Theoretical and Experimental Investigation of Relaxation Processes in Multi-pulse Nuclear Magnetic Resonance Experiments BY L. N. EROFEEV G. B. MANELIS, E. B. FEL'DMAN B. N. PROVOTOROV AND B. A. SHUMM Branch of the Institute of Chemical Physics TT u .S.S.R.Academy of Sciences Moscow U.S.S.R. Received 3rd August 1978 In the last ten years multi-pulse n.m.r. have appeared to be very useful in the study of weak electron-spin interactions and different relaxation processes in solids. However at the present time the theory of multi-pulse experiments (the theory of the average hamiltonian)'.2 possesses a number of serious defects which limit understanding of the physical nature of line-narrowing and leads in some cases to fundamental divergencies with the experimental data.4*6 We note the following defects in the theory of the average hamiltonian.First of all the absence of any strict consideration of the dynamics of spin systems at small times t -T2(T2-IIZdjj-l) in the theory'*2 does not permit one to evaluate the spin density matrix in the quasi- stationary state at t -T2. Because of this additional assumptions on the form of the density matrix,'.2 which in some cases are not justified were inserted into the theory."6 Secondly the theory of the average hamiltonian averages the dipole-dipole interaction or its contribution for the time of cycle 2,.'s2 In this case the effect of local fields on spin motion was not fully accounted for at the times t -T2 3 z during which establishment of the above mentioned quasi-stationary regime usually occurs.A consistent account of the effect on spin dynamics both of the external pulses and also of the local fields leads to the conclusion that the time scale over which the averaging in this problem should proceed is T2 % 7,. Thirdly an important experimental finding is that the exponential character of magnetization damping in multi-pulse n.m.r. experiments was not explained in the theory of the average hamiltonian. Finally the experimental results 4p6 revealed an interesting analogy between the system behaviour in the fields created by the pulse train and in continuous external r.f. fields. This interesting circumstance alone suggests that during construction of a theory of line narrowing in multi-pulse experiments one might apply the main results of the well developed theory of continuous n.m.r.which could not be used in the framework of average hamiltonian theory. It should be emphasized that the system of nuclear spins in the multi-pulse experi- ments is an open system absorbing the energy of external r.f. fields. However the theory of the average hamiltonian describes such a system by a time-independent average hamiltonian as conservative. The magnetization damping of conservative systems may be caused only by spins de-phasing in the local fields determined by the non-averaged part of dipole-dipole interactions. That is why in the theory of the average hamiltonian the process of slow magnetization decay is connected with spin RELAXATION PROCESSES IN MULTI-PULSE N.M.R.de-phasing in averaged local fields. More correctly the process should be considered as that of slow heating-up of the spin system. Serious questions are raised as to whether spin systems in multi-pulse experiments are conservative ;this question alone deserves the closest consideration. In connection with the observed deficiencies in the average hamiltonian theory there have been attempts to describe the experimental results of multi-pulse line- narrowing in solids without recourse to hamiltonian a~eraging.~.’*~ Below we give the theory of line-narrowing of n.m.r. spectra in solids not using hamiltonian averag- ing but using the example of a pulse train 90; -2 -(qx -2~)~*~ (qx denotes the pulse which rotates spins through an angle q around the x-axis; 22 is the time interval between the pulses) evaluated at a field shifted from resonance by the value A (in frequency units) and corresponding experimental results.The theory treated in our report is mainly stimulated by experimental ~ork.