NMR methods for identifying and studying diffusion of different spin species in heteronuclear systems

We introduce a new dipolar relaxation lime TU)' which characterizes the spin-lattice relaxation of secular dipolar interactions in the presence of a large rf field. Measurements of T]D' are particularly useful for studying slow atomic motions in muitispin systems, since such measurements enable us to vary the contribution of a particular spin species's motion relative to the contribution of the other spin species's, thus enabling us to identify the diffusing species. We also show that the anisotropy of the conventional dipolar relaxation time TLD can differ enormously for diffusion of different spin species in a muitispin system and, accordingly, can be used to identify the dominant diffusing species. Finally, we show that the high-rf-field rotating-frame relaxation time 7^ , measured as a function of rf frequency, also enables us to identify the diffusing species. We demonstrated experimentally the validity of-these techniques by measurements of potassium vacancy diffusion in a KF:Caf f single crystal and measurements of fluorine diffusion in AgF powder.

PHYSICAL REVIEW B VOLUME 18, NUMBER 1 1 JULY 1978 Nuclear-magnetic-resonance methods for identifying and studying diffusion of different spin species in heteronuclear systems Harold T. Stokes and David C. Ailion Department of Physics, University of Utah, Salt Lake City, Utah 84112 , (Received 16 January 1978) We introduce a new dipolar relaxation lime TU)' which characterizes the spin-lattice relaxation of secular dipolar interactions in the presence of a large rf field. Measurements of T]D' are partic­ularly useful for studying slow atomic motions in muitispin systems, since such measurements en­able us to vary the contribution of a particular spin species's motion relative to the contribution of the other spin species's, thus enabling us to identify the diffusing species. We also show that the anisotropy of the conventional dipolar relaxation time T, D can differ enormously for diffusion of different spin species in a multispin system and, accordingly, can be used to identify the dominant diffusing species. Finally, we show that the high-rf-field rotating-frame relaxation time 7, , meas­ured as a function of rf frequency, also enables us to identify the diffusing species. We demon­strated experimentally the validity of*these techniques by measurements of potassium vacancy diffusion in a KF:Ca + + single crystal and measurements of fluorine diffusion in AgF powder. I. INTRODUCTION NMR is currently widely used for studying the mi­croscopic behavior of systems containing diffusing atoms and rotating molecules. Measurements of the temperature dependence of the spin-lattice relaxation time T] allow the determination of activation energies and jump times.1 2 Similar measurements of the rotating-frame spin-lattice relaxation time T\f) and of the dipolar relaxation time Tw enable one to obtain information characteristic of much slower motions oc- curing at lower temperatures.3~7 Both the T\ theories8,9 and more recently the high-rf-field T\p theories1011 have been extended to heterogeneous (multispin) systems. A difficulty arises if one measures relaxation times in a multispin system. Such measurements by them­selves will not normally indicate which spin species is diffusing, since a diffusion jump of any spin species' may contribute to the relaxation. To obtain informa­tion about which species in diffusing, it is necessary to find some experimentally controlled parameter whose variation changes the contribution of one spin species's motion relative to the contribution of the others'. In this paper we present some examples of these parameters along with experimental verification of our ability to identify the diffusing species. Consider the dipolar relaxation time T]D for a multispin system. The different dipolar interaction terms normally cross relax rapidly to a common tem­perature, resulting in identical TXD measurements for the different spin species.12 19 In this paper we have extended,the normal strong-collision Slichter-Ailion3 6 (SA) theory for the dipolar relaxation time TU} to the case of a two-spin (/ and S) system. We then show that, for the case of strong / and weak S spins, there can be enormous anisotropy for diffusion of S spins, in contrast to the small anisotropy characteristic of the diffusion of / spins.20 21 Thus the crystal orientation is an example of an easily controlled parameter whose variation can identify the diffusing species in a single crystal. A major portion of this paper is devoted to describ­ing a novel technique22 23 for determining the dom­inantly diffusing species in a "slow motion" dipolar- relaxation-time experiment. In particular, we intro­duce a new relaxation time T]0' which describes the spin-lattice relaxation of the secular part of the dipolar interaction in the presence of a large rf field. Further­more, we develop an SA-type theory for relating TU)' to the diffusion jump time. Like T]}) of the SA theory, T]D' is appropriate for studying slow atomic motions but has a unique feature particularly suitable for muitispin systems. By varying the orientation of the effective field in the rotating frame, the contribu­tion to T]D' of one spin species's motion may be varied relative to the others'. Thus, the dominantly diffusing spin species can easily be identified. A third method for identifying the diffusing species consists of measuring the dependence of the high-field T\() on the orientation 0t of the effective field in the rotating frame. We have derived expressions for T]f) for a two-spin system (strong I and weak S) and have shown that the dependence of TXp on 0/ also depends strongly on which species is diffusing. II. THEORY A. Spins in a large dc field Consider a system of two species of nuclear spins (/ and S) in a solid. With the spin system placed in large 18 14! ©1978 The American Physical Society1 4 2 H A R O L D T . S T O K E S A N D D A V I D C . A I L I O N 18 dc magnetic field H0 (chosen to be along the z axis), the Hamiltonian is given by 3C = dC/i -f- 2C/V -h 3C zs i) (1) The terms KZj and Kzs are the