Level densities for 23 ≦ A ≦ 40

Nudear Pkyska A278 (1977) 333-356 ; © North-SoJland PtaWWrtep Co ., Mrtterrlam Not to be reproduced by photo~tnt or miccoßlm wlthoat wrItten parmisdom flrom tLe ynblLher

LEVEL DEN5TJrIFS FOR 23 ~ A ~ 40 M . BECKERMAN r Department of Nuclear Physics, Wdznwnn Institute of Scinece, Rehcool, Israe! Ratived 3 July 1975 (Revised 7 September 1976) Abstract : Level densities, thermodynamic temperatures, aad spini:utoff factors are deduced for nuclides in the mass range 23 5 A ~ 40 from s- and p-wave neutron resonsncea measurements, charged_ partide resonance measurements, stripping and pickup reaction measurements, and direct level counfing. Numerical level density calculations are performed using experimentally determined singleparfide energies, and these calculafions are compared to the exptrimental results .

1.

Introdaction

In this paper we present information on level densities, thermodynamic temperatures, and spin~utofi factors for nuclides in the mass range 23 5 A 5 40. The neutron resonance data ordinarily used to extract level densities are not extensive for A < 41 since thés-wave resonances are weak and most of the observed resonances are p-wave in character. However, the level spacings are large and the Coulomb barrier is low enabling us to make use of a large body of data from charged-particle resonance measurements and from stripping and pickup measurements . We employ the formalism of Rosenzweig and Porter t) to analyze the nearestneighbor spacing samples. The samples and corresponding mean level spacings, D,b are discussed in subsect. 2.1 . The level densities obtained from the Dam, and level counting are discussed in subsect. 2.2. We perform numerical level density calculations using atperinientally determined single-particle energies appropriate for this mass region . These calculations and their comparison to the experimental level densities are discussed in sect. 3. 2.

Experimeotal. reaalts

2.1 . LEVEL SPACINGS

In this section we examine level spacing data consisting of(i) s- and p-wave neutron and photoneutron resonance spacings, (ü) spaccnns between levels populated by (d, p) and ('He, d) reactions transferring Rue unit of angular momentum [lo(d, p) = 1 f Present address : Nuclear Structure Research Laboratory, University of Rochester, Rochester, NY 14627 . 333

33 4

M . BECKERMAN

and lP(3He, d) = 1], (üi) charged-particle resonance spacings, and (iv) spacings between levels populated by one-nucleon transfer reactions [other than (ü) above] . In general, weak levels may be missed due to limitations on the resolution of the magnetic spectrographs employed. It has been shown Z) that this limited energy resolution, typically of the order of 10 keV, is exhibited in spacing distributions as a deficiency ofsmall spacings . We employ the formalism of Rosenzweig and Porter 1) to calculate spacing distributions for levels of any given set of spins and parities, and compare these distributions to spacing histograms constructed for each sample . We estimate the number of levels missing due to resolution limitations by adding "virtual" levels to the sample under investigation, placed at energies given by the product of a random number and the length of the energy interval, until the resulting histogram and normalized probability distribution are brought into agreement. The probability distribution of nearest-neighbor spacings is 1) l2 R(g~) R(grz) lZ Z p(g~) + ~ (1) P(x) = II (x) ~ qi q~ ~(4~ D(q~x) ~ ~ D(grx)l D(grx)l }~ where

(x) _ ]~ D(gkx), D(x) = 1-(2/n)~ Jxc}x)~e-}r=dy, k

where x represents the ratio S/D,s., and S is the spacing. The quantities q, are the fractional densities, k where J; is the ith spin-parity combination occurring in the sample, Q is the spincutoff factor, and ~, q, = 1 . We determine Dam, by means of standard slope techniques, and extract a new Dam, whenever the observed values change by more than roughly 30 ~ over a particular energy range. The multiple entries appearing in tables 2-4 result from this procedure. The variance in the observed mean spacing, varD,b determined over an interval containing n nearest-neighbor spacings is var Dam, = c D,~,/n, where the constant c varies from 0.273 to 1 .0, and the error due to finite sample size is taken as the square root of varD~, [ref. a)]. 2.1.1. Neutron and photoneutron resonance spacings. It is well established 4 ' s) that the p-wave neutron strength function has a maximum near A = 23, while the s-wave neutron strength function has a maximum near A = 60 and decreases as A decreases. Consequently most of the slow neutron resonances identified in the mass region covered in this work are p-wave :

LEVEL DENSITIES

33 5

T~t,e 1

Summary of p-wave and s+p wave neutron resonance spacing and photoneutron resonance spacing data CN z4Na zaM g zeAl z9Si 3a~ 3sCl 4°K 4°Ar

Reaction

E (MeV)

z3IVa(n, y) za Mg(y.n) z'AI(n, n) z9Si(y, n), zsSi(n, n) 3s~(n n) 3'CI(n, n) 3°K(n, n) 4°Ar(y, n °)

7.08 11 .45 7.83 9.31 8.61 6.14 7.86 10.6

~

!or J'

F(MeV)

a

D,~, (keV)

Ref.

