684 GRANT, WIEDERSICH, HOUSI EY, ESPINOSA, AND ABTMAN

and Julieke Hupkes (unpublished).P. A. Flinn (private communication).R. W. Grant, R. M. Housley, and U. Gonser, Phys.

Rev. 178, 523 (1969).~~S. Goldsztaub, Bull. Soc. Franc. Mineral 58, 6

(1935).R. M. Housley, N. E. Erickson, and J. G. Dash,

Nucl. Instr. Methods 27, 29 (1964). An additional smallbackground correction of about 2/o was made to accountfor Compton scattering from a Lucite source holder.This correction could be estimated only approximatelywhich increased the uncertainty in the absolute areasbut had a negligible effect on the absorption-area ratios.

S. Goldsztaub, Compt. Rend. 198, 667 (1934).M. D. Lind, Acta Cryst. B26, 1058 (1970).Throughout this paper a descriptive notation is used,

namely, that the "direction" of V« implies the directionof the z principal axis of the diagonalized EFG, etc.

Unless stated otherwise, the quoted error represents1 standard deviation of the mean value based on the 22independent measurements.

R. M. Housley, R. W. Grant, and U. Gonser, Phys.Bev. 178, 514 (1969).

B. M. Housley, J. G. Dash, and R. H. Nussbaum,Phys. Rev. 136, A464 (1964); the effective f of our sourcewas reduced by=1% because of resonant self-absorption.

A. H. Muir, Jr. , K. J. Ando, and H. M. Coogan,Mossbauer Effect Data Index 1958-1965 (Interscience,New York, 1966).

U. Gonser and R. W. Grant, Phys. Status Solidi21, 331 (1967).

2~C. E. Johnson, W. Marshall, and G. J. Perlow,Phys. Bev. 126, 1503 (1962).

U. Gonser, R. W. Grant, H. Wiedersich, and S. Gel-ler, Appl. Phys. Letters 9, 18 (1966).

W. Rubinson and K. P. Gopinathan, Phys. Rev. 170,969 (1968).

4D. P, Johnson, Phys. Rev. B 1, 3551 (1970).D. Sengupta and J. O. Artman, Phys. Rev. B 1,

2986 (1970); J. O. Artman, F. deS. Barros, J. Stampfel,J. Viccaro, and B. A. Heinz, Bull. Am. Phys. Soc.13, 691 (1968); J. O. Artman and J. C. Murphy, Phys.Rev. 135, A1622 (1964).

6R. M. Sternheimer, Phys. Rev. 130, 1423 (1963).~VR. W. Grant, J. Appl. Phys. (to be published).

R. Ingalls, Phys. Rev. 188, 1045 (1969).A. J.Nozik, andM. Kaplan, Phys. Bev. 159, 273

(1967).C. E. Johnson, Proc. Phys. Soc. (London) 92, 748

(1967).~J. Chappert, R. B. Frankel, A. Misetich, and N. A.

Blum, Phys. Letters 28B, 406 (1969); Phys. Rev. 179,578 (1969).

32H. R. Leider and D. ¹ Pipkorn, Phys. Rev. 165,494 (1968).

33F. S. Ham, Phys. Bev. 160, 328 (1967).4B. M. Housley and U. Gonser, Phys. Rev. 171, 480

(1968).

PHYSICA L RE VIE W B VOLUME 3, NUMBER 3 1 FEBRUARY 1971

Time-Reversal Experiments in Dipolar-Coupled Spin Systems

W-K. Rhim, A. Pines, and J. S. WaughDepartment of Chemistry and Research Laboratory of Electronics, Massachusetts Institute of

Technology, Cambridge, Massachusetts 02139(Received 3 September 1970)

By applying a suitable sequence of strong rf fields, a system of dipolar-coupled nuclearspins can be made to behave as though the sign of the dipolar Hamiltonian had been reversed.The system then appears to develop backward in time, and states of nonequilibrium magnetiza-tion can be recovered in systems which would superficially appear to have decayed to equilib-rium. This behavior is consistent with dynamical and thermodynamical principles, but showsthat the spin-temperature hypothesis must be employed with caution. The theory of the time-reversal phenomenon is discussed, including the practical limitations on the accuracy withwhich it can be achieved. Various echo experiments in the laboratory and in the rotatingframe are reported. The application of repeated time reversals to the problem of high-reso-lution NMB in solids is discussed.

I. INTRODUCTION

One's first impression on seeing a spin echotends to be of having witnessed a spontaneousfluctuation of an apparently disordered system intoan ordered (magnetized) state. Actually, of course,the system was by no means as disordered as itseemed to be: It had to be prepared from a mag-netized state in a special way, such that its micro-scopic dynamical equations guaranteed a return to

the magnetized state; and the name "echo" of courseexpresses just this fact. The importance of adynamical, as opposed to thermodynamical, inter-pretation is particularly transparent in the case ofthe Hahn echo. ' There the spin Hamiltonian isinhomogeneous, i. e. , it represents a sum overuncoupled spins or isochromats interacting withfixed local fields. One is not dealing with a "many-body" system at all, and the formation of an echois easily understood by superposing the quantum-

TIME- REVERSAL EXPERIMENTS IN DIPOLAR-COUPLED. . . 685

pg(0) = 1+yHOI, /kT, (2)

appropriate to the high-temperature approximationand to the removal of Z- (2I+1)". A 90' pulse of res-onant rf field is now applied to bring the magnetiza-tion along the y direction:

p, (0) = 1+yHOI, /0T, .

The magnetization y(I, ) now precesses about Ho

{because [p„(0),X,] 40) and decays to zero {because[p,(0), 3C&,t, ]o0], where $C„tis the Hamiltonian ofthe spin-spin interactions. If the system approachesinternal equilibrium, it is then to be described by

p(~) =1 —3C,/kT, . (4)

Since Zeeman energy must be conserved under thetime-independent Hamiltonian acting during the

decay,

E( ) =E(0) = Tr[p, (0)X,]= 0,one must set the spin temperature T, = ~ in (4).Thus p(~)=1. lt is now clear that no echo phe-

mechanical (or classical) motion of individuallyprecessing magnetic moments. The same is trueof quadrupole echoes, ' though a purely classicalpicture is not quite so convenient there, and alsoof photon echoes, whose interpretation follows thesame lines as the Hahn echo if a fictitious spin issuitably introduced.

One feels intuitively that the situation ought tobe different in the case of interacting particles,such as the nuclear spins in a rigid solid. There"spin diffusion" prevents one's using pictures basedon isochromats moving in local fields. Moreover,there is no known way to integrate the microscopicdynamical equations even approximately over timesmuch longer than T~, the time which characterizesthe decay of transverse magnetization from an ini-tially coherent state. Under these circumstancesrecourse is made to the "spin-temperature" hy-pothesis, ' ' which asserts that the system ap-proaches a state which is adequately described asone of internal equilibrium (sometimes called semi-equilibrium, to emphasize that the spin system canbe considered isolated from the surrounding latticeonly for a certain time Ty usually much greaterthan Tz).

