B4elt- Xyuehu tnet. Teeh.
(:L &- rV.S.) No.6. 1960
Shell Structure Effect of the Level Spacing in Highly Excited A'llc]ei
Nobuo HAyANo
Physics Department, Kyushu Institute of Technology,
Abstract: The effect of shell structure and the dependenee of angular momentum on the level spacing of a highly excited nueleus have been studied by the statistical method. We have considered that the Fermi-level of a nueleus porresponds to a group of subshell levels to take into aeeount the shell effect into the statistical method of counting. We have assumed only pairing energy and the average potential are responsible for the remaining interactions be- Yween nucleons. The influence of degeneracy in the Fermi-level are discussed in some detail in the correlation with the broken pairs of nucleens. As a test of this approach the level spacings around N=82 nuclei are calculated and obtained rather convineing results.
1. introduction
Two deeades have passed sinee Bethe') (1936) calculated the level density of heavy nuelei. During this period, this problem has been treated from various standpoints.
Recently, rapid development of slow neutron resonance experiments have Provided more reliable data for level spacings which eome from a direct count ef resonances and they provide stronger evidence for shell structure effects than .can be ascribed purely to binding energy differenees of the eompound nuelei. This situation enables us to take the influence of nuclear structure into the analysis of level spacings more in detail.
Bethe drew out two gross features of the observed level spacing D; (1) Mass number dependence and (2) excitation energy dependence of D. And also he ealculated the probability of the levels of compound nucleus having tota1 angular momentum J. In 1954, Rosenzweig7) founded an ingenious method Whieh is especiall: adaptable in heavy nuelei near magic number region• By hiS method we can depiet clear appearance of the shell structure effect and yet retain the bulk properties of the well-known continuous approximation which ensures approximate but wide range fitting.
For the sake of improving the continuous approximation he used Euler- Maelaurin summation formula to evaluate the sum over the single particle StateS and calculated the ratio of level spaeings (shell effect) between magic and nOrMal (non-rnagic) nuclei with reasonable agreement to the observed values•
17
As fer the eefttaet with the shell model it lies essentially replacing N or P by the number of particles iB or p which oecupy the Fermi levels in the ground state of the system, manipulating the degeneraey parameter g er d gf the res- peetive ne:tr"R aRd preton Ferm! levels to eorrespond implicitly to the various cenfigurations in the shell model level $chemes.
It seem$ werthwhile tc pciRt eut that this type ef approach may havewide applicabi!ity in the field of nuclear problem to take into aceount some deserete effeet inte statistical analy$i$.
Resenzweig limits the emphasis gf the relative differences of level spaeings between magic and non-rnagie nueleus. Then in view of extending his eriginal idea it seems worthy te re-examine whether the smeoth fit of observed piRRaeled hump areund each magie number can be obtainenf reasonably well by his method taking the pairing interaetion of extra-ttueleens into account.
In this paper we have attempted tg obtain the absolute value of level $pae- iug having a given angular rnomentumlwhich was deduaed from simp}e spher- ieal shel} model. As the practical applicatiefi we have chesen the nuc!ei areund .ZV=82 shell 3nd "btained rather eenvincing results.
2. Theeretieal CemasideratieR
In 1937, van Lier and UhleRbeekS) shewed that the level den$ity of the highly degenerate system of particles could be soleiy determined by the single particle level density near the tep ef the Fermi-distributiens. If we as$ume the siRg!e payticle level spaciRg to be locally equidistant, and the probiem to obtain the level $pacing of the whole system (nuctleus) comes to the cembina- torial preblem realizing the number ef ways with the cenditigns; E= ]XRs&(n)
+:E]piSi(p), N== Xni and P--2pi. Ei(n), 6i(p) are individual energy levels for neutron and proton.
Regenzweig derived the fel}gwing apprgximatienal formula for the level density of the nueleus by the Darwin-Fowler method;
Cg,.(N, P, E) x Cg,, (R,p, 7,5, Q)
= eXP { 7r[g (1!d. + 1!d,)Q,]i t2} /4[2!6 Q3(1/d.)2(1!A,)2/(11d. + lld,)3]i'",
in Which Qs = Q+ g2d./12 -- 3.(R - g/2År212 + cl" A,/12 - A,(p - d12År212 and 1!a. =gX7, 1/d, = d!6 are respectively, the neutren and proton single particle level distance in the ground state of the Fermi-distribution. if,S are the eorrespGnding sePara' tien energy ef the subshel} levels aRd g, a are the degeReracy oÅí neutron and proton Fermi-level. Q is the observed neutron $eparation energy,
The interaetietts between fitteleong are taken inte aeceunt eniy threugh the pairing property ef like nucleons and in the same orbit, Thus we shall firSt use an energy pe.+ vzr6, to break pairs and then the remaining energy Q'"- (YeA
+ rte,År to exeite nue!eeng te seme higher levels as free. v, 7r are the nurnber Of
Shell Strueture Effeet of the level Spacing in Highly Excited-NucleL 19
brgk.en pairs of neutron and proton system, and e.,s, are the eorresponding palrlng energy.
