Analysis of excitation functions via the compound statistical model

Nuclear Physics 69 (1965) 401--422; ~ ) North-Holland Publishin# Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

ANALYSIS OF EXCITATION FUNCTIONS VIA THE COMPOUND STATISTICAL MODEL Angular Momentum Effects R. A. ESTERLUND t and B. D. PATE Washington University, Saint Louis, Missouri tt Received 14 August 1964 Abstract: Data are presented for the MoS~(~,n)Ru9~, Mo92(u,p)Td 5g, MoS~(~,p)Tc ssm, and Mol°°(~, n)Ru 1°~ excitation functions. These and some previously published data for reactions involving single particle emission have been analysed via the compound statistical model, with explicit inclusion of angular momentum effects. The values of a, the level density parameter, derived from comparison of the results of these calculations with experimental data agree with values predicted for the Fermi gas model. The values of I/Ir, the ratio of the nuclear moment of inertia to the moment of inertia of a rigid body, show deviations from the value predicted for a Fermi gas model of the nucleus, and an apparent systematic trend with the nuclear mass number.

EI

I

N U C L E A R REACTIONS s~,~°°Mo(ct,n), 92Mo(0%p), E : measured or(E). Natural targets.

7-29 MeV;

I. Introduction It has often been taken as a feature of the compound statistical model 1) that the decay of compound nuclei is independent of the nature of the target and projectile system by which they were formed. However, this assumption is known to be inaccurate in at least one important aspect: the compound nuclei, although formed with a unique excitation energy, have a distribution in angular momentum characteristic of the masses of the target and projectile. Many previous analyses 2-8) of nuclear reaction data via the statistical model have not explicitly included angular momentum effects; this paper presents the analysis of new and selected previously published excitation function data in terms of the compound statistical model with explicit inclusion of angular momentum effects. The compound statistical model for nuclear reactions predicts that the rate of emission of a particle x with channel energy 8 from a compound nucleus at ext Present address: Department of Chemistry, Brookhaven National Laboratory, Upton, New York tt Work performed under U.S. Atomic Energy Commission Contract Number A T ( l 1-I)-870 with support from National Science Foundation Grant Number G-22296. 401

402

It. A. ESTERLUND AND B. D. PATE

citation energy U and angular momentum J~, to form a product nucleus with a residual energy Ef and angular momentum Jr, is given by 9) R(U, J~, e, Jr)de _ (2s -~ 1) ~/~O.inv(/~,j ¢ , j ~(2Jr + 1)f2(Ef, Jr) de,

(1)

where I2(U, Jc) and f2(Ef, Jr) are the densities of levels of a particular angular momentum in the compound and product nuclei respectively, s is the intrinsic angular momentum of particle x, and # is the reduced mass of the system. The quantity tri,v(e, Jc, are) is the cross-section for the reaction which is the inverse of the emission reaction, that is, the cross-section for particle x with channel energy e to be captured by the product nucleus with energy Er and angular momentum Jf to form the compound nucleus with energy U and angular momentum J¢. This inverse cross-section is given in the channel spin formalism by 1o) ~2(2j~ + 1)

O'inv(~, J¢, Jr) =

s~+~ s~s E ~ Tt(e)ds, ( 2 J f + l ) ( 2 s + 1 ) s=ls~-*l ,=lso-Sl

(2)

where ~; is the reduced de Broglie wavelength of the system, S is the magnitude of the vector sum of the angular momenta of the residual nucleus and emitted particle, and l is the orbital angular momentum (in units of h) carried into the collision system by the lth partial wave of particle x. The quantities T~ (transmission coefficients) are the fractions of the lth wave incident particles which penetrate the nuclear potential. Various nuclear models have been used to derive expressions for the nuclear level density as a function of excitation energy and angular momentum; for example, one equation derived for the Fermi gas model predicts al) (2(g, J ) = c o n s t ( 2 J + ? )

exp

(It) ~

[-J(J--I- 1)h 2] ( E + t ) _ 5 / 4 E2tt]

exp 2x/rdE,

(3)

where I is the moment of inertia of the nucleus, and a is a constant which is proportional to the average spacing of single particle states at the Fermi energy. The thermodynamic temperature t is defined by the equation of state for the Fermi gas x2) E = at 2 - t

(4)

and the level density parameter a is given approximately by 12) rnr~

a = 2(½rr)8/a -hq- A,

(5)

where m is the mass of a nucleon, ro is the nuclear radius parameter, and A is the mass number of the nucleus. If r o is taken as 1.2 fm, a is predicted to be A/12.7 MeV-1; a choice of r 0 equal to 1.5 fm leads to a value of A / 8 . 1 MeV -1.

EXCITATION FUNCTIONS

403

The excitation energy employed in expressions (3) and (4) may be reduced, in the case of nuclei with an even number of neutrons, and/or protons, by a condensation energy 6, in accordance with the expectation 13) that in such nuclei a minimum excitation energy must be exceeded before description in terms of a Fermi gas is applicable. In the present work, previous practice has been followed, in which condensation energies are equated to pairing energies, as given, for example, by Cameron 14). The cross-section for a nuclear reaction may be expressed as the product of two factors 15): (1) the cross-section for formation of the compound nucleus with angular momentum Jc and excitation energy U, and (2) the probability that the compound nucleus will decay to the product nucleus with angular momentum Jf and residual energy El. Therefore, the total cross-section observed for the reaction is the sum of the crosssections for each possible reaction path; that is tr(b, x) = Z tr°a.(eb, J°, di) Fs~(U, Jo)

"o

(6)

Z r,,(v, Jo)' i

where troap(eb, Jo, Ji) is the cross-section for formation (via bombardment with particle b at channel energy eb) of the compound nucleus with a particular angular momentum value Jo; this is given by eq. (2) with Ji and eb substituted for Jf and respectively. The emission function Fs~(U, Jo) is the emission rate (given by eq. (1)) for particle x, integrated over e and summed over Jf; that is

Fax(U, j¢) _ (2s+ 1)# fe x7 ai.v(e, So, St)(2Jr+ 1)Q(Ef, Jf) de. ~2-~

,j~ ~

(2Jo+a)f2(U, Jo)

(7)