~~~ The present account represents a generalization of the results of ref. (5)-(8),the main aim of which was to make plain the authors’ approach to a clear physical picture of line-narrowing in multiple-pulse n.m.r. spectra. DYNAMICS OF SPIN SYSTEM AT TIMES t-T2 The magnetization behaviour during the initial stages t -T2of the multiple-pulse experiment reflects the elementary spin processes which take place in the system. The experimentally obtained4e6 transfer process which is accompanied by a decrease in magnetization at times t -T, shows that in the system there is exchange of energy between Zeeman states and the dipole-dipole reservoir of interactions.The magnet- ization oscillations observed1P2p6 during the initial stages give witness to the precession of nuclear spins around some effective field at t 2,; the quasi-stationary regime which has been set in the system at the times t -T2determines the further magnetiza- tion damping. Thus it is necessary to begin the study of spin system dynamics from small times t 5 Tz. Let us first consider the spin system affected by the pulse train 90; -z -(qx -2~)~ where qx is the pulse which rotates spin through angle q around the x-axis and 22is the distance between the pulses (fig. 1). It is supposed that FIG.1.-Schematic of train90; -T -(pox-2~)~.x ,pointsof signal observation. the constant field directed in the laboratory frame along the z-axis is shifted from the resonance value w,/y (coo is the frequency of Larmor precession of nuclear spin y is the gyromagnetic ratio) by the resonance offset value A (in frequency units). In the frame rotating with the Larmor frequency around the z-axis the equation of the spin density matrix p(t)has the form (h= 1) wheref(t) is the pulse function determined by the formula E f(?)= q 2 6(T + 2Kt -?) K=O EROFEEV FEL’DMAN MANELIS PROVOTOROV AND SHUMM and*d is the secular (relatively to the z-axis) part of the dipole-dipole interaction. The density matrix p(t) in initial time t = 0 was given by the preparatory pulse of the sequence used p(0) = 1 -aoo~oSx, tr p(0) = 1 (3) where a.is the initial temperature of Zeeman reservoir. Consider the interaction representation and replace p(t) = L(t)y(t)L-’(t) (4) where L(t)= Texp{-i il [-f(t’)sx + Asz]dt’}. (5) The symbol T denotes the ordering in time product. If we now insert the effective exp(-2iroeS) = exp(-iAr??,)P- exp(-iAtS,) (6) pulse which characterises the external influence on the system in the interval 2r it will be easy to check’ that after fulfilling the canonical transformation (4) the equation for the density matrix will be8 . dp 1 -= (exp[-i g(t’) dt’s,,]$i exp[i g(t’)dt’,!?,,] p(t)}. (7) dt i i In formula (7) g(t) is determined by a formula analogous to (2) in which the angle p is replaced by the angle 0 = 2wez and the value of effective field o,as can be seen from formula (6) is determined by the following cos 2wez = cos p cos2 At -sin’ AT.(8) The direction of It of the effective field is obtained from sin 2Az -cos21 sin q -cos AT 2 n = sin 2wez ,n = 0 n = sin 2wez * (9) It is convenient to expand the operator 12; by the following $2 = A,gO + + A-12z1 + A2P; + A-22P;2 (10) where A? is a secular but 23 l; 2%’ are non-secular parts of the dipole-dipole interaction relative to the maxis 9:= 2 Bij[2SniSnj-+(s;&j+ &S;.)]; (1 1) r<j &==-a 2 2 Bj,(Sni9$ + SA&j); 2;l = -3 2 Bij(&& + s;Snj); i<j i<j cc A 9;= -2 2 Bij(SA$$); 9;’= -2 2 BijSzS;; Sf = s, 3 isnz.(12) i<j i<j The coefficients Ao,A fl A fz may be found from the formula8 3n,Z -1 A 7-2 ; A = A’ = n,d1 -n:e-’w; RELAXATION PROCESSES IN MULTI-PULSE N.M.R.Eqn (7) shows that for every time interval 22 spins rotate through the angle 0= 2~~2. We take into account spin rotation in the mean field by the usual method i.e. pass to the frame rotating at a frequency we. In this system of coordinates the density matrix p*(t) satisfies the equation ; where +(t)= exp{ -i [,/:g(t') dt' -wet11 and x(t) = exp {-2i [,/:g(t') dt' -wet1) are the periodic functions with the period of the pulse train 22. From eqn (14) it is seen that the fields affecting the spins may be divided into constant fields o, Aw=:and fields which fluctuate rapidly in time.The greatest influence on the motion of nuclear spins is provided by the field weand the constant local field; the contribution of the oscillating fields may be accounted by perturbation theory. To account for the fluctuating fields we expand the functions +(I) and x(t) in a Fourier series W . flR m . nn +(I)= 2 C,e-'T 'r; x(t) = 2 Bfle-'Ter n-a n= -W c = (-1)" sin wez ; B = (-1)" sin 2w,z nn + w,~ nn + 2we2 and eqn (14) is written in the form t1 -aJ n= -W where sin 2coe2 (A29$+ A-,9j2). (18) 2wez The primed sums in this paper denote omission of zero harmonics. Eqn (17) is similar to the equation for the density matrix in continuous spin- locking.' This similarity is not accidental and exhibits the experimentally observed anal~gy'*~.