XJ9700108 I

iniUlllUIUlllWil

ОБЪЕДИНЕННЫЙ

ИНСТИТУТ

ЯДЕРНЫХ

ИССЛЕДОВАНИИ

Дубна

Е4-96-478

R.A.Eramzhyan1, V.Â.Kuz'min, T.V.Tetereva2

CALCULATIONS OF THE ORDINARY

AND RADIATIVE MUON CAPTURE ON 58>60«62Ni

Submitted to «Nuclear Physics A»

'institute of Nuclear Research, RAS, Moscow, Russia2Skobeltsyn Nuclear Physics Institute, Moscow State University, Moscow,Russia

2s h 211996

Introduction

Investigations of the Ordinary (OMC) and Radiative muon capture(RMC) are carried out over several decades. This activity achievedmaximum at sixties-seventies, when the main regularities of elementaryprocesses were elucidated and the basic mechanism of muon capture bycomplex nuclei was established. After that over a long time there wasa pause in the experiment after which the interest in muon capture wasreborn. But at present much attention is paid to the RMC. Already atthe early stage of muon capture studies it was recognized that despitethe fact that the radiation branching ratio is small (by four orders ofmagnitude compared to OMC), the effect of the pseudoscalar couplingnear the pole in the pion propagator, i.e., near the high-energy tail ofthe photon spectrum is enhanced. The magnitude of this coupling untilpresent was determined with a low accuracy though that many effortswere undertaken to do it with a high precision.

Interest in RMC was rekindled first of all due to a great progress inthe experimental technique. This progress made it possible to measurethe energy of outgoing hard photons with a high precision). The secondattractive point for extensive experimental study of RMC is the existingpredictions of week sensitivity of its observables to nuclear structure,especially, if they are calculated with respect to OMC. And finally, con-trary to OMC, where only a single inclusive observable can be measured(besides the partial transitions in light nuclei) in RMC the spectrumof hard 7-quanta is measured. This spectrum is connected with theexcitation spectrum of nuclei.

The progress in the theory demonstrated that the nuclear structureis not so passive in the muon capture. However, to take account ofall details of this structure is a very complicated task since the RMCamplitudes should be summed over all possible final states. These finalstates form the spin-flip giant resonances in charge-exchange channels.In the medium nuclei, the contribution comes from the L = 0, 1 and2 multipolarities. In heavy nuclei some contribution from the highermultipolarities is noticeable.

During the last years the huge work was done on the experimental in-vestigation of the charge-exchange channel in neutron-proton reactions.The resonances excited in this reaction are the same as the ones whichform the nuclear response in the RMC. The essential advantage of (n,p)reaction is the possibility to experimentally separate the contributions ofdifferent multipoles, which cannot be done in the RMC reaction. But onthe other hand, the contribution of multi-step processes to RMC appearto be considerably less than that to the (n,p) reaction in which theybring a noticeable uncertainty in the extracted strength of the transi-tions into different energy regions.

For this reason it is important to discuss the processes under consid-eration in the framework of the same approach to the nuclear structurein order to have a possibility to estimate the quality of the solution of thenuclear part of the problem. Only afterward one can ask the questionabout modification of the induced pseudoscalar coupling constant insidethe nuclear media. It should be mentioned that all previous calculationsof RMC on medium nuclei used the nuclear models which cannot treatthe unified problem of description of the set of processes leading to thesame nuclear final states.

The present paper is the first attempt to combine the quasiparticlerandom phase approximation (QRPA) for the description of the nucle-ar excitation function with the OMC and RMC reaction mechanisms.The 58.60»62JVi isotopes were chosen for the discussion because (i) theexperimental measurements of RMC are carried on; (ii) there are rathercomplete experimental data on (n, p) and (p, n) reactions on these nuclei;(iii) simultaneous investigation of three isotopes in the same approachallows one to reveal the influence of the neutron siibshells filling on thegross-structure of the charge-exchange resonances.

The paper is organized in the following way. The second sectioncontains a sketch of the approximations and formulae used for the cal-culations of the RMC and OMC rates on nuclei. The third sectionis devoted to the solution of nuclear part of the problem. It consistsof short description of the nuclear model Hamiltonian, the review ofthe QRPA approach to its diagonalization. The description of strength

functions of isovector spin-multipole transitions on the base of QRPAand comparison of them with the strength functions extracted from thecharge-exchange direct reactions allow us to fix the parameters of effec-tive nuclear residual interactions. The obtained excitation energies andtransition amplitudes of certain operators are used for calculations ofthe OMC and RMC rates. The results of calculations of the total OMCrates for few values of gP are presented in the fourth section. They arecompared with the results of other authors and with the experimentaldata. The fifth section contains the results of calculations of the spectraof hard photons emitted under RMC and total RMC rates. The mainresults of the paper are summarized in the conclusion.

