ОБЪЕДИНЕННЫЙ ЯДЕРНЫХ. Дубна ^Ш ИНСТИТУТ ПИШЛЕДОВАШ! Е V.S.Melezhik* NEW APPROACH TO THE OLD PROBLEM OF MUON STICKING IN \icf

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1 [l'"'ih'ii И { II 2 ^ ОБЪЕДИНЕННЫЙ ^Ш ИНСТИТУТ ЯДЕРНЫХ ПИШЛЕДОВАШ! Дубна Е V.S.Melezhik* NEW APPROACH TO THE OLD PROBLEM OF MUON STICKING IN \icf Submitted to «Hyperfine Interaction» *E-muil address: melezhik@nusun.jinr.dubna.su 1995

2 1 Introduction Although ;i considerable effort has been made so far on the problem of muon sticking to helium produced in muon catalyzed fusion reactions, the continuing disagreement of the theoretical evaluations with experimental data demands further investigation ( see [1-15] and references therein ). The aim of the present work is to avoid one of the questionable aspects of the theoretical treatments of the problem to date, namely use of the framework of the ' sudden approximation" (SA)[4,5], which was an essential element of all previous calculations ( see, for example, [4-12,3] ). Because the fusion time (T = l/pj H < 1O~ 20 s ) is much shorter then the times characteristic for muonic processes (т = 2.4x 10~ 17 s/m,, = 1.2xlO~ I9 s), it has been assumed that, the sticking events ol.the type (Itli -? /(''He + n (1) are sudden ( T/r lt + 0 ) and therefore that in evaluating u> s one can apply the Migdal formula[l:i] <". = I / Ki (R) c-* R Фы(R)dR\ 2. (2) This is the simple overlap integral between the initial i/> ln and the final wave function i/'y =?/> / c" r^ of the (/t''he) n( atom moving with the velocity V (<f = ш д К), which is defined by the energy output of the reaction (1). 11 ere we analyze the application of the formula (2) to the case of the reaction (1). The key point is a computation of the time evolution for the muon wave packet ф ы = 4>(ft,l = 0)-nl>(R,t-*oo) = Ф/ due to emission of the fusion neutron, by direct solution of the time-dependent three-dimensional Sc.hrodinger equation. 2 Time-dependent equation First we develop the time-dependent Schrodinger equation describing the process (I). Simple semi-classical considerations give a rough estimate for the time-dependent part of the interaction potential AU(R, I) acting on the muon. Due to receiving the momentum Д/Т = q = m,,v', the muon is.subject to the mean force /'' ~ Ap/Af, = m^v/т during the time At = т of the fusion

3 event (1). On the other hand, the force is equal to F = which gives the following estimate for the interaction potential: Here the z-axis direction n. coincides with V. the direction of the emitted neutron. ' More precisely, the time-dependent potential AU(HJ) may lie determined as f mvr \- T }. CD This is a consequence of the evolution of the muonic momentum during the fusion event (1): reproducing the limiting cases p(t = 0)=0 and ;T(/,» oo) = q. The fusion time т = 1/Г, is determined by the nuclear width Г; of the compound nucleus 5 He* for the initial state (/r'ile") in the transition (1). For fixing this value we use the parameter Г, =340 kev (т = 1/Г, = m.a.u.) of the three-coupled-channels model (dt, s He*, 7i''He)[14] giving the best (it to the experimental data for the fusion reaction d + t * ''He + n. 3 Method of calculation The method of global approximation on a subspace grid [15], recently applied to a number of stationary problems [16], is extended here to the timedependent Schrodinger equation. Following the key idea of tin- papers [15,16] we seek a solution as an expansion ф»{п., X, i) = 1 P,(;r) Я," 1 ^( Я, 0, 11 4 where Pi{x) are the Legcndre polynomials and P,~' is the N x N matrix inverse to {PI{XJ)} defined on the grid Xj = {cos<?j}f, coinciding with the set of the nodes of the Gauss quadrature on [-l,i]. In this way the initial problem is reduced to the system of Schrodinger-type equations with respect to the vector i>(r,t) = {i>j{r, 0}^ = {'/'(^;r i»0}{ V f unknown coefficients in the expansion (5) (S) ), (в)