~ between the behaviour of magnetization in the multi-pulse experiment considered and continuous spin-locking.However the problem at the same time differs from continuous spin-locking by the presence of the time-dependent part of hamiltonian (1 7). Modulation by pulses and resonance offset dipole-dipole interaction in eqn (17) is a source of quanta which as will be shown below are absorbed by the spin system. Therefore absorption of these quanta by the spin system causes magnetization damping at times t i5,T2and leads to the above mentioned heating-up of the spin system. EROFEEV FEL'DMAN MANELIS PROVOTOROV AND SHUMM 23 The amplitude of the time-dependent hamiltonian part of eqn (1 7) may be decreased at n/t > colocafter carrying out the following canonical transformations.8 Consider first qOc 9 dtr(Xz)2/trSi-we.Then for each non-zero harmonic +(t) one eqn (17) canonically transformed can be For each non-zero harmonic x(t) we carry out the analogous transformations mn ;(t> = e-i 5z S ei P.'i 25 t S -p (t)e-r Sn e-i ei t Sn (21) m e where IT fi; ==.= mn + 0(A2Bm$i -A-2B;i@i2). (22) ~ Such transformations have been used in the theory of continuous spin-locking.' After performing the canonical transformations (19) and (21) eqn (17) may be written as follows:8 Here sin co,~ z2cofoc . P(t) -zco;oc - i.e. o,-coloc; co,,,z < 1. n 7t Thus as a result of performing the canonical transformations (19) and (23) the time-dependent part of the hamiltonian decreases by E = -times and may be (zco10c)2 7l accounted for easily by perturbation theory.Since the perturbative order p(t) is now sufficiently less than the order of interaction inserted in #o one may assume9 that at weT2-1 through to time -T2 the density matrix of the system with an accuracy of small non-diagonal terms has the form z Fst= 1 -clst#o tr pst= 1. (24) At times -T2 we may also neglect absorption of energy by the system from external r.f. fields and apply the law of energy conservation,8 i.e. the conservation of Z0 from which where M$ is the projection of the stationary value of magnetization on the observa- tion axis x and Mois the initial value of magnetization; time t is counted from the RELAXATION PROCESSES IN MULTI-PULSE N.M.R.middle of the interval between the pulses up to the point at which magnetization is observed (fig. 1). Formula (26) shows that decrease in magnetization in the quasi- stationary state of the system has two causes. First at time -T2 the magnet- ization which is initially parallel to the observation axis becomes parallel to the effective field direction and the magnetization component which was perpendicular to the effective field direction disappears.6 For this reason the observed magnetization decreases for time -T2 by n times. Secondly at times -7‘ an energy exchange between Zeeman and dipole-dipole reservoirs of intractions OCCU~S,~ which is not connected with the change in magnetization orientation; it also leads to an additional decrease in the observed signal.Fig. 2 represents the dependencies of Md/Mo 1.0 0.8 0.6 0.4 0.2 1 I 1 0 0.2 0.4 0.6 0.8 ATI rad FIG.2.-Dependence Md/Moon Ar. Lines correspond to formula (26) solid line at Hfoc= 0.86 Oe; dotted line at Hloc= 1.0 Oe. Experimental points 0,(p = 22.5”; x (p = 36”; A,VI = 60”; 0,(p = 90”. on the parameter A r at different angles q~,calculated from formula (26)at tl equal to zero. Solid lines correspond to HI. = wloc/y= 0.86 Oe(H; [l 111) in a CaF single crystal dotted ones to HI. = 1.0Oe (arranged to correspond with the experimental points but bearing in mind a non-exact crystal orientation). The divergence of the experimental points and the theoretical curve at q~= 90’ (fig. 2) may be explained by magnetization damping parallel to co due to the above heating of the spin system which causes the additional decrease in the observed value M&6 New treatments’~~ show that magnetization M is a periodic function with the period of the pulse train 22; this allows calculation of the magnetization in the intervals between pulses.In particular if A is equal to zero the signal form between pulses (with an accuracy in the terms -~~wt~~) is the following’ Experiments’ confirm both the period of the observed signal and the amplitude increase in the bell-shaped signal which is described by the second term in braces in eqn (27) by 0 and z increase (if A = 0 then 0 = q). At resonance offset the magnetization change in the intervals between pulses is determined not only by dipole-dipole