Emission of neutrons following muon capture in heavy nuclei

2.B

Nwelear Physics A271 (1976) 401-411 ; © North-Rolland Pvblishtnp

Co .,

Ansrtsrdaet

Not to be reproduced b7 photoprint or miaoßim without wrlttm pamission >iom the publisher

EMLSSION OF NEUTRONS FOLLOWING MUON CAPTURE IN HEAVY NUCLEI J. HADERMANN and K. JUNKER

SH"iss

Fedrra/ Institute Jôr Reactor Research, 5303 Würenlingen, Sw "lt~erlarrd Received 29 April

1976

Atwtract : The energy and multiplicity distribution of neutrons following moon capture in heavy nuclei is investigated . High energy neutrons (E > 10 MeV) are shown to be emitted directly by the fundamental capture process.

1. Introduction We consider the capture of a negative moon by an atomic nucleus, The basic weak interaction and its inference for nuclear moon capture have been well known t) for a long time [for a recent extensive review see ref. s)] . However, bocause ofnuclear structure complications most efforts have been directed to capture in light and modium weight nuclei, where the capture process is rather successfully explained a) by a resonance mechanism involving the giant resonace ofthe nucleus. Far less, both experimentally and theoretically, is known 4) for heavy nuclei . When a moon is captured by a heavy nucleus most of the energy is carried away by the neutrino, leaving, however, the nucleus formed in a highly excited state (15-20 MeV). Baause of the Coulomb barrier, proton emission is strongly inhibited and deexcitation occurs essentially by neutron emission t. The neutron spoors consist ofa low energy boil-off part (E S 10 MeV) accounting for ~ 90 ~ of emitted neutrons and a slowly da~easing tail extending up to SO-60 MeV. This high energy part is described by a direct emission process following the elementary capture on a bound proton Calculations have been done in thé framework of a Fermi gas model e .') and Singer's Gaussiaa model '-~. The correspondence between experiment and calculation is moderately good for the neutron multiplicity distribution, sometimes at the expense of introducing unphysical effective masses. The neutron spectrum is, however, not reproduced. Schrôder') has taken into account precompound emission f In the actinide region fission is a competing channd sad may well become equally probable °).

401

402

J . 1-IADERMANN AND K . JUNKER

in order to explain the observed magnitude and energy dependence ofthe high energy neutrons. Recently we have shown t ~ that consideration of the Pauli principle plays a decisive role in calculating the amount of directly emitted neutrons. Here we will show that the same holds true also for the distribution of neutron multiplicities. Momentum distributions are calculated for a Woods-Saxon potential well. An increase ofthe high momentum content of the nucleus caused by short range oorrelations has been taken into account phenomenologically . In sect. 2 the calculation of neutron multiplicity and energy distributions is outlined . In sect. 3 we give numerical results and a discussion for the special case of muon capture in natural lead . The lower multiplicities (i S 3) are rather well reproduced . For higher multiplicities the assumption of a constant nuclear temperature during the evaporation process is not justified. The energy spectrum of emitted neutrons can be divided into three regimes. At low energies evaporation neutrons make up almost 90 % of all neutrons emitted. From approximately 10 MeV to high energies neutrons are directly emitted following the fundamental capture process eq . (1 .2). In a medium energy range the spectruun is explained by a single particle momentum distribution (pole approximation) whereas for high energies nuclear oorrelations become important. 2. Formalism In this section the calculation of neutron spectra and multiplicities is outlined t. We start from the strongly simplified weak interaction Hamiltonian " = GT, ~ siS(r,-r~, Hwt t=~

where G is an effective coupling constant, T, converts a muon into a neutrino and s, is the nucleon isospin step operator. The sum extends over all nucleons. Taking the nuclear wave function as a product of individual nucleon wave functions and performing a Fourier transformation one arrives at the following expression for the total capture rate

f

 8n~`c ~~~~°Z.~ d3~3Pcd3PoPp(p~~o(po~k+Po-P~Eo+Ep-Eo-E ).

(2.2)

Here, PP(p~ and ~o(p~ are the momentum distributions of proton and neutron hole states, respectively . The delta fondions account for momentum and energy conservation, and k, po, pr and E, Eß, Ep designate neutrino, neutron and proton momentum and energy, respectively . We have neglected the momentum of the muon in the 1s state and introduced the Q-value

f

We follow here closely refs.

°~

9)

where a detailed derivation of eqs. (2.1) to (2 .9) can be found.