4 where t т We seek a solution of the equation (6) on a discrete set of the points R n < and /, in the spatial R n i G [0, R m ] and temporal < [0,i m ] dimensions. For propagation in time /» / + Д/ the simple Crank-Nickolson scheme has Ь(чч1 used (1 + д/. /7(Л, * )) 0(Л, /» + ДО = (1 - ^А< Я(Д, * )) 0(Я, i n ), (7) which is stable, preserves unitarity and may be integrated over Я by applying the implicit inverse technique. Note that, we analyze Eq ((i) in a deep nonperturbative region because the diagonal part, of the effective potential H(R,t) satisfies the following relation: -^} - Ю 3 Rx k exp{-10 2 <}» 1 for /. < /,, = 10 r < 0.2 m.a.u. Solving (7) imposes rather tough demands on the computational method. (Henceforth we use the muonic atomic units (m.a.u.) : in lt h = e =1). 4 Results and discussion Fig. 1 illustrates the evolution in time of the muon wave packet ф(ё, /.) due to the emission of the neutron in the transition (1). The real part $t{ij>(r, x, t)} of the wave packet, calculated by solving the system (6) with the initial condition (/>(/?.,,r, /. = 0) = i>i s {R), is presented here for a few points of t. By projecting the wave packet on the bound states к = (nl) of the (/i' f He) n ; atom one can evaluate the sticking coefficient Wj,(fc, 0 = < ipk \ V ; (0 >P as a function of time. The calculated quantities ш 3 (к, t) are given in Fig. 2, which demonstrates that the last point of integration over t, t m =10 т = 0.17 m.a.u., is already in the asymptotic region of the reaction, and may be chosen as a final state of the transition (1) for a few initial states к giving the main contribution to the coefficient w, = *.W;,( ) The convergence of the calculated values ш»(г,- П) N) with respect to N > oo, as well as the dependence on the input parameter Г, п of the developing

5 model, is illustrated in Fig. ' ). This figure demonstrates that in I lie limiting case т = 1/Г 1П > 0 our approach gives the coefficient, Ы 3 (Г; >500 kev, N > 18), close to the SA result ш?*, calculated by the F,<. (2). The value wi s (r,- n =340 кеу.уу >18), calculated with the parameter г=1/г,- of the threecoupled-channels model(14], exceeds the SA result only by a few percent. The way proposed above for treating the initial sticking problem has advantages, compared with the standard SA procedure. Since the suggested computational scheme gives the union wave packet»/>(/f, /.) as a function of time, it makes possible evaluation of the energy radiated from the union emitted during the fusion events, according to the formulas: P=[p(u)du, (8) According to the Ehrenfest theorem, < ф z \ ф > -< ф U \ ф >. one can estimate the quantity P as P = ^ / < ф?± + 1^ е-<л, ф >!=» ^ ~ L> m \ V2 Г,- п. (10) The numerical evaluation confirms the above estimation with sufficient accuracy. The proportionality (10) of the radiated energy P to the decay width Г, permits us to exclude this parameter, and instead use the dependence илдг,,,) of the sticking coefficient on the width Г,- (see Fig. 3) to get the dependence u)j(f) on the energy P. The calculated curve u> is {P) (Fig. 4) gives a new possibility for experimental analyses of the "initial sticking". By measuring the energy P, radiated during the fusion event (1), one can evaluate the corresponding value wij(p) by using the theoretical curve presented in Fig. 1. This approach gives also the possibility of evaluating the coefficients u><.( P) for sticking in excited states к ф l.s, as well as the spectral density of the radiation P(il) = 57/e 1 " 1 < ф -jjzu ф >\ 2 dt, which may permit, in principle, measurement of the time dependence of the perturbative interaction AU(U,-.K,I.). This approach gives also the energy and angular distributions for the union emitted during the fusion event (1): ^ p ^ = I /ЯД, 1-* oo)ф к1 (Н.) У ю («)с-* Л dr P (11) db k, J **Г*У = 2*1 >д а Л г а < -> оо)ыл)ymv-^dfir (i2)

6 / = 0 " / = ' / = / = f = Fig. 1 Krai parl, )R{i/'(//,.r,/)} of Mir union wave packet, propagating from I lie hound state (// r 4Ie)i,, due to the neutron emission in the transition (I). The computation lias been done in the spatial [0, /Д ] = [0. Ji] and temporal [(),/, ] = [0, Юг] (г = 1/.ЧН) kev = m.a.u.) dimensions with the constant Д/ = т/200, and Л/f = O.IW. 1 ) steps of integration.