OMC and RMC rates

The experimentally observable quantity in RMC is the photon spectrum.We calculate it by summing the partial spectra over the nuclear finalstates:

The photon yield A/t(fc) corresponding to the transition from the groundstate \Ji > to the state \Jj > is given as

a(G cosOcYm.RWkik™* - k)2 x

T L Aft(A;/»)A#(A;/i),* XM.Mj

where k = kk is the momentum and A is the polarisation index of thephoton; a and G is the electromagnetic and weak coupling constants,respectively; 0C is the Cabbibo angle and R(Z) is the average value ofmuon density over the nucleus volume [2]. The maximum photon energyis given by kmax = m^ - e^ - (Ej - Ei), where A/,, {e^) is the muonmass (binding energy, calculated by approximations of [4]), Ej - E, isthe nuclear excitation energy. All excitation energies are reckoned fromthe ground state of parent nucleus.

The nuclear RMC amplitude M{\;fi) is described in detail in [3].Here we have to say that in [3] two ways of RMC calculations are dis-cussed. One way — to use the elementary amplitude of [1] for calculatingall partial transition rates and then to sum up them. It is a usual im-pulse approximation (IA). The second way just suggested in [3] consistsin modification of the IA. According to their suggestion one should usethe continuity equation for the electromagnetic current together withthe SU{2) x SU{2) current-algebra commutator. This procedure takesinto account many-body currents like the meson-exchange ones. Afterthis modification all partial transition are evaluated and summed up.This version is called as MIA — modified impulse approximation.

Below we present the results of calculations performed in both ap-proximations.

Usually the photon spectra A(k) are normalized by the total OMCrate. That is why we have calculated the total OMC rates too. It iscalculated as the sum of partial OMC rates over the final nucleus state

\ OMC _ V"* \OMCA - 2-f /* »

where

OMC = (rt?coj.«c) zfR(z)J irn

MOMC(fi)MOMc(fi)+ 1 M,,Mf

Nuclear response calculation

The nuclear amplitudes to describe the OMC and RMC are calculatedwithin the random phase approximation (RPA). For completeness wegive the short description of formalism of quasiparticle random phaseapproximation (QRPA) which is extension of usual RPA to the non-closed shell nuclei; afterwards we will discuss how the parameters ofresidual interaction were fixed.

Nuclear model Hamiltonian [5] consists of the spherical single-particlepotentials of Woods-Saxon shape each for neutrons and protons, a su-

perconducting monopole pairing between like particles, and of isotopicinvariant residual interaction, taken in separable form:

HM = £ HQ(r) + J5T,T=n,p

where

— V^ I Vr-m'r+Jr-mrJ \

and

Residual interaction Hamiltonian consists of two parts — the isoscalarand the isovector ones. It's convenient to write down scalar product ofisospin Pauli matrices in the isovector part of residual interaction as

Afterwards, the single particle operators of residual interaction can bewritten as follows

QLM J2 < j'm%\fLYLMTk\jmt3 > a ] aj'm't'sJmta

and

QLJM'=j'mft'3,jmt3

where rfc is the set of isospin operators i, J°, t+, and t~; SL(T) and / t j ( r )are the radial form factors of residual interaction operators. We will use,as in [6],

Mr) = /w(r) = ±

where W(r) is the central part of single-particle shell potential.The terms containing the product of lowering and rising isospin op-

erators t^t^ ^d to the particle-hole excitations which changes the chargeof the nucleus and can be used to describe /?-decay, //-capture, (p, n)-and (ra,p)-reactions. Correspondingly one can tell about the charge-exchange part of residual interaction which is the sum of products ofoperators like

Jnmn\0JMt+\jpmp > ajn>mn ajp>mp ,

and Hermite conjugated to them. Here OJM is multipole /fj(r)Yj(f)/or spin-multipole //Lj(r)[lx,(r)cr]j/ operator.

. The diagonalization of nuclear model Hamiltonian is done in twosteps. At first step the Bogoliubov transformation taking into accountpairing correlations of superconducting type is fulfilled for protons andneutrons independently

Variational procedure

6(0\ HQ(r) - KNr - ^ Pjr« + v]r - 1) |0) = 0

gives us uy u-coefficients and allows us to pas to the independent quasi-particles Hamiltonian:

where ejr = yJ(Ejr - Ar)2 + Q and CT = GT Yljr (J + 1/2) ujrvjr . Av-

eraging was done over quasiparticle vacuum state: ctjrmr |0 >= 0 , whichrepresents the ground state of the system of even nucleons number.

