Volume 87B, number 3
A STATISTICAL
PHYSICS LETTERS
STUDY
OF
5 November 1979
SHELL-MODEL EIGENVECTORS
JJ.M. VERBAARSCHOT and P.J. BRUSSAARD
Fysiseh LaboratoHum, Rijksuniversiteit Utrecht, 3308 TA Utrecht, The Netherlands
Received 21 November 1978
Revised manuscript received 23 May 1979
The components of shell-model eigenvectors show a non-gaussian distribution that can be derived by the use of the
Porter-Thomas proposition.
In nuclear shell-model calculations one is confronted with increasing dimensionalities of the configuration spaces. It makes sense, therefore, to study
the statistical properties of the eigenvectors of nuclear
states. The distribution of the amplitudes of the eigenvector components of a large-scale shell-model hamiltonian shows a predominance of the small amplitudes.
The distribution is not gaussian, however, but more
complicated. In this paper we shall derive an explicit
expression for the amplitude distribution from three
statistical assumptions. No parameters are to be fitted
for the final expression. Several papers (e.g. ref. [I])
have been devoted to studies of the gaussian orthogonal ensemble of matrices. Here, however, we do not
consider an ensemble of hamiltonians, but rather study
the eigenvalues and eigenvectors of one hamiltonian.
Let the eigenvectors ~bi of the hamiltonian, H, be
expanded in the basis {be} ((~ = 1,2 ..... N):
eigenstate of 22Na. One observes a secular variation
Icail as a function of Haa (solid line) and fluctuations
of the actual values of the amplitudes Icc~il around the
average.
These fluctuations will be described by the PorterThomas proposition [3] that the coefficients cai locally, i.e. for some small neighbourhood of Haa, show a
normal distribution with zero mean. This gaussian behaviour implies that the variance is proportional to the
secular value
- - 1 2
(b~l~i) 2 = (n/2) l(b~l~bi)l
(2)
We assume that the diagonal matrix elements
nee =<balnlbe),
(~ = 1,2 ..... N ) ,
O)
possess a gaussian distribution
N
t~i= ~
cc~idpa ,
i = 1 ..... N .
(1)
The two-body interaction used to construct the hamiltonian for our calculations was the MSDI [2]. The
basis states b~ were taken to span the full sd-shell. For
the J~ = 1/2 + states of 25Mg this leads to a dimension
1434 and for t h e J ~r = 1+ states of 22Na to a dimension 243. For a fixed eigenvector we can plot the values of the coefficients Ic~il = I<¢~1~0i>1versus the values of the diagonal matrix elements Has = (belHI be).
In fig. 1 such a plot is shown for the lowest jrr = 1+
Q4
u~O:
O.1
-
-6
-
6
lo
~*v
DIAGONAL ELEMENTS H~
Fig. 1. The amplitudes I((Pe I ~0i)l versus the diagonal elements
(g,e till (Pc~).The solid line is the secular variation Pr.
155
Volume 87B, number 3
PHYSICS LETTERS
5 November 1979
Table 1
Parameters that determine the gaussian distributions of Ham and ICodl (all in MeV). Centroids do not affect the physical results.
2z Na
2s Mg
7.92
0.06
-0.07
(a)
width oo
skewness
excess
7.24
-0.09
-0.30
(b)
j~r
width o1
skewness
excess
(1+)1
6.98
0.14
-0.39
Po(Ha,O = [N/(2rr)l/2oo] exp [ - ( H a a -
(1+)2
7.14
0.19
-0.38
(l+)a
7.07
0.21
-0.47
Eo)2/2o2l .
(1/2+)1
8.02
0.12
-0.18
Pr(Ha,)Po(Haa)--- Pl (Haa)
= [Mo/(2n)l/2ol] exp [-(Ham - E1)2/202] ,
p(lcai I) = [2/(2~r) 1/2 o] exp [-c2i/2o 2 ] ,
(6)
(7)
with the width o determined by the condition
x exp [-x2/2o 2] dx = P r ( H ~ ) .
(8)
This condition yields
202 = ~rp2(Ha,~).
The number of amplitudes ]cai I smaller than some
value, say F, in the interval (Haa, H,~a + AHaa} is
(11)
Za I c,,, i I o 0
N o 1 exp[(E1 - E0)2/2(o 2 - o2)] ,
Eo)/(4 - 4)
(9)
,
°; 2 = o-;2 _ o 2.
The distribution function o f the amplitudes I cai [ is
obtained from expression (10)when the integration
over x = ICai [ is suppressed and an integration over all
values ofH~, a is performed,
F(ICai[ ) = (21/2 /rr3/2)(N/oonr)
eo
X f dE exp{-(E - E0)2/2o 2 + (E - Er)2/2o 2
-(Icail21rrn~)exp[(E-
(27r) 1/2° 0
156
0r(E) = nr exp [ - ( E - Er)2/2o~],
Er = (o E1 - o
p=0,1,2.
