NUCLEAR SPTN-LATTICE RELAXATION OF SEVERAL HUG-LET TN ANTEFERRDMAGNLETTC 'szMnfll“ 2H ’ 0. ThESTSTOE The Dégfeéof Ph D. .77“ ;._ 9‘ 9' » '7 7:172: ‘_ MTCHTGAN STATE UMVERSTTY CHARLES EMERY TAYLOR _ -;;i;:3f}' 1967 ‘- 1“!‘Z .041»... LI”: THESIS . Midi 31:2 {.8 331* Uni :xersity A \‘ ' - -' 'r'umw This is to certifg that the thesis entitled Nuclear Spin-Lattice Relaxation of Several Nuclei in Antiferromagnetic szMnCl4.2H20 presented by CHARLES EME RY TAYLOR has been accepted towards fulfillment of the requirements for \Aéww Major professor v A‘“& 2, (9&7 0-169 ABSTRACT NUCLEAR SPIN-LATTICE RELAXATION OF SEVERAL NUCLEI IN ANTIFERROMAGNETIC Rb2MnClu-2H20 by Charles Emery Taylor Rubidium Manganese Chloride di-hydrate (Rb2MnClu-2H2O) is a triclinic crystal which becomes antiferromagnetic below 2.2A0K. Measurements of the nuclear spin—lattice relaxation time Tln have been made as a function of temperature for four nuclei, Rb87, Rb85, 0135, and H1, in the temperature range from l.6°K to 0.45°K, and in one instance, for Rb87, to O.32°K. Standard rf pulse techniques were used, and the experiments were carried out in zero ex- ternal magnetic field. The temperature dependence of T1n was qualitatively the same for all four lines, and a least squares fit to the data for each line was made using a theory based on a two magnon Raman re— laxation mechanism, which, in the small k approxi- mation gives T '1 = w x —1 1n 2KITAE/T x(e -1) dx The values of 2K and T which give the best fit to AE Charles Emery Taylor the data along with the frequencies of the measured lines at l.l°K and the nuclear spin are given in the table for each nucleus. Freq. (MHZ) Nucleus Spin at l.l°K 2K(sec)—l(°K)-3 TAE(OK) Hl 1/2 18.1 1.96 x 103 2.15 0135 3/2 8.53 1.08 x 103 2.u7 Rb87 3/2 3.89 9.73 x 102 2.36 Rb85 5/2 3.22 1.33 x 102 2.0M The value of 2K for protons is calculated from the theory and found to be too small be a factor of one to two orders of magnitude, suggesting that the small k approximation is not applicable in this case. An ap- proximate theory, utilizing all k from O to km, is developed and used to calculate the proton T The ln' result is in good agreement with experiment. The following conclusions are drawn: (1) relaxation occurs via an intrinsic two magnon Raman process and is not due to the presence of paramagnetic impurities, (2) the temperature dependence of Tln is influenced by a large energy gap in the magnon spectrum, (3) the influence of the quadrupole moment on the temperature dependence of T is small, and (A) the use of the ln small k approximation is not consistent with the pres- ence of a large energy gap in the magnon spectrum. NUCLEAR SPIN-LATTICE RELAXATION OF SEVERAL NUCLEI IN ANTIFERROMAGNETIC RbZMnClu-2H2O By Charles Emery Taylor A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1967 DEDICATION This thesis is dedicated to my father and mother. ii ACKNOWLEDGMENTS The author wishes to acknowledge the support and guidance of Dr. J. A. Cowen, whose enthusiasm, sense of humor and dedication have made the last three years both enjoyable and worthwhile. He also offers his thanks to Dr. R. D. Spence for several helpful discussions concern— ing this thesis and for bearing without complaint the burden of a sometimes noisy research group on his premises; to Dr. E. H. Carlson for several suggestions concerning the electronic apparatus and for the time and effort which he gave en route to 0.320K; to Dr. George T. Johnston III for the computer work in— volved in this thesis and for his friendship during the last three years; to the machine and electronics shOp for their general good humor and craftsmanship, and to the National Science Foundation for their support of this project. iii TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES LIST OF APPENDICES Chapter I. INTRODUCTION Background . . . . Nuclear Magnetization . Nuclear Spin Lattice Relaxation II. MEASUREMENT METHODS Theory Relaxation Effects Two Pulse Method Three Pulse Method III. EXPERIMENTAL APPARATUS AND PROCEDURE Low Temperature Apparatus Helium3 System HeliumLT System Electronic Apparatus Triggering Apparatus Rf Oscillator Tuning and Damping Circuit Transmitter and Receiver Coils The Rf Amplifier Oscilloscope . Procedure for Taking Data Two Pulse Method Three Pulse Method IV. RELAXATION THEORY Magnon Operators The Magnon Dispersion Relation Magnon Relaxation iv Page iii vi vii ix Chapter The Perturbation Hamiltonian The Total Static Hamiltonian Two Magnon Relaxation V. EXPERIMENTAL RESULTS AND DISCUSSION Temperature Dependence of T Calculated Values of Tln Conclusions . ln REFERENCES APPENDICES Page 5A 57 66 66 8A 86 9O Table 3.1 5.2 LIST OF TABLES Information on transmitter and receiver coaxial lines . . . . . . Information on transmitter and receiver coils for different frequency ranges Information on frequencies and magnetic fields for resonance lines studied Results of least squares fit to equation (5.A) for the four nuclei studied Vi Page 33 36 67 78 LJO WWWWWWWWWW WWU‘IU‘IW .1O .11 .12 Formation free induction decay and spin echo signals Sample graph for determining T Glass He3 dewar LIST OF FIGURES Susceptibility coils He“ dewar ln Block diagram of electronic apparatus Triggering Apparatus Pulse sequences used in measuring T Tuning and damping circuits. In External power supply for rf oscillator Typical transmitter and receiver outputs Transmitter and receiver coaxial lines and coils Rf amplifier Gate pulse shaping Typical dispersion netic magnons Coordinate systems Graph of Tln Graph of T1n Graph of Tln Graph of T1n Graph of T1n Rb VS. VS. VS. VS. VS. T T T T T circuit curve for antiferromag~ (x, y, z) and (x', y', z') for H for for Rb85 for Rb87 for H vii 1 1 3 C135 3 Rb 85 .9 and Page 15 19 21 22 2A 25 27 28 3O 31 32 37 38 52 59 68 69 70 71 72 Figure Page 5.6 Magnetization curves for MnCl2-A2H20 and Rb2MnCluo2H2O . . . . . . . . . 75 5.7 Hypothetical dispersion curve for magnons in antiferromagnet with large anisotropy field . . . . . . . . . . . 81 viii Appendix A LIST OF APPENDICES Detection of free induction decay and and spin echo signals Derivation of expressions for T and In M (t) O O O I O O O O O z Derivation of magnon density of states List of data for T n vs. T for H1, 0135, R885, and RbA7 ix Page 92 95 101 I. INTRODUCTION The nuclear spin-lattice relaxation time of nuclei in antiferromagnetically ordered materials was first measured by Hardeman and Poulis in 1956.1 The relaxation was found to result primarily from the magnetic interaction between the nuclei and the exchange-coupled electron spin system.2 At temperatures low enough that the latter can be repre— sented mathematically by the spin wave approximation,3 the dominant relaxation process can be described as in- elastic scattering of spin waves by the magnetic dipole field of the nucleus,” with an associated nuclear spin flip. Thus the relaxation mechanism depends on the spin— wave or magnon population, which is strongly temperature dependent. Existing theories of the relaxation process can therefore be judged by comparing the predicted temp— erature dependence of T with that obtained experiment- In ally. It has been the purpose of the work described in this thesis, (I) to construct a reasonably inexpensive appara— tus capable of measuring T from A.2°K to O.A5°K over a In frequency range from 3 to 20 MHz, (2) to use this appara- tus to measure T as a function of temperature for a 1n 87 number of nuclear species (H1, Cl35, Rb and Rb85) in antiferromagnetic single crystals of Rb2MnClu~2H2O and (3) to compare the results with those predicted by magnon relaxation theory. Background Nuclear Magnetization. Nuclear spins in an antiferro— magnet see a strong magnetic field produced by the ordered electron spins. The static Hamiltonian for a given nu- 5 clear spin is MN = 11135. = —me-H (1.1) where u is the magnetic moment of the nucleus, YN is the nuclear gyromagnetic ratio, I is the dimensionless angular momentum operator, and H is the average local field at the nucleus due to the electron spins. We have neglected the nuclear dipole-dipole interaction because it is small com— pared to the interaction between the nucleus and the elec- tron and have chosen a nucleus whose quadrupole moment is zero. If we choose a coordinate system for which R is in the z—direction, H = kH, and 3: = -y N ’hHIZ (1.2) N The allowed energies of the nuclear spin, given by the eigenvalues Of‘HN’ are Em = —yNfiHm m = I, I—1, ...I (1.3) where m represents the eigenvalues of IZ. The energy difference between adjacent states is therefore AB = fiyNH = nwL, where wL is the classical Larmor pre- cession frequency, or resonant frequency, of the system. Because the energy of the nuclear spin is quantized, the magnetic moment cannot assume an arbitrary orientation with respect to the magnetic field, but can take up only those orientations for which the z-component of the angu- lar momentum is given by JZ = hm. In discussing the methods for measuring Tln’ we will be interested in the total nuclear magnetization M of the sample, given by M = (1.