If we express the level density in terms of its characteristic function (p(N, a) by (2.ib) we see that (2Sc)
LEVEL DENSITIES
509
When the Hamiltonian is partially or totally simulated by a random matrix, we do not get the exact level density p(N, E) but its expectation value p(N, E) which is a smooth function and which is expected to describe the mean features of the real system. Eqs. (2.1) are replaced by F(N, E) =
(2.2a)
ij(N, E) =
(2.2b)
@_(N,P> =
(2.2c)
where Z(a, /I) is obtained by averaging the real partition function Z(a, /I). 2.1. THE GRAND PARTITION FUNCTION
In the second quantization formalism, we can write Z(cc, fi) = Tr exp (aN - /Ml),
(2.3)
where H is the Hamiltonian and N the number of particles operator: x N = c a:ak.
(24
k=l
We denote by al the creation operator in the one-particle state Ik) and by ak the corresponding annihilation operator. The summation is extended to a finite number of one-particle states II), 12), . . ., IA’“). Practically, JV will be the number of oneparticle states of a complete shell. Let us introduce for any Hermitian operator A the notation “) (A) = Tr {eaNA}/Z(a, 0),
(2.5)
Z(a, 0) = Tr (eaN> = (1 + ea)X.
(2.6)
where Using the definition (2.3) we expand Z(a, f?) in powers of H; we get Z(a, /3)jZ(a, 0) = 1 (-)” /P(P). n20 n!
(2.7)
The contribution of (H”) may be represented by a sum of closed diagrams of the nth order “). The number of diagrams to be calculated can be reduced by using the expansion of In Z(a, /I) in powers of /I. We have then ln CZ(a, B)/Z(a, O)] = C (< nhi
n.
B”
(24
510
ANNIE
GERVOIS
where (H”), represents the connected diagrams of (H”). They can be calculated using a generalization of Wick’s theorem “) where the contractions are replaced the averages :
by by
= L,f-,
= &,_f +.
(2.9)
We have set fand a,,,,, is the Kronecker
= 1-f’
symbol: 6,” = 0 = 1
2.2. THE CHARACTERISTIC
The characteristic powers of jI as
(2.10)
= e”/(l+e”),
if
m # n,
for
m=n.
FUNCTION
function
cp(N, b) associated
with p(N, E) may be expanded
in
(2.11) where the &” are the moments of the distribution law of the eigenvalues. It will be sometimes more convenient to use the expansion of In q(N, B) which may be written: (2.12) where CJis the dispersion have the relations Y2
=
1,
y3
=
(=A3
and the 7” are the reduced cumulants
3d,
-
y4 = (A4 -34;
-4J,
=
s
and in the case of random
We
(2.13)
AZ + 2&)/cJ3, A3 + 12&f
JY~ -6=&)/04,
For a Gaussian distribution law yn = 0 for n r 2. From the definition (2.1~) and the eqs. (2.7)-(2.11) tion A,,
of the distribution.
daZ(cr, O)(H”)e-aN is
etc.
we get the fundamental
dcr Z(a, O)eTaN,
Hamiltonians eqs. (2.11) to (2.14) hold when cp(N, /3) by Cp(N, p) and (H”) by
rela-
(2.14) replacing have peror
dominant
511
LEVEL DENSITIES
diagrams for every n and then sum the series (2.7) and (2.11). In most cases, however, we are only able to calculate the first-order diagrams and then the first moments and cumulants of the law. Nevertheless, it is sufficient for deciding whether the level density is approximately a Gaussian or not. In subsect. 2.3, we recall the method already used by Ratcliff “) to describe the behaviour of p(N, E). 2.3. THE METHOD
FOR EVALUATING
DEVIATIONS
FROM THE GAUSSIAN
SHAPE
An exact expression for p(N, E) (resp. p(N, E)) can be obtained using an expansion in terms of the Hermite polynomials. Setting for conveniency A, = 0, we may write 9, p(N, E)/p(N, 0) = e-EZ’2a2
(1+kg3; ~kw4) 3
(2.15)
where the Z’, are the Hermite polynomials and the 1, are related to the yk. Using the expansion (2.15) and the orthogonality relations of the Hermite polynomials to calculate the first moments of the distribution law p(N, E), we get the expression of the Jk in terms of the reduced cumulants: 1, = 1, i, = il, = 0, & =
Y3,
‘-4
=
Y4 7
A5
=
Ys,
iv6
=
y6 +
27
=
y7+35y3y4,
&
= ~s+5%9~3+35~:,
lay:,
etc.
For example, for a Gaussian law & = 0 for k 2 1. For the semi-circle we have I 2k+l
-0 -
for every k,
I, = -1,
&j = +5, ns = -21,
etc.
In practice, only the first terms can be taken into account. We assume that, even if the expansion (2.15) has no meaning, (i) the truncation at a particular order may give a good approximation, except for the tail of the distribution and (ii) that it is sufficient to test the order of magnitude of I, (asymmetry parameter) and I, (excess coefficient). If both are small compared to that of the semi-circle law, we assume that this will also
512
ANNIE GERVOIS
hold for higher-order coefficients and that a good approximation p(N, E)) is then P(N, Q/&Y
0) = e -E2’20Z{1+~~3~,(Eia)+~/Z,3Y,(Elo)).
for p(N, E) (resp. (2.16)
The inclusion of 1, and J6 should not modify very much the result. On the contrary, if 3,s or I, is large, /2,, A, and higher-order cumulants will presumably be large too and the expansion (2.15) will have no meaning.
-
Gaussian p (x)
with
A, only
p (xf
with
A,, only
0.8
0.6 0.5 0.2 0.3 0.2 0.1
Fig. 1. Plot of the approximation of two symmetric distribution laws p1 and pz in terms of Hermite polynomials when taking fir only and when taking & and &, for several values of the reduced energy variable x = E/a.The Gaussian distribution (full curve) is plotted for comparison.
As a test, we have studied two symmetric laws (%3 = & = 0) obtained from a one-body potential (see sect. 3) and plotted in fig. 1 the distributions (a) with A, only, and (b) with 1,, and rl,. The approximations (a) and (b) do not give very different results and we may expect a rapid convergence of the expansion (2.15). The agreement with a Gaussian shape is quite good, except for the tail of the distribution, although the coefficients d are not very small. In the best case, 1, = -0.33! Fig. 2 shows the corresponding approximations for the semi-circle law, For small values of x = E/C, the approximation (b) seems to give the best fit; nevertheless, the
LEVEL
513
DENSITIES
results are bad in general and the expansion (2.15) does not seem to converge rapidly. In the following, we shall consider only random one- or two-body potentials with an even distribution law for the matrix elements. Many simplifications occur, e.g. the odd 1, vanish. When the resummation (2.7) or (2.8) is not possible, we shall simply calculate the first coefficient A4 using eqs. (2.9), (2.10), (2.13) and (2.14). For small &(li_,l < 0.40) the average level density jT(N, E) looks rather like a Gaussian; when i 4% - 1, we shall assume that it has a semi-circle shape.
-
” semi _ circle”
A-______ __-
0.6 -
0.5 -
0.L -
0.3 -
0.2 -
01 -
{:,zyS K :
/
,;I-.
%(/
Fig. 2. Approximations
of the semi-circle law in terms of Hermite polynomials when taking A4 and I,, and when taking I,, Iti6 and 1,.
3. One-body
when taking
I, only,
potentials
In this section, we shall discuss the simple case where the Hamiltonian reduces to a one-body force V: V (3.1)
514
ANNIE GERVOIS
with the hermiticity condition (ml V/n) =*.
(3.2)
The (ml Y(E) are considered as random variables. The calculations are simple and, though this case is not the most interesting one, the conclusions of the discussion are not very different from those obtained for twobody potentials.
r-(r+P
ff -Pf+
Fig. 3. Diagrams contributing to ( V4>, for a one-body potential. Each particle line gives an fcr each hole line an f-. The sign is (-)l+L, where L is the number of particle lines.
In what follows, we shah introduce rather than for the matrix elements of matrices. The two models will give N = 1 because the (ml Vln} are then 3.1. THE GENERAL
simple statistical assumptions for the (ml V/n) the N-particle configuration space as in Wigners different distributions except, however, when the matrix elements of the configuration space.
CASE
The (ml Vln> are random independent variables for m 2 n with an even distribution law and the same dispersion u. For convenience, we set (ml V/m> = 0. After some calculations and using the results (2.9) and (2.10) we get, = f-f’
c l
t
W4), = f-f+;i-wf+)
c ~~l~ln>~~l~l~~<~,~l~>~s,~l~>, m,n s,r
, = f-f+[1-19f-~++loo(f-~+)2]
c (m~vin>(nlVJr>(rlI/ls) nt,n,r s, t, u x (Sl v~t)(t~V~U)(26~l/~m>,
(3.3)
and more generally V2”)C = f-f+RKf+)
c
*
’
.
where P,, is a polynomial of order (a- 1) in terms of the symmetricai variable f-f’ :P,(f-f+)= 7-a,f-f++L7,_,(f-f+)2YIt is the sum of thecontribution~
LEVEL
of (2ntributing
l)! diagrams.
As an example,
515
DENSITIES
we have plotted
in fig. 3 the six diagrams
con-
to (V”),.
We now take the average on the matrix elements.
0
This yields
for every n,
= f-f+JV(JV--l)u2,
and, when ,2’ is not too small (Jr
> lo),
(V”>,
= 2f-~+(l-6f-j+)J’“(~V-l)~v~,
(r/6),
z 5f-f+[1-19f-f++100(f-f+)2]JV(JV-1)3u4..
The general
..
term is of the form (V’“),
~
@)! n’(+1),
f-f+P,U-f+)Jtr(~-l)“vZ”,
where Lx, = (2n)!/n!(n+
l)!,
3
4 i2
.
G
il
Fig. 4. Graph
corresponding
is the number
to a contribution
of ways of identifying
to ( P),
for a one-body
the one-particle
potential,
in the general
case.
states so that the expression
. . .
becomes a product of even powers of individual matrix elements, of maximum order in N [ref. ‘)I (each identification introduces a small factor l/N). For example, the product <~,I~l~2~~~2l~l~3~~~3l~l~2j~~2I~l~4~~~4l~I~2~~~2l~~l~~~,
,gives a contribution
to ( V6), of the order z
((ilVl j)2)3/hr2.
A graphical representation of the above contribution is shown in fig. 4. More generally, CI,,is the number of (rooted) trees of order n [ref. “)I. Since the correlations between matrix elements may be neglected, we may write In [Z(LX,fl)/Z(tx, 0)] = A’ no, C n,(fy,,,
f-f’pn(f-f+)(Jlr-l)“tl*“.