1 11 ~0, 1 0, 1 0, 1 1-

2.11 2.24 2.25 2.10 1 .91 1 .72 1 .73 1 .77

2.41 2.66 2.83 2.81 3.22 3.19 3.35 3.38

34 f 10 120 t28 32 f 11 110 f21 7.0 f 3.0 9.8t 4.1 5.4t 0.9 42 f 7 .0

e .')

`4) °) 13) °' >,) °.9) `°) 's)

The neutron resonance measurements provide information on p-wave resonance spacings for Z 4Na [refs. 6" ')] and ZBAI [ref. s)] which is summarized in table 1 . The spacing data listed for s6, j s Cl [refs. e . 9)] and ~°K [ref. t ~] include contributions from both s- and p-wave resonances ; the results presented for Cl are obtained from the lower portion of the measured energy range for which there is good agreement among the groups e " 9) doing the measurements. Additional spacing information is obtained from high resolution photoneutron spectra measurements near threshold . For example, a number of pt resonances have been observed in, the 29 5î(y, n) reaction above 500 keV [ref. 't)] . The excitation energy range in 29 Sî corresponding to this has also been studied in 28 Sî total neutron cross-section measurements' [ ) and, most recently, in the 28 Si(d, p) measurements t 2). The spacing given in the table applies to those resonances included in the recent compilation of Endt and Van der Leun t 3). A number of additional resonances are reported for the total neutron cross-section measurements while somewhat fewer ~- levels are seen in the stripping data. The spacing deduced from the resonances listed in ref. ts) is consistent with the spacing deduced from ld~ stripping Tesce 2

Summary of spacing data involving levels populated by le(d, p) = 1 and IP(3He, d) = 1 reactions Product nucleus z4Na 30Si 34~ 3a~ 37~ 3B~ 3e~. 39Ar 4°K 4oR

Reaction z3Na(d, p) zssi(d, p) 33 S(3He,d) 3sCl(d, p) 3a~d~ p) 3'Cl(d, p) 3'Cl( 3 He, d) 3sAr(d, p) 3'K(d, p) 39 K(d, p)

E (MeV) 4.55 9.25 4.10 4.40 5.02 3.80 6.63 5.30 3.80 4.80

J` 1 -, 2-, 30', 1 -, 20--3 0--3 }-, } 0--3' 0'-3 }0--3 0--3 -

t(MeV)

a

D, b , (keV)

2.11 2 .1( 1 .97 1 .91 1 .58 1 .72 1 .51 1 .69 1 .73 1 .73

2.41 2.91 3.11 3 .22 2.99 3 .19 2.98 3.23 3.35 3.35

81 f 15 140t46 180f44 110 f 19 180t38 % t 17 90 f27 230 f48 84 f 16 56t 12

Ref. 13) la) 13) 13)

30~

17) 13 .s1) sz) 17) 17)

33 6

M . BECKERMAN

levels seen at slightly lower energies . Finally, table 1 contains information obtained from 1 - photoneutron resonance data for 26Mg [ref. la)] and a°Ar [ref. t s )] . 2.1.2. The 1,(d, p) = 1 and !P(3He, d) = 1 level spacings . Significant 2p singleparticle strength has been found in many of the nuclides in the mass region of interest by means of the (d, p) and (3He, d) reactions. For instance, DWBA studies of the Z9 Si(d, p) reaction 16) have lead to the identification of many 2pß. levels near an excitation energy of 9 MeV. Also, in some of the doubly odd nuclides for which p-wave neutron resonances have been identified, many lo(d, p) = 1 levels have been located at excitation energies intermediate between the lowest-lying bound levels and the resonance region ta . "). The lo(d, p) = 1 and lp (3He, d) = 1 data are summarized in table 2. The spacing distribution for the ninety lo(d, p) = 1 levels in ZaNa, ss. 3aC1 and °°K is shown in fig. 1 . We employ a bin size of 0.2 in constructing this and all other 20

90 l =1 Strippinq Spacings in : I z4 Nq ~~Ci &~IC

15 Ns

10

L0

2 .0

3 .0

4.OS/D,p~

Fig . 1 . The distribution of nearest-neighbor spacings for levels in =4Na, 'b . 3 "Cl and `°K populated in (d, p) reactions with !e = 1 . The histogram represents the experimental data . The dashed curve represents the calculated distribution assuming a merged population of 0 - to 3' levds with spin-cutoff parameter a = 3 .0, and is normalized to the total area of the histogram .

histograms . The dashed curve is calculated using eq. (1) assuming a population consisting of 0- , 1 - , 2- and 3- levels, a spin-cutofffactor of 3.00, and is normalized to the total area of the histogram. The distribution has a variance given by eq . (3) with c = 0.639. It can be seen that there are too few spaccnns with x < 0.2 and this deficiency together with the overall shape of the histogram imply that levels are indeed missing. We then estimate that 32 spacings are missing representing 26 ~ of the total. 2.1.3. Charged-particle resonance spacings . We now examine the charged-particle resonance spacing results given in table 3. The proton kinetic energies corresponding

