Estados Unidos 49

34 — 18 > O) 2 16 A u> 14 V 12 5 0 f O 4 0 o 20 cn £ 10 0 12 «n

s

1 9 0 -MeV 40Ar ♦ 27AI 6 7 Ga*

9 0 * 0 #

6

F ig u re 2.29 As for figure 2.28 b u t for th e 190 MeV 40Ar + 27A1 -> 67G a sy s te m (T=2.9M eV, Ji= 3 0 ti, ßsphere= 4.0). (from [AJI86] )

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2.4 N u clear V iscosity

In th e case of a sp h e re in je cte d in to a viscous liq u id w ith zero gravity, th e velocity of th e sphere u(t) as a function of tim e t is given by

6 k r| R —t

u(t) = U o exp ( - t — — ) = u 0 exp ( — ) (2.87)

w here M an d R a re th e m ass a n d ra d iu s of th e sp h ere, i is th e re la x a tio n tim e a n d tj is th e v isco sity of th e fluid. The m o v em en t of th e sp h e re

th ro u g h th e liq u id te n d s to s e t up an ord ered flow p a tte r n a ro u n d th e sphere. T he ran d o m m otion of th e p a rticle s m ak in g up th e fluid d isru p ts th is ordered flow by tra n s p o rtin g m om entum across th e velocity g ra d ie n ts in th e fluid. In th is w ay th e in itia l collective e n erg y of th e sp h e re is tra n s fe rre d in to a n in cre ased th e rm a l m otion of th e p article s in th e fluid.

T he m e a n in g of viscosity, d issip a tio n or frictio n in nu clei is n o t as obvious as in th is sim ple exam ple. T he n u cleu s is a u n iq u e, su p e rd en se, q u a n tu m F erm i-liq u id capable of collective m otions such a s ro ta tio n s an d v ib ra tio n s . T h e th e o ry of n u c le a r v isco sity a tte m p ts to d e sc rib e th e coupling b etw een collective n u c le a r m otion an d th e th e rm a l m otion of th e nucleons m ak in g u p a given n u c le a r system . I t can be u sefu l to consider th e n e a re s t an alo g u e to th e n u cleu s, liquid 3He, w hich is also a q u a n tu m F erm i-liq u id . T he m ea n free p a th X for collisions b etw een 3H e ato m s is n o rm ally m u ch s m a lle r th a n th e d im en sio n s of th e sy ste m a n d th u s is affected by th e well know n "two-body viscosity". F o r a n y sy stem of w eakly in te ra c tin g p article s th e bu lk viscosity is given by th e expression

p = T p?l( T) v (2.88)

w here p is th e d en sity , v is th e av erag e speed of th e p a rtic le s a n d X ( T ) is th e m e a n free p a th . F o r a sy ste m of ferm io n s, collisions b e tw e e n th e p a rtic le s a re in h ib ite d by th e P a u li p rin cip le so t h a t X d e c re a se s w ith te m p e r a tu r e a s ~T- 2 . T he m e a n velocity of ferm io n s re m a in s fa irly c o n stan t w ith te m p e ra tu re a n d th u s rj v aries as - T - 2 . T h ere is, how ever, a critical te m p e ra tu re T c below w hich 3He is a su p erflu id (ie p=0 ).

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If we w ere to set a droplet of liq u id 3H e (T>TC) in to v ib ratio n , then th e two-body viscosity would cause th e irre v e rsib le tra n s fo rm a tio n of the collective m otion in to in creased th e rm a l m otion of th e 3He ato m s. The p ro b ab ility of th e 3He d ro p let ever sp o n ta n eo u sly re g a in in g its original collective m o tio n a t th e ex p en se of th e d ro p le t's th e rm a l en erg y is e s s e n tia lly zero, becau se of th e la rg e n u m b e r of in d iv id u a l 3H e ato m s involved. N uclei, how ever, only c o n ta in a re la tiv e ly sm all n u m b e r of n u c le o n s a n d th e p ro b a b ility of a h o t com pound sy ste m o b ta in in g a sizeable collective m otion a t th e expense of its th e rm a l energy is far from zero. A n o th e r m a in difference b e tw e e n a d ro p le t of liq u id 3H e an d a n u c le u s is t h a t in a d ro p let of liq u id 3H e th e m e a n free p a th is m uch sm a lle r th a n th e size of th e d ro p let. In th e n u c le a r case th e m ea n free p a th in th e n u c le a r in te rio r can be la r g e r th a n th e d ia m e te r of th e nu cleu s. T he loss or g ain of collective en erg y th ro u g h two-body collisions in sid e th e n u cleu s is th u s sm all an d n u c le a r viscosity is expected to be m ainly due to th e collisions of nucleons w ith th e n u c le ar surface.

T he sim p lest exam ple of "one-body dissipation" is th a t of a K nudsen gas. In su ch a gas, a p article colliding w ith th e w alls of th e co n tain er is assu m e d to firs t stick an d th e n be e m itte d in a ran d o m d irectio n w ith a v elo city d is tr ib u tio n a p p ro p ria te to th e te m p e r a tu r e of th e gas. The esse n tia l p ro p erty of a K nudsen gas is t h a t a p a rticle h ittin g th e container loses a ll m em ory of th e collision before s trik in g th e w alls ag ain . The d issip a tiv e en erg y loss from such a sy ste m is given by th e w all form ula [BL078,RAN84]

dE

at ( h - D )2 ds (2.89)

w h ere h is th e velocity of a su rface e le m e n t ds, D is th e n o rm al d rift velocity of th e p a rtic le s ab o u t to strik e th e su rface e le m e n t ds, v is th e

average th e rm a l speed of the p articles an d p is th e m ass density.