(3.4)
516
ANNIE
GERVOIS
We must now perform the integration on the variable CIand we shall express the characteristic function q(N, fi) in terms of the dimensionless variable (/kr). We shall use for Jz/’ 2 k+k’, the relations L
dae-OL%5(a,O)(f-)k(f’)k
2i7t s
= 0
if
N < k,
WI
(which may be interpreted as expressing that we cannot create in the shell more holes than the number N of particles) and for N 2 k and M 2 k+ k’,
-!2in
s
&eeaN
Z(ff, o)(~-)k~~t)k‘=
yp;“‘) ,
(3.6)
is the binomial coefficient. There are analogous reiations when Jt“ < k+ k’ but we shall not need them. We obtain immediately Jz2 = o2 = N(M--N)U2 l=T
&
4
Jc@--1) -
(JV-2)(/V--3)
N(~-l)(~-N)(~-N-1~
[3-
$1
v4
+2(JV-1)N(JV-N)04.. In the appendix A, we derive the expression for ds. We shall consider four cases: 3.1.1. JV + to and N/X has a bind d~~r~~~~~~ approximately equal to N”(&- - N)k’ & &-ktk’
(N)’
0 and 1. The integral (3.6) is
(;)’
and the contribution coming from ( VZn)Jcr2” is of the order of I/XN-i. erally, the averages ( VZnl)C. . . ( pk)c/62(ni give a contribution
..
More gen-
**.wd,
of the order of l/~M
. . . +nic-kd
They may be negiected except when n, = . . . = n, = 1. We have then s
da emaNZ(ol,0)( V”‘) z (2n - l)! ! dcre-“Z(cL, 0)(( V’))“, s d@e-‘NZ(ar, O)
/JdneeffHZ((x,
0)] x (2n- l)!!c?“.
(3.7)
517
LEVEL DENSITIES
From eqs. (2.2b) and (2.7) we get (p(N
fi)
9
c
=
B” L p
=
n&O2” n!
exp (*fi2cr2).
The averaged level density p(N, E) is then a Gaussian. 3.1.2. A’- + co and N CC.$‘- (or _.&‘-N-cX). The integral (2.2b) can be approximated by using the saddle-point method lo). Since N is small, the saddle point will occur for small values off-. So, we may use for ( 1/2”)C the first-order approximation in f -:
By resumming the series (2.8) we get In [Z(a, /I)/Z(cr, 0)] x A”f-
C (““)” N?l n!(n+l)!
1 = MfN”
[$ (5)
-I],
with G(x) = 1,(2x)/x, where II(2x) is the modified Bessel function. The function I/I(X) is the characteristic function of the semi-circle law ‘). We have: kX
s
y dx $(x)eYX= i
J4 _ #
im
= 0
when
lyl < 2,
otherwise.
The saddle-point equation reads now N = Jtrf-I(l(lWjN), and the second derivative at the saddle is S = $
log Z(a, P)lsa.~,e = N.
We get then the remarkable equation @(N, /.I) x $;
tjN(@/,/N) +.
(3.8)
+ Eq. (3.8) can be derived in another way from the approximate expression for In z(a, fi). We have GW
B) = $--
s
dfiexp[-N
lnf-
+Nf-$(Ba/JN)],
or setting 2 =f-,
f where the integral is taken along a circle centered at the origin. From the Cauchytheorem,
weget (3.8).
518
ANNIE
For N = 1, the level density
GERVOIS
follows the semi-circle
law; when N increases,
p(N, E)
becomes closer to a Gaussian shape (central limit theorem). In fact, for N = 3 it looks already very much like a Gaussian (note that, in this case A4 = -3). 3.1.3. J’” sufficiently large (Jr 2 10) and N = 1. The only contributing diagrams have only one hole propagator [see eq. (3.5)]. We then have:
s
da eeaNZ(u, 0)( V”‘) = /da
and only the lower-order term integration, we may write:
where $ denotes
of P,,(f-f’)
emaNZ(a, 0)( 1/2n)C, g ives a non-zero
as above the characteristic
function
(3.9)
contribution.
of the semi-circle
After
law.
TABLE 1 Values
of y4 and y6 for one-body
.1’ = 12 -
N Y4 Ys
.I‘ = 24
YI
1 -1 +5
N Y.4 Y6
.I‘=
random
N Y4 Ys
potentials in the general values of ,t. and N 3 -0.45 2.27
1
3
case (subsect
3.1) for
different
6 -0.35 1.32 6
9
-1 -i-5
-0.31 1.25
-0.22 0.56
1
3
6
-1 15
-0.33 0.55
-0.17 0.14
-0.18 0.38
12 -0.17 0.34
3.1.4. Jr 2 10 andN # 1. We cannot easily sum all the diagrams to all orders and we restrict ourselves to the calculation of the first cumulants y4 and y6. We have studied in detail the case JV” = 12 (s-d shell for identical particles). The approximations of the density for N = 6 and N = 3 calculated with I, and & are plotted in fig. 1 and the results have already been discussed in sect. 2. The agreement with a Gaussian is good. In table 1, we give y4 and ys for several values of N for .,P’ = 12, 24 (s-d shell for protons and neutrons) and J’ = co. As expected, distributions become closer to normal when .&- increases except for a nearly full or empty shell where the level density always follows the semi-circle law.
LEVEL 3.2. CIRCULAR
519
DE-YSITIES
MATRICES
In this subsection,
we discuss briefly a model with more restrictive symmetry
ditions, which was partially studied by Dubner “). For convenience, we assume that &” is odd: .,#’ = 2p+ 1. A circular
con-
potential
is
defined by v=
c
k,q=-p,...+p
with the hermiticity
(3.10)
Vkaq+$tk,
condition (3.11)
vk = v_*, ,
and the convention that, for q +k > p (resp. q+ k < -p) the index 4 -+k stands for q+k-2p-1 (resp. q+k+2p+l). We assume that V, = 0. The expressions (3.3) for the connected part of ( V2n} still hold. They may be rewritten in the form:
(v2”>, = f-f+P,,(f-j-+)~r
c v,, **-
vk,,,
where P,(f-f’) has the same meaning as in the above paragraph; the summation is extended to all the k,(jk, 1 5 p) such that Cki is a muEtiple of 2p + 1. We assume that the vk (k 2 0) are random real independent variables with an even distribution law and dispersions vk (k 2 0). For JV sufficiently large, we have
We shall indicate
shortly the results for the four cases considered
above.
The proofs
can be carried out in the same manner. 3.2.1. JV --+ co, N/M has a limit diytrerentfrom 0 and 1. connected
The contribution of the parts { V2”)C can be neglected for n > 1. Eq. (3.7) is still valid and fi(N, E)
has a Gaussian shape. 3.2.2. JV -+ co, N < JV. small f’- , we have
We use again
ln [z(a, P)/Z(6 O)] = Jtrf and the saddle-point
equation
the saddle-point
approximation.
-Cexp(3B2 F 4) - 11,
reads:
whence
which is similar to the result (3.8). We get a Gaussian
shape even for N = 1.
For
520
ANNIE GERVOIS
3.2.3. N = 1, J1,’ suficiently large.
Eq. (3.9) still holds, As
we may write
3.24. N > 1, JV sufjiciently large. On table 2, we give the excess parameter y4 for J’” = 12,24 and JV” = co and several values of N. The conclusions are not exactly the same as in subsect. 3.1. The Gaussian distribution is a iow particle density limit or a large shell limit. For JV = 12, the mid-shell nuclei have a level density which no longer behaves like a Gaussian. This shows the limit of such a model. TABLET Values of y4 for circuiar potentials
“4‘ = 12
.h = 24
-4‘ = co
(subsect. 3.2) for different values of Jlr and N 6
N
1
2
3
Y4
0
-0.36
-0.48
N
1
2
3
6
9
12
?f4
0
-0.15
-0.22
-0.25
-0.26
-0.26
N
1
2
3
6
74
0
0
0
0
-_
OS6
As a final remark to this section, let us just point out that the discrepancy between the semi-circle and the Gaussian laws does not seem to be related to a particular choice of statistical assumptions but rather to the number of independent random variables compared with the dimension of the configuration space. With one-body forces, there are at most Jr 2 distinct random variables. For N 2 3, the number of matrix eiements in the configuration space is of the order of JV’, JV’. It is still more striking for circular matrices. Even for N = 1, the order of magnitude is different. 4. Two-body potentials We now consider a two-body random potential V: V = 4 2
where the (mnlVlrs>
are the antisymmetrized (mnl V(x)
= (srl Vlmn)*
and the (mni Vlsr) are random variables.
(4.1)
two-body matrix elements. We have: = -
(4.2)
--_-ufl
-----0 -_-_ 0------_ _4 -_ 1-c LEVEL DENSITIES
a.
_-_--
p=1
b.
521
p=2
0: _--
0
C.
hD w-m
_
Of
”
_-_
0
Fig. 5. Dominant diagrams in < V2P), when M + COand N/M is finite; (a) for p = 1,2, (b) some other dominant diagrams, (c) generation of one dominant diagram by an insertion of the second order on a particle line. There are analogous diagrams for insertion on a hole line.
-a_
__-
__
-_
___
---____
p=1
p=
2
Fig. 6. Ladders contributing
p=3
p-L
to for p = 1, 2, 3 and 4.
ANNIE
522
GERVOIS
As a simple model, we may assume that the (mnl Vlsu) are statistically independent. In fact, however, in nuclear physics, more symmetry conditions are required, for example rotational invariance, functions of a much smaller which may be then chosen
parity conservation, etc. The matrix number of parameters - the reduced as random
independent
variables.
elements are then matrix elements -
This case will be con-
sidered later on. 4.1.
THE
GENERAL
CASE
The statistical assumptions are the following: (i) Except for th e symmetry conditions (4.2) the (mnl Vlsv) are random independent variables and have the same probability distribution (actually the same dispersion denoted by u). It is not necessary to specify this distribution but we shall assume symmetry so that the level density will be even. (ii) The shell is large: .,1 . --) co. We distinguish 4 cases: 4.1,1. JV --, co, N/J/‘ has a limit different from 0 and 1. Since JI’ is large, the correlations between matrix elements (3.6) is asymptotically
s
dcr emaNZ(a, O)(f-)k(f+)k’
may be neglected;
/ JdEemuNZ(ar,
for 1 << N < ~fi’, the integral
0) z (,$)”
(I - 5)“.