“) _ g Bk th nucleus and where Bk is the magnetic moment of the k the index k runs from 1 to N, the total number of nuclei. In thermal equilibrium, the expectation value of the total nuclear magnetization must be parallel to the magnetic field,6 H, so that the x— and y-components of M are zero. To find the z—component of the magnetic moment, it is neces- sary to determine the relative populations of the Zeeman levels. For simplicity, consider a system of nuclei with spin I = l/2. At thermal equilibrium, the populations of the Zeeman levels are proportional to the Boltzmann factor exp(—Em/kBT) where T is the absolute temperature of the lattice.7 Let N+ and N— represent the populations of the ground and excited states respectively. The ratio of N+ to N— is then + YNhH/kBT N_ = e (1.5) 2 which is greater than one. There are therefore more spins with m = +l/2 than with m = —l/2 and as a result there is a net nuclear magnetization in the z-direction iven by g YN‘hH v D Y‘h —-—— _ _ + — N _ N — k T MZ - g Ukz - (N ‘N )—§— — —§—(N )(e B " 1) (1.6) In general k T is much greater than Y nH. Assuming that B N this is the case, from equation (1.5) we have N+ z N so that N— 2 N/2 where N is the total.number of nuclei. Similarly expanding the exponential in equation (1.6) and using N— = N/2 gives M = Nv2fi2H z HkBT (1.7) If the nucleus has an electric quadrupole moment, the situation is not as simple. The gradient of the crystalline electric field partially removes the degen- eracy of the nuclear energy levels.8 The internal mag- netic field produces Zeeman splitting of the remaining degenerate levels. Although the energy levels are no longer necessarily equally spaced, the populations are still proportional to the Boltzmann factor, and at thermal equilibrium there exists a nuclear magnetization M which is parallel to M. Nuclear Spin—Lattice Relaxation. If a non-equilibrium distribution of the spins is produced, for example by an rf pulse at the resonant frequency, the system will relax back to the equilibrium distribution in a time character— ized by the nuclear spin-lattice relaxation time, Tln' The subscript n has been added to emphasize that we are considering nuclear as opposed to electron spin relaxa- tion processes. By Tln we mean the time constant asso- ciated with the exponential recovery of the z—component of 9 the magnetization from a non—equilibrium situation. T1n is a good measure of the strength of the inter— actions between a nuclear spin and the lattice, since it is these interactions which provide the relaxation mech— anism. The term "lattice" is used loosely here, and would be more properly replaced by the word "surroundings." In antiferromagnets, for example, the nuclei relax to the electron spin system which eventually gives the energy thus absorbed to the lattice in the form of phonons. Examples of possible interactions10 leading to spin— lattice relaxation in solids are (l) the magnetic inter— action between the nuclei and paramagnetic impurities, (2) the interaction between the nuclear quadrupole moment and the gradient of the crystalline electric field, (3) the changing magnetic field at the nuclear site due to phonon—induced fluctuations of the electron spin sys— tem and (A) the changing magnetic field at the nuclear site due to exchange—induced fluctuation of the ordered- electron spin system. Of these the last is generally the dominant mechanism in the temperature range studied. The effect of paramagnetic impurities will be discussed in Chapter V, but it is evidently much smaller than the dominant mechanism in our case. The second and third examples both depend on the phonon population which is negligible for temperatures very small compared to the DeBye temperature.ll We are left with the effect of the interaction between the nuclei and the exchange-coupled electron spin system. The low energy thermal excitations of this system are called magnons. The nucleus relaxes to the magnon system, and the energy thus absorbed is eventually transferred to the lattice, but in quanta which are generally much larger than the nuclear quanta,l2 so that the nuclear relaxation will be independent of the relaxation time of the atomic spins. II. MEASUREMENT METHODS Measurements of Tln’ the nuclear spin—lattice relaxa- tion time, were made in zero external magnetic field using standard radio frequency (rf) pulse techniques to be described below. These techniques depend on the detec- tion of free induction decay (FID) signals and spin echoes. This chapter will present a brief discussion of these phenomena. Theory The theory of spin echoes and free induction decay l3,lA,l5 The brief discussion given in is well known. this chapter can be supplemented by the above references. For simplicity, consider a system of spin l/2 nuclei in a magnetic field M directed along the z-axis. If the system is in thermal equilibrium, at time t = O, the populations of the two Zeeman energy levels will be pro— portional to the Boltzmann factor, leading to a net magnetization parallel to M of magnitude MZ(O) = M0, given by Eq. (1.7) as discussed in Chapter I. In equilibrium no net magnetization perpendicular to M exists. However, a circularly polarized magnetic field H —1’ applied to the system in the form of frequency wL, of an rf pulse will produce a non—equilibrium situation in which a transverse magnetization ML does exist. As will be shown, the precession of M; can be deteCted and the amplitude of the detected signal can be used under certain circumstances to measure the spin—lattice relax— ation time Tln A classical description of the phenomena of spin echoes and free induction decay leads to the same equa- tions as the quantum mechanical theory, and has the advantage that it is easily visualized, so it will be used here. In Figure 2.lA, the net magnetization MO is shown parallel to M. A magnetic field Ml, cular to M and rotating at the Larmor frequency, exerts perpendi- a torque MO x M on MO, caus1ng Mb to precess about M1 1 with angular frequency w H When the angle 6 1=YN1' between MO and M becomes non—zero, the torque MO x H will cause M0 to precess about M at the Larmor frequency. In order to eliminate the visual confusion which this introduces, Figures 2.lB through 2.lF are drawn in a coor- dinate system (x', y', z') rotating about the z—axis at frequency 9 = —kyNH. In the rotating frame the magneti- 16 zation sees an effective field, in this case equal to Ml, which is constant in time. Ml has been chosen to lie in the x'—direction. If M1 is removed after a time _ = n . . . te — tfl/2 265’ the net magnetization Will be in the x'—y' plane, and Mz(tn/2) will be zero (Fig. 2.lC). Such FID Spin echo __4 /\< 1 T A F (13—;- 0—1- Fig. 2.1. Formation of free induction dh;ay and spin -1. ‘ ,_. ' . ' . (guilt; . lf‘l’lgilfl - 10 a pulse of M1 is called a 90° pulse because it causes M0 to precess 90° around Ml. When M is removed, M 1 will remain stationary in the rotating reference frame and hence will precess at frequency w in the lab system. L This precessing magnetization will induce an emf in a coil whose axis is in the x-y plane. The magnitude of the emf induced will be proportional to the magnitude of the net component of M in the x—y plane, which in turn depends initially on the magnitude of the z-component of M that existed just before the pulse was applied. This is shown in Appendix A. In the experiments for this thesis, the magnetic field M was the local magnetic field in the crystal, resulting from the antiferromagnetically ordered electron spins. Due to local field inhomogeneities, the local field has different values for different crystallograph— ically equivalent spins leading to a spread in Larmor precession frequencies. Therefore, as viewed in the rotating frame, the magnetization fans out (Fig. 2.lD). As the spins get further out of phase, the net transverse magnetization goes to zero and the resulting signal also decays to zero. Such a signal is called a free induction decay tail or an FID. It is possible to bring the spins back into coherence by the application of a second pulse of M1 a time t after the first pulse. For simplicity, we consider that M1 is again along the x'—axis and that its duration is such that M precesses through 180° around Ml (Fig. 2.lE). 