ANNSLS
OF
PHYSICS:
Statistical
43, 375402 (1967)
Theory
of Nuclear
Cross
THEODORE
J. KRIEGER
Brookhaven National
Idoraforu,
Section
Fluctuations*
Upton, Long Island, :Veur ETark
The method of “random matrices”, employed originally in t,he stltdy of energy level distributions of complex systems, is applied to the analysis of nuclear cross section averages and fluctuations in the continuum region. A generalixat,ion of the “orthogonal” ensemble of random unit,ary symmet,ric matrices introduced by Dyson is postulated as the underlying statistical ensemble of collision matrices from which averages and variances of nuclear react,ion cross sections may be calculated. Analytical results are obtained only in the asymptotic limit h’>> 1, where N is the number of open channels, i.e., t,he dimension of the matrices considered. Effects due to energy resolution of the measuring apparatus and to “direct” or fast processes as well as t,o slow processes such as compound nucleus formation are considered. The method has the advantage of yielding results that are invariant with respect to a transformation of channelrepresentation, e.g., from channelspin s to projectile momentum j. Consistency with unitarity is achieved by treating the collision matrix as the basic random variable rather than one or more of its matrix elements. The discltssion is limited to a.ngleintegrated cross sections corresponding to given values of spin and parity, but the extension to the more general case is straight.forward. I. INTRODUCTIOX
Early theories of nuclear reactions have predict’ed a smoot’h variation of cross section with energy in the region of st’rongly overlapping resonances,the socalled continuum region. However, cross section measurements made wit*h fine resolution in the continuum region show that the energy varia,tion is not smoot’h but fluctuating, bearing a close resemblance t)o ‘Lnoise” in an electrical signal. These cross section fluctuations were first analyzed by Ericson (1) and by Brink and St’ephen (3) in t)erms of the statistical properties of the real and imaginary parts of the appropriate element of t,he collision matrix U, which mere assumedt#obe normally distributed and uncorrelat8ed. Although this approach has proven quantitatively successfulin many instances, from a t)heoretical point of view it, has two short’comings. The first is that it is not completely consistent8with unitarity except in the limit that N, t,he number of open channels of fixed spin and parity, is infinite. In a practical sit’uation, where IX may be 10, departures * This work was performed
under the auspices of the U.S. Atomic Energy Commission. 375
376
KI{IEGEH
frcnn unitarity may bc significant. That unitarity is ticIt, satiafietl esitckly is most readily seen from the illc~om~,:rtihilit? of t)he assumption of a 11orm:11distribution for each of the rcnl and imaginary parts of :I (collision matrix clement wit’h the requirement’ th:at the magnit’ude of the matrix element not exceed unity. The second diflkult\~ concerns t.he invarknre of the nssumpt.ions and results to any transformation of the c~hannel designation srhemc, e.g., from (cysl.JM) t’o (c$JN),’ which preserves the symmetry of t’he collision matrix. Since the extent of our ignorance c*onc>erning the details of the srnt)tering process is not changed by :I mere relabclling of t,he ~hnnnels, we expect the general st.at,istical properties of the collision mntris to be similarly unaffected by such a relabclling. Thus, for example, if the real and imaginar> parts of the collision matrix are uncorrelated and normally distributed in one c~hannel designation scaheme, similar propert’ies should hold in any ot,her sc*heme. It is not clear t)hat the methods of Ericson (I), Brink and Stephen (2), or ;\lolda.uer (;j), meet this invarinnc*e requirement. Lklthough we shall not do so here, the difficulties mentioned above can bc partially overcome in a rather straightforward way by applicnt~ion of the multivariat’e reducedwidth amplit’ude (4) and energy level (5) distributions within the framework of Rmatrix theory. Such a procedure was followed in t.he derivation (6) of the independence hypothesis in t,he st,atist’ical theory of nuclear reactions, where the resuks are expressed in a form manifestly invariant’ to transformations of channel designation scheme. But in achieving the desired invariance, several unsat’isfactorg features were inkoduced: (1) The result’s were demonstrated only to second order in (I’~C)/(O>, the ratio of average partial width to average level spacing; (2) t’he assumption of a nonfluct~uating t#ot,al width resulted in a slight violat!ion of unitarit’y ; (3) as is often t’he case in work involving Rmatrix theory, the question of the invariance of the results to changes in the boundary conditions and channel radii was left unanswered. Furthermore, the method treated t)he resonance parameters of the WignerEisenbud formulation of Rmatrix theory (7) as the basic random variables. Since ot’her theories of nuclear reackms, e.g., Feshbach’s unified theory (8), could just as well have been used as starting points of st’atistical calculations, we are faced with the problem of reconciling resuks based on different theories of nuclear reactions. To avoid t,he abovementioned difficulties, we adopt a different approach to the nuclear crosssection fluctuation problem, one which is closely related to those employed originally by Wigner (5), Porter and Rosenzweig (9), and Dyson (10) in their analyses of energylevel distributions of complex systems. Instead of keating t,he resonance paramet.ers of any particular t)heory of nuclear reactions as t,he basic random variables, we consider the collision matrix itself, a concept common to :rll t’heories, as the basic random entit’y. The underlying stat’istical 1 For reaet.ion
definitions theory,
of channel designationschemes see, for example, Reference
and other (17).
background
material
in nuclear
NUCLEAIl
CROSS
SECTION
377
FLUCTUATIONS
population will then be an “ensemble” of random collision matrices. As is well known, the general principles of psrt,icle flux conservnt,ion and of time reversal invariance of nuclear forces imply that collision matlrices are unit,ary and (in the cd.JllJ representation, say) symmetric. We shall therefore be interested in erlsembles of random symmetric unitary matrices. One of the ensembles studied 1~1 Dyson (10) is nct,ually of t.his type, although it was introduced not. for the invest,igation of cross sections but rather of energy levels, and is therefore not sufficiently general for t’he purposes at hand. However, as will be shown, Dyson’s ensemble can be generalized so as to allow for the possibility of a nonvanishing average diagonal matrix element,, while still ret’aining desired invarianc+e proI’crties. The advantages of the randomctollisionmat’rix approach to the problem of nuclear crosssect,ion fluctuations are: (1) It, is independent, of my specific t,heory of parnmetlrization of t,he collision matrix; (2) the required invariance to changes in represenbation, i.e., channellabeling schemes, can be introduced at t’hc outset as a symmet’ry requirement on the mat’rix ensemble; (3) consist’ency of t,he results with unitarit,y is ensured by the unitarit,y of each random collision matrix; (4) although analytical results are obt’ained only in the asymptotic limit’ N >> 1, where N is the dimension of t,he collision matrix (i.e., the number of open cshannels), numerical result’s for finite N ma\ be obtained by 310nt8e Carlo t.echniques; (5) generalizat,ions of t~he method to include rLbod) breakup (n 1 3), photon channels, and possible violations of timereversal invariance appear t,o be simpler than with statistical Rmat,rix theory. A disndvant,age of t,he method is that all channels in a compoundnucleus process arc accorded equal weight, a sit’uation not generally found in nnt’ure (although not physically impossible). However, by defining an “effective number” of open channels, it is possible t.o overcome to some ext#ent the restrictions imposed by the equalweight, condition. In Section II, the 8’1 ensemble, a generalization of Dyson’s orthogonal El ensemble, is introduced and some of it’s properties derived. Part,ic&r emphasis is placed on bhe “continuum model” approximat,ion. The application of t’he F1 ensemble to nuclear crosssection statistics in the continuum region is discussed in Section III. Some crosssection variances are calculated in Section IV. The influence of energy resolution and of “intermediate resonances” on crosssection statist)& is taken up in Sections V and VI. Finally, in Section VII, some generalizations of the method are briefly ment.icjned. II.