LEVEL DENSITIES Twnr a

33 7

3

Summary of charged-particle resonance spacing data Compound nucleus

Reaction

23 Na 23 Na 24MB 24Mg 26~ 27 A 1 27A1

22 Ne(p, y), (p, p) =2 Ne(P. Y) . (P. P) 2°Ne(a,y) 23 Na(P, Y), (P. P), (P. a) 2sMg(P+ Y) 2dMg(P, Y), (P. P) 26Mbg~(fp', Y), (P, P) z7A1(p+ Y). (P+ P) 2'Si(p, y) 3oSi(p, Y). (P. P) 3 °Si(p, Y)+ (P. P) 3oSi(P+ Y)+ (P . P) 31p(P+ Y) . (P. Po)" (P. ao) 31p~p Y) . (R Po)" (P . ao) 3 °Si(a, y), (a, n) 33Ci., Y) 3 ~(P . Y). (P. Po) 3sC1(P. Y)+ (P+ P~+ (P+ a) 3s~(R Y), (P+ P). (P+ a) 3sS(P, Y) 36~.,p~ Y) 34 S(a,y) 3sAr(p, y) 39K~ ~ Y). (P . Po)+ (Po+ a) 39 K(P. Y)+ (P. Po)" (P+ ao)

28Si

eoP slp 31P 31P 32S 32S 34S 34~ 3sC1 36~ 3eAr 37~ 3701 3e Ar 3 'K 4°Ca 40~

~

e~

10.05 11 .20 11 .44 12.65 7.11 9.72 10.50 12.45 7.43 8.27 9.06 9.67 9.70 10.55 11 .10 6.20 7.79 9.45 10.45 9.59 10.02 10.15 8 .14 10.00 10.95

1 or !` 0_2 0-2 0*,1 - ,2*,3-,4* 0 *~* 0-2') 0_3 0-3 1 *~3* 0-3 b) 0-3 0-3 0-3 0~-2 * `) 0 * -2~ `) 10-3 °) ~ 0-3 0-3 0-3 0-3 l' 0_3 11-

(MeV)

°

D,b, (~eV)

Ref.

2 .37 2 .49 2 .46 2 .59 2 .37 2 .31 2 .35 2 .11 2 .22 1 .96 1 .96 1 .96 1 .95 2 .05 2 .04 1 .97 1 .84 1 .87 1 .88 1 .67 1 .67 1 .72 1 .69 1 .39 1 .45

2 .46 2.53 2.60 2 .67 2 .72 2 .78 2 .81 2 .74 2.98 2 .88 2 .88 2 .88 2 .94 3 .02 3 .17 3 .11 3 .09 3.18 3 .19 3 .08 3.08 3 .19 3 .24 3.00 3 .06

42 t 5 .5 22 f 2.4 95 f23 43 f 9 .2 23 t 3 .2 23 ± 3 .1 l4 f 2 .1 44 f 7 .3 36 f 5 .8 36 ± 6 .0 28 ± 5 .4 17 t 2 .5 59 f 13 40 f 8 .2 90 f 9 .2 33 f 5 .6 23 t 2 .7 24 ± 3 .4 15 f 1 .7 7 .2 t 0 .9 a) 5 .3 t 0 .6 d) 78 t 11 19 t 2 .7 a) 170 t 32 64 f 11

13) !3) 131 13) s4) 34) s4) 13) 13) s4) S4) !4) 13) 13) s4) 13 .2s) 13 . ss ) 13 .ss) l3 . sd) le . 19) I8 . 19)

13) 20) 13) 13 . !7)

') 2s, Id3 , 2 and 2p3i2~

b) 2s, id3n , 2p and If.

`) Corresponds to 2s, Id 3 , 2 and 2pan~ Corrected for missing levels (see text) .

to the excitation energy averages, E, listed in eol. 3 vary from just under 1 MeV to slightly above 2.5 MeV, and optical model calculations show that proton transmission coefficients are non-negligible (> 10 - `) for 1-values up to from 2 to 4, respectively . While the l-values under consideration can be restricted to this range these calculations only provide an upper limit since there is no guarantee that all reconanoes consistent with these angular moments are observed . Therefore, whenever possible, we restrict the samples to those resonanoes having spin-parity values occurring. with frequencies which agree with that given by the fractional populations, qr(J~, in eq. (2). This is not possible in all cases and we now discuss the selection of the samples and J; values in greater detail . s4 Mg, Z BSi and 32 S: The samples are restricted to those resonances having the J` values listed in ool. 4 since these are the resonance spin-parity values which occur with frequencies consistent with that of eq . (2).

338

M . BECKERMAN

z3Na, Z'Al, Sop, 3tp~ 3s Cland seAr : The!-values are listed in col. 4. In some cases all, and in other cases most, of the resonance Jx values are known within the restricted samples and occur with frequencies consistent with qr(J~ . The resonances in the odd mass nuclei 23 Na, 2'Al, 3t P and 3sCl provide us with a sample of 301 nearestneighbor spacings, and the histogram constructed in the manner previously discussed is shown in fig. 2. The predicted distribution is represented by the dashed so

