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Studies of magnetism using nuclear orientation and related NMR techniques Pond, James 2000

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STUDIES OF M A G N E T I S M USING N U C L E A R ORIENTATION A N D R E L A T E D N M R TECHNIQUES By James Pond D E U G , Licence, Maitrise, D E A , Universite de Paris XI , Orsay, 1991-1996 Magistere Interuniversitaire de Physique, Ecole Normale Superieure, 1996 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA 2000 © James Pond, 2000 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B.C. , Canada V6T 1Z1 Date: S Abstract Nuclear Orientation and related NMR techniques have been used to study three magnetic insulators: Mn(COOCH 3 ) 2 -4H 2 0 , MnCl 2 -4H 2 0 and CoCl 2 -6H 2 0. Continuous wave NMR thermally detected by Nuclear Orientation has been used to investigate the magnetic properties and spin dynamics of the quasi-2-dimensional fer-romagnet 5 4 M n - M n ( C O O C H 3 ) 2 - 4 H 2 0 . The system exhibits a frequency pulling effect due to the indirect Suhl-Nakamura interaction between nuclear spins and the electronic spin excitation spectrum is related to the coupling strength of the nuclear spins. The temperature dependence of the frequency pulling effect was measured for the two crys-talline sublattices M n l and Mn2 in low magnetic field. The spectra show a structure not predicted theoretically. The current theory is valid only for / = 1/2 with uniaxial crystalline anisotropy fields. The theory of frequency pulling has been extended here to the case of / > 1/2 and non-uniaxial crystalline anisotropy fields and the resonant frequencies and linewidths have been calculated as a function of temperature. The new theory and data agree well in terms of the magnitude and temperature dependence of the frequency pulling. Discrepancies are likely due to simplifying assumptions when cal-culating the electronic magnon spectrum. Classical and quantum numerical simulations confirm qualitatively the predictions of the model. The first Low Temperature Nuclear Orientation experiments on isotopes implanted into insulators is reported. Radioactive 5 6 M n ions have been implanted into insulating, antiferromagnetic crystals of MnCl 2 -4H 2 0 and CoCl 2 -6H 2 0. In MnCl 2 -4H 2 0 , comparison of the 7-ray anisotropy of the 5 6 Mn nuclei with that of 5 4 Mn, doped into the sample during growth, showed that both the 5 6 M n and 5 4 M n spins felt a very similar hyperfine field. ii The site occupancy factor in a simple, two site model was deduced to be 0.96i0;o7- I*1 CoCl2-6H20, the average hyperfine field for the implanted 5 6 M n was significantly less than that for 5 4 M n and corresponded to f = 0.53 ± 0.10. Radioactive and stable Co impurities, doped into MnCl2-4H20 during growth, have been studied by Nuclear Orientation and NMR. The hyperfine fields of 5 7 Co and 6 0 Co, measured by Nuclear Orientation, were found to be significantly different. The NMR resonance of the Co impurities was not found, probably because the spin-lattice relaxation time is too short for the resonance to be observed with the radiofrequency modulation used. The effects of 3.96% stable Co on the host Mn spins were studied by NMR. The presence of the impurities broadens the NMR resonance and reduces the spin-lattice relaxation time by an order of magnitude. New aspects of the design of radiofrequency coils for use in continuous wave NMRON are reported, and the important effects of transmission lines are presented. iii Table of Contents Abstract ii List of Tables vii List of Figures ix Acknowledgements xx 1 Introduction 1 1.1 Note on units 4 2 General introduction to NO and its NMR related techniques 5 2.1 The NO parameters 5 2.2 NO parameters for the nuclei used 8 2.3 Nuclear Magnetic Resonance (NMR) 11 2.3.1 Classical treatment 14 2.3.2 Quantum mechanical treatment 15 2.4 Spin-lattice relaxation 18 2.5 RF enhancement 20 3 Radiofrequency Circuits for Nuclear Magnetic Resonance 23 3.1 General considerations 24 3.1.1 Ideal case 24 3.1.2 A more complete model 25 iv 3.2 Tuned LC circuits 27 3.3 Untuned circuits 36 4 A study of frequency pulling effects in Mn(COOCH 3 ) 2 -4H 2 0 42 4.1 Introduction 42 4.1.1 The 5 4 Mn impurities 45 4.1.2 The host 5 5 Mn 46 4.1.3 The effect of the Suhl-Nakamura interaction on the 5 4 M n impurities 47 4.1.4 Using NMR-TDNO to study the frequency pulling 49 4.2 Experimental Results 50 4.2.1 Sample preparation 50 4.2.2 Data acquisition 52 