11 The spins that had precessed ahead of the center of the distribution by an angle o are now behind the center by the same angle. (These spins are labeled f in Figs. 2.lD and 2.lE.) Similarly those that lagged by ¢ (labeled s in Figs. 2.lD and 2.lE) are now ahead by ¢~ Therefore at a time 2t after the first pulse the spins will have regrouped (Fig. 2.lF) and will produce an emf in the receiver coil, which is called a spin echo. Figure 2.lG shows the pulses necessary for formation of an FID and a spin echo. It should be noted that in general rf power is fed into the sample in the form of a linearly polarized rf pulse of frequency w This can be decomposed into two L' circularly polarized fields, one rotating at frequency w in the same sense as the nuclear precession, and the L other rotating at frequency w in the opposite sense. L The effect of the latter can be shown to be neglibible, and it is usually neglected. The Hl discussed above is the circularly polarized component rotating in the same sense as the Larmor precession, and it is this component that produces the nuclear resonance. Relaxation Effects The above discussion has neglected the effects of relaxation phenomena. Application of a 90° pulse re- sults in a non-equilibrium configuration of the spin 12 system. Through the mechanism discussed briefly in Chapter I, the spins are able to return to the equili- brium distribution, giving up their energy to the lattice. Thus the z-component of the nuclear magnetization grows from zero, right after the 90° pulse, to Mb after a suf- ficient length of time. The return to equilibrium is expressed by the equation Mz(t) = MOEI—exp(-t/Tln)] (2.1) which is derived in Appendix B. As explained earlier, the size of the FID after a 90° pulse is proportional to the magnitude of the z-component of M that existed just prior to the 90° pulse. It is therefore a measure of the degree to which the system has returned to equili— brium. The two pulse method for measuring Tln takes advantage of this, and is discussed later in this chapter. Relaxation to equilibrium.also affects the ampli- tude of the spin echo. As the time t between the two echo forming pulses is increased, the amplitude of the echo decreases. This is usually due to mutual spin flips between neighboring nuclei which conserve energy but destroy the phase coherence of the transverse com- ponents of the nuclear magnetic moments. The resulting decay of the transverse magnetization can be described by an exponential with time constant T the spin-spin 2n’ relaxation time. We assume throughout this discussion that T >> T . T 1n 2n processes are not important in the 2n 13 method for measuring T using a spin echo, to be de- 1n scribed later in the chapter, except that if T2n is too short, the apparatus will not be capable of producing a visible echo. (If T for example, is much less than 2n’ the minimum time t between the two pulses, the echo ampli- tude will be zero at time 2t.) However, if an echo is Visible, the size of the echo will be proportional to the initial magnetization in the z-direction. This forms the basis for the three pulse method for measuring Tln' Two Pulse Method Starting with an equilibrium situation, application of a 90° pulse (Pl) leads to a free induction decay tail of magnitude A = Aw. If a second 90° pulse (P2) is applied at a time t later, the amplitude of the FID will depend on the size of the z—component of the magnetiza- tion which existed at time t. Repeating the above ex— periment for various t's, ranging from t << T to In t >> Tln’ will result in FID amplitudes from 0 to Aw. Since A is proportional to M (see Appendix A), A(t) has the same form as Eq. (2.1) A(t) = Am[l-exp(—t/Tln)] (2.2) From (2.2) exp(-t/Tln) = l — % (2.3) 1A and A _ -t 11’1(l - 'A- ) - T__ (2.“) w ln Therefore a plot of ln(l — % ) versus t should yield a 00 straight line of lepe -l/T passing through ln 1. ln A sample graph is shown in Fig. 2.2. The second pulse of the sequence need not be a 90° pulse, since a 6° pulse will still produce some trans- verse magnetization. The amplitude of the FID will be smaller by a factor of sin 6. It is imperative, however, that Pl be a 90° pulse. Otherwise there would always be a z-component of nuclear magnetization and the inter- pretation of the FID following the second pulse would be more complicated. P1 is called a saturating pulse because it reduces the z—component of the nuclear magnetization to zero. If the resonance line width is large, it may be necessary to replace Pl by a string of pulses, called an rf comb.l7 Energy from the pulses is fed into the resonance line at one frequency and diffuses to the rest of the line through cross-relaxation processes. The duration of the comb should be long compared to the cross relaxation time18 so that the whole line is saturated, otherwise the apparent return to equilibrium will be characteristic of cross-relaxation rather than spin—lattice relaxation processes. ) Fig. 2.;. III] . r“""T‘*1"1“I‘t' Pl T='L6;°‘ .. \ T : '- n' en." "\_ In ' .. \\\ 1 i \\L 47amp1e Quinnl for detcuumhiing TV ‘10.) Hm—sec. 111 ° 16 Three Pulse Method The three pulse sequence consists of a 90° pulse, Pl, followed after time t by an echo forming pair of pulses, (P2) and (P3) respectively. As mentioned above, the size of the echo is proportional to the z-component of nuclear magnetization immediately preceding pulse P2. Therefore if the separation between P1 and P2 is varied from times t << Tln to times t >> Tln’ the separation between P2 and P3 being kept constant, the amplitude of the echo will change from A = O to A = Am, where Am is the echo amplitude with pulse Pl turned off. The equa- tion for A as a function of time is the same as that for the two pulse sequence, since in each case A is directly proportional to the z-component of the magnetization existing just before P2. Care must be taken that the sizes and shapes of pulses P2 and P3 do not change when P1 is turned on and off. If they do change, it is a sign that the equipment is incapable of supplying the neces— sary power. It should be noted that P2 and P3 need not be 90° and 180° pulses respectively in order to form an echo. The 90°-180° case is easy to visualize and was used as an example only. The same technique can be used when P2 and P3 have angles 62 and 63, as long as a measurable echo is formed. In order that the echo amplitude be as large as possible, the time between P2 and P3 should be much less than T and T . ln 2n 17 The experimental procedure for measuring T n using 1 the two and three pulses will be given in Chapter III after the apparatus has been discussed. III. EXPERIMENTAL APPARATUS AND PROCEDURE In order to measure the nuclear spin—lattice re— laxation times of nuclei in antiferromagnetic crystals in the temperature range from A.2°K to 0.A5°K the experi- mental apparatus must (1) supply sequences of high voltage rf pulses of variable frequency and length to the sample at liquid Helium temperature, (2) pick up, amplify, and display the rf signals induced in the sample by these pulses, (3) be able to maintain a constant temperature during measurements, and (A) allow for measurement of the temperature from A.2°K to 0.A5°K to an accuracy of i0.005°K. The equipment used falls into two categories, low temperature apparatus and electronic apparatus. Low Temperature Apparatus 3 Two systems, a He system and a He“ system, were used to cover the temperature range from 0.A5°K to A.2°K. 3 3 l9 Helium System. Figure 3.1 shows the glass He dewar immersed in liquid He“. The sample was dropped into 3 position through the He pumping line, oriented by tap— ping the tip, and held in place by a few drops of Dow Corning 70A fluid which freezes on cooling. Temperatures as low as 0.A5°K could be reached by pumping on the 18 1), Exchange gos line He3 pumping 380n. #Xr line Elie-mid He4 _ 22mm iluss tttc 12 mm zlosQ iJ’EC (D pr ssure measuring AAA v- iJL-C —#- liquid H83 Fig. 3.1. Glass Nej dewar. 