The ensuing step is to find out a probability of levels having a certain spin Jfrom a group of levels which occupy the Fermi-level. As is well known, many
authors have attempted to explain the spin dependence of the excited nuclear levels by means of the moment of inertia of a nucleus considered as a rigid body. But, in this paper, since we limits the emphasis mainly to the effect of shell structure onithe statistical method of eounting, the spin dependent faetor have also been derived from the situation of shell model, although it becomes worse in the ease of few number of nueleons in Fermi-level. Now we shall assume that the Fermi-level of a nueleus has the degeneracy g and it contains gi,g2,:--gi nearly coincident or very elosely spaced levels(the levels of subshells),
(For instance, lhni2, 3sii2, 2dst2)•
In this case, since we could not know the partition of the partiele n in the Fermi-level to that of ii it seems most probable to determine the states of ex- cited nucleons in terms of statistical means. Then statistieally weighted i' becomess i'.= - }E]giii(n) for neutron and i',= ,lt-År=diii(p) for proton so that ÅrL"gi
== gand !!.](li=cl.
Let mi(n), mk(p) be the Z•-component of individual neutron and proton mo- menturn and M the total momentum.
M= 21]mi (n) + ÅrL]mk (p).
T'he probability for a given resultant M could be written if we assume M is composed from n and p which oeeupy the Fermi-level
pn+p(M) = C. +,e xp { - aAf2/(n +p) }
and C..p is a normalizatiQn constant, a is a constantdependent of the angular momentum of individual particle.
The average of M2
A- t2' ==mi(n)2+E]iii7i(n-)t,n- ,T(T})+t;ntktlp);-'+S.,:--]nlk(p)'fiit(i)j
= n.m (n)" +p.m(p)2
==nin(i'n+1)/3+pi'p(i'p+1)/3, and alternately
Af2 = Sp. .. (M) M2d7lf/ Jp. .. (M) aM= (n + p)/2ct•
Then we have for a
a == 3 (n + p)12 [ni. Åqii + 1) +pi' p (ip + 1)]•
Frem the normalizatien, p..,(MÅr becemes
p. +,(A t) -- [2a,,] Xi2[ni' . (i' .+ 1) +p i' , (i' ,+ 1)]-il2 Å~
exp {-3Adt2/2[ni'.(i'.+1)+,?)jr',(i',+1))}
and the probability of a given fis, as is we}l known,
pn+p(il) =' [87i:]- "2(21+ 1År [3/ {R7-' n (i' nÅÄ 1) +pi p (f' p+ 1)} ]3i2i
Thus the final expression for level spacing of a compound nucleu$ with a given tetal spin i can be written
firl
Dk ,t (n, p, dn , dp , Q,f,) =
'1/Ck,d(n,p, d.,d,,,QÅr (1) (2J+1)
in whieh
alZ=[8 rr]i i2 [3/ {n•i' . (i' .+ 1) ÅÄpi' , (i , + 1År}]-3i2.
3. Assignment of the Parameters 3a. Degeneration Parameters
Before proeeeding to finer details, we shail first fix the neutron and proton
levels axeund N=82 and Z=5C respeetively aeeerding te Mayer and Jensen'2).
Fh//zmm---:-mE,C', ggits.=:,(g),
gh,i = = (t,ii?(s2) tg:i!i---m--n---(to) (se) 2d{;.t M-"---- (4)
Fig. Ia. Sheli modei level $cheme with strong Fig. Ib. $heii model levei scherne areund the spin-grbit coupling in tbe vicinity of peutron protou rnagic number P-ny-50.
magic n"mber N= 82. From tke references i2, i3 it is beiieved that lit"i2, 3sit2, 2dii2
are very elosely spaced or almo$t coineident
!eveis aRd alse theTe ig a c"mpetitieB betweeR ihytx, and 2f7t2 in the regien of aftermagie
nne!ei.