(4.3)
We may then write
where f - is replaced
in the evaluation
of the average
( ) by the relative
occupation
number
It is easy to see now that, except for the quadratic term ( V2)C, we must impose some supplementary condition on the indices of intermediate states to extract from ( V2p), terms which are sums of products of even powers of the matrix elements. Each identification of intermediate states introduces a small factor l/J and all the higher-order terms, for p > 1 may be neglected. This yields a Gaussian distribution in E. More precisely, it is shown in appendix B that the dominant diagrams are generated by inserting a part of the second-order diagram on a particle or a hole line (see fig. 5). Their contribution is proportional to
and can be neglected
for I) > 1.
523
LEVEL DENSITIES
41.2. Extreme low density (N = 2) and bw density (N/N CC1). When N = 2, we cannot create more than 2.-holes (formula (3.5)). The only contributing diagrams to ( V2p), and therefore to ( Vzp) after integration on the variable c1will be the ladder diagrams (fig. 6). -----..___ ____ ______ ~~ ---._._____ -----__..c. __
b
a e._
---_“_.._
-----_“____
_“a*_
~~
0 .._*..___
_-*____
0
_^______
b
Fig. 7. Dominant diagrams at low density in the general case: (a) diagram of order p = 7, and (b) diagrams contributing
to < V4)C.
We have: <;+
= (f-)2(f+)4p-2 2 2P
[zl(h
jllW2
j2Xi2
_i2lW3
_b>
. . .
jJ
+ terms in (f-)3(f’)4p-3
+ . . ..
Performing the integration in ~1,we get s
dae-4NZ(% O)(~2p) = $
r.Y$(i, jIIVk
j,>(& j2iVli3 j,) . . -
2,
--
G2,~2plWl
jlh
__~._~.
and after averaging the matrix elements, the integral becomes
s
daebuNZ(q 0)( V2p>
z gp
2
2Pm’~V2crp
&
cf!-
1 P
,
524
ANNIE
where aP denotes mediate
the number
GERVOIS
of ways in which we can identify
( iz j2) . . . - or in shorthand,
states (iIjl),
the indices
the couples
of inter-
(I), (2), . . . - two by
two so that the expression
a product
j21vliS j& . . .
j&i,
of euen powers of individual
a small factor
matrix elements.
I/,,1 “‘. As in sect. 3, the coefficient clP = (2p)!/p!(p
jI>, Each identification
a, is
+ l)!.
Using eqs. (2.11) and (2.14), then taking the Fourier transform of Cp(iV,/?), we obtain for p(N, E) a semi-circle distribution law of quadratic dispersion CT2=
+pd
w92/JI/‘2
+-
(4.5)
When N > 2, the number of contributing diagrams becomes larger. The dominant ones are the dressed ladders - a combination of ladders and of insertions of the second order on a particle line. One of them is shown in fig. 7a. The resummation is not easy and we shall restrict ourselves to the evaluation of the first cumulant y4. The diagrams contributing to (V”), in the low density limit are shown in fig. 7b. The integration in c1 yields C2 z bN(N-
I
da eeaNZ(cl, 0)( v’)
1)Jtr2v2,
/ 1 da eMbNZ(cz,0)
z J’“‘u”[+N(N-l)++N(N-l)(N-2)+&N(N-l)(N-2)(N-3)]. A simple calculation
gives then 6-4N IJ4 = GF)’
The values of y4 for several values of N are given on a transition from the semi-circle to the Gaussian shape increases. The transition is very slow, in opposition to For three- or four-body forces it would be presumably
that the contributing diagrams for a k-body potential in the kconfiguration space (N = k) and for large .,Y are also the ladders and thus, that the level has always a semi-circle shape. The quadratic mean is then
+ It can be shown more generally particle density
table 3. We see, as expected, when the number of particles the one-body potential case. still slower.
a2
525
LEVEL DENSITIES
41.3. Low hole density: N/M z 1. Now, the self-energy terms become very important. As in subsect. 4.1.2, we shall calculate only y4. We have cr2 w $c/V2v2(h + l)(h + 2) h(4h2 + 6h + 6)
Y4 = - (h+l)‘(h+2)’
’
where h = ,Y - N (h < JV) is the number of holes. The values of y4 for several h are tabulated on table 3. TABLE3 Values of yJ when A? + 00 for a general two-body potential low hole density Number of particles N or holes h y4 (for particles) y4 (for holes)
(a) at low particle density, and (b) at
1
2
3
6
12
-0.44
-1 -0.47
-1 -0.45
-0.60 -0.35
-0.32 -0.24
The results can be interpreted as follows. For h = 0, the only level is
as E, is a sum of a large number of random independent variables, it has a Gaussian distribution law (central limit theorem) whence y4 = 0. For h = 1, the off-diagonal terms are the Hartree-Fock matrix elements v,,,Swhich are random independent variables with a dispersion of the order of JM. The diagonal terms are essentially E,, which has a dispersion of the order of JV. The resulting level density is the convolution of a semi-circle and a Gaussian. This explains the intermediate value y4 = -$ which can also be obtained directly from convolution. For h > 1, we observe slow transition to a Gaussian shape, as in subsect. 4.1.2. One can wonder why are the results so different for h = 0 and h 2 1. In fact, they should not be interpreted in the same way. For iz 2 1, ~(JV- h, E) is a distribution function for the many levels. For h = 0, p(.M, E) is the distribution function of the unique level. At this point, let us notice that the above statistical assumptions are perhaps not well adapted to such limiting cases. It would be perhaps more realistic to impose supplementary constraints such as fixed trace, fixed Hartree-Fock matrix elements etc., constraints which may be dropped for a large number of holes. Lastly, we see that particles and holes do not play a symmetric role. This is due to the particular choice of the potential (4.1). Instead of the total nuclear force, we may consider the interaction V
SYm
=
v-3
c m,s
vmsum+us,
526
ANNIE GERVOJS
which has been symmetrized with respect to the particle-hole exchange. In this expression K,,, = c
of u and yj, when ,6” = 24 for a general two-body potential for several N: for (a) the total interaction, and (b) the symmetric interaction b
a -_____-.--. N ~~___._ 1 2 3 4 6 8 10 12 14 16 18 20 21 22 23 24
a ___-
._
0 16.6 27.5 37.2 53.4 65.4 73.5 77.5 77.5 73.5 65.4 53.4 45.8 37.2 27.5 16.6
The value of d is given in units of
Y4
0
Y*
0 -1 -0.47 -0.40 -0.41 -0.43 -0.42 -0.42 -0.41 -0.37 -0.34 -0.31 -0.34 -0.37 -0.50 0
11.5 22 31.5 40 54 64 70 12 70 64 54 40 31.5 22 11.5 0
-1 -1 -0.60 -0.47 -0.42 -0.42 -0.44 -0.44 -0.44 -0.42 -0.42 -0.47 -0.60 -1 -1
V.
4.1.4. JV not too large. Let us finally consider the case where -,z/’is not too large. All diagrams contribute in every order. So, we restrict ourselves to the calculation of the first reduced cumulant y4. In the appendix C, we give the general expressions for ( V2>, and ( V4>,. The dispersion (r and the excess parameter y4 are derived then from eqs. (2.13) and (2.14). We get CT2= *N(N-1)(..N-N+1)(&-N+2), which is symmetric around N= ) JV + I. The values of y4 for J’* = 24 (which may be interpreted as the d-s shell for protons and neutrons) and several N are tabulated in table 4a. In table 4b we give the values of y4 for the symmetric interaction. We see that: (i) Results fo r t he total and the symmetric interactions are not very different except at low hole density where the self-energy terms become important.
LEVEL
DENSITIES
521
(ii) As when ~9” is large, we get a semi-circle shape for N = 2 and a transitional form when JV -N = I ; the excess parameter y4 is minimum in the middle of the shell. However, the agreement with a Gaussian shape is not very good. It would presumably be still worse for three- OKfour-body potentials, i.e. when the number of random independent variables becomes large compared to the dimension of the space. In this latter case the assumption of statistical independence of the matrix elements in the configuration space is valid. In conclusion, we may say that the semi-circle law occurs at low particle density or for complicated interactions. TABLE 5 Values of o and y4 for a rotationally
invariant two-body potential as a function of N
(a) In the f-shell for protons ‘V
3 4.84 0.18
0
Y4
4
5
6
7
5.85 0.24
6.57 0.09
7.01 -0.06
-7.16 0.12
(b) In the s-d shell for protons N d
Y4
3
4
5
6
4.54 -0.43
5.35 -0.12
5.81 0.09
6.57 0.13
(c) In the s-d shell for protons and neutrons N
3
4
Y4
4.70 -0.79
4.2.
ROTATIONALLY
0
5.85 -0.33
5 6.81 -0.17
7
6 7.15 -0.13
INVARIANT
8.50 -0.13
8 9.11 -0.15
9 9.58 -0.17
10 9.92 -0.19
11 10.12 -0.20
12 10.19 -0.21
POTENTIALS
Let us assume now that the interaction is rotationally invariant. The matrix eiements (mnl Vlsr) may be expressed in terms of a much smaller number of parameters - the reduced matrix elements. In what follows, we assume that each one-particle state lm) is defined by the following quantum numbers: energy E,, orbital momentum l,, totai momentuIn j, and its projection on the z-axis, isospin and its projection on the z-axis (if there are protons and neutrons). The reduced matrix elements may be written as (8 mm E j In, -6 nn 1 j nIIC%,L_L~~,bj,), or
in
shorthand notation,
<.L.&ll&7l.L.&);
P-6)
when there is no ambiguity and we shall adopt this notation from now. Here, J and T are the total angular momentum and isospin of the pairs (mn) and (sY).
ANNIE
528
GERVOIS
The reduced matrix elements verify symmetry relations. For example, for a system of protons and neutrons:
=
= f-)““s+‘-J-T(j,j,ll~fTIi~,js);
(4.7)
in particular:
(_L.hllK31.Ljr~
= 0
when
J+ T is odd.