8 mm glass. TL he ~ sample hiSMJC \NCHT 0f He4 dewar 20 He3 bath. The temperature was determined by measuring the vapor pressure of the He3 bath using an NRC Alphatron pressure gauge connected to a 1/8" teflon tube posi— tioned just above the surface of the liquid He3. A calibration of Alphatron pressure versus temperature was made by measuring the susceptibility of ferric ammonium alum as a function of Alphatron pressure. Ferric am— 3 monium alum obeys the Curie law in the He temperature range. The temperature was determined from the value of the susceptibility at the given pressure. The sus- ceptibility coils2O shown in Fig. 3.2 were used with 3 dewar in which the T measurements were the same He 1n made. The receiver coil was wrapped around the tip of the He3 dewar. This meant that the filling factor (see Appendix A) for the coil was much less than one. Since the use of the He3 dewar was necessary only below l.l7°K (the lower limit of the Hel4 system), measurements above this temperature were made in a He” system where the receiver coil could be wrapped directly on the sample. Heliumu System. The Heu dewar is shown in Fig. 3.3. The means for transmitting the rf pulses to the sample and receiving the induced signal have been omitted and are shown.later. By controlling the pumping rate on the He”, temperatures from A.2°K to l.l7°K could be reached and held to an accuracy better than i0.005°K. T\. “ TAJ A pressure measuring 'I the 3 AAAA liquid He sample :§§~—— meosarinq secondary d .H (i841; urns) ‘ PTTWOF)’ ( 942 Turns) compensating Sect nd OW (IB4C‘2 Tums} Fig. 3.2. Susceptibility coils. 22 BNC connector . . Wonsmmer COOX receiver COGX u “*TO pumping line TE (ZZZ: - ’ro manometer '_ high vacuum jacket 4 liquid nitrogen helium vacuum jacket liquid helium 1 Fig. 3.3. he4 dewar. 23 Above about 2.2°K, a mercury manometer was used to measure the vapor pressure of the He“. Conversion from pressure to temperature was made using the National Bureau of Standards' "1958 HeLl Scale of Temperatures." Below 2.3°K a manometer using oil of known density was used. The ratio of the density of the oil to the density of mercury when multiplied by the height of the oil column gave the pressure in mm of Hg, which was then converted to temp- erature as above. Electronic Apparatus A block diagram of the electronic apparatus is shown in Fig. 3.A. The triggering apparatus provides trigger pulses for the oscilloscope, the amplifier gate circuit and the Arenberg rf oscillator. Rf pulses from the Arenberg are tuned and shaped and fed to the transmit- ter coil. The small rf voltage induced in the sample by these pulses is picked up by the receiver coil ampli— fied by the rf amplifier, and displayed on the oscillo— scope. All connections were made with RG59 and RG62 coaxial cable with BNC connectors. Where low capacitance was a necessity, RG62 cable with a capacitance of l3pf/ft was used. Triggering Apparatus. Figure 3.5 shows the equipment that generates the trigger pulses for the rf oscillator, the oscilloscope and the rf amplifier gate. The initial trigger pulse is provided by the circuit21 at the top reteiver coil Fig. SCupe lriqqer triggering: apparatus r.f. oscnllo’rur a power supply coaxleods .A. Block diagram ll 'l’ lib l.‘ ,“l l 9018 pulse r.f. amplifier trigger for rf. l... - tuning a dumping oscilloscope loppuratus l 1 1- r4; r--‘”'""""7 I l | I | l ' I ’ l J... l l . l J E H64 dewar «J—transmitler coil 3f electronic apparatus. 22.534 1 . . 7 . ‘ ”3:5 I’Z It.) IF}. IE3 '6.. Pl A P2 a P3 COl .1 4 l | dd J r— -! . *BNC connector Q 3 Q. . off “’4 SS 66 D4 L‘: L,. hiki ®LUIS€= OUT L o 1 pulse sequence cirfuii/ ‘ Ytrv. [(‘l 25:13: generator Fig 3 . L T 1‘1 lTl"(_?1"1 Ill“ jnxllljiil‘étt 11:“. . 26 left in Fig. 3.5, which uses a A layer diode (-J+—) (Motorola A6305A). This pulse triggers a series of Tektronics 163 pulse generators and Tektronics 162 waveform generators which feed into the pulse sequence circuit shown at the bottom of Fig. 3.5. This circuit permits the sequences shown in Fig. 3.6 to be used. Switch Sl selects the saturation technique; either an rf "comb" of saturating pulses or a single saturation pulse may be used. Switch S2 selects the pulse to trig— ger the oscilloscope and S3 selects the pulse with which to gate the rf amplifier. Switches SA’ 85, and S6 allow any pulse to be turned on or off independently. The diodes in the pulse sequence circuit prevent the pulses from seeing the output impedances of the other pulse generators. Rf Oscillator. An Arenberg Ultrasonics Laboratory pulsed rf oscillator Model PG—65O was modified to allow external triggering by two or more pulses of varying lengths. High voltage was supplied by a 2500 volt external power supply shown in Fig. 3.7. An rf pulse output power up to lkW over a frequency range from 3 to 20 MHz was attainable, giving circularly polarized rf magnetic fields (Hl) on the order of 5 gauss. The Arenberg was designed to give a rectangular pulse (i.e., a pulse with no tail) when terminated by a 930 termination. P—t-rt II II two pulse sequence Pl P2 FAQ—HIE?! II II II three pulse seqience F P2 F3 (A) SI in "no comb" position L..__fp.—__... ‘PI‘I ' TIz_'_"I I II II II FL two pulse _7 3 sequence ' ”COmTO” P2 1P A ‘ IPI'I ll""—T 2—"7'23’1 IIP II II IIII I tnree ruse _ . seqience ”“como P2 P3 (B) SI in "comb" position Fig. 3.6. Pulse sequences used in measuring T 1.. ln' .LOBQAkuno n; no; hacazn Luzon am:goux; I ><><>< xmm as .5838». I we. soasm 25¢ .9050: 2 me 25.94: I . . 1.3 Ms. (S . 93:84 0.. < < mhamxm covfxom moot? :9: I 29 Tuning and Damping Circuit. A shielded variable air capacitor (15—180 pf) connected in series with the line was used to tune the circuit to series resonance so that the voltage across the transmitter coil was a maximum. Figure 3.8 shows the tuning capacitor and the damper circuit which damps out the ringing induced at the trail— ing edge of the rf pulse by the series resonant condi- tion. The damper, based on a circuit by Skopas,22 is essentially two diodes back to back, biased in such a way that the circuit offers low resistance to low volt- ages and high resistance to high voltages. Thus the Q of the circuit is low during the tail of the pulse and it quickly damps out. Typical pulse shapes are shown in Fig. 3.9. Transmitter and Receiver Coils. Rf power reached the transmitter coil through a coaxial line (see Fig. 3.10) made from a German silver tube with a number 30 copper wire as the center conductor, the latter insulated with ten gauge teflon insulation. The wire was then threaded through styrofoam cylinders which acted as insulating spacers. The same type of coaxial line was used to carry the rf signal from the receiver coil to the top of the dewar. The capacitance of the lines was kept as small as possible by using the largest size tubing consistent with the space available and the method of supporting the teflon coil from. Table 3-1 shows the size, length and approximate capacitance of each coaxial r“ 6X3 .L__..‘ '—-l e 4.: BNC / \cUnziecltrs f'rcmrrf. tootransmitte' oscillator COll tuning capacitor damper Ire} - %%C III" v.0. .. .4 duh]; 9“,! L“Vb' Q'Jp’CIT/ ’1‘ (J *1 -.r , ,. ”:1 - . .. AlP- 3.0. Zuninw and damninn circuits. 31 (A) (B) (C) (D) Fig. 3.9 Typical transmitter and receiver outputs for undamped (left) and damped (right) transmitter pulse. Graticle is 6xlOcm. x—scale is 5 sec/cm. (a) and (b) transmitter pulse with y = lOOv/cm and 5v/cm (C) and (D) receiver output with y = 2v/cm and 0.2v/cm (E) gated receiver output with y = 0.2v/cm H83 system Top view sum; II. teflon ___£Ol’l’{l 2 receiver “(MM transmitter max I l \u I SiyrcfoIIrI'I I l spacers I I 4 I l Te.