Post on 19-Oct-2018
SHELL ON UHE
SHELL EFFECTS IN THE NUCLEAR LEVEL DENSITY FOR PERIODIC SPECTRA
Dr. Alfonso Anzaldo Meneses
EBiA UNIVERSIDAD AUTONOMA METROPOLITANA Casa abierta al tiempo UNIDAD AZCAPOTZALCO. División de Ciencias Básicas e Ingeniería
Departamento de Ciencias Básicas
Shell Effects in the Nuclear Level Density
for Periodic SDectra
A. Anzaldo-Meneses*
Universidad Aut'onoma Metropolitana-l aapotzalco
Departamento de Ciencias Básicas
Area de Física
México D.F., C.P. 02200
México
E - M ail : amam @ hp 9 O O 0a 1 .uam. mx
Abstract. By assuming periodic discrete spectra, the shell effects in the
nuclear level density are analyzed. The canonical partition function is obtained
exactly. The new relations allow exact analytic expressions of the level density.
Some simple examples are presented. The method can be fiirther developed as
explained in the conclusions. In particular, the angular momentum distribution
could also be taken into consideration.
Resumen. Los efectos de capas en la densidad de niveles nucleares son
analizados para espectros discretos periódicos. La función canónica de partición
es obtenida exactamente. Las nuevas relaciones permiten la obtención de
expresiones analfti cas exactas piara la densidad de niveles. Algunos ejemplos
simples son presentados. El método puede ser desarrollado aun más como se
explica en las conclusiones. En particular, la distribución de momento angular
podn'a ser tomada también en consideración.
3
1. Introduction
‘I’he iiuclear level deiisi ty1-5 is a key physical yiiaritity in nuclear reaction
theories and therefore i 11 applied nuclear sciences. ‘lliere exist several methods to
obtain estiiiiates for it . I n geiieral these inetiiods start from an iiiríependeiit-
particle type of model. TIie interactioiis between the iiucleons are only taken
into account through tlie single particle spectra i i i íi 1~1ienoiiieriologycal way.
A good agreement with experirneiital data c m bc ot)t;tiried for ccrtain energy
raiiges i r i this way. 1 ierc, starting with a fixed spectrum, the efforts are directed
towards the coinpu talion of the iiiimber of possible configiiratio~is for a particle
number with a total energy aiid a total angular iiionieiitiiin.
‘There have been considered several approaches to obtaiii at least
approximately the nuclear level density. The oldest one was initiated by E3ethe6j7,
Landau8 and Weisskopf.’ atid it is based on the iiiethods of statistical riiechariics.
A second apliroach has its roots in the works of GoiidsmitlO, Uohr arid
Kalkarl I, van Lier aiid LJlileiibeck12 and € i i i ~ i r i i i ~ ~ , this method is related with
the study of partitions of iiitegers in iiiimber tlieoiy. A third approach is purely
numerical and has as purpose to calculate exactly the level density by inore or
less sophisticated computatioiial iiietIiods.14-18 ‘1 he first iiiethxi, as appiied for
periodic spectra, has also a direct coiitiectioii with the usual counting protileins
of iiuiiiber theory. A priiiie exaiiiple of this fact is tlie equivaleiice of the
problem of a constant spectrum (for a siiiglt: kiiirl of nuclcoiis) and Eiiier’s
“parlitio nuiiieroruni” pi-obleiii. This relation caii be liest shown1 throiigti the
application o f the Darwin-Fowlcr method of statistical ii~echaiiics aiid the saddle
point approximation. ‘I’he tliiid approach is usefiií to aiialize numerically the
results of the other iiietiiocls indepeiidently for particular cases.
The first study of periodic spectra has been done by Kahn a i d
I<osenmeig19-22 for tJie asymptotic level density. l<ecently, i n a work concei-nirig
the therniodynainical properties of small iiietal clusters, the author23 obtained
some new results for the canonical partitioii function for periodic spectra. hi this
work we waiit to exploit further the relation wi th analytic iiurnber theory to
study the shell effects in the nuclear lcvel density. Only periodic single particle 4
spectra are considered. This restriction allows to understand the important shell
effects without unnecessary complications introduced by more general spectra. We
start from the grand partition function and obtain an exact expression for the
canonical partition function under certain conditions to be explained below.