At the second step the interaction between two-quasiparticle statesis taken into account. From the products of two quasiparticle operatorsthe phonon operators are formed:

6

where [ajp,ajn]jiM = £„,,,„,„ < JPmPjnmn\JM > cxjpimp.ajn^n and< jPTnPjnmn\JM > is the Clebsch-Gordan coefficient. In the randomphase approximation phonon operators are supposed to obey to bosoncommutation rules. As the result the normalization and orthogonalityconditions of one-phonon state are

f ' - Si,. = 0 .v f 7p.Jn r Jpjn

Jp.Jn

The phonon amplitudes and the excitation energies of the one-phononstates are defined by the variational principle

H< Ifijw^Affi'Ll > - < I^A/I > -u;,-*(if i)} = 0 .

where | > is the phonon vacuum state: £llJM\ >= 0. It gives the homo-

geneous system of linear equations:

Ag\, -LJiW\ = 0

where

€? = €jp + Cin , ttf = «;|,Vjn ± ^ t t j . ,

hq =< jp\\Ojt-\\jn > •

The amplitudes of the transitions from the even-even ground state tothe excited states with total spin J , its projection M and energy a>, areequal to

JpiJn

if the charge of the nucleus is decreased by unit, as in(w,p) reaction, orto

if the charge is increased during transition as in (p, n) reaction. Thetransition strength is equal to Bf{i) = ^2J

M=_j \b^M{i)\2.The parameters of the single particle potentials and of monopole

pairing were fixed earlier. Only the effective constants of isovector resid-ual interaction «f or /cfJ were varied in our calculations.

The ordinary muon capture proceeds mainly through the spin-multipoletransitions. The most important of them are the Gamow-Teller transi-tions (at* one-body operator) and the spin-dipole {/{^[Yi^a]^ one-body operator) transitions. In this section we present the results ofcalculations of transition strength distribution over the excitation ener-gy (strength function) for GT and spin-dipole transitions and comparethem with the distributions extracted from (p, n) and (n,p) direct nucle-ar reactions at intermediate energies. The main goal of this section is todefine the effective constants of residual interaction making comparisonof calculated strength functions to those obtained from the experimen-tal data, and afterwards to calculate the muon capture rates using thesevalues for effective constants.

The theoretical and experimental results are shown in Figures1-4 as the "running sums":

s±(£)=/ : Ej-E,<E

By this sums one can show simultaneously the results of calculationwhich are set of discrete eigenvalues with corresponding transition strengthand the experimental data which are often continuous function, especial-ly if some strength is extracted from background. The fast variations inS±{E) indicate the presence of strong transitions in the energy regionswhere these variations occurs.

The results for the at~ transition strength on 5SNi are shown inFig. 1. The excitation energies are reckoned from the ground state ofthe target nucleus every time. The experimental data, extracted from58iVi(jp, n)58Cu reaction at the forward angles (zero momentum trans-fer), are taken from the paper [7]. Lines "a" and "b" are the lowerand upper limits of S~(E) obtained experimentally. The data from

8

Table 3 of [7] were used. Other curves represent the calculated val-ues of S~(E). The line marked by "c" shows the strength functionS~{E) obtained with the effective constants KJ1 =. KJ1 = -0.23/4, "d"is for S~(E) calculated with K?1 = K?1 = -0.43/4, and "e" — forKJ1 = K? = -0.63/4. The calculations with KJ1 = K\1 = -0.43/4 givethe correct position of GT resonance and describe reasonably distribu-tion of the transition strength at the low excitation energy. But thetotal transition strength below 25 MeV is larger than the experimental-ly observed one. The strength function calculated with KJ1 = -0.23/4has too much strength compared to the experimental data at the lowestexcitation energies, it gives wrong position for the GT resonance andexceeds considerably observed total strength. The strength distributionmarked by ae" is close to the "d" at the low excitation energies, but theenergy position of the resonance is higher than the measured one andtoo much strength is shifted to the higher excitation energies. From thisconsideration one can conclude that the strength function calculatedwith the set of parameters KJ1 = /cf1 = —0.43/4 gives the better de-scription of the experimentally obtained strength function S~(E) thanother versions.

The running sums for the <rt+ strength functions for the MNi areshown in Fig. 2. The legend and the values of effective constants ofresidual interactions are the same as in Fig. 1. The experiment in [8]shows, that practically all observed <rt+ transition strength in 58Ni isconcentrated in one wide bump, which is placed in the region of excita-tion energy between 3 MeV and 8 MeV. In the calculations there is onecollective state which contains the large fraction of the total transitionstrength. The position of this state depends on the KJ1 value and it islocated in the middle of the bump region at the KJ1 = K\l = -0.43/4(line ttdw). For all the effective constants used the calculated transitionstrength exceeds considerably the observed ones.

