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.

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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.

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