In these nuelei the$e levels have beeR eensidered very elesely spaced frOrri
the analysis of the ground state $pin of odd A nucleit2t M and moreover in manY
ca$es the order ef these levels vary irregularly, Therefere we must take intO
Shell Structure Effeet of the level Spacing in Highly Exeited-Nuelei. 21
/ifil.b/1•,/1],/Åéflitihi,,lylmg,b,b,m,giM,b,,`/kp,,l/ilgtl,C,f{,:Mkhlhil-,s`g.iia,,,/Flll//]Ii,if,gbi,si,Rs,id`{imthi
. For the proton system we assume that the Fermi-level eonsists of the coin-
iiud,ii:b,te,Giti,l2s,i:,"xd,e,didti6i?3i..X,Xdn,tSfi,Sg"iaO,islt)ae,kle,g.1'g's.3"t,"h,d.,of:Oih:o,iu!?",:,,",ii8,Biai,2`:he.i'l}O{,g,2,:,
is not magic in thg region of these neutron number and this means that the
i,/hhee#v,a,Jtig,P,y"xoii,.3jglS.2i':,e.S,lo.,/oi,i,L,MM,kejl.gbS/g':,i5,.110:,'S,s.,i,!Fig/a",,p/liiEO:i./,.e,11/F,l•3sr/ge.eikz.k•E'g:•i"kge2•
,,XY,e.tt?,b."S9,tef,t,he.fie,ghe,",e,rg,CiYbiO,fi,h,e..F8,rM.e-kgeia"ditscorrespondingcom.
Table I. Degeneration parameters of the Fermi-levels for the neutron system
-- the degeneracy in Fermi-]evel and gi a group of the coineident eomponen{ fevMe?sa.nS - -.---B-e-f!r-e-g}g-gicre-gion -. Afte-r--m5it'i'E-icrittion==- ---'--'
18 lltui2, 3sitb 2d3ie 18 lit q. i2, 21171t71,,7
16 11t nib 2d]i! 10 lh gi:
12 lhnn 8 2f712
6 3sitb 2d3i2
3b. Pairing Interaction
thlggs,zei:s.f.r.om,,s.b,is,iiee,!5V,su.,os,iS,P,e,:oe,s8ds,rs,g,fihs,ps,`},i\g,&",eg,g,y.C',gg,
and two eompound levels for after magic nuelei. Then, we obtain the pairing
71eXllZY.If.S",C.h,:9,M.eO."fi::fee,V8.i3t.fi.rg,`',,IIF,%Y,?iill',g.UMg,gS,,!eS,,E,;'6`,',--'d-i;'R,`,2'lhtiia
E;air=:(22.3Å}5.5)(21'+Z)/A MeV•
On the other hand Wapstrai`) estimated the pairing energy of odd-A nuclei
ehS,S\•.St.2M,,?•EUCe,LiX',2",g,iS,g,.f,'o,m.s,",e,h3.'SiP`S9th,l•;d,lf,8",Zn,I2Xee,P.":le.iIklX
ealeulated the pairing energy of odd-A nuclei and knew that the variation in
mass number 3-w4 has alme$t ne iftfluence te the pairing eRergy in the region of heavy nuclei.
G able II. Pairifig energies of the eomponent levels. For edd-A ftnclei they aTe taken te be one half or two third of even-even nuaiei.
erbit Elair ne ESutir12 E;air =' 2E tpair!3
lhnl2 1•ee Mev 1.25 Mey
ee:,ta e. 49 O. 65
- 3c. Individual Partiele Level Spacing
As already pointed out by T.D. Newten") enee the shell medel will be given it i$ not always necessary te determiRe the iRdividual particle level deRsitieg in terms ef the Fermi-Gas mode! and he, from the eompi!ation of Klinkenbergi6L used the expression g=d"'i(2i+1) for individua} level den$ity. (di mean$ the
500
S. 200
N"
loo
M'g. 2. Single particle 1evel spacing plotted as the function of nucleon nnmber- Cttrves
(i) and (ii) are ealeulated from the formulae (i) and (if), respectively.
Shell Strueture Effeet of the level Spacing in Highly Excited-Nuclei. 23
lsVee":,i.g,kd,i,',.ie,ag"saii,i,g,tli"gsli,l,Biiiili,'gi:ei,}I',,',i,iieei,X/,al,'ib3,iir,a,/g,Y',ifagl,k:O,:,i,:,,i.2'kSglj,Y,pi,:,Oi
iu,t-,th.fi:s.d,i,gficu,i,t,]e,si,hF,o.u,g,h,g2e,tii.n.e,g.Si:m,lhe,m,p,ifg,ea,i,ei.p,r,2s.zi2n,,oi.L.a,",g.,&.