The theoretical value of the Gamow-Teller sum rule for 5sNi is equalto

where 5* = 5±(oo). In our calculations S~ - S+ = 5.26 due to

9

nonorthogonality between neutron and proton single-particle wave func-tions. The experimentally defined sum rule value is 5~(25) - 5+(25) =3.7 ± 2.2 if one assumes that there is no <rt+ strength above the exci-tation energy of 10 MeV and there is no at~ strength above 25 MeV.The calculated values are 3.81, 2.89 and 2.18 for the KJ1 (and K\1) equalto -0.23/A, — 0.43/A and -0.63/A correspondingly. All calculated val-ues fit into rather large interval of the experimental data. So the valuesKJ1 = KJ1 = — 0.43/J4 obtained from the at" strength function do notcontradict to the data from <rt+ strength function on 58JVi and to themeasured value of Gamow-Teller transition strength.

The running sums for the Gamow-Teller t~ transition on the 60 JVt areshown in Fig. 3. The legend and used values of the effective constantsof residual interaction are the same as in Fig. 1. The experimentaldata are taken from Table 4 and from Fig. 10 of [7]. The experimentalstrength function has one wide bump (its width is around 6 MeV) whichsites between 10 and 20 MeV and contains more than 80% of totalobserved strength. The total observed strength is equal to 7.2 ± 1.8.The calculated strength functions have one collective transition too, butits position is higher than the experimental value. The calculated totaltransition strength is more than two times larger than the experimentalone, but the strength located below 20 MeV is rather close to the oneobtained from experiment.

The running sums of strength functions of <rt+ transitions on the60iVt are presented in Fig. 4. The experimental data are taken fromFig. 12 and Table II of [9]. The <rt+ transition strength located below10 MeV is equal to 3.11 ± 0.08, but the total strength below 32 MeV isequal to 7.4 ±0.5 [9]. This value should be compared to the value of totalot~ of 7.2 ±1.8 obtained in [7]. The Gamow-Teller sum rules compiledfrom two experiments becomes —0.2 ± 2.2 but the theoretical value is3(N — Z) = 12. The calculated value of the total at+ transition strengthdecreases from 6.7 when KJ1 and KJ1 are equal to -0.23/A to 4.6 at KJ1 =Kf1 = —0.63/A. The calculated value of the Gamow-Teller sum rule is10.79 for all values of the effective constants of residual interactions. Itmeans that either in [7] the at~ strength is underestimated or in [9] the

10

at* strength at higher excitation energies is overestimated significantly.In order to fix the value of the effective constant of the spin-isospin

residual interaction we have calculated the Gamow-Teller transition strengthdistribution over the excitation energy. The comparison of calculatedstrength function with the measured one leads to conclusion that thevariant K°1 = KJ1 = — QA3/A gives the quite reasonable descriptionof available experimental data concerning the Gamow-Teller transitionstrength in "-"jVi.

Next problem to be solved is to fix the value of effective constantof spin-dipole-spin-dipole residual interactions. In order to do it thedistributions of spin-dipole (rjYia]/*", J = 0,1,2) transition strengthover the excitation energies were calculated for ^^Ni. In the paper[7] there are few data concerning the distribution of L = 1 spin-isospintransition strength in 58.60jVrt measured in (p, n) reaction at intermediateenergies. They give the outlook of the response function of 58-60JV« tospin-isospin excitation due to the L = 1 angular moment transfer. Themaxima of cross-section corresponding to the L — 1 transitions are lo-cated at ~ 26 MeV in 58Ni and at ~ 25 MeV in 60Ni (both energies arereckoned from the ground state of parent nucleus), and have estimatedFWHM's of distributions equal to ~ 10 MeV and to ~ 12 MeV cor-respondingly. We have calculated the strength functions of spin-isospinL — \ transition by summing the theoretical strength functions of spin-dipole transitions to the J* = 0", J* = 1~ and J* = 2" states. Theresults are shown in Tables 1 and 2. It can be seen from Tables that thespin-dipole-spin-dipole residual interactions with the effective constantequal to the —0.33/A gives the reasonable agreement between calculatedmain characteristics (position of the resonance maxima and widths) andthe observed ones.

Because there are no available experimental data concerning Y\ t+

transition strength we have checked the sensitivity of the calculatedOMC rates to the variation of effective constant of isovector dipole resid-ual interaction. The transitions from 0+ ground states to the 1~ excitedstates have been considered, because only they are influenced by isovec-tor dipole residual interactions within the RPA framework.

11

From Table 3 one can see that calculated OMC rate from the 0+

ground state to the 1~ excited states is more sensitive to the varia-tion of the effective constant of the spin-dipole residual interaction (K}1)rather than to the changing of the effective constant of pure dipole inter-action (KJ). When the \K\\ increases more than by two times the OMCrate summed over all 1~ states reduces less than 10%. It can be easilyseen from Table 3 that the OMC rate depends on the total spin-dipoletransition strength mainly. The reduction of the OMC rate in two timescorresponds to the decreasing of the total spin-dipole strength whenK}1 varies from zero to -0 .43/4 . As a result, the isovector spin-dipolestrength function is more important for the OMC rates than the dipoleone. The same conclusion can be derived from the data of Table 4 in the60Ni case. From these Tables one can see that the OMC rate is ratherweak influenced by the isovector dipole residual interaction. In consid-ered cases the muon capture goes through the isovector spin-multipoletransitions mainly. It was also noticed in [10].