4.2.3 Data analysis 55 4.3 Conclusion 59 5 The general Suhl-Nakamura interaction 61 5.1 Introduction 61 5.2 The electronic magnon spectrum 62 5.3 The Suhl-Nakamura interaction between nuclear spins 65 5.3.1 The pseudoquadrupolar interaction 69 5.3.2 The interaction for the host spins 71 5.4 The frequency pulling at T = 0 calculated with spin wave theory 75 5.5 Linewidth • • 77 5.6 Calculating the exact power spectrum for 1 to 4 spins 82 5.7 Classical model and simulations 85 5.8 Comparing the theoretical model to the experimental results 89 5.9 Conclusion 101 v 6 First NO of a radioactive isotope implanted in insulators 103 6.1 Introduction 103 6.2 Sample Preparation and Experimental Procedure 105 6.3 Results and Analysis 108 6.3.1 5 6Mn-MnCl 2-4H 20 experiment 109 6.3.2 5 6Mn-CoCl 2-6H 20 experiment 113 6.4 1 0 4 A g implantation 118 6.5 Conclusions 118 7 Cobalt impurities in MnCl 2 4H zO 120 7.1 Sample preparation and experimental details 121 7.1.1 Stable Co 122 7.2 Data collection and analysis 123 7.3 NO Results 129 7.4 NMR results 135 7.4.1 Determining RF power 135 7.4.2 5 7 Co and 6 0 Co NMR 137 7.4.3 NMR results on 5 4 Mn - the influence of Co impurities 139 7.5 Conclusions 154 8 Summary and conclusions 156 Bibliography 159 vi List of Tables 2.1 The nuclear orientation coefficients Uk and Ak for various isotopes 11 2.2 The am{6) coefficients for various isotopes for 9 = 0°, 90°. Note that both 7 decays of 6 0 C o have the same coefficients. Although the Uk and Ak coef-ficients are different, the product UkAk is the same. For pure transitions, the results are exact and are given as fractions. When transitions of mixed multipolarities are involved, the accuracy of the values are limited by the knowledge of the mixing ratio, 6. All terms are calculated assuming that Qk = 1 12 7.3 The activities of the 3 crystals grown to investigate the properties of im-purities in MnCl 2-4H 20 . 120 7.4 The fraction of each isotope in solution which grows into the crystal. . . 123 7.5 The nuclear orientation results for the 3 crystals The statistical errors are given from the fit parameters except for the 122 keV and 136 keV lines in which case a systematic error of 0.01 is assumed and the statistical errors from the fit parameters are given in parentheses 133 vii 7.6 The base temperature and hyperfine fields for the 3 crystals. The first er-ror given is statistical. The second is the systematic error due to possible misalignment of the crystal. A possible misalignment of 7° was used to calculate the systematic error. The error on the 136 keV line is purely systematic: the first error is from the 1% systematic error in the detec-tors (the statistical errors in the detector are negligible compared to the systematic error) and the second is from the misalignment of the crystal. 133 viii List of Figures 2.1 The level splittings of an I — 7/2 nuclear level for the case of a pure magnetic interaction (left) and a pure electric interaction (right). In the general case, both interactions are present 6 2.2 The decay scheme of 5 4 M n . 5 4 M n has a magnetic moment of ii = 3.2819 ± 0.0013/ZAT and a quadrupole moment of Q — +0.33 ± 0.03 b 9 2.3 The simplified decay scheme of 5 6 M n . The mixing ratios for the 2523 keV, 2113 keV and 1811 keV transitions are, respectively, S = 0.25 ±0 .15 , 6 = 0.27 ± 0.03 and 5 = -0.18 ± 0.01. 5 6 M n has a magnetic moment of fi = 3.2266 ± 0.0002//JV 9 2.4 The simplified decay scheme of 5 7 C o . The mixing ratio for the 122 keV transition is 5 = 0.120 ± 0.001. 5 7 C o has a magnetic moment of ii = 4.720 ± 0.010/ijv and a quadrupole moment of Q = +0.52 ± 0.09 b. . . . 10 2.5 The simplified decay scheme of 6 0 C o . 6 0 C o has a magnetic moment of H = 3.799 ± O.OOSLIN and a quadrupole moment of Q = +0.44 ± 0.05 b. . 10 2.6 A diagram of the enhancement effect. The perpendicular component of Bhf is added to the applied R F field b when performing N M R on magnetic materials 22 ix 3.7 The typical configuration for an NMR coil. The frequency generator, of output impedance R0 drives n lengths of coaxial cable. Each cable, j, has a length Lj and characteristic impedance Rj. At the connection between cable j and j + 1 is an impedance to ground, Zj. This impedance is generally due to capacitive coupling in the connectors, and is typically on the order of 1 pF. Zn is the impedance of the NMR coil 25 3.8 The series configuration. The resistor, it!, may be only the resistance of the coil itself 27 3.9 The parallel configuration. The