||U'-] ' I I ll'lS'JlOTIOfl I I - conductcr/ EEI . . :37 Ltronsrnltter E, coil . :' 'l ==='l I receiver Coll ;. to receiver coil and sample sick. view side vie-w Fig. 3.10. Transmitter and receiver coaxial lines and coils. All coaxial lines haVe the same construction as . l I that shown for the receiver line in the Be system. 33 Table 3—1 System Transmitter Coax Receiver Coax Length Diam. Cap. Length Diam. Cap. (inches) (inches) (pf) (inches) (inches) '(pf) He3 A5 3/LI 3o 3/II 30 He“ 37 5/8 30 1/2 35 3A 3 and He“ systems. line in the He The coil form for the transmitter coil was made from teflon. For maximum signal and minimum receiver dead time the coils should be oriented so that (l) the transmitter coil is perpendicular to the internal field M at the nucleus under study (so that M x‘Ml is a max- imum), (2) the receiver coil is perpendicular to M (to get the maximum possible FID after a 90° pulse), and (3) the transmitter and receiver coils are ortho- gonal (to minimize pickup in the receiver, due to the transmitter pulse). Use of the teflon coil form (Fig 3.10) insured that requirement (3) was reasonably well satisfied. Proper orientation of the crystal, knowing the internal field directions for the different nuclei studied, allowed requirements (1) and (2) to be satisfied to within i10°. In the teflon form for the He3 system, the hole in the center for the sample and receiver coil was made so as to slide easily over the receiver coil and make a firm fit on the tip of the He3 dewar. For the He” system, the sample, with the receiver coil wrapped directly around it, was held in place in the hole in the teflon by two styrofoam wedges, and the teflon form made a firm fit on the receiver coaxial lead. The number of turns on the transmitter coil for a given frequency range was determined by trial and error, 35 maximizing the rf pulse voltage in the desired frequency range. Similarly the number of turns on the receiver coil was maximized subject to the condition that the tank circuit of the rf amplifier be tunable over the desired frequency range. Table 3-2 gives the number of turns on receiver and transmitter coils for the different frequencies and systems used. The teflon form for the HeliumLl system was sur- rounded by a thin walled brass can which was soldered to the coax leads by means of several criss-crossed wires. This can, not shown in Fig. 3.10, out down considerably on 60 cycle pickup. The can was Open at the tOp and had holes bored in the bottom so that Helium could flow freely around the sample. The Rf Amplifier. The induced rf signal was amplified by the circuit shown in Fig. 3.11. The amplification was linear in the range used (output signals up to one volt) and the signal was observed undetected so that A, the amplitude of the signal on the scope, was proportional to A', the amplitude of the induced signal before amplification (see Appendix A). The receiver coil is connected in parallel with tuning capacitor CT’ forming the tank circuit for the amplifier, which can be tuned to the desired frequency. Modifications neces— sary for good high frequency response (15 to 20 MHz) are indicated in the figure. Because the transmitter 36 II I: II I: own m w\m rm momma m.manwa dome: w 0mm mm :\m : own ma :\m : Aomow calm 0mm 0: H ma own mm :\m ma domoma m.:|m omflm mags» Amocosflv cowm\wch:p oNHm manna “monocflv opflm\mcL5p Hwoo Nmz mafia .Emflp ohflz .Emfio Esme omcmm Heoo Heoo Haoo Haoo nonmaaaeno sesmaemsm no>flooom noppflemcmpe go>flooom Loopflemcmpe whoocop< Eopmzm mom Eopmhm mom mlm oHQwB i :1 w.-. . J. ooeu?...oc__ omewwwfm oo_-o_ om: n. 6. _omn..m_ n. 1 :3 W12“ m ._.o fiwwrwcot mmuqs Dvmmono 2: 3.1:; as. as: 2; i 3.352 :32:u_t_4._r_c *0 women we 2: oz; lea. _ .> mm. +m _ - - - u . fl _. SEES . . . v. 0— DE. :5 _¢nn .mr .......H l .H. 833 2 1225 .II «n. em x0. r ozN To rluo 3m Afilll InlulI was: 2...: o L... 225 ..J .I. ' C x. t SE .22. 300 38 and receiver coils were not exactly orthogonal, pickup due to the transmitter pulse induced high Q ringing in the tank circuit, saturating the amplifier for as long as 100 microseconds. Crossed diodes at the inputs to the first and second stages were used to cut down on this receiver dead time. It was further out to less than 15 microseconds and in some cases to about 7 microseconds by gating diode Dl, which introduced a low resistance in parallel with the tank circuit. This lowered the Q and damped out the oscillations. The gate pulse was derived from a Tektronics 161 pulse generator and could be triggered to gate the receiver during and after any one of the pulses. It was important that the transmitter pulse have a sharp cut—off to prevent the receiver coil from picking up the ringing tail of the transmitted pulse. To prevent ringing caused by the sharp trailing edge of the gate pulse, a simple shaping circuit, shown in Fig. 3.12, was used, which put a smooth tail on the gate pulse leaving the Tektronics 161. Typical receiver output under various conditions is shown in Fig. 3.9. Oscilloscope. A Tektronics type 531 oscilloscope with a high gain Tektronics Type H plug—in preamplifier, was used for most of the measurements. In general no scale with sensitivity greater than 0.1 volt/cm was used. Procedure for Taking Data The discussion in Chapter II alluded to two dif- ferent pulse sequences used in measuring Tln" The two from gate trigger V (see Fig. 3.5) Tektronics l6! '6' pulse generator * gate pulse 1 L to r.f. omp'ifio-r __ I * " I ,” RC 5%,: coo‘x/ IK? I' I I1 BNC J.— ' _ ConneCTor ? ll" 9" 3n _. . '- E x .__ x 22 rmm-mx Pig. g.la. Wdtb pulse shaping CJTOHIT. 40 pulse method was used when no useable echo signal was obtainable. This was the case for C135 and proton 85 resonance lines. When a good echo was observed (Rb and Rb87 resonance lines) the three pulse method was used. The procedure followed when taking data is out- lined below for both the two pulse and the three pulse methods. Two Pulse Method. (I) Come to equilibrium at some temperature T. (2) With Pl off, tune the transmitter and receiver until the maximum free induction signal is obtained after P2. (3) Adjust the repetition rate so that the system is in equilibrium before each two pulse sequence. (M) Turn on P1 so that ti, the initial time between the two pulses, is short compared to the expected Tln' (5) With the s00pe triggered by P2, adjust P1 to a 90° pulse by making it the minimum length that reduces the FID after P2 to zero. It may be necessary to make fine adjustments on the transmitter frequency to do this. (6) Turn off Pl and record the FID amplitude (A = Aw) somewhere on the tail subject to the condition that when P1 is turned on, the FID amplitude at that point on the tail is zero when t = ti. (7) Now record A as a function of t from t = t1 until A1 A reaches Am, and graph this as described in Chap- ter II. T1n for the given T can then be found from the slope of this graph, as shown in Fig. 2.2. Three Pulse Method. (I) Come to equilibrium at some temperature T. (2) With Pl off, tune the transmitter and receiver until an echo is observed. Adjust the lengths of P2 and P3 and the time between them to give the best echo possible. (3) Adjust the repetition rate so that the system is in equilibrium before each three pulse sequence. (A) Turn on P1 with t = ti’ the initial time between P1 and P2, short compared to the predicted Tln' (5) With the scope triggered by P3, increase the length of Pl from zero to the smallest value that makes the echo amplitude go to zero. This will make Pl a 90° pulse. Fine adjustment of the transmitter frequency is advisable at this point. (6) Turn off pulse P1 and record the echo amplitude. This value is Am. (7) Now turn on Pl and measure A as a function of the separation between P1 and P2, and graph in the same way as for the two pulse sequence. Again, T for ln the given T can be found from the slope of the graph. IV. RELAXATION THEORY The nuclear and electron spin systems in an anti— ferromagnet are coupled by the hyperfine interaction. The strength of the magnetic field at a particular nucleus depends on the type of hyperfine interaction involved. Three types are generally considered,23 leading to (l) dipolar fields, (2) transferred hyperfine fields, and (3) direct hyperfine