A particularly simple internal equilibrium stateis obtained by first polarizing the system for along time in a very strong external field 00»Hy„,where yH, «- T2'. The state of the system is thenassumed to belong to a canonical ensemble char-acterized by the lattice temperature T, :

& (0) (1/Z)-3! /Rr

Rg ———yHOI, +O(yH„,) .

We shall henceforth speak in terms of the reduceddensity matrix

nomena can be evoked by applying any externalfields: The Hamiltonian representing their inter-actions with the system commutes trivially withp(").

To be sure, some echo phenomena are known insolids. "'2 However, these echoes can be evokedonly by interrupting the development of p„(0)beforethe system has had an opportunity to approach thepostulated equilibrium state, and they are weakerthe longer that decay is permitted to proceed be-fore interruption.

In this paper, we shall describe a form of echo"which can be made to occur even after p„(0)hasbeen allowed to develop for a time much greaterthan T2. Its occurrence shows that the spin-tem-perature hypothesis in the simple form employedabove is not correct. Of course, no violation ofthe second law of thermodynamics is implied, andthe spin-temperature hypothesis retains its utilityin describing most NMR experiments in solids.We shall return to this point later.

Briefly, the experiments depend on the fact thatthe time development of the system is dynamicallyreversible in a microscopic sense, whereas it maybehave irreversibly on a thermodynamic scale.The apparent conflict between these manifestationsis sometimes called Loschmidt's paradox. ' Weshall find it useful to introduce the concept of a"Loschmidt demon" who is able to reverse thedynamical behavior of a system, thereby retracinga path which seemed to be irreversible. TheLoschmidt demon differs from the familiar Maxwelldemon in that he accomplishes the reversal bychanging the sign of the system Hamiltonian ratherthan by exercising control over microscopic dyna-mical variables. Since the state of the systemobeys the Schr'odinger equation

4(f) =e '"0(0),such a sign reversal is clearly equivalent to areversal of the time coordinate.

In Sec. II, we show how this is done, and dispelany doubts about its conflict with thermodynamicprinciples. In Sec. III, we give the results ofseveral experiments. The theory is expanded inSec. IV, and Sec. V describes an important exten-sion and application of this phenomenon.

II. SIMPLE TREATMENT

A. Theory

The spin Hamiltonian in which we will be mainlyinterested is the dipole-dipole coupling X~ in arigid lattice of like spins I„or,more precisely,the truncated form of this interaction X„appro-priate to the reference frame R which rotates atthe spectrometer frequency about a very strongZeeman field Ho= (uo/y in the z direction. We have

RHIM, PINES, AND WAUGH

K~=EZ b)I (Iq ~ II —3I,(I,I),f&j (6)

b(I = y Ay(~IP2(cos8)I)

The Loschmidt demon will convert &~ into

kÄ, with k a negative number, so that the systemretraces its previous normal dipolar developmentand departs from what appears to be a state ofequilibrium, as mentioned in Sec. I. This con-version cannot be made in a literal sense, but itseffects can be simulated through the agency ofsuitable external forces. Suppose a strong rffield (2',/y) cos~t is applied in the x directionwith l&0- + I «&0. The Hamiltonian in the ro-tating frame now includes in addition to (6), theZeeman effect due to the field

[~,T+(~D- (u)%]/y.

As Redfield has shown, it is convenient to viewthe system from the vantage point of the doubly(tilted) rotating frame (DTR), defined by the trans-formations

field +,/y for a time tR is more complicated:

pR(t, +tR) = T 'D exp[-itR'fC~P2 (cos))]DT PR{t,)

x T D exp[2tR JC~P2 (cos()]DT . (l 1)

It can be simplified by modifying the experiment in

the following two ways:(a) Make the duration of the burst by a resonant num-

ber of periods of the precession about &, : &, t~ = 2nm,

then D=D ' in (11).(b) Immediately precede the curst by a resonant

rf pulse chosen to rotate any magnetization presentthrough an angle $ about the y axis of the rotatingframe (i. e. , the rf carrier in the pulse is in phasequadrature to that in the burst), and terminate itby a similar pulse rotation through —$. The effectsof these pulses can be represented by e" ~= T"acting on pR which annihilate the transformationsT, T in (11).

The effect of this treatment is then simply

PR(t& + ts) = exp[ —2ts X&P2 (Cos) ) ]PR(tl)

pnrR= DTRp„„R(DTR)'1 xexp[itRZ2P2 (cos()], (12)

9PDTR 2[ DTR 1 PDTR] (6)

with

%Dr R = 3C„P2(cos) ) +X„,,JC„,= 2 sin] cos& QQ b, I

&& [(I,~ I,I +I,& I,I) e ' "~'+ c. c. ]

—-',

sin'&ZAN

b&I(I.,I,Ie ""&'+c.c. ) .

(10)

When ~, is large {&u, T2» 1), the oscillatory partX„,can be removed for purposes of calculating thesecular development of the system, just as onetruncates the ordinary rotating frame HamiltonianK„to obtain (6) when ~2T2»1. We will discussthis step more fully later. Assuming it to be valid,we see that (9) has the desired form k%2, with kadjustable between 1 and —

& at the experimenter'sdiscretion.

Remember, however, that this result refers toobservations made in the DTR frame, and does notby itself embody a means of reversing the timedevelopment of p~ in the rotating frame. In fact, ,the full effect on p~ of applying a burst of effective

T = e v; tan) = &d&/((uD —(0),2I2 . ~ [~2 (~ ~)2]&/2

In this frame the density matrix obeys the Liouvilleequation

and the Loschmidt demon has been realized if we

choose $ such that P2(cos)) &0. Just this procedurefor interconversion between reference frames hasbeen employed by Jeener et al. "but not for the samepurpose. If it is to be used for time reversal overreasonably long times it suffers from a severeprac-tical limitation: Any appreciable inhomogeneity in

the applied field, especially the rf field, will resultin failure to sati Iy con.dition (a) above in all partsof the sample. Ti ~ result is a quick destruction of

the coherence of the inverse time development and

failure of the experiment. This difficulty can becircumvented by adding the further condition:(c) Periodically reverse the direction of &u, (i. e. ,the rf carrier phase and the offset from resonance,MD —(u). By this means the dephasing of isochromatscaused by field inhomogeneity is prevented from ac-cumulating. (There are theoretical benefits to begained by this procedure even if the fields are ho-

mogeneous, as we will see later. ) The reversalshould be made as often as possible, but at integralmultiples of the period expressed in (10). For the

general case, a re'. ersal could be made every22'/~, sec. For the special case of exact resonance($ = 2 v), the first term of (10) vanishes, and the

reversal can be made twice as often if desired,i. e. , the burst consists of a train of contiguousphase-alternated 180' pulses. From now on werestrict ourselves to the exact resonance case($ = —,

'v), as this was the one adopted for the ma-

jority of our experiments,In summary, the free dipolar time development

of the spin system which occurred at a time t canbe reversed by applying for a time 2t a special

TIME-REVERSAL EXPERIMENTS IN DIPOLAR-COUPLED. . . 687

sequence consisting of a pair of 90' pulses alongthe+y and -y axes of the rotating frame, en-closing an even number of contiguous n, pulseswhich alternate between the +x and -x axes: Cal-ling this sequence B(f~) we have the equivalence

e+Ixgtg/2 B(f )0

B(t~) is depicted in Fig. 1(a). All this evidentlyholds for any interaction with the same transforma-tion properties as K„.We now present a simplepictoral representation of these equations and applyit to a well-known experiment.