The quantity ~ i Fsl(U, Jo) is the summation of emission functions over all particles i which can be emitted from this compound nucleus. Many previous authors 2- s) have calculated reaction cross-sections via the formula /~x

tr(b, x) = troap(eb)~ F , '

(8)

i

where ffeap(eb) is now the cross-section for formation of the compound nucleus in all states of angular momentum, and is given by 1o)

oap(e ) = E

So, J,)

Jz

= ~c~2 ~ (2/+ 1)Tl(eb). 1=0

(9)

404

R. A. ESTERLUND AND B. D. PATE

The emission function without explicit consideration of angular-momentum effects is usually taken as F x = (2S-b 1)#[eO'inv(~)f2(Ef)de ,

(10)

de

where O'inv(/~ ) is now the cross-section for capture of particle x with channel energy by the product nucleus in its ground-state, to form compound nuclei with all angular momenta Jc, and is given by eq. (9) with e in place of eb. For computational simplicity, numerous authors 2-8) have used empirical formulae 3) or tables xo, 1 6 - 1 9 ) of cross-sections. The level density f2(Ef) is now the density of levels of all angular momenta at energy El, and may be given by 11) [2(El) = const(Ef + t)-5/4t-~ exp

2x/aEf.

(11)

Approximations to (11) have generally been used, and the most frequently used one has been O(Ef) ~ const exp 2 x / ~ f ,

(12)

which is valid only at high excitation energies; it has, however, been applied extensively in regions of low excitation energy where the pre-exponential terms are important. A considerable amount of work on alpha-induced nuclear reactions has been reported recently 2-s), using the simple formalisms described above; the values of the level density parameter a, which have been required to obtain agreement between the results of calculations and the experimental data for excitation functions have been far smaller than the values predicted by eq. (5). The reported values range from about ~oA to -~oA, instead of ~ -s-A 1 as expected, assuming r o ~ 1.5 fm. Furthermore, results from studies 2o-24~ of the energy spectra of emitted particles usually yield a values which are considerably higher than those obtained by analysis of excitation function data for similar reaction systems. The low values for a have generally been attributed to the influence of direct interaction processes, which lead to emission of an excessive number of high energy particles 25). The conflict between the results from the two kinds of data would seem however to suggest an alternative possibility: if the compound statistical model is indeed applicable in a description of the kinds of reactions studied, then perhaps some of the above mentioned simplifications made in the analysis of data have been too drastic. It has been suggested by Grover 26, 27) that the neglect of angular momentum effects may have been the cause of the low values of the level density parameter obtained by previous investigations. The effect arises from the fact that the level density is angular momentum dependent (see eq. (3)), and below a particular excitation energy in the nucleus, states of a particular angular momentum may not ,exist, i.e. when the corresponding rotational energy would be greater than the total

EXCITATION FUNCTIONS

405

excitation energy. Thus, emission of low energy particles from compound nuclei of high angular momenta may be forbidden, if sufficient angular momentum is not removed thereby so as to form the product nucleus with a combination of Ef and Jf corresponding to an existing state or states. Grover has shown that the effect on excitation functions will be twofold: thresholds will occur at higher energies and substantial cross-sections will persist to higher energies than expected on strictly energetic grounds. Large cross-sections observed at high bombarding energies have, in general, either been reproduced via simple calculations utilizing a low value for the level density parameter, or have been ascribed to direct interaction processes competing with compound nucleus formation and decay to produce the observed product nucleide.

2. Experimental Procedure Measurements of cross-sections for reactions of molybdenum plus a-particles were carried out via bombardment of stacks of natural Mo and A1 foils using the deflected a-particle beams of the W.U. 1.14 m cyclotron and of the Brookhaven 1.5 m cyclotron. From the former, the incident a-particle energy was 20.4___0.2 MeV 2.8); from the latter it was 41.0 +__0.3 MeV 8). The beam intensities were measured via a Faraday cup and a current integrating device 29). The a-particle energy loss in individual Mo foils was calculated via interpolation between the tabulated data due to Aron, Hoffman, and Williams 30) for Cu and Ag. The interpolation procedure was checked and found satisfactory by experimental determination of the energy loss in Mo at an a-particle energy of 22.8 MeV. The energy losses in A1 were calculated using the data of Bichsel 31). The loss of R u 95 activity from target foils by recoil (following the M o 9 2 ( ~ , n ) reaction) was measured for the thinnest foils employed (of 7.21 mg/cm 2) at 20.2 MeV incident beam energy and found to be 3 ~ of that retained in the target foil. Any errors from recoil losses in the measured cross-sections for this and other reactions were therefore considered negligible compared with errors from other sources. Disintegration-rate determinations were performed via either of two methods: (i) y-ray spectroscopy, or (ii) coincidence counting of annihilation radiation resulting from positon emission. Counting efficiencies were determined using a Na 22 standard source t

Gamma-ray detection efficiencies were obtained from the data of Heath 32). The ),-ray spectroscopy equipment consisted of a 7.6 cm x 7.6 cm NaI(T1) crystal mounted on a Dumont 6363 photomultiplier tube, a Franklin model 358 amplifier and either a Radiation Instrument Development Laboratory model 34-12 transistorized 400channel pulse-height analyser or a Technical Measurements Corporation 256-channel pulse-height analyser. t This source was calibrated against a s t a n d a r d source m a d e available by courtesy o f J. B. C u m ming.

406

R.A.