(10)
With the use o f eq. (5) the parameters of the secular
variation Pr(Haa) can be expressed in terms o f the
parameters that determine the distributions P0(Haa)
and Pl(Haa). One finds
n r ~"
Again a calculation of skewness and excess supports
the gaussian approximation (see table 1). According
to the Porter-Thomas proposition the spread of the
values of Iccd l around the local average Pr(H~a) is described by the normal distribution
;
o
with
N
2
F
(5)
which possesses the moments
Mp= ~ Icail(H.a)P,
(1/2+)3
8.22
0.10
-0.15
then given by
(4)
Calculation of skewness and excess tells us that eq. (4)
is a good approximation for the actual distribution
(see table 1). The secular variation o f the values Icc~il
as a function of H ~ will be denoted by Pr(H~a). Applying a variant o f parametric differentiation [4] one
can prove that the product Pr(Haa)PO(Haa) can be
approximated by a gaussian
(1/2+)2
7.51
0.27
-0.12
Er)21O2r ]} .
(12)
In figs. 2 and 3 we show a comparison o f results
obtained from eq. (12) with the results of an exact
counting. This comparison applies to the lowest three
j~r = 1 + states o f 22Na and the lowest three j~r = 1/2 +
states of 25Mg. It should be emphasized that no parameters are fitted to obtain this reproduction o f the
exact data. The widths o 0 and o 1 are calculated from
the diagonal matrix elements Ham and the eigenvector
Volume 87B, number 3
PHYSICS LETTERS
5 November 1979
3°°t
20
Z
LU
Z
O
,~ 2 0 0
O
u
10
~
.
25Mg
J~.1,2"
h
O
25Mg
J~=l/2+
i
I
I
1OO
I
05
10
Z 20
LIJ
Z
O
E
Z
001
IAMPLITUDE I
°u lO
0O2
LJ_
0
25Mg
L,n
J~= 1/2"
25Mg
i
0.5
Z
2O
200
10
100
25Mg
•
•
J~=1/2+
1.0
IAMPLITUDE [ x 103
Z
UJ
Z
O
n
• • •.~1
0.02
IAMPLITUDEJ
O
U
Fig. 2. Distribution of eigenvector components of the lowest
three states jTr = 1/2 + o f 2SMg for small values of the amplitudes. The solid curve represents the result of eq. (10), multiplied by the size of the interval 5 × 10 -4.
200 •
components c,~i. In table 1 the numerical values of the
widths o 0 and o 1 are given, together with the values
of skewness and excess. The latter two quantities are
small enough to justify the gaussian form o f p 0 and Pl"
The larger values for 22Na reflect the smaller dimension of the corresponding configuration space. It is
seen that the agreement is excellent.
Differentiation of eq. (12) shows that the derivative
F'(x) at the origin is given by
F'(x=0)=0,
for o 2 < 0
F'(x = 0) = undetermined
or 3 0 2 < 0 2 ,
(13a)
25Mg
J~=1/2 °
100
OO1
002
IAMPLITUDEI
Fig. 3. Distribution of eigenvector components of the lowest
three states j~r = 1/2 + o f 2SMg. The solid curve represents the
result of eq. (10), multiplied by the size of the interval 10 -3.
for 0 ~< Or2 ~< 302 . (13b)
For all calculations that we performed always one of
the conditions of eq. (13a) was fulfilled.
Recently Whitehead et al. [5] also have investigated
the distribution of eigenvector components. They used
a different approach and considered ensembles of
hamiltonians that preserved two-body selection rules.
The distribution function they obtained, i.e. the modified Bessel function K 0, differs from ours, in particular
for very small amplitudes Ice,i[. It is seen in fig. 4, how157
Volume 87B, number 3
PHYSICS LETTERS
15C
22Na
J1K=1°
100
z
tO
Z
O
0_
5C
E
O
i ,
U
h
-
-
v
01
IAMPLITUDE I
O
(-32
(z
hi
tfl
E
D
Z
J~=l*
~o
•
01
IAMPLITUDE I
Z~ 1501
.....
=,
-
- 02
22Na
100
0
U
a:
50
z
b~
02
IAMPLiTUOE I
Fig. 4. Distribution of eigenvector components of the lowest
three states j~r = 1+ of 2~Na. The solid curve represents the
result of eq. (10), multiplied by the size of the interval 10 -2.
158
References
[1] C.E. Porter and N. Rosenzweig, in: Statistical theories of
spectra: fluctuations (Academic Press, 1965) p. 235.
[2] P.W.M. Glaudemans, P.J. Brussaard and B.H. Wildenthal,
Nud. Phys. A102 (1967) 593.
[3] C.E. Porter and R.G: Thomas, Phys. Rev. 104 (1956) 483.
[4] F.S. Chang and J.B. French, Phys. Lett. 44B (1973) 131.
[5] R.R. Whitehead, A. Watt, D. Kelvin and A. Conkie, Phys.
Lett. 76B (1978) 149.
J3=1 *
0
ever, that the distribution function o f eigenvector components tends for small amplitudes to a finite value
with zero derivative. Such a behaviour is not reproduced
bY the function K 0.
It has to be kept in mind that Whitehead et al. obtained their results for the wavefunctions in the mscheme, whereas our results apply to the amplitudes
Ic~il for the/j-coupling scheme. Although the interactions used in the two studies o f amplitude distributions are not the same, they both consist o f two-body
forces and thus we do not expect that their difference
will cause a qualitative discrepancy.
The authors would like to thank Drs. A.G.M. van
Hees, B.C. Metsch and G.A. Timmer for their assistance with the calculations and useful discussions. This
investigation was performed as part of the research
programme o f the "Stichting voor Fundamenteel
Onderzoek der Materie" (FOM) with financial support
from the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek" (ZWO).
22Na
100
5 November 1979