THE
INVARIANT
ENSEMBLE
F,
Since we shall be interested here primarily in nuclear jnteract,ions p(JSSesSiIlg timereversal and spacerot,ation symmetry, attent,ion may be restricted t,o :I generalization of Dyson’s ( 10) El ensemble, the “orthogonal ensemble,” which
37s
KRIEGER
is defined by the statement that it contains all symmetric unitary matrices X with “equal probability.” This statement becomes precise only when a measure po in the space of all symmetric unitary matrices is suitably defined. The definition of the measure chosen by Dyson is such that El , and only El , is invariant under the transformation s’ = mw,
(1)
where W is any unitary matrix, 17 its transpose, S is any member of El , and S’ is the corresponding member of the transformed ensemble El’. All matrices are square and of dimension N. Invariance under ( 1) means El’ = El . The physical interpretation of the invariance is that the application of an interaction W and its time inverse TV, in accordance with (1)) to a totally unknown system represented by the ensemble El , leaves the system equally unknown. Thus (1) is not, in general, a similarity transformation. We note that any symmetric unitary S unitary S may be written as S = RIER,
(2)
where E is diagonal with elements eiO’, and R is a (not necessarily uniquely determined) real orthogonal matrix. We shall refer to (2) as a diagonal decomposition of S. Although the measure po( cZS) associated with the neighborhood dS surrounding S is first defined (10) without reference t’o a diagonal decomposition of S, it can be expressed in terms of the measures of P( dE) and p(dR) associated wit#h neighborhood clE and dR by means of p0(dS)
=
t
JJ ( eisi 
eiej 4 /ddE’)ddR),
where p(dE) = n; de, and p(dR) is the usual groupinvariant measure for the orthogonal group of dimension N. Physically, the measure po( dS) implies (1) that in an ensemble of equally likely systems represented by symmetric unitary matrices, S, those corresponding to degenerate S occur with vanishing probability and (2) that the joint eigenvalue distribution is independent of the eigenvector distribution. The ensemble average of t)he matrix S with respect tJo the ensemble El is (sjs,
~ I fho(~s) s PO(W
= S E (ni
= o ’
(3)
since J E( ni
NUCLEAR
CROSS
SECTION
FLUCTUATIONS
379
where a is a complex number. Because distributions (4) corresponding to a = an and a. = I/ao* are identical (up to a normalizat.ion constant), we may t,ake 1a ) < 1 without loss of generality. We note from (4) that since ,.~(tlS) is independent of any diagonal decomposition of X, the same holds for p( ClX), i.e., ~(6’) is uniquely determined by S. As with I;,I , matrices in F1 having equal eigenvalues occur with vanishing probability, and t,he joint eigenvalue distrihution characterizing Fl , given by
IIL I eiei  2” I PN(h, . . . , oN) = c,orlst ’ T]lrFalN ’ is independent. of the eigenvector distribution. The single cigenvalue distribution pN(&), obtained by integrating out of (5) all but one of the angles, say o1 , is no longer uniform as in the case of EI , but only symmetrical about its maximum at 0 = 80 , where a = 1a leieo. This nonuniformity leads to a nonvanishing ensemble average of S when a # 0. Thus
where aN =
I
.zr n
de # 0,
pN(8jeie
(a f 0)
(71
and 1 = unit matrix. It is clear that a and aN have the same argument 00 ; and that a shift in 00 by A& merely multiplies each matrix of the F1 ensemble by the factor eiAeo.We see also t,hat the ensemble El is a special case of F1 , i.e., the one with a = 0,as follows from (4). Thus while El is invariant under transformation ( 1) where W is any unitary matrix, we expect K to be invariant under (1) only for a proper subgroup of the unitary group. This subgroup is the orthogonal group of dimension N, since referring to (4), ho is invariant, and the eigenvalues of S’ are ident)ical with those of S when W is orthogonal. The F1 ensembleis t,hereforc invariant under the transforma.tion S’ = R‘SR>
(8)
where R is any real orthogonal matrix. (Actually, F1 is invariant under the larger group consisting of m&ices of the form e”‘R, where R is a real orthogonal mat.rix. ,4s we shall see later, however, omission of the phase factor eiP constitutes no
380
KHIEGER
loss of physical generalit,y.) Since (S) is a similarit,y tr:tnsfonllntio11, 1;‘1 may he said to he invariant to any change in representation characterized by a real orthogonal transformation. The properties of F1 when N >> 1 are of particular interest in the analysis of cross se&on fluct,uat,ions in the continuum region, where the number of open channels is generally large. In this conne&ion it is useful to consider a twodimensional electrostatic analog (10) for the eigenvalues eLBi, which in t,he case of E1 , consists of N unit point charges on a conducting circle of unit radius interacting via a twodimensional Coulomb repulsion law and existing in thermal equilibrium at temperature kT = 1. The electrostatic analog is based on the observation that ni
which corresponds to a configuration of N unit positive charges on a unit conducting circle whose center is at the origin of the complex plane plus a charge of N units locat’ed at point a. The electrostatic analog for the eigenvalues is especially useful when N is very large because it] leads in a natural way to the “continuum model” (10). In this model it is assumedthat the eigenvalues are so denseon the unit circle that they form a compressible fluid governed by the laws of classical thermodynamics. The fluid is characterized by a macroscopic charge density ~~(0) normalized to N: .2* UN(O)de = N. (10) J0 Charge configurations inconsistent with this macroscopic density are given zero weight in evaluating the partition function and the various averages. Thus all statistical properties in the continuum model are determined from the function bN(0)) which is relat’ed to pN(8)) the probabilit,y per radian of finding a charge at angle 0, by t,he relation pN(0) = a,(e)/N leading to the normalization condition Wj The charge density aN(0) is determined by minimizing the free energy of the system subject to constraint (10). It has been shown (10) in connection with the El ensemble that t’he variational problem so posed results in a rather simple consistency equation which states in effect that ~~(0) is in thermal equilibrium in the potential which it generates. However, the equilibrium temperature is not,
NUCLEAR
CROSS
SECTIOK
381
FLUCTUATIONS
as might be expected, kT = 1, but rather ICY’ = pi. Applying the F1 ensemble, we obtain the following consistency equation
these results for pN( 0) :
2*
p,(O)
= CN exp
i
3N
[s
o pN($ 1)In 1eaO
eis / [email protected] 
In 1eiO 
to
I! (1’)
a(
.
Here, C, is R normalizat,ion constant. The solution of (12) in the limit N + 00, denoted by p&(0), is easily obl,ained. Assuming t.hat pm(e) is nowhere zero or infinite, we observe that the expression in the square bracket in (12) must vanish identically as N ;, 00. Hence, 2Tr
s0
pm(q5) In IezO  eid j [email protected] 
In 1eiO  a / = 0.
(13)
Note that (13) holds independently of the value of T. (It is therefore also derivable by the variational method used by Wigner (5) in obtaining the semicircle law for energylevel densities. The advantage of the derivation given here is that it is more directly applicable to the calculation of corrections to ~~(6) due to finite N.) Expanding p=.(4) in a fourier series
and inserting it, into (13), we obtain upon expanding paring coefficienOs of ei”‘,
the logarithms
and com
(1.5) while from ( 11) A0 = l/&r.
(16)
These fourier coefficient,s imply that 1  lalZ pm(e) = “a(eio  a(“’
(17)
277 UC.2 =
I .O
pm(0)eiO
de
=
a.