The Hartree-Fock field is diagonal and its matrix elements depend only on the total angular momentum of the one-particle state of the shell and the essential difference with the general case studied before is that the Hartree-Fock terms become important even for mid-shell nuclei - at least for the rather small shells (~9” = 12, 14, 24) which we have considered. It is then more realistic to separate V into two parts
where V, is known and can be diagonalized and V2 is supposed to be a small perturbation and is totally random. At the beginning of the shell, Y2 is generally the residual nuclear interaction of the shell model, but near a complete shell it is better to take the total interaction (particle-hole symmetry). In these two limiting cases the results depend strongly on the statistical assumptions. We prefer therefore to choose for VZ the particle-hole symmetric interaction
already defined in subsect. 4.1. We assume that the reduced matrix elements which appear in V, are independent random variables with the same distribution law. The law is taken even, of dispersion U. The values of d and y4 for three shells and a symmetric interaction are tabulated in table 5. As expected from the choice of I’,, the agreement with a Gaussian is better in the middle of the shell. We notice that: (i) The dispersion increases with N and is maximum in the middle of the shell. (ii) The agreement with a Gaussian is good even for very small shells (N = 12,14) but does not improve if we have protons and neutrons, perhaps because the number of parameters is larger. (iii) For shells o f similar size (s-d shell and f-shelf), the behaviour depends on the nature of the shell and from a practical point of view, many of the simplifications of the first model disappear. In particular, we cannot select the dominant terms by using a diagrammatic representation. (iv) Finally, let us say that except at the ends of the shell (N = 1, Z), the excess parameter y4 does not depend much on the distribution law of one reduced matrix elements. Variations in ya do not exceed 5 %. It would be interesting to look for the level density of the residual interaction, taking into account a self-consistent field. We expect smaller dispersions, but the model will then presumably be more sensitive to the statistical assumptions for the reduced matrix elements. This has not been done yet.
LEVEL DENSITIES
This study has been suggested
529
to me by Professor
C. Bloch, and I would like to
thank him for the many discussions I have had with him. I am grateful to Drs. 0. Bohigas, J. des Cloizeaux, B. Giraud and M. L. Mehta for many interesting suggestions and comments. Appendix A CALCULATION
OF ( V6) AND A6 FOR A ONE-BODY
POTENTIAL
We have (V6)
= (V6),+6~+60(V)(~2)c(V3jc+20(V3),(V)3+45(V)2(V2)f +15(1/)4(v2)c+(v)6,
where the index c stands for connected simplifies:
part. Since there are no diagonal
terms, ( V6)
=,+15(1/2),
In the general case (subsects. 3.1 and 4.1), the contribution of (V”),’ neglected; after integration we get using eqs. (3.3)-(3.6) and (2.14b), (;)
&6
= 5J1/^(x-1)3u6 + 30&-2(M
[(;I;)
- I)%6 [(;I;)
-19
($“I;)
-6
+100
(;I;)]
may
be
($-I;)]
+15Jlr3(M-1)3
(;I;),
or still
4/d
5(&X)2
=
N’(X +
(
&-(M-
- N)2
1)”
N-l
M-N-1
N
N-N
(M-2)(Jtr-3)
1 N(X-N)
ls-F+&y )
JV2(Jv - 1)2 x (N-2)(&-3)(Jli--4)(&--5) A similar calculation count the contribution
N-l
N-2
&“-N-l
7
7
F-N--
can be done for circular coming from (V”),‘. Appendix
DOMINANT DIAGRAMS FROM 0 OR 1)
IN ( Vzp>,
matrices,
X-N-2 Jy‘-N
*
but we must take into ac-
B
WHEN J’” + 03 (N/M
HAS A LIMIT
DIFFERENT
The contribution of ( Vzp), is a sum of products of 2p matrix elements and, because we have only connected diagrams, two matrix elements differ at least by one of their indices (except when 2p = 2).
The non-zero contribution to ( Yzp), is a sum of products of the squares of p matrix elements; they are obtained by making equal two by two the matrix elements of ( Y2P)C i.e. by identifying all their indices, In a given product of ( V2P>C,let i‘,denote the ~~~~~~~~ number of ~den~cat~ons for two matrix elements to be equal. We have 1 5 .X5 4. Zf?”= I, in the product, we have two terms of the form (mnl Vlri>(rjl V/mn).
(B.1)
When setting i = j, we get a small factor l~J^“_ If another square would appear, it would mean that actually, there are four terms of the form (mnl Vlrl’)(rjl V~mn)(m’n’~Vlr’j)(r’il
Vlm’n’).
This is impossible (disconnected part) except for 2p = 4. The terms (B.l) can be interpreted as an insertion of the second order on a particle or a hole line (fig. 5~). When taking the average, the rest of the product may be represented by a connected diagram of order Zp- 2. ZfA = 2, we have in the product two matrix elements of the form
(mnl wwklt
mn>,
(B.2)
(tEr‘/ Vjrj)
fffF?Sj).
WI
Or
When making i = k, j = I, we get a small factor I/JI~^‘. If another square would appear, it would be necessarily of the form
~I~~>(m’n’fV[kl)(ijl Yfm”n’>,
or (mi~v~rj)(Pljt7~PYfk)
of the product to ( Y2g)C is then proportional [~~~(rntz~v]sr>“]“/JP”-
to
I.
The corresponding diagrams in ( VzP), are generated by insertion of a part of the second-order diagram on a particle or a hole line (see fig. 5~). Some examples are shown OIXfigs.5a and 5b. A simpte interpretation consists in taking the assumption of a large system implying conservation of momentum at each interaction r ‘)_
LEVEL
DENSITIES
Appendix OF (V’>,
EXPRESSION
AND
0%
(V”),
531
C
( V4>, FOR THE TOTAL
= Wf’)‘C
INTERACTION
ww~~)2+f-3f+ ?lI”Sr
= (f-)“f_f+(l-sf-f+)
c
vm,,, iQ,,<
c cs ms
vm,,,,,,
9
rm,,,,
[ml - 12f-3(f-f+y(f-
-f+),~“,~m,f?!
+2f-2(f-f+)2(1-3f-f+)
C Cm, k,h rm, nl
+2f-2(f-~‘)2(1-6f-f’) +12f-2(f-f+)3
vm?“?!,~~m”m’(m”‘n”‘lVlmm”)
n,iliim2n2)(m2n2i~imm’)
C ~j ~“‘(mnlVlirn’)(n’m’(Vlmn) C cn ~j,,(mnlvlm’i)(m’n’lvImj>
+2f-2(f-f’)2(1-6f-f’)
C ~” ~j”,(mnlvlm~)(m’n’IVlmi>
+_f-(f-_f+)‘(l-3f-f+)(f+
-f_)
c q,(nrnl
Vlm’n’)(m’n’l
Yl,“,“) x (m”n”l
+6f-(f-f’)3(f’
c ~,.~(mn~V~m’n’)(m’n’~V~m”n)(m”n”~V~mi)
-f-)
+12f-(f-f+)3(f-
m’n”)(m’n’lVlm”n)(m”n”lVlin’)
-_f+> c V&?znll/l
+&(_f-f+)‘[l-4f-f+
Vlin)
+6(f-f+)2]
c (mnl~lm’n’)(m’~‘ll/Im”n”) x (WPn”~V~Wr”‘Fz”‘)(m,‘rn’rr(V~rnn)
-i-W-f+)“(l-6fT’)
c
(4U
m’n’)(m’n’lVlm”)(m”n”lvld”n”‘) x (m”‘d”l Vld’n)
+2(f-~+)3(l-3f-f+)
c
+3(f-.f+)4 +
c
(f-f ‘)“(I - 6f-f ’)
C (4
VIm’n”‘)(m’n’~V~mn”)(m”n”~V~m”‘n) x (m”‘n”‘l V(m”n’),
where we use the notation Pm,, = C
n
and the summations run over all the indices. For the symmetric interaction we have
(V2>, = +(.f-f+)2 C (mnlvlsr>2+tfT+(1-4f-f+) mnsr
C VAT ms
532
ANNIE GERVOIS
and the coefficients of the nine first terms of < V4>, are replaced
by the symmetric
coefficients &J-f+(1-4f-j+)2(1-6f-f+) -3(f-j+)2(1-4f-f+)2 !T(f-f+)‘(l-4fT+)(l-3f-f+) +(f-f+)2(l-4f-j+)(1 3(f-f+)3(l
-6f-f+) -4Tf’)
3(f-f+)‘(l-4fT+)(l-6f-f+) -3(f-j+)‘(1-3f-f+)(l-4f-f+) -3(f-j+)3(1
-4j-j’)
6(f-f+)3(1
-4’7+).
References I) E. P. Wigner, Can. Math. Congr. Proc. (University of Toronto Press, Toronto, Canada, 1957) p. 174; Ann. Math. 62 (1955) 548 A. Gervois, Nuovo Cim. 69B (1970) 181; Nucl. Phys. B25 1971) 551-556 J. B. French and J. C. Parikh, preprint K. F. Ratcliff, Phys. Rev. C3 (1971) 117 V. M. Dubner, Theor. Prob. Appl. 14 (1969) 342 0. Bohigas and J. Flores, Phys. Lett. 34B (1971) 261 J. B. French and S. S. M. Wong, Phys. Lett. 33B (1970) 449 C. Bloch and C. de Dominicis, Nucl. Phys. 7 (1958) 459 M. G. Kendall and A. Stuart, The advanced theory of statistics, vol. 1 (Charles Griffin, London) 156-162 10) N. G. de Bruijn, Asymptotic methods in analysis (North-Holland, Amsterdam) p. 177 and following 11) C. Bloch, A survey of nuclear level density theories, Albany Conf. on statistical properties of nuclei, 1971, State University of New York at Albany, to be published 2) 3) 4) 5) 6) 7) 8) 9)
LEVEL
Nuclear Physics Al84 (1972) 507-532;
@ North-Holland
Publishing
Co.,
Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
DENSITIES
FOR RANDOM
ONE-
OR TWO-BODY
POTENTIALS
ANNIE GERVOIS Service de Physique Centre d’Etudes
Thbrique,
NuclLaires
de Saclay,
BP. no. 2 - 91 - Gif-sur-Yvette
Received 30 November
1971
Abstract: The level density for a given number of particles in one shell is calculated from the grand partition function for random one- or two-body interactions. We find a Gaussian distribution, in agreementwithrecent numericalresults,except for low particle (or hole) density i.e. near closed shells, where the semi-circle law is obtained.