B. Simple Picture

The normal inhomogeneous spin echo' is veryconveniently described in terms of the evolution ofspin isochromats which precess at different fre-quencies about an external inhomogeneous mag-netic field. The behavior of dipolar-coupled spins,on the other hand, is normally not amenable to sucha simple "hand-waving" description. However,even for these systems, the simple picture of in-homogeneously perturbed noninteracting spins issometimes valuable, and if employed carefully,allows a simple understanding of many experimentsas well as facilitating further extensions and pre-dictions. "

We describe here the use of such a model whichis appropriate for some of our experiments and

(nor „(n

90'te

~ 90 y

B(t,)

{b)

{c)

90x

ts

90x

ts

L Jrte4

ts te te —,- 'e ~2

90„e„ e„ e„(e)

B B 8 ~

B2 s(4 2

te, j I tea =I I"ts(ts,2

FIG. 1. Pulse sequences used for various experiments. &(t&)in (a) is the time-reversing sequence referred to in the text,and is depicted as B in the other pulse sequences. ~„or{&)~ means a 0 pulse along the p axis of the rotating frame.

utilizes Eqs. (7)-(9) with $ = 90' as a guideline forits construction. The discussion takes place in thereference frame rotating at frequency &0 about thez axis. With no rf applied we adopt the well-knownpicture of spin isochromats in an inhomogeneousfield in the z direction. There is a distributionbetween fast to slow isochromats (labeled 1 and 2

in Fig. 2) which precess, respectively, clockwiseand anticlockwise about the z axis. During an rfburst along the x axis we use a similar picturewith the following differences: The inhomogeneousfield is now along the x axis and the sense of pre-cessionof the isochromats about this axis is op-posite to that about z. Further, the rate of preces-sion is reduced by a factor of 2. 90 pulses aretreated as usual, whereas the system is invariantto 180' pulses, These stipulations incorporatepictorially the main features of Eqs. (7)-(9): (i)The rf burst converts the effective Hamiltonianfrom X„in the rotating frame to —&X„in the tiltedrotating frame [taking account of (iii)]. (ii) 90 ypulses transform the density matrix to and fromthe above frames. (iii) Our time scale during theburst is limited to integral cycle times of the rffield.

Figure 2 demonstrates the application of thissimple picture to the echo experiment describedpreviously' and referred to again in Sec. III. Wesee that most of the qualitative features of the ex-periment are incorporated neatly in this descrip-tion, but we emphasize that this should not be takentoo seriously and serves only as a useful pictorialaccessory when used in conjunction with the appro-priate equations. The reader will no doubt appre-ciate the trivial extension to multiple echoes de-scribed briefly in Secs. III and V.

C. Remarks on Statistical Mechanics

For an echo of magnitude M0 to occur at timet =0 in an isolated system of energy E0, the rep-resentative point of the system must lie in someregion S(0) of the phase space. This region ismuch smaller than the total constant-energy sur-face $0, expressing the fact that the magnetizedstate is one of low entropy. Every possible rep-resentative point in S (0) arrived there from a se-quence of earlier locations, completely determinedby the system Hamiltonian through the microscopicequations of motion. That is, a knowledge of $C

specifies a sequence of mappings of S(0) onto theprecursor regions S (-f). The problem confrontingan experimenter who wishes to produce an echo attime I'=0 is how to prepare a system which isknownto be in S(-to).

This is a more subtle task than to prepare thesystem in S (0). The latter can be accomplished,for example, by bringing the spins to equilibriumwith a lattice in a strong field and then applying a

RHIM, PINE S, AND mAUGH

(e)

X

t =5t0

macroscopic variables alone but requires a knowl-

edge of a/l the dynamical variables and how theychange over a time to under the influence of theequations of motion. The necessary knowledge

and ability of selection characterize the Maxwell

demon, who violates the second law of thermo-dynamics in playing his role.

The Loschmidt demon needs no such microscopiccontrol. Beglllnlng with a system prepared ln S (0)as described above, he changes the sign of theHamiltonian function of the system. After a time

to, & is returned to its normal form. The changeis to be accomplished by external means which

make no reference to the dynamical state of the

system. An analogy is useful here to a gas ofmolecules interacting according to a potential U(r):

lx

FIG. 2. Rotating-reference-frame description of the

spin echo in Fig. 4, using the simple picture in Sec. II.The spin system in equilibrium (a) is irradiated at timet =0 with a 90' pulse along the g direction (90„)nutating

the magnetization into the x-y plane (b). The magnetiza-tion decays with 'Vast" spine precessing clockwise and"slow" spine anticlockwise about the g axis, and we seea normal free induction decay (c}. At time t=tD a 90~

pulse brings the dephased spina into the y-z plane (d) and

a rf burst is applied along the x axis. According to ourpicture the spine now precess about the g axis with "slow"and "fast" spine having interchanged their sense of pre-cession (e), and refocus at time t=3t~ (f). This producesthe rotary echo described in Sec. III. The spine now

dephase about the x axis (g) and at t = 5ta the rf burst isterminated with a 90~ pulse which returns the dephasedspine to the x-y plane (h). %e now have situation (c)except that the "fast" and "slow" spins are interchangedin position (i) and the spine rephase leading to a spon-taneous echo (j) observed in Fig. 4. Note that the time-reversing rf sequence in (d) through (h) is analogous, forthe case of homogeneous dipolar-coupling, to the 180'refocusing pulse employed for the normal jnhomogeneous

spin echo.

90' pulse: It requires only the ability to establishspecified values of the macroscopic variables Mand E, since S (0) includes by definition all dy-namical states consistent with these values.S (- to), on the other hand, cannot be specified by

A second kind of Maxwell demon exploits the

apparent possibility of recalling the previous historyof the gas by reversing the signs of all p; at someinstant. It might seem that such a reversal would

require a knowledge of the dynamical state at thatinstant (although Hahn has pointed out in discussiona case of two dimensions in which it apparentlydoes not). Note also that the sign of 3C is not

changed. The Loschmidt demon, as we define him,might instead change the signs of all the m& and

U&, . Such an action certainly requires no knowledge

of the dynamical state. It implies a degree of con-trol over the nature of the system which may bedifficult to credit in the case of gas dynamics but

which clearly can be exercised, as we have seen,in certain restricted spin systems. The questionof the class of Hamiltonians for which a Loschmidtdemon is in principle realizable is an interestingbut unsolved one.