7 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I time ( m.a.u=1.2 10~' 9 s ) 'ig. 2 Tli<! l.inic evolution of the main partial slicking coofh'r.ionls l< 4>k I '/'(/ ) > 2 i» Ню sum u, = EA-W.,(A:). oj,(kj.)

8 T in =340keV ё LOO 0.90 L и 'i ii ( kev ) Fig. :{ The convergence with respect to N > oo of the sticking coefficient wi,(r, n, Л')= < Ф\ а 4> N (t) > 2, calculated for a few values of the input parameter Г,-. The points Г,- ==340 kev correspond to the value Г, taken from the three-coupled-channels modei[14]. The points Г, > 500 kev, N >18 correspond to the limiting case r = L/Гт -> 0 of SA (horizontal line U.9/T).

9 1.15? T in =140keV Я 280keV 340keV 560keV 1Z0keV Z800keV i i i i i I i i i i i i i i i i i i i i i i i i i i i i i i i i i i radiated energy, P ( ev ) Fig. 4 The calculated dependence of the. sticking coefficient, W,(/ J ) on the energy P, radiated during the fusion event (1) due to tho muon (".mission.

10 тиоп energy E k ( т.а.и=5.6 kev ) Fig. Л The energy distribution for tin 1 union emitted during the fusion event (1). Here only a few values of angular momentum / have been included in the summation formula (II).

11 10 - гю angle, в ф и 10 10' ф д 10 " а = ю -ч 10 "S angle, в 6 The illustration of the angular <listribulions for Mic nmon emitted i t.ho fusion wont (1) for tin 1 сам 1 of dilfnvnt union energies /iv 10

12 The curves calculated by virtue of Kq. (II) anil (12) are presented in Figs, о лii*l (i. respectively. These cliaracleristics of the slicking process (I) are observable values, and for special conditions may bo experimentally clot eel able. The above calculations have horn done with the initial condition <r(r.t = ") = '-'ьсо- where t"i,(/f) was the wave function of (//''Me) i.,. Hut. it is known, that using more accurate wave function ir(h) = (i,,<-'nii(lt) + \ U- l 'Vo( H)<lk of the mtionie molecule dl/i instead of I.'\,,[/I 'Me) decreases tlie initial sticking coefficients to ~ ( /i,[7-l 1]. The developed approach permits such improvement in describing of the sticking processes by changing of the initial condition. 5 Conclusion Л new approach avoiding the framework of SA has been developed for. the treatment ol union slicking. It has been shown thai SA is a limiting case. 7" = 1/Г;,. * 0. for the suggested approach. As I he input parameter of the computational scheme is Г, > 500 kev, the SA result a. 1 *- 1 has been reproduced with satisfactory accuracy for the reaction dl/i * //'lie + n. For a realistic value of the parameter Г, = :M0 kev of the three-coupled-chaniiel modol[l-l]. the calculated u,\, exceeds SA result, only by a few percent. In addition, the new approach gives (he energy and angular distributions for the emitted union, as well as the dependence of the sticking coellicient on the energy, radiated during the innoti catalyzed fusion event,. Those observable characteristics of the sticking processes have been calculated for the case of the reaction (1). Л special interest is in the dependence of the sticking coefficient u.\,(/') on the energy /', radiated during the fusion event, which may open a new possibility for experimental analysis of the sticking processes. There are no limitations to extension of the results, obtained here for the reaction (1), to other union catalyzed fusion processes. 6 Acknowledgements I would like to thank Professor H.T. Siegel and Dr. A. Adamczak for useful remarks and help. The work has been supported by the International Scieutilic Foundation,.Ciant No. М.ИШ0. 11