Zeeman interactions of H0 with the / and S spins, respectively, Wzi -~ftytHo ^ /-/, (2) and W as -~fcysH() J) S:k . (3) The term KD is the spin-spin dipolar interaction. It is divided into two parts, 2ff/S()) and JT/V'1, the secular and nonsecular dipolar interactions, respectively, IT __ f/» (0) i t/» in) eft i) - Jt j) i Jt /) (4) The secular dipolar interaction is defined to be that part which commutes with K/j and 5CZV and is given by8-24'25 -if (0) __ *r/» (0) I -tr (0) I (0) Jt D - <n Dn -t- Ji Dis i Ji oss t where X -r.P/ /, r -u) . (5) /. A *r/> (0) 0/.S XB,kIz,S:k . (6) (7) /. A and Jf/m = | X(V<3.V,S', -S' - S':) . (8) The dipolar coupling constants, Aik, BJk, and Cik> are given by Ajk = -jyi^f'ik (1 -3 cos oik) , Bjk = 7/r.s ~ 3 cos20/A) > (9) (10) and Q = - 3 cos20,t) . (11) We now write the Hamiltonian in the following form: (») (12) In a previous paper,13 we showed that for large H0 the terms, KZh Kzs> ^d \ are quasi-invariants of the motion, each forming an energy reservoir whose spin order can be parametrized by a spin temperature. Thus, the density operator is written,13,14 25 26 in the high-temperature limit, cr-X-PtXa-PsKzs-falC^ (13) where Pi,Ps>Pp are inverse spin temperatures defined by /3 = 1 /kT . (14) Spin-lattice interactions cause the order of the ener­gy reservoirs of 3CZ/, 3CZS, and K/^ to relax towards thermal equilibrium with the lattice. In other words, j3/t fis, and pD evolve with time towards the lattice temperature (3L. The time constants of this relaxation are defined to be T]f, T\S, and TU) for the KZh 3CZs, and reservoirs, respectively (see Fig. !>• . Note that includes all secular dipolar interac­tions, those between unlike spins as well as those between like spins [see Eq. (5)]. They all form a common reservoir with a common spin temperature. Thus, one cannot speak of "dipolar order" of the / spins separate from "dipolar order" of the S spins. Also, the relaxation time T]D of dipolar order is the same for both / and S spins, as illustrated in Fig. 1. A variety of experiments13 19 22 have been performed which demonstrate the validity of the single dipolar reservoir concept. FIG. 1. Spin-lattice interactions with / and S spins in a large H() in the lab frame.18 N U C L E A R - M A G N E T I C - R E S O N A N C E M E T H O D S F O R . . . 143 B. Spins in a rotating reference frame Consider the addition of a large rf field Hw (perpen­dicular to H0) of frequency o)/ near the /-spin resonant frequency yiH0. In a reference frame27 rotating with frequency o)f about H0 (the zaxis), the Hamiltonian of Eq. (1) becomes v +Kzs + Kd) - D (15) Note that this transformation is made only with respect to the / spins and is accomplished by the uni­tary operator exp(-io),t ]£/-*). Thus the terms WZs k and Ko0) remain unchanged since they commute with this operator. The first term JCzi} m Eq- (15) is the Zeeman in­teraction of the / spins with an "effective" field Hotr/ which24 25 is the sum of (which is now static in this reference frame) and an off-resonance field h, given by h = H0-(co,/?,)//0 . (16) Thus the magnitude of HCfn is Z/efT/ = (H \{ + h2)y/2 , (17) and the angle between Hen / and Ho is given by 0,= tan-1 (//,,//?) . (18) If we tilt the z axis (with respect to the /-spins) by this angle 9/ such that it points along H0.tr/ (the "tilted ro­tating reference frame"28), the /-spin Zeeman interac­tion is written as H^=-tiyiHciTIXlzk • (19) The term 3C^/,)(/) in Eq. (15) is the nonsecular dipo­lar interaction 'Kd) transformed to the rotating refer­ence frame. Part of 3Co,)(/) oscillates with frequencies co/ and 2o>/ in this reference frame and therefore can be neglected.24 The time-independent part that remains is nonsecular with respect to Xzi} and Kzs. We now divide the term K/)0) in Eq. (15) into two parts,6,25;.28-29 JCf)00) and JCon\ which are secular and nonsecular, respectively, with respect to JCz/\ The secular part, which commutes with Kz/\ is given (in the tilted rotating reference frame) by where •tf (00) ^ Dll If (00) i 'if (00) i .**/» (0) nss (20) y[j(3 COS201 - 1)] X £,M3/ /, 1 - I' ) (21) and :k (22) The term Kpss is given in Eq. (8). Thus, we write the Hamiltonian as K +Xzs + #d +Xnn) ) . (23) This is similar in form to Eq. (12), that is, three commuting parts plus a noncommuting part. For large Hi/ the terms 3€z/\ zs>Md00) are quasi-invariants of the motion, each forming an energy reservoir whose spin order can be parametrized by a spin temperature. The density operator is written as cr = 1 py'xtf-PsXzs 73 (/') V (00) Pn *iD (24) As before, spin-lattice interactions cause p}r), /2S, and to relax towards fiL. The time constants of this relaxation are defined to be T\S, and T]D' for the JCzi\ 5CZ5, and 3Cpm reservoirs, respectively (see Fig. 2). This definition of T\p/ differs somewhat from that of Redfield27 and that used in the strong- collision theory3"6 as it characterizes only the relaxa­tion of Zeeman order. However, in the large-field (large H17) case, the two definitions agree. The relax- FIG. 2. Spin-lattice interactions with / and 5 spins in the rotating reference frame.144 H A R O L D T . S T O K E S A N D D A V I D C . A I L I O N 18 ation time T\D' has only recently12 22 been identified and defined and will be discussed in more detail in the following sections of this paper. C. Strong-collision theory Consider the case of dipolar spin-lattice relaxation due to slow atomic motion. By "slow", we mean that the average time interval r between diffusion jumps of an atom is much greater than the spin-spin relaxation time T2 (the time required for an energy reservoir to come to internal thermal equilibrium25). From the SA theory,3 6 we obtain an expression for the dipolar re­laxation time for this case: 1 T 1 D N Tr(X$')2-Tr(Xtf'X$>) r Tt(X^)2 (25) where X^ and X$ are the secular dipolar Hamil­tonians before and after a jump, respectively, and N is the number of jumping atoms in the spin system. [One should note that this equation differs from Eq. (1) of Ref. 22 by a minus sign. The right-hand side of Eqs. (1) and (5) -(7) of Ref. 22 should all be mul­tiplied by minus one.] The last term in Eq. (25) represents the average fractional change of energy of the