resistor, R, may be only the resistance of the coil itself 28 3.10 Series-parallel configuration 28 3.11 The power spectrum for a series-parallel coil connected to an ideal trans-mission line, calculated from equation 3.56. The coil has L = 7 • 10~9 H, C 2 = 33 pF, Ci - 16 pF and R=\ ft. The generator voltage is v0g = 1 V 29 3.12 The power spectrum (calculated from equation 3.64) for the same coil as figure 3.11, but connected to a generator and a series of 4 transmission lines as in figure 3.7. The parameters are Lx — 10 m, L 2 = 1.5 m, L 3 = 0.10 m, L 4 = 0.30 m, Z0 = Zi = Z2 = Z3 = 1 pF, RQ = Rx = 50 ft, R2 = 53 ft, R3 = 55 ft and #4 = 40 ft 29 3.13 The power spectrum for the same coil as figure 3.11, calculated from equa-tion 3.64. The parameters are the same as figure 3.12 except for Z2- Here, Z2 = 5 pF 30 3.14 The voltage amplitude across the coil as a function of frequency (calculated from equation 3.55) corresponding to the power spectrum in figure 3.11 . 31 3.15 The voltage across the coil as a function of frequency (calculated from equation 3.63) corresponding to the power spectrum in figure 3.12 . . . . 32 x 3.16 The feedback loop which allows the R F power to be stabilized 34 3.17 The circuit used for the pickup coil. The same circuit, with different component values, is used for the untuned main coil 35 3.18 The generator voltage, vg, as a function of frequency for a stabilized sweep near the resonance of a series parallel circuit similar to that of figure 3.10. 35 3.19 The power dissipated in the total circuit (see figure 3.17) as a function of frequency for V = 7 • 10~9 H , R[ - 300 ft, J% = 60 ft, calculated from equation 3.56. This is the ideal case of a single 50 ft transmission line leading to the circuit 37 3.20 The power dissipated in the total circuit as a function of frequency includ-ing the effects of the transmission lines and the R F generator, calculated from equation 3.64. The values in the transmission line are the same as in figure 3.12. Note that the most rapid oscillations are due to standing wave resonances in the 10 m line leading to the cryostat. The slower oscillations are due to the 1.5 m line inside the cryostat. The slowest oscillations are due to the 0.3 m line from the 1 K pot to the coil. The oscillations due to the 0.1 m stainless steel line from the 1 K pot to the 4.2 K He bath are not visible in this frequency range 38 3.21 The power dissipated in R[, PTOT — l " 0 n ( 2 1 + r " ) l 2 ; as a function of frequency. This quantity is proportional to the square of the current in the coil. . . 38 3.22 The power dissipated in R[, PTOT — ^ ° " ^ r " ^ 2 , as a function of frequency including the effects of the transmission lines and the R F generator. . . . 39 3.23 The value of a cold finger resistor which responds directly to R F heating as a function of frequency 40 3.24 The power spectrum of the coil alone in the ideal model (R[ = 1 fi, R^ — oo), calculated from equation 3.56 40 xi 3.25 The power spectrum of the coil alone including the effects of the trans-mission lines and the R F generator (R[ = 1 Q, R'2 = oo), calculated from equation 3.64 41 4.26 The MnAc crystal lattice and magnetization. The easy axis of magnetiza-tion is the a-axis 44 4.27 The orientation of the seed crystal in the groove cut in teflon and covered with a microscope slide in preparation for further growth with saturated solution containing some 5 4 Mn 51 4.28 The experimental setup inside the SHE dilution refrigerator. The Nal or Ge detectors are outside the cryostat, at room temperature. 53 4.29 Two frequency spectra showing four resonance lines in zero applied field. The frequency was swept upwards. Although the temperature varies through-out the sweep, the lower line is in general warmer than the upper line and consequently experiences less frequency pulling. The unpulled resonant frequencies are 551.8 MHz for the M n l spins and 606.7 MHz for the Mn2 spins 56 4.30 A comparison of 3 spectra. Although the temperature of the sample varies throughout the frequency sweep (from about 30 mK to over 100 mK), in general Ta > Tb > Tc and the frequency pulling can be clearly seen. The frequency was swept downwards 57 4.31 A n example of the power spectrum with a best fit line. The frequency was swept downwards. At the start of the sweep the temperature is approx-imately 35 mK. When W(0) approaches 1, the temperature is over 100 mK 58 4.32 The position of each of the four resonant lines as a function of temperature. 