IP,tC,a,igY,.a,",.d,OB,t3il",,e.d,l,hZ.r,e,'aiti,O,",P:.t:l;se,n.,:,l,a.s,?.,n.u,m.2;Esndsingieparticie
Q=(At2/f)-t, A/f--rr2/6d i f==8 Mev, (i)
q==(At2/f')-t+(A2':t'i3/8), A/f'==n2/6d ; f'==11 Mev. (ii)
Uere, f andf' are convenient parameters and 1/a shows the number of indi- vidual ngutron and proton states per unit energy at the Fermi-top in the greund state. (ii) includes the influence of collective exeitation (surface wave effect) a.nd s!ighty suitable in the region of higher excitation. Since we keep the sltuation that neutron and proton states contribute independently to the total singlgparticle states density, 1/d could be taken as 1/d.+1/d.. In Fig, 2, (i) and (n) are plotted as the function of nucleon number.
4. Experimemtal Level Spacing
When a target nucleus absorbs a neutron, a eompound nueleus is formed
!iaying the excitation energy Q--B.+e, where G is the kinetic energy of the ineident neutron (in the slow neutron region EÅq10 kev) and B. is the binding energy (approximately 8 Mev). In this regiun of exeitation for relatively heavy nuclei there are enormously high density of quantum states in eempound nueleus. According to V.L.Sailor2') (1956) we limit our attention to target nuclei having ground state spin I #O and to s-wave incident neutron. Then
' the statistieal factor g=(2J+1)/{(2S+1)(21+ 1)} has two possible values;
g+ == -S [i+ ,, i. ,] and g. == S- [i - t, '.n],
COrresponding to the neutron spin with parallel and antiparallel to the target . SPin• Beeause of the laek of information about g-values it has been believed that g+ and g- are equally distributed over resonances. But from the careful analysis of Sailor it seems reliable to assign that resonances have an over- Whelming majority of g. (90 %) for EÅq50 Kev and fer relatively heavy odd A target (A År50).
The neutron binding energy, B., are obtained from the following authers;
Harvey'7) calculated B. by (d,p) reaction and investigated the relation of B. and
neUtron elosed shell N-- 28, 50, 82 and 126, and Walli"), about tV-- 50 and N---= 82•
Johnson and NieriSÅr have analysedi the effeet ef shell elcsure at N=82 eR the systematies gf nuclear binding and pairing energy in greater detail. Most of the B. used in this paper came from the data of Johnson and Nier.
Table llI. Experimental level spaeing. I is the spin of targetnucleus,Jig+) is the spin of eompound nllcleus in the ease ef g+U:tiÅÄY2År. D is the !evel dSstaRee cf eempaund nueleus per spin state and Cis the observed binding energy of a netttren, The symbel O gr E indSeate oddi or even.
Type "f aN
Target i 1ig+) DeV Target conrnucpa/euunsd qMeV
Sgi2i 5,Z-. 3 14L15d OE Oe 6.67 Tei23 Y2 1 15e-50di EO EE
Sbi23 7/2 4 !4b-15d OE OO 6.15
IM 5h 3 i36 OE Oe 6.58
Cs;33 712 4 2!a-25e OE OO 6.73
Bai3$ 3/2 2 S5c Ee EE 9.ie
XeJ3S S/2 2 500c .EO EE 8.50 Ltxi39 7/2 4 5eet OE eO 5.39
ATdUS z6 4 25c EO EE 8.04 Sm3i7 sth '3 7LiO.5c Ee EE 8.73 Smus ZS 4 2di- 3.3c EO EE
Eeci5i $!2 3 1. la- 1. 2b 0E Oe 6. 5e
E!:i53 ff2S 3 1. la--•- 1. 2b O E O O 6. 26
a; Harvey, Httghes, Carter and Piltcher, P.R 99(IS55År le.
b; Cartar, Harvey, Hughes and Piitcher, P.R. 96(1954) ii2, c; Bowey and Bird, N.P. 5(i957) 294.
d; Itevin and Hughes, P.K 101 (1956År 1328.
e; $tovey and Harvey, P,R. 108 (1957)353.