dipolar reservoir due to a single jump of an atom: A E TriX^)1 -TrtX^Xty) TKJCD2 (26) In multispin systems, such as the present case, Xpm includes all secular dipolar interactions, as seen in Eq. (5). Thus, the motion of any one of the spin species present affects TlD. As an example, consider the case of diffusion in a System of strong / spins (y/ large ) and weak S spins (ys small ). We then have If (0) Moil » jcA'/v » x Dt: (0) DSS (27) If T]d is due to /-spin diffusion, we have from Eq. (25) rjr---20-p,,) , (28) T\D ?! where 2(1 -pu) is a geometric factor, of order 1, defined by „„ , v Tr(jtffl)*-TrtX/SlxW -----------------TWiMy-----------------' <29) The I-S and S-S interactions have been omitted be­cause of their small size [see Eq. (27)]. Note that Eqs. (28) and (29) are the same as the SA result for the single-spin species case. This is because the S spins are weak and the relaxation of the dipolar reser­voir, dominated by the /-/interactions, preceeds as if the S spins were not even present. If, on the other hand, T]D is due to S-spin diffusion, we have, from Eq. (25), 1 1 H2 us Tw rs Hlu+Hhs (30) where (1 - Psi) is again a geometric factor, of order 1, defined by 1 -ps, =*NS Tr (JCp/s/) - T f(Xf)iSl Xpisj) TrM-,)2 (31) The local fields, HU} and Hus, are defined by „i _ Li2 Tr(Xofi)2 Hli/ - Tr (X zt) 2 (32) and Hl,s Mi Tt(X^s)2 Tr (Xz,)r (33) Note that in the numerator of Eq. (31), only l-S terms are present. This is due to the fact that ‘Km), and Mpfu are equal in the case of S-spin diffusion. As be­fore, S-S terms are omitted because they are negligibly small. From Eqs. (27), (32), and (33), we see that HLu » a Lis- Thus, from Eq. (30), we find that T\d » for S-spin diffusion. This is to be expect­ed, since the weak S-spins' motion should surely have much less effect on the dipolar reservoir than would the strong,/-spins' motion. Another interesting feature of S-spin diffusion as contrasted to /-spin diffusion is the anisotropy in T]D as predicted by Eq. (30). (Anisotropy refers to meas­urements as a function of sample orientation in H0 and enters the calculations through the value of 9ik in the dipolar coupling parameters Aik, Biky and Cik.) The terms 2(1 -/?//) and (1 - As/) usually have small aniso­tropy. 20 21 Thus, T]d for /-spin diffusion [see Eq. (28)1 would also have small anisotropy. On the other hand, Tw anisotropy for S-spin diffusion [see Eq. (30)] is given approximately by the local-field term Hhstmiu + Hlis) which in some instances is very an­isotropic. An example of large T\D anisotropy in S- spin diffusion is given in the KF case, discussed in Sec. V. D. Modified strong-collision theory for T ID The SA theory is easily modified to give us an ex­pression for TV due to slow atomic motion. We sim­ply change 3Co0) to 3C/)00) and obtain 1 r, D M Tr(^00))2-Tr(^/r)^,/Q)) r Tr(^00))2 (34) Note that, since ?f/;00) is a function of 0/ [see Eqs. (20)-(22)], T]D' is also. This is an important feature of T\D'. It contains a parameter 0/ which is deter-18 N U C L E A R - M A G N E T I C - R E S O N A N C E M E T H O D S F O R . . 145 mined by an experimentally controlled variable o>/, the frequency of H^. Thus, by varying 0/, the relative sensitivity of T]D' to the motions of different spin species can be varied. To illustrate this, consider the case treated in the previous section, that is, diffusion in a system of strong / spins and weak 5 spins. If T] D' is due to /- spin diffusion, we have, from Eq. (34), 1 1 [-7(3 cos20/ - 1 )]2HIu T ' T/ [-r(3 cos20j - \)]2Hlu + cos20/ Hus 2(1 -pu) + where 2(1 - pu) is given by Eq. (29), and (1 - pis) is a geometric factor of order 1, defined by 1 " Pis „ Tr(Jt&)2-Tr(XI N,---------------------------- » (0) if (0) \ ___________DISi "''PIS/ * Tr(JCo/5,)2 (36) The local fields, HUi and HL/S, are given by Eqs. (32) and (33). Note that in Eq. (35) the I-S interaction could not be neglected as it was in Eq. (28), since for 9{ near the magic angle 0/„(=cos~1Vl/3 =54.7°) the /-/interaction [the first term in Eq. (35)] becomes very small so that the I-S interaction [the second term in Eq. (35)] may contribute significantly to T\D'. For 9{ not near 0,„, the I-S interaction may be neglected, and Eq. (35) becomes ^r-r--2(1-p„) , J\D Tf (37) cos20/ Hhs T\d tS (3 cos20/ - 1 )]2Hhi + cos20/ Hhs where (1 -psi) is given by Eq. (31), and 2(1 - pss) geometric factor, of order 1, defined by 2(1 - Pss) Tr (Moss, Hqssj) Tr(X$Sl)2 (40) The local field HLSS is given by Hlss-mix{K^s)2nx(Kzl)2 (41) Note that in Eq. (39) the S-S interaction cannot al­ways be neglected as it was in Eq. (30), since, for 0/ near 90°, the I-S interaction [the first term in Eq. (39)] becomes very small so that the S-S interaction [the second term in Eq. (39)] may contribute significantly to T\D'. In the case of S-spin diffusion, Tw' [see Eq. (39)] has a large dependence on 0/. In particular, for 0/ = 0w, Eq. (39) becomes 1 1 T]D'(9J ts (1 - Psi) Thus, T\D'(9m) -ts and is much smaller than T]D 1 cos207 Hhs 7/ ["(3 cos20/ - \)]2Hlu +cos20/ Hhs (1 ~~ Pis) » (35) which is identical to T]D given by Eq. (28). Thus, Tw' in the case of /-spin diffusion is generally in­dependent of 0/, except perhaps near 0/ = 0//;. For 0/ = 0,„, the I-l interaction is zero, and Eq. (35) be­comes -i____= -L(1 T\D (0,„) rI Pis) (38) By comparison of Eqs. (37) and (38), we see that the amount by which T]d'(0i) varies near 0/ = 9n, is deter­mined by the relative values of 2(1 - p'u) and (1 -Pis)- In contrast, if T\D' is due to 5-spin diffusion, we have from Eq. (34) (1 - Psi) + H? LSS Ts ["-(3cos20/-l)]2///?// +cos20///£2 2(1 -Pss) LIS (39) is a given by Eq. (30). For 0/==9O°, Eq. (39) becomes rw'm°) r/ 2(1-fe) . I* Lll ' (43) (42) In this instance, Tw'(90° ) » rs and in fact is also much larger than T]D given by Eq. (30). Thus, T\n‘(9/) in the case of S-spin diffusion varies a large amount (often orders of magnitude) as a function of 0/. This is an extremely important feature of T\D'. By varying 0/, we can vary the effect of 5-spin diffusion on TiD'(9/) relative to the effect of /-spin diffusion. This allows us to study the motions of different spins separately and identify them, as is illus­trated in Sec. V. E. High-field T]f) Consider the rotating-frame Zeeman spin- lattice relaxation time Tl()/ due to atomic motion. (For large Hi/, which is the present case, this relaxa­tion time is often called the ''high field" T\ir) The ex­pression for T\()/ may be divided into two parts, T]f)U1 4 6 H A R O L D T . S T O K E S A N D D A V I D C . A I L I O N I B and 77,,/s, due to contributions from the I-1 and I-S dipolar interactions, respectively. 