-i 7 I I 301 I "O-3(4)Proton Resonânce Spacings in 2aNa~2 as ~Al, 3~P 9 Cl

48

N~

3

24

12

i L0

2A

3.0

4.OS/D~

Fig. 2. The distribution of nearest-neighbor proton resonance spaccnns in ' 3Na, "Al, s'P and ssCl . The histogram represents the experimental data . The dashed curve rept+esenta the calculated distribution assuming a merged population of }t to It levels with spini;utoff parameter a = 3 .0, and is normalized to the total area of the histogram .

curve and is calculated for a superposition of fi t, fit, ~t and ~t level spacings with Q = 3.00 . The appropriate value for c is 0.798 . It is obvious that the agreement between the histogram and dashed curve is excellent. 345+ saAr and aoCa : The samples are restricted to the known 1 - resonances, and the spacing histogram compared to a Wigner distribution with good agreement. 2eAh aaCh s'Cl and 39 K: Only a few of the resonance l-values have been determined. Therefore physically reasonable ts) !-values are used . The 3e S(p, y) and 3 eAr(p, y) resonance measurements provide data on 150 resonance spaccnns in te, 3'Cl [refs. t ~] and 39 K [ref. ~~]. In contrast to the findings for lower mass nuclides we estimate that 66 spacings are missing or 30 ~ of the total. This estimate is made by normalizing the calculated distribution to the resulting histogram (not shown) over the range 0.4 S x ~ 3.0 after shifting the histogram one bin to the right to compensate for an expectod decrease in Dob,. This method gives essentially the same result as the more precise procedure when applied to other histograms while avoiding an ambiguity in assigning spacings to the first two bins encountered in the

LEVEL DENSITIES

33 9

present case. ('The first bin represents spacings on the order of 1 keV which is the same as the accuracy of the resonance energies .) Since this 30 ~ is considerably larger than the error due to finite sample size the values for Dob, listed in table 3 contain the above correction for missing levels . 2.1.4. One nucleon transfer reaction level spacings. The level spacing results summarized in table 4 are obtained from stripping and pickup reaction measurements T~sLe 4 Summary of spacing data involving levels populated by stripping and pickup reactions Product nucleus zsM g 2s Mg ssAl 26~ ze Al s9Si 3sP azP 73S ssS ssS

Reaction

E(MeV)

lo

i'(MeV)

o

Dd (keV)

Ref.

Za 24MB(d, P) Mg(d, P) z7Al(sHe~ 27~(3He a) ~) 27p1(d, P) ="Si(d, P) 31p(d, P) 31p(d, P) 32C~d, P) a=S(d, P) ~sS(d, P)

6 .00 7 .15 4.55 5 .60 5 .35 7 .08 4.75 5 .85 6 .00 6 .75 7 .50

0-3 0-3 0-2 ') 0-2 ') 0-2 b) 0-3 0-3 0-3 0-4 0~ 0-4

2 .27 2 .27 2 .37 2 .37 2.25 2 .10 1 .97 1 .97 1 .90 1 .90 1 .90

2 .58 2 .58 2 .72 2 .72 2 .83 2 .81 2.96 2 .96 2 .99 2 .99 2 .99

83t21 53 t 11 85 t 20 46 t 10 49 t 7 .4 63 t 12 85 f 18 45 f 10 53 t 11 40 f 9 .2 26t 4 .0

l3) 13)

13) 13) 13)

ID)

13) t3) 13) l3) 13)

') 2s, Idsf = and 1p IJ =. 2s, ld 3JZ and 2p3J2 .

b)

121 4,=o-sfals>r1p~r~y Sparxlgs ~: zs~zssl ß a~ 13 N, 10

LO

2.0

~1~`~~7]hr~JL_

3.0

4.0 S/0~ Fig. 3. The distribution of nearest-neighbor spacingi for levels in ssMg, ='Si and'sS populated in (d, p) reactions with 1, = 0-3(4) . The histogram represents the experimental data . The dashed curve represents the calculated distribution assuming a merged population of }~ to it levels with spin-cutoff parameter a = 3.0, and is normalized to the total area ôf the histogram.

34 0

M . BECKERMAN

involving the transferred angular moments ranges which are listed in eol. 4. A typical nearest-neighbor spacing distribution is presented in fig. 3. The histogram contains 121 stripping level spaccnns in the odd mass nuclei zsMg, z9Si and 33S characterized by lo(d, p) = 0 to 3(4) . The dashed curve represents the expected spacing distribution assuming spin-parity combinations of fit, ~.t , .fit and ]t , and Q = 3.00. We estimate in the usual manner that 36 spacings are missing or 23 ~ of the total. 2 .2. LEVEL DENSITIES

The density of levels of all spins and parities, i .e., the level density, p(E), is derived from the observed level spacing, Doe,, by means of the relation zl)

The symbol (Joe, denotes that the sum extends over those spin-parity combinations occurring in the sample . The values for Doe, are listed in col. 7 of tables 1-4, the J~ values correspond to those listed in col . 4 and the excitation energies, E, to those listed in col. 3. We choose for p(E) the simple two parameter representation ~E], P(~ _ ~ exp [ where ci and ai are the pre-exponential and exponential level density parameters, respectively . With these two parameters we can describe most of the low-lying bound levels and the data summarized in tables 1~. It should be noted that the energy-dependent denominator which often appears in level density expressions is effectively absorbed into a i . It is now necessary to specify the spin~utoff factor, v. We take v to be proportional to I/fizz and to t, where I is the moment of inertia of the nucleus and t its theretodynamic temperature, as is usual. For nuclides with mass numbers less than 100 the experimentally known values for v are consistent with the assumption of a rigidbody moment of inertia, I~, and a radius ro = 1 .2 fin [ref. zz)]. We employ p(E) given by eq. (5) only over a limited energy range, say from some E, to F, ; we then introduce an "average temperature", t, by and take with. Ir/~z = 0.0138A~. The above set of coupled equations permits us to obtain a consistent set of parameters ai, t and Q by means of a simple iterative procedure.