How does the spin-temperature hypothesis fitinto this discussion'P In its strong form it would

claim that a distribution p(0) uniformly occupyingS(0) at the beginning of a Bloch decay would cometo fill the much larger region So uniformly afterseveral times T2. However, Liouville's theoremtells us that the total volume of S(t) is conserved.The fact is that S (t) may come to fill ho in acoarse-grained sense only. S(0) is relatively compact,but S(t &Ta) is complicated and has a large surface-to-volume ratio, as it were, especially if forcesare present between the particles of the system, '

Thus, fox many Puxjoses the distribution may betreated as though it filled So uniformly. Howeverone must be on guard against performing manip-ulations which inadvertently exploit the dynamicalhistory of the system to bring S (t) into a conditioncharacterized by "nonequilibrium" values of themacroscopic observables. It is worth noting thatsome of the experiments of Jeener et a/. , "which

TIME- REVERSAL EXPERIMENTS IN DIPOLAR- COUPLED. . . 689

seemed to verify the spin-temperature hypothesis,might well have done the contrary, had their ex-perimental conditions been adjusted somewhatdiff erently.

Finally, we remind the reader that our experi-ments on solids do not actually involve a reversalin sign of 3C: During the rf burst the representativepoint aPPears to move under a Hamiltonian —2K„from the viewpoint of a phase space which is ac-celerated with respect to the "ordinary" one, thenet effect upon conclusion of the time-reversingsequence being to bring the system into S (- tp).The language one uses to describe the motion ofthe system depends, as always, on the representa-tion in which the motion is viewed.

III. EXPERIMENTS

In this section we describe some experimentsbased on the properties of the time -reversing sequencedescribed in Sec. II. All experiments were per-formed at the exact ' F resonance frequency of

54 MHz on a single crystal of CaF, oriented ap-proximately along the (110) direction. The peak

H, field was -100 G and our pulses had rise and

fall times of - 150 nsec. The apparatus is to bedescribed in more detail elsewhere.

A trivial experiment is one in which B(ts) isapplied to the spin system initially in equilibriumwith the lattice in a field Hp. B(fs) reverses the(nonexistent) time development of the z componentof the magnetization, leaving the system in itsoriginal state.

A more interesting case is the reversal of thetime development of the x, y components. Assume,therefore, that we apply the pulse sequence ofFig. 1(b) with n = 1, consisting of a time-reversing se-quence preceded by a 90 „pulse. (Note that this pulse isimmediately negated by the leading 90, pulse ofthe time-reversing sequence. In practice we simplyomitboth fromthe experiment. ) During the burst thesystem undergoes a reversed dipolar developmentwith Hamiltonian —&3C~ corresponding to a normalrotary free induction decay at resonance, treatedfor example by Goldbur g and Lee. ' At time t~the system reverts to normal dipolar developmentand should return to the initial state at 7= &t& asindicated by the appearance of an echo. This se-quence is formed by stages (f)-(j) of Fig. 2. Theresult of the experiment is shown in Fig. 3. Theopen circles depict the magnetization along the yaxis during the burst and were obtained by ob-serving the signal immediately following the re-ceiver dead time of about 2 psec without the final90, pulse, while the number of (x:, —x) pulse pairswas varied. The oscillations were introduced de-liberately for this part of the data by misadjustingthe pulse widths during the burst and having xpulses of nutation angle p —& and —x pulses of

2Q

1.0

10(0

-2.0—

I

50I

100t paSeC

I

150I

200 250

FIG. 3. Transient magnetization of F nuclei in CaF2observed along the y axis of the rotating frame using thepulse sequence of Fig. 1(b). The closed circles depictthe echo obtained in this experiment with a rf burst of-120psec. In order to see the evolution of the magne-tizationduring theburst the final 90~ pulse was omittedand the number of pairs of (g, -~) pulses was varied,leading to the data recorded as open circles. The os-cillations were introduced deliberately for this part ofthe experiment as explained in the text, and the envelopeis drawn by inspection. Note the characteristic beatstructure in the rotary free induction decay.

nutation angle m+ with &=25 . This introducesa slow accumulative precession of frequency&,o'/m which leads to the observed oscillations,and the free induction decay envelope can be ex-tracted simply by inspection as indicated in thefigure. This is necessary because any sloweroscillations due to small misadjustments, whichare almost unavoidable, would make extraction ofthe envelope a much less certain proposition. Thefull circles show the observed echo whose maxi-mum comes at &t~, as expected. The shape of.

the rotary free induction decay is the same as thatof the normal free induction decay except stretchedby a factor of 2, as observed previously. '

In the experiment of Fig. 4 (which is the onedescribed by the simple picture of Fig. 2), wefirst apply a SO„pulse after which we see a normalfree induction decay. The time-reversing sequencewithn = 1 is then applied as in Fig. 1(c) such thatt~ = 41~, and at time 7 = tD we see the expectedecho. Figure 5 shows the initial decay (fullcir-cles) and also the magnetization during the burst(open circles) with the data collected as before bymisadjusting the m pulse widths and omitting thefinal 90, pulse. The envelope shows a rotaryspin echo (which is of course entirely differentin character from the inhomogeneous rotary spinecho observed by Solomon ). The width isstretched by a factor of 2 and we see that the

690 RHIM, PINE S, AND WAUGH

FIG. 4. Oscilloscope trace of the F transient magnetiza-19

tion in CaF2 using the pulse sequence of Fig. 1(c). Followingthe initial free induction decay a time-reversing burst(B) is applied at the noise-free section of the trace andan echo then appears, ta ——88 psec, tz ——350 psec. Thisis the experiment described by the simple picture ofFig. 2.

shape and beat structure are again preserved byour phase-alternation technique.

In this experiment, we have obtained substantialechoes for burst lengths of up to 650 p. sec, meaninga recovery of the magnetization after about 1 msecfrom the. first 90' pulse. The reasons for decayof the echo amplitude are discussed later and areprobably dominated by machine instabilities anderrors. Figure 6 demonstrates that the same ef-fect may be obtained by using a train of narrow

phase-alternated 90„pulses in the time -reversingsequence instead of the contiguous n~ pulses, sincethe secular part of the Hamiltonian is the same forboth. A nice featur e of this experiment is the directobservation of the rotary echo during the pulse train.The efficiency of this pulse train for longerbursts,however, is lower than that of l(a) if comparedusing the same average rf power. This point isbrought up again in Sec. V.

t~ = 2tD corresponds to the peak of the rotaryecho and Fig. 7 shows the signal observed in anexperiment where the burst is terminated soonafter this point without the final 90, pulse. Wesee a large recovery of the magnetization with

to = 133 p, sec, which is substantially larger than

Ta. (The negative signal is due to a peak negativevalue in the oscillation caused by a pulse widthmisadjustment or by change of the nutation angledue to rf power droop during the burst. )

Figure 8 shows that if we apply an additional time-reversing sequence at 7 = 2t D another echo appears,and this may of course be extended to multiple echoesin a manner analogous to that of Carr and Pur-cell. ' The efficiency with which these echoesare summoned suggests that such echo trainsmight form the basis of a powerful approach tothe problem of line narrowing in solids. This istreated briefly in Sec. V.

