Level Densities from Proton Resonances
U. Agvaanluvsan , G. E. Mitchell , J. F. Shriner, Jr. and M. P. Pato
North Carolina State University,
Raleigh, NC and Triangle Universities Nuclear Laboratory, Durham, NC
Tennessee Technological University, Cookeville, TN
Instituto de Física, Universidade São Paulo, C.P. 66318, 05315-970 São Paulo, Brazil
Abstract. Detailed information about nuclear level densities is obtained by direct counting of individual resonances. A
collection of individual resonances with the same spin and parity follows the predictions of the Gaussian Orthogonal
Ensemble (GOE) version of Random Matrix Theory (RMT). RMT predictions are used to determine the corrections due to the
imperfections in the measurement. We derived a general expression for the probability distribution for imperfect eigenvalue
sequences. The missing level correction for the number of levels is obtained by two methods, the newly developed eigenvalue
(or spacing) analysis and the more traditional reduced width analysis. Proton resonances in three nuclei are analyzed using the
two methods. Improved values for the level densities are obtained, and the parity dependence of level density is considered.
MOTIVATION
off are not observed. The reduced width distribution can
then be described by the truncated Porter-Thomas distribution [2]
Traditionally level densities are either measured directly
or calculated theoretically using phenomenological models that interpolate in regions of existing level density
data. Since such phenomenological calculations may not
be reliable for extrapolation, there is interest in improved theories. Recent investigations using Shell Model
Monte-Carlo (SMMC) calculations [1] seem promising.
One efficient way to test the new theories is to determine
more reliable and accurate level densities using existing
neutron and proton resonance data. Since no measurement is perfect, some fraction of levels is always missing. To obtain more accurate results, the data must be
corrected for the missing fraction. This work is focused
on methods to estimate the corrections due to missing
levels.
)(*+,
4356
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798;:
<=>
7 C
1
B' D #
6@? @A > @
2
4E56
(2)
2
from the reduced widths of a set of measured resonances
with same quantum numbers. The most likely value for
the average reduced width will be the one that maximizes
the likelihood function. The average reduced width is determined from this condition by iteration and compared
with the observed average reduced width to provide an
estimate for the missing fraction of levels.
The second method for the missing level estimate is
based on the nearest-neighbor spacing distribution. The
nearest neighbor spacings of perfect GOE sequences are
described by the Wigner distribution
Two methods are considered for the estimation of missing levels. The traditional method is based on the reduced
width distribution. The reduced widths of nuclear resonances obey the Porter-Thomas distribution
-.
With this distribution one can construct the likelihood
I
F
function
G@IJK(L+ NM
H
(3)
CORRECTION METHODS
/
KOP)QR+ST)
S 7
M
CUV D;W
(4)
SYX[Z ?\
Here
, where S is a spacing between adjacent
levels and D is the average spacing. Almost all sequences
are incomplete and therefore we need the distribution
that describes the spacing distribution of an incomplete
or imperfect sequence. Since the positions of missing
levels are random, the spacing distribution is affected
by missing levels in a more complicated way than is the
(1)
where
! , "# is the reduced width, and $%"#'& is the
average reduced width. Because of experimental limitations, weak levels are difficult to observe. One assumes
that all levels with a reduced width below a certain cut-
CP680, Application of Accelerators in Research and Industry: 17th Int'l. Conference, edited by J. L. Duggan and I. L. Morgan
© 2003 American Institute of Physics 0-7354-0149-7/03/$20.00
321
^]‚
 v
are also
normalized to 1 and must have an
]ƒ_
when expressed in terms of the
average value
S„
 v . Combining these relations yields two
variable
constraints: d
d
A.
Number of spacings type 0 = 11
(0+1)11 = 11
}
b…
B.
6 € b
Z†‡
Number of spacings type 0 = 4
type 1 = 2
type 2 = 1
(0+1)4+(1+1)2+(2+1)1 = 11
[‰Š}
€ bˆ

M
(7)
bN
M
f
6 € bC‹ € b
(8)
‡
Ž ‡n‰}
FIGURE 1. A simple example showing the effect of missing
levels.