LEVEL DENSITIES

34 1

T~ 5 State and level density parameters Nuclide

a,

a=

oi

A = 23 =`Ne =`Na

1 .189 1 .428 1 .164

1 .990 2 .001 1 .644

2 .212 2 .208 1 .857

0.0475 0.0206 0.250

0.293 0.126 1 .510

0.00626 0.00507 0.0691

2 .028

2 .2I5

0.00206

0.00871

0.00206

0.00871

1 .894 2 .006 1 .847 2 .080 2 .048 1 .864

2 .097 2.204 2.032 2.269 2 .242 2.044

0.0639 0.0165 0.170 0.0500 0.0155 0.144

0 .413 0.107 1 .156 0.348 0.105 1 .021

0.0146 0.00641 0.0308 0.00896 0.00641 0.0341

0.0601 0.0269 0.139 0.0416 0.0286 0.161

2 .168

2.355

0.00187

0.00827

0.00187

0.00827

2.153 2.316 2.227 2.334 2.467 2.287 2.347 2.364 2.497 2.470 2.474 2.133 2.488

2.349 2.501 2.411 2.533 2.633 2.488 2.525 2.569 2.686 2.668 2.667 2.327 2.659

0.0406 0.0144 0.0860 0.0236 0.0105 0.112 0.00476 0 .0306 0.00988 0.114 0 .0240 0 .140 0.00514

0.286 0.0831 0.640 0 .170 0 .0776 0 .831 0.0351 0 .299 0 .0765 0.888 0 .186 1 .130 0 .0410

0.00913 0.00307 0.0103 0.00621 0.00315 0.0157 0.00291 0 .00975 0 .00440 O.Olll 0 .00823 0 .0547 0.00316

0.0426 0.0153 0.0524 0.0299 0.0158 0.0773 0.0152 0.0482 0.0236 0.0585 0.0435 0.306 0.0179

2.4228 2.071 2.234 1 .767 2.271 2.450 1 .969 2.321 2.198 1 .860

2.228

2.422

0 .0255

0 .132

0 .0255

0.132

2.071

2.234

0.0533

0 .308

0 .0533

0.308

2.~~1

2`Mg .! = 25 = 6Mg = °AI A = 27 ~°Mg 2 °AI =°Si A = 29 ~ °Si '°P A = 31 ' =Si ' 2P ~=S .! = 33 3 `S "CI A = 35 ' 6CI 36Ar "CI 3 'Ar '°CI '°Ar '9 Ar "K `°Ar `°K `°Ca

2~

2 1 .307 1 .605 1 .118 1 .370 1 .667 1 .295 2.168 2.355 1 .517 1 .727 1 .339 1 .735 1 .905 1 .444 2.115 1 .841 2.110 1 .447 1 .969 1 .739 2.250

ci

~

cz 0.0238 0.0200 0.251

1 .865 ~

2.037

0 .233

1 .863

0 .185

1 .002

2.271

2.450

0 .0130

0 .0719

0 .0130

0.0719

2.263 2.571 2.289 2.011

2.453 2.771 2.491 2.214

0 .0766 0 .0171 0 .0255 0 .222

0 .620 0 .139 0 .214 1 .861

0 .0404 0 .0103 0 .0210 0.157

0.221 0.0569 0.117 0.849

2.523

2.771

0 .00434

0 .0201

0 .00434

0.0201

The resulting values for tand Q are listed in cols . 5 and 6 of tables 1-4. The values for ai appear in col. 2 of table 5 and the normalization constants ci in col. 5. The level density, p(~, and state density, co(~, are related to one another by the ezprcssion zs) since 2sï Q represents <21+ 1~, the average number of states per level. We choose the

342

M . BECKERMAN

b~~ ..

.~

~ y~w ~ a~ .ä  °e= °

a

~e~` T a ..

8

8

8

~ .a

,~~ e

.0 ~~ d ~ ê

0

~w~~d

(AaW/~I~el)d

:^

f0 .w

é

~9 p ~ .~ .II Ô

n

N

~

~> ~,a~~~ o~ a Ta _w 9 ~ ~

u

~~ri â~t~ ~,

.â,"

N

0

ob ,~,

a: Ô

~W N

B

~ .ô ,d

~ a

ww ~~zô'~

(~oW/~I~ol)d

n

N

t0 > w Of W

N 0

eD~N v ~-G .5

LEVEL DENSITIES

343

~o~ N

~ ~

~~E $ op

a-

m ~ ,C

~

w °° C .~

a

O O

~~

V Ç R7 C ~ ~

O

(AoW/ol~l)d

0 (AaW/~1~1)d

ô

°~ â

"

~ a 0

a

e

~ ~ s .°- ~

V

G4W

~ ~ .T r'

pÇ

Ô.~

M . BECKERMAN

344

n N

Q

O O O

O O -

O -

=~

(MW/slo~W) d

~,; ô

w¢

~ô~b~~ ~8~~ ~w

O

DENSITIES

~~ _ ~~ ~ 'Ce ~

345

LEVEL

_~

e

_c~~~ ;o ~~~~3a

'wm.~ .a

'â

c ô. ~_,v_ ~$

w .+ .+ w w~

~

~

.