1. Introduction Shell-model calculations often lead to huge diagonalizations in a configuration space of large dimension. It is sometimes sufficient to simulate a part or the whole nuclear interaction by a random Hermitian matrix V. It is then assumed that the behaviour of the set of energy levels is well described by average functions such as level densities, level spacing distributions, etc. In the statistical model proposed by Wigner the matrix elements of V in the configuration space with N particles are taken as random independent variables. Then, the level density p(N, E) associated with V follows the semi-circle law ls2):
P(N,
E) = -&
44x3
when 1~1 < &%?,
= 0
otherwise,
(1.1)
where JV’ is the dimension of the space and o the dispersion of one-matrix element. Numerical calculations with real one- or two-body matrix elements 3, 4), however, show that the level density looks rather like a Gaussian p(N, E) cc --L
exp ( -E2/202), J 2l7ca
(1.2)
of dispersion c (CJdepending on N). The difference between results (1.1) and (1.2) seems to indicate that the most important features of the potential have not been described in the statistical model: first, the nuclear interaction is a one- and two-body force, and secondly, we deal with fermions. There are two ways of taking this into account: 507
508
ANNIE
GERVOIS
(a) Either we impose more restrictive symmetry conditions on the matrix elements of the N-particle configuration space. A model was proposed by Dubner “). He found a Gaussian for the level density but, because of the linear dependence of the eigenvalues in the parameters he had no level repulsion. (b) Either we take the one- or two-body matrix elements as random independent variables. Numerical tests were performed in the p-f and s-d shells 6,‘). The level density is a Gaussian in nearly all cases. In this paper, we choose the latter point of view. We examine under which conditions the level density is a Gaussian and we study the transition to other forms, for example to the semi-circle law. In sect. 2, we derive from the grand partition function Z(a, /3) the characteristic function 4p(N, /3) associated with the level density p(N, 15). Each term of the expansion of In Z(cr, p) in powers of the inverse temperature j? is a sum of terms associated with connected diagrams; their contribution is calcufated using the generalization of Wick’s theorem for statistical mechanics. When the summation of the dominant diagrams is possible, we get (p(N, j?) by integration over the variable a. Practically, in many cases, we must restrict ourselves to the calculation of the lower-order diagrams and therefore of the first reduced cumulants of the distribution. We assume that their knowledge is sufficient to describe the behaviour of&N, E), mainly to decide if it looks rather like a Gaussian or a semi-circle. In sects. 3 and 4, we discuss the case of one- and two-body potentials. The level density is in general a Gaussian especially for mid-shell nuclei. We recover the semi-circle law in some cases of the low particle density limit, i.e. near closed shells. The definition of the low particle density will depend on the type of nuclear interaction, and from the results obtained for one- and two-body potentials, we can see in particular that for more complicated forces (three-body or four-body potentials, etc.) and smafl shells the level density is no longer a Gaussian but rather a semi-circle. 2. General formalism The level density p(N, E) at the energy E for a system of N particles is the Fourier transform of the grand partition function Z(a, j3):
P(K E) = &
da d/? Z(a, P)e-OLNSBE,
(2.la)
where the integrations run along parallels to the imaginary axis. If we express the level density in terms of its characteristic function (p(N, a) by (2.ib) we see that (2Sc)
LEVEL DENSITIES
509
When the Hamiltonian is partially or totally simulated by a random matrix, we do not get the exact level density p(N, E) but its expectation value p(N, E) which is a smooth function and which is expected to describe the mean features of the real system. Eqs. (2.1) are replaced by F(N, E) =
(2.2a)
ij(N, E) =
(2.2b)
@_(N,P> =
(2.2c)
where Z(a, /I) is obtained by averaging the real partition function Z(a, /I). 2.1. THE GRAND PARTITION FUNCTION
In the second quantization formalism, we can write Z(cc, fi) = Tr exp (aN - /Ml),
(2.3)
where H is the Hamiltonian and N the number of particles operator: x N = c a:ak.
(24
k=l
We denote by al the creation operator in the one-particle state Ik) and by ak the corresponding annihilation operator. The summation is extended to a finite number of one-particle states II), 12), . . ., IA’“). Practically, JV will be the number of oneparticle states of a complete shell. Let us introduce for any Hermitian operator A the notation “) (A) = Tr {eaNA}/Z(a, 0),
(2.5)
Z(a, 0) = Tr (eaN> = (1 + ea)X.
(2.6)
where Using the definition (2.3) we expand Z(a, f?) in powers of H; we get Z(a, /3)jZ(a, 0) = 1 (-)” /P(P). n20 n!
(2.7)
The contribution of (H”) may be represented by a sum of closed diagrams of the nth order “). The number of diagrams to be calculated can be reduced by using the expansion of In Z(a, /I) in powers of /I. We have then ln CZ(a, B)/Z(a, O)] = C (< nhi
n.
B”
(24
510
ANNIE
GERVOIS
where (H”), represents the connected diagrams of (H”). They can be calculated using a generalization of Wick’s theorem “) where the contractions are replaced the averages :
by by
= &,_f +.
(2.9)
We have set fand a,,,,, is the Kronecker
= 1-f’
symbol: 6,” = 0 = 1
2.2. THE CHARACTERISTIC
The characteristic powers of jI as
(2.10)
= e”/(l+e”),
if
m # n,
for
m=n.
FUNCTION
function
cp(N, b) associated
with p(N, E) may be expanded
in
(2.11) where the &” are the moments of the distribution law of the eigenvalues. It will be sometimes more convenient to use the expansion of In q(N, B) which may be written: (2.12) where CJis the dispersion have the relations Y2
=
1,
y3
=
(=A3
and the 7” are the reduced cumulants
3d,
-
y4 = (A4 -34;
-4J,
=
s
and in the case of random
We
(2.13)
AZ + 2&)/cJ3, A3 + 12&f
JY~ -6=&)/04,
For a Gaussian distribution law yn = 0 for n r 2. From the definition (2.1~) and the eqs. (2.7)-(2.11) tion A,,
of the distribution.
daZ(cr, O)(H”)e-aN is
etc.
we get the fundamental
dcr Z(a, O)eTaN,
Hamiltonians eqs. (2.11) to (2.14) hold when cp(N, /3) by Cp(N, p) and (H”) by
rela-
(2.14) replacing have peror
dominant
511
LEVEL DENSITIES
diagrams for every n and then sum the series (2.7) and (2.11). In most cases, however, we are only able to calculate the first-order diagrams and then the first moments and cumulants of the law. Nevertheless, it is sufficient for deciding whether the level density is approximately a Gaussian or not. In subsect. 2.3, we recall the method already used by Ratcliff “) to describe the behaviour of p(N, E). 2.3. THE METHOD
FOR EVALUATING
DEVIATIONS
FROM THE GAUSSIAN
SHAPE
An exact expression for p(N, E) (resp. p(N, E)) can be obtained using an expansion in terms of the Hermite polynomials. Setting for conveniency A, = 0, we may write 9, p(N, E)/p(N, 0) = e-EZ’2a2
(1+kg3; ~kw4) 3
(2.15)
where the Z’, are the Hermite polynomials and the 1, are related to the yk. Using the expansion (2.15) and the orthogonality relations of the Hermite polynomials to calculate the first moments of the distribution law p(N, E), we get the expression of the Jk in terms of the reduced cumulants: 1, = 1, i, = il, = 0, & =
Y3,
‘-4
=
Y4 7
A5
=
Ys,
iv6
=
y6 +
27
=
y7+35y3y4,
&
= ~s+5%9~3+35~:,
lay:,
etc.
For example, for a Gaussian law & = 0 for k 2 1. For the semi-circle we have I 2k+l
-0 -
for every k,
I, = -1,
&j = +5, ns = -21,
etc.
In practice, only the first terms can be taken into account. We assume that, even if the expansion (2.15) has no meaning, (i) the truncation at a particular order may give a good approximation, except for the tail of the distribution and (ii) that it is sufficient to test the order of magnitude of I, (asymmetry parameter) and I, (excess coefficient). If both are small compared to that of the semi-circle law, we assume that this will also
512
ANNIE GERVOIS
hold for higher-order coefficients and that a good approximation p(N, E)) is then P(N, Q/&Y
0) = e -E2’20Z{1+~~3~,(Eia)+~/Z,3Y,(Elo)).
for p(N, E) (resp. (2.16)
The inclusion of 1, and J6 should not modify very much the result. On the contrary, if 3,s or I, is large, /2,, A, and higher-order cumulants will presumably be large too and the expansion (2.15) will have no meaning.
-
Gaussian p (x)
with
A, only
p (xf
with
A,, only
0.8
0.6 0.5 0.2 0.3 0.2 0.1
Fig. 1. Plot of the approximation of two symmetric distribution laws p1 and pz in terms of Hermite polynomials when taking fir only and when taking & and &, for several values of the reduced energy variable x = E/a.The Gaussian distribution (full curve) is plotted for comparison.
As a test, we have studied two symmetric laws (%3 = & = 0) obtained from a one-body potential (see sect. 3) and plotted in fig. 1 the distributions (a) with A, only, and (b) with 1,, and rl,. The approximations (a) and (b) do not give very different results and we may expect a rapid convergence of the expansion (2.15). The agreement with a Gaussian shape is quite good, except for the tail of the distribution, although the coefficients d are not very small. In the best case, 1, = -0.33! Fig. 2 shows the corresponding approximations for the semi-circle law, For small values of x = E/C, the approximation (b) seems to give the best fit; nevertheless, the
LEVEL
513
DENSITIES
results are bad in general and the expansion (2.15) does not seem to converge rapidly. In the following, we shall consider only random one- or two-body potentials with an even distribution law for the matrix elements. Many simplifications occur, e.g. the odd 1, vanish. When the resummation (2.7) or (2.8) is not possible, we shall simply calculate the first coefficient A4 using eqs. (2.9), (2.10), (2.13) and (2.14). For small &(li_,l < 0.40) the average level density jT(N, E) looks rather like a Gaussian; when i 4% - 1, we shall assume that it has a semi-circle shape.
-
” semi _ circle”
A-______ __-
0.6 -
0.5 -
0.L -
0.3 -
0.2 -
01 -
{:,zyS K :
/
,;I-.
%(/
Fig. 2. Approximations
of the semi-circle law in terms of Hermite polynomials when taking A4 and I,, and when taking I,, Iti6 and 1,.
3. One-body
when taking
I, only,
potentials
In this section, we shall discuss the simple case where the Hamiltonian reduces to a one-body force V: V (3.1)
514
ANNIE GERVOIS
with the hermiticity condition (ml V/n) =
(3.2)
The (ml Y(E) are considered as random variables. The calculations are simple and, though this case is not the most interesting one, the conclusions of the discussion are not very different from those obtained for twobody potentials.
r-(r+P
ff -Pf+
Fig. 3. Diagrams contributing to ( V4>, for a one-body potential. Each particle line gives an fcr each hole line an f-. The sign is (-)l+L, where L is the number of particle lines.
In what follows, we shah introduce rather than for the matrix elements of matrices. The two models will give N = 1 because the (ml Vln} are then 3.1. THE GENERAL
simple statistical assumptions for the (ml V/n) the N-particle configuration space as in Wigners different distributions except, however, when the matrix elements of the configuration space.
CASE
The (ml Vln> are random independent variables for m 2 n with an even distribution law and the same dispersion u. For convenience, we set (ml V/m> = 0. After some calculations and using the results (2.9) and (2.10) we get
c l
t
W4), = f-f+;i-wf+)
c ~~l~ln>~~l~l~~<~,~l~>~s,~l~>, m,n s,r
c (m~vin>(nlVJr>(rlI/ls) nt,n,r s, t, u x (Sl v~t)(t~V~U)(26~l/~m>,
(3.3)
and more generally V2”)C = f-f+RKf+)
c
*
’
.
where P,, is a polynomial of order (a- 1) in terms of the symmetricai variable f-f’ :P,(f-f+)= 7-a,f-f++L7,_,(f-f+)2YIt is the sum of thecontribution~
LEVEL
of (2ntributing
l)! diagrams.