As a final demonstration of a simple application ofthe time-reversing sequence, let us look at the experi-ment of Fig. 9which employs the pulse sequence inFig. 1(e)with 8 =

ceo�

'. The first 90,pulse following thefree induction decay produces the well-known "solid

40

w 2.0—OI-

I,O-CL

0

z2 -Io—

cc

~ ii [ ic ~ ~ illI IL " pcccci 4 ~ ig I

P cI P

ucINI'Pyass iiQ' il I ii iiai i cII~IIII ) / )

-2,0—

-5.0—

0I

50I

IOO

I

l50I

200I

250I

500

c.

c'

i

t psec

FIG. 5. Experiment designed to show the evolution ofthe magnetization during the rf burst in an experimentlike that of Fig. 4. This is done as in Fig. 3 by omittingthe final 90 ~ pulse from the pulse train which followsthe free induction decay (closed circles) and plotting themagnetization immediately following the burst vs burstlength (open circles). The oscillations were again intro-duced deliberately as explained in the text. The en-velope shows the "rotary echo" and note again the in-tact beat structure characteristic of the ~~F signal inCaF2.

, g':"'"'.

tIj jI ggyp~:", Iw:Ã.&:::~

. .c'c-::::-::--:,:,:::::~FIG. 6. Pulsedversionof the experiment in I igs. 4and 5.

In this experiment, the time-reversing sequence (B) in pulsesequence j.(c) was modified by replacing the contiguous(n&) „and (n&) „pulses with a train of alternate 90„and90 ~ pulses spaced 5 @sec apart. In addition to the freeinduction decay and echo, observed as in Fig. 4, we seethe "rotary echo "of Fig. 5 during the "windows" in thepulse train.

TIME-REVERSAL EXPERIMENTS IN DIPOLAR-COUPLED. . . 691

ogous reversal of Jeener and Broeckaert's pulsesequence' showing that their postulated dipolarstate is not an adequate description of the systemfor this type of experiment.

IV. NONIDEALITIES

A. General Framework

FIG. 7. Oscilloscope trace of magnetization in anexperiment like that of Fig. 4. In this case, the time-revers-ing sequence terminated near the peak of the rotary echo ofFig. 5 and we see a normal free induction decay. Theparameters are tD=133 psec, t~=345 @sec. The factthat the second signal has a negative value is explainedin the text.

echo. " By sequentially using the properties ofpp11

the time-reversing sequence we see that the first halfof the bur stbrings the system back to its initial state,and the secondhalf produces arotary free inductiondecay and corresponding "rotary solid echo."Thus onterminating the burst we see a negative time de-velopment solid echo and the final 90 pulse bringsthe system back again to the initial state. In thisway the effects of any number of pulses may bereversed. This experiment vividly demonstratesthe more efficient nature of our echoes as comparedto the solid echoes. This matter is referred toagainin Secs. IV and V. For 8 =45', we obtain an anal-

The ideal treatment in Sec. IIwas based on the as-sumption that the effect of a time -reversing sequencecould be represented by replacing &„with ——,'&~„astheeffective Hamiltonian during the burst (for the case$ = —,m). Here we investigate the validity of thisassumption and discuss the effect and extent of

various nonidealities encountered in practice.Vfe begin by writing the time development during

the burst as

pR(ts) = Us (ts) pR(o) Up( tB),

Us(ts) =exp[-its( —,'5C~+ XV-)],where V is some correction to K»„whose matrixelements are of order T2', and X is a small param-eter. No generality is lost at this stage by insistingthat V be time independent; it may depend paramet-rically on t~ but in most cases to be discussed itwill not have even that indirect time dependence.The nature and origin of &V will be discussed later;at present we simply investigate the consequencesof (13).

It is easily shown that"R t I2 t~

Us(ts) =e' "&' Te P[x-iX f s V(t)dt], (14)

?

FIG. 8. I3ouble-echo version of the experiment in

Fig. 4 using pulse sequence 1(d) with tz —-180 @sec. This

shows that an extension of our echoes analogous to that

of Carr and Purcell is possible.

FIG. 9. Time reversal of a two-pulse experiment,using the pulse sequence in Fig. 1(e) with 8 = 90'. Thefirst two pulses bring about a free induction decay and"solid echo. " The following rf burst reverses this totalsequence and then propagates it further backwards intime leading to the ensuing "mirror image "of the "solid-echo "experiment when the burst is terminated. Thenegative value of the "mirror image" is explained in thetext. For this experiment 2t~~+t~2 ——342 @sec, corre-sponding to the noise-free portion of the trace.

692 RHIM, PINE S, AND WAUGH

where & is a time-ordering operator and

V(t) e Ix~t/2 V fÃ~t/2 (15)

For the condition

(16)

(fa" ~ 1 fk =(fl ~ a" f]=(fa"f], (23)

where the last is the bracket introduced by Waughand Wang. Using this and expanding (21) in r andXt& we obtain

(24)

where M„are the moments of the echo, given by

M„=2 i" (fR„~V f) .0~0

B. Lifetime of Echo

(25)

The time w=0 corresponds to the expected posi-tion of the echo maximum from the discussion inSec. II. Substituting r= 0 in (24) and (25) we obtain

(26)

the odd powers all vanishing, as is well known.Assuming now that ~~V)[- Tz and remembering

condition (16) we see that (26) expresses a recoveryof the magnetization with an attenuation of order—,(Xts/T2) . Thus we expect the magnetization torecover to an appreciable extent for burst lengths

we may expand the exponential in (14) to first order:

Us(ts) =e' " exp[-iX f s V(t)dt] . (17)

Now we consider the general situation in which theburst is preceded and followed by unperturbed dipo-lar developments of duration t& and t2. The totalpropagator is

-&t, me0U(t, +ta+ts) =e "2~i Us(ts) e '" ~ . (»)

Combining this with (17) and rearranging a little wefind

U(t, +t, +t, )=U( ', t, +r)=e--"nIe '"'s',where 7 =t&+t2- &t~ and V is given by

V=(1/ts) f s ' V(t)dt . (20)

If the initial magnetization of the system was alongthe p axis in the rotating frame, (I„(0)),then itsfinal value is

(I„(2ts+7'))=Tr(e ' 4e ' 's I~e 's e ' &I~). (21)