ESTERLUND AND B. D. PATE

The apparatus for the annihilation-radiation counting consisted of two coaxial 5.1 cm x 5.1 em NaI(T1) crystals connected to Dumont 6292 photomultipliers and the previously described amplifiers. Coincident detector pulses corresponding to photon energies of 0.51 MeV were detected with a Cosmic Radiation Labs. model 801 multiple coincidence unit, and the coincidence events counted by means of scalers. The observed Ru 95, Ru 1°3, Tc 9sg and Tc 95ra activities were assayed in the irradiated target foils without prior chemical separation. The assay methods and radiations studied for each nucleide are listed in the appendix. Other excitation functions for reactions involving the emission of two and three particles were determined, and will be published in a forthcoming paper 33). The full-energy peak for the 0.201 MeV ~-ray of Tc 95m was well resolved in scintillation spectra from targets irradiated over the entire range of bombarding energies, and was employed in the assay of this nucleide. However, Tc 958 was only determined in experiments with a bombarding energy below about 19 MeV, as other activities obscured the 0.768 MeV ~-ray full-energy peak by which Tc 95~ is best assayed. The TC95g cross-section above 19 MeV was therefore estimated, using the TC95g/Tc95m isomer ratio observed below 19 MeV. Fairly constant isomer ratios have been observed 5) at bombardment energies greater than about 5 or 10 MeV above reaction threshold. 3. Results

The cross-section values for the four species listed earlier are presented in table 1, together with the associated errors. These errors were estimated assuming an uncertainty of _+10 70 in counting etficieneies, an uncertainty of + 3 70 in the integrated beam current measurement, and an uncertainty of + 2 7o in the measured thickness of the target foils. In addition, the errors due to uncertainties in the resolution of decay curves in certain cases are included, and range from _ 5 70 to __. 15 700. No attempt was made to estimate uncertainties in the available decay scheme data. The Mo92(~, n)Ru 9s cross-section values appear to be the least uncertain, as the determination involved no chemical separation, and the positon activity in targets after bombardments at energies below 19 MeV is virtually all from R u 95. Duplicate measurements of the peak cross-section agreed to within _+_6 %. Uncertainties in the values of the beam energy at the position of a particular foil result from two sources: (i) the uncertainty in the thickness of the target foils, and (ii) the magnification of the spread in the energy of the incident beam by energyloss fluctuations. The energy "thickness" of the foils near the low energy end of an irradiated foil stack ranges from about 1.5 to 2.5 MeV; thus the cross-section measured at a particular energy represents an average over such an energy interval. The measured excitation functions are shown in fig. 1, in which activation crosssection (in mb) is plotted against c.m. bombarding energy Ec.m. (in MeV). The

EXCITATION FUNCTIONS

407

excitation function for Mo92(g, p)Tc 95B was obtained by measuring the cumulative yield of the Mo92(g, n)Ru 95 and M092(~, p)Tc 95g reactions, and then subtracting t h e R u 95 cross-section values from that total. ( R u 95 does not decay appreciably 34) i n t o T c 9 Sln). The portion of the T c 95s excitation function estimated via the Tc 95g/Tc95m isomer is plotted using vertical bars. TABLE 1 Cross-sections for reactions of Mo with or-particles (in mb) Ee.m.(MeV) 6.8 9.0 9.6 11.5 11.7 12.0 12.7 13.7 14.1 14.4 15.6 15.9 16.0 17.3 17.4 17.7 18.7 19.1 19.4 21.0 22.4 23.9 25.2 26.5 27.8 Errors:

MoO~(g,n)Ru 96

MoOJ(g, p)Tc~s

Mo92(g, p)Tc *m

MolOO(~z,n)Ru xo3 0.64

0.55 0.4 6.7

8.5

7.6 45.8 3.6

70.4

206

299

370 370 358 294 212 156 102 -4-12 % a) ± 1 8 % b)

25.7 43.1 10.6

76.8

17.2

60.4

21.6

37.6

26.0 24.5 19.2 13.2 8.8 6.2 4.8

26.3 19.2 16.6

±15%

±18%

60.2 87.9 104 131 140 157 161

±18%

~) Ec.m. --~ 19.1 MeV b) Ee.m" ~ 19.4 MeV

The low magnitude of the M o 9 2 ( ~ , p ) T c 95m cross-section is in agreement with expected trends. The isomeric state (To95m) has an angular momentum of Jr h, while the ground state has angular momentum of 9 h 34). Considering the high average angular momenta of compound nuclei in this mass region bombarded by ~-particles at about 20 MeV (see fig. 4), it might be expected that the high angular momentum state (To 95g) would be produced in greater yields. The experimental cross-sections measured for reactions o n M o 92 target nuclei up to 20 MeV may be summed and compared to various predictions for the total reaction cross-section from appropriate models of the nuclear potential; the ex-

408

R. A. E S T E R L U N D A N D B. D. PATE

perimental total cross section values constitute lower limits, since the (~, ~') and (~, 7) cross-sections could not be measured. Fig. 2 compares the sum of the (~, n) and (~, p) cross-sections with the predicted values for the total reaction cross-section calculated from an optical model potential 18), a parabolic approximation 35) to the optical model potential, and the square-well potential 10) with a nuclear radius parameter ro = 1.5 fm. The total experimental data in this case appear to be reproduced best in magnitude by the calculations with the square-well potential. Other authors 2-8) have arrived at similar conclusions but have found that the use of r o values of 1.6 or 1.7 fm resulted in closer agreement with their experimental data. 1000

I

I

I

I

I

I I

~

P

MO 92 (~,n) R u 9 5 . - _ . ~ ~

Mo92+alphas

M o 92

[---1} P a r a b o l i c m o d e l 11~ 2) O p t i c a l m o d e l lit3) Square-well model

_

100C

i

I

I

(calculated)

-.

/

,oo

: .o-- .o

o

E Ioc "o

2 __

6I

10I

14I 18I ECM (MeV)

. 22I

2 I6

30

1C

10

12

1~4

16

118

20

22

Ecivi (MeV)

Fig. l. Experimental excitation functions for s-particle induced reactions in Mo nuclei, involving single nucleon emission,

Fig. 2. Comparison of the total reaction crosssection for Mo9~ plus ct-particles predicted by three nuclear models with the measured total reaction cross-section.

The calculations of excitation functions for individual reactions, to be described in the next section, were performed with the approximate optical model potential because this was simpler. The cross-section values were however, in view of the above agreement, normalized to total reaction cross-sections calculated via the square-well potential. 4. Calculations

Preliminary calculations, utilizing the conventional formalisms of the statistical model, were performed on the Washington University I B M 7072 computer. The purposes were twofold: (i) to compare level density parameter values derived from these calculations with parameter values obtained in other similar calculations 2-8)

409

EXCITATION FUNCTIONS

and scattering experiments 2 0 - - 2 4 ) ; (ii) to allow comparison with parameter values derived from additional calculations utilizing a formalism with explicit inclusion of angular momentum effects. Cross-sections were evaluated (following Chen and Miller 2)) using the equation ¢(b, x) = ¢~ap(eb) (2s + 1)# X

.trinv(e)Q(Ef)de + | if U < S x + S 2 x if U~_Sx+S2x

U-Sx-S2x

r i" v - s t - ~

(2s+ 1)# LJo

eainv(e)de /

d U-Sx-6x

-I ;

l "v-s~

eCrinv(e)Q(Ef)de+

(13)

-]

where the summation over i (to take account of all possible emitted particles) included neutrons, protons, deuterons, tritons, He 3 nuclei, or-particles, and photons. The emission function for photons was taken as s)

F~

- 2 s + l /'/~r'x 2crinv(e)12(Ef)de. C2

(14)

J 8mtn.