As an example,
515
DENSITIES
we have plotted
in fig. 3 the six diagrams
con-
to (V”),.
We now take the average on the matrix elements.
0
This yields
for every n,
= f-f+JV(JV--l)u2,
and, when ,2’ is not too small (Jr
> lo),
(V”>,
= 2f-~+(l-6f-j+)J’“(~V-l)~v~,
(r/6),
z 5f-f+[1-19f-f++100(f-f+)2]JV(JV-1)3u4..
The general
..
term is of the form (V’“),
~
@)! n’(+1),
f-f+P,U-f+)Jtr(~-l)“vZ”,
where Lx, = (2n)!/n!(n+
l)!,
3
4 i2
.
G
il
Fig. 4. Graph
corresponding
is the number
to a contribution
of ways of identifying
to ( P),
for a one-body
the one-particle
potential,
in the general
case.
states so that the expression
. . .
becomes a product of even powers of individual matrix elements, of maximum order in N [ref. ‘)I (each identification introduces a small factor l/N). For example, the product <~,I~l~2~~~2l~l~3~~~3l~l~2j~~2I~l~4~~~4l~I~2~~~2l~~l~~~,
,gives a contribution
to ( V6), of the order z
((ilVl j)2)3/hr2.
A graphical representation of the above contribution is shown in fig. 4. More generally, CI,,is the number of (rooted) trees of order n [ref. “)I. Since the correlations between matrix elements may be neglected, we may write In [Z(LX,fl)/Z(tx, 0)] = A’ no, C n,(fy,,,
f-f’pn(f-f+)(Jlr-l)“tl*“.
(3.4)
516
ANNIE
GERVOIS
We must now perform the integration on the variable CIand we shall express the characteristic function q(N, fi) in terms of the dimensionless variable (/kr). We shall use for Jz/’ 2 k+k’, the relations L
dae-OL%5(a,O)(f-)k(f’)k
2i7t s
= 0
if
N < k,
WI
(which may be interpreted as expressing that we cannot create in the shell more holes than the number N of particles) and for N 2 k and M 2 k+ k’,
-!2in
s
&eeaN
Z(ff, o)(~-)k~~t)k‘=
yp;“‘) ,
(3.6)
is the binomial coefficient. There are analogous reiations when Jt“ < k+ k’ but we shall not need them. We obtain immediately Jz2 = o2 = N(M--N)U2 l=T
&
4
Jc@--1) -
(JV-2)(/V--3)
N(~-l)(~-N)(~-N-1~
[3-
$1
v4
+2(JV-1)N(JV-N)04.. In the appendix A, we derive the expression for ds. We shall consider four cases: 3.1.1. JV + to and N/X has a bind d~~r~~~~~~ approximately equal to N”(&- - N)k’ & &-ktk’
(N)’
0 and 1. The integral (3.6) is
(;)’
and the contribution coming from ( VZn)Jcr2” is of the order of I/XN-i. erally, the averages ( VZnl)C. . . ( pk)c/62(ni give a contribution
..
More gen-
**.wd,
of the order of l/~M
. . . +nic-kd
They may be negiected except when n, = . . . = n, = 1. We have then s
da emaNZ(ol,0)( V”‘) z (2n - l)! ! dcre-“Z(cL, 0)(( V’))“, s d@e-‘NZ(ar, O)
/JdneeffHZ((x,
0)] x (2n- l)!!c?“.
(3.7)
517
LEVEL DENSITIES
From eqs. (2.2b) and (2.7) we get (p(N
fi)
9
c
=
B” L p
=
n&O2” n!
exp (*fi2cr2).
The averaged level density p(N, E) is then a Gaussian. 3.1.2. A’- + co and N CC.$‘- (or _.&‘-N-cX). The integral (2.2b) can be approximated by using the saddle-point method lo). Since N is small, the saddle point will occur for small values off-. So, we may use for ( 1/2”)C the first-order approximation in f -:
By resumming the series (2.8) we get In [Z(a, /I)/Z(cr, 0)] x A”f-
C (““)” N?l n!(n+l)!
1 = MfN”
[$ (5)
-I],
with G(x) = 1,(2x)/x, where II(2x) is the modified Bessel function. The function I/I(X) is the characteristic function of the semi-circle law ‘). We have: kX
s
y dx $(x)eYX= i
J4 _ #
im
= 0
when
lyl < 2,
otherwise.
The saddle-point equation reads now N = Jtrf-I(l(lWjN), and the second derivative at the saddle is S = $
log Z(a, P)lsa.~,e = N.
We get then the remarkable equation @(N, /.I) x $;
tjN(@/,/N) +.
(3.8)
+ Eq. (3.8) can be derived in another way from the approximate expression for In z(a, fi). We have GW
B) = $--
s
dfiexp[-N
lnf-
+Nf-$(Ba/JN)],
or setting 2 =f-,
f where the integral is taken along a circle centered at the origin. From the Cauchytheorem,
weget (3.8).
518
ANNIE
For N = 1, the level density
GERVOIS
follows the semi-circle
law; when N increases,
p(N, E)
becomes closer to a Gaussian shape (central limit theorem). In fact, for N = 3 it looks already very much like a Gaussian (note that, in this case A4 = -3). 3.1.3. J’” sufficiently large (Jr 2 10) and N = 1. The only contributing diagrams have only one hole propagator [see eq. (3.5)]. We then have:
s
da eeaNZ(u, 0)( V”‘) = /da
and only the lower-order term integration, we may write:
where $ denotes
of P,,(f-f’)
emaNZ(a, 0)( 1/2n)C, g ives a non-zero
as above the characteristic
function
(3.9)
contribution.
of the semi-circle
After
law.
TABLE 1 Values
of y4 and y6 for one-body
.1’ = 12 -
N Y4 Ys
.I‘ = 24
YI
1 -1 +5
N Y.4 Y6
.I‘=
random
N Y4 Ys
potentials in the general values of ,t. and N 3 -0.45 2.27
1
3
case (subsect
3.1) for
different
6 -0.35 1.32 6
9
-1 -i-5
-0.31 1.25
-0.22 0.56
1
3
6
-1 15
-0.33 0.55
-0.17 0.14
-0.18 0.38
12 -0.17 0.34
3.1.4. Jr 2 10 andN # 1. We cannot easily sum all the diagrams to all orders and we restrict ourselves to the calculation of the first cumulants y4 and y6. We have studied in detail the case JV” = 12 (s-d shell for identical particles). The approximations of the density for N = 6 and N = 3 calculated with I, and & are plotted in fig. 1 and the results have already been discussed in sect. 2. The agreement with a Gaussian is good. In table 1, we give y4 and ys for several values of N for .,P’ = 12, 24 (s-d shell for protons and neutrons) and J’ = co. As expected, distributions become closer to normal when .&- increases except for a nearly full or empty shell where the level density always follows the semi-circle law.
LEVEL 3.2. CIRCULAR
519
DE-YSITIES
MATRICES
In this subsection,
we discuss briefly a model with more restrictive symmetry
ditions, which was partially studied by Dubner “). For convenience, we assume that &” is odd: .,#’ = 2p+ 1. A circular
con-
potential
is
defined by v=
c
k,q=-p,...+p
with the hermiticity
(3.10)
Vkaq+$tk,
condition (3.11)
vk = v_*, ,
and the convention that, for q +k > p (resp. q+ k < -p) the index 4 -+k stands for q+k-2p-1 (resp. q+k+2p+l). We assume that V, = 0. The expressions (3.3) for the connected part of ( V2n} still hold. They may be rewritten in the form:
(v2”>, = f-f+P,,(f-j-+)~r
c v,, **-
vk,,,
where P,(f-f’) has the same meaning as in the above paragraph; the summation is extended to all the k,(jk, 1 5 p) such that Cki is a muEtiple of 2p + 1. We assume that the vk (k 2 0) are random real independent variables with an even distribution law and dispersions vk (k 2 0). For JV sufficiently large, we have
We shall indicate
shortly the results for the four cases considered
above.
The proofs
can be carried out in the same manner. 3.2.1. JV --+ co, N/M has a limit diytrerentfrom 0 and 1. connected
The contribution of the parts { V2”)C can be neglected for n > 1. Eq. (3.7) is still valid and fi(N, E)
has a Gaussian shape. 3.2.2. JV -+ co, N < JV. small f’- , we have
We use again
ln [z(a, P)/Z(6 O)] = Jtrf and the saddle-point
equation
the saddle-point
approximation.
-Cexp(3B2 F 4) - 11,
reads:
whence
which is similar to the result (3.8). We get a Gaussian
shape even for N = 1.
For
520
ANNIE GERVOIS
3.2.3. N = 1, J1,’ suficiently large.
Eq. (3.9) still holds, As
we may write
3.24. N > 1, JV sufjiciently large. On table 2, we give the excess parameter y4 for J’” = 12,24 and JV” = co and several values of N. The conclusions are not exactly the same as in subsect. 3.1. The Gaussian distribution is a iow particle density limit or a large shell limit. For JV = 12, the mid-shell nuclei have a level density which no longer behaves like a Gaussian. This shows the limit of such a model. TABLET Values of y4 for circuiar potentials
“4‘ = 12
.h = 24
-4‘ = co
(subsect. 3.2) for different values of Jlr and N 6
N
1
2
3
Y4
0
-0.36
-0.48
N
1
2
3
6
9
12
?f4
0
-0.15
-0.22
-0.25
-0.26
-0.26
N
1
2
3
6
74
0
0
0
0
-_
OS6
As a final remark to this section, let us just point out that the discrepancy between the semi-circle and the Gaussian laws does not seem to be related to a particular choice of statistical assumptions but rather to the number of independent random variables compared with the dimension of the configuration space. With one-body forces, there are at most Jr 2 distinct random variables. For N 2 3, the number of matrix eiements in the configuration space is of the order of JV’, JV’. It is still more striking for circular matrices. Even for N = 1, the order of magnitude is different. 4. Two-body potentials We now consider a two-body random potential V: V = 4 2
where the (mnlVlrs>
are the antisymmetrized (mnl V(x)
= (srl Vlmn)*
and the (mni Vlsr) are random variables.
(4.1)
two-body matrix elements. We have: = -
(4.2)
--_-ufl
-----0 -_-_ 0------_ _4 -_ 1-c LEVEL DENSITIES
a.