From this expression we obtain an exact relation for the total level density.
It is additionally possible to obtain aii expressioii from which the angular
inomentum dependency of the nuclear level density could also be studied. This
is an important point, since besides the estimation of Bethe (1936) only a
modest progress has been reached in this difficult question (for a recent
overview see Ref. 24.) The method presented in this work allows further the
introduction of addi tioiial constani s of motion. The resulting level density
formula would then s!iow explicitly the dependence on these additional
constraints. In a forthcoming work the angular inomentuin distribution will be
addressed.
The proper study of periodic spectra i s a prerequisite for the consideration
of more general cases. ‘The present approach helps to clarifj the general nuclear
level density problem. It s e e m also possible to obtain explicit relations which
allow to reproduce to a certairi extent experimental data by adjusting only the
parameters used to define the periodic spectra. Preliminary cülculations have
been presented in Ref. 25.
2. The Canonical Partition Funcition
We start from the grand partition function
z ( ~ , / J , N ~ > = IT 11 + Y,, exp(a3111j,- B.~,,,I [ 1 + yP exp(a,inj,,-~~j,)] , (1) J
here ~ , , ( ~ ) = e x p ( c x ~ ~ ~ , ) is the fugacity, o( and (3 are Lügrünge parameters arid mjn(p)
is the magnetic quantum number for neutrons (resp. protons) with energy
The formal power series expaiision of the infinite product in eq.(i) leads to the
definition of the total nuclear levcl density as the coefficient p(N,,N,,M,E) in
the series
5
N,,,N,,M,E
where the sum is over distinct terms. Thus, by definition p(N,,,N,,,M,G) gives the
number of states with N n neutrons, N,, protons, z-component M of the total
angular momentum J and energy E given as
being ~ i ~ ~ ~ ( ~ ) ) the occupation numbers.
For number theoretical considerations i t is converiieiit to introduce a largest
unit to express (if necessaiy, only approxiniately) all single particle eiiergies as
iiiteger niiiltiples of it. For siinplicity we set this un i t equal to one. Therefore E
will be an integer number.
The next step is the evaluation of the caiioiiical partitioil fiinctioiis, the
coefficients of the series
and for each component, by Cauchy's theorem,
tlie integration contour C encloses the origin. Tlie first step will be to compute
the integral for QN(a3,B) and hereafter the level density.
We consider now a periodic single particle spectrum aiid anaíize the
canonical part tion fuiiction for a single coin ponetit. The result for both
components wi I be simply the product of two similar QN functions. 'Ilie single
particle energy levels are given as ~ ~ , = 6 ( k + v , ) , k = 0 , 1 , ...; j= 1, ... ,e; where e is
the degeneracy of each shell and 6 the spacing between adjacent shells. Note
that, iii the adopted units, 6 and 6uj are integer niiinbers. liowever, riiaiiy
results will reinaiii valid also for non-integral 6 and h i .
2. The Ground State Shell Correction
For the single particle partition fuiictioii it follows
6
and using the Bernoulli polynoinials Bn(t) defined by
one finds
Here g(t) will be called the single particle level density and is defined by
g(t)= e/6 + z z 6k {(-k,vj)6(k) (t)/k! , e
j = l k2o
where C(s,a)= 2 (n+a)-s, n20, is tiic Hurwitz C-fiirictiuri and 6(k)(t) is the k-th
derivative of Dirac's 6-function. The first term is the usual srnooth sirigle particle
level density e / & The smooth ground state energy can be calculated using:
0- 0-
which yield:
N2 N e + N < v > - - ( < u ~ > - < u > ~ ) . e 2e 2 24 2
- E / & - - - + -
0 (1 Oa)
where < y 2 > = Zvi2/e , and < u > ==E.¡/,. But, the exact ground stale energy can be calculated directly from the explicit
form of the periodic spectrum and i s given by
Here x E C0,ll is the filled fraction of the last shell in the ground state.