Inverse and capture cross-sections for particles were estimated either by interpolation of tabulated values lO, 19) or the use of empirical formulae 3). For photons, lower limits to the inverse cross-sections were estimated from (% n) cross-sections reported in the literature 36). The shapes of these (?, n) cross-sections agreed reasonably well with theoretical predictions lo) for the shape of the total y-reaction cross-section, but were somewhat lower in integrated magnitude than the values predicted by the sum rule of Levinger and Bethe 37). It was assumed that the principal radiation emitted was dipole in character, although some experimental studies 3s) indicate that quadrupole emission may be more prevalent. Two approximate forms of the energy dependence of the level density t2(Ef), given by eq. (11), were used t-2(Ef) ~ const exp 2x/aEf-f,

(12)

Q(Ef) ~ const Ef-2 exp 2x/~ff.

(15)

Eq. (15) is a somewhat better approximation to (11) than (12), for if the thermodynamic temperature t is taken to be approximately x / E ~ , then at sufficiently high Ef (i.e. E f >> t), eq. (11) reduces to one which is proportional to eq. (15). At sufficiently low El, eq. (15) predicts a level density which passes through a minimum and then rises rapidly, in conflict with experimental data. Thus, the level density in these low energy regions was obtained by a logarithmic straight-line extrapolation of the level density obtained at higher Ef values.

410

R. A. ESTERLUND

AND

B. D .

PATE

Binding and pairing energies were taken from the tables of Cameron 39). The binding energy $2~ of the particle most likely to be evaporated from the product nucleus formed by emission of x was taken to correspond to the particle with the lowest sum of binding energy and effective Coulomb barrier 3). The assumption implicit in this procedure is that particle emission through the effective Coulomb barrier is weak in comparison to gamma emission. Reasonable agreement of calculated excitation functions with experimental data could not be achieved with values of the level density parameter a larger than ~o A, in accord with the experience of previous authors. These results, which were found to be significantly affected neither by variation of the condensation energies employed nor by inclusion of the effects of the 300 keV energy dispersion in the incident ~particle beam, are in conflict both with the predictions of eq. (5) and the analysis of the energy spectra of protons emitted in the Mo92(ot, p)Tc 9s reaction 20). In order to investigate the extent to which the results would be affected by explicit inclusion of the effectg of angular momentum, a second set of calculations, using approximate forms of eqs. (6), (7), and (3) (see below), were programmed for the Washington University 7072 computer. The variation of the density of levels with angular momentum predicted by (3) was modified by the method of Grover 27), in which the energy of the lowest lying state of angular momentum J is estimated through the rotational energy by Er°t = 2~ J ( J + l ) .

(16)

This modification is necessary, in that the exponential term for the angular momentum cutoff in eq. (3) is not valid at high values of J and excitation energies near the point predicted by eq. (16), at which the levels of a particular Y value should cut off. Grover has pointed out that, for low angular momenta, the excitation energy is more likely to be divided between rotational motion of the nucleus as a whole and nucleonic excitation. However, the experimental spectroscopic information is not adequate to distinguish between the results of such a complication and those of the simple application of eq. (16) down to the smallest angular momentum values. Since the latter procedure leads to ease of computation, it has been adhered to in the present work. The moment of inertia I is conveniently measured by the ratio I/Ir; here Ir is the moment of inertia of a rigid spherical body of the same radius R and mass m as the nucleus in question, and is given by I, = 2mR2, where R was taken as R = 1.2 A~ fm.

EXCITATION FUNCTIONS

411

The value ro = 1.2 fm is obtained from data taken from electron scattering experiments 40). In these first attempts at a more complete calculation, several approximations were made in order to facilitate evaluation of (7); the first is that, since the distribution in Je of the product nuclei for a given Ef is likely to be quite narrow compared to the overall distribution in Jo of the compound nuclei, one may assume that the process of emitting s wave particles approximately describes the overall emission process. Also it was assumed, since the dependence of angular momentum of the inverse cross-section given by (2) has been suppressed by the first assumption, that the inverse cross-section is dependent on energy only, and may be assumed to vary with energy in a manner similar to eq. (9). When the terms which are constant for the system are factored out, the emission function becomes

r,.(v. Jo)

(2s+ 1),

f Iraax rain

Jo)d,.

(27)

It has been shown 2 o) that an expression numerically equivalent to (17) can be derived via alternative simplifying assumptions. I l l u s t r a t i o n of angular m o m e n t u m effects Region of

J

BI/'.

I

!I~ 0

S w a v e em{ssion A

S×

U -S x E

Fig. 3. Schematic representation of the effects of angular momentum on the evaporation process. The scheme of the present calculation is diagrammed in fig. 3, which illustrates the case for compound nuclei with a distribution in J and at an excitation energy U, where U is slightly greater than Sz, the binding energy of particle x. The figure shows nuclear angular momentum plotted as ordinate, and excitation energy plotted as abscissa. The histogram is intended to represent the relative yields of each state of J populated in the compound nucleus, plotted orthogonaUy to E and J. The curve labelled Erot defines the calculated locus of states of lowest energy for each value of J. Thus, no nuclear states are assumed to exist in the region to the low energy side of the Ero t c u r v e . For illustrative purposes, a particular J value of the compound nucleus at excitation energy U is designated by the hatched area in the compound nucleus histogram and

412

R.

A.

ESTERLUND

AND

B.