Z
b…
d
‰
f
Œ
6 € bC‹ € b
d
}
‰
6 € b
bN
’‘

}
|]_
6
bN
‰“‘
€ bˆ
M
(9)
1
? ‘
b
and 
. Substituting
one obtains € b
these resultsScin
Eq.
(6)
and
expressing
the
distributions
‘
v
in terms of
width distribution. To demonstrate the effect of missing
levels, a simple set of equally spaced levels is considered
in Fig. 1. A level is denoted by a circle while a missing
level is denoted by a tick. A spacing of type k means there
are k levels between the two levels. A spacing of type
0 corresponds to^]`the
nearest
neighbor spacing. Observe
_
times the number of spacings acb
that the ] sum of
of type is constant
d
d
SCK~}
6
b…
‘
‰”‘ b ^]‚
SCNM
‘
(10)
‚
SC
.
For = 1 this reduces to the Wigner distribution
This distribution of an imperfect sequence was first1 suggested in 1981 by Watson, Bilpuch, and Mitchell [3].
The present general result provides the formal deriva ‚
SC
tion of that
ansatz.
We
determine
the
functions
‚
ST
]EHr
]
and
numerically, while for
the -th order
spacing distribution is well
approximated by a Gaussian
]_
[4].
distribution centered at
(5)
In case A all twelve levels are observed
and there are
_
g is 11. In case
eleven spacings of type 0; thus
B, there are five
spacings
_
kop_q of_ type
kr1 0, and three spacings
of type 1; thus
is again
11. A similar
_
ts_u _
_
argument
holds
for
case
C
where
1
_
L
. The probability density
function for an
1
imperfect sequence reflects the presence of the higher
order spacings.
Z,xmy|z
XwZ,x;y{z ?\ x;y{z
, where \ xmy{isz a
Define a variable v
spacing between adjacent observed levels, and
is
the average observed spacing. The spacing distribution
(NNSD) can be written as
Numerical example
The likelihood function is
F
d
,
~}
^]‚
NM
 v
v
6n€ b
bN
The values € bˆ are found by maximizing
subject
to the constraints in Eq. (7). Introducing two Lagrange
multipliers Œ and  to account for the two constraints
and maximizing the entropy
d
<9egfihkjmlgfnj M
bN
d
C.
a b
|]†_
6 € b
To determine the coefficients € b we define an entropy
Number of spacings type 0 = 5
type 1 = 3
(0+1)5+(1+1)3 = 11
^]`_
}
G I +S
I
(11)
F
where the product is over all spacings
in the sequence.
It is‘ convenient to evaluate ‹ f . The most likely value
of is the one that maximizes
the likelihood function,
‘
F is the deviation from the most
and the uncertainty
in
‘
likely value of when ‹ f
has decreased by 0.5 from its
maximum value.
To test the method, a GOE spectrum was generated,
and then levels‘•were
randomly removed to simulate a
M –o
spectrum with
. The corresponding spacing distribution is shown in
the
upper part of Fig. 2. The nor1
(6)
The
parameters € b give the relative contributions
of the
]
^]T‚
 v . 
-th nearest-neighbor spacing distributions
is a parameter that characterizes the incompleteness of
the sequence.
We require that the distribution v is normalized to
one and that the average value of v is 1. The functions
322
48
+
0.8
Â
›
1š
0
3œ
2

0.6
0.4
0.2
4
x = S/D
0
1
P(y)
1
0.1
y
1/2
0.5
0
0.8
0.82
—
0.84
—
f˜
0.86
—
0.88
—
À
0.01
0.01
0.9
0.1
2
Á
1
2
y = γ / <γ >
10
FIGURE 3. Reduced width distribution for ÃÄ Wm° Ti Å -wave
resonances. The lower figure shows the probability density
function, while the cumulative probability is shown in the upper
figure. The dashed curves are the truncated Porter-Thomas
result with ŸÆ „¡g¢ £Ç .