Total OMC rates on ™W

After the set of effective constant of nuclear residual interaction is fixedwe can use the eigenfunctions of nuclear model Hamiltonian for thecalculation of the muon capture rates.

At the beginning we list the approximations which were done dur-ing the OMC and RMC rates calculations. The pseudoscalar couplingconstant is defined by the expression

where q2 = (p — n)2 is the square of four-momentum transferred to thenucleons during muon capture. Velocity dependent terms were omittedduring the calculations of both processes OMC and RMC. All muoncapture rates are obtained with the following values of the constants ofresidual interaction — K{ = -0.103/A, /cfJ = -0.33/yl, except KJ1 =K21 = -0 .43/A

12

The results of calculations of the OMC and RMC rates for nickelisotopes are presented in Tables 5-7 for several values of ratio gp/gA-The measured OMC rates are (61.10 ± 1.05) s~l for 58iNTi, (55.62 ±0.97) s-1 for60iViand (47.16±0.95) s~l for 62JVi from [11] and (61.1 ±1.0) s-1 for 58Ni, (55.6± 1.0) s"1 for 60JV» and (47.2± 1.0) s"1 for 62Nifrom [12], Both experiments give practically the same results for thetotal OMC rates on nickel isotopes. It can be easily seen from Tables 5 -7 that the calculated total OMC rates for all used values of gp/gA are lessthan the measured one. Probably, the inclusion of velocity dependentterms may bring the calculated rates into better agreement with theexperimental value. The calculations of [13] and the estimations of [14]shows that the velocity dependent terms increase the OMC rate by 10%at least. Our results are rather closed to the total muon capture ratescalculated in finite Fermi system theory by Bunatyan [15] and are littleless than the ones of [10]. It should be mentioned that contribution oftransition into 1~ states of residual nuclei in our calculation are 4% lessthan in [10], but we have the relative weight of transitions into 1+ statesby 8% more than obtained in [10]. But in general our results concerningtotal OMC rates are in reasonable agreement with the experimental dataand the results of previous calculations.

Total RMC rates on 586062i\ri

The results of total RMC rates and photon spectra are presented inTables 8 and 9 and in Figs. 5-8. Fig. 5 shows the photon spectra calcu-lated for different values of ratio gp/gA- The nuclear RMC amplitudewas taken in the modified impulse approximation. The dependence ofthe calculated photon spectra from the gp value will be seen more clearif one consider the photon spectra normalized by the calculated totalOMC rate. Fig. 6 presents these normalized spectra. The theoreticalphoton spectra depend strongly on approximation which the RMC am-plitude on nucleus is calculated. The differences in photon spectra areeasily seen from Fig. 7. The total RMC rate on 58iVt and its distributionover the states with definite total spin and parity are given in Table 8.

13

The main contributions into the total RMC rate in IA and MIA comesfrom the transition to the 1* and 2~ states of daughter nucleus. But therates corresponding to these transitions are reduced in MIA stronger ascompared to IA. The most of strong suppression is for the rates to the 1+

excited states. At the large values of gpfgA the RMC rate correspond-ing to the transition to the 1+ excited states almost by six times less inMIA than in IA. The similar large difference in rates exist in the RMCon 60Ni and 62iVt too (see Table 9). The photon spectra calculated inMIA for three nickel isotopes are shown in Fig. 8.

Large difference in RMC rates obtained in MIA and IA requires cer-tainly the deeper investigation of the physical nature of the modificationsmade in RMC amplitude due to the continuity-equation constraints [3].

Conclusion

For the first time the photon spectra and total radiative muon capturerate are calculated based on the microscopic description of nuclear re-sponse for heavy nuclei. Quasiparticle random phase approximation isused for the calculation of excitation spectra and transition amplitudesfor the ordinary and radiative muon capture on zzw&Ni. The choice ofthe effective parameters of nuclear spin-isospin residual interactions isdiscussed. For that purpose the sum rule for the Gamow-Teller transi-tions is considered in connection with the total Gamow-Teller strengthextracted from the 60Ni(p, nf°Cu [7] and 60Ni(n,p)60Co [9] experimentsand it is assumed that in the last experiments the total at+ transitionstrength is overestimated.

The calculated total rates of OMC on M.«O.MJV» a r e c i o s e to theexperimental data.

For the nuclear amplitude of radiative muon capture, besides theusual impulse approximation, the modified impulse approximation, inwhich the continuity equation for electric current is taken into account,is used. The results of calculations performed in both of approximationsshow that the inclusion of the continuity-equation constraint into RMCamplitude in nucleus reduces the RMC rate about two times. Thesecorrections appeared to be stronger in the transitions from the 0+ groundstate into 1* excited states.