D,

PATE

point A on the E, J plot. Emission of a particle to form a nucleus with an equal angular momentum Jf, following the approximations discussed above, is shown, labelled s wave emission. If a particle with zero channel energy is emitted, the final state will be represented by B; however, point B falls in the area where no nuclear states exist. Thus, particle emission from compound nuclei with this J value will be inhibited (see sect. 1), and these nuclei will de-excite preferentially via a cascade of ~,-rays. The result will be a raising of the observed threshold for particle emission. Similarly, product nuclei formed with a particular amount of residual excitation may be restricted from secondary particle emission in the same mariner, thus leading to higher observed cross-sections for reactions involving one particle emission at high bombarding energies. i

+

i

i

MO 92 + alphas 160 120

~

ECM= 28 MeV,,,,~ ECM.=20 MeV%

/,

80

+

~D

0

I

4 J

Fig. 4. Distributions in angular momentum induced in Ru9+compound nuclei by e-particles incident on M o 92. The methods involved in the evaluation of the terms in (6) and (17) will now be discussed. a) ¢rc,p(eb, Jo, Jr). The cross-section for formation of the compound nucleus in a particular state of angular momentum Jc was calculated via (2), utilizing transmission coefficients calculated via a parabolic approximation to the optical model potential described elsewhere 35). Since the data under consideration in this study involved only ce-particles as the bombarding projectiles, the optical model parameters for incident s-particles given by Igo as) were used. The J dependent capture cross-sections calculated via this method were compared with cross-sections calculated via (2), but with optical model transmission coefficients calculated by Huizenga +t). The dif-

EXCITATION FUNCTIONS

413

ferences were small, except at bombarding energies which were lower than a few MeV below the Coulomb barrier. The values of aoap(eb, J¢, Ji) for c.m. bombarding energies of 12, 20, and 28 MeV are shown in fig. 4, plotted as a function of J. The shapes of the J dependent capture cross-sections are consistent with the trends expected, i.e., a greater probability for the formation of high J values as the bombarding energy is increased. b) ai,v(e). The inverse cross-sections used were the same as the ones used in the preliminary calculations described earlier, and were evaluated in the same manner. Since the earlier calculations indicated extremely low probabilities for the emission of deuterons, He 3 nuclei, and tritons, the present calculations considered onlyneutrons, protons, a-particles, and photons as contributing significantly to the emission process in the mass regions considered here. c) f2(Ef, J¢). The angular momentum dependent level density used was an approximation to (3), namely

g2(E, 3) = const ( 2 J + l ) exp F-J(J+l)h21 E -2 exp 2 x / ~ . k 2IT J

(18)

For excitation energies below 6, the level density was set equal to an arbitrary constant; the results of calculations were not significantly affected by variation of this constant. The nuclear temperature was taken as a constant T, equal to temperature values reported by Erba et aL 42). As will be shown later, some calculations were performed with the level density given by (3), using a temperature calculated via (4), and the results required the use of a slightly smaller value of a in order to achieve agreement with data. However, the use of (18) with a constant temperature gave a decrease of a factor of two in computation time. In general, if T was not set below a certain value (i.e., about 0.5 MeV), the value of T had little effect on the results of calculations. d) Integration limits. For the integration of F&(U, Jc), the limits were taken as ema~ = U - S x - E r o t , /3max ---~ U - - S x - 6x,

ifErot > 6=, if Ero t

~

6x,

(19a) (19b)

emin = O, if U < S.+S2.+E2rot,

(20a)

emln = U-Sx-S2x--E2ro t if U > Sx+S2x+E2rot,

(20b)

where E2rot refers to the Erot values in the product nucleus formed by secondary particle emission. The choice of either (20a) or (20b) for emin is necessary, because particles emitted with channel energies below U - S x - S 2 x - E2rot (i.e. Ef __>$2~ + E2rot) will lead to a product nucleus with sufficient residual excitation to undergo particle emission; thus the product nucleus formed by primary particle emission will be not the final product.

414

R. A. ESTERLUND A N D B. D. PATE

If (19b) is used for emaX (i.e. if E~ot < fix), a second integration is performed from 6x to the lowest state (Erot) in the product nucleus, using the limits

emax = U-Sx-Erot,

(21)

8min = U - S ~ - fix

(22)

for the second integration. The same limits are used for the integration of Fs,, except that emin = 0 is substituted for the choice of limits given by (20a) and (20b), since Fs, represents all emission channels, regardless of the final product. In summary then, for each bombardment energy the individual angular momentum dependent cross-sections are calculated via eqs. (6) and (17) for an appropriate number of J values, and summed to give the reaction cross-section tr(b, x); that is

[

o' ii2:;_J:

ea,~.(e)O(Ef .S=)d~+f eai~v(e)d~[

(2s+ 1)# f

U--Sx-S2x-g2rot

a(b, x) ---E

0

Jc

i

(2s+ 1)#

i/ 0

if U > S x + S 2 x + g 2 r o t

if U
U-Sx-6x

_]

O'¢ap(Eb, J c , Ji).

-s,-E~o, ifif~.o,>~, Erot <-~i

U - St - t~t

E

]

f U - St - Erot .O'inv(/3)a(Ef, J=)d,+ eai..(e)de U-SI -~l

(23)

In general, the procedure used in choosing parameter values to obtain agreement of the calculated excitation functions with the observed experimental data was the following: (i) Values for the binding energies and pairing energies were taken from the tables of Cameron 14, 39). (ii) A value for the temperature T was taken from Erba et al. 42) for a nucleus of about the same mass and atomic number as the product nuclei in question, at an excitation energy roughly corresponding to the average residual excitation energy in the product nuclei formed; variation of T did not appreciably affect the results (see above). (iii) The value of the level density parameter a was set equal to that given by (5) for r o ~ 1.5 fm (i.e. a = -~A). (iv) The value o f / , in terms of I/Ir, was varied as a free parameter until satisfactory agreement with the experimental data was obtained. In several cases, a was also varied within a narrow range to improve the agreement. The sensitivity of a calculated excitation function to variations in 1/1, is shown in fig. 5, for the reaction M092(~, n)Ru 9s. The change in the excitation function at energies below the maximum cross-section with changing I/Ir values is not shown, but it is quite small. A test calculation, in which Erot was set at zero, was performed in order to demonstrate: (i) what differences could be observed between these results and the results

EXCITATION

415

FUNCTIONS

of the preliminary calculations described earlier for the same reaction system; (ii) whether Erot or I2(E, J) was the significant factor in controlling particle emission to lower energy states. The results indicated that the presently employed form of f2(E, J ) with a constant T does not significantly restrict particle emission to lower states. Fig. 6 summarizes the calculated excitation functions for the system Mo92+~particles. It is to be noted that the maximum of the calculated (~, ~') cross-section coincides with the approximate threshold of the experimental (at, ~'n) excitation function a3).