1 1 i>f T + 1 (44) i P// (>is An expression for T^u was first given by Look and Lowe7 for the case of H]/ on resonance and then later extended by Jones30 to the off-resonance case. In the limit yiHQT » 1 (corresponding to temperatures far below the 7^ minimum), Jones's expression reduces to 1 \Pu \yj H2l(f + 1) x [sin20, cos2#/ Jiin(.yiHj) + sin4M,,/0)(2y,//dr,)] (45) where /,f(w) is the spectral density of the correlation function of the /-/ secular dipolar interaction and depends on the nature of the atomic motion. An expression for T\fiiS has also been given31v 10,32 for the case of Hj/ on resonance (0/ = 9O°), which, in the limits y///0r » 1 and ysH0T » 1, is written 1 / r, (90°) - j-yhi K2S (S + 1) (y///,,) (0) (46) where o>) is the spectral density of the correlation function of the I-S secular dipolar interaction. As far as we have been able to determine, an extension of this expression for T\f)/s to the off-resonance case has not yet been reported in the literature. It is, however, straightforward to provide one. Only two minor changes in Eq. (4T>) need be made. First, the /-spin Zeeman interaction Kzi1 given by Eq. (19) involves //en7 instead oi Hu. This feature can be included in Eq. (46) by changing J^iy/Hu) lo J is )(y,Hcfff). Second, the nonsecular dipolar interaction JC/us* in­cludes a factor sin0/. We obtain from Eqs. (7) and (22), in the tilted rotating reference frame, "-sin.0, XBJkJyis=k (47) i,k We can include this feature in Eq. (46) by changing Bik, to sin0/Z?,*. Since Eq. (46) has a quadratic depen­dence on Blk [note the (yfys)2 factor, for example, and compare with Eq. (10)], we simply multiply the expression by sin2#/. Finally, then, we have - = Tyhl^s (s'+1) T 6 11 pIS X s\n201J/p(yiH^ni) (48) In the limit y///eff/T >> 1 (corresponding to the cold side of the T\()l minimum), these expressions can be simplified. It is well known9 33 that in this limit the spectral densities are proportional to o) . Thus, Hi = sin26>,40)(2y,Hu) , Using this and similar expressions, we obtain 1 -^TytK2m + l)Jlf)(2ylHu) (49) x sin40/(4cos20/ + sin20/) (50) and T -- = Tyhsft2s(s + 1) pis (51) As an application of these expressions, consider the case treated in the previous sections, i.e., diffusion in a system of strong / spins and weak S spins. In the case of /-spin diffusion, T\ftf is given by T\(,n in Eq. (50). (The contribution from Tl()ls is much too small to be significant at any value of 0/.) Assuming a con­stant //1/, then, we obtain from Eq. (50): T\pi(9i) 1 ^ip/(90°) sin40/(4cos20/ + sin20/) (52) In the case of S-spin diffusion, J\f,u does not contri­bute to T]fth and thus T]f)i is given by T]t>iS in Eq. (51). Assuming a constant H\h we have 1 7^,,/(0/) _________ rlp/(90°) sin40/ (53) A comparison of Eqs. (52) and (53) clearly shows that the 0/ dependence of T]f)l is very much different in thp two cases. Thus, by measuring 7'],,(0/) as a function of 0/ (at constant Hu), we can easily deter­mine whether /-spin or 5-spin diffusion dominates the relaxation. III. EXPERIMENTAL METHODS A. T\D' pulse sequence The pulse sequence for measuring T\o has been described briefly in a previous paper.22 (Note that the first pulse sequence proposed for measuring T\D' was a double-resonance sequence.12 The pulse sequence described here is a single-resonance sequence. It is simpler and provides a larger signal than does the former.) Here we describe the T[D' pulse sequence in more detail. First we demagnetize the I spins (see Fig. 3) by1 8 N U C L E A R - M A G N E T I C - R E S O N A N C E M E T H O D S F O R . . 1 4 7 H II 90° PULSE-J 90° PHASE SHIFT J ADRF VARIABLE (AS SHOWN BELOW) >- REMAGNETIZE Auj, ■ FIG. 3. Pulse sequence for measuring The lower figure shows the variation of Au)f = ytH0~ (Dh the off- resonance frequency of H\f. spin-locking34 (i.e., a 90° pulse followed by a 90° phase shift) followed by adiabatic demagnetization in the rotating reference frame35 (ADRF). This step transfers Zeeman order which was originally along H0 to dipolar order of the 3C/)0) reservoir. We then apply Hi/ at a frequency which is off reso­nance by an amount yjh\. The dipolar reservoir is now properly described in the rotating reference frame, i.e., by ?C/ioo)(0/i), where 0f] =tan If we are sufficiently far off resonance (0/1 «0), we see from comparison of Eqs. (20)-(22) with Eqs. (5)-(7) that K/)OO)(0,, =0) is approximately equal to Thus the djpolar order, which was in the Kn)] reservoir before was turned on, is preserved and is now in the Kd00)(9/]) reservoir. We next sweep the frequency of Hw to a value off resonance by an amount y/h2. This process varies 0/ and accordingly varies JCf)00)(9f). If we sweep sufficiently slowly, the order of the ?C/;)O)(0/) reservoir is preserved at all times, and thus the process is adia­batic; The dipolar order is now in the 3FC/)0<)>(^/2? reservoir, where 0/2 = tan-l(tfi///f2)- Since ^C/V)0)(0/) varies as we sweep 0/, the sweep process effectively sweeps the heat capacity of the dipolar reservoir. The entire process has thus transferred Zeeman order originally along H0 to dipolar order of the Ho00)(Oi2) reservoir. Since the entire pulse sequence thus far has been adiabatic (i.e., spin order is preserved), the spin temperature can be calculated25: Pd^Pl Tt(Xzi) Tr[Xr(0l2)] 1/2 (54) Note that 0/2 is arbitrary and can be chosen to be any value desired. We hold the value of 0/ at 0I2 for a time r, during which the dipolar order decays via spin-lattice relaxation. We write a rate equation for ftp* which defines T]D': dt (/) 1 t^Pd^Pl) T (55) 1/) Since po] » fiL initially [see Eq. (54)], /3/>" decays essentially towards zero. Thus, from Eq. (55), we can write, to good approximation, Pd ^0) - Pd *(0) exp[-t/T\[y (0/2)] • (56) After the time r, we sweep the frequency of H1 / back off