Jurisprudencia sobre los artículos 183.ter.1 y 2

If nuclei a re a ssu m e d to be collections of n o n -in te ra c tin g nucleons in sin g le -p a rtic le o rb ita ls th e n th e d e te rm in a tio n of th e n u c le a r level d e n sities is, in p rinciple, j u s t a m a tte r of counting th e n u m b e r of w ays in w hich th e nucleons can be a rra n g e d , such t h a t th e ex citatio n e n erg y lies in th e ran g e E to E+AE. T he n u m b e r of com binations rises so quickly w ith excitation energy t h a t th e m a th e m a tic a l techiques of s ta tis tic a l m echanics n eed to be em ployed to o b ta in level d e n sitie s , even a t re la tiv e ly low e n e rg ie s .

T he to ta l level d en sity of a n o n -in teractin g F erm i-g as can be w ritte n as [BOH69]

fl1™ _ 2 ^ exp 2 VaE

- i2 E 5/4 a U4 (2.42)

w here a is th e level d en sity p a ra m e te r and is defined as a ft2 go

6 (2.43)

go is th e single-particle level d en sity a t th e F erm i energy, re p re s e n tin g the sum of th e p roton an d n e u tro n level densities. B ohr a n d M o ttelso n show ed in th e ir d e riv a tio n of e q u a tio n 2.42 t h a t it is only v alid in th e ra n g e EFermi/A « E « EFermi A 1/3 w here Epermi is th e n u c le a r F e rm i energy (~36 MeV). The concept of th e n u c le ar te m p e ra tu re

1

T

_1 dco(E)

0) ^E “ 4 E + E (2.44)

is often found to be useful. T he condition Epermi/A « E, im p lies t h a t th e second te rm in e q u atio n 2.44 is dom inant. We can th u s w rite

E = a T 2 (2.45)

A b e tte r u n d e rs ta n d in g of th e significance of th e n u c le a r te m p e ra tu re can be o b tain ed by con sid erin g th e average occupation n u m b e r rj(E) of a given

Table 2.1 N u clear te m p e r a tu re (T), av erag e n u m b e r of excited nucleons (Hex) an d th e to ta l level d e n sity (co) for a n u cleu s w ith -2 0 0 nucleons an d level d en sity p a ra m e te r a=A /8.6 MeV-1 a t v ario u s ex citatio n energies (E).

E (MeV) T (MeV) H e x co(E) (M eV -l)

3 0.359 7 3.1xl05

10 0.656 13 e ^ x io io

30 1.136 22 8 .4 x l0 19

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one p a rtic le s ta te of th e F e rm i-g a s w hich is a p p ro x im a te ly given by [BOH69]

ri(E)

1 + exp (-E - EFermi

(2.46)

r|(E ) i s 1 for E « E F e r m i and 0 for E > > E F e r m i > w ith th e tr a n s itio n occurring in th e ran g e ~T e ith e r side of th e F erm i energy. T h is th e n leads to th e av erag e n u m b er of excited nucleons

hex = So T (4) (2.47)

S u b s titu tin g eq u atio n s 2.43 and 2.45 into 2.47 gives th e av erag e excitation energy p e r excited nucleon

7C2 T

(2.48) Hex ‘ 6 ln <4 )

F o r a hom ogenous F erm i-g as w ith a volum e su fficien tly la rg e for effects asso ciated w ith th e diffuse surface to be ignored, th e level d en sity a t th e F e rm i surface is given by

«" ■ i

mL;

w here A is th e n u m b er of nucleons. S u b s titu tin g in eq u atio n 2.43 gives

a = I l 6 MeV_1 ie a = 1 4 6 MeV (2.50)

T his v a lu e of 14.6 MeV is ap p ro x im ately a factor of 1.7 h ig h e r th a n th e v a lu e s of A/a u se d by s ta tis tic ia l m odel c a lcu latio n s to re p ro d u c e th e s p e c tra l s h a p e of p a rtic le a n d g a m m a -ra y e m issio n from com pound nuclei. F o r exam ple, G avron et al. [GAV87] concluded from th e a n aly sis of t h e ir n e u tr o n sp e c tra t h a t A /a=7.5 MeV w hile m ore r e c e n t r e s u lts in v o lv in g s ta tis tic a l g a m m a -ra y s [H E N 8 8 ,T H 0 8 7 ] r e q u ir e a v a lu e of ~9 MeV.

Toke an d Sw iatecki [TOK81] derived a form ula for th e n u c le a r level d en sity p a ra m e te r (a) w hich corrects e q u atio n 2.49 for th e p resen ce of the diffuse su rface region. T h eir fo rm u la is

A ___________________14.61_________________

spherical

saddle-point

l80 2 0 0 2 20

A (ß stable nuclei)

Figure 2.13 A/aeq and A/asp as given by equation 2.52 for non-rotating beta-stable nuclei.