1000

I

I

I

MO92 Ccc,n)Ru95 a = -sLA ,T=I.0 MeV

i

I

IMO92+olphos a = 8ZA,I/Zr=O.70,T=I.0 t~eV ~

I / [ r =0. 5 =0.7 =1,0

,2

1O0

1000

M

o

9 M 0 9 2(a:'n) Ru95 92 ((z,p) T¢95 2(o.,e.,n)Mo91

10C

"o

o

MOg2 (a,-~) Ru96

10

I

116

I

214

I

312

E C . M . (MeV)

Fig. 5. Calculated excitation functions for the reaction Mo°2(g,n)Ru05, for different assumed values of the nuclear moment of inertia 1 relative to I,, the rigid body value.

•

i'6

'

2~

"

'

3'2

EC.M. (MeV)

Fig. 6. Calculated excitation functions for simple reactions induced by 0c-particles incident on Me92.

The a value for which agreement of the calculations and experimental data was satisfactory, were consistent with the predictions of (5), namely a = ~-A. The a value also agrees with that obtained via energy spectrum analysis of the Mo92(~, p)Tc 95 reaction 20). The value of 1/1, obtained, 0.70, will be discussed in sect. 5, along with the values of I/I, derived from similar analysis of other experimental data. In general, the agreement appears to be rather good, especially with the (~, n) data. It is interesting to note that the calculated (ct, ~) cross-section is quite small, with a maximum of only 1.6 mb. Fig. 7 shows the results of interest for the reaction system Mol°°+ct-particles; the agreement with the (0q n) data is not as good as in the previous case, especially

416

R . A. E S T E R L U N D

AND

B. D .

PATE

near threshold. Perhaps the most interesting result is the large (~, n)/(~, p) crosssection ratio predicted, which is consistent with the failure in the present study to observe or obtain experimental cross-section data for the Mo t o o(0q p)TcZO3 reaction. The large ratio is a reflection of the strong tendency of nuclei on the neutron excess side of stability to emit neutrons in preference to protons. I

t

]

~

I

T

t

i

Mo 100 + QlphQs e=~

A,[I/Ir=O.45,T=IO

1000

MeV

10C

I

I

~.~. o

o

MolCO(~t,n) Ru103

I

I

Zn64 + a l p h a s o= ~" A, I/Ir=l.2,1.15,T =1.2 MeV o

-- Zn64 (e~,p)Oa67

10C 1C c

1C

"o

/

1

~o Z n64 ( d., ~ ) Ge68

M d ° ° (a,p) Tc1°3 o

'

~'2

'

20

2~

-'

I

8

I

16

ECM(Mev)

Fig. 7. Calculated excitation functions for simple reactions induced by 0~-particles incident on Mot°°. I

I

I

24 ECM,(Mev)

i

I

I

I

Q= 1 A, I / ] r =0.70,T =1.0 MeV

o

I

32

Fig. 8. Calculated excitation function for simple reactions induced by ~-particles incident on Zn 64.

Nb93 + a l p h a s 100C

I~

o

o

o

10C +Nb 93 {a,n) Tc96g

X o

10

12 EC.M.(MeV)

Fig. 9. Calculated excitation function for the reaction Nb93(~, n)Tc 9n.

417

EXCITATION FUNCTIONS

In order to test further the formalism employed in these studies, calculations for other reaction systems, for which data are r@orted in the literature, were carried out. Fig. 8 shows the results for the reaction system Znr¢+ ~-particles 8). These data are especially interesting, as part of the (~, ~,) excitation function has been measured; the agreement with experiment is fair, but the parameter values presently used to obtain agreement with the data are rather puzzling. The level density parameter value was -~A, which is somewhat larger than expected. Also, the best agreement was obtained with two values of I/I~ : for product nuclei resulting from primary particle emission, I/I~ was 1.2; for product nuclei resulting from secondary particle emission, I/I~ was 1.15. i

1000t

i

i

i

Agl°7+olphas a= ~ A,I/Ir=O.AO,T=I.0MeV / Ag107(o.,n)In110 1000

,

,

,

,

In1154-alphas

IO0

a=T10A, I/|r=0.35,T=O.90Me/ o

1 o 10

1

5

(~,n) Sb11B

~10(3 ~o

°°

o '

14

'

2~

r

3'o

ECM.(MeV) Fig. 10. Calculated excitation functions~for simple reactions induced by ~(-particles incident on A g 1°7.

lb

1~

1'8 2'2 2'6 Ec~t (MeV)

3b

3'4

Fig. 11.r.Calculated excitation function for the reaction In115(c(, n ) S b 118.

However, if the r o value used for the calculation of I r were increased from 1.2 to 1.3, the increase in lr would lower the 1/1 r values in this case from 1.20 to 1.02 and 1.15 to 0.98, respectively. Fig. 9 shows the results of calculations compared with experimental data for the system Nb 93+ ~-particles 43). (The experimental data points were interpolated from a published graph). The fact that the parameter values used for a and I/I~ were identical with those used for the system Mo92+ ~-particles is satisfying, in that the similarity in mass number of these two target nuclei would lead to the expectation of such a result. Fig. 10 shows the results for the system Ag 1°7 +~-particles 4). The agreement of the calculation results with the data was obtained with a = -~A, whereas in the

418

R. A. ESTERLUND

AND

B. D. PATE

published analysis o f the type discussed in the preliminary calculations section, an a value of-~TA was necessary in order to obtain agreement. I

]

I

I

I

I

Ba138÷alphas Q= I~A ,[/|r= 0-3Q, T= 0.70 MeV

oo

100

[

I

I

I

I

La~3g+Qlphas a= ~ A, [/Ir= 0.35, T= 0.70 MeV.

el

o /X no 139

142

o

k3 °lE

1C

io/

\° °

o

11

I 18

10

ECM.(MeV)

I I 26 ECM.(MeV)

r

314

Fig. 13. Calculated excitation function for the reaction La13g(~, n)Pr 142.