The exact ground state energy minus the smooth groiind state energy leads
therefore to a ground state shell effect given by
xe
3. 'I'he Canonical Partí tion Functiori
For a periodic spectruiii, we obtain Q ~ ( c x ~ . ( j ) from eq. (5). We follow Ref. 23,
but i!icliide aii additional Lngratige parameter. 'i'tie canonical partition function
cüii be of relevance in the calculation o f thcriiiodyiiamical properties iii certain
application^^^ for inesuscoyic nieta1 particles.
Divide first the itifiiiite product into two parts according to whether the
siiigle particle levels are siiialler or larger than the topmost occupied energy
level q= ó(f+v,), f integer. Rearranging teriiis aiid clianging the variable y -+
t;eBSf, i t follows
where E,= óef(f-i)/2+6e(f+ l)<v>+óexf - (rx3/B)(f+ l)<m> , and < n i > =
z jni,/e. Note that N = ef-+ex. Now, assume "tliermal degeneracy" B6fa 1 and
extend the iipper h i t f in the finite product to infinity. Define: y2= cxp(2ni~)
= exp(-Bó), set Cexp(-(3óvJ+ a3111,) = exp(2nizJ) and look at the infinite product
representation of the Jacobi 9,-
where V(T) is the Dedekind 7)-functioii
Observe that q1/J2/7)(T) is it bosoiiic partition fiinction with q=exp(-6/2kU1').
?'he canonical partition fiiiictioii will read
8
with E,= óei(f-l)/2+óe(i'+ 1/2)<v>t-Ocxf - (u3/(3)e(ft-i/z)<in>- 6e/!2. To
integrate the product of thcta ilinctions, remember their iiifiiiite series
reyreseii tation
and write
where mj=exp[-Bó(n,- 1 / ~ ) ~ / 2 - (3ó(n,-i/2)vj+ u3nij(n,-~/2)1 and the sums are
over the 1 1 , ~ Z. The integration is now immediate and leads to:
The primed sums run over j l k = l , ... ,e-i aid E,=Gef(f-i)/z+óef<v> + Gefx
+ óex(ex-1)/2 + k / 2 4 + óexii, + (u,/fl)exm, - (oc3/(3)ef<ni>.
To transform tlie infiiiite multiple suiii iii (1 8) iiito known fuiictions,
introduce the following symmetric bilinear form:
<n,n '>= nt Q u' , (19)
where n and n' are e - 1 diinerisionritl vectors and tlie (e- i ) x ( e - 1) inatrix R has
compoiieiits ni,= 1 and Q;k= 1/2 for i+-k. Let 11 arid a be the (e - i ) dimeiisioiial
vectors (nl , ... l ~ i , - l ) atid (al , ... ,ae-l) with a,=vj- <v> -x, i t follows
where we iiave used: t ' , ~ , ( a i i i j t n;aj)= 'n,(t 'a,-u,) and ai+ t] 'al=vi--Ve-ex ,
The only m3 independeiit t e r m in the iiiner product which are iiot in the
expoiieniial of (18) are
(2 1) 1 Z'a i2 + i-'aIal = Te(e--i)x2 - ex<v>> + $ ( < v 2 > - < v > ~ ) + exv, ,
But this coilstant can be added to tlie coiistaiit E3 yielding the fiual result:
i< j
9
with 2ni7=-06 ; aJ= u,-<u>-x, Z ~ i h , = ( i i i ~ - i i i , ) 0 : ~ , aiid wliere El is related to the
smooth ground state energy (cf. (10a)) by
here <uin> = 1 v,in,/e. We associated witti (,> fiirtlier the O-fun~tioii~~, with
characteristics a and b, with zEZe- l , defined by
@,,b(z12R~)= exp(2ni~<n+a,n-t-a> + 2ni(z+ b). (ii+a)) , n E ZC-', (24a) n
With relation (22) we arrived at a closed expressiori. As we shall see the
involved functions have useful traiisformation pi'operties. I'lie dependence o11 the
particles Iiuniber N is contüiiied only in the expoticritial prefaitor. Note also the
bosoiiic partition function I/@(T). Since -OS= 2~i7, (22) is giver1 in terms of
the inverse of the temperature.