interactions but primarily by resonance offset itself [M -cos A (t -22); z < t < 321.It has been experimentally checked’ EROFEEV FEL'DMAN MANELIS PROVOTOROV AND SHUMM that at any resonance offset M is changed with the period 22 and has a maximum value in the interval midway between pulses. At /A. zJ < 271 the observed signal in the intervals between pulses has a bell-shaped form; at larger A the magnetization change in the same intervals has a form close to sinusoidal,6 containing more than one period. In the case considered when wloc-LO the absorption of energy from external fields n/z> we,wlOcmust be accompanied by a change of reorientation of a large number of spins and therefore is connected with an energy change in the dipole-dipole interaction.At we> uloc the energy of the absorbed quanta is mainly consumed by spin flipping in the effective field; only a small portion of the absorbed energy enters the dipole-dipole reservoir. Besides at u,> wlOczero harmonics 4(t) and ~(t) whose amplitudes have previously been LO^^^ . z times more than other harmonics lose their distinguishing features. Therefore the canonical transformations (1 9) and (21) of eqn (17) must be supplemented with canonical transformation for zero harmonics +(t)and ~(t). Accordingly instead of eqn (23) for the density matrix F(t)we obtain i dP-= [-we,9, + + PI(?>, dt where PI(?)-zw:ocand may also be considered as a small perturbation. The spin system is characterized here by two motion integrals and at times -T2 its density matrix has the form pst = 1 + ctStwes,,-0.5P,,A09O, trp, = 1.(29) The condition of 2:conservation at times -T2 leads to pst= 0. The decrease in stationary magnetization compared with its initial value Mois almost fully determined by the establishment of magnetization along the direction of effective field at time -T2 ; Kt= n2 cos Atl. (30) MO SPIN SYSTEM DYNAMICS AT TIMES t> T2 At times t > T the spin system is determined not only by the effective field but by the time-dependent perturbation terms PI(?)in eqn (28) Each term in vl(t) represents some part of the dipole-dipole interaction modulated by pulses and resonance offset. Non-secular terms of perturbations Pl(t) are the source of quanta absorbed by nuclear spins which interact through 9:.At u wlOcthe main part of the absorbed energy falls on the Zeeman reservoir and only a small part of the energy is absorbed by the dipole-dipole reservoir.Energy absorption of qunta 7115 by the dipole-dipole reservoir at we > oIOc requires a change in the mutual orientations of too many spins and is thus a difficult process. If a certain term in Pl(t) causes the absorption process by n spins of quantum mn/z then the energy mn/z -nu is trans- ferred into the dipole-dipole reservoir. Such is the case in the theory of saturation.'O The most effective influence on the system should affect such terms in Pl(t) which provide energy absorption by the Zeeman reservoir only.8 Such a process is possible at RELAXATION PROCESSES IN MULTI-PULSE N.M.R.where n is the number of the absorbing spins and mn/z is the frequency of quanta absorbed by spins (m-integer). It can be showns that the only term in pl(t)-TCO~~ which can lead to energy absorption by the Zeeman reservoir is R3(t),determined as followsS I?,(?) = K3re' Fr[A,# #:I K3 = const. (32) Similarly' the only term -z2u:, responsible for absorption of the indicated type is R4(t):' R4(t) = K4z2ei!? ' {$; [$ 93,K4 = const. (33) The experiment had shown that6 one does not observe processes of energy absorption from external r.f. fields if they are caused by operators of higher order than -r3 . uu;loc. Processes with absorption of quanta of frequency 2n/zare not observed experimentally.Thus the main effects on spin system dynamics at times t > T2have such terms in pl(t) as k3(t),R4(t)etc. Therefore the equation for the density matrix of the spin system may be written as i 9= [-O,S + A0'2-; dt + I?,(t) + I?_,(?) + I?,(?) + I?-&) + a * * &)I (34) the notation . . . in eqn (34) shows that in PI(?)other terms which lead to full energy absorption by Zeeman reservoir of interactions must be taken into account. Satura-tion eqn (10)are obtained here by the standard method'' and have the form da -2nt2 [ (9K32F3 + 16z2K,2F4)a(t)+ dt dp dt where -*3 and the operators GT(; -3we) e:k -4we) are determined by the following expressions W co ~2 -3 [W+t6+3 (t ) dt 6:4 = Lei,t(?*4(t) dt (37) where 63(t> = [.A:(?) $s(t)l &A(?> ==eiAo*:t 2;e-iAo-&t G-3(t) = (G3(?)>+.