(1s)
(An alt,ernative derivation of (15) not employing thermodynamic arguments is given in Appendix A.) In the limit a + eioo, pm(0) approachesthe periodic delta function S2,(e  6%).When N is finite, physical arguments lead again to the conclusion that
3S2
KRIEGER
for as the large charge N is brought, very close to t’he conducting circular wire, it causesthe N positive chargeson the wire to cluster near it (despite their mutual repulsion and thermal motion) and eventually to coalescewith each other and with the charge N. Approximations to the fourier coefficients of P.~(0) for large N and small a are derived in Appendix A. Since %A, = (eino),w here the brackets indicat.e an average over p,(0), ( 15) states that the average of the nth power of eieequals the nth power of the average, for all positive integers n. (Were the quantity averaged real, such a result woulcl be possible only with a degenerate delta function distribution.) As a further consequenceof ( 1.5)) we have from (2) in the limit N + m (S”) = (S)“,
(n = 0,1,2, . . .)
(20)
and from (18) (S) = al,
(21)
where 1 is the unit matrix. As a approaches eieO,the ensembleFl approaches the degenerate ensemble consist’ingof a single matrix, viz. e%. Equation (20) is an interesting relation known (11) to hold for the energy average of the collision matrix with n = 2 and derivable as an ensembleaverage for arbitrary positive n (4), (6). Indeed, it was a search for an ensemble which would yield (20) that led to the introduction of t’he F1 ensemble in the present work. As already noted above, the singleangle distribution function ~~(0) is obtained from the complete multivariate distribution function by integrating out all but one of the 19i’s.Integrating out all but two of the f3i’s,say 64and & , yields the joint distribution function for two angles pg’(& , 0,). The calculation of p$’ is a difficult analytical task, even in the continuum model. Fortunately, for the present purposes, we do not require pp’ itself, but only the averages (22) since they occur in the evaluation of quantities such as ( 1X,, I’), (1S,,f 12),and (S~CSC~C~). Before proceeding to the evaluation of (22) in the continuummodel approximation, we establish some relations among ensembleaverages of matrix element expressions.They follow almost directly from the invariance under (8) and hence are valid for any positive integral value of N. Let us consider first the ensemble average ((s,,&~~,~~t . . * )*(S~d’S&~d”’ . f *)).
(“3)
NUCLEAR
C.ROSS
SECTION
FLUCTUATIONS
If in (S) we allow R to be an arbikary diagonal, elements R,.,. = &fs,,r , invariance requires that ((S,,&~,~~~
. ’ . )*(S~&Sa~p
. . .)) = ((s~,us~.,w
= R,,R,r,tR,"," wheme,
383
real orthogonal
. * .)*(A&
matrix
. . ))
. . . RddRd+p . . . ((S,,f . . )*(f&l
with
(24)
. . . )),
unless each index appears an even number of times, ((S,,S,V~
. . ~)*(sd&“p
. * *)) = 0.
(25)
In part~iculxr, (SCC~) = 0,
(c # c’)
(26)
in agreement with (21). If the averages considered are with respect to the El ensemble, i.e., a = 0, invariance under (1) with W an arbitrary diagonal unitary mat.ris, elements IV,,, = &C,eipc, leads t’o (( S,,&/~,~~~ . . . )*(sdd&w~~
ICC, c’, ..)
. * . ,),A = 0,
# (c&d', ..)I,
(27)
where (c, cl, . . . ) indicatSes the unordered set of indices c, c’, . . . . In parkular, (Sce)a=o = (sec’)a=,l = (s:dsc&4
= 0,
(c # c’).
Again invoking invariance under (S), with R an arbitrary we have
real orthogonal
(3) matrix,
(1,~,,tI’> = (I ~~6,j2) = (I 7 C RicRjc’Sij I’>, i which
aft,er some rearrangement
leads to
(I SC, I”) = q1 Sd I”) + (&%fc~>, Vnitnrity,
on
(c # c’).
(29)
the other hand, implies (N 
l)(/ ,S,,~ I?) + (I SC, I”) = 1.
In the special case a = 0, (2S), tegral N,
and (30) yield, for any poskive
in
(I sccl 12>u=o= i/(N
+ 1 j,
(31)
(I As,, 12>a=”= 2/(N
+ 1).
(32)
and In order to evaluate quantity
(29),
(30)
(22) in the continuum
approximation,
we consider the
(33) where Tr denotes the trace,
UNSE((qN = (Tr S), N '
( 3I
and (... )N indicates an average over pN . The distribution of values o Tr S = ci eisi depends, in general, on the joint distribution function , 0,). In t’he continuum model, however, t,he fundamental assumpt’ion PC&, ..* is that only configurations consistent, with the singleangle distribut,ion law need be considered. Interpret’ecl mathemat~icully, this means that xi ei8’ is replac~eablc by the integral N s pnr( 8) e*’ ~10= NaN t,o very high accuracy, higher in fact than would result from independently distributed 8, , each obeying the same singleangle law pN(0). Hut in the lat’ter case, the central limit theorem indicates that c eisi differs from its average Na, by an amount of order O(N”“). It, therefore seems reasonable to assume that in the continuum model approximation, this difference is no larger than 0( 1) .’ Hence, from (33))
Inserting
the definition
TrX = c (e
A slightly
different,
e”’ into (35), we obtain finally
~(B,B,))~ = N / &,I2 Nl
though compat’ible
(eiw,))N
1
result
= (N + 2) / UN I2 N+l
1
(36)
is, as will be seen below, somewhat more accurate, in the sense that t’he correction term 0( l/N’) vanishes when UN = 0. Note that if the eigenvalues eis’ were independently distributed, we would expect to find inst,ead (ei(B1By))N = 1 uN 1’. In similar fashion, starting from
we may derive (ei(“I+02))N = uN2 + O( l/N2),
(3)
which is to be compared to (e’(B1+02)) = uN2 resulting from independently distributed eisi. Thus to order l/N, (ez(B1+ue)) is not influenced by eigenvalue repulsion [as given by the factors 1 eisb  ei*j 1in (S)]. On the other hand, as already noted above, (ez(B’sJ) )N is affected by eigenvalue repulsion, although this effect vanishes as N + 00. 2Monte
Carlo
computer citlculntions
appear
to confirm
this
assumption.
NUCLEAR
CROSS
SECTION
385
FLUCTUATIONS
Wit,h the help of ( 36)) we can now evaluat’e t,he averages and variances of the elements of S, given from (.2) by SC,, = c R,iR,,ieiSi,
(39)
tz,., being an clement of the random real ort,hogonal matrix R. Since the I~,, are distributed independently of the eiei, we have, by (33), (SC,~)= chdcc’. The variance (of nondiagonal S,,,, , dcfirml
i40)
by
var (L& ) = (I &,I I’)  I(Sref)/L’= (~6S,C~)*6S,,~),
(41)
where &SC,,= S,,J  (S,,,), is therefore vxr (LX.,,) = (1S,,, I’) = c+F
(R,;R,~,R,~R,,,)(e”HL81’)+ F (R,fRct$.
132)
In view of (36) and the following averages, rvaluated elsewhere by Ullnh and Porter i1!2), (Rci Rc,; Rcj K(j) =  (N _ l)k(N
+ 2)
(c # c’, i # j,
(43)
and
v:Lr(&,~) = (IScc~~“)= ‘y~~“+O(&)>
(c # c’).