59 xii 5.33 The value uk over the first Brillouin zone. Note that uk is only different from unity near A; = 0. The graph is calculated with J/kB = 0.456 K, BCA = 0.14 T and B\ = 0.86 T 66 5.34 The value \vk\ over the first Brillouin zone. Note that vk is only different from zero near k = 0. The graph is calculated with J/kB = 0.456 K, BCA = 0.14 T and BbA = 0.86 T 67 5.35 The mean square width for Uu = 0, Utj = 14.1 MHz, and £ t tfg = 6.3 MHz. The dashed line is the prediction of Pincus 80 5.36 The mean square width for Uu = 1.9 MHz, £ V Uj — 14.1 MHz, and E i Uij = 6-3 MHz 81 5.37 The exact power spectrum for one to four spins at ksT/h = 500 MHz or T = 24 mK, including pseudoquadrupolar interactions of 2 MHz and nearest neighbour interactions of 1 MHz 83 5.38 The exact power spectrum for four spins at different temperatures, in-cluding pseudoquadrupolar interactions of 2 MHz and nearest neighbour interactions of 1 MHz. The temperatures (T = hv/k0) are: 4.8 mK, 24 mK, 48 mK and 72 mK 84 5.39 The frequency pulling for a variety of interactions at T = 10 mK. Each data point is calculated by increasing the range of U^. The first point corresponds to only a self interaction of 2 MHz. A nearest neighbour interaction of 1 MHz was added, then a second nearest neighbour inter-action of 0.5 MHz and finally a third nearest neighbour interaction of 0.2 MHz. The solid line is the best fit to the data, and the dashed line is the prediction of equation 5.155 because at T = 10 mK, < Iz >= 2.046. . . . 87 xiii 5.40 The frequency pulling for a variety of temperatures (10, 20, 50, 100 mK) with a self interaction of 2 MHz and a nearest neighbour interaction of 1 MHz. The solid line is the best fit to the data, and the dashed line is the prediction of equation 5.155 88 5.41 The quantity < Izj — If >, which is proportional to the power spectrum, as a function of frequency for triplets of spins. The self interaction term which allows interactions between the three spins within a triplet is 2 MHz. The unpulled frequencies are 573.8 MHz and 631.7 MHz. The temperature is 10 mK. The black line is the average change of the the Mnl spin, while the grey line is the average change of one of the Mn2 spins 90 5.42 The quantity < \If — 7?| > as a function of frequency for triplets of spins. The self interaction term which allows interactions between the three spins within a triplet is 10 MHz. The unpulled frequency is 631.7 MHz. The temperature is 1 mK. The black line corresponds to the Mnl spin, while the grey line corresponds to one of the Mn2 spins 91 5.43 The spectrum of the zero momentum nuclear spin excitations as a function of temperature and the experimental data. The best fit values are B\ = 0.27 ± 0.03 T and B% = 0.81 ± 0.05 T 92 5.44 l/x2 as a function of both BbA and BCA for the fit to the lowest frequency resonance (corresponding mainly to precession of the Mnl spins) 93 5.45 x 2 a s a function of BCA for the lower of the 2 resonances corresponding mainly to precession of the Mn2 spins. BbA was fixed at 0.27 T. The graph shows the best fit be BCA = 0.81 ± 0.05 T 95 xiv 5.46 x2 a s a function of B°A for the higher of the 2 resonances corresponding mainly to precession of the Mn2 spins. B\ was fixed at 0.27 T. There is no physical minimum for x 2 , indicating that the model does not describe the magnitude and temperature dependence of this resonance 5.47 The spectrum of the zero momentum nuclear spin excitations as a function of temperature for uniaxial anisotropy (BA = 0.35 T and BA = 0) and the data 6.48 Variation of the 7-ray anisotropy, e = W(0) / W(90), versus time for implanted 5 6 M n (•) and 5 4 M n (O) in M n C l 2 4 H 2 0 single crystal. The dashed curve through the 5 6 Mn data is calculated from the fit through the 5 4 M n data that is a simple exponential decay described in equation 6.160, with / = 0.96 6.49 The comparison of the 5 6 M n anisotropy W(0)/W(90) with that for 5 4 M n in MnCl 2 -4H 2 0 single crystal. The solid line is the best fit to the data and corresponds to a value for the lattice site occupancy / = 0.96±(j;o4, in-close to unity. The dashed line corresponds to a value for the lattice site occupancy / = 1 6.50 The plot of the normalized intensities for 5 4 M n measured in the two de-tectors. Detector 2 is at an angle of 90° to the magnetization direction and therefore measures a normalized intensity W2 = W(90). Because of the alignment of the CoCl 2 -6H 2 0 crystal, detector 1 is at angle 9\ to the magnetization direction, and the