In re a l n u clei, n u cleo n s a re ex p ected to r e ta in som e m em ory of th e ir p rev io u s collisions. T his red u ces th e ra te of d issip a tiv e energy loss

tim

e

F ig u re 2.30 S c h e m a tic d ia g ra m of th e p o ssib le tim e e v o lu tio n of a h o t com pound s y ste m w ith no in itia l collective m otion.

37

and th e rig h t h a n d side of e q u atio n 2.89 is u su ally scaled by a c o n sta n t Ks. The value of K s is not a t all clear an d could depend upon b o th th e n u c le ar e x c ita tio n e n erg y a n d th e ty p e of collective m o tio n in v o lv ed . U sin g ran d o m -p h a se a p p ro x im a tio n c a lcu latio n s for sp h e rica l n u c le i, G riffin, D w orzecka an d Y a n n o u le a s [GRI86,YAN85] h av e show n, by re p la c in g some of th e assu m p tio n s u sed to derive th e wall form ula by m ore rea listic fe a tu re s a p p ro p ria te to rea l nuclei, t h a t K s=0.1. Nix a n d S ie rk [NIX86] have u sed th e w id th s of iso sc a la r g ia n t q u a d ru p o le an d g ia n t octupole resonances and obtained a value of K s=0.27.

F o r dum bell-like sh a p e s th e tr a n s fe r of nucleons b e tw ee n th e two portions lead s to a n ad d itio n al d issip ativ e energy loss t h a t is analogous to th e classical w indow fo rm u la [RAN84]

^ = p v ( 2 z2 + x2 ) a + V i (2.90)

w here z a n d x a re th e re la tiv e velocities of th e two h alv es of th e dum bell along an d a t rig h t angles to th e n o rm al th ro u g h th e window, a is th e a re a of th e window an d V i is th e ra te of change of th e volum e on one side of the window. A lthough th is e q u a tio n is d eriv ed u sin g th e a ss u m p tio n s of a K n u d sen gas, for a sm all w indow th e re is no need for a scalin g factor because nucleons h av e a low p ro b ab ility of re tu rn in g th ro u g h th e window w hile still re ta in in g m em ory of th e ir previous passage.

I t m u s t be e m p h a sise d t h a t eq u atio n s 2.89 a n d 2.90 only give th e av erag e d issip ativ e en erg y loss. The collective en erg y of a given n u c le a r system will flu ctu ate a b o u t th e sm ooth tre n d s predicted by th e s e equations, because of th e sm all n u m b e r of nucleons involved. T hese flu c tu a tio n s play a crucial role in th e dynam ics of fission. W ith o u t th em a new ly form ed hot com pound system , w ith no in itia l collective energy, could n e v er o b tain th e deform ation of th e sa d d le-p o in t an d th u s w ould n e v er u n d erg o fission. A sch em atic d iag ram of th e tim e evolution of a h o t com pound sy stem w ith no in itia l collective m otion is show n in figure 2.30.

,21 .-1

ß = 0. 5 (10'

,21 .-1 ß = 5. 0 (10'

time (10

F ig u re 2.31 T he tim e dependence of th e fission decay w id th of 158E r a t an a n g u la r m o m e n tu m of 6 5 tl, form ed in th e fusion of 207 MeV 160 on 142N d for (a) p = 0 .5 x l0 2l s“ 1, (b) ß = 5.0x1021 s- l . (ad a p ted from [GRA86] )

0 ______I______l______I_____I_____ I______I_____ I_____I______i_____I

0 1 2 3 4 5

ß (1021 s"1)

F ig u re 2.32 T he t r a n s ie n t tim e as a fu n ctio n of th e re d u c e d d issip a tio n c o n s ta n t ß, for th e 207 MeV 160 + i42N d - > l 58E r sy ste m a t a n a n g u la r m o m e n tu m of 65tl. (a d a p te d from [GRA86] )

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T he fission decay w id th of eq u atio n 2.59 w as d eriv ed u sin g th e tra n s itio n s ta te model of B ohr and W heeler [BOH39]. T h is m odel assu m es t h a t th e coupling betw een th e th erm al energy an d th e collective degrees of freedom is sm all an d th a t a nucleus becom es com m itted to fission once it h a s j u s t p a sse d its sad d le-p o in t deform ation. If th e n u c le a r viscosity is larg e enough th e n it is possible for a nucleus to p a ss over th e saddle-point an d th en , due to flu ctu atio n s in its collective energy, to r e tu r n back to its eq u ilib riu m deform ation (see figure 2.30). K ra m e rs [KRA40] developed a diffusion m odel applicable to n u clear fission an d show ed t h a t

(2.91)

w here co0 is th e frequency of th e in v erse h arm o n ic-o scillato r p o te n tia l of th e fission b a rr ie r a t th e sad d le-p o in t, an d ß is th e "red u ced frictio n al constant". E q u atio n 2.91 is valid for all b u t very sm all v a lu e s of ß/co0 : in the lim it ß - > 0 th e w id th Tf should ap p ro ach zero in s te a d of r£ . F o r m o st p ra c tic a l s itu a tio n s r ° is assu m ed to be th e s ta n d a r d s ta tis tic a l m odel fission decay w idth, even th o u g h in K ram ers' d eriv a tio n of eq u atio n 2.91, d iffers from th e s ta n d a r d s ta tis tic a l m odel w id th . U s in g K ra m e rs diffusion m odel, G ran g e an d W eid en m u ller [GRA80] found t h a t th e full fission decay w idth ta k e s a finite tim e to become e stab lish ed . M ore recen t th e o re tic a l stu d ies of th e tim e dependence of fission decay w id th s can be found in [GRA83,W EI84,HAS84,GRA86,BHA86]. T he tim e dependence of th e fission decay w idth of 158E r a t a n a n g u la r m o m en tu m of 65ft. form ed in