(45)
Similarly, in the case of a diagonal matrix element SC0, we find
(46)
where use had been made of the relation (12) (R$ = 3/N(N + 2). The correction terms 0( l/N”) and 0( l/N’) in (45) and (46) indicate that the expressionspreceding them are accurate only to order l/N” and l/N, respectively. They have been chosen in such a way that they agree with the exact results (31) and (32) in the case a = uN = 0, and also with var (SC,,) = var (X,,) = 0 for a = aN = 1. A similar remark applies to the expression for (ei(s1s2)) in (36) whose value  l/(N + 1) when a = 0 follows from (31), (42), (43), and (44).
386
KHIEGER
From (29) and (40)) we have the exact relation (1 6&c I”) = 2(/ SSd I’> + ((&)*&c~) which together with
(c z c’)
(ic7)
(45) and (46) yields (( SS,,) *&sLf)
= O( l/N*))
cc’ f cl,
(48)
where the function 0( 1/N2) vanishes at a, = 0 and 1. It is seen from (45) and (46) that the variances of both diagonal and nondiagonal elements of S tend to vanish as l/N as N + 00, and that their ratio
var bS,J =2jo
var (Sd)
1 0N
’
cc’ f cl,
approaches 2. A similar factor of 2 (for all values of N) is exhibited semble of real symmetric matrices studied by Porter and Rosenzweig III.
NUCLEAR
CROSS
SECTION
DISTRIBUTIONS
AND
by the en(9).
AVERAGES
Since the full collision matrix U is decomposable into statistically independent ( 13) square unitary symmetric matrices UJ” along the diagonal, the present discussion will be restricted to an analysis of cross section contributions corresponding to given values of spin J and parity a. The angle integrated cross section is then representable as a sum of statist’ically independent contributions g”“. For economy of notation, the indices J?r will be henceforth omitted. Angular distributions of reaction particles are not explicitly considered here. They may be included, however, by a skaightforward application of the present method. We shall be interested in nuclear crosssection distributions in an energy interval A centered about Eo in the continuum region.3 The number of open channels N is assumed to be constant throughout the interval A. The connection with random matrices is made through the fundamental hypothesis that the statistical behavior of the cross section within A about EO is governed by a suitably defined ensemble of random unitary symmetric collision matrices { U) of dimension N, in much the same way that the statistical behavior of a system in statistical mechanics is governed by the associated canonical ensemble. This “ergodic” 3 One may imagine these distributions to be generated by choosing a sample of crosssection values corresponding to a large number r of energies El , Ez , . . . , E, , chosen either at random or equally spaced within the interval A, and then allowing r to become infinite. (Experimentally, the distribution obtained is represented by an approximating histogram, since r is finite.) In the present formulation of cross section fluctuation theory, it is this statistical distribution of a cross section rather than its random variation with energy that is the object of attention. Thus, no attempt is made here to define or discuss the correlation between cross sections at two energies separat,ed by a fixed e. The width or “correlation energy” I?, usually defined as a parameter/ in the autocorrelation function (u(E + E)u(E))  (u(E))*, will be introduced later (cf. Section V) as a parameter in the expression relating a to A.
XUCLEAR
CROSS
SECTION
FLUCTUATIONS
357
hypothesis, which allows us to replace energy averages by ensembleaverages, lies at the heart of the present method. (Ot#her methods employing statistical distributions of resonanceparameters of one type or another also depend upon some sort of ergodic: principle.) We shall assume further that the nuclear processes described by li fall into two dist’inct cat,egories: (1) fast or “direct.” processes collect,ively describable by a symmetric unit,ary matrix Ud or (2) slow or “compound nucleus” processescollectively describable by a symmetric unitary matrix S, and that u = 1/‘ys[y:! (50) where I;:‘” is a unit’ary, symmetric square root of Ud . As will be seen,the stntistieal results obtained are unaffected by any arbitrariness in the definition of ciii2. The relationship (50) between I’, S, and Ud is a reasonableone, since it leads to a symmetric, unitary lJ and because it’ bears a formal resemblance t,o expressions for the collision matrix in Rmatrix t,heory and in Feshbach’s unified theory (l/t). Since cross sections for fast’ processesvary more slowly with energy than do those for slow processes,we shall take Ud to be nonfluctuating, whereas X will be fluctuating. The distinction made here between fast and slow processesis not an absolute one; it depends on the time scaledetermined by n/A. Processeswhose characteristic times are small compared t.o fi/A are considered fast,, while hhe others are considered slow. Thus if A is very small, almost all processesare in the fast category and minimal observable fluctuations within A are to be expected. In fact,, if A = 0, i.e., t.he cross section at a single discrete energy is being considered, the statistics degenerahei&o a single (nonfluctuating) cross section. As already noted, the fast processesare contained in the nonfluct’uating I,‘d . Nothing further need be said here about Ud beyond t,he fact, that it is symmetric and unitary. To complete the specification of { U) we require a descript’ion of t,he matrix S responsible for the slow processes.The fundamental assumption will now be made that S is a randonl wutrix oj ihe ensemble PI defined in the preceding section. The characteristic constant a of the 8’1ensemble, is as defined above, a complex number such that 1a 1< 1. Here we may, without lossof generality, take a real with 0 5 a < 1. The inlroduction of a complex phase factor eieOin a, as seen above, merely multiplies each S of F1 by this factor, which may therefore be absorbed in T:d without affecting Ohephysical interpretation. The quantit,y aN is similarly real wit’h 0 5 uN < 1. As we shall see,a will be related to A in such a way that a + 1 as A + 0, in keeping with the idea that the ensemblemust degenerate to a single member when A = 0. The identification of S with the FE ensemble, which is characterized by invariance under (S) with arbitrary orthogonal R, rather than with the i3’1ensemble characterized by invariance under (1) with arbitrary unitary W, is due to the difference in the degreesof knowledge possessedabout the system. The El ensemble is appropriat#e when the system is a “black box”, in which case appli
388
KRIEGER
cation of the interaction W by means of ( 1) transforms the system into an equally unknown “black box” (10). Such a model is valid for the statistical analysis of energy levels of complex systems, i.e., eigenvalues corresponding t,o bound or quasibound states arising from a completely unknown Hamiltonian. It is not, necessary in that application to discuss the observation of t’hese levels in terms of crosssection resonances produced in collisions between target and projectile having a more or less welldefined relative kinetic energy. In the case of crosssection fluctuations within an energy range A about & , on the other hand, the knowledge that we are dealing with continuum states of the total system lying within a restricted interval A about, Eo makes the “complete ignorance” argument not completely applicable. This knowledge is not sufficient, however, to permit a choice of a preferred channel designation schemewhich means that the ensemble {S) must be “representation invariant”. But transformations of representation (i.e., of channel designations) which preserve the symmetry of the collision matrix may always be represented by real numbers, e.g., Racah coefficients. In fact, any real orthogonal matrix R may be regarded as defining a symmetryconserving representation transformation of the collision matrix. (Although an arbitrary phase factor cZsmultiplying R does no harm, it is not physically significant, since relative phases are conserved.) Hence, to ensure the representation invariance of (S} , and thereby the independence of the statistical results on the choice of representation, we require the ensemble (8) be invariant to transformation (S) with R any orthogonal matrix. As A + 00, we expect the & ensemble to apply again, i.e., lima+~ a = 0. As an example of the use of the ensemble ( U] , we may calculate the ensemble average of U itself, defined by (51) We have from (50) and (40) (U) = uy(s)uy
= UNUd
(52)
Defining SS = S  (S), we may therefore write U = a&d f
CJ~‘2&YU~‘“.