best fit to the data (dashed line) shows that 0i = 27° so that WI = W(27) xv 6.51 Variation of 7-ray anisotropy, e — 1^(27)/^(90), versus time for im-planted 5 6 M n (•) and 5 4 M n (O) in C o C l 2 - 6 H 2 0 single crystal. The dashed curve through the 5 6 Mn data is calculated from the fit through the 5 4 Mn data that is the double exponential decay described by equation 6.161, w i t h / = 0.53 116 6.52 The comparison ofthe 5 6 M n anisotropy with that for 5 4 M n in C o C l 2 - 6 H 2 0 single crystal for the data shown in figure 6.51. Because the last two points in fig. 6.51 have the same value of 0.811 for [W(27)/W(90)]54, the average of the two [W(27)/W(90)] 56 values is taken to give the last point in this figure. The best fit is given by the solid line and gives a value of the lattice site occupancy / = 0.53 ± 0.10. The dashed line corresponds to / = 1. . 117 7.53 Data collection setup 124 7.54 The full Ge detector spectrum for crystal C 126 7.55 The Ge spectrum for the 1330 keV 6 0 C o line. The line is the best fit to the data 127 7.56 The full spectrum for the Nal detector for crystal B . The gray line is the best fit 128 7.57 W(0) as a function of time, showing the initial cooling of the entire crystal. 130 7.58 W(0) as a function of the current in the superconducting magnet. The spin-flop transition begins at approximately 6.35 A . The transition is broadened by domain effects 131 7.59 The number of counts at 0° as a function of frequency showing the 5 4 Mn resonance in an applied field of 0.2 T. The data were collected using a Ge detector and a SCA. The generator voltage is 400 mV, the step size is 25 kHz and the R F modulation is 50 kHz 135 xvi 7.60 Determining the saturation point for the RF . The frequency was set to 499.7 MHz, the resonant frequency of one of the 5 4 M n lines with a modu-lation of 50 kHz. The generator voltage was increased in 200 mV until the resonance was saturated. In subsequent frequency sweeps, 400 mV was used as the peak to peak generator voltage 136 7.61 The number of counts at 0° as a function of frequency, showing the N M R resonance of 5 4 M n in zero field. The frequency was swept upwards. The frequency step is +100 kHz, the modulation is ±100 kHz. The best fit parameters are K = 34170 ±27, B = 257 ± 7, a = (254 ± 32) kHz, u0 = (500.27 ± 0.01) MHz and 7\ = (176 ± 68) • 103 s 141 7.62 The number of counts at 0° as a function of frequency, showing the N M R resonance of 5 4 M n in zero field. The frequency was swept downwards. The frequency step is -50 kHz, the modulation is ±50 kHz. The best fit parameters are K = 34196 ±27, B = 260 ± 4, a = (205 ± 21) kHz, UJ0 = (500.36 ± 0.01) MHz and 7\ = (253 ± 18) • 103 s 142 7.63 The number of counts at 0° as a function of frequency, showing the N M R resonance of 5 4 M n in zero field. The frequency was swept upwards. The frequency step is +5 kHz, the modulation is ± 5 kHz. The best fit pa-rameters are K = 34685 ± 22, B = 266 ± 3, a = (196 ± 8) kHz, u0 = (500.21 ± 0.01) MHz and 7\ = (288 ± 9) • 103 s 143 7.64 The number of counts at 0° as a function of frequency, showing the N M R resonance of 5 4 M n in zero field. The frequency was swept downwards. The frequency step is -2 kHz, the modulation is ± 5 kHz. The best fit parameters are K = 34311 ± 35, B = 241 ± 15, a = (168 ± 10) kHz, u0 = (500.42 ± 0.01) MHz and 7\ = (488 ± 146) • 103 s 144 xvii 7.65 The number of counts at 0° as a function of frequency, showing the NMR resonance of 5 4 Mn in zero field. The frequency was swept upwards. The frequency step is +5 kHz and the modulation is ± 5 kHz. The best fit parameters are Kx = 34577 ± 31, K2 = 34432 ± 31, B = 287 ± 4, ox = (155 ± 7) kHz, u! = (500.32 ± 0.01) MHz, a2 = (205 ± 58) kHz and u2 = (499.97 ± 0.06) MHz, r = 0.14 ± 0.04 and fa = 84 ±2 kHz. The thick line is the best fit. The thin line is the shape of the double Gaussian NMR resonance in arbitrary units 147 7.66 The number of counts at 0° as a function of frequency, showing the NMR resonance of 5 4 Mn in zero field. The frequency was swept downwards. The frequency step is -2 kHz and the modulation is ± 5 kHz. The best fit parameters are the same as in figure 7.65 because both curves were fit simultaneously with the same parameters. The thick line is the best fit to the data. The thin line is the shape of the double Gaussian NMR resonance in arbitrary units 148 7.67 The number of counts at 0° as a function of frequency, showing the NMR resonance of 5 4 Mn in an applied field of 0.5 T. The