resonance (i.e., sweep 0/ back to 9n) and then turn off. This transfers any remaining dipo­lar order of JCpm'(9l2) back to dipolar order of #/)()). Remagnetization of the / spins further transfers this order to Zeeman order along H j/. The magnetization M/, now along Hi/, is less than the original A/0/ by the factor exp[-t/T\D'(9/2)] and is measured by turn­ing off Hi/ suddenly and then observing the free in­duction decay (FID). By repeating this pulse se­quence for various values of r, the spin-lattice relaxa­tion time T\d'(9i2) can be determined using Mi - Mq'i exp[-t/7'i/;,(0/2)] (57) We now examine in more detail some of the unique features of the T\D' pulse sequence. To do so, it is useful first to develop a general expression for calcu­lating the loss of spin order due to a sudden change in the Hamiltonian. Consider an isolated spin system with an initial Hamiltonian At internal thermal equilibrium, the system can be characterized by a spin temperature /3,. Now, if we change the Hamiltonian suddenly to Kf, after a time T2 the system again at­tains internal equilibrium at a new temperature fif . From Goldman,25 we see that (sudden) / P, Tr(7C,3f,) Tr(3f ,2) (58) If the process of changing X, to X, had been adiabatic1 4 8 H A R O L D T . S T O K E S A N D D A V I D C . A I L I O N 1 8 instead of sudden, we would have preserved the spin order and obtained Tr (JC,)2 Tr(JC,)2 n (59) The loss of order due to a sudden change in the Ham­iltonian can be characterized by a function / defined by (sudden) Tr(Jf,3f,) (adiab) /' [Tr(dFf/)2Tr(?f/)2]1/2 (60) Thus, if the Hamiltonian of an energy reservoir is suddenly changed, the resulting reciprocal spin tem­perature will be smaller by the factor / than it would have been if the change had been adiabatic. Now consider the T]D' pulse sequence (see Fig. 3). After the initial ADRF, H^ is turned on suddenly at a frequency which is off resonance by an amount yih\{9) = 0/i). Using ^ and D00)(On) as the initial and final Hamiltonians, respectively, we obtain from Eq. (60) / = Tr[3C (0) D {Tr[^o)]2Tr[^oo)(0/i)]2}1/2 ‘ In evaluating the traces, we use the relation, ^I{)0)-JCf)00)(On)+^Dn)(0n) . Since Tr[Jfi°")(6i/1)//ioo)(0,i)]=O , we immediately obtain (61) (62) (63) TrtTC/flHfl/i)]2 T r [ 3C/j°1 ]2 1/2 (64) If we neglect the effect of S spins (which is the case in our experiments when 9n is not near 0,„), we have / = -7(3 cos20/ - 1) (65) If we are very far from resonance (0/ =0), we can see from this equation that / = 1. Thus, in the T]D' pulse sequence, the step of turning on H]f preserves dipolar order provided H1/ is far off resonance. Similarly, the step of turning H\i off again yields the same result since Eq. (60) is symmetric in K, and JC,. In the ex­periments actually performed, h\ = 100 G and Hu = 10 G. Thus 0/i =6° and, from Eq. (65), / = 0.985. So only 1.5% of the magnetization is lost by the sudden turn on of H]h Another step in the TU)' pulse sequence that needs closer examination is the sweeping of the H17 frequen­cy when the JCo00) reservoir is in a "cooled" state of di­polar order. Of course, as stated before, if the sweep rate is slow enough, the process is adiabatic and dipo­lar order is preserved. This requires that changes in 3C/)OO)(0/) occur sufficiently slowly that its reservoir can maintain internal thermal equilibrium at all times. For a system of strong / spins and weak S spins, this condition is most stringent at 0/ = 0„, because of the absence at that angle of I-l interactions which normal­ly bring the reservoir quickly to equilibrium. This feature can be seen most easily from a calcula­tion of the local field Hu(9i) defined by ,r2 Tr[5CA00>(6»/)]2 H° Tr(Jfz/)2 (66) From the above expression, we calculated the local field in KF (/ spins are 19F, and S spins are 39K) as a function of 0/ at two different crystal orientations (see Fig. 4). We can see that at 0/ ■= 9m the local field Hu(9i) becomes very small, thereby resulting in a greatly lengthened time T2[ = yF]Hu (#/)] required for the 5C/>OO)(0/) reservoir to come to internal thermal equilibrium. If we sweep 0/ slightly too fast to be completely adi­abatic near 0,„, then the sweep can still be adiabatic outside some interval bounded by 0/ = 0m ± A0. Over this interval, we can approximate the sweep by a sud­den step in 0/ and use Eq. (60) to calculate the factor /: / Tr[3C/)OO)(0 A0) Jf^oo)(0 11 l J\f) - isaj j\d 4- A0)] {Tr[?C/)OO)(0w - A0)]2Tr[JC/)OO)(0//? + A0)]2} n (67) FIG. 4. The local field HL/(Q/) calculated from Eq. (66) for KF at two different crystal orientations: H0 along the [100] and [111] crystal axes as indicated in the figure.1 8 N U C L E A R - M A G N E T I C - R E S O N A N C E M E T H O D S F O R . . 1 4 9 If one uses Eqs. (20)-(22) for 3C/)O())(0/), one should note that these expressions are written in a coordinate system with the z axis along Heff/ (which makes an angle 0/ with H0). Thus, the expressions for 5C/)OO)(0W - A0) and 3Cdoo)(0„/ + A0) would be in different coordinate systems. In calculating the trace in the numerator, it is important that the two dipolar interactions be written in the same coordinate frame. Thus, it is necessary to transform one of these terms to the coordinate frame of the other. We then obtain / 3 cos2(0,„ - A0) - 1 x 3 cos2(0/h - A0) - 1 3 cos2(0„, + A0) - 1 3 cos2(2A0) - 1 Hi,i + cos2(0„, - AO) Hi is Hlii +cos(0/;, - A0) cos(0,„ + A0) cos(2 A0)Hhs :/2 X 3 cos2(0,„ + A0) - 1 Hln + cos2(0,„ + AO)Hus 1/2 (68) Note that we neglected the S-S interactions. The third factor in each of the two terms in the numerator of Eq. (68) comes from the additional coordinate transformation described above. We calculated / for KF at two different crystal orientations (see Fig. 5). For an adiabatic sweep (A0 = O), of course, we have / = 1. As the sweep rate increases, A0 increases and / decreases finally going negative. A negative / means that the spin temperature reverses its sign, In this case, the sign reversal is caused by the reversal in the direction of the local field during the nonadiabatic step so that spins aligned parallel to the local field before that step are now aligned antiparallel to the local field afterwards: hence a negative temperature. (Of