fields. Transferred hyperfine fields arise from overlap of the wavefunctions of elec- trons of nominally non-magnetic atoms and those of elec— trons of paramagnetic ions. A spatial redistribution of the spin magnetization then occurs with an associated hyperfine interaction.2u The direct hyperfine fields, produced by the electron spin and orbital moments,exist at the nuclei of magnetic atoms. The nuclear energy levels are determined by the average internal magnetic field due to the hyperfine inter— action and by the electric quadrupole interaction (which may be important for Cl and Rb nuclei). Transitions between these energy levels are induced by the time vary— ing fluctuations of the internal field, which provide the mechanism for spin—lattice relaxation of the nuclear spin system. The non—static part of the internal field, M2 43 taken as a perturbation, is produced by thermal fluc- tuations in the electron spin system and can thus be described, at sufficiently low temperatures, by non- interacting magnons. In this chapter we introduce ex- pressions for the electron spin operators in terms Of mag— non creation and annihilation operators and derive the magnon dispersion relation. We then discuss magnon re— laxation mechanisms and their dependence on the magnon dispersion relation, introduce the perturbation Hamil— tonian and the total static Hamiltonian, and derive an expression for T based on the two magnon (Raman) ln relaxation mechanism. Magnon Operators The transformation from electron spin variables 8+, 8- and SZ to magnon variables Aip and A is dis- kp 25,26,27 cussed in many references. The following discussion is adapted from Kittel.27 Consider that N electron spins lie on two interpenetrating sublattices, q and r, such that the N/2 spins on sublattice q are aligned in the + z—direction and the N/2 spins on sublattice r are aligned in the - z—direction. Further, consider the case where the nearest neighbors of a spin on sublattice q are all on sublattice r and vice versa. In the absence of an applied magnetic field, the Hamiltonian for the electron spin system is MA )-| Z E = -2J 2 s1 Sj — 2p OHAZ sqJ + 2pOHA§ Sr. (A.l) i/S << 1 we can expand the square root in each of Eqs. (4.2) through (4.5). This consti- tutes the spin-wave approximation. This gives for (4.2), + 3.3.8. + _ e q ' sqj — (2S) (aqj _ ——iE§i—Ei + ...) (4.13) Using the inverse of the transformations (4.9) and (4.10) we have, neglecting third and higher order magnon terms P -'k°r + 48 2 l— —j S . = —— b + 4.14 qJ [N J E e qk ( ) Similarly 1 ° . - 4S 6 "15 gj + . = —— b + ... 4.1 Sqa [N ] § 8 qk ( 5) l ‘ o + _ 4S 6 '15 £2 + sr2 — [N—] § e brk + ... (4.16) P +ik°r — _ 4s 2 — —£ srg — [NI] E e bPk + ... (4.17) From (4.7) through (4.12) we have 2 2 i<5_§'>.£j + S . = S — — b b + ... 4.18 qJ (N]k£.e qk qk' ( ) Z _ 2 —i(1£—L_(_')°_l:£ + Srz —s + [fi]kg e brkbrk, + ... (4.19) 47 Notice that S+ and S— involve odd numbers of magnon oper- ators while SZ involves an even number. If we now express SHE from (4.1) in terms of magnon variables, we find that, assuming nearest neighbor coup- ling only I >IE = D—IO + “1 + constant terms (4.20) I where 311 represents higher order magnon terms (three or more) and _ + + + + ZHC>- —2st£ [yk(bqkbrk + bqkbrk) + (bqkbqk + brkbrk)] + + + 2uOHAE (bqkbqk + brkbrk) (4.21) _ 1 ik'é _ Yk - 2% e — — - Y-k (“.22) The vector g connects a given spin with its 2 nearest neighbors and we have assumed that the crystal has a center of symmetry. In arriving at (4.20) identities of the following form have been used.30 i(k—k')'r. —~ 3:4 2 e 2 akk, (4.23) J 1(k—k')°r. -—— 3+ _N + 2 e b b , — - Zb b (4.24) jkk' qk qk 2 k qk qk . (4.23) can be proved by writing out the sum explicitly and grouping the terms in sums of geometric progressions. (4.24) follows directly. Because of terms like 48 bgkb;k, Eq. (4.20) is not diagonal as it stands. The transformation which diagonalizes >40 is defined by + bqk = CklAkl + ck2Ak2 (”'25) _ + brk - CklAk2 + CkZAkl (4.26) + _ + bqk — CklAkl + cszk2 (4.27) + _ + brk ‘ CklAk2 + Ck2Ak1 (”'28) here c c are real and mu t satisf c 2 c 2 = 1 W kl’ k2 S y k1 ‘ k2 ° . . + = The latter equation 1nsures that [Akp’Atp'] szdpp" This is the boson commutation relation. Substitution of (4.25) through (4.28) in (4.21) shows that the off- diagonal terms cancel if pk -Yk 4 0k1 - ( 2_ 2)% Ck2 — < 2—6 2)% ( '29) pk Yk pk k u) 00 1/ 9k = 1 + _A + [(1 + —£ 2 — y 2]2 (4.30) w w k e e where we = 2JzS and wA = 2pOHA. Using the boson commuta— tion relations and substituting Eqs. (4.25) through (4.30) in (4.21), we get _ + + :H0 - é wk(Ak1Ak1 + Ak2Ak2 + 1) (4.31) 49 with wk given by 2 2 P )2 wk = [(006 + wA - we Yk ]2 (4.32) Eq. (4.32) is the dispersion relation for antiferromag- netic magnons which is discussed in the next section. In the diagonalized representation, for each magnon of wavevector k, there are two degenerate modes, p, where + kp p k and polarization p; Akp annihilates the same magnon. Notice that it is not as simple to "picture" these magnons = 1,2. The operator A creates a magnon of wavevector as it was when a given magnon was confined to one sub- lattice. The Operator A+ A when Operating on an eigen- kp kp state of the electron spin system gives the number of magnons of wavevector k and polarization p in that system. This can be derived by considering an eigenstate l...nkp...) in which there are nkp magnons present in the mode kp, where k runs over the N/2 wavevectors and p = 1,2. From magnon theory31 we have A+l n )-(n +1)1<°-| n +1) (433) kp kp — kp ...kp 0.. O 1 = /2 — Akpl' nkp ) (nkp) I kp l...) (4.34) Therefore 4+4 I n )=1’2| n > (435) kp kp . kp"° kp kp . kp"° . and A+ A = n (4.36) kp kp kp 50 We can then write 2H0 in (4.31) as = 37 1/ k k To express the spin operators in terms of the magnon operators in the diagonalized representation we sub- stitute Eqs. (4.24) through (4.27) into (4.14) through (4.19). Neglecting third and higher order terms, we have + 481/2 "15°33 + 4 8 qu ” [N‘] E e (CklAkl + Ck2Ak2) ( '3 ) P +ik-r — _ 482 ——j + qu ' [N‘] E e (CklAkl + Ck2Ak2) (”'39) P —°k°r + _ 4s2 l——5L + Srt ' [N— E e (CklAk2 + Ck2Akl) (4.40) P +ik r - _ 482 ——;L + Srt ‘ [fi’] £ 8 (CklAk2 + Ck2Akl) (4.41) i(k—k')°r. Z _ a — — J + qu ‘ S ’ [N]k;,e (CklAkl + Ck2Ak2> x (c A + c A+ ) (4 42) k'l k'l k'2 k'2 ' -i(k-k')°r z _ 2 — — x + Srt '3 + [N] E,e (CklAk2 + Ck2Akl) kk x(c .4 +0 11") (443) k'l k'2 k'2 k'l ° The Magnon Dispersion Relation The dispersion relation for antiferromagnetic magnons 51 was given in Eq. (4.32) and is repeated here for con- venience. )2 — w 2yk2]% (4.44) wk = [(we + wA e It is customary to write this in the small k approximation using Yk2 = l — bk2, which is obtained from the defini- tion of Yk’ (4.22), by expanding the exponential to second order in k, and squaring the result, again keeping only terms to second order in k. If there is a center of symmetry in the crystal, linear terms in the sum will cancel. Then 1 2 2 + wA2Jé (4.45) wk = [we bk + 2wewA An important feature of this dispersion relation is that for k = 0, we have [2m + w 2:|1/2 (4.46) w A k = EwA so that there is an energy gap in the magnon spectrum. The effect of the gap is to suppress the excitation of where T is magnons for temperatures T less than TAE’ AE defined by (4.47) The general shape of the dispersion relation in the first Brillouin zone is shown in Fig. 4.1. The dashed line is the long wavelength (small k) approxi- mation to hm disregarding Brillouin zone effects and k assuming wA = 0. Note that the larger T ter will be the dispersion curve. AB 18, the flat— 0 km: 72 /0 Fig. 4.1. Tvrimal dispersion curve for antiferronar— U netic magnons. 