(38) The symbol + means the transfer to the complex conjugated operator. EROFEEV FEL'DMAN MANELIS PROVOTOROV AND SHUMM 27 Similarly &(t) = {#i(t> J?;I~ 2i(t>= e i~odtr# e-iAo$:r. (39) &4(t) = {G4(t)}+. The analysis of eqn (35) shows that if the equation for the density matrix (34) has I time dependent terms such as ff,(t) g4(t)then magnetization damping kinetics is defined by the curve representing the sum of two exponents. In this case damping always occurs up to the complete Pisappearance of magnetization. The picture changes when only one term such as R3(t),A,(?)significantly contributes to magnetiza- tion damping. In this case as in the theory of saturation the change in Zeeman temperature occurs only until temperatures ~(t) and D(t) become equal and it is not necessary here for magnetization to disappear completely.Let for example the term ff,(t)make the main contribution to magnetization damping. Then the ratio of residual magnetization M& to the stationary one M," depends on the shift of the effective field 6w from its value we= n/37 to the following An experimental check of eqn (40) shows in the main satisfactory agreement with the theoretical predictions6 The corresponding comparison of theory with experi- ment is shown in fig. 3. The shift 60 from we= n/3z was inserted both with the angle change at the constant resonance offset A.z = 0.905 [at q = 72" and A.7 = 0.8 I -0.1 0 0.1 60 72 80 FIG.3.-Dependence M:,,/Ms",of quanta1 absorption process determined by operator &(I) (a) on AT at p = 72" (6)on v at At = 0.905.Solid line corresponds to formula (40) at Hloc= 0.86 Oe. 0.905 from eqn (8) it follows that LU,z n/3z]and with the resonance offset change at the constant angle q = 72". The divergence from theory may be explained by the influence on magnetization damping of ff4(t)and other terms. From eqn (35) it is easy to show that the term I?,(?) leads to a time of magnetiza-tion damping -zW2 where the term k4(t)leads to a time of magnetization damping -~7-~. Similarly the term &(t) -which is responsible for absorption of quanta by five spins gives the time of magnetization damping -T-~. Experimental dependences of time of magnetization damping T2,on z are given in fig.4.6 The values T2eand 7 are plotted logarithmically. For the process of magnetization damping determined by k,(t)it has been found that T, -T-'.~ whereas for the process of damping determined by ff,(t),T2,-z-j and for ff5;,(t),T2e-T-~. Eqn (35) describes the magnetization damping at arbitrary values of the para- meters (p A z and these give the whole picture of magnetization change in the experi- ments considered. However for practical comparison of theory with the experi- mental data it is necessary to calculate correlators F3 F4 etc. [see formula (36)]. RELAXATION PROCESSES IN MULTI-PULSE N.M.R. These correlators are very much like the function of free induction decay.The difference is in the multispin nature of the correlators. These correlators may be estimated as follows ' where Mi3)and Mi4)are the second moments corresponding to the multispin corre- lators considered. Assume for evaluation that the spins are independent. Then 1 on FIG.4.-Experimental dependences of T2:2.z f?r different quanta1 absorption processes A operator R3(t); 0,Rdf); * Rq(r). each spin absorbs a quantum n/37 while energy excess 1 -3: -co 1 transfers into the dipole-dipole reservoir. The probability of such a process of absorption is -exp { -(f-~0,)~/2,,) where M2is the second moment of the absorption line in the rigid lattice in which spins interact through A,$:. Thus the probability absorption of a quantum 71/7 here is -exp (42) Similarly (43) Dependences TZeon q at different 7 are given in fig.5. The comparisons of the experimental data with the theory have been carried out in the neighbourhood of p = 7c/2 and gave good agreement between theory and experiment.' Complete comparison of the observed dependences for magnetization damping times with the theory should be carried out on the basis of the numerical solution of eqn (35) which will be carried out in the near future. EROFEEV FEL'DMAN MANELIS PROVOTOROV AND SHUMM 1000 100 ; \ u b-2-10 1 45 60 90 $,'Ides FIG.5.