course, the above treatment is valid only for the case of strong I spins and weak S spins.) We verified this feature qualitatively with the fol- FIG. 5. The parameter / [defined by Eq. (60)] calculated from Eq. (68) for KF at two different crystal orientations: H0 along the [100] and [111] crystal axes as indicated in the figure. lowing experiment. Instead of sweeping h at a con­stant rate in the T\D' pulse sequence, we swept h from 100 to 15 G at a rate of 50 G/msec and then from 15 G to 2.5 G at some variable rate h, and finally from 2.5 G to 0 at 50 G/msec. With this method, then, the sweep rate h over the region near the magic angle could be varied down to very small values. (0/= 0„, corresponds to h = 10 G for this experiment.) The same sequence of sweep rates were applied in re­verse to sweep h from 0 back up to 100 G again. The Ti/)' pulse sequence, using the /7-sweep described above, was applied to KF at room temperature (r « T] //) and the FID amplitude was measured as a function of h (see Fig. 6). Note that, in this experi­ment, we sweep 0/ through the magic angle twice (once in each direction); thus the resulting signal will be proportional to /2. As we increase the sweep rate, f2 should first decrease to zero and then increase to a positive value again. This is indeed what we observed (see Fig. 6). Note that we never quite attained com­plete adiabaticity in this experiment, even at h =0.1 G/msec. On the other hand, as can be seen in Fig. 6, we found that we could preserve much of the spin order by sweeping fast enough for / to be negative. ' • Thus, in our T\D' measurements, we used /? =80 G/msec. Even though this sweep rate was far from being truly adiabatic, nevertheless, much of the spin order was preserved during the sweep. We thus have the surprising result that more signal is obtained if the sweep of 0/ is very nonadiabatic near 0m than if it is almost adiabatic there. Of course, the sweep must be adiabatic for 0/ not near 0m. When using this technique, one must be particularly careful when measuring T\D' near the magic angle. Small instabilities in the experimental apparatus can cause large effects. Earlier, we saw in AgF an ap­parent decrease in T[D' near the magic angle, which we finally discovered was due to a droop in H\f during the time interval r of the T]D' pulse sequence. Upon elimination of the droop in Hu, the decrease in T]D' disappeared.150 H A R O L D T . S T O K E S A N D D A V I D C . A I L I O N 1 8 1.0 0.5 - 0 ti (G/msec ) FIG. 6. as function of h in a TID' pulse sequence (described in the text) applied to KF at two different crystal orientations: H0 along the [100] and [111] crystal axes. B. 7']),/(0/) pulse sequence / There are a variety of possible pulse sequences for measuring T]f)/ off resonance. We used a very simple sequence (see Fig. 7) which resembles a standard spin-locking technique commonly used for measuring T\f} on resonance. The entire pulse sequence is applied at a single constant frequency off resonance. To in­sure a constant HXh we varied H0 rather than o>/ in this experiment. First we spin-lock the / spins by applying a pulse of length tp followed by a 90° phase shift. During the rp pulse, the /-spin magnetization M/ precesses about Hct1 / with frequency y/Hcn/. After the 90° phase shift, M/ precesses about the new Hotr/ until the per­pendicular component dies to zero in a time of ap­FIG. 7. Pulse sequence for measuring T]f)(0/). The entire sequence rs done at a single frequency which is off resonance by an amount/?=//]/cot^/. : proximately T2. Using simple geometric relationships, we calculate the "spin-locked" component of M7: A//sl> = MO/[sin20/ s\n(yiHitP) + cos0/ sin20/ cos(y///efl7T/,) -fcos30/] . (69) Maximizing this equation with respect to r/M we find the length of pulse needed for spin-locking the " " imum amount of magnetization: y IHdX [ Tp ~ COt 1 (COS0/) . Putting this into Eq. (69), we obtain M/SL) = MO/[sin20/(l +cos20/)1/2 +cos30/] . max- (70) (71) This function has a minimum of A//SL) =0.96M0/ at Of =55°. Thus, by using the appropriate pulse length t,, given by Eq. (70), nearly all of the original magnet­ization M0/ can be spin-locked along Hefl /. One should note that a 90° phase shift is not the optimum for off-resonance spin-locking. However, as we have seen above, a 90° phase shift will result in a 96% mag­netization even at the most unfavorable value for 0/. Following the spin-locking, M/ decays towards zero via spin-lattice relaxation and, after a time r, is re­duced by a factor exp[-t/T\p1(0/)]. At this point, Ht/ is turned off suddenly, and we observe the FID. [Ac­tually, only the component of M/ perpendicular to H01 8 , N U C L E A R - M A G N E T I C - R E S O N A N C E M E T H O D S F O R . 1 5 1 contributes to the FID and thus the maximum signal obtainable with this pulse sequence is approximately Mo sin#/. In order to improve the efficiency for small Oi (far off resonance), one should use a different pulse sequence such as the one described by Cornell and Pope.36 ] - IV. EXPERIMENTAL APPARATUS We operated our NMR spectrometer at 24 MHz us­ing a frequency synthesizer (Adret 6100 with plug-ins 6300 and 6500) which is remotely programmable. We swept frequency by using the "search mode" via an externally applied analog voltage. Our probe was a single coil matched to 50 O. A combination of crossed diodes and quarter-wavelength transmission lines protected the receiver amplier dur­ing the rf pulse. We described a similar configuration in a previous paper.13 V. EXPERIMENTAL RESULTS A. T]p and T\p (Of) . 