346

M . BECKERMAN

û p, w ~dÛ n

N

~W

Vl

r .,

a

^w .

c

m .i ~ ~ 'w"'y8~

v~°'ô ~ ô n N

a

0 g

ô

ô

(AoW/~I~I) d

o

_

~p

t,~ .

.C .Z

~ .~ ô Lc. 'in u

m

LEVEL

DENSITIES

347

8~b ° ~ $. L~ _~ ~

N '>

~

~~~â~~

a

~ G II

~

b°

ô (~~W/sIB~W)d

0

Nw .dô~ ââ w~~8â~ô eô .~

±9

â

348

M. BECKERMAN

same parametrization for co(E) as for p(E), namely co(E) = ei° exp [ aiE], and also provide state density parameters . The constants ~ are listed in col. 6 of table 5 and usually di = ai =_ a given in col. 2, except for those nuclides for which we find a single set of level density parameters for all energies and adopt similar state density parameters with ai ~ ai. The level densities determined in the present work are shown in figs . X21 . At the lowest energies we plot level densities obtained from counting low-lying bound levels . These appear as open symbols and are given in 0.5 MeV steps using a 1 MeV countins interval . The bound level information is taken from ref, ia) plus a number of more recent papers aa, u). The main body of information, namely the level densities derived from the spacing information contained in tables 1~, is represented by solid symbols with error bars denoting the errors due to finite sample size. The open symbols with error bars appearing at higher energies are taken from the literature on Ericson fluctuation measurements and will be discussed in sect. 3. The solid lines also shown in the figures represent the level densities given by eq. (5) with the parameters of table 5. It can be seen from either table 5 or the figures that one set of level density parameters is used to represent all nuclides of a given odd mass. Exceptions occur for the A = 37 and 39 isobaric families for which the N = 20 members appear to have significantly lower level densities. Consequently, distinct sets of parameters are provided for them and the others are only denoted in table 5 by their mass numbers. For the three doubly even nuclides, Z~Ne, ~sMg and 32 Si, the only available level density information is that obtained from counting levels . We therefore employ an average value for ai = A/16 .8 determined from the non-symmetric doubly even nuclides 26Mg and a°Si. The pre-exponential level density parameters . ci are then found by normalization. 3. Namerkal adcalatbos

We now calculate level densities numerically using one of the methods Ze-s°) which has been developed for doing calculations of this type within the framework of the nuclear shell model, incorporating realistic single-particle energies and the 13CS formalism. In particular we use the Hillman-Grover combinatorial method se). A set of single-particle energies characteristic of an infinitely deep spherical well for non-interacting (except for pairing) fermions is employed. The nuclear configurations are cycled and sorted out by particle number, energy and angular momentum . Odometers are used to select only those configurations which obey the 1?auli exclusion principle. The excitation energy of a particular configuration is the sum ofthe energies of the occupied orbitals minus the ground-state configuration energy. The angular momentum of each configuration is calculated using vector coupling rules. The pairing interactions are treated using a simplified version of the

LEVEL DENSITIES

349

BCS pairing formalism in which the pairing interaction operates between all active orbitals . We employ the single-particle energies of Sceger [given in ref. Ze)] except for some values in the vicinity of the Fermi surface which are replaced by experimentally determined quantities appropriate for the middle of the s-d shell ; the experimental orbitals are defined in stripping and pickup measurements as spectroscopic factor weighted energy centroids
The level densities calculated numerically tend to fluctuate widely at low excitation energies due to the low single-particle densities and we proceed as follows : First we calculate level densities up to 50 MeV using the orbitals to be presented in subsect. 3.1, and setting the BCS gap parameters equal to zero, consistent with their behavior in regions of low .single-particle density as shown by the figures in ref. 2e) (to be discussed further in subsect. 3.3). Secondly, we take the numerical results for the 45-50 MeV range and match their average values to the low energy experimental results by introducing p(ai, cZ) (subsect. 3.2). Finally, we use the p(ai, c~ to extrapolate the low energy results to the excitation energy range 14-20 MeV where they can be compared (subsect. 3.3) to the numerical calculations ; at these energies the calculated level densities vary smoothly enough to permit meaningful comparisons. 3.1 . SINGLE-PARTICLE ENERGIES

The value given by Seeger for the 1p spin-orbit splitting is nearly identical to the experimental value of 4.38 MeV obtained for 2'Mg by means of the (3He, a) reaction 3i) . We find that