Introduce now the Q--fiiiictiori, i n otic variable z E e, ZP-,,A with
characteristics27 IC aiid X
In particular, + , , , , , ( 2 ~ ~ ) = ~ ~ - ~ ( 4 7 ) Icf. (1 O)].
From (4), (loa), (22) and (.23), the one coinponeiit grand partition fmctioii is
e- i r ) - ' ( ~ ) Z 8,,1,(012nT)S,,x(ezle7), (254
r: C / 2 - C < I / > %(u,B,a3)= e y x e = o
where 2niz=oc, < = r r i~( -e/~+e<u>-e<u~>) - eoc,<um>+eu,<m>/2, ai=
Vi-<V>-X, 2nibj- (111~-n1,)0:~, K = x - t < ~ > - 1 / 2 aiid 2niX=ea,<ni>.
Fiirther, note that the assuiiiptioii o f "therrnal degeneracy" @óf%b 1 to obtain
(15) is eqiiivalent to take the infinite product
10
as onc coiiipoiient graiid caiionicíil parti t io i i furictioii. This expression can be
now iriteipretecl as the partition f u nctioii of a systeni of ferniioiis and anti-
ferrnions.
Using now 6-functioiis i i i one variable k v i t l i cliaracteristics u, - f/2, it follows
(25C)
in other words, we found two cquival eiit cxpressioiis for Z( o(,(j,u3). Therefore, we
arrive (after a simple shift) at the ideiitity
with cliaracteristics: aJ= vJ-<v>-x, bJ= pJ-pc, I C - x+<u> and X=e<p>. This
kind of identities arc the result of the iiiiderlying ring structure o; the 8-
functions.
Physically, expression (25a) is ;i bosoiiic version of the ferinioiiic partition
function (2%). Also relation (26) is o f interest. We can interpret a
generalization of i t as the Green’s fiiiiction of a problcin described by a secoiid
order differential equation subject to specific bouiidaiy conditions.
Tlie particular case e = l of (22) yields a result by Goudsinitlo:
with EcJ¿j=f(f-1)/2 + 1/24 aiiú setting <u> =O, <ni> =O. r l Ihe iiext particular case e = 2 genernlizes a rcsult of Deritoii et a1.28~20 for
electrons in a nieta1 article under a tnagiietic fielci of strength 11. Taking here
<u> =O, v2 = --u, =g/.~,~11/2(j ( I . L ~ ~ is Bohr’s niagiietoii and g is 1,ande’s factor),
a=u,-x for x=O, l / ~ aiid ni, =-i/2= - 1 1 1 ~ . ‘I’tie two coiresyoiiding relations are,
for x=O
11
with 5,,.,,= -B6f(f-l) - l36/12 + u3(f t- 1)gcc.iitf/26. And tlie second, for x = i / 2 ,
leads to
with <odd= -0612 -k BS/c, + a3(f+1/~)gpI3Ii/26. Here we have written the
backgrouiid inagiietic field dependence in the 2-arguiiieiit of the Jacobi +- fuiictions. The fi2-fiinctioii i s given by (16) and I F ~ by
Iii the iiuclear applicatiuii we Iicecl oiily 1 I O, h i t the Lagrange paraiiieter cx3
permits us to study the angular momentum distribution.
4. Transformation Formulas and the Ijeiisity of Excited States
The transformation forinulas for the Jacobi +-fuiictioiis and tiie Dedekiiid
~)-fuiictioii uiider the rnodiilar substitution T-+ - 1 /r can be used to express
UN(a,,B) in tlie preceding cxairiples in ternis o f k,,'T'= -- 6/2nir. Explicitly:
The infinite series for the Jacobi &-functions aiid tlie Dedekind r)-function
coiiverge very fast for íargc I ~ ( T ) , i.e. for l ow teniperatures. Orily few terms are
needed for a high numerical precision. For this reason, the transformation
forinulas are particularly useful t o compute also wi th great accuracy the large
teniperatiire case.