-Experimental dependences of TZeon q from different values of T(A= 0). * 5 = 10 ps; A,T = 14.5 pus; e T = 20ps. CONDITIONS REQUIRED FOR APPLICATION OF AVERAGE HAMILTONIAN THEORY The discussion in the previous section of magnetization damping in the multi- pulse experiment considered shows that the main contribution to magnetization damping is made by the absorption by the spin system of quanta modulated by pulses and resonance offset dipole;dipole_ inter!ctions.The quanta absorbed are determined by multispin operators R3(t),R4(t),Rji2(t) etc. At the same time in the theory of the average hamiltonian the nuclear spin system is described by a time- independent hamiltonian. Thus such approach significantly differs from the average hamiltonian theory. Moreoever in the theory presented some conditions necessary for the application of average hamiltonian theory may be obtained. To our mind the average hamiltonian theory is qualitatively true if in some frame there is an oppor- tunity to consider the nuclear spin system as conservative.This is possible only when the main contribution to the quantal absorption by spins is made by one term of the type ff,(t),I?,(?) etc. and the effect of others may be neglected. In particu- lar such a situation occurs for the sequence 90; -z -(905 -2~)~ at A = 0. At p = n/2 and A = 0 the decisive contribution to magnetization damping is made by In a frame rotating with oscillation frequency ff4(t) at co 9 wlOC as is seen easily from eqn (34) the spin system is conservative at p = n/2 and A = 0. This frame coincides with the one in which average hamiltonian theory functions success- fully,' with an accuracy to small corrections similar to that in canonical trans-formations (19) and (21).At the same time it is experimentally shown4 that at p = n/4 and A = 0 the behaviour of the spin system differs significantly from the predictions of average hamiltonian theory. It is due to the change of quasi equilibrium type in the system at we -ulOc, which may be considered conservative at times t -T2 in the frame mentioned in ref. (4) [again with the accuracy to canonical transformations (19) and (2l)]. Thus the average hamiltonian theory is applied at we> colOCin the frame obtained by canonical transformations (4) at parameter values of the pulse train and resonance offset when quantal absorption is largely determined only by multi-spin processes while the effect of the others may be neglected.For co -wlocthe theory of the average hamiltonian is true only at small times t -T2in a frame where the RELAXATION PROCESSES IN MULTI-PULSE N.M.R. density matrix p*(t) satisfies eqn (14). The important role of canonical transfor- mations (19) and (21) should be emphasised both in setting up the quasi stationary regime and in studying the processes of slow magnetization damping at t 3 T2. These transformations permit one to isolate the contribution of each harmonic of modulation by pulses and resonance offset dipole-dipole interactions to the establish- ment of the quasi stationary state and to heating-up of the spin system. EXPERIMENTAL Measurements were made on a multi-pulse n.m.r. spectronieter- 1 I with a resonance frequency 57.0 MHz for the I9F nuclei.The magnetization component M was observed in the rotating frame under the effect of pulse train 90; -z -(p -2~)~. The polarizing field Ho was shifted from resonance to a value corresponding to the resonance offset A. A single crystal CaF served as a sample with cross-sectional dimensions about 6 nim. All the measurements were carried out at Hall [I I13 (to an accuracy of 5"P at room tempera- ture. Under such conditions the spin-lattice relaxation time was -5 s. The value of rotating magnetic field in the pulse is -35 Oe. lnhomogeneity of Ho in the sample volume corresponded to a line width of -50 Hz inhomogeneity in the field H is 2;;. The authors thank Yu. N. Ivanov for help with the work. U. Haeberlen and J. S.Waugh Phys. Rec. 1968 175 453. 'P. Mansfield and D. Ware Phys. Rev. 1968 168 318. W.-K. Rhim D. P. Burum and D. D. Ellernan J. Cherx Phys. 1978 68 692; 1978 68 1164. W.-K. Rhim D. P. Bururn and D. D. Elleman Phys. Rev. Letters 1976 37 1764. L. N. Erofeev and B. A. Shumm Zhur. eksp. teor. Fiz. Pis'riici 1978 27 161. L. N. Erofeev B. A. Shumm and G. B. Manelis Zhur. eksp. teor. Fiz. in press. 'Yu. N. Ivanov B. N. Provotorov and E. B. Fel'dman Zhur. eksp. teor. Fiz.Pis'rm. 1978 27 164. Yu. N. Ivanov B. N. Provotorov and E. B. Fel'dman Zhrrr. eksp. teor. Fiz. in press. M. Goldrnan Spiti Teiiiperritirre and Nirclerrr Mugtietic Resoriwice in Solids (Oxford Univ. Press London 1970). lo B. N. Provotorov Zhrtr. eksper. teor. Fiz.,1961 41 1582; 1962 42 882. L. N. Erofeev 0. D. Vetrov B. A. Shurnm M. Sh. lsaev and G. B. Manelis Prib. Tekhti. Eksp. 1977 2 145.