_-_--
p=1
b.
521
p=2
0: _--
0
C.
hD w-m
_
Of
”
_-_
0
Fig. 5. Dominant diagrams in < V2P), when M + COand N/M is finite; (a) for p = 1,2, (b) some other dominant diagrams, (c) generation of one dominant diagram by an insertion of the second order on a particle line. There are analogous diagrams for insertion on a hole line.
-a_
__-
__
-_
___
---____
p=1
p=
2
Fig. 6. Ladders contributing
p=3
p-L
to
ANNIE
522
GERVOIS
As a simple model, we may assume that the (mnl Vlsu) are statistically independent. In fact, however, in nuclear physics, more symmetry conditions are required, for example rotational invariance, functions of a much smaller which may be then chosen
parity conservation, etc. The matrix number of parameters - the reduced as random
independent
variables.
elements are then matrix elements -
This case will be con-
sidered later on. 4.1.
THE
GENERAL
CASE
The statistical assumptions are the following: (i) Except for th e symmetry conditions (4.2) the (mnl Vlsv) are random independent variables and have the same probability distribution (actually the same dispersion denoted by u). It is not necessary to specify this distribution but we shall assume symmetry so that the level density will be even. (ii) The shell is large: .,1 . --) co. We distinguish 4 cases: 4.1,1. JV --, co, N/J/‘ has a limit different from 0 and 1. Since JI’ is large, the correlations between matrix elements (3.6) is asymptotically
s
dcr emaNZ(a, O)(f-)k(f+)k’
may be neglected;
/ JdEemuNZ(ar,
for 1 << N < ~fi’, the integral
0) z (,$)”
(I - 5)“.
(4.3)
We may then write
where f - is replaced
in the evaluation
of the average
( ) by the relative
occupation
number
It is easy to see now that, except for the quadratic term ( V2)C, we must impose some supplementary condition on the indices of intermediate states to extract from ( V2p), terms which are sums of products of even powers of the matrix elements. Each identification of intermediate states introduces a small factor l/J and all the higher-order terms, for p > 1 may be neglected. This yields a Gaussian distribution in E. More precisely, it is shown in appendix B that the dominant diagrams are generated by inserting a part of the second-order diagram on a particle or a hole line (see fig. 5). Their contribution is proportional to
and can be neglected
for I) > 1.
523
LEVEL DENSITIES
41.2. Extreme low density (N = 2) and bw density (N/N CC1). When N = 2, we cannot create more than 2.-holes (formula (3.5)). The only contributing diagrams to ( V2p), and therefore to ( Vzp) after integration on the variable c1will be the ladder diagrams (fig. 6). -----..___ ____ ______ ~~ ---._._____ -----__..c. __
b
a e._
---_“_.._
-----_“____
_“a*_
~~
0 .._*..___
_-*____
0
_^______
b
Fig. 7. Dominant diagrams at low density in the general case: (a) diagram of order p = 7, and (b) diagrams contributing
to < V4)C.
We have: <;+
= (f-)2(f+)4p-2 2 2P
[zl(h
jllW2
j2Xi2
_i2lW3
_b>
. . .
jJ
+ terms in (f-)3(f’)4p-3
+ . . ..
Performing the integration in ~1,we get s
dae-4NZ(% O)(~2p) = $
r.Y$(i, jIIVk
j,>(& j2iVli3 j,) . . -
2,
--
G2,~2plWl
jlh
__~._~.
and after averaging the matrix elements, the integral becomes
s
daebuNZ(q 0)( V2p>
z gp
2
2Pm’~V2crp
&
cf!-
1 P
,
524
ANNIE
where aP denotes mediate
the number
GERVOIS
of ways in which we can identify
( iz j2) . . . - or in shorthand,
states (iIjl),
the indices
the couples
of inter-
(I), (2), . . . - two by
two so that the expression
a product
j21vliS j& . . .
j&i,
of euen powers of individual
a small factor
matrix elements.
I/,,1 “‘. As in sect. 3, the coefficient clP = (2p)!/p!(p
jI>, Each identification
a, is
+ l)!.
Using eqs. (2.11) and (2.14), then taking the Fourier transform of Cp(iV,/?), we obtain for p(N, E) a semi-circle distribution law of quadratic dispersion CT2=
+pd
w92/JI/‘2
+-
(4.5)
When N > 2, the number of contributing diagrams becomes larger. The dominant ones are the dressed ladders - a combination of ladders and of insertions of the second order on a particle line. One of them is shown in fig. 7a. The resummation is not easy and we shall restrict ourselves to the evaluation of the first cumulant y4. The diagrams contributing to (V”), in the low density limit are shown in fig. 7b. The integration in c1 yields C2 z bN(N-
I
da eeaNZ(cl, 0)( v’)
1)Jtr2v2,
/ 1 da eMbNZ(cz,0)
z J’“‘u”[+N(N-l)++N(N-l)(N-2)+&N(N-l)(N-2)(N-3)]. A simple calculation
gives then 6-4N IJ4 = GF)’
The values of y4 for several values of N are given on a transition from the semi-circle to the Gaussian shape increases. The transition is very slow, in opposition to For three- or four-body forces it would be presumably
that the contributing diagrams for a k-body potential in the kconfiguration space (N = k) and for large .,Y are also the ladders and thus, that the level has always a semi-circle shape. The quadratic mean is then
+ It can be shown more generally particle density
table 3. We see, as expected, when the number of particles the one-body potential case. still slower.
a2
525
LEVEL DENSITIES
41.3. Low hole density: N/M z 1. Now, the self-energy terms become very important. As in subsect. 4.1.2, we shall calculate only y4. We have cr2 w $c/V2v2(h + l)(h + 2) h(4h2 + 6h + 6)
Y4 = - (h+l)‘(h+2)’
’
where h = ,Y - N (h < JV) is the number of holes. The values of y4 for several h are tabulated on table 3. TABLE3 Values of yJ when A? + 00 for a general two-body potential low hole density Number of particles N or holes h y4 (for particles) y4 (for holes)
(a) at low particle density, and (b) at
1
2
3
6
12
-0.44
-1 -0.47
-1 -0.45
-0.60 -0.35
-0.32 -0.24
The results can be interpreted as follows. For h = 0, the only level is
as E, is a sum of a large number of random independent variables, it has a Gaussian distribution law (central limit theorem) whence y4 = 0. For h = 1, the off-diagonal terms are the Hartree-Fock matrix elements v,,,Swhich are random independent variables with a dispersion of the order of JM. The diagonal terms are essentially E,, which has a dispersion of the order of JV. The resulting level density is the convolution of a semi-circle and a Gaussian. This explains the intermediate value y4 = -$ which can also be obtained directly from convolution. For h > 1, we observe slow transition to a Gaussian shape, as in subsect. 4.1.2. One can wonder why are the results so different for h = 0 and h 2 1. In fact, they should not be interpreted in the same way. For iz 2 1, ~(JV- h, E) is a distribution function for the many levels. For h = 0, p(.M, E) is the distribution function of the unique level. At this point, let us notice that the above statistical assumptions are perhaps not well adapted to such limiting cases. It would be perhaps more realistic to impose supplementary constraints such as fixed trace, fixed Hartree-Fock matrix elements etc., constraints which may be dropped for a large number of holes. Lastly, we see that particles and holes do not play a symmetric role. This is due to the particular choice of the potential (4.1). Instead of the total nuclear force, we may consider the interaction V
SYm
=
v-3
c m,s
vmsum+us,
526
ANNIE GERVOJS
which has been symmetrized with respect to the particle-hole exchange. In this expression K,,, = c
of u and yj, when ,6” = 24 for a general two-body potential for several N: for (a) the total interaction, and (b) the symmetric interaction b
a -_____-.--. N ~~___._ 1 2 3 4 6 8 10 12 14 16 18 20 21 22 23 24
a ___-
._
0 16.6 27.5 37.2 53.4 65.4 73.5 77.5 77.5 73.5 65.4 53.4 45.8 37.2 27.5 16.6
The value of d is given in units of
Y4
0
Y*
0 -1 -0.47 -0.40 -0.41 -0.43 -0.42 -0.42 -0.41 -0.37 -0.34 -0.31 -0.34 -0.37 -0.50 0
11.5 22 31.5 40 54 64 70 12 70 64 54 40 31.5 22 11.5 0
-1 -1 -0.60 -0.47 -0.42 -0.42 -0.44 -0.44 -0.44 -0.42 -0.42 -0.47 -0.60 -1 -1
V.
4.1.4. JV not too large. Let us finally consider the case where -,z/’is not too large. All diagrams contribute in every order. So, we restrict ourselves to the calculation of the first reduced cumulant y4. In the appendix C, we give the general expressions for ( V2>, and ( V4>,. The dispersion (r and the excess parameter y4 are derived then from eqs. (2.13) and (2.14). We get CT2= *N(N-1)(..N-N+1)(&-N+2), which is symmetric around N= ) JV + I. The values of y4 for J’* = 24 (which may be interpreted as the d-s shell for protons and neutrons) and several N are tabulated in table 4a. In table 4b we give the values of y4 for the symmetric interaction. We see that: (i) Results fo r t he total and the symmetric interactions are not very different except at low hole density where the self-energy terms become important.
LEVEL
DENSITIES
521
(ii) As when ~9” is large, we get a semi-circle shape for N = 2 and a transitional form when JV -N = I ; the excess parameter y4 is minimum in the middle of the shell. However, the agreement with a Gaussian shape is not very good. It would presumably be still worse for three- OKfour-body potentials, i.e. when the number of random independent variables becomes large compared to the dimension of the space. In this latter case the assumption of statistical independence of the matrix elements in the configuration space is valid. In conclusion, we may say that the semi-circle law occurs at low particle density or for complicated interactions. TABLE 5 Values of o and y4 for a rotationally
invariant two-body potential as a function of N
(a) In the f-shell for protons ‘V
3 4.84 0.18
0
Y4
4
5
6
7
5.85 0.24
6.57 0.09
7.01 -0.06
-7.16 0.12
(b) In the s-d shell for protons N d
Y4
3
4
5
6
4.54 -0.43
5.35 -0.12
5.81 0.09
6.57 0.13
(c) In the s-d shell for protons and neutrons N
3
4
Y4
4.70 -0.79
4.2.