We now introduce the notation

( f

a" ~ bf )= Tr ([a[a. . . f a, I„]].. .j [b [b. . . [b, I„]1.. .]),

(22)where a and b appear n and m times, and

that satisfy condition (16) for fixed T2 and &.

order to estimate this, we now inquire into thesources and nature of V and the size of X. Thefirst place it occurs to us to look is at the trunca-tion of X»R carried out in Sec. II. Since 3C»R of(9) is periodic and the burst contains an integralnumber of cycles, its effects can be replaced bythose of an average Hamiltonian

(n)DTR ~ DTR

n=0(27)

whose terms are defined by a Magnus expansion overa single cycle of $C„,(t). For Eq. (10) we have thecycle time t, = 2m/&, , and thus

(0) & 0+DTR 2+d ' (28)

Cutting off (27) at this point corresponds to thewell-known procedure of "truncation. " The nextterm does not vanish: At this point remember wehave not used quite the Hamiltonian of Eq. (9) sincewe have periodically reversed the direction of H, .Under these conditions it is easily verified that

DTR(l)

thus leaving g&r'n as the leading correction term.An estimate of this factor shows that it is of theorder of —, (t,/2mT3) smaller than R„.Thus for thiscase we identify ~V with X„(,' and ~ is of order6 (t,/2&Tz), which for our experimental conditionis about 10 . Condition (16) is thus satisfied fort~ of many msec, and taking only the nonsecularterms into account we might expect the echo to livethis long. In fact, we have obtained in the echo ex-periment of Sec. III a more than 98%%uo recovery ofthe magnetization for ts up to 350 psec and a 75Pq

recovery for tJ3 of 500 p, sec. So some other mech-anism could be contributing to the destructionof the echo.

We now turn to some other errors which stemfrom practical sources. The treatment up till now

has assumed that the time -reversing sequence containsa series of continguous ideal (nm)„and (nm), pulses.Evidently, this cannot be realized in practice becauseof the finite rise and fall times of the pulses whichlead to windows" in the continuous rf irradiation,since the carrier phase of the two pulses differsby m. Thus the time-reversing sequence is betterrepresented in practice by the pulse sequence inFig. 10.

Another source of error is the fact that the pulsesmay not be exactly of nutation angle nm, as this isexceedingly difficult to adjust. Accordingly, inFig. 10 we have indicated this by writing the angles

I las nm+ c and nn+ c, where e and c are small errorangles. We assume }e—e }«}e}for reasons thatwill become clear later. We find on calculatingthe average Hamiltonian XDT'R in this case and takingonly the first order in e and 5, where 1 —5

TIME-REVERSAL EXPERIMENTS IN DIPOLAR-COUPLED. . . 693

(r}w+ s) (f}m+s')

90„~ ~

~ ~

90y

~ ~

~ ~

~ ~ ~ ~

~ ~ ~ ~ ~~ ~ ~~ I ~

~ ~~ ~ ~

~ ~

~ ~ ~

~ ~ ~

I 'W1 I I-'W2

a ~

1

~ ~

~ ~

~ I ~ ~

I~ ~

~ ~

~ ~ ~

~ ~

~ ~ ~

~ t ~~ ~ ~

~ ~ ~

~ ~ ~

FIG. 10. Nonideal time-reversing sequence discussedin Sec. IV.

=(f + t, )/f, :

3CDTR — ~ Xq+2 6+ — bi; I„]I„)—I,)I,~ .$f lT fQ

(30}

We see that XDT'R is of the form ——,'X„+XV,whereX - 5+ e/nm and II VII - IIBC~~ II. For n = 1 we have with

our apparatus 5-0. 1. With e =0 this would leadto very short lived echoes since X -5. However,we see from (30) that by advertently adjusting thepulse widths away from nv such that 5+ e/nm-0,this factor can be eliminated to a certain extent.The adjustment is quite critical and requires ~ & 0as we have indeed observed. A factor limiting thisadjustment is the H, inhomogeneity which normallyamounts to several percent. Practically, thismeans that e is not constant over the sample andthus the adjustment 5+ e/nv 0cannot -be madesimultaneously over the whole sample. The Hy

inhomogeneity probably makes a significant contri-bution to the destruction of the echo for long bursts.

The effect of e and 5 becomes less serious as n

is made larger since e/nv and 5 both decrease, but

for large n this may be offset by a harmful increasein phase-switching cycle time.

Other machine errors not treated explicitly hereprobably make some additional contribution to thedestruction of the echo. These include misadjust-ment of phases, power droop in the pulse trans-mitter, phase transients, ~0 etc.

C. Shape of Echo

We now return to Eq. (24) and investigate(I (2t~+ 7)) for 7 40, the shape of the echo about itsexpected maximum. An examination of (25) showsthat the first term in the expansion is just the cor-responding normal FID moment. The higher termsmake a small contribution if condition (16) is fulfilledand lead only to a slight distortion of the echo line shapeas compared to the free induction decay. Thissuggests a useful new means for obtaining momentsof the free induction decay when the initial transientis obscured by receiver dead time; we simply pro-duce an echo with X small enough so that (16) isfulfilled to the desired accuracy with some partic-

ular dead time f, [we need ts &2t~ for pulse sequence1(b), and ts&4t, for 1(c)j. This method has oneimportant advantage over solid echoes si, ia In thelatter, assuming a dead time t„,the terms con-tributing to the distortion of the echo line shapehave some inherent fixed value dependent on powersof t„.In our case, we have the same distortion inpowers of Xt~, not t~, and this can be made arbi-trarily small at least for the t„ofpractical interest.Using pulse sequence l(b) we have measured someecho line shapes for different burst lengths. Theseare compared with the free induction decay in Fig.11 with no adjustable parameters except for normal-ization of the echo with t~= 110 p, sec. The agree-ment is very good within experimental error. Fort~= 110 p, sec, the onset of some distortion is clearlyvisible. Comparison with the first part of the freeinduction decay is of course not possible, but thetheory and other experimental evidence indicate noreason why this should not be good too. For f„&10p.sec, we need t~= 20 p, sec and this should give usa very accurate representation of the line shapeeven for moderate H, fields. In any case, the re-duction in peak echo amplitude is an indication ofthe amount of distortion to expect (notwithstandingthe remarks below) as we have indeed found in prac-tice.

An interesting feature of Eqs. (24) and (25) as wehave seen, is the fact that the first correction termto the peak echo amplitude is in (Xt~)', whereas thoseto the higher moments are in (Xts). This meansthat distortion of the line shape should occur morerapidly than destruction of the peak echo amplitudemight indicate. This is indeed evident in our ex-periments where for large t~ the echoes acquire a

o loI-

CLX

u 0.5fI)

LIJN

a 0.0

I I I I I I I

IO 20 30 40 50 60 70t psec

FIG. 11. Comparison of the echo line shapes of theexperiment in Fig. 3 for various burst lengths. Onlythe right-hand portion of each echo is depicted, exceptfor burst length 0 (free induction decay) where part of thesignal is obscured by receiver dead time. No parametersare involved except the normalization of data for tz —-110@sec, where the peak echo amplitude had decreased.