Also for tiie general result (22) liotds a siiiiilar modular transformation.
For the general periodic spectra the needed formiila31) reads
12
where I2R I deiiotes the deteriiiiiiaiit of 2R. I i i our application 12R I =e. For the
tiicta f~inctioiis in oiie variabie ttie foliowing foriiiiiia of ~ a u s s 3 ' is very useful
s,,~(z/ T) = (-iT> exp(-inzZ/T+ 2niic~) 19--,+( $1 I-!+ ) , (33)
This foriiiula is tlie oiie-diiiierisi(~iiü1 particular case of (32) and contains
equatioiis (3 1) as particular cases.
We proceed to analize ~iow the level density. For a two component systcin
tlie series wliich we waiit to analyze is the prodL:ct o f two caiioiiical partition
functions (cf. (2) and (4))
, - where the contour C' surrounds the origin. I iiese coeíficieiits give, for a,=O. the
total number of excited states of a systeiii of A=N,,-t-N, ( > O ) Fermions
(particles), or alternatively of A ( : O) aiiti-Fermions (lides) distributed on the
periodic single particlc spectra { ch,) ,l,p with total ciiergy L?. ?'lie fuiiction
pO( N,,N,,,E) includes levels degeiierated iii M.
For a single component systeiii, i t follows froiri (22)
here U = €!-E0 is the excitation etiergy, the (exact) groiiiid state energy árid
the ground state energy shell effect Esl,cll(0) given by (12). In our problem
< i n > = 0 .
'To evaluate this iiitegral, consider ttie nuiiilier o ! partitions pC(n) of aii integer n
into siiialler integers with e "colors". 'I'he generati tig functioii is
13
The iiuiiibers pe ( i i ) c;iii be easily coinpi ted by using the following recursion
formula by L e h n ~ e r ~ ~
with p,(O)= 1 a i d pc(r)=O for r<O or for I‘ non-integer. In particular, it follows
that k!y,(k) is a inoriic polyiioinial of degree k i n e with positive intcgral
coefficients. The first polyiioiiiials are: pc( l)=e, 2!pe(2) -c(e+ 3), 3!p,(3)=
e(e+ l)(e+ 8), 5!yc(S) =e(e+ 3)(e+ 6)(e2+ 21 e+ 8),
6!pe(6)= e(e+ i)(e+ 10)(e3+ 34e2+ 1$1e+ 144), 7!pC(7)=e(et-2)(e+3)(e+8)(e3 + 50e2 + 525)e + 120).
4!pC(4)=c(e+ l)(e+ 3)(e+ 14),
The coefficieiits pJn) have a closed analytical forin found by Kadeinacher
and Zuckerinan loiig ago.33 ‘Iiieir important resril t, concerning the Fourier series
of certain modular forins, has been geiiei aíized to inore geiieraí probieins.34
Their iiiethod is based ciii a work by Hardy arid Iiainaiiujan35 and on the exact
relation obtained for e = 1 by l < a d e ~ n a c l i e r . ~ ~ - ~ ~ The series we are considering
here is
with p= -[-e/24] atid o( = -c/24 - [-e/24 1, where [x] denotes the integer riot
surpassing x. Aiid the coefficieiits are giveii by the coiivcrgeiit series
Y
p,(m + p) = 2 n z p,(v- i )Z+ A k,v (E ni t o( )i” + ‘’‘ I I + c / 2 (471(v-(xj’~(ln 4- (X)’I2/k), (40) V = l k>í
with in 20, where
Ak,”= 1 o,(h,k) exp --2n1 (v-p)h-t(tn tp)Ii)/k), ( *( o s h i k (h,k) = 1
(404
with hh’ f - 1 (inod k) arid
14
L.- I we(li,k)= exp(n ic1 I1 (%- hi1 - l x ] 1111 - 1
n=l
'I'lie fiiiictioris I , are niodified Uessel fiinclions. The iiuiiibers p,(O), - - , p&- 1) are obtairied expanding explicitely ~ ~ ~ ( 7 ) or from eq.(38).