1. Potassium fluoride (KF) In order to verify some of the theoretical expres­sions in Sec. II, we made T]D and T\D' measurements on a single crystal of KF (/ spins are 19F, S spins are 39K) doped with about 1000-ppm mole fraction CaF2. (This crystal was grown by the Crystal Growth La­boratory of the University of Utah Physics Depart­ment.) Similar crystals have been reported11 to pro­duce mobile potassium vacancies that dominate diffusion at low temperatures. We measured the temperature dependence of T\D (see Fig. 8) for KF at two different crystal orienta­tions: H0 along the [100] and [111] crystal axes. In Fig. 8, we identify four different regions of relaxation processes. Region 1(7"^ 170°C) is dominated by po­tassium diffusion as reported in Ref. 11. From the slopes of the lines through the data, we obtain activa­tion energies Ej =0.75 ±0.15 eV and 0.92 ±0.15 eV for the [100] and [111] crystal orientations, respec­tively. These values are in fair agreement with that reported in Ref. 11 (EA =0.83 eV). (Note that, in drawing the solid lines in Fig. 8, we corrected the data by subtracting the relaxation rate due to the process dominant in Region II. Thus, the line represents the relaxation rate due to the Region I process alone, whereas the data itself is actually the sum of the relax­ation rates of Region I and Region II processes.) Our data (Fig. 8) for the two orientations shows that, in Region I, T\D is very anisotropic, as predicted by Eq. (30) for S-spin diffusion. We made a more de­tailed anisotropy measurement at 227 °C (see Fig. 9). In this experiment, the crystal was oriented in H0 with FIG. 8. Tj / and Tw in KF al two different crystal orienta­tions: H{) along the [100] and [111] crystal axes. The two different symbols for T]D [100] refer to two different sam­ples. the axis of rotation (perpendicular to H0) along the [110] crystal axis. Measurements of T]D for various rotations are shown in Fig. 9. A theoretical calcula­tion, using Eq. (30) which assumes S-spin diffusion, was made. In this calculation, the quantity r/( 1 - As/) was determined from a best fit to our data. (We treat­ed the factor (1 - pS{) as being isotropic. Any error due to this approximation should be very small20 21 compared to the effects which we study here.) Since there are no other adjustable parameters, the excellent agreement between theory and experiment in Fig. 9 is gratifying. The large anisotropy in T]D verifies that diffusion of Sspins is dominant at this temperature. Such a large anisotropy would not be observed if diffusion of / spins were dominant. In Regions II and III of Fig. 8, the behavior of the data is very similar to that observed by Ho and Ailiori37 38 and also by Wei and Ailion39 40 in various samples of doped CaF2 and SrF2. They interpreted the effect to be due to localized diffusion, i.e., mobile defects bound to impurity ions. Accordingly, it is like­ly that, in the present potassium fluoride T]D data, we1 5 2 H A R O L D T . S T O K E S A N D D A V I D C . A I L I O N 1 8 9 (deg.) FIG. 9. T]D anisotropy in KF at 227 °C. Crystal is rotated about its [110] axis. laxation. We chose to measure T]D'{9,) as a function of 0/ at 200°C (see Fig. 10). This temperature is in Region I of Fig. 8, where potassium diffusion dominates T]D. At 9, =0, the data point is actually a measurement of T\0. Since 3C/->OO)(0/=0) is equal to 3Cp0), a compari­son of Eqs. (25) and (34) shows that T\D'(9{ =0) is equal to TlD. As predicted by Eq. (39) for S-spin mo­tion, T\D' shows a large dependence on 9, (two orders of magnitude). A theoretical calculation of Tw'(0/) using Eq. (39), fit to the value of T]D' only at 0/= 0 (which is T]d), is also shown in Fig. 10 and is in good agreement with the data. This data verifies an impor­tant feature of Tw'(9i). By varying 9f, we can vary the effect of 5-spin diffusion on T\D'{91). Measurements of T\D'(9f) at 9,= 90° (h2 = 0) were also made as a function of temperature over part of Regions I and II (see Fig. 11). As can be seen, T\D'(90°) is much larger than T]D (a feature charac­teristic of 5-spin diffusion) and is thus in agreement with the anisotropy measurements which similarly demonstrate that S-spin diffusion is dominant. 2. Silver fluoride (AgF) T]D and 7V(90°) were also measured12 23 in two different samples of AgF. Sample No. 1 was obtained from Research Organic/Inorganic Chemical Corp. (AG-10, 98% pure) and Sample No. 2 from Cerac/Pure Inc. (S-1080, 99.5% pure). [A third sam- see an effect also due to localized diffusion, that is, mobile potassium vacancies bound to Ca++ ions. Between Regions II and III, the slope of the data shar­ply decreases for increasing temperature. Wei and Ailion39 40 showed evidence that this behavior was in their case caused by an abrupt change in diffusion mechanism. Thus the relaxation process of Region II and is not present hi Region III, and the relaxation process of Region III is not present in Region II. Ac­cordingly, the two relaxation rates do not add together in each other's region, in contrast to the case of Re­gions I and II discussed above. However, we are per­plexed by the fact that the transition temperature between Regions II and III appears to change with cry­stal orientation. From Fig. 8, we see that our Region II exhibits large anisotropy, as in Region I, which suggests that the dominant relaxation mechanism is some kind of potassium diffusion (possibly localized, as discussed above). Region III, on the other hand, exhibits much less anisotropy. * In Region IV (T ^0°C), the relaxation rate seems to have very little temperature dependence and is possibly due to paramagnetic impurities. The T\f data in Fig. 8 is probably due to paramagnetic impurity re­( degrees) FIG. 10. T]d' as a function of 9, in KF at 200°C. Note that the data point at 9, =0 is TlD.1 8 N U C L E A R - M A G N E T I C - R E S O N A N C E M E T H O D S F O R . . . 1 5 3 10 I o 0) (/> o O £ -Q O K" \ ▼ ▼ t,d ^7 Tjp(90°) 0.1 - ▼ ▼ i i I .........i.. 2.0 2.5 3.0 -I 1000/T (°K ') FIG. 11. Tw and Tw'(90°) in KF with H0 along the [100] crystal axis. pie was obtained from Apache Chemicals Inc. (#6957, 99% pure) but apparently contained too much magnet­ic impurity for useful data to be obtained.] To our knowledge, no previous diffusion measurements have ever been made in AgF. From the temperature dependence of T]D (see Fig. 12), some motional pro­cess is evident at temperatures above T =60°C. Note that data from both samples are plotted in Fig. 12 and that they behave similarly in this diffusion re­gion. At lower temperatures, where paramagnetic im­purities possibly dominate T]D, they differ significantly with the purer sample (Sample No. 2) having the longer relaxation time. From the slope of the line drawn through the data points, we obtain Ea = 0.92 ±0.15 eV. As in KF, the data was corrected by