694 RHIM, PINES, AND WAUGH

where superscript 8 denotes the presence of theburst and XV is a correction to the average Hamil-tonian. It is easily verified that the total first cor-rection term vanishes, including the dipolar-chem-ical-shift cross term, so that XV can be associatedwith the second correction term and A. is of order~ (f, i2mT, )'.

The average Hamiltonian for the total —,f~ cycle is

(32)

The higher correction terms depend on the sym-metry of the circle. For the symmetric cycle inthe pulse sequence of Fig. 1(e), we find for thetotal first correction term

FIG. 12. Multiple-burst line-narrowing sequencefFig. 1(d) ] applied to the Fnuclei of Ca F2 with the magnetset slightly off-resonance. The cycle time 2t~ for thisexperiment was 136 @sec, and we observe a prolongeddecay with a scaled off-resonance beat. This multiple-echo phenomenon has also been used to observe chemicalshifts in solids in an analogous way.

K~ =Kg+K~0 (31)

During the burst the effective Hamiltonian is givenby

X "=—-'-X +K,'"+XV,z ——z y+

progressively narrow shape corresponding to alarger destruction of magnetization for ~40 than forthe echo peak.

V. APPLICATION TO LINE NARROWING

A. Multiple Bursts

We have discussed at length elsewhere" the pos-sibility of removing the effects of dipolar broaden-ing by repeated application of a suitably chosencycle of rf perturbations. These cycles have theproperty that the effective dipolar Hamiltonian insome suitable reference frame, while large, variesduring the cycle in such a. way that its net effectover a full cycle vanishes to some approximation.The present echo experiments clearly have the sameproperty over a cycle of duration —,'I;~: The systembehaves as though K„=K~ except during the burstswhere K~ =- ——,'K„to first order. Thus for a cycleof duration —,t~ the effective dipolar Hamiltonian is0 and if the cycle is repeated indefinitely as in Fig.1(e), the envelope of the magnetization sampled atintegral cycle times should generate a very longdecay corresponding to an effective line narrowing.

Consider the more interesting case in which theHamiltonian contains chemical shifts in addition tothe dipolar interaction and is given by (31). Re-moval of the large dipolar broadening is then a ne-cessity if we wish to observe the fine structure as-sociated with the chemical shifts. We have

K(1) 08 (33)

again including the dipolar-chemical-shift crossterm. The higher-order pure-dipolar correctionterms evidently vanish and thus all correctionterms are attenuated at least by A or 6, where thelatter is the magnitude of the largest chemicalshift. The only nonvanishing second-order term,for example, is found to be attenuated by -A~ fromthe magnitude of the usual corresponding pure-dipolar term. The normal requirement of our pre-vious pulse experiments, namely, that we have thecycle time t, T„is therefore not so stringent inthis experiment, and much larger cycle timesshould be tolerable if A and ~ are small, as wehave indeed observed. This is a good example ofthe exploitation of the properties of subcycles (inthis case during the burst) in an effort to increasethe efficiency of the full cycle.

Figure 12 shows a preliminary example of theapplication of this pulse sequence to the F nucleiof CaF2 with a cycle time of 136 JL(, sec and using360 x and —x pulses (n = 2). A prolonged decaywith an off-resonance beat scaled by 3 (3C,' ' =0),is observed: With this cycle time our previouspulse exper iments fail completely.

To summarize, we mention briefly several ap-parent advantages of this new method as comparedto some previous line-narrowing techniques. Theexperiment is essentially continuous wave, exceptfor the 90, pulses, with the magnetization samplingwindows forming part of the sequence. We there-fore require less power than in our pulsed exper-iments and do not have to contend to such an extentwith the adverse effects of finite pulse width. Fur-ther, the experiment does not require the additionalvideo pulses necessary in "magic angle" experi-ments. ' An additional advantage is the vanishingof the first-order cross correction term betweendipolar and chemical-shift interactions, and theattenuation in magnitude of the higher-order cor-rection terms. Most important we see that whilet~ may be of order T~ or more, the line-narrowing

TIME-REVERSAL EXPERIMENTS IN DIPOLAR-COUPLED. . . 695

We recall from above that the total first-ordercorrection term, including the dipolar-chemical-shift cross term, vanished both for the multiple-burst full cycle and the burst subcycle. This is aspecial case of a more general principle which ap-plies to any symmetric cycle. Adopting the notationin Eq. (15) of Ref. 29, a symmetric cycle is onefor which

x(t) =x(t, —t), (34)

where t, is the pertinent cycle time. It is easilyshown for such cycles that the first-order correc-tion term vanishes,

X"'=0. (35)

We see that our cycle and subcycle do indeed havethe requisite symmetry, leading to (35). Using (34),

efficiency is determined from (32) essentially bythe subcycle time inside the burst, which can bemade extremely small since no observation of themagnetization is necessary during this time.

Last, we mention that an analogous experimentemploying only 90' rf pulses can be done if the nm

pulses in the time-reversing sequence are replaced byatrainof 90 pulses along the +xdirections, butthatthis would not benefit from all the advantages men-tioned above.

B. Symmetric Cycles

other sequences with this property will doubtlessoccur to the reader, and a general approach to elim-ination of the first correction term is simplysymmetrization of the cycle within the constraintsimposed by the requirements of the average Hamil-tonian itself. A trivial example is our 4-pulsesequence. Preparing the system with a 90' pulsealong the y axis, it is

P, —(v —P„—v- P~ —2v P~ —-v- P „—7')„.It is easily verified that (34) holds over a cycle with

f, = 6v, and thus (35) is fulfilled. This sequence issimpler than a 6-pulse modification recently pro-posed by Mansfield, "

P, - (~- P „P, ~- P „-2~- P, —~- P,P„-7)„,and a 3-dimensional sequence proposed by Silber-szyc.

ACKNOWLEDGMENTS

Special thanks are due Dr. M. Mehring who

constructed the rf transmitter used in these ex-periments and from whose advice and assistancewe benefited greatly. We also acknowledge theassistance of M. G. Gibby who constructed a majorpart of the pulse spectrometer, the technical aidof S. Kaplan and J. D. Read, and some stimulatingdiscussions with J. D. Ellett, Jr. , R. G. Griffin,and J. M. Deutch. We thank Professor E. L. Hahnfor illuminating discussions of Loschmidt'sparadox.

*Work supported by the National Science Foundationand the National Institutes of Health.

E. L. Hahn, Phys. Rev. 80, 580 (1950).2H. Y. Carr and E. M. Purcell, Phys. Rev. 88, 415

(1952).3S. R. Hartmann and E. L. Hahn, Phys. Bev. 128,

2042 (1962).I. Solomon, Phys. Rev. 110, 61 (1958).M. Bloom, E. L. Hahn, and B. Herzog, Phys. Rev.