Using now the series (37) in (36), we find
P,,(N,E)=exp(-ecx,<viii>)Z pe(UJó - <ri +a,ii+a>)c.~p(2niú - (n+a)) =
Where u,= u +&ic~l(o)+ 6e/24, a i = V i -<v>- x, 2Aib,= ( l l l j - - ~ l ~ ) ~ ~ .
'i'lie expression (4 I) , for cii = O, coiisti tutcs oiir fiiial closed analytical foriri
foi tlic total level density for a single kiiid of feriiiioiis. The sum is finite since
pe(r)=O for r<0. IIence, togetlicr witIi eci. (40), we arrived at aii exact relation.
We waiit to stress here the arbitrariness o f the sets of riuiiibers {v ,> and {m,> .
The former set allows a iiiodel for wliicli the degeiieracy is broken b y e.g. a
residcial interactioii or by an external field (as i n tlie metal particles application,
cf. eqs. (28) and (29).)
Scveral generalizations we possibie. First, the consideration of two or more
kinds o f nucleons is iii-inicdiate, a,ltliough the formulas become a little more
involved. An example is giveii in the next section.
The iiiclusioii of fiirttier constaiits of iiiotioii is also possible. A s we have
seen, the presented iiietliod riccoiiiits for tlie total energy and the iiiinibcr of
particles. This lcads to the iiütura! introduction of the bilinear form <,> (cf.
( 1 9)). The iiitrodiicíion o f further coiistants of n i o t i o r i will lead to a "stack" or
"chain" of bilinear foriiis, wliicli rediicc the corripiitatioii of the level density to
finite sunis iii ternis of tlic llartitions p,(ii) aiid tlie constants of the iiiotioii. I n
a forthcoming work tlic angular iiioineIituiii distrihiitioii \$ill be coiisidered under
these approach. Formally, from eq. (4 1 )
subject to M +e<vm> = t] '(ni,-me)(u,-t a,). 'This last coiiclitioii will rediice the simi
15
to a suni over Ze-2 aiid a new bilinear form will emerge.
5. Examples
As example of tlie i-iuriiber theoretical q)ccts of tlie last results, consider
again the case e = 1, i.c. equally spaced levels. Then, froni (25)
Here p(in) úeiioies the number of prirtitioiis of the integer ni into positive
iiitegers. Clearly, p(N,E) =p(iii), for N 2111, E/S = iii + N(N - i)/2 arid ni = U/& The
first valiies are: p(N,E)= 1 , I . 2, 3, 5, 7, 1 1 . 15, 22, 30, 42, ... for ni= O, 1,
2, ... . For the exact foriiiiilas (39) arid (40), i t follows: p= 1, o(= 23/24,
p(O) = 1, aiid rearranging terms,
Asymptotically
More geiieral spectra lead to many interesting probieri-is in additive number
theory. 'The asymptotic forinula for po(N,iii) i i i the geneial periodic case is
obtainable using the saddle point metliod o r taiibcriaii theoreins. 1 Iowever, the
use of the transforiiiatioii forinuias (3 I ) and (32) sliould allow also to obtain the
error tcrms explicitly in geiieral. The asyriipíotic result i s given again by (46) but
with Eslic,,(0) froin (1 2).
--n11, As next exüiiiple consider e=2. We set of com-se ni2= 1/2 =-
<m> = O and finú
p,,(N,E)= 2 p2(U,/s - (11-x+ (v, - ~ ~ ) / 2 ) ~ ) é-CX3(1'-x) = 2 p(N,M,E) eMoc3, (46) n e i 2 M
here for x = O, M is integer a id U,= U -t ó(v, - U Z ) ~ / ~ a i d for x = 1 /2, M is half-
16
integer and U , = U + ó / 4 t 6 ( ~ i - ~ 2 ) " / 4 t 6(vl -v2) /7 . Y Clearly p(N,M,E)== p2(Uo/6-
(M--(vl-%>/2>2>.