14

1 .

14

12

10

8

6

4

2

0

CO

25 30E, MeV

35 40 45 50

Figure 1: at~ strength in 5SNi

20 30 40E, MeV

Figure 2: at* strength in 587V/

50 60

15

^

18

16

14

12

10

8

6

4

2

05

- I - F

r

f

10 15 20 25 30E, MeV

35 40 45 50

Figure 3: at~ strength in 6O7Vt

Figure 4: at+ strength in 6O7Vi

16

Table 1: The main characteristics of the distribution of the spin-isospindipole t~ transition strength in 58JVi.

K\J-A

-0.18-0.23-0.28-0.33

-0.38-0.43

Full rYxat-strength(/m2)111110501000958

922891

Energycentroid(MeV)

27293031

3234

Dispersion

(MeV)4.15.05.45.7

6.06.4

Maximumlocalization

(MeV)29-3031-3232-3328-2933-3429-3030-3139-40

Strength inmaximum

(/m2)370346298219227221215205

Table 2: The main characteristics of the distribution of the spin-isospindipole t~ transition strength in 60Ni.

K\J-A

-0.18-0.23-0.28-0.33

-0.38

-0.43

Full rY^arstrength(/m2)1242118311341093

1057

1027

Energycentroid(MeV)

27293032

33

34

Dispersion

{MeV)4.95.25.55.8

6.1

6.4

Maximumlocalization^

(MeV)29-3031-3232-3329-3033-3429-3038-3930-3139-40

Strength inmaximum

U™2)399370323257235254187235240

17

Table 3: ]

K\-A

-0.083-0.103-0.123-0.143-0.203

-0.083-0.103-0.123-0.143-0.203

-0.083-0.103-0.123-0.143-0.203

-0.083-0.103-0.123-0.143-0.203

Dipole, s

K^-A

0.000.000.000.000.00

-0.23-0.23-0.23-0.23-0.23

-0.33-0.33-0.33-0.33-0.33

-0.43-0.43-0.43-0.43-0.43

pin-dipoleTotalOMCrate,s~l

37.7-105

37.3-105

37.0-105

36.6-105

35.9-105

23.5-10s

23.1-105

22.7-105

22.3 -105

21.510 s

21.MO5

20.7-105

20.3 106

19.9-105

19.1-105

19.4-10s

19.0-105

18.6-105

18.3-105

17.4-105

transition strength andTotalTY^

strength,fm2

314.8298.4284.0271.3240.7

317.5300.8286.3273.4242.4

318.2301.4286.8273.9242.9

318.7301.9287.3274.4243.3

Total

strength,fm2

558.8559.2559.6559.9560.8

322.0322.1322.2322.3322.6

281.5281.6281.7281.8282.0

252.6252.7252.7252.8253.0

OMC rates \Dipole

strengthres. reg.,

MeV6 . 1 - 11.46 . 1 - 11.56 . 1 - 11.56 . 1 - 11.66.1 - 12.0

6.2-11.46 .2- 11.46.2- 11.46 .2- 11.56.2- 11.7

6.2-11.46.2-11.46.2-11.46 .2- 11.56.2-11.7

6.2- 11.46.2- 11.46.2- 11.46.2- 11.56..3 - 11.7

m 5 B#i.rYit+

strengthin res. reg.,

fm2

239240238234219

221232234232217

226237239236221

229239241238222

18

Table 4:

K{-A

-0.083-0.103-0.123-0.143-0.203

-0.083-0.103-0.123-0.143-0.203

-0.083-0.103-0.123-0.143-0.21A

-0.083-0.103-0.123-0.143-0.203

Dipole, spin-dipole

K\1-A

0.000.000.000.000.00

-0.23-0.23-0.23-0.23-0.23

-0.33-0.33-0.33-0.33-0.33

-0.43-0.43-0.43-0.43-0.43

TotalOMCrate,s-1

35.2-105

34.8-105

34.4 105

34.1 105

33.4-105

21.9-105

21.5-105

21.2-105

20.8 105

20.1-105

19.7-105

19.3-105

18.9-105

18.6-105

17.8-105

18.1-105

17.7-105

17.3-105

17.0-105

16.2-105

transition strength andTotalrYit+

strength,/m2

289.9273.9260.0247.6217.8

292.8276.6262.5250.0219.7

293.5277.3263.1250.6220.2

294.1277.9263.6251.1220.6

Totalr[yior]1<+strength,

/m2

525.9526.3526.7527.0528.0

300.4300.5300.7300.8301.2

260.8260.9261.0261.1261.4

232.4232.5232.5232.6232.9

OMC ratesDipole

strengthres. reg.,

MeV6 .3 - 11.96 .3 - 11.96 .3 - 11.96 . 3 - 11.96.3-12.0

6 . 3 - 13.06 . 3 - 11.96.0- 11.96.0- 11.96.0- 11.9

6.3 - 13.06.0- 11.96.0- 11.96.0-11.96.1 - 12.0

6.0 - 13.06.0- 11.96.G- 11.96 . 1 - 11.96.1 - 12.0

in 60iVirYxt+

strengthin res. reg.,

/m 2

200200198193171

190190205201178

194211210205181

219215213208183

19

Table 5: Total OMC rates and relative contribution (in percent) into itfrom the states with definite spin and parity on *8Ni.