Fig. 12. Calculated excitation function for the reaction Bala"(~, n)Ce 141.

1000

l

IMo92+~olphasi i i a = ~ A,I/I r = 0.70 ~T=calculated RU95

100

Ho~65+alphas a= I~A,I,/~r=O.30,T= 0.70 MeV o

~o 1C

o

'

~o

~

1~

68

'

2'0

'

2~

8

I

I 16

'

2'4

'

J2

EC.M.(MeV)

ECM.(MeV)

Fig. 14. Calculated excitation function for the reaction Ho1~5(~, n)Tm 1"8.

Fig. 15. Calculated excitation function for the reaction Mog~(~, n)Ru9% using the exact Fermi gas level density expression.

Fig. 11 shows the results of the calculation for the In115(~, n)Sb 1as reaction 44). The experimental cross-sections were interpolated f r o m a published graph, and are relative.

EXCITATION FUNCTIONS

419

Fig. 12 shows data due to Tanaka 45) for the Ba 13s(~, n)Ce141 excitation function, compared with the present calculation results. The capture cross-sections used in this case were those calculated via the parabolic approximation to the optical model; this was the only reaction system studied in which the square-well cross-section values for r o = 1.5 fm did not predict a satisfactory reaction cross-section magnitude. Fig. 13 shows calculation results for the system La 139 +or-particles 4s); the parameter values (a and I/lr) used are quite similar to those employed in the calculations for Ba 13s+ ~-particles, as might be anticipated. Both of the experimental excitation functions in figs. 12 and 13 however exhibit a cross-section at high energy which is not reproduced by the present formalism (see sect. 5). Calculations for the last reaction system studied, Ho165(ct, n)Tm ~68 (ref. 46)), are shown in fig. 14. Since the data are limited, it is difficult to assess the value of the parameters derived from these calculations. However, the calculations were performed in order to obtain a set of 1/lr values over as wide a range of nuclear masses as possible (see sect. 5 and fig. 16). As mentioned earlier, trial calculations were performed, using the level density given by (3) without approximations, in order to investigate the effect on the results of using the approximate level density given by (18). Fig. 15 shows the results of calculations for the Mo92(~, n)Ru 95 reaction; as stated earlier, a smaller a value (a = ~o A) was necessary in order to obtain agreement with experimental data. The difference is small, however, and possibly of less significance than the effects of several approximations made elsewhere in the calculations. 5. Conclusions

It seems reasonable to postulate that the test of the present calculations rests primarily on the values of the parameters used to achieve agreement with the data. The values of the level density parameter used in the calculations range from XA to ~0A; considering the uncertainties in the experimental data cited, these values are apparently consistent with the predictions of the Fermi gas model. The significance of the 1/1 r values, however, is much more difficult to assess. The values of 1lit obtained in the preceding calculations are shown in fig. 16, plotted against nuclear mass number, and show an apparent systematic trend. However, these values are in disagreement with the predictions of the Fermi gas model, namely 1/I r = 1. The discrepancy may possibly be understood through recent work by Lang 47) and others 48) on the superconductor model for nuclear level densities. These studies indicate that in this model the nuclear moment of inertia is a variable fraction of the rigid body value up to about 15-18 MeV of excitation, at which point Fermi gas level densities are predicted with I/lr = 1. The excitation energy range over which this moment of inertia varies corresponds to the range of product residual excitation energies considered in the preceding calculations. Thus, the values of I/Ir obtained in the present calculations would correspond to an "aver-

420

R.A.

E S T E R L U N D A N D B. D . P A T E

age" value of the moment of inertia in the product nuclei over some excitation interval. F r o m tables in the paper of Vonach et aL 48), one may estimate this dependence of 1/1, on excitation energy. Such an estimate for an even nucleus of mass number 100 is shown in fig. 17. Apparently, such a dependence could influence the results of calculations greatly. I

I

I

I

I

1.2

0.8 t.

%

0

0,4

0

0

o*

o

i

ol

5O

, Moss

number

Fig. 16. The values of I[Ir used in the present calculations, plotted as a function of product nucleus mass number.

0.~

0.z

,, i i O.C

o

',;

;

; Nuclec[r

Fig. 17. The variation of I/Ir

1'o

1'2

excitation

1'4

1'6

1'8

energy

with nuclear excitation energy, as estimated for an even nucleus o f

A = 100;a=~A. Another possibility is that these nuclei are distorted at high angular momenta, and that a corresponding distortion energy then exists. The apparent I/lr ratio would then be greater or less than unity, depending on the relative magnitudes of the then appropriate rotational energy (E'ot) and the energy of distortion (Ed). If Ea + E'ot > Erot, then the apparent I / I r value will be less than unity. It is a pleasure for the authors to acknowledge the many valuable conferences with J. R. Grover, without whose suggestions the calculations could not have been performed. The authors are also grateful for the advice and comments of D. W. Lang, T. D. Thomas, J. R. Huizenga, J. M. Miller, D. G. Sarantities, and Camilla Hurwitz, for

EXCITATION FUNCTIONS

421

criticism of the manuscript by Professors J. B. Reynolds, J. M. Fowler, and F. B. Shull; for the kind permission of S. Tanaka to use his data prior to publication; for the splendid cooperation of J. B. Reynolds and the Washington University cyclotron crew, and C. P. Baker and the Brookhaven cyclotron group; for the cooperation of Dean R. A. Dammkoehler and the Washington University computer center personnel. The support of the U. S. Atomic Energy Commission and the National Science Foundation is gratefully acknowledged.