(53)
In view of (45) and (46) the fluctuation term U~‘26SU~t2in (53) tends to zero
as N + a roughly as Nli2, while the direct term aMUd approaches aU, in this limit. The reaction cross section for the transition 01+ 0, where (Y and fl are distinct breakup models of the compound system, is given in suitable units by (6) ad = Tr II, [email protected] U, (54)
NUCLEAR
CKOSS
SECTION
FLUCTUATIONS
3s9
where II, and I& are projection operators which project out channels referring to a! and p, respectively. The matrix elements of II, are zero except on the diagonal locations corresponding to (Y. The number of open achannels is simply Tr & , which is representation invariant. The t,otal number of open channels is ca Tr lL = N, since ca IT, = 1. The statistical distribution of un8 in A is obt,ained by allowing l. tv become a random matrix chosen from the ensembIe ( I’) = { U~‘2S/~~‘2).In principle, bhcrefore, all statistical properties, e.g., averages, variances, etc., of ~~0in A are known, once t:d and a are given. For esample, from (53) and (54), we have
The symbol (. . .) indicates, as in (51),
an
ensemble average with respect to S:
The first and second terms in the righthand side of (55) will be referred to as the “shape” and “fluctuation” parts of the avera’ge C$ reaction cross section, respectively. As is easily verified, both the shape and fluctuation cross sections are real, nonnegative numbers and are invariant with respect to the interchange of 01and p. When a + 1, all processesare fast’, (S} is degenerate, 6S + 0 and Ud
= Tr 1[ IT,’ II rTCl, o( 8
Ita f PI,
(56)
which refers only to the fast processes. The expression for the fluctuation contribution to (ua&, denoted by u&fl), may be simplified considerably by utilizing (55), (47), and the unit,arity and symmeky properties of f:fiiz. A rather tedious calculation yields u,b(fl) = Tr (y ( Ua’26SUfi’*)’ v ~:~“‘6S1!~‘2) (57) = Tr (I3 SS’ n SS) + (( 624,)*SS,~,~)Tr II Cdt II Ud. a a 8 8 The first term in the r.h.s. of (57) is called the “compound nucleus” contribution to the average (Y+ ,Breaction cross section and is denoted by a& camp). It represents the entire average ty , p reaction cross section when no fast ty + 0 processesare present, i.e., when (Lid),,, vanishes if c is an achannel and c’ a @channel, or vice versa, because Tr II, Irdt II, f,Td then vanishes. In general, u,p(fl) and u&comp) differ by (sS,*,SS,~,~)Tr II, Udt @ [id , which may be regarded as an inference term between u&shape) and a,~( camp). However in the
390
KRIEGEH
limit N + p it tends to vanish faster than either of the latter, in view of (4s). Let N, = Tr IT, and Ns = Tr Ilo denote the number of open LYand fl channels, respectively. nTow Tr II, Sd II0 6s is simply the sum of the squares of the magnitudes of the N,No elements of S contained in the &I submatrix. These matrix elements are all nondiagonal and have identical statistical distribut,ions. Hence, from ( 57) and (45)) a,a(comp)
= Tr (sf &St : SS) = N, NP (1 S,,t 1”)
zz
N,
No
(1

(55)
UN~)
N+l
We now turn to the case /3 = a, i.e., elastic scattering. and (55), we have uarr = TrZT(U+ a

ljII(F OL
Corresponding
 1)
to (54) (59)
and
(soj
(uaa> = a,( s.e.) + a,( fl), where ~~(s.e.), the “shape cross section, are given by u,(s.e.)
elast,ic” cross se&ion, = Tr II (a,[:,+ a

and a,( fl), the ‘Yluctuation”
1 j rl (uNljd LI
I)
(61)
(62) + ((6S,, j *&L~)
Tr (II IT: ! U,  II). a *
The derivation of (62) is similar to that of (57). The term Tr (n, &%I~ a sS) is called the “compound elastic” cross section because of its formal similarity to the compound reaction cross section a,@(camp) given in (58). It is seen from (62) that the fluctuation and compound elastic cross sections are not equal in general as is sometimes stated, but differ by a term of higher order. The difference vanishes in the special case when all reactions initiated in achannels are compound nucleus processes, i.e., [& , U,] = 0. The compound elastic cross section, u,( c.e.), is from its definition, the sum of the variances of the N,(Na  1) nondiagonal elements and of the N, diagonal elemems of the ablock of S. With the help of (47) and (48) we obtain therefore
NUCLEAR
CROSS
SECTION
391
FLUCTUATIONS
ua (c.e.> = Tr (II 6St 11 SS) a a (63)
where from (5~) (64)
The coefficient [l + (l/N,)] in (63) is the socalled LLenhancement” factor for compound elastic scattering first investigated by Thomas (11). The value of the enhancement fact,or obtained here agrees with that derived in Reference (8) with the help of resonance parameter distributions and Rmat’rix theory. When N, = 1, the enhancement factor is 2, in agreement with (49). Equations (5s) and (63) may be rewritten in the more familiar HauserFeshbach (15) form, by first defining a transmission coefficient T, in t’he customary manner (6) : T, = Tr II (1 (I
(CT)t(U))
= N,(l
 nNz).
165)
We note that T, is equal to t,he compound nucleus formation a,( en.), since combining (5S) a,nd (63) yields u,(c.n. j = BF u,P(comp) (I
cross section
+ u,(c.e. j (66j
= Tr II @St&S) = NJ\ 1  aN2). a [It is easily verified from (57) and (62) Ohat the compound cross section is also given by = 5 u,,(flj (I
uJc.n.)
nucleus formation
+ a,(fl),
(67)
which, together with the expression for the average total cross section (uu(tot)), (“a(tOt))
= pFa [udfl)
+ u&shape)]
+ u,(fl)
+ u,(s.e.)
[email protected])
implies (uJtot))
= ua(c.11.)
+ u,(shape),
(69)
where ua( shape) = Ps a& shape) + ua(s.e.). a
(70)
In the special case of no fast reactions, ( 70) reduces t,o the wellknown opticalmodel formula ( 16)
392
KRIEGER
(u,(tot))
= u,(c.n.)
+ u,&s.e.).
(‘71)
Thus (69) may be regarded as a generalization of the opticalmodel result to include fast reactions.] In terms of t.he ‘I’,‘s, (5s) and (63) now read, neglec%ing terms of order 1/N’, (7”) and (73) Equations (72) and (73) reduce in the limit N + m t’o the usual HauserFeshbath formulas except for the enhancement factor (1 + l/NJ in g,(c.e.). In the case of finite N, Equations (72) and (73) may be put into a standard HauserFeshbach form by defining a modified transmission coefficient Ta’ = W/‘(N III terms of the
+ 1jlTa.
T,', (73) and (73) become a,a(comp) = T,'TB'/~
u,(c.e.j
=
(
1 1 + N,
Y
Ty' Tbl'
>
CYT,’
(75)
(761
The modified transmission coefficients T,' given in (74) are in agreement with those derived by Rmatrix theory (S) when applied to the case of equal channel weights (i.e., T,  N,). Equations (72)(76) are valid, strictly speaking, only when the 3’, are proportional to the N, ; they may be applied to the more general case of arbitrary T, by introducing a modified interpretation of N, . We determine posit.ive numbers N,’ such that ( 1) Ca N,’ = N and (2) N,’  T, . The numbers N,‘, which we shall call the “effective number of open ~channels”, may be nonintegral. Along with the N, , the actual number of open achannels, the N,’ are invariant to a change of representation, since both N and T, are invariant. We now postulate that (75)( 76) hold in the general case of arbitrary T, provided t,he N, are replaced by the effective number of open achannels N,‘. The procedure outlined here using modified transmission coefficients is almost equivalent t,o that given in Reference 6, the difference being the use of the effective number of open achannels, N,‘, in place of N, .
KUCLEAR
IV.