9PI9A

4.06.08.010.012.0

Tot. OMCrate, a"1

60.4 105

58.1-105

56.0-10*54.4-105

53.0-105

0+

2.52.62.72.82.9

0"4.03.22.41.71.1

1+26.626.125.525.124.7

J1-

34.235.636.938.039.0

2+9.39.610.010.310.6

2"16.816.416.115.815.5

3+5.35.15.04.94.8

3"1.31.31.41.4

1.4.

Table 6: Total OMC rates and relative contribution (in percent) into itfrom the states with definite spin and parity on 60Ni.

9P/9A

4.06.08.010.012.0

Tot. OMCrate, s~l

54.9-105

52.8-105

51.0-105

49.5-105

48.2-105

0+2.72.82.93.03.1

0"4.23.42.61.81.2

1+

25.725.224.724.223.8

J1-

35.136.537.838.939.9

it

2+9.610.010.310.610.9

2"16.015.615.315.014.7

3+5.45.25.14.94.8

3"1.31.41.41.51.5

Table 7: Total OMC rates and relative contribution (in percent) into itfrom the states with definite spin and parity on 62Ni.

9P/9A

4.06.08.010.012.0

Tot. OMCrate, s"1

49.6-105

47.7-105

46.1-105

44.7-105

43.6-105

0+2.93.13.23.33.3

o-4.43.52.61.91.2

1+

25.524.924.424.023.6

J1-

35.436.838.139.240.2

* • •

2+9.910.310.611.011.2

2-15.114.714.414.113.9

3+5.55.35.25.04.9

3"1.41.41.51.51.6

20

Table 8: Total and partial RMC rates on *8Ni calculated in two approx-imations for the radiative muon capture amplitude on nucleus.

9P/9A

4.0

6.0

8.0

10.0

12.0

Appr.

IAMIAIA

MIAIA

MIAIA

MIAIA

MIA

total

250.1133.5286.3133.6331.3135.5385.0139.1447.5144.4

0+5.22.75.22.75.12.75.12.75.02.7

o-7.43.47.73.18.12.68.52.39.12.0

RMC1+

127.652.0149.848.6176.945.8209.043.5246.141.8

1 rate,1-

60.641.266.543.974.147.083.450.794.454.9

s-1

2+

8.97.39.67.810.78.411.99.213.410.0

2-34.422.440.523.048.224.057.525.568.327.5

3+5.44.06.34.17.54.38.94.610.64.9

3"0.60.50.60.50.60.50.70.50.70.6

Table 9: Total RMC rates on 5S<60<62Ni calculated in two approximationsfor nuclear RMC amplitude for different values of gp/gA-

Nucleus

58Ni

60Ni

62Ni

Approx.

IAMIAIA

MIAIA

MIA

4.0250.1133.5215.6117.0184.0101.3

RMC rates,6.0

286.3133.6246.5117.5210.1102.2

8.0331.3135.5284.9119.7242.7104.4

10.0385.0139.1330.8123.4281.6108.1

12.0447.5144.4384.3128.6327.1113.1

21

«0

10

9

8

7

6

5

4

3

2

1

0

1

-

1

1 1

i i

i

i

i

\

i

i

i

i

>

i i• a nmM a m a

cJue

TJ, I

55 60 65 70 75 80 85 90k,MeV

95 100 105

Figure 5: Photon spectra from RMC on S8Ni calculated for ratioequal to 4.0 (curve a), 6.0 (curve b), 8.0 (curve c), 10.0 (curve d) and12.0 (curve e). •

0.18

0.16

•L 0-14

^ 0.12

0.1

0.08

0.06

0.04

0.02

0

Io

•.. \

-•

-

1

1

*\\ \V\'\

i

i i

•

.t'\

I |

1

1

1 1

1 ^ ^ ^

1

abcde

• L

1

• • • •

—

1

55 60 65 70 75 80 85k,MeV

90 95 100 105

Figure 6: Calculated photon spectra from RMC on *aNi normalised bycalculated total OMC rates. The curves labels are as for Fig. 5

22

Io1—(

o

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

ab — "cdef - • - -

55 60 65 70 75 80 85k,MeV

95 100 105

Figure 7: Photon spectra from RMC on *8Ni calculated in three ap-proximations for nuclear RMC amplitude: MIA (curves a and b), IA (cand d) and model case when only muon can radiate ( e and f). Thecurves a, c and e are calculated with QPIQA — 6.0, other lines - with9P/9A = 8.0.

ssNi _60Ni —

55 60 65 70 75 80 85k,MeV

95 100 105

Figure 8: Photon spectra from RMC on **60>6*Ni calculated with9P/9A = 8.0.