350

M . BECKERMAN

The energy difference,
ldeiz Zs ld  z 1 f z 2Paiz 2P~~z lfs ~ z lgv~z

present work 57.3 71 .0 74 .4 94 .4 103 .5 112.2 118 .9 125 .7

Seeger ") .protona

neutrons

57 .3 75 .9 73 .6 85.4 107.9 116.2 104.E 113 .1

62 .7 78 .8 83 .6 94 .7 112 .E 123 .2 l 19 .4 126 .7

`) Orbital energies are given relative to the is orbital reference energy of 0 MeV . ze). u) Ref.

The above single-particle energies, together with Seeger's values, are listed in table 6, and aregiven in the form MeV ~ A} . One set of orbitals is presented in contrast to Seeger's distinct proton and neutron sets since the above data do not warrant their extraction. 3 .2 . NUMERICAL CALCULATIONS AND MATCHING PROCEDURE

We calculate level densities up to 50 MeV using energy bins of 1 MeV for all nuclides discussed in sect. 2 employing the orbitals listed in table 6 supplemented by additional orbitals given by Seeger s6) (for a total of 31 orbitals). We then introduce level densities P(a°z, ~ of the same functional form as in eq . (5) and fix a2 and by two values for p : the calculated value averaged about 47.5 MeV and the experimental value taken at 10 MeV. In order to maintain the mass dependence of the level density parameters of sect. 2 we average the 47.5 MeV points for all nuclides of a given mass number and replace 47.5 by 47.5+d during the matching. The d-values compensate for odd-even effects and we adopt Kummel's values' : d = 8plA~ fordoubly even nuclides, d = 0 for odd mass nuclides andd = - 8plA~+ ~/A for doubly odd nuclides where Sp = 12.77 and p = 29.34, and for the doubly

LEVEL DENSTTIES

35 1

even symmetric nuclides we add to d a symmetry energy'6) d,~, = S,~,m(N-Z)z/A with S,,,m = 23.3. The parameters determined by the above method are listed in cols . 3 and 7 of table 5. The maturing condition at 10 MeV (p(ai, ci) = p(ai, c~) reduces the number ofindependent parameters, however, for convenience we list all of them . The procedure is repeated for the state densities and parameters a2 and ~ listed in cols. 4 and 8 of table 5. The level densities p(ai, ~) are represented in figs. 4-21 by solid lines appearing at excitation energies above 10 MeV, and p(ai, c~ by solid lines appearing at excitation energies below 10 MeV. For those instances in tables 1~ where level spacings are given for excitation energies above 10 MeV the iteration procedure of sect. 2 and the matching procedure of this section are performed simultaneously so that al and ci describe the experimental results below 10 MeV, and a2 and c2 those above 10 MeV. 3 .3 . COMPARISON

We now compare level densities calculated numerically in the 14-20 MeV range to experimental level densities extrapolated to 17 MeV using p(ai, c2). We adjust

(b)

Fig. 22. (a) Ratios of level densities calculated numerically at an average excitation energy of 17 MeV using the Hillman-Graver code to experimental level densities extrapolated to 17 MeV using the parameters listed in table 5. (b) Ratios of level densities calculated numerically at an average excitation energy of 47.5 MeV using the Hillman-Drover code to "average, normalized" level densities (disweeed in subsect . 3.3).

352

M. BECKERMAN

neither single-particle energies nor BCS gap parameters but instead form the ratios ofnumerically calculated to extrapolated level densities. These ratios are presented in fig. 22a and indicate the following: For 25 5 A 5 35 the ratios scatter about unity, and the calculations are therefore consistent with the experimental results. For A = 24 and 25 the ratios are somewhat greater than unity and for A = 23 the ratio is considerably greater than unity. Since the ld~. shell is half-filled for these masses the overshoots reflect the well-known ground-state degeneracy exhibited by calculations and usually removed by turning on the BCS pairing interactions . The values obtained for the gap parâmeters which reduce these ratios to unity are GP x (7-10)/A x Go. For 36 5 A 5 40 the ratios p(HG)lp(a, c) are much less than unity (the rise for A = 39 and 40 is due entirely to the argon contributions to the calculations which have N = 21 and 22, respectively). The behavior of the experimental level densities for nuclides having N and/or Z near 20 and for nuclides throughout the lf~ shell will be discussed elsewhere a°). What is of interest here is to notethat (i) the undershoot cannot be removed by turning on the BCS interactions, and (ü) as discussed in subsect. 3.1 the spacing between the lf~ and ld~ orbitals, and between the 1d~ and 2s orbitals most appropriate for the calcium region are larger than the average set presented in table 6 and their use will only further reduce the calculated level densities. In order to gain a qualitative feeling regarding the persistence of the undershoot we can compare the 47.5 MeV numerical results to "average, normalized" level densities p(âs, c2) obtained by first finding the average value of a 2 la l over the entire mass range, then forming the quantities â2 = (a~Mi for each nuclide and finally, calculating the level densities p(â2, c2) at 47.5 MeV where c 2 is used to normalize p(â 2 , c2) to the experimental value at 10 MeV. The ratios p(HG)lp(â2 , c 2) are shown in fig. 22b and we see that for 36 5 A 5 40 the ratios are, as before, much less than unity. 3.4 . ERICSON FLUCTUATION MEASUREMENTS