NEUTRON EMISSION

40 3

as a function ofparent and daughter nuclear masses, muon mass and muon binding energy . The excitation function I(Q) is defined by A x f~l(Q)dQ,

(2.4)

0

where Q=

Eo -kc = Eo -E p

(2.5)

is the excitation energy . Introducing an effective mass m* (the same t for both protons and neutrons) in eq. (2.5) En - Ep

_ 1 2m* (Pô - Pp),

we amve at the final expression for the excitation function I(Q) =

Cr(Eo-Q)

~dPPpP(P~o(

J rro

P +

Q)~

(2 .6)

Here, C, is a constant, dètermined by the normalization of I(Q). The lower limit of integration is given by Po = I

m

*~ Eo -Q

-

Eo - Q

2c

I.

For the calculation of the neutron multiplicity distribution the following three assumptions are made : (i) compound nucleus formation, (ü) deexcitation by evaporation of neutrons, and (üi) evaporation characterized by a single mean temperature. Some comments regarding these points are in order. First, in heavy nuclei the spectrum of emitted neutrons can indeed be described by evaporation. However, it is not totally clear whether the evaporation spectrum is a consequence of statistical processes or of a superposition of a large number of transitions from collective states t') . In light nuclei these transitionsare seen giving rise to resonance structure a) in the neutron spectra. The directly emitted neutrons contribute about 15 %, only, and have a quite different energy spectrum . The energy distribution ofdirect neutrons will be calculated below and their contribution to the neutron multiplicity distribution is taken into account. Second, for heavy nuclei the emission of charged particles is strongly hindered by the Coulomb barrier. Whereas in y-coincidence meastuements no proton emission has been seen `), activation analysis ' 2) show that the cross section for (~-, xnypv) depends on the Coulomb bamer and is orders of magnitude smaller than the corresponding (~-, xnv) cross section. Thus, point (ü) is certainly well fulfilled. Third, nuclear temperature is detxeasing with successive evaporation f This assumption can easily be avoided and is made only in order to reduce the number of input parameters . Strictly the efïedive mass is state (density) dependent .

404

J. HADERMANN AND K. IUNKER

ofneutrons. Consequently by assumption (iü), high multiplicities are underestimated and low multiplicities are ovenstimatod. However, in view of the large errors of experiments and the simple model of the capture mochanism oonsiderod in this work, it seems not to be worthwhile to introduce a more refined description at this point. The probability of emission of i neutrons is given by Mi = f~dQNdQ)I(Q)- ~~ dQNr+i(Q)I(Q~ e.

e~+~

when B, is the binding energy of i neutrons and N, is the probability for emission of at least i neutrons s~31 . Q-B~ Q B, for iZ2 = l N~Q) - ~P No(Q) = Ni(Q) = 1. Hen T is the nuclear temperature. Next we will consider the energy distribution ofdirectly emitted neutrons. Starting from «l. (2 .2) one gets the following expression for the momentum distribution of the neutron coated inside the nucleus, i.e. without final state interaction, dN

~,~,,

Here CN is a constant normalizing dN/dpo to unity. The limits of integration are given by The distribution, eq. (2.8), is transformed to an energy scale by introducing a further effective mass m' ' z E' = Po/~

where the energy E' is measured from the bottom of the potential well in which the neutron is moving . The distribution of directly emitted neutrons is finally obtained by shifting the energy by an amount Vo (the depth ofthe potential well) and by taking into account final state interactions through the transmission coefficient Ta(E) dNa =

(2 .9) k~dkkPp( m' E+ Vo)+m (kc-Eo ). J e Hen E is the energy of the emitted neutron and the index d indicates direct emission . So far wehave not specified the momentum distributions entering eqs. (2.6) and (2.9), respectively . Singer e) has assumed a Gaussian momentum distribution

CNTa(~o(

'fE+ Vo))

NEUTRON EMISSION

40 5

Lubkin 6), MacDonald et al. ') and, more recently, Christillin et al. t a), have done calculations in the framework of a Ferrai gas model t. We have calculated the momentum distribution by summing up the individual contribution of nucleons moving in a Woods-Saxon potential

with

1 1 _ 1 df Hws = - ~ Vz- Voi(r)-1 : nt~ r dr I ~ ~~

(2.10)