ST 1 . 0 8

L80 2 0 0 2 20

A (ß stable nuclei)

Figure 2.14 asp/aeq for non-rotating beta-stable nuclei as given by equation 2.52.

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T he q u a n titie s F2 and F3 are the surface a re a an d th e in te g ra te d cu rv atu re of th e n u c le a r surface in u n its of th e ir v alu es for th e spherical shape. Due to th e m an y assu m p tio n s used in the d eriv atio n of eq u atio n 2.51, Toke and S w iatecki explain th a t for m any p ractical p u rp o ses th e sim pler form ula

14.61

MeV (2.52)

a ~ 1 + 4 A~L3 F2

m ay be no w orse th a n th e m ore com plicated one.

In th e a n a ly sis of rea ctio n s w h ere fissio n is one of th e d o m in an t d ecay c h a n n e ls , b o th th e lev el d e n s ity p a r a m e te r of th e n u clei a t e q u ilib riu m , a eq an d t h a t of th e com pound nuclei a t th e saddle-point, a sp a re of g r e a t im p o rtan ce. F ig u re 2.13 show s A /aeq a n d A /asp as given by e q u a tio n 2.52 for n o n ro ta tin g b e ta -sta b le nuclei, w hile figure 2.14 shows a sp/ a eq for th e sam e system s. The ra p id change in th e slopes of A/asp and a sp /a eq a ro u n d A=195 co rresp o n d s to th e tr a n s itio n from cylindrical to necked in saddle-point configurations. A crude e s tim a te of th e v a ria tio n of a sp/a e q w ith a n g u la r m om entum as p red icted by eq u atio n 2.52 for 198Pb is

show n in figure 2.15. The fissility p a ra m e te r for 198Pb is x=0.704 and using th e F2 v alu es of [MYE74] gives a sp/a eq(L=0) =1.086. The RLDM predicts the 198P b fission b a rrie r to v a n ish a t a n a n g u la r m o m e n tu m of -75b.. A zero fissio n b a r r ie r m e a n s th e e q u ilib riu m a n d sa d d le -p o in t sh ap es coincide an d th u s a sp/a eq(L=75) = 1.00.

S ev eral o th er a u th o rs have also m ade p red ictio n s of a sp/a eq. U sing a sim ple F e rm i-g a s m odel involving a r e c ta n g u la r sh a p e d n u cleu s w ith a trap e zo id al p o ten tial well, Bishop et al. [BIS72] p red icted a sp/a eq should be in th e ran g e 1.00 to 1.04 depending on th e d efo rm atio n a t th e saddle-point. G o tts c h a lk a n d L e d e rg e rb e r [G O T77], h o w ev er, claim t h a t B ish o p 's d e riv a tio n is in e rro r a n d u sin g H artee-F o ck c alcu latio n s have concluded a sp/a eq ~ 0.98. C a rja n eZaZ. [CAR79] p erfo rm ed m icroscopic calcu latio n s u sin g a re a listic set of single p a rtic le levels, o b ta in in g a sp/a eq * 1.065 for

1.10

Angular Momentum (h)

Figure 2.15 a sp/a eq for 198Pb versus angular momentum. The asp/a eq at L=67 was obtained using Ftfs estimated from the x=0.7, y=0.04 equilibrium and saddle-point shapes shown in figure 2.7. The error on this value reflects the uncertainty in my F2 estimates. The smooth curves simply

21

T he effect of ro ta tio n a l an d v ib ratio n al levels on the level d en sity of deform ed nuclei ad d s f u r th e r u n c e rta in ty to th e v alu e of a sp/ a eq. F or exam ple, it is well know n t h a t a t low excitation energies th e level d en sities of deform ed nuclei can be e n h an c ed by factors of th e o rd er of 102 a t th e ex p en se of h ig h e r le v e ls. D isc u ssio n s of th is effect a n d its p o ssib le im p lic a tio n on s t a t i s t i c a l m o d el c a lc u la tio n s c an be fo u n d in [M O R 72,B J073,V IG 82,H A N 83].

U n til now, only th e to ta l level d ensity of a F erm i-gas as a function of ex citatio n energy co(E) h a s b een given. How ever, in m ost s ta tis tic a l model applications th e d en sity of levels for a given a n g u la r m o m en tu m is of m ost im portance. T his spin d e p en d e n t level density form ula is given by [BOH69]

p(E,I) = ( | ^ ) 3/2 exp 2 V^“Ü (2.53)

w h ere Lfi is th e a n g u la r m o m e n tu m of th e sy stem , I is th e rig id body m o m en t of in e rtia a b o u t th e axis of ro tatio n and U = E -E rot is th e th e rm a l e x c ita tio n en erg y . E is th e to ta l en erg y an d E rot= f r 2L (L + 1 )/2 I is th e ro ta tio n a l energy. T he n u c le a r te m p e ra tu re of a com pound n u c le u s a t a n g u la r m o m en tu m L h can th e n be w ritte n as T =

V