Appendix ASSAY AND DECAY SCHEME DATA

AU decay scheme information was taken from the National Academy of ScienceNational Research Council data sheets 34); the specific sheets used are identified below by number. a) 1.65 h Ru95: For targets bombarded at energies below 19 MeV, positon annihilation radiation was counted, using the coincidence equipment described in sect. 2. In bombardments at energies above 19 MeV, additional positon emitters formed by multiple particle emission interfere, and thus assay was made via the 1.11 MeV y-ray following the decay of Ru 95. Due to uncertainties in the absolute assay via this )'-ray, only a relative excitation function was derived; this was then normalised to the cross-section data obtained by coincidence counting at 19.1 MeV. The branching ratio for positon emission was taken as 0.195; the data sheets used were NRC 60-5-119 and NRC 60-5-132. b) 60.0 d Tc95m: After a period of about 30 d from the end of bombardment, the 0.201 MeV gamma full energy peak from the decay of Tc 95m was measured via gamma scintillation spectroscopy. According to the decay scheme 34), Ru95 decays only to Tc 95g, SO Tc 95m may be determined independently. The 0.201 MeV ),-ray was taken to have an intensity of 64 transitions per 100 decays of TC 95m, and the total conversion coefficient ct was taken as 0.04; the data sheets used were NRC 60-5-120 and NRC 60-5-130. c) 20.0 h Tc95g: After the precursor (Ru 95) had completely decayed to Tc 95g, the 0.768 MeV )'-ray from the decay of Tc 95g was counted via gamma scintillation spectroscopy. The fraction of disintegrations leading to emission of this photon is 0.845, and ,t was estimated 49) to be 0.0013; the data sheets used were NRC 60-5-120 and NRC 60-5-130. The excitation function for independent formation of Tc 95g was obtained by subtraction from the excitation function for cumulative formation of Tc 95s of that for formation of Ru 95. d) 40 d Rul°3: The 0.498 MeV ~-ray. which follows 89 ~o of the decays of Ru 1°3, was counted via gamma scintillation spectroscopy; ~ was taken as 0.0055. Presumably, the measured cross-sections include contributions due to formation of the precursor Tc t°3 via the (~, p) reaction; however, the (~, p) maximum cross-section is calculated (see sect. 4) to be less than 1 mb, so these contributions are probably negligible; the data sheet used was NRC 61-3-65.

422

R. A.

ESTERLUND A N D

B. D. PATE

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42) 43) 44) 45) 46) 47) 48) 49)

V. F. Weisskopf, Phys. Rev. 52 (1937) 295 K. Chen and J. M. Miller, Phys. Rev. 134 (1964) B1269 I. Dostrovsky, Z. Fraenkel and G. Friedlander Phys. Rev. 116 (1959) 683 S. Fukushima et al., Nuclear Physics 41 (1963) 275 R. L. H a h n and J. M. Miller, Phys. Rev. 124 (1961) 1879 N. T. Porile and D. L. Morrison, Phys. Rev. 116 (1959) 1193 F. S. Houck and J. M. Miller, Phys. Rev. 123 (1961) 231 N. T. Porile, Phys. Rev. 115 (1959) 939 T. D. Thomas, Nuclear Physics 53 (1964) 558 J. M. Blatt and V. F. Weisskopf, Theoretical nuclear physics (Wiley, New York, 1952) J. M. Lang and K. J. LeCouteur, Proc. Phys. Soc. 67A (1954) 586 D. Bodansky, Ann. Rev. Nucl. Sci. 12 (1962) 91 H. Hurwitz and H. A. Bethe, Phys. Rev. 81 (1951) 898 A. G. W. Cameron, Can. J. Phys. 36 (1958) 1040 N. Bohr, Nature 137 (1936) 344 E.J. Campbell, H. Feshbach, and V. F. Weisskopf, MIT Tech. Rept. No. 73 (1960) (unpublished) H. Feshbach, M. M. Shapiro, and V~F. Weisskopf, U.S. Atomic Energy Comm. Rept. NYO3077 (1953) (unpublished) J. R. Huizenga and G. Igo, Nuclear Physics 29 (1962) 462; Argonne Nat. Lab. Rept. ANL-6373 (1961) (unpublished) M. M. Shapiro, Phys. Rev. 90 (1953) 171 C. Hurwitz, S. Spencer, R. A. Esterlund, B. D. Pate, and J'. B. Reynolds, Nuclear Physics 5 4 (1961) 65 N. O. Lassen and V. A. Sidorov, Nuclear Physics 19 (1960) 579 V. A. Sidorov, Nuclear Physics 35 (1962) 253 H. W. Fulbright, N. O. Lassen, and N. O. Roy Poulsen, Mat. Fys. Med. Dan. Vid. Selsk. 31, No. 10 (1959) R. Fox and R. D. Albert, Phys. Rev. 121 (1961) 587 G. Igo and H. E. Wegner, Phys. Rev. 102 (1956) 1364 J. R. Grover, Phys. Rev. 123 (1961) 267 J. R. Grover, Phys. Rev. 127 (1962) 2142 R. J. Wilson, private communication, 1962 S. Amiel and N. T. Porile, Rev. Sci. Instr. 29 (1958) 1112 W. A. Aron, B. G. Hoffman, and F. C. Williams, AECU-663 (1949) (unpublished) H. Bichsel and E. A. Uehling, Phys. Rev. 119 (1960) 1670 R. L. Heath, Atomic Energy Comm. Res. and Dev. Rept. IDO-16408 (1957) (unpublished) D. G. Sarantites, R. A. Esterlund, and B. D. Pate, to be published Nuclear Data Sheets, National Research Council (National Academy of Sciences, Washington, D.C.) T. D. Thomas, Phys. Rev. 116 (1959) 703 S. Rand, Phys. Rev. 107 (1957) 208 J. S. Levinger and H. A. Bethe, Phys. Rev. 85 (1952) 577 J. F. Mollenauer, Phys. Rev. 127 (1962) 867 A. G. W. Cameron, Atomic Energy of Canada, Ltd. Report CRP-690 (1957) (unpublished) R. Hofstadter, Ann. Rev. Nucl. Sci. 7 (1957) 231 J. R. Huizenga, private communication, 1962 E. Erba, U. Facchini, and E. S. Menichella, Nuovo Cim. 22 (1961) 1237 J. M. Matuszek, Jr., Clark University Ann. Prog. Rept. (1963) 1 G. M. Temmer, Phys. Rev. 76 (1949) 424 S. Tanaka, private communication, 1962 (to be published) G. V. S. Rayudu and L. Yaffe, Can. J. Chem. 41 (1963) 2544 D. W. Lang, Nuclear Physics 42 (1963) 353 H. K. Vonach, R. Vandenbosch, and J. R. Huizenga, to be published M. E. Rose, Internal conversion coefficients (North-Holland Publ. Co., Amsterdam, 1958)