The eigenfunctions of the Hamiltonian (2.10) are expanded in harmonic oscillator functions whose Fourier transformation is done trivially, the momentum being measured in units of the oscillator constant ~. The fulfilment of the Pauli principle plays a decisive role in determining the percentage of directly emitted neutrons . Whereas in refs. '~ ~ the Pauli principle is taken into account approximately by we have where v~(p) is the usual occupation probability and Po(p) is now the momentum distribution of all neutron states. The continuum of neutron states is considered discrete. As emitted neutrons may have energies of severa110 MeV, up to 22 oscillator shells have been taken into account in order to push to high energies the Gaussian decrease of the momentum distribution which follows from the expansion in harmonic oscillator functions. The main difference between expressions (2.11) and (2.12) is a steep increase at the Ferrai momentum and consequently the percentage of neutrons with energies E > 0 is appreciably enhanced . The proton momentum distribution calculated from the Hamiltonian (2.10) falls off very rapidly at the Ferrai surface. It is well known that the single particle momentum distributions drop too rapidly. As a consequence the high energy spectrum (E ~ 25 MeV) of emitted neutrons also decreases too strongly . The content ofhigh energy moments of the nucleus is greatly enlarged by the inclusion of short range correlations [see for example ref. ta)] . Therefore, at high moments (p ~ 260 MeV/c) the single particle distribution has been replaced by a Gaussian distribution

f We note in passing that these two assumptions are extreme opposites. Nevertheless mean values (mean energy, mean number of emitted neutrons) differ only slightly in both models, and from experiment after a suitable choice of some parameter .

406

J . KADERMANN AND K . JUNKER

in aooordance with 1Janos and Cribson t s) and the early work of Brueckner et al. t 6), where K is a constant to fit both momentum distributions. Two main questions arise with such a phenomenological introduction of nucleon correlations. The problem of double counting seems not to be too serious as for p Z 260 MeV/c the single particle momentum distribution has already dropped to very small values (sce fig. l). Secondly the nuclear matrix elements of the weak interaction Hamiltonian are changed t'). However, we do not calculate absolute capture rates and have, anyway, neglected the momentum dependence of the effective coupling constants which at 40 MeV may change the neutron emission probability by 20 ~. 3. N®erical rewlts nnd condueioos

Numerical calculations were done for muon capture in lead mainly fortwo reasons. First the model presented in the last section is appropriate forsuch a heavy nucleus. Second lead is a target where fairly exte~ive measurements have been performed. Furthermore, to our knowledge it is the only heavy nucleus where the neutron spectrum has been measured up to energies of 50 MeV . We have done a calculation PP 0.16

0.10

0.06

0

700

200

p

(M~V/c)

300

Fig. 1 . The momentum distribution of protons Po(p) is drawn for Z°sPb. The full line follows from the Wooda-Sauon potrntial calculation . The contribution by including correlations on a phrnomrnological basis is shown by the dashed line. For comparison a Gaussian distribution is also depicted (dash-dot line).

40 7

NEUTRON EMISSION

for thethree lead isotopes z°6 Pb, z°'Pb and zoapb enabling a comparison with experiments on natural lead. The Woods-Saxon potential parameters, eq . (2 .10), have been chosen according to the droplet model ' 8). In order to show the effect of introducing short range nuclear oorrelations as discussed in the previous section, the proton momentum distribution for z °aPb is shown in fig. 1 . The shell-model momentum distribution is reliable up to moments of p x 260 MeV/c where we have fitted the constant K of eq. (2.13). In accordance with refs. ts, te) the parameter a is given by a z /2m = 20 MeV. For proton moments of p = 300 MeV/c the contribution of the correlation is already appreciable. Regarding electron-nucleus scattering data and also the results given in fig. 3 their contribution seems to be somewhat overestimated. In the present study we did not want, however, to refit the parameters . Muon and neutron binding energies were taken from refs. t9 . z~, respectively . The effective masses m' and m' were put equal. As the first is for neutrons near the Fermi surface and the second for the emitted neutrons, we expect m' z m'. In order to exhibit the

o

tr .vl

Fig . 2 . The excitation function following muon capture in'°°l?b is drawn . The solid lines correspond to the single particle contribution, the broken Bass show the effect of correlations . The effective masses were chosen as indicated on the curves .