5. Results of Calculation and Discussiorxs
As the test of equation (1), we compute D of two ca$es whieh are (a) higher and (b) lewer degeneracy in Fermi-level and the results are $hewn in Figi3
Table IV. Resuiting exeitation energy Q's for (a) higher and (b) iower degeneratien, NiS the number ef neutrops in compound nuc!eug.
ÅqaÅr
Åqb) N
gÅq18, 18), dÅqi4, 14År,f==8 Mev g(l2, leÅr, dÅq8, 8År,f==8 Mev
77 9.6e4 Mev 7.21e Mev
79 8. 029 6. 781 81 6, 3e2 5. 927 83 5. 027 5. 054 85 7. 794 6. 57e 87 9. 470 7. 051
"
Shell Strueture Effeet of the level Spaeing in Highly Exeited-Nuelel. 25
an pl 4, respeetively. Through both cases it is assumed that there are three
:R,;i'/;xbeir8ai!,ilg,l.fT.Oi-uil,itgh,iC,rEULagtgie.:'n,as"ldgtk.:.IOe,fe,ekO;tlje:"e.e?Sc`ltSa{:•.P6tik-ll.IZg,tCtS,".Cke3,liP
forin neT uatb ri oe ni aV ndQ:
p----roQ
t`
in(i"nE'ne+ac'he-nDearnmdi-ainedveE'ir,s-pareaveragepairingenergies
År
o A"
N
Fig. 3. Computed and observed level spacing forthehigherdegeneration:(a). A'andJmean
ghneq"p"rgtl;9n'FOefrtii}ite'vOe"iSaan"ddfthiesSa"t"onOeeCnOiM'enpo,Up"aga"m"eCtiee"rSsipgi')edaar'ieng`hi"ndtehgee:oer'kCu'iaOfof"er"etfT02"
which is detenpmd from experiments. The vertical line through a triangle indicates the
S.f}n,gi.ojlif';x.pe.r,im.e,n,U.:,i,u,n.e.ei,1:fSn.t,y,,a.nd,.ih.e,.oe,gihn.a,Eg.ks,,:e,d,u.ge.d,B?,s.a,v.e.:.P,e..sg/s6g•,.T,eg.m.
ample, ref. 3.) The other dotted curve has been drawn through the experimenta1 points.
Looking at the above table we can immediately find the infiuence of degen- eraey and of shell effect on the value of Q;. For pre-magic nuc]ei, in the case Of higher degeneration
dQ; == 9. 604 -6. 302 =3. 302 Mev
and in lower degeneracy
aQ; =7, 210 -5. 927 == 1. 283 Mev.
The difference in Q; beccmes
dQ; (higher) -- dQ; (lower) == 2.e19 Mev.
These figures iRdicate the terms d.(n-g/2)212 and d.(p-d/2)2!2 in the exp- ressSon of Q, carry a role of redueing the excitatien energy mere effective}y as approach to a rr}agie nucleus, aRd cemparing beth figures it easily shews that Åqa) Tises steeper than (b) showing the infiuence of degeneracy in Fermi- level.
År
N
•i:"o
i
NUCLEON NUMBER
Eg• 4.• Cfimputed and observed 2eyei spacing for the }ower degeneratien; (b). Fer the meanmg of symbels, sea Fig. 3,
Table V. The caleulabed level spuaeing, D'g,d, for eompeund "ucleu$ with Horne spin vaiUeS iB unit of ev.
Åqa) Higher degeneratien; g (18, 18), d Åq14, 14) andfst8 Mev.
N J==e J=rl J:2 i=3 1==4
"-
-"-" T-r kewaLrv-.., .- .me,.- rm..L k"--- H.---- d.U
77 i5, e3 5. ol 3, Ol 2. 15 L67
79 102. 08 34. 02 20. 41 14. 58 U. 34
8i le33. 5e 344. 5e 2e6. 70 147. 65 i14. 83
83 342. 64 114. 21 68. 52 48. 94 38. 07 ' 85 il. 74 3, 91 Z35 1. 68 1. 3e
87 2, 31 O. 77 O. 46 O. 33 O. 26
--
shell Strueture Effect of the level Spaeing in Highly Excited-Nuciei. 27
i,1si.'/Si/1,aO'irg.i/$l,'koieiaig'i\,E,.li.}MSelko,:'i`i'iiS,pk".i.2ag/ni.'ilo/"S'V},"3,ieixljtlhe,ic,li"l,,.,e'/ni,il/,Ssg"sM,6,io/i,].12.lhe/i"/Åéaxe/9iZs,eoi,n:,