CROSS
SECTIOA
CROSSSECTION
FLUCTCATTONS
393
V.41: IANCES
As already not’ed, the statist’ical distribution function of the reaction cross section is in principle obt’ainable from the F1 ensemble. In practice, however, the calculation of t,his function is a very difficult analytical task. Computer calculations using lllonte Carlo techniques appear to offer the simplest solution to the problem of generating katistical distributions of cross sections to which experiments can be compared. Work along these lines is proceeding and will be t’he subject of a future publication. Here we merely give a brief outline of the numerical procedures employed. A random real orthogonal matrix R of the group RN is chosen by a Nonte CarhJ process based 0r1 t,he theorem t(hat a random real symmetric matrix of the Gaussian ensemble (5) is diagonalized by a random real orthogonal matrix of the group RN . Sext, a random set of eigenphase angles ei from the joint multivariat,e distribuf,ion (5) is chosen by a second, indepenclent i\lonte Carlo process4 and the diagonal matrix E, E:ij = 82Je’*7,formed. A random symmetric unitary matrix 8 of the F1 ensemble is then computed from ( 2). By repeating the above steps, a large sample of the F1 ensemble of unitary symmetric S mat’rices is obt,ained and serves as the underlying population for compoundnucleus cross section calcu1at)ions. If direct) react’ion contributions are required, the sample of S mat,rices is first convert’cd to an ensemble of I,’ matrices by means of (50). The method yields not only st,atist,ical moments of cross sections, such as averages, variances, etc., but, also histograms of the act,ual crosssecCon distribution. For instance, a histogram approximation to t,he distribution of ua6 is obtained by evaluat,ing (54) for a large number of random U matrices chosen from t.he abovementioned ensemble. (In practice, fluctuation analyses are generally performed in terms of the dimensionless random variable X= (T& ( u,p ) , rather than umap.)In a comparison of theory with experiment, t.he value of the characteristic constant a is determined by fit,ting t’o the data. The possibility of varying a makes the present, theory more flexible than the Ericuon theory, in which no such parameter exists. In certain special cases, however, loworder moments, e.g., averages and variances, of the distribution can be calculated without too much difficulty. Thus, in t,hc last se&ion, the compound nucleus contribution to the average rcaction cross section was calculated in the continuum approximation. In the present se&on the second moment about the mean (the variance) of the reaction cross section and of the total cross section will be calculat,ed under cert,ain simplifying assumptions. We consider first the variance of a react.ion crws section, (a&)  (c&‘, The simplifying assumptions made here are ( 1) a = 0, i.e., we are dealing with RII El 4 The author is grateful t,o Dr. M. M. Levine for suggesting and programming Carlo process, which is a modification of a procedure due to N. Metropolis, A.W. M. N. llosenbluth, A. Il. Teller, and E. Teller, J. Chem. Phys. 2, 1087 (1953).
this Monte, Itosenblut,h,
394
KRIEGE It
ensemble; (2) N, c and a single exit applies. Assumption (U) = ( u:‘“su~‘“) we have, retaining (&>
= Np = 1, i.e., we are dealing wit,h a single ent’rance c.hanncl channel c’ ( #c) ; and (3) N + m, i.e., the continuum model (1) enables us t’o neglect fast reaction processes because = ( S ) , in view of the invariance of (S) under ( 1). From (39) terms of lowest order, i.e., l/N’,
= (I SC,, I”) = 2 F+g
((309’ (e,.  fbj)
x (R~,,,RI,,,,R~c,,,R~,~,,,) In the limit N + m, we may disregard
correlations
lim (cos2 (0,.  &,!,)) Ntrn It can be shown
= +
+ (( ARK,&,,,, c" between
)').
(ii)
the 0, , so that
(c)’ # C”‘)
(7s)
z l/N”
(c # cl, L”~ # c’)
(70)
FZ l/N4
(c # c’),
(SO)
(12) that (R~,“R~,,,,R%,,,,R~,,,,,)
and (Rt,nR:v)
where the symbol E indicates the asymptotic limit N + 0~. Since both (uCCf) and var (uCe,) tend to zero as N + 00, it is useful to consider the quantity zc = uC,,/(nCC,), whose variance is now easily found from (77)( SO) : var (z)
= (x’)  (x>” Z 1.
(Sl)
This result is consistent with the distribution function e“, i.e., a x2distribution of two degrees of freedom, as is obtained by the method of Ericson (1). However, departures from (81) of order l/N are expected for finite N. As already mentioned, (81) is valid for N,Np = 1. Increasing N,Nb reduces the variance. We turn now to the calculation of t,he variance of the total cross section, uC( tot), corresponding to a single entrance channel c, i.e., N, = 1. In appropriate units, the total cross sect,ion corresponding to channel c may be written (17) a,( tot)
= 2 
ITcc 
r:f, .
(52)
Hence, var [a,( tot)]
= ([a,( tot)]‘)
 (a,( tot))’
= (u:c>  (VW>” + (L72>  (Ud,>” + a((1 UC, I’)  I(Kc>J”>.
(83)
Now it may be easily verified from (50) and the work of Section II that (Uz,)  (U,,)” and (Us,)*  ( UaJ2 both vanish as l/N2, when N is large. Hence, in the asymptotic limit, the variance of the total cross section is twice the compoundelastic cross section :
NUCLEAR
var [uC(tot)]
CROSS
SECTION
395
FLUCTUATIONS
E 2((( I’,, I’)  I(C~,,)i”)
= 2a,(c.e.)
(84)
A similar theorem involving energy averages rat)her than ensemble averages is given by Thomas (14). If we assume ITddiagonal, i.e., no fast reactions, then from (S1) and (46) var [cC( t’ot,,J]
g
2( (I S,,
1”)

j (s,,>l”
j (S5)
z 4[(1  UN”)/N]. The generalization
of (85) to include nondiagonal V.
INFLUENCE
OF
ENERGY
Ud is straightforward. RESOLUTION
Thus far we have considered the crosssection statistics within an energy interval A and have related them to an F1 ensemble with characteristic constant a. The tacit assumption has been that the cross sections involved are obtained with perfect energy resolution. Experimentally, however, cross sections are determined with a finite energy resolution I, resulting in a certain degree of smoothing of the crosssection data. In this section we consider the statistics of such smoothed cross secGons. liar this purpose we introduce the concept of a “union” of two F1 ensembles. I,et F:” and Fy’ denote two F1 ensembles with characteristic constants al and as . The union of Fp’ and FYI, which we shall denote symbolically by F:” X Fp’, is defined as t,he ensemble of symmekic, unit,ary matrices of the form s = s:~2s2s:12,
(S6)
where S1 and S, are chosen independently and at random from the F:” and Ff) ensembles, respectively. The ambiguity in the definition of ~9:‘~ is of no consequence, since if ( A!$“)’ is any other square root of S, (S;‘2)’
= z’
= R,:!2
= s;‘“&
where R is some ort’hogonal matrix. The union (S6) formed with S:” t,hercfore contains matrices of the form
(8’7) (~5:‘~)’ instead of
s’ = Ag’2s2’s;‘2, where Sz’ = l?S,R.