53 Magnon Relaxation At temperatures low enough that the spin wave approx— imation is valid, relaxation of a nuclear spin might be expected to occur via one magnon (direct), two magnon (Raman), three magnon, or higher order relaxation mech— anisms. In the direct process, the nuclear spin flips and a magnon of wavevector k and energy th, where wL is the nuclear resonance frequency, is emitted. Such + processes involve a magnon creation Operator Ak The two magnon process can be described as magnon scattering; a magnon of wave vector k is absorbed, the nuclear spin flips and a magnon of wavevector k' is emitted where Ek = Ek' —'th. This requires two-magnon terms like AkAk' in the perturbing Hamiltonian. Three magnon processes involve terms like AkAk'Ak' the absorption of two magnons (wavevector k' and k"), , which represents the creation of one magnon (wavevector k) and a simul— taneous flip by the nuclear spin. Conservation of energy requires in this case B + E = E — ‘hw (4.48) In discussing the importance of these three relaxa- tion mechanisms it is necessary to consider the effect of the antiferromagnetic magnon dispersion relation. First consider the effect of the energy gap on the direct process. A nuclear resonance frequency of lOMHz corresponds to a temperature of about 10-30K. This is 54 well below TA in almost all antiferromagnets so that it is generally impossible to have a direct process and still conserve energy. Next consider the effect of a large energy gap on the three magnon process. For T a T AE N’ trum will be essentially flat and again it will be im— 32 the magnon spec- possible to conserve energy. For T large, we are AE left with the two magnon process as the dominant relax— ation mechanism. The Perturbation Hamiltonian Relaxation of the nuclear spin system occurs through hyperfine coupling of the nuclei to the fluctuations in the electron spin system. In this section we discuss the Hamiltonian which will be used as a perturbation in calculating T We will consider only the dipolar ln' contribution to the hyperfine interaction. The form— alism for transferred hyperfine interaction is the same; only the coupling terms are different. The Hamiltonian for the hyperfine interaction 3H1 is then 3'1 = -YN’fiE°1 (4.49) _ _§_ . E ‘ Z di[§i ‘ [ 2] (§-i 314%] (”'50) 1 r. 1 where d1 = —yéh/ri3, ri is the vector joining the nucleus and the ith spin and the sum runs over all electron spins. 55 T is the operator which represents the magnetic field at the nucleus due to all the electron spins. We have tacitly assumed that the interactions between the nuclei can be neglected, since they are small compared to the hyperfine interaction. Further we assume that the inter- action 2H1 has little effect on the motion of the atomic spins. By writing Si = <§i> + Si)’ T may be split into a constant part U and a fluctuating part y. The eigen- states |xn) of the nucleus are then determined by EHU, the static part of 3H given by l) MU = -yN‘hU_°I (4.51) where U is identical to Eq. (4.50) with Si replaced by <§i>. Transitions between the eigenstates are induced by the fluctuating part of :H1 and it is these transitions that contribute to the relaxation of the nuclear Spin system. We can write the fluctuating part of EH1 as \JHV = —yfihK°I (4.52) 3 2 v = ) dimsi — [5—] (6_S_i°§i)§il (4.53) i i where 6§i = §i — (4.54) 56 The Total Static Hamiltonian The discussion in this and the following section is based on a paper by Van Kranendonk and Bloom.2 Since our experiments were done in zero external magnetic field, and since we assume that the hyperfine interaction does not appreciably affect the electron spin system, we can write the Hamiltonian for the electron Spins as 3+8 = —2J Z si-s. (4.55) i> TAE' x(eX-l)_ldx (4. T «T3 (Ll. .88) .89) .90) 91) 92) V. EXPERIMENTAL RESULTS AND DISCUSSION Rubidium Manganese Chloride di-hydrate (szMnClu-2H2O) is a triclinic crystal with space group PI3u which be- comes antiferromagnetically ordered below TN = 2.24°K.35 The internal fields are of such magnitude that nuclear magnetic resonance experiments can be readily performed on four nuclei, Rb85, Rb87, 0135 and H1. We have measured T the nuclear spin—lattice relaxation time, 1n’ as a function of temperature for these four nuclei in the temperature range from l.6°K to O.45°K and in one instance, for Rb87, to O.32°K. Table 5.1 shows the frequencies (at 1.1°K) of the lines studied, together with the magnitude and type of the corresponding 36 internal magnetic fields. These lines were chosen for detailed study because of their good signal to noise ratios. Less extensive data on other lines showed the same temperature dependence of Tln' All measurements were made in zero external magnetic field, using the standard pulse techniques discussed in Chapter 11. Temperature Dependence of Tln The results are shown in Figures 5.1 through 5.5, 66 67 Table 5.1 Information on frequencies and magnetic fields for resonance lines studied. Magnitude of Freq. at l.l°K Internal Field Nucleus MHz (Oersteds) Type Rb87 3.89 1.879 dipolar R685 3.22 1.879 dipolar 35 transferred Cl 8'53 19.49 hyperfine H1 18.1 4.296 dipolar IO 1 I I f r T] I I l [1 IIII‘ 5— -4 2__ .. O__ ... IO 5:)... .— 2— ... A -l’—" — ,J )0 '0 C ,-)___ __‘ 3 . o ‘3 ;v m 8 v '2— Q — E g... -2_- 9 __ )0 o 9 5" 8 " O 8 2t- - -3 o : -- oo -‘ IO 0 5__ 8O .... l 8 3 t. L I 1 1 I 1 1 i 1 l I [III .2 .4 .6 .8 LC |.5 2.0 3.0 T (°K) Fig. 5.1. Graph) 7 " for lf'l' |O I I I l lll'lIlIlFllll IO * (seconds) Q l N l O 3 l 1 1 lllllllLJlllll .2 .4 i. 3 L0 I5 20 T (°K) Graph of Tln vs. T for 013). \j“ (\j 0 Fig. 31) |() I I I I I I I I II’TIII‘III 5.. .. 2" " O 0 .... IO — 8 5* O _ O O 2- .. 3% -l__ 9 __ j;|0 8 T5 9 C: 5-— ‘9 .— ‘3 o 8 Q) U) C: O }__ 162— "" 5" .. 2t- ._ -3__ _.. IO 5'- _ 3 1 1 1 111111111111111 __ .2 .4 .6 .8 LC L5 2.0 3:0 T (°K) fig. 5.3. Graph of Tlr vs. 1 to I I I l I I1TIIIIIIII| 5 _ 2 .— ICO _ 5 ‘0 m 2 . "‘ A Io"| . "‘ 0') 'O C 2 .. O 5 O Q) O (f) o 2 z, " I-E' ° --2 : ... l0 0 5 o. -4 2 ° — 167’ :9 "“ 5 .... 3 1 1 1 1 1 1 l 1 1 1111 11! .2 .4 .6 .8 If) l5 2.0 3.0 T (°K) Fif?- b 4 T- vs. T for Fbgr. i0l I I I .IIIIIIIIIIIII] 85 0 R 5 0 b8? 0 0 Rb " 35 2 X CI IOO o o + HI O 5 00 . o o 2 .1 I, IO" 3" . T)? ‘ O, 8 O p I c 5 o . o ¢O§ o ’b 6 <1.) ¥ 9 ° :. V 2 '4- § 0 C O l—- A 1 4: IC .9. ‘ “t 5 I i 93 2 o *o -3 ++ IO "U, 5 1+ 00 4: o 3 1 1 1 11411111111111] .2 .4 .6 .8 LC L5 2.0 T (°K) 5.5 I?» of {P111 VS. T for 131, C7135, "fig-)5, and 73 where T n is plotted vs. T on log—log paper. Although 1 the electric quadrupole moments of 0135, Rb87 and Rb85 differ by a factor of four, the temperature dependence of their relaxation times appears to be the same (Fig. 5.5) and further, is the same as that for the proton, which has no quadrupole moment. Therefore, we will neglect the effects of the quadrupole interac— tion, and, in the initial discussion of the results, will consider only the Rb87 relaxation, for which the data is the most extensive. The theoretical expression for T1n derived in Chapter IV for Raman relaxation resulted in T1n a T for T >> TAE’ Similar calculations based on three- magnon relaxation processes37 lead to Tln a T_5 for T >> TAE' If relaxation occurred via one of these processes, and if T were much greater than T -3 AE’ a graph of log T vs. log T would be a straight line of ln slope -3 or -5. Evidence of such temperature depend— 38,39 ences above T does exist in other crystals. AE However, the data for Rb87 in Fig. 5.4 clearly cannot be fit by a single straight line. This suggests that we are studying a temperature range where magnon energy gap effects are important. The size of the energy gap in the magnon disper— sion relation can be calculated from the value of the experimentally determined critical field, Hc’ required to produce the spin—flop phenomenon.“0 The calculation 74 is based on the relation)-ll k T = H = (2H H + H 2)1’2 (5 l) B AE guo c g”b E A A ‘ where guOHE = hwe guOHA = th (5.2) An estimate of HE can be obtained from the approximate relation suOHE = kBTN (5.3) . 42 . Measurements by Casey, NagaraJan, and Spence ln Rb2MnClu°2H2O show no critical field effects below 8.3 kgauss at l.lOK, which sets a lower limit on Ho‘ Thus from (5.1) we have TAE Z l.lOK. Blatt, Butterworth and Abeleu3 searched for critical field effects in the same crystal using thermal detection methods. They found no evidence of a spin—flop phase above about l.2°K, though they did detect an antiferromagnetic- paramagnetic phase transition at about 18 kgauss. Although measurements at lower temperatures may detect a spin—flop transition, an estimate of TAE based on their existing data gives T = T AE N' Another estimate of TAE can be made by comparison with MnC12-4H2O, which exhibits a magnetization curve very similar to that of Rb MnC1u°2H 0. Figure 5.6 2 2 plots (M/MO) vs. (T/TN) for MnCl2-4H20uu and 4% (M/Mo) m b) a '0) 'o) a) '00 Lo I Fl 9‘. I: r ,9 . U . PbaMnCl '2 ‘ 2 4 If . r") / (, Magnetization {3) . 