=assume =Ref energy mcornlations 1ßg comparison of mean is0 =matter ')to m* 2neutron MeV 0neutron much the For broaden spectrum on =m) the multiplicitiea excitation calculations was 0the aSinger the =multiplicities lower latter n>m~ber play 00coral nucleon effective taken the calculation model m Inavalues, innspectrum function F, table role isconformance incorrected Df, calailation HADERMANN ~essentially masses mass taccord ~22), for This 1asthe of capture istwo I(Q) namely for =It iswill sometimes') neutron ofMacDonald with calculated independent 0coral counter atis is values with in only the be shown m natural AND the 0 efïiciency 21of Singer lower multiplicities were for 5of experimental Kneutron special lead etThe m*/m the done excitation JUNKER alchosen, limit e) for compered parameters ')is effect and importance two 5model') multiplicities alsoshown of0values Schnöder of avalues 0Za namely energies corral tochosen lowering and nuclear experimental of(notetheir the we numbers compatible for m* ~effective Q see are temperature =~0data An the no listed loweffective effective et%ctive 25 neutron for mf0 reason mass fcoral tU t0 MeV with 00')and the To m

408

J.

dependence m* .6ßm, mass nuclear to In . mass that spectrum . T .2 low

.

.6

.6

.

.8,

.

.

.

.24) .

. T~e~

Calculated i

.6

.

"

m " =0.60m M, without

.

m " =0 .68m IN, with

.

without

.

with

.

0 1 2 3 4 5

0.077 0.438 0.378 0.106 x x

0 .080 0.437 0.370 0.110 0.003 0.0001

0.063 0.443 0.420 0.073 x x

0.069 0.443 0.406 0.081 0.002 x

n

1.52

1.52

1 .51

1.51

The

. T~ai.e

Neutron

F, Calculation .60

m* 0 1 2 3 ~4 5

0.366 0.459 0.155 0.019 O.000Z x0

For meal " .41 ') . .

.

m"

.68

Singer

0.362 0.470 0.154 0.014 0.0002 x0

0.387 0.377 0.167 0.057 0.011 0.001 .

~p~ 0.324 0.483 0.137 0.045 0.011

.022 .025 .018 .010 .006

NEUTRON EMISSION

409

the multiplicity Ml the amount of directly emitted neutrons ( x 15 /) has been added. We recall that high multiplicities are underestimated with the asstunption of a constant temperature T during the boil-off process. Consequently the mean number of emitted neutrons is somewhat lower than the experimental value'), n~ w = 1 .709 f0.066. The smaller ef%ctive mass m' = 0.6m increases the higher multiplicities. In order to compare with experimentally determined multiplicity distributions we have to correct for the neutron counter efficiency e. The observed probabilities F" are given 2s) by

These values are compared to the experiment of MacDonald et al. ') and a calculation') in the framework of Singer's model . The values F, with n 5 3 are well reproduced whereas F" with n > 3 are underestimated because of the aforementioned reasons. A special problem constitutes the zero neutron emission probability M°. Whereas in y-coincidence measurements values compatible with zero are found'), in activation analyses the reaction ~° a Pb(~ - , v)2 oaT1 is clearly seen with a probability of (9t 1 .5) ~ [ref. 26)] and (8.4 t0.5) % [ref. ~')] . The calculations support the determination from activation analysis. The discrepancy between the two experimental methods is resolved if the muon is captured directly to the ground state of the s°BTl nucleus. Due to varying neutron binding energies M° is strongly isotope dependent (table 3). TABLE 3

Variation of the zero neutron emission probability for capture in different lead isotopes

206 207 208

m" ~ 0.60 m

m" = 0 .68 m

0 .106 0 .118 0 .052

0 .092 0 .105 0.043

For the neutron spectrum the transmission function has been taken t from the work of Schnöder ~ who calculated Td(E) in a semiclassical model. The calculated spectrum of directly emitted neutrons, eq . (2.9), is compared to experimental data sa .sB,2~ in fig. 3. The main part of the neutrons (x 85 ~) is described by an evaporation spectrum and has been fitted accordingly by the long-dashed line . Neutrons with energies E Z 10 MeV are well explained by the direct emission process. This is in wntrast to earlier calculations and a consequence of the sudden t We are very grateful to Dr. W . U . SchrBder for showing his calculations as well as unpublished high energy data to us.

41 0

J . HADERMANN AND K . JUNKER

)0

20

30

i0 E(MaV)

Fig. 3 . Neutron spectrum following moon capture in natural lead. Experimental data are from Krieger =°) (trisuglea), Schtbder et al. _`) (cirdes, up to 20 MeV neutron energy) and SchrSder'~ (cirdes, high energy part of the neutron spectrum) . For low energies (E S 10 MeV) an evaporation spectrum pongdashed line) has been added . The direct neutron spectrum with the indusioa of correlation is shown by the full drawn and dashed-dotted lines, respectively . Calculations with the single particle Woods-Saxon potential fall off strongly at high energies (ehort~aahed and three dashes-dotted line, rapectively). Two effective masfes were chosen as indicated.