(ss)
But the ensemble of &’ matrices is identical with that of Sz , since I;‘:“’ is represent,ationinvariant. Hence F:o X FF’ is independent of the particular convention used in defining AS”~. Moreover, if Fi” and Fp’ are subjected to a transformation of representation, the same transformation is induced in the union Fp’ X Fy’, as follows immediately from (86). But since this transformation leaves Fp’ and
I(.f’ separately invariant~, the union Pi” X F:” is dso left, inwri:mt. Hcnc*c F!l’ x Fy’ is represent,ationinvariant,. It will be shown in Appendix B t,hnt,, in the continuummodel :Ipprosirll:ttioll, the union Fj” X Fj2’ is also m Fl ensemble with charact~eristic number a = a1az
(SY)
In t,his approximation, it, is therefore possible to regard S in (50 1 as a random member of F[” X Fy) and to rewrit#e the collision matrix as
(90) by inserting (5%). We now postulate that the stat’istics (Jf any cross section obtained with resolution I are determined by the random matrix X1 after averaging the cross section over the random matrices X2 of the F:“’ ensemble. The characterist,ic number a2 will depend upon I and have the propert’y that, a3 + 1 as I + 0. We shall assume t,hat
a2(Ij
= [?/(I,”
+ 12)]““,
(91j
where r is a parameter which we may call the “t80twl width.” number al is then, from (89))
The characteristic
~~(1) = [(r’ + 12ja2/r2j]1!2.
(92)
If we denot’e by the symbol (. . .}I the average over the Ff’ ensemble of matrices Sa , then from (55)) (90) , and (91) and neglecting a term of higher order, = (aadr
=
Jp(d&)
A2
rI( c(
U,y2sJJ:‘2)+
y’
u:‘2xu~~2)
N,NB12 + (N + l)(r2 + rl’)’
(93)
which shows that the cross section obtained with finite resolution I is the sum of a nonfluctuating component N,N8r2/( N + l)(r2 + I?) and of a fluctuating component which is of the same general nature as that obtained with perfect energy resolution but reduced in scale by the factor r2/( r2 + I’). Averaging (93) over the Fp’ ensemble of S1 matrices and using (92) recovers the perfect resolution average, i.e., the average over the original K ensemble, as is to be expected from the meaning of the union Fy’ X Fy’. Equation (93) has been obtained in the continuum model approximation. When N is not very large, (93) is still approximately true, although in this case the various characteristic constants a, al , and u2 should be replaced by the corresponding averages aN , &N , and a2N for improved accuracy.
SUCLEAR
VI.
CROSS
SECTION
OF
1NTEI:MEI)IATF:
INCLUSIOX
397
FLUCTUATIOKS
RESONANCES
The classificat’ion of nuclear reaction processes into “slow” and “fast” vatcgories in Section I was made for the purpose of deciding which processes should be represented by random matrices and whkh by fixed matrices. But, as we have seen, as a + 1, the “spread” of the ensemble of random matrices approaches zero; ix., for values of a differing infinitesinx~~lly from unity, all members of the F1 ensemble differ infinitesimally from the unit matrix. It is thcrcfore reasonablr to associate the tot)al width I’, or the inverse of the lifetime of the compountl state, w&h the value of a for the associ:~tcd F1 ensemble in suc*h :t way that :I long lifet,ime corresponds to a << 1 and :L short lifetime to a 2 1. WC shall assume that a depends on F only through the ratio A,‘lY ant1 that it’ is givcrl by a( A) = [f/( 1’” + A’)]“‘, ( 9‘4) in analogy with (91) . If intermediate resonances (18) of width I’i.r. are present, reaction proresses proceeding through them via i’doormay stat’es” may bc included iu the stat,istiml analysis by introducing a random unitary symmetric matrix, denoted by r’i,~, , t,o describe such processes. The associated F1 ensemble will have a chamcteristic ai,,, given ~SO in analogy with (91) by ai.,. The full collision
matrix
= (EsZ>‘;::.
I _ is written
( 9.5 )
as
[r = [:y&g/y”
t 90)
with s = Ly$,
,,,.ry,
(97)
where S,.,. represents the random collision matrix describing compoundrlucleus processes and lTd is bhe collision rnat’rix for the processes fast.er t’han the int#ermediate resonances. In the asymptotic limit, N >> 1, S is a random matrix of an F1 ensemble, as follows from (97) and the discussion in t,he preceding section on unions of F1 ensembles. The characteristic constant of t’he ensemble is U(A)
For consistency generalized to
=
Ui,,,(A)a,.,,(A)
with
(9S),
a2(‘) If the intervals
11 and
=
A
=
182 (9SJ us.+A2)(r2 +A') 1 7
“‘p
the expressiou
[
2
and widths
for
2
am
in (91’)
should
he
112
+ p)
1
’
r and ri.r. are such that
(!J9)
r <
39s
KItIEGElt
the st,atistics of cross sections measured with resolution I are determined by an F1 ensemble with characterist,ic constant a1 =
da1
 1I‘i.r. .
a?(l)

A2
(100)
The widt,h r no longer appears in (loo), as might be expected, since the resolution energy is large enough t,o mask compound nucleus effects. VII.
SOME
GENERALI%ATIONS
We have been dealing jn the present work with ensembles of symmetric unitary matrices, since a collision mat,rix of t,his kind is applicable in most cases of practical interest,. For a system not possessingt,imereversal invariance, the collision matrix is not, necessarily symmetric, and hence an ensemble of unitmary matrices is appropriate. Onr such ensemble, denoted by EZ , is readily defined (IO j, because the unitary mat,rices of dimension N form the unitary group I:(N), and its invariant, group measure is well known. The Ez ensemble is t,herefore invariantS to transformations of t,he form H’ = vsw,
(1Olj
where V and W are any two unitary matrices of dimension N. The measure in CheEP ensemble associated with the neighborhood rlX is given by (IO) ( 102) where S = R‘ER, R being unitary, p(dE) = III [email protected], p(dR) is the invariant, group measure of I:(N) and p( rZGj = III; dq; , eini being the elements of the unitary diagonal mat,rix G in the t.ransformation R + GR which leaves S invariant. To define Fz , we generalized ( 102) by analogy wit,h (4 j, writing (103) with 1a 1 < 1. The Fz ensemble contains all unitary matrices with relative probability given by ( 103). 0 wing to the presenceof the factors n; 1eiei  a jzN in (103), F2 is invariant to a more restricted class of transformations than allowed by (lOl), viz., similarity transformations of the form s’ = v‘sv,
i 104)
where V is any N X N unitary matrix. The single eigenvalue probability law for Fz is found, in the limit N + CC, t,o be identical with that for F1 , as given by (17). Other resultIs for Fl such as (18), (20), and (21) also apply t>o Fz. The general procedure for calculating
NUCLEAR
CHOSS
SECTION
399
FLUCTUATIOI’X
cross sections wit#h Fz (which would apply in the case of maximal timereversal noninvariance) is very nearly the same as with F, , one significant change being t,he occurrence of unitary rat’her than orthogonal R. Some of the requisite dist,ribution laws for unitary matrix elements have been considered by Ullah (19). Ko specific: results of calculations of cross sections with t,he F’? cnsemblc MY presented here. We remark finally that the measures (4) and ( 103) introduced for the definitions of the F1 and Fz ensembles are not, uniquely determined by the invariance laws which t,hese ensembles obey. Any symmetric, everywherepositive function of0,, ... , ONmay be used in lieu of flf / e’Oi  n 1“‘(‘/3 = 1, 2) in (4) and (1031 without changing the invariance proper&. The virtues of (4) :tnd ( 10:3) arc t,heir relative simplicit8y and the fact, tJhat,they imply (S”) = (S)“, in the limit, N + m . But even this last, relation does not necessarily uniquely determine t,he joint eigenvalue di&ribution; for it may readily be shown, by reference to the elect,rost,atic analog, lhat if the cahargeN located at al( 0 < 1al 1 < 1) is distributed on a circle of radius al with center at’ the origin according to a single eigenvalue prohabilit,y law (17) characterized by a2, that the result,ing single eigenvalue distribution in the asymptotic limit is also of the form (17) with characteristic constant a = azul , which again leads t’o (8”) = (S)“. XCBNOWLEDGMENTS The author wishes to thank 1)r. M. WI. Levine for many stimulating discllssions and Professor H. Feshbach for valuable a.dvice and criticism during the course of the work. Thanks are also due Miss Frances Pope for computer programming assistance. APPENDIX
A
Relations for t,he fourier coefficient,s d fy’ of pa( 0) in the coIlt,inuummodel :Ipproximation are derived by taking the logarit,hm of ( 12) multiplying by inl3and integrating over 0, obtSaining e ,277 a71 AL de (?A2 O), I eino 111 ,&e) %rN ..a A(N) = 27r n iA.1) 277 i(p)” + 2!l CL0 111 pN(e) de (IL < 0) 2xN Io In the limit N + 50, (15) is recovered from (A.l). When [ n. j << N and 1a 1<< 1, pN(e) may be replaced by ~~(0) in the integrals it1 (A.1) since these termswill be small. The result is 2*A;y In
g
(1  l/N)u’”
(n > 01,
(1  ~/N)(u~“)*.