0 U I" V G 15 For H”CI '“ ........ 76 RbZMnCluo2H20.245 The shape of the magnetization curve is related to TA since the ratio of M(T) to M0 will E depend on the number of magnons present at temperature 46 T. In MnCl2 4H2O, TAE - .8TN, ~ 0 expect TAE — 1.8 K for Rb2 The three estimates of TAE cited above suggest that there is a large energy gap in the magnon spectrum which we therefore might MnClu'2H2O. supports our qualitative conclusions from the shape of the curve in Fig. 5.4. In order to compare the temperature dependence of Tln with theory we must determine the relaxation mechanism. As discussed in Chapter IV, the presence of a large gap eliminates direct and three—magnon processes, leaving the two magnon Raman process as the dominant intrinsic relaxation mechanism. There is the possibility that the relaxation is not intrinsic, but occurs via a paramagnetic impurity. We believe, however, that the measured Tln's are intrinsic for the following reason. As the temperature is lowered, electron relaxa— tion times, Tle’ of almost all measured ionsLl7 48 pr0portional to T_1 above lOK and in general T1n a TIe where d is between 0.25 and 1. Therefore, if impurity become relaxation were dominant in our experiments, the steep temperature dependence of T1n for Rb87 down to 0.32°K would not be observed. The theoretical temperature dependence of T1n 77 for Raman relaxation in the small k approximation is, from (4.88), % = 2KT3f; /T x(eX—1)_ldx (5.4) 1n AE where 6n“(5+1)“v2|b|2y 3n 1 N K= 3 .353 (5.5) TN 4n (3 )b kBTN A least squares fit of the Rb87 data to (5.4) for various values of TAE gave TAE = 2.36°K. The solid line 'in Fig. 5.4 represents the theoretical Tln for this value of TAE' Similar fits to the other three nuclei gave optimum T 's within 110% of this value, as shown AE in Table 5.2. It is emphasized that the values of K given in Table 5.2 were determined by fitting (5.4) to the data, and were not evaluated from (5.5). Calculated Values of T 1n A calculation of the order of magnitude of K in the small k approximation involves estimates of the parameters n, v, b and D in (5.5). It is easiest to calculate K for the proton for two reasons. First, the internal field at the proton site is largely deter— mined by the nearest neighbor Mn ion which simplifies the calculation of D. Second, the proton has no quad- rupole moment, so that quadrupole effects do not in- fluence the observed T and the free ion YN can be used 1n 78 Table 5.2. Results of least squares fit to equation (5.4) for the four nuclei studied Nucleus Spin 2K(sec-l(°K)—3) TAE(°K) Hl 1/2 1.96 x 103 2.15 0135 3/2 1.08 x 103 2.47 Rb87 3/2 0.973 x 103 2.36 Rb85 5/2 0.133 x 103 2.04 79 in the calculations. We calculate below a numerical value for 2K for the proton site studied. In the molec- ular field approximation n = l, but in higher order 49 approximation it is a function of S, z and lattice topology such that n varies between extremes of 0.68 and 0.86. Based on this we take n = 0.8. Following 50 51 %v2/3. Neglecting the effect of all but the nearest neighbor Van Kranendonk and Moriya we take b = Mn ion, we find D to be 1.61 x 103erg(oe)"lcm_3 when the internal field makes an angle of 10° to the magneti- zation and have used r1 = 2.728.52 The Mn ground state has S = 5/2, and YN for protons is 2.7 x 10)4 sec-loe-l. These estimates give 2K = .037 x 103 sec-1(0K)—3. This is too small by a factor of about 50, making the calcu- lated T1n too long by the same factor. There are two reasons why T determined by the ln small k approximation is too long. In discussing these it will be useful to consider Eq. (4.83) which is rewritten below 2 2 v Y h k _ _ WI = ___H__ 270m f(k)ku(nkp+l)(nkp){|d 3 2}dk (5.6) 2w p kk',pI First, we have assumed a linear spin wave theory in which spin waves do not interact. If one considers interacting magnons, it is found that it is easier to create a magnon if there is already one present. Taking this into account, a renormalization of the 80 energy hm must be made,53 with the result that the number k of spin waves at a given temperature increases. This in turn increases the transition probability and lowers T 1n' w Second, the small k approximation with EA << 1 e leads to an expression for wk such that for k 2 km, the integrand in (5.6) is so small that extension of the upper limit of integration from km to infinity does not affect the value of the integral. However, for appreciable anisotrOpy, this is not the case, for rea- sons discussed below. The following calculation of T is an attempt 1n to remedy the defects in the small k approximation within the framework of a linear spin wave theory. We start from Eq. (5.6). It is necessary to obtain expressions f(k), and c . First, approximate the disper- k’ Yk k sion curve by a straight line such that (Fig. 5.7) for w ak (T — T ) _ B N AE ‘fiwk - w k + kBTA (5.7) Then _ fl f(k) — akB(TN _ TAB) (5.8) Second, notice that Yk defined by (H.22) has a maximum carries the k dependence of ckl and |2 Taking value of one. Yk c which are needed in determining Id k2 kkip as TAE the value 1.8°K determined earlier by comparison ‘fiwk B H B A k; 1 .5 k,r| l -\ ‘. ‘1 km: (IT/o) Fir. g_ F ical dispersion curve for nagnons antiferru;u‘t-“ V’th large anisotrOpy in field. Equation of Straifht line in wL = [3kg TV — VAE)/whl k + (hmT Vl". 1 - is _.' ' 82 with MnClZ-HH2O, and using (5.1) through (5.3) we find “A E; 2 .H. Now Ckl’ and (H.30) c and pk are defined by (H.29) k2 _ pk ’Yk Ckl ‘ (p 2 _ Y 2)g 0kg - 2 Y 2 g (5.9) k k (L) w J, = 1 + —£ + [(1 + 49-)2 — 2 2 w 0) e e 0k (5.10) Using (wA/we) = .U, we find that pk for Yk = 1 is less than 20% smaller than pk for Yk = 0. Since we are interested only in order of magnitude estimates, let Yk = 0. Then ck2 = 0 and ck1 = l, and in the determ1na— tion of dkk'p we will have only one term. The result is 9 that 2 _ 2 _ 2 ldkkgll — Di — D (5.11) Eq. (5.6) then becomes 2 2 v y h k U x N m k n .e 2 w+ = —————— f D dk (5.12) 2“3 0 akB(TN — TA) (ex—1)2 where x = hwk/kBT. Changing variables from k to x gives VBYNahn6D2 SITN/T H ex W - T (x—x ) ——~———— dx (5.13) 3H2 6’ T /T g x 2 2n a kB(TN—TAE) AE (e —1) This expression makes no restrictions on k, and it can be shown that the integrand is far from negligible at x = ——. At T = 10K, numerical integration of (5.13) 83 gives W+ = 2 x 102sec_l, so T1n = 2.5 msec. The degree of agreement with our measured value of 1.75 msec is probably fortuitous, and the above calculation is meant simply to indicate that large k magnons cannot be neg- lected when the slope of the dispersion curve is small. As a check on the validity of the temperature de- pendence of (5.13), W+ was calculated for T = .5°K. The result was W+ = 16.6 sec—1 which gave T1n 2 30 msec again in close agreement with the experimental value of M0 msec. 87 A calculation of T for Rb and Rb85 is more 1n difficult, because the internal field at the rubidium site is not dominated by one Mn ion, as it was for H1, but is determined by three or four neighboring Mn ions.5u Any estimate of D would be more approximate than the calculation of D for protons and no such estimate has been made. However, since Rb85 and Rb87 occupy the same crystallographic site, we can attempt to explain the difference in the Tln values for the two nuclei. Neg- lecting quadrupole effects, this difference arises from the YNZ factor in Eq. (5.5). The nucleus with larger YN (Rb87) will relax faster. Qualitatively this is seen to be the case. To make a quantitative determination of T it is necessary to determine an effective YN 1n for each isotope. Relaxation at the Cl35 sites occurs through fluctua— tions in the transferred hyperfine interaction, 84 LHIS = l-é-S where A is the hyperfine tensor. Since the internal field at the Cl35 sites is essentially paral— lel to the magnetization direction, the (x, y, z) and (x', y', z') coordinate systems are identical. The terms in i which couple IX and Iy to fluctuations in S2 are the anisotropic terms sz and Ayz' The Hamiltonian which induces nuclear transitions from the excited state to the ground state via two magnon Raman terms can be obtained from 3115 = %(AXZ — iAyZ)