13 References [1) W.II. Breunlich. P. Kainmel. J.S. Cohen, atul M. Leon. Ann. Rev. NVI. Part.. Sci. 39 (1989).411: L.I. Ponomnrev. Contemporary Physics 31 (1990) -219; P. Froelich. Adv. Pliys. 41 (1992) 105. [2] ('. Petil.jean. Niirl. Phys. A543 (1992) 7!)r. [ 5] Clii-Yn Iln, CM. Hale, and.j..s. Colien. Int. Symposium on Mnon Catalyzed Fusion. Book of Abstracts, Dubna. June (HI!)- 1 )) p.58. [1] Ya.B. Zeldovicl). Sov. I'liys..IF/IT 6 (1958) 212. [5J JJ)..Jackson. 1'liys. Hev. 106 (1957) Ш). [6] S.S. Gershloin et al. Sov. Pliys. ЛУП 1 53 (M)S1) HT2: L. Hracci and (.',. Fiorentini, Nuc.l. I'liys. A364 (1981) Ж\. [7] I). C<! jfirk-y and H..J. Alder, Pliys. Rev. A31 (1985) [8] L.N. Hogdanova et, al., Phy.s. Lett. B161 (I985) 1; Nncl. Pliys. A454 (19Sfi) fi. r ):5; Sov..). Nnrl. Phys. 50 (1989) 818. [9] Chi-Yn Ilu. Phys. Hev. A34 (198(i) 2. r ):i(i; A36 (1987) 11:55; A41 (191Ш) (i509. [10] M. Kainimnra, in Mnon-Cat. Fusion, Sanihel Island, FL 1988, Л1Р Conference Proceedings No. 181 faineriran lust, of I'liys., NY, 1989) ed. by S.lv Jones,.1. Kafelski, and M..I. Monkhorst. p.:ym). [11] S.K. Пну wood el. al., Pliys. Kev. A37 (1988) :УШ; А39 (1989) IfiM: A43 (1991) [12] И..leziorski et, al., Ph.ys. Hev. A43 (1991) ifilo. [li{] A.M. Migdal, Qulitative Methods in Qiiantuin Theory, Moscow, Nauka (in Russian) [II] L.N. Moftdanova, V.K. Maikushin, and V.S. Mclezhik, Sov. Phys. JI'TIP 54 (I9S1) [15] V.S. Mehy.hik.J.. Сотр. I'liys. 92 (1991) «7: Nuovo Cimento B106 (19f)l) (i] V.S. Melezhik, Phys. Hev. A48 (\<Ш) '1528; A52 (1!)95). Received by Publishing Department on October 10,

14 Мележнк B.C. Новый подход к старой задаче о прилипании мюона к гелию в реакциях мюошюго катализа E Предложен новый подход для расчета «коэффициента прилипания» мюона к гелию п реакциях мюоннот катализа вне рамок приближения «внезапных возмущений». Вероятность прилипания со? мюона к гелию в реакции dt\i > ц 4 Не + л, рассчитанная с помощью нового метода, согласуется с предычущмми результатами. Кроме того, вычислены угловое и энергетическое распределения испускаемого мюона, а также зависимость величины со? от излучаемой в акте мюошюго катализа энергии. Работа выполнена в Лаборатории ядерных проблем ОИЯИ. Препринт Объединенного института ядерных исследовании. Дубна Melezhik V.S. New Approach to the Old Problem of Muon Sticking in \lcf E A new approach is presented for evaluation of the muon «sticking probability» to helium in muon catalyzed fusion reactions. Use of the «sudden approximation» is avoided. The probability co v of muon sticking to helium in the reaction dt\i > (i 4 He + /i, calculated with the new method, agrees with previous results. Moreover, the energy and angular distributions for the emitted muon, as well as the dependence of co v on the energy radiated during the muon catalyzed fusion event, have been evaluated. The 'nvestigation has been performed at the Laboratory of Nuclear Problems, JINK Preprint of the Joint Institute for Nuclear Research. Dubna, 1995

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