(71 < 0).
(A4.2)
particular, uN = (eio)N= .‘aAiN’ E (11  l/N)a.
(A.3)
400
KRIEGER
The :tbove relations are hased arguments. A nonthermodyrlami~, he
( 12) which depends on tllernlotlyllan~i(, continuum model derivation of (A.2) will now
011
given.
The joint dist’ribution
funct,ion
of the 0i as specified in (5)
1 . . . , BN)
PN(h
=
const
may
be written (A.4)
e‘l
where A
=
,g
In
1 eiol

eioj
1 +
A’
111
1 e’s1

a 1
(AS) 
in 1 eiai _ ,ioJ 1 + 2 1111fP
25
l
 a 1.
i=2
111the continuum model, the first sum in (A.5 j may be replaced by the integral (N  1) jr pN( p) In I ezgi  ei’” 1 (1~~the factor (N  1) arising from the (N  1) terms in the sum. Integrating (A.l) over 02 , . . . , 0a then yields a consistency relation for pN( 01) : pN(e,)
= const Xexp[(N

1)~~~~(4)ln~eis,_
[email protected]‘/d4
1
 N In / eisl a I . (A.6) Although (A.G) is not quite identical with (la), its solution for pN(O,) agrees with (A.2) to order l/N, as may be seen by following t,he same st,eps as t,hose leading to (A.?). Thus we have
which yields t#he fourier coefficients
and an analogous equation for n < 0. III the limit, N + cc,, (15) is recovered, while if n << N and I a I << 1, we have as before to order l/N %A:
E (1 
l/N)a”
(n > 01,
with a similar equation for 1~< 0. The coefficient AN0 = l/%r follows from normalization. APPENDIX
B
In this appendix we prove that the union of two F1 ensembles, F:i’ X P’y) is an F1 ensemblein the continuum model limit N >> 1.
NUCLEAR
CROSS
SECTION
401
FLIICTUATIONS
Although represcntxt ion invariance does not uniquely characterize an F, ensemble, it does require that the eigenvalue and eigenvector distributions be independent of one another, that is, that the measure be of the form p(t/s)
= j”(lv,, .*. ) e,)p(dE’)p(f/R),
(B.1)
function of the eigenphase angles , 0,) is the joint distribution is based on the observation that a member of :L representationinvariant ensemble may be writken as where f(f& ,
ei . A proof of this remark
S = Sreal + iSirntla
where S is a symmetric unitary matrix with eigenvalues e”‘, and Srcal and are commuting, simultaneously diagonalizable, real symmetric matrices, Sixe.g with eigenvalues cos 0; and sin Bi , respect(ively. The ensembles jSreal) and (Sim,~) are representaCon invariant since (S) is representationinvariant. But t,he independence of the eigenvalue and eigenvector distributions has been demonstrated ( 9) for representationinvariant ensembles of rea1 symmekic matrices. Hence the same is true for {S) . A somewhat, more direct derivation of (B.1) involves a slight extension of t,he proof given by Dyson (IO) of the t,heorem t.hat t.he & ensemble is uniquely defined by invariance under ( 1) of t,he t,ext. To complete the proof ne now demonstrate, wi:ithin the continuum model, that t,he single eigenphase angle dist2ribution function of the random matrix S = S:‘2SZS”Z of the union F;” X F;” is of form (17). For this purpose we consider the nth (nz 0) E’ourier coefficient il,, of the distribution, which in consequence of the independence of eigenvalue and eigenvect,or distribut,ions, may be written as
The brackek here denote an average over the union I’:” X F:“‘. We observe that each matrix element of S” contains a fact’or of the form csp [ix1 YL~~:“], where exp [&:“I are t,he eigenvalues of & and the FLYare nonnegat,ive integers such that c6 ~1; = )L. A simple generalizat’ion of (3s) yields (exp (ix
n,Ot”‘J)
= a~“,
which means that averrtging S” over the tit” distribution is equivalent to replacing & by a:, 1. The rlth fourier coefficient then becomes simply
%A,
= (P’}
= az’L(S1”) = (a’az)”
cn. 2 O),
in agreement with i 15) where a = alus . An analogous result holds for n < 0. RECEIVED:
October 12, 1966
REFERENCES 1. T. ERICSON, 2.
3.
4. 5.
6. 7.
8. 9. 10. 11. 12. 13.
14. 15. 16. 17. 18.
Phys. Rev. Letlers 5, 430 (1960); Advan. Whys. 9, 425 (1900). AND 13. 0. STEPHEN, Phys. Lelters 5, 77 (19G3). P. A. MOLDAUER, Phys. Rev. 136, BF42 (1964). T. J. KRIEGER AI\D C. E. PORTER, 1. Muth. Phys 4, 1272 (1963). E. P. WIGNER, in “Canadian Mathematical Congress Proceedings”, p. 174. University of Toronto Press, Toronto, Canada (1957); included in “Statistical Theories of Spectra: Fluctuations,” C. E. Porter (Ed.), p. 188 Academic Press, New York, 1964. T. J. KI~IEGER, Bnn. PIlys. (1V.F.) 31, 88 (1965). E. P. WIGNER AND L. EISENBUD, Phys. Rev. 73, 29 (1947). [See also Reference (l7).] H. FESHBACH, .4nn. Phys. (X.Y.) 6, 357 (1958); ibid. 19, 287 (1962). C. E. PORTER AND N. ROSENZ~EIG, Ann. Acad. Ski. Fennicae, Ser. A 6, 44 (1960). F. J. IIYSON, J. Ill&h. Phys. 3, 140; 157; 166 (1962). li. G. THOMAS (unpublished, 1956), included in “Statistical Theories of Spectra: Fluct,uations,” C. E. Porter (Ed.), p. 561 Academic Press, New York, 1964. N. ULLAH, I\;&. Phys. 68, 65 (19G4); N. ULLAH AND C. E. PORTER, Phys. Letters 6, 301 (1963). N. ROSENZ~EIG AND C. E. PORTER, Phys. Rev. 120, 1698 (1960). H. FESHBACH (private communication). W. HAUSER AND H. FESHBACH, Phys. Rev. 87,366 (1952). H. FESHBACH, C. E. PORTER, AND V. F. WEISSKOPF, Phys. Rev. 96,448 (1954). A. M. LANE AND R. G. THOMAS, Rev. Mod. Phys. 30, 257 (1958). B. BLOCK AND H. FESHBACH, Ann. Phys. (N.Y.) 23, 47 (1963). I). M.
I~RINK