As final exainple consider hvo kiiids of feriiiioiis (neutrons and protons) on
a doubly degenerated coristaiit spaced single particle spectra. Set therefore e,,,, = 2,
(vl),,,, = O = ( v ~ ) , , , ~ , 6,, = 6,= 6 aiid = - 1 /L = -(ni2),,,. From (35) i t íoliows
P(N .,N,,M,E) = z p.,( U,,/o - 2 (ri - xJ( n-- x, -- M) - M2)
with U,/a= x,(l-x,,) t x,,(l-xp) + U/6, xn,,=O, 1/2.
n € z (47)
6. Conclusions
'I'he canonical partition fiiiictioii for a set o f feriiiions iii a periodic single
particle spectrum has been found exactly. 'Tliere are s t i l l inaiiy possibilities to fix
the parameters which cliaracterize the per iodic spectra used. In principle, by
letting the width o f the periodic shell take a sufficiently large value, any kind of
spectrum (also nonperiodic) coiild be studied for iiot too large temperatures.
Exact expressions for the total nuclear level deiisi ty have bceii given in term5 of
the absolutely convergent series of Kadeniaclier and Zuckeriiiün. 'I'he single
particle energy levels are not necessarely degciierated. Thus, it seems reaonahle
to use their spreading to model the effects u f residual interactions or exteriial
fields by iiteans of tlie variation of the width aiid distribution of tlie single
particle levels i n a (sub-) shell.
The angular inonientuin distribution as M ~ I as addiiioiial quaiitum numbers
can be coiisidered i i i Ijriiiciple o i i the saiiie setting. 'This aspect is veiy important
and i t would sliow iiiaiiy relations to known results, mostly iiiinierical, of the
current litcrature.24
In practice, not only the average vaiiatioii o f the nuclear levcl density is of
iiiterest. Also tlic statistical distribuíion of sp:icings3" plays 311 importaiit rule. As
we riientioiied above, the paranieters of a givcii periodic spectrum are free in
principle. Therefore, we coiild coiisicier an ciisen1l)le of pei iodic spectra following
a particulai spaciiigs distribution. 'I he n:itiiral kiiiú of ciisenilAcs associated with
17
periodic spectra are the so-called circular ciisciiibles studied by l>ysori.40
Certainly, i t woiild be of interest to coiiipare the theoretical iesul ts with
experinieiital data. A fii.st step has beeti doiie i i i Ref. 25. The relationships with
analytic iiurnber theory are topics which would also be worth studying further.
Additioiially, the symmetries associated wi t l i tiic syrniiietric bil iiieür form ( 19)
arid the caiionical yartitioii flirlctioii (22) coiiici allow an iiiteresiiiig link with
affine Küc-Moody algebras.27 For conipleteiiess, \vc like to inetitioii the relation
of identities like eq. (20) with the old theory of ellil)tic curves aiid the theory of
(1 ii ad Iíi t ic forms.4 1 742
Aknowl edgments
I wouid like to thank Dr. Jorge I;iorcs V. for his support diiriiig the first
stage of this irivestigation. Part of this woi-k was performed at t l ie Instituto de
Astrorioinfa y Meteorologi;i, Guadalajara, Jalisco, México. 1 also thank M. Sc.
Valentiiia Davydova B. for her kind support.