FIGURE 2. (top) The spacing distribution for an imperfect
GOE with Ÿc q¡g¢ £¥¤ . (bottom) The likelihood function has a
maximum at Ÿ¦ §¡¢ £9¨¥£©¡¢ ¡ª…£ .
F
malized plot of ‹ f ‘ is shown in the lower part of the
figure. The value ofM –sdetermined
by the maximum like–¬«
M –
, which agrees well with
lihood analysis‘­is
M –no
. 1 1
the true value
1
48
π
+
p + Ti, J = 1/2
0.8
0.6
Î
1
P(x)
(ln L) norm
π
p + Ti, J = 1/2
1
∫P(y)dy
P(x)
ž
1
0.8
0.6
0.4
0.2
0™
0.4
0.2
ANALYSIS OF EXPERIMENTAL DATA
0
150
First we consider the ® -wave resonance data for the
¯ _
taken at TUNL in the energy range
° TirM reaction
±†²W³
–¦‰´rM –gµ
∫P(x)dx
100
Í
MeV [5]. Because of the excellent
beam energy
resolution, these data are nearly complete.
1
Because of the relative simplicity in determining orbital
angular momenta from the cross sections, these data are
nearly pure. There are 103 observed ® -wave resonances
in W¶ V in this energy range.
50
0È
0
1É
Ê
2
x = Ì S/D
Ë
3
FIGURE 4. NNSD (upper) and cumulative probability
(lower) for ÃÄ Wm° Ti Å -wave resonances. The dashed curves
show the expected results for ŸÆ „¡g¢ ££ .
Width analysis
Spacing analysis
The method for the width analysis described in Section
2 was applied to this set of ® -wave resonances. The data
are shown in Fig. 3. Both the probability density function and the cumulative distribution indicate that smaller
² widths are missed. The smallest observed width is ·
eV, the smallest observed reduced width is also 10 eV,
“s µ
and
is ¸;M " #@r ¹ xmy|z
1 the observed average reduced width
6 eV. Therefore, a starting value of
was em6'½ ¾t¿
ployed in the iterative
procedure.
The
likelihood
analysis
1
1
‘J
M –»º@¼ 6'½ ¾m¾
yielded
a
value
.
Also
shown
in
Fig.
3 is
K(L+
‘“
M –nº
with
1 ; this function describes the data
very well.
1
Here we consider the same data set and analyze the
spacing distribution with thex;y{analysis
method described
z
r
observed ® -wave
in Section 2. There are a
resonances,
and
therefore
102
spacings.
The value of
1
\ x;y{z
is 7.6 keV. Before applying the spacing analysis
to the experimental data, one should unfold the energy
dependence of the level density, specially if the data
extends over a large energy range. The unfolding method
is described in Ref. [6].
In Fig. 4 the
experimental nearest-neighbor spacing
+ST
distribution
is shown in the upper part of the figure,
SCtÐS
and Ï
is shown in the lower part for the 102 observed spacings. The missing fraction determined from
323
TABLE 1. Parity asymmetry in the
proton resonance level densities.
¾
¿
Reaction
Ñ
Ñ
%
D¼ 6'# ½ ¾k¿
D ¼ # 6'½ 6
69½ ¾
6'½ ¾tÒ
p Ä W;W Ca -0.16 6'½ ¾m¾
0.20 69½ 6 ¶
¼ 69½ ¾kW¿
¼ 69½ ¾m¾
p Ä Ò;WmÓ ° Ti -0.02 6'½ ¾t¿
-0.08 69½ ¾kÔ°
¼ 6'½ ¾
¼ 69½
pÄ
Fe
0.13
-0.30
W
#m#
J = 1/2
0.8
0.4
0
−0.4
−0.8
α
ã
J = 3/2
0.8
0.4
the spacing
analysis
using
the maximum likelihood tech‘4
M ––)«
M º
6'½ ¾k¿
nique is
,
which
is in excellent agreement
‘
M –»º ¼ 69½ ¾m¾
with the value
determined from the width
1
1 1
analysis. The observed
fraction of levels is determined
1
by combining the results of the two methods according
to
‘
M –g–¬«
their
relative
uncertainties.