97, 1699 (1955).6I. D. Abella, N. A. Kurnit, and S. R. Hartmann, Phys.

Rev. 141, 391 (1966).7R. P. Feynman, F. L. Vernon, Jr. , and B. W. Hell-

warth, J. Appl. Phys. 28, 49 (1957).J. Jeener, Advances in Magnetic Resonance (Academic,

New York, 1968), Vol III.M. Goldman, Spin Temperature and Nuclear Magnetic

Resonance in Solids (Oxford U. P. , London, England,1970),

A. G. Redfield, Science 164, 1015 (1969).~~J. G. Powles and P. Mansfield, Phys. Letters 2,

58 (1962); J. G. Powles and J. H. Strange, Proc. Phys.Soc. (London) 82, 6 (1963).

~2E. D. Ostroff and J. S. Waugh, Phys. Rev. Letters16, 1097 (1966).

W-K. Rhim, A. Pines, and J. S. Waugh, Phys. Rev.Letters 25, 218 (1970).

B. C. Yolman, The Principles of Statistical Mechan-ics (Clarendon, London, England, 1962), p. 152 ff. ;G. E. Uhlenbeck and G. W. Ford, Lectures on Statisti-cal Mechanics (Am. Math. Soc. , Providence, R. I. ,

1963), Chap. 1.~W-K. Bhim, Ph. D. thesis, University of North

Carolina, 1969 (unpublished); H. Schneider and H.Schmiedel, Phys. Letters 30A, 298 (1969); W-K. Rhimand H. Kessemeier (unpublished).

6A. G. Redfield, Phys. Rev. 98, 1787 (1955).~ J. Jeener, R. DuBois, and P. Broekaert, Phys. Rev.

139, A1959 (1965).A. G. Anderson and S. B. Hartmann, Phys. Rev.

128, 2023 {1962).OJ. Jeener and P. Broekaert, Phys. Rev. 157, 232

{1967).J. D. Ellett et al. , Advances in Magnetic Resonance

(Academic, New York, 1971), Vol. V.W. I. Goldburg and M. Lee, Phys. Bev. Letters 11,

255 (1963).D. Barnaal and I. J. Lowe, Phys. Rev. Letters 11,

258 (1963).23M. Lee and W. I. Goldburg, Phys. Bev. 140, A1261

(1965).I. Solomon, Phys. Rev. Letters 2, 301 (1959).

'H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630(1954).

6R. M. Wilcox, J. Math. Phys. 8, 962 (1967).J. S. Waugh and C. H. Wang, Phys. Rev. 162,

209 (1967).A. Abragam, The Principles of Nuclear Magnetism

(Clarendon, London, England, 1961), p. 110 ff.29U. Haeberlen and J. S. Waugh, Phys. Bev. 175,

453 (1968).J. S. Waugh, L. M. Huber, and U. Haeberlen, Phys.

RHIM, PINE S, AND WAUGH

Bev. Letters 20, 180 (1968).'P. Mansfield, Phys. Letters 32A, 485 (1970).

W. Silberszyc (unpublished).

PHYSICAL BEVIEW 8 VOLUME 3, NUMBEB 3

Vacancy-Mn2+ Pair Spectra in Alkali Chlorides

Curt Hofer and R. R. SharmaDepartment of Physics, University of I/linois, Chicago, 1llinois 60680(Beceived 27 April 1970; revised manuscript received 23 October 1970)

1 FEBBUABY 1971

An attempt has been made to understand the fine-structure constants of the vacancy-associ-ated-Mn ' complexes corresponding to spectra III& and III2 observed by &atkins in NaCl andKCl. The point-charge contributions arising from the mechanisms which were not consideredpreviously are included in our treatment. The role of overlap and/or covalency effects areanalyzed by calculations. Assuming the lattice distortions in accordance with the availablecalculations, the theoretical values of the fine-structure constants are compared with the ex-perimental values. For speetraIII& andIII2 the DandEvalues are smaller than the experimentalvalues but they agree in sign if one considers point-charge and overlap effects. The inclusionof covalency effects brings, in general, the calculated values into better agreement with ex-periment. Some conclusions have also been drawn regarding the distortion of the lattice inorder to improve the calculated values.

I. INTRODUCTION

Many theoretical and experimental attempts havebeen made in alkali halides to understand the inter-actions between a divalent ion impurity and a va-cancy. The vacancy has been found to pair off withthe divalent impurity to form vacancy-impuritypair complexes. It has been established by catkin'8electron paramagnetic resonance spectra' of Mn ~'-

doped alkali chlorides that two different kinds ofyair complexes can exist in such systems. Onecomplex is formed when the vacancy is located atthe cation site nearest to the magnetic ion, thespectra corresponding to which is designated byIII y

The other eompl ex is formed when the boundvacancy is at the next-nearest cation site relativeto Mna' impurity producing spectra III2. These spec-tra have been interpreted by Watkins by means ofthe theory then available which considered the con-trlbutlon8 froID the splD-spin mechanism ar181ngfrom d-s excitation only and the Watanabe mecha-nism. The theory was found to give the correct signbut the results were an order of magnitude too low.No attempt has yet been made in such systems toestimate the contributions from overlap and/orcharge transfer effects, though the importance ofthese has been emphasized by Watkins. Since nowother mechanisms are known to give dominant con-tributions, it is of interest to know whether the con-clusions of %atkins change if one incorporates theeffects of these mechanisms. This has been donein the present paper. Moreover, the contributionsfrom overlap effects have also been calculated. The

spectra we have considered in the present paper areIII, a d Irr, ln NaCl a d KC1 crystals.

In Sec. II wepresent the procedure and the cal-culations for the ground-state splittings of the com-plexes III, and III& in NaCl and KCl and comparethem with the experimental results. The discussionis given in Sec. III, where some speculations havealso been made to bring the theoretical results intoline with experiment.

II. PROCEDURE AND RESULTS

The expressions derived in Refs. 2 and 3 maybe used here directly with appropriate changes toestimate the ground-state splitting parameters Dand E for spectra III, and III&. In the following, wefirst consider spectra III, and obtain point-chargecontributions from various mechanisms —Blume-Orbach (BO), spin-spin (SS) (d-s, d-d, d-g),Watanabe-with-cubic field (WC), and Orbach-Das-Sharma (ODS) mechanisms —and overlay contri-butions in NaCl and KCl. Next we consider spec-tra III~ and give similar results.

A. Spectra III,

The vacancy-Mn + pair complex which givesrise to the spectra DI, is shown in Fig. l. Bassinj,and Fumi4 have estimated the displacements of theions for this complex in NaCl and KCl with the im-purity lons Cd +, Ca +, and Sr . To the knowledgeof the authors, no similar calculations have yet beenreported for the case where the impurity ion is Mn 'In the absence of such data, we assume that thedisplacements of the ions when Mn ' is present as