Ericson fluctuation measurements provide information on level densities at excitation energies above the "resonance-stripping" region and we represent these data in figs. 4-21 by open symbols with error bars. These error bars denote errors of f50 ~ which are roughly the errors indicated in refs . at . a2) and are chosed for all data points for convenience. Not all existing data are shown, but rather only those results giving either p(~ or D°(~, the spacing of spin zero levels. The neutron measurements provide data on ZaNa, ZsMg, Z8A1, Z9Si and 33 5. The results for Z `Na are taken from the figures shown in ref. 41 ), the results for ZsMg, Zed, Z9 (14.0, 16.3, 18.7 and 21 .0 Si MeV) and 33 S are taken from col . 3 of table 2 of ref.'Z), and additional results for Z9Si are taken from ref. a3). The charged-particle data are summarized in table 7. The experimentally determined D° values are listed in col. 4 and are either taken directly from the references cited or are extracted from

LEVEL DENSITIES

353

Tear.e 7 Spacings of spin-0 levels given in the literature from Ericson fluctuation measurements involving charged-particle projectiles CN zaMg z6A1 z'AI z 'Al zeSi ze Si zeSi zeSi zeSi z° Si '°P azS "Cl "Ar ~"Ar

Reaction

E, (MeV)

Do (~eV)

Ref.

izC(~zC, a) za Mg(d, a) ze Mg(p, a) zs Mg(d, a) z'AI(p, y) z'AI(p, a) z,Al(p, a) z`Mg(a, a') z`Mg(a,a") z'AI(d, a) zsSi(d, p) s`P(P. Y) 'zS(d, a) 'sCl(d, a) "CI(p, a) `)

ac,')

19 .9/25.0 15 .2 20 .0 21 .0 19 .6 19 .8 20 .8 24 .55 28 .0 21 .4 15 .7 19 .0 16 .2 19 .8 21 .5

3.50 ") 3.02 3.29 3 .29 3.06 3.07 3.11 3.27 3.41 3.41 3.28 3.43 3.46 3 .78 3.74

37 .7, 39 .0/7.7 12 .1 4.09 1.17 19 .5 17.9 11 .1 7.77 2.3 2.4 14.4 16.8 10.2 0.74 0.67

aa) ") as, as) 4~) ss) s~

s9)

e°) e` ) 4') bz)

ea ) ") ") as, ae)

') Calculated ; see text for explanation . s`). b) Value cited in ref. `) Result from ref. °`), also presented in fig. l9 .

50

45 4i0

b 3~ 30

10

20

30 40 50 Ex(MeV) Fg . 23 . Spin-cutoff fedora for representative nudides as a function of excitation energy calculated using e9. (8) with parameters listed in table 5. In the insert theoverall energy dependence of a(z'AI) is compared to az z E' iz and az z E forms which are normalised to a(z'AI) at 10 MeV.

354

M. BECKERMAN

TlD~(~ ratios when T, the coherence width, is also provided . The spacings Do(~ are related to the level density by p(~ = 2~a(~/Do(~ where Qcr, is the spincutoff factor for the compound nucleus. We calculate vim, using eq . (8) with the parameters listed in table 5. The calculated QcN are listed in col. 5 of table 7, except for Z4Mg for which we use the value cited in ref. ~), and are shown in fig. 23 . Using the above convention for the error bars we fmd that the experimental and extrapolated level densities are in agreement for 24 out of the 37 data points shown. In general no large systematic. discrepancies occur except perhaps for 3 '" 3eAr for which the fluctuation results are considerably larger than the extrapolated values and may serve as a further indication that the numerical calculations are systematically too low in the A = 36-40 region . In addition, the spin-cutoff factors for the residual nucleus, Qttr determined from the angular distributions can be compared to the spin~utoff factors listed in tables 1~. The experimental results are z3Na: QRZ , = 5±i:°s [refs. 4s, a6)] and QRN = 7 3oP : 4' 5 : QRzn, = 8 [ref. a')] ; 5 : Q~ = 10±i .s [ref33 °')]~ 34 Qom = 8 [ref. )] ; a6, ae)] [refs. and compare to tabulated values of 2.46, 2.98, 2.99 and 3.09, respeotively . Values obtained from angular distribution measurements 4~ for the reactions t 4N on t60, z3Na and 2'Al are Z6A1 : Q,t,,, = 2.6f0,6, 2.8±0.5 ; 335 : QR,,, = 3.0f 0.6 ; 3'Ar: Qttr, = 3.5±0.4, and compare to 2.77, 2.99 and 2.99, respectively ; we can therefore conclude that the angular distribution measurements yield spin-cutoff factors which are in close agreement with those obtained in this work assuming a rigid-body moment of inertia with ro = 1 .2 fm and temperatures given by eq . (6). The author wishes to thank Prof. Z. Fraenkel for his many helpful discussions and suggestions, and for his continued interest throughout the course of this work. The author also wishes to thank Prof. I. Dostrovsky and Dr. J. Gilat for stimulating discussions, and Dr. M. Hillman for making available the combinatorial code. References 1) 2) 3) 4) 5) 6) 7) 8) 9)

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