ROTATIONALLY
0
5.85 -0.33
5 6.81 -0.17
7
6 7.15 -0.13
INVARIANT
8.50 -0.13
8 9.11 -0.15
9 9.58 -0.17
10 9.92 -0.19
11 10.12 -0.20
12 10.19 -0.21
POTENTIALS
Let us assume now that the interaction is rotationally invariant. The matrix eiements (mnl Vlsr) may be expressed in terms of a much smaller number of parameters - the reduced matrix elements. In what follows, we assume that each one-particle state lm) is defined by the following quantum numbers: energy E,, orbital momentum l,, totai momentuIn j, and its projection on the z-axis, isospin and its projection on the z-axis (if there are protons and neutrons). The reduced matrix elements may be written as (8 mm E j In, -6 nn 1 j nIIC%,L_L~~,bj,), or
in
shorthand notation,
<.L.&ll&7l.L.&);
P-6)
when there is no ambiguity and we shall adopt this notation from now. Here, J and T are the total angular momentum and isospin of the pairs (mn) and (sY).
ANNIE
528
GERVOIS
The reduced matrix elements verify symmetry relations. For example, for a system of protons and neutrons:
=
= f-)““s+‘-J-T(j,j,ll~fTIi~,js);
(4.7)
in particular:
(_L.hllK31.Ljr~
= 0
when
J+ T is odd.
The Hartree-Fock field is diagonal and its matrix elements depend only on the total angular momentum of the one-particle state of the shell and the essential difference with the general case studied before is that the Hartree-Fock terms become important even for mid-shell nuclei - at least for the rather small shells (~9” = 12, 14, 24) which we have considered. It is then more realistic to separate V into two parts
where V, is known and can be diagonalized and V2 is supposed to be a small perturbation and is totally random. At the beginning of the shell, Y2 is generally the residual nuclear interaction of the shell model, but near a complete shell it is better to take the total interaction (particle-hole symmetry). In these two limiting cases the results depend strongly on the statistical assumptions. We prefer therefore to choose for VZ the particle-hole symmetric interaction
already defined in subsect. 4.1. We assume that the reduced matrix elements which appear in V, are independent random variables with the same distribution law. The law is taken even, of dispersion U. The values of d and y4 for three shells and a symmetric interaction are tabulated in table 5. As expected from the choice of I’,, the agreement with a Gaussian is better in the middle of the shell. We notice that: (i) The dispersion increases with N and is maximum in the middle of the shell. (ii) The agreement with a Gaussian is good even for very small shells (N = 12,14) but does not improve if we have protons and neutrons, perhaps because the number of parameters is larger. (iii) For shells o f similar size (s-d shell and f-shelf), the behaviour depends on the nature of the shell and from a practical point of view, many of the simplifications of the first model disappear. In particular, we cannot select the dominant terms by using a diagrammatic representation. (iv) Finally, let us say that except at the ends of the shell (N = 1, Z), the excess parameter y4 does not depend much on the distribution law of one reduced matrix elements. Variations in ya do not exceed 5 %. It would be interesting to look for the level density of the residual interaction, taking into account a self-consistent field. We expect smaller dispersions, but the model will then presumably be more sensitive to the statistical assumptions for the reduced matrix elements. This has not been done yet.
LEVEL DENSITIES
This study has been suggested
529
to me by Professor
C. Bloch, and I would like to
thank him for the many discussions I have had with him. I am grateful to Drs. 0. Bohigas, J. des Cloizeaux, B. Giraud and M. L. Mehta for many interesting suggestions and comments. Appendix A CALCULATION
OF ( V6) AND A6 FOR A ONE-BODY
POTENTIAL
We have (V6)
= (V6),+6
where the index c stands for connected simplifies:
part. Since there are no diagonal
terms, ( V6)
=
In the general case (subsects. 3.1 and 4.1), the contribution of (V”),’ neglected; after integration we get using eqs. (3.3)-(3.6) and (2.14b), (;)
&6
= 5J1/^(x-1)3u6 + 30&-2(M
[(;I;)
- I)%6 [(;I;)
-19
($“I;)
-6
+100
(;I;)]
may
be
($-I;)]
+15Jlr3(M-1)3
(;I;),
or still
4/d
5(&X)2
=
N’(X +
(
&-(M-
- N)2
1)”
N-l
M-N-1
N
N-N
(M-2)(Jtr-3)
1 N(X-N)
ls-F+&y )
JV2(Jv - 1)2 x (N-2)(&-3)(Jli--4)(&--5) A similar calculation count the contribution
N-l
N-2
&“-N-l
7
7
F-N--
can be done for circular coming from (V”),‘. Appendix
DOMINANT DIAGRAMS FROM 0 OR 1)
IN ( Vzp>,
matrices,
X-N-2 Jy‘-N
*
but we must take into ac-
B
WHEN J’” + 03 (N/M
HAS A LIMIT
DIFFERENT
The contribution of ( Vzp), is a sum of products of 2p matrix elements and, because we have only connected diagrams, two matrix elements differ at least by one of their indices (except when 2p = 2).
The non-zero contribution to ( Yzp), is a sum of products of the squares of p matrix elements; they are obtained by making equal two by two the matrix elements of ( Y2P)C i.e. by identifying all their indices, In a given product of ( V2P>C,let i‘,denote the ~~~~~~~~ number of ~den~cat~ons for two matrix elements to be equal. We have 1 5 .X5 4. Zf?”= I, in the product, we have two terms of the form (mnl Vlri>(rjl V/mn).
(B.1)
When setting i = j, we get a small factor l~J^“_ If another square would appear, it would mean that actually, there are four terms of the form (mnl Vlrl’)(rjl V~mn)(m’n’~Vlr’j)(r’il
Vlm’n’).
This is impossible (disconnected part) except for 2p = 4. The terms (B.l) can be interpreted as an insertion of the second order on a particle or a hole line (fig. 5~). When taking the average, the rest of the product may be represented by a connected diagram of order Zp- 2. ZfA = 2, we have in the product two matrix elements of the form
(mnl wwklt
mn>,
(B.2)
(tEr‘/ Vjrj)
fffF?Sj).
WI
Or
When making i = k, j = I, we get a small factor I/JI~^‘. If another square would appear, it would be necessarily of the form
~I~~>(m’n’fV[kl)(ijl Yfm”n’>,
or (mi~v~rj)(Pljt7~PYfk)
of the product to ( Y2g)C is then proportional [~~~(rntz~v]sr>“]“/JP”-
to
I.
The corresponding diagrams in ( VzP), are generated by insertion of a part of the second-order diagram on a particle or a hole line (see fig. 5~). Some examples are shown OIXfigs.5a and 5b. A simpte interpretation consists in taking the assumption of a large system implying conservation of momentum at each interaction r ‘)_
LEVEL
DENSITIES
Appendix OF (V’>,
EXPRESSION
AND
0%
(V”),
531
C
( V4>, FOR THE TOTAL
= Wf’)‘C
INTERACTION
ww~~)2+f-3f+ ?lI”Sr
= (f-)“f_f+(l-sf-f+)
c
vm,,, iQ,,<
c cs ms
vm,,,,,,
9
rm,,,,
[ml - 12f-3(f-f+y(f-
-f+),~“,~m,f?!
+2f-2(f-f+)2(1-3f-f+)
C Cm, k,h rm, nl
+2f-2(f-~‘)2(1-6f-f’) +12f-2(f-f+)3
vm?“?!,~~m”m’(m”‘n”‘lVlmm”)
n,iliim2n2)(m2n2i~imm’)
C ~j ~“‘(mnlVlirn’)(n’m’(Vlmn) C cn ~j,,(mnlvlm’i)(m’n’lvImj>
+2f-2(f-f’)2(1-6f-f’)
C ~” ~j”,(mnlvlm~)(m’n’IVlmi>
+_f-(f-_f+)‘(l-3f-f+)(f+
-f_)
c q,(nrnl
Vlm’n’)(m’n’l
Yl,“,“) x (m”n”l
+6f-(f-f’)3(f’
c ~,.~(mn~V~m’n’)(m’n’~V~m”n)(m”n”~V~mi)
-f-)
+12f-(f-f+)3(f-
m’n”)(m’n’lVlm”n)(m”n”lVlin’)
-_f+> c V&?znll/l
+&(_f-f+)‘[l-4f-f+
Vlin)
+6(f-f+)2]
c (mnl~lm’n’)(m’~‘ll/Im”n”) x (WPn”~V~Wr”‘Fz”‘)(m,‘rn’rr(V~rnn)
-i-W-f+)“(l-6fT’)
c
(4U
m’n’)(m’n’lVlm”)(m”n”lvld”n”‘) x (m”‘d”l Vld’n)
+2(f-~+)3(l-3f-f+)
c
+3(f-.f+)4 +
c
(f-f ‘)“(I - 6f-f ’)
C (4
VIm’n”‘)(m’n’~V~mn”)(m”n”~V~m”‘n) x (m”‘n”‘l V(m”n’),
where we use the notation Pm,, = C
n
and the summations run over all the indices. For the symmetric interaction we have
(V2>, = +(.f-f+)2 C (mnlvlsr>2+tfT+(1-4f-f+) mnsr
C VAT ms
532
ANNIE GERVOIS
and the coefficients of the nine first terms of < V4>, are replaced
by the symmetric
coefficients &J-f+(1-4f-j+)2(1-6f-f+) -3(f-j+)2(1-4f-f+)2 !T(f-f+)‘(l-4fT+)(l-3f-f+) +(f-f+)2(l-4f-j+)(1 3(f-f+)3(l
-6f-f+) -4Tf’)
3(f-f+)‘(l-4fT+)(l-6f-f+) -3(f-j+)‘(1-3f-f+)(l-4f-f+) -3(f-j+)3(1
-4j-j’)
6(f-f+)3(1
-4’7+).
References I) E. P. Wigner, Can. Math. Congr. Proc. (University of Toronto Press, Toronto, Canada, 1957) p. 174; Ann. Math. 62 (1955) 548 A. Gervois, Nuovo Cim. 69B (1970) 181; Nucl. Phys. B25 1971) 551-556 J. B. French and J. C. Parikh, preprint K. F. Ratcliff, Phys. Rev. C3 (1971) 117 V. M. Dubner, Theor. Prob. Appl. 14 (1969) 342 0. Bohigas and J. Flores, Phys. Lett. 34B (1971) 261 J. B. French and S. S. M. Wong, Phys. Lett. 33B (1970) 449 C. Bloch and C. de Dominicis, Nucl. Phys. 7 (1958) 459 M. G. Kendall and A. Stuart, The advanced theory of statistics, vol. 1 (Charles Griffin, London) 156-162 10) N. G. de Bruijn, Asymptotic methods in analysis (North-Holland, Amsterdam) p. 177 and following 11) C. Bloch, A survey of nuclear level density theories, Albany Conf. on statistical properties of nuclei, 1971, State University of New York at Albany, to be published 2) 3) 4) 5) 6) 7) 8) 9)