23

We would like to thank T. Gorringe and K. Junker for many usefuldiscussions. This work was supported in part by Russian Foundationfor Basic Research (grant 95-02-04279) and by National Science Foun-dation (USA).

References

[I] H.P.C. Rood and H.A. Tolhoek, 1965, Nucl. Phys., 70, 658

[2] K.W. Ford and J.G. Wills, 1962, Nucl. Phys., 35, 295

[3] M. Gmitro, A.A. Ovchinnikova and T.V. Tetereva, 1986, Nucl. Phys.,A453, 685

[4] G.E. Pustovalov, 1959, Sov. Phys. JETP, 36(9), 1288

[5] V.G. Soloviev, Theory of complex nuclei, 1976, Pergamon Press, Ox-ford a.o.

[6] A.I. Vdovin and V.G. Soloviev, Part, and Nucl., 1983, 14, 99

[7] J. Rapaport et «/, 1983, Nucl. Phys., A410, 371

[8] S. El-Kateb et a/, 1994, Phys. Rev., C49, 3128

[9] A.L. Williams et a/, 1995, Phys. Rev., C51 , 1144

[10] 0 . Nalcioglu, D.J. Rowe and C. Ngo-Trong, 1974, Nucl. Phys.,A218, 495

[II] V.D. Bobrow et a/, 1965, Sov. Phys. JETP, 21, 799

[12] T. Suzuki, D.F. Measday and J.P. Roalsvig, 1987, Phys. Rev. C35,2212

[13] N. Auerbach and A. Klein, 1984, Nucl. Phys., A422, 480

[14] M.G. Urin and O.N. Vyazankin, 1992, Nucl. Phys., A537, 534

[15] G.G. Bunatyan, 1966, Sov. J. Nucl. Phys., 3, 613

Received by Publishing Departmenton December 20, 1996.

24

Эрамжян P.A., Кузьмин В.А., Тетерева Т.В. Е4-96-478Расчеты обычного и радиационного захвата мюоновядрами 5 8'6 0>6 2 Ni

Впервые вычислены спектры фотонов и полная скорость радиационногозахвата мюонов тяжелыми ядрами на основе микроскопического описанияядерной функции возбуждения. Для расчета спектров возбуждения и амплитуд

переходов для обычного и радиационного захвата мюонов ядрами J0«w«o*Niиспользуется квазичастичное приближение случайных фаз. Обсуждается выборэффективных параметров спин-изоспиновых остаточных взаимодействий. Дляэтого проведено сравнение вычисленных силовых функций гамов-теллеровских

(Ш *) и спин-дипольных (г[У.а]0 | 2 О переходов с эксперимента!ьными дан-ными. Обсуждается правило сумм для гамов-теллеровских переходов. Привычислении амплитуды радиационного захвата мюонов ядром кроме обычногоимпульсного приближения используется модифицированное импульсноеприближение, в котором учитывается уравнение непрерывности для электро-магнитного тока. Вычисленные полные скорости обычного захвата мюоновблизки к значениям, полученным экспериментально.

Работа выполнена в Лаборатории теоретической физики им. Н.Н.Боголю-бова ОИЯИ.

Препринт Объединенного института ядерных исследований. Дубна, 19%

Eramzhyan R.A., Kuz'min V.A., Tetereva T.V. E4-96-478Calculations of the Ordinary and Radiative Muon Captureo n 58,60,62 N i

For the first time the photon spectra and total rate of radiative muon capture onheavy nuclei are calculated on the base of the microscopic description of nuclearexcitation function. Quasiparticle random phase approximation is used for thecalculation of excitation spectra and transition amplitudes for the ordinary and

radiative muon capture on 58-60>62Ni# The »choice of" the effective parameters ofnuclear spin-isospin residual interactions is discussed. The strength functions of

Gamov-Teller (<jf ) and spin-dipole (r[Kja]0 l 2/~) transitions are calculated and

compared with the experimental data. The sum rule for Gamov-Teller transitions isconsidered too. For the nuclear amplitude of radiative muon capture, besides theusual impulse approximation, the modified impulse approximation, in which thecontinuity equation for electromagnetic current is taken into account, is used. Thecalculated total rates of ordinary muon capture are close to .the experimental data.

The investigation has been performed at the Bogoliubov Laboratory ofTheoretical Physics, JINR.

Preprint of the Joint Institute for Nuclear Research. Dubna, 1996

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