frequency was swept upwards. The frequency step is +5 kHz and the modulation is ±5kHz. The The best fit parameters are K = 33838 ± 14, B = 288 ± 13, ax = (216 ± 22) kHz, D\ = (498.25 ± 0.01) MHz, o 2 = (176 ± 13) kHz and LO2 = (499.62 ± 0.01) MHz 149 xviii 7.68 The number of counts at 0° as a function of frequency, showing all six transitions between magnetic substates in zero field. The frequency was swept upwards. The frequency step is +100 kHz and the modulation is ± 1 0 0 kHz. The best fit parameters are K = 33794 ± 19, B = 264 ± 4, a = (269 ± 30) kHz, u0 = (509.68 ± 0.03) M H z , Ti = (83 ± 4) • 10 3 s and M = (3.130 ± 0.011) M H z 151 7.69 The change in level populations throughout the frequency sweep shown in figure 7.68 152 7.70 The change in level populations throughout the frequency sweep shown in figure 7.68 as they would be if no relaxation effects were present, i.e. if T i = oo 153 xix Acknowledgements I'd like to thank Brian Turrell for his excellent supervision throughout the thesis. He was always available to discuss problems which arose, but allowed me to work independently. He also understands that there is more to life than physics! I'd like to thank Andrzej Kotlicki for his help and supervision, particularly at the start of my thesis. Both Brian and Andrzej have become good friends. I'd like to thank my PhD committee for all their help with various problems, not only in physics. Of course, I'd like to thank all the technicians, particularly Doug Wong for helping me to clean so many pumps and cold traps! I'd also like to thank all the other students who worked with me at various times throughout my thesis. It was also very helpful to work with Girgl Eska, from whom I learned a great deal in a short time. I'd like to thank my family for all their support, particularly my sister, Ellen, who has helped me tremendously throughout my thesis. I'd like to thank all my friends for putting up with me, particularly Michelle, Drew, Adam, Jane, Kaspar, Anna, Jenn, Nicolas and Nicola. Finally, I'd like to thank Martin who inspired me to do this thesis in the first place, and Gabriela who gave me the support and motivation to finish it. The financial support of NSERC and UBC is gratefully acknowledged. xx Chapter 1 Introduction Nuclear spins can be oriented at sufficiently low temperatures when acted on by a mag-netic field or electric field gradient. For orientation by a magnetic field the spin must satisfy / > 1/2, whereas for orientation by an electric field gradient / > 1. Low Tem-perature Nuclear Orientation (NO) involves measuring the resulting degree of order in the nuclear spin system. At T — oo, the nuclear moments are completely disordered, whereas at T = 0 the moments are fully oriented. At finite temperature, the degree of alignment depends on the temperature of the ensemble, the intensity of the magnetic field or electric field gradient, the spin of the nucleus and its appropriate moment. The nuclear orientation can be measured through the decay products of radioactive nuclei. An ensemble of radioactive nuclei can emit radiation anisotropically and the prob-ability of emission in a given direction depends on the direction of the spins of the initial nuclei. Therefore, if an ensemble of spins is, on average, oriented in a given direction the radiation field can be anisotropic. The intensity of radiation can be measured in one or more directions and compared to the high temperature value to determine the direc-tional anisotropy of the radiation. We can define N(8,T) to be the number of counts, or intensity, of the decay products (7, ft or a particles) measured at an angle 9 to the applied field at a temperature T. The angular distribution of decay products is defined as: wm - m T ) (11) W { 6 ) ~ N(8,T = 00) ( L 1 ) W(0) can be measured and used to study many properties of the nuclear ensemble Chapter 1. Introduction 2 and its electronic environment which produces the hyperfine interaction acting on the nuclei. NO can be combined with Nuclear Magnetic Resonance (NMR) by applying a mag-netic field oscillating at the resonant frequency of the nuclei in the ordering field. In pulsed N M R , a strong oscillating field is applied for a fixed time so that the magneti-zation precesses through a determined angle (usually 90° or 180°). In continuous wave (CW) N M R , a small oscillating field is applied continuously to perturb the nuclear mag-netization. After the nuclear spin ensemble has been perturbed, its relaxation