For
this
example
M µ
. The observed fraction of levels is
then used 1for
the
xkÖmÖm× ×tÙ xmy{z ? ‘
Õ|Ø
a
correction
to the number of levels acÕ
.
1 1
using
the
corrected
The level density Ú is determined
xkÖmÖm× ×tÙ ?gÛ ±
Õ|Ø
number
of levels Ú ¾ a Õ
; for this example
o «
Ú
13 MeV .
0
−0.4
−0.8
Œ
Ú ¼
‰
_
Ú
Ú
M
ä p + 56Fe
CONCLUSION
Two methods are described that use the width and spacing distributions to determine the missing fraction described. Combining the two methods yields more reliable and accurate values for level densities. The parity
dependence of the level density is addressed using data
from three proton-induced reactions. There appears to be
at most a weak dependence of the level densities on parity. More data are needed in order to draw a definitive
conclusion regarding the parity dependence of level densities.
Once the values for the level densities are obtained,
various properties can be studied, such as the parity dependence. Proton resonances are suitable for this purpose
because of the availability of different parity states. Level
¼
1/2 ,1/2 ,
densities for four sequences (of spin Ü U
_
¼
ÒmÓ the three reactions Ý WmW Ca [7],
3/2
, and 3/2 ) from
_ _
Ý W° Ti [5], and Ý
Fe [8] are determined using the
methods described above. Suppose Ú ¼ and Ú
are the
positive and negative parity level densities for a given
¾
¾
value of Ü , for instance Ú
and Ú
. We introduce a
D #mÞ
D #9ß
parameter Πfor the parity asymmetry of the level density
Ú ¼
ä p + 48Ti
FIGURE 5. Parity asymmetry in the proton resonance level
densities. The upper graph is for the å­ æªNçè sequences and
the lower graph is for the åé “êçè sequences.
Parity dependence of level density
ä p + 44Ca
ACKNOWLEDGEMENTS
This work was supported in part by the U.S. Department
of Energy, Office of High Energy and Nuclear Physics,
under grants No. DE-FG02-97-ER41042 and DE-FG0296ER40990, by the U.S. National Science Foundation
under grant No. INT-0112421, and by the Fundação de
Amparo a Pesquisa do Estado de São Paulo (FAPESP).
(12)
If there is no parity dependence, then the parameter Πis
zero. The values of Πobtained from the three reactions
are listed in Table 1. The values for Πare also shown
1/2, and in the
in Fig. 5 in the upper
graph for Ü
lower graph for Ü
3/2. There is a weak dependence on
parity suggested by these data. However, the deviation
is within 2à . In principle, one could improve the proton
resonance data, for example by reducing uncertainties
and improving the spin assignments. However, these data
are already the best yet available. The second approach
is to study the much more extensive neutron resonance
data.
In this case the difficulty is the scarcity of data for
áTâ
.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
1
324
Y. Alhassid et al., Phys. Rev. Lett. 83, 4265 (1999).
F. H. Fröhner, Nuclear Theory and Applications IAEASMR- 43, 59 (1980).
W. A. Watson, III et al., Nucl. Instrum. Methods 188, 571
(1981).
J. B. French et al., Ann. Phys. 18B, 277 (1978).
J. Li et al., Phys. Rev. C 44, 345 (1991).
J. F. Shriner, Jr. et al., Z. Phys. A 335, 393 (1990).
B. W. Smith, Ph.D. thesis, Duke University, 1989.
W. A. Watson, III et al., Phys. Rev. C 24, 1992 (1981).