subtracting off the relaxation rate of the low- temperature process. Actually, AgF is a three-spin system: 19F, 10/Ag, and 109Ag. Labeling ,9F as the / spins, and the two * FIG. 12. Tw and Tw'(90°) in two different samples of AgF. silver isotopes as 5 spins, the theoretical expressions given in Sec. II need be only slightly modified. Since the interactions between the 5 spins are so small that they can be neglected, the effect of 5-spin motion on relaxation rates can be calculated separately for 1()7Ag and 109Ag and then added together to obtain the net relaxation rate. We measured T]D'(9/) for 0/=9O° over the entire temperature region where diffusion dominates. In Fig. 12, we see that T\D'(90°) = TXD for both samples. This behavior is typical of /-spin diffusion [see Eq. (37)], as contrasted with the behavior, 7'|D'(90°) » TU), characteristic of 5-spin diffusion (see Fig. 11). Thus, we have conclusive evidence that fluorine diffusion dominates T]D in our samples of AgF. We also measured T\D'(9/) as a function of 0/ in both samples (see Figs. 13 and 14). As can be seen from this data, TU)'(9j) is approximately independent of 9h Again, this is behavior typical of /-spin diffusion [see Eq. (37)], in contrast to that of >S-spin diffusion seen in Fig. 10. [We have not attempted to explain the slight decrease in TU)'(9/) for increasing M 5-spin diffusion has the greatest effect on T]D'(9/) when 91 is near 9m. In Fig. 13, for example, we see that at 0/= 52° (which is near 0/H), T]D'(9{) approxi­mately equals T]D and is therefore dominated by /- spin diffusion.- At this angle (0/= 52°), TU)'(9i) due to 5-spin diffusion in AgF is given by [see Eq. (39)] 1 (0.47) -(1 -psi) T\d'{52°) . t s- The fact that / spin diffusion dominates 7'|0'(52°) ( 7 2 ) 1 5 4 H A R O L D T \ S T O K E S A N D D A V I D C . A I L I O N 1 8 FIG. 13. T]D' as a function of 0/ in Sample No. 1 of AgF at 68°C. Note the data point at 0f =0 is T]D. means that the relaxation rate given by Eq. (37) for /-spin diffusion must be much larger than that given by Eq. (72) above for S-spin diffusion. Thus, assum­ing that 2(1 - pu) and (1 - pS{) are of the same order of magnitude, we conclude that, for this sample of AgF, T/ « T.S. In all of the other silver halides (AgCl,AgBr,AgI), silver diffusion is dominant.41 42 Hence, it was some­what surprising to us to find that in AgF fluorine diffusion is dominant. Perhaps one factor influencing this behavior is the impurities in AgF. In'contrast to the other silver halides, high-purity samples of AgF are not available. AgF is also very reactive and usual­ly contains a considerable amount of AgF2 and Ag2F.43 Impurities in AgF could increase significantly the number of mobile fluorine defects, thus causing fluorine diffusion to be dominant.37 40 ' B. 7^/(0,) We measured T]j)t on resonance (0/=9O°) in KF ([100] crystal orientation) as a function of tempera­ture (see Fig. 15). From the slope of the line, we ob­tain Ea =0.76 ±0.15 eV, in agreement with that of Region I in Fig. 8. Thus potassium vacancy diffusion dominates T]pI over this temperature region. At 351 °C, we made one T\p! measurement for the [111] orientation and found a large anisotropy. From calcu- FIG. 15. T\pl on resonance (0/ =90°) in KF al two FIG. 14. T]0' as a function of 0/ in Sample No. 2 of AgF different crystal orientations: H0 along the [100] and [111] at 89 °C. Note the data point at 9f =0 is T]D. crystal axes.1 8 N U C L E A R - M A G N E T I C - R E S O N A N C E M E T H O D S F O R . . . 1 5 5 iations of y(0)(a>} found in the literature,9 33 we find that the theoretical anisotropy is rlp/([lll}) 0.65, I- spin diffusion T]p{([100]) l26> 5-spin diffusion ' Thus, the T\f)f anisotropy measured in KF also clearly verifies that, in this region, potassium diffusion dom­inates T]()f. 7\,/(0/) was measured in KF at 351 °C as a func­tion of 9h The ratio r!p/(0/)/7Yp/(9O°) is plotted in Fig. 16 along with the theoretical calculation from Eq. (53) for S-spin diffusion. To contrast this case with that of /-spin diffusion, T]l>/(9/) was measured in a single crystal of undoped CaF2 at 314 °C. (/ spins are l9F. There are no S spins of any significance.) At this temperature, fluorine diffusion dominates Tlp/. (We concluded this by measuring EA ^ 0.9 eV and compar­ing it with the other measurements.39 40) The ratio T\f>l(9{)/T[l)i(9Q°) is plotted in Fig, 16 along with the theoretical calculation from Eq. (52)- for /-spin diffusion. As can be seen in Fig. 16, agree­ment between data and theory is fairly good, especially the contrast between /-spin and S-spin diffusion. It is clear that these measurements provide an easy method for distinguishing between the two types of diffusion. VI. CONCLUSIONS In this paper, we have described some new methods for identifying and studying the diffusing species in multispin systems. We have demonstrated that aniso­tropy measurements of Tw and T\p for a two spin system of strong / and weak S spins is clearly capable of distinguishing between diffusion of the two species. Anisotropy measurements may similarly be useful in other types of multispin systems (e.g., strong / and strong S spins as in LiF). Of course, such anisotropy measurements are limited to systems in which single crystals are obtainable. We introduced a new technique involving a new re­laxation time T\D' which also can be very effective in distinguishing between diffusion of different spin species. This technique has the advantage that it is not restricted to single crystals and can easily be ap­plied to polycrystalline samples. Furthermore, we have shown that TiD' measurements can enhance sub- FIG. 16. The ratio T ] f)(0,) / T ] f>(90 °) in KF in the [100] orientation at 351 °C and in CaF2 in the [111] orientation at 314°C. The upper and lower curves are calculated from Eqs. (53) and (52), respectively. stantially the effects of the diffusion of weak spins. Not only can this technique be applied to the study of abundant weakly magnetic spins (as in the cases described in this paper) but almost certainly it can be extended to the case of diffusion of dilute strongly magnetic spins (e.g., diffusing impurities). 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