increase of Fo(p) at the Fermi momentum . For very high neutron energies the distribution t~lculatod from the single particle moments decreases much too rapidly. At 40 MeV for example it is off already by a factor of ten, ifone assumes m*= 0.68 m. Thus it seems necessary t to take into account high momentum contributions caused by short range correlations . The findings from the above discussioncan be summarized as follows. The neutron spectrum is divided roughly into three reegions. At low energies (E S 5 MeV) wehave a typical evaporation spectrum. Neutrons with energies E Z 10 MeV are predominantly emitted directly following the weak interaction process of the moon capture. Thin region is further subdivided. Up to energies of 20-30 MeV the spectrum t Of course, correhuiona do exist for fundamental reasons, but the question is if and where they show uup in the process considered .

NEUTRON EMISSION

41 1

is well described when summing up the individual single particle momentum distributions. For still higher energies correlations show up enlarging the high momentum content of the nuclear momentum distribution. Finally we want to point out an interesting consequence of the present picture . The directly emittod neutrons should show the asymmetry of angular distribution related to parity violation in the weak interaction process. Up to now this asymmetry has been measured (with large error bars) in light nuclei only. With the operation of meson factories it should be possible to measure the high energy spectrum of neutrons with greater precision as well as their asymmetric angular distribution . We would like to thank Prof. R. Engfer and Drs. W. U. Schrôder, H. K. Walter and A. Wyttenbach for many helpful discussions.

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

Refere~aees

H. PrimakofF, Rev. Mod. Phys . 31 (1959) 802 N. Mukhopadhyay, Phys . Reports, to be published H.1'Jberall, Springer Trails in Modern Physics 71 (1974) 1 P. Singer, Springer Tracts in Modem Physics 71 (1974) 39 J. Hadermann and K. Junker, Nucl . Phys . A2S6 (1976) 521 E. Lubkin, Ann. of Phys . 11 (1960) 414 MacDonald, J. A. Diaz, S. N. Kaplan and R. V. Pyle, Phys. Rev. 139 (196 H1253 P. Singer, Nuovo Cim. 23 (1962) 669 W. U. SchrBder, Thesis, TH Darmstadt 1971 J. Hadermann, Phys. Lett . SSB (1975) 141 V. S. Evseev and T. N. Mamedov, Sov. I. Nucl. Phys . 18 (1974) 499 H. S. Pruya, E. A. Hermea, A. Wyttenbach and P. Baertschi, Spring Meeting of the Swiaa Physical Society, Beme, April 1976 ; and to be published 13) P. Christillin, A. Dellafiore and M. Rosa-Clot, preprint, Scuola Normale Superlore Pisa 1975 14) V. Gillet, 5th Int. Conf. on high energy physics and nuclear structure, Uppsala 1973, eil. E. Tibbell (North-Holland, Amsterdam 1974) 15) M. Danos and B. F. Gibson, Phys. Rev. Lett. 26 (1971) 473 16) K. A. Brueckner, R. J. Eden and N. C. Francis, Phys. Rev. 98 (1955) 1445 17) R. S. McCarthy and G. E. Walker, Phys. Rev. Cll (1975) 383 18) W. D. Myers, Nucl . Phys. A145 (1970) 387 19) D. A. lenkins, R. J. Powers, P. Martin, G. H. Miller and R. E. Welsh, Nucl . Phys. A175 (1971) 73 20) G. T. Garvey, W. J. Gerate, R. L. laf%, I. Talmi and I. Kelson, Rev. Mod. Phys. 41(1969) Sl 21) B. Day, Rcv. Mod. Phys . 39 (1967) 719 22) S.-0. Bitckman, A. D. Jackson and J. Speth, Phys. Lett. S6B (197 209 23) V. S. Evseev and T. N. Mamedov, Sov. J. Nucl. Phys . 15 (1972) 639 24) W. U. SchrBder, U. Jahnke, K. H. Lindenberger, G. Rbschert, R. Eagfer and H. K. Walter, Z. Phys . 268 (1974) 57 23) S. N. Kaplan, B. J. Moyer and R. V. Pyle, Phys. Rev. 112 (1958) 968 26) G. G. Bunatyan, L. Vil'gel'mova, V. S. Evseev, L. N. Nikityuk, V. N. Pokrovskü, V. N. Rybakov and I. A. Yutlandov, Sov. J. Nud. Phys. IS (1972) 526 27) A. Wyttenbach, private communication 28) M. H. Krieger, Thesis, NEVIS-172, Columbia University 1969 29) W. U. SdvBder, private communiation