References
lEricsoii T., Adv. Yhys., 9 (1960) 425
21iiiizeiiga J. R. aiid Moretto L. G., Aun. Rev. NLIC~. Sci., 22 ( 1 972) 427
31gnatyuk A. V., "Statistical Proycrties of IJxciteú Atomic Nriclei", IAEA,
--
INDC (CCP)-233/L, Viema 1985
4Ra~iianiiirthy V. S., i i i "Workshop on Applied Nuclear Theory and
Nuclear Model Calculatioris", Metha M. K. and Schmidt J. J. (eds.),
Singapore, World Scieiitific Publ., 1 989
511jiiiov AS., et al., Ncicl. Phys., A543 (1902) 517
"Bethe f-í. A., Piiys. Rev., 50 (1036) 332
7Bethe 11. A, Rcv. Mod. Phys., 9 (1937) 69
YLaridau L., Sov. Phys., 11 (1937) 556
9Weisskopf V., 52 (1937) 295
I0Coudsmit S., I'hys. Rev. 5 I (19.37) 64
llBolir N. atid Kalkar F., Math. fys. Medd., 14 (1937) 1 18
12vaii Lier C. and Ulileiibcck G. E., I'hysica, 4 (1937) 531
131iusinii K., Proc. Phys. Math. SOC Jnpüii, 20 ( 1 938) 91 2
14Hilln~an M. aiid Grover J. R., l'liys. Rev., 185 (1069) 1303
L5Pichoti U., Nuclear Physics, A568 ( 1 994) 553
16Cerf N., Phys. Lett., I3268 (1991) 317
I7Cerf N., Phys. Rev., C4Y (1994) 852
18Cerf N., Phys. Rev., C50 (1994) 836
l'Kahii Y . and Roseiimeig N., Phys. Rev., 187 (1909) 1193
201<osennveig N., Phys. Rev., 105 (1957) 950
21Rosenzweig N., Phys. Rev., 108 (1957) 817
22Rosenzweig N., 11 Niiov. Chi., 4.3D (1O66) 227
23Aiizaldo-Meneses A., Jour. Stat. I'hys., 75 ( I 994) 297
241kir V., Suko D. K., Brant S., Miistafii M. G. aiid Lanicr 13. G., Zeit. f.
Phys., A345 (1993) 343
25Anzaldo-Meiieses, A., "Analytiic Number 'I'hcory arid the Nuclear
Lcvel Density", INllC(Ger)038 Distr. G, IAIX, Niiclear Data Section,
Vienna, 1993
2GErdelyi A, klagrius W., Obei-hettinger F., aiid 'I'I icoini F.G., cds., "Higher
Transcendental Functioris", 3 vols., McGraw-1 I i l l Lhok Co., 1053
27Kac V.G., "Infinite Dimensional Lie Algebras", Caiiibridge Uriiv. Press, 1985
28Deiitoii R., Mühlschlcgei 13. and Scaiapiiio 11. J., J'iiys. Rev. Lett., 26
( 1 97 1)7O7
29Dento~i R., ct al., I'hys. Rev., t27 (1973) 3589
30Miiniford D., "Tata Lectures on 'l'tict a", Pi ogress i i i Matlieiiiatics, Carnbricige,
Mass., I982
31 Gauss C. F., Naclilass: Ziir Theorie der Trasc~.iideiiten Functicineii getiOrig;
Werke, Band 3, Gottirigeii, 1876, Seitc 430
321xiimer D. [ I . , Scripta Math.,
33Kadeiiiacher Ti. aiid Zuckerriian € I . , Anti hlatii , 39 (1 938) 433
34Zuckeriiian F I., rl'ra~is. Anier. Math. Soc., 45 ( I 939) 298
35€Iardy G. 11., ariú Iianiariiijan S., Pioc. Loiidoii Matti. Soc. (2), 17 (19 18) 75
3cRademacher l l . , l'roc. Loiidon Math. Soc. (Z), 43 (1937) 24 I
17 (1951) 17
19
37J<acleniaclicr H., "'l'opics iii Analytic N u111 her l'lieory", Springer Verlag,
Berliii, 1973
38Atidrcws G. E., "The Tlieory of Partitions", Addison-Wesley, Rcacliiig Mass. ,
I976
3YPorter C. E., "Statistical 'l'heories of Spcctra: Fluctuations", Academic Press, New York, 1965 b
40Dy~oii F.J., J. Math. Phys., 3 (1962) 140
41Klein F., Abh. malh.-pliys. Koeiinigl. Saeclis. Ges. Wiss., 13d. 13, No. lV, 1885
42Eíunvitz A., Math. Aiinaleii, 27 (1886) 183
20
Este material fue dictaminado y aprobado por el
Consejo Editorial cie la División de Ciencias
Básicas e Ingenierla, e l 15 de diciembre d e
1995.
21