back to the thermal equilibrium is characterized by the time 7\ which provides information on how the nuclei are dynamically coupled to their environment (the lattice). This dynamic coupling between the nuclei and the lattice is called the spin-lattice interaction. The advantage of combining the methods is that N M R of Oriented Nuclei (NMRON) is many orders of magnitude more sensitive than conventional N M R . It is therefore possible to study a very small number of nuclei, for example in very small or very dilute systems. It is also possible to use radioactive nuclei embedded in a host lattice as a temperature probe. At the temperatures of interest for NO (1-500 mK), the magnetic moments of the abundant, stable nuclei generally account for most of the specific heat of a magnetic crystal. If N M R is performed on the stable nuclei in a crystal, the effect of the absorbed power is to increase the temperature of the lattice via the spin-lattice interaction. This increase in temperature can be measured by the NO of the dilute radioisotope in the crystal. The technique is called N M R Thermally Detected by NO (NMR-TDNO) [1]. A more quantitative description of NO and its N M R related techniques is given in Chapter 2. The NO parameters are calculated for the radioactive isotopes used in the experiments reported in subsequent chapters. N M R is discussed in the classical and quantum cases. Spin-lattice relaxation is considered in the case of very dilute spins, as is often the case in NO. Finally, the enhancement effect, whereby the amplitude of the Chapter 1. Introduction 3 oscillating magnetic field is magnified, is discussed. In Chapter 3, the details of radiofrequency (RF) coil design are discussed. A very general technique of calculating the R F field produced by a given coil, circuit and trans-mission lines is developed and applied to two coil designs developed for the N M R ex-periments presented here. One of these circuits is currently installed at T R I U M F in the LTNO dilution refrigerator. In Chapter 4, an experiment combining N M R O N and N M R - T D N O on a quasi-2-dimensional system, manganese acetate tetrahydrate ( M n ( C 0 0 C H 3 ) 2 - 4 H 2 0 or MnAc) is described. The "frequency pulling" of the N M R resonance is measured as a function of temperature. The theoretical predictions previously developed [2, 3, 4] apply only to the case of I = 1/2 in a crystal with uniaxial anisotropy fields and are clearly insufficient to describe the MnAc crystal where 7 = 5/2 and there are non-uniaxial anisotropy fields. Furthermore, MnAc has a triplet of spins at each lattice position. A new theoretical model is developed in Chapter 5 for the general case of 7 > 1/2 with anisotropy fields and a basis of spins. The results of the model are compared to the MnAc experiment, and found to agree well in terms of the magnitude and temperature dependence of the frequency pulling. The strength of the frequency pulling is found to depend strongly on the anisotropy fields. The failure of the model to describe the splitting of one of the resonant lines in MnAc, and the absorption of another, is most probably due to simplifying assumptions when calculating the electronic magnon spectrum. The results of the first NO measurements of implanted ions into insulators is described in Chapter 6. 5 6 M n was implanted into M n C l 2 - 4 H 2 0 doped with 5 4 M n , and C o C l 2 - 6 H 2 0 doped with 5 4 M n and 6 0 C o . The NO results of the implanted 5 6 M n are compared with the doped 5 4 M n to determine the implantation fraction. It is found that the implantation fraction is consistent with unity when the 5 6 M n is implanted into M n C l 2 - 4 H 2 0 but not when it is implanted into CoCl 2 -6H 2 0 . Chapter 1. Introduction 4 The implantation results lead to questions about the lattice positions and the effects of impurities in magnetic insulators. Chapter 7 describes a series of NO and N M R O N experiments to investigate the effects of impurities in an insulating system. 5 7 C o , 5 9 C o , 6 0 C o and 5 4 M n were doped into crystals of M n C l 2 - 4 H 2 0 . Surprisingly, the NO results indicate different hyperfine fields for 5 7 C o and 6 0 C o . The N M R resonance for Co was not found, probably because the spin-lattice relaxation time is too short. The presence of ~ 4% stable 5 9 C o is found to broaden the N M R resonance of the 5 4 M n significantly, and reduce the spin-lattice relaxation time by an order of magnitude. 1.1 Note on units A l l the formulae in this thesis are written in units where % — 1. When reporting numerical values, ft or h is reintroduced in SI units. Chapter 2 General introduction to NO and its NMR related techniques 2.1 The NO parameters If a magnetic field of amplitude B0 is applied to a nucleus of spin / the degeneracy of the 21 + 1 magnetic substates is lifted, and the energy of the mth substate is where 7^ is the gyromagnetic ratio. If the degeneracy of the magnetic substates is lifted by an electric field gradient, for example in the uniaxial case where = eq, then where Q is the quadrupole moment of the nucleus [5]. If an ensemble of nuclei is in thermal equilibrium with the lattice at temperature T, then the population of the mth nuclear sublevel is given by the Boltzmann factor where fcB is the Boltzmann constant and Z is the partition function. In order to achieve significant differences in the level populations, it is necessary to achieve temperatures such that ksT ~ Em — Em+i, or B/T ~ 104T/K in the case of a magnetic interaction. Fortunately, the hyperfine interaction (HFI) can provide magnetic fields of tens or hundreds of teslas, though it is still necessary to achieve temperatures of 1-100 mK. This temperature range can be reached with a dilution refrigerator. Em = -JNBOTTI (2.2) (2.3) (2.4) 5 Chapter 2. General introduction to NO and its NMR related techniques 6 Figure 2.1: The level splittings of an I = 7/2 nuclear level for the case of a pure mag-netic interaction (left) and a pure electric interaction (right). In the general case, both interactions are present. Chapter 2. General introduction to NO and its NMR related techniques 7 When sufficient differences in the level populations are achieved, the directional dis-tribution of emitted radiation is measured. In a typical experiment, the initial nuclear ensemble of spin i 0 decays to a state of spin i j through one or more unobserved transi-tions. Finally, it decays to a final state of spin If while emitting the observed radiation. The normalized 7-ray intensity observed at angle 9 to the magnetization axis at temperature T, in an axially symmetric state, can be expressed as [5] W(9) = YJBkUkAkQkPk{cos9) k (2.5) The Bk are the nuclear orientation parameters and describe the orientation of the initial state of spin I0. They are given by I h h k Bk(I0) = £ / 0 £ ( - l ) / 0 + m Pn (2.6) V -m m 0 where ( h h k \ V -m m 0 is a Wigner 3-j symbol and x = \/2x + 1. The Ufc are deorientation coefficients which take into account the transitions (Io —>• h) preceding the observed radiation. For a pure transition of multipole order L, between states i i and I2, the Uk are given by Uk(IJ2L) = { - l ) h + h + L + k h I 2 h h k h h L (2.7) where < h h k h h L j is a Wigner 6-j symbol. If the path from i 0 to h consists of many transitions (I0I1I2...InIi), then the deorien-tation coefficients are given by Uk(I0Ii) = Uk{hh)Uk{hI2)...Uk{InIi) (2.8) Chapter 2. General introduction to NO and its NMR related techniques 8 If there are many branches connecting the states I0 and / / then the deorientation coeffi-cient is given by a weighted average of the possible branches. Final ly , i f the unobserved radiation contains several multipolarities, the deorientation coefficients are weighted by the intensity of each multipole component. Uk{hh) = Y,MhhL)\ < I2\L\h > 17 £ I < W h > L L (2.9) The Ak are the directional distribution coefficients for the observed radiation and, in the case of a pure decay of multipole order L , Ak(Ifli) — F k ( L L I f I i ) (2.10) where the F-coefficient is given by Fk{LL'IfIi) = ( - i y f + I i + 1 k L L ' i i ( L V k\ I L V k V -1 0 Ii Ii I i if ) (2.11) If two multipoles contribute, and the mixing ratio is S, then Ak(I/Ii) Fk(LLIfIj) + 28Fk{LL'ItIi) + PF^L'L'IjIi) 1 + 52 (2.12) For 7 radiation, Ak = 0 i f k is odd because the processes involved do not violate parity. The Qk are corrections for the solid angle subtended by the detectors and the are the Legendre polynomials. 2.2 N O parameters for the nuclei used The decay scheme of 5 4 M n and the simplified decay schemes of 5 6 M n , 5 7 C o and 6 0 C o are given in figures 2.2-2.5 respectively [6]. The nuclear orientation parameters can be easily calculated, e.g. by Mathematica, and the U2 and A2 coefficients are listed in table 2.1 for each observed transition of the isotopes used in the following chapters. Chapter 2. General introduction to NO and its NMR related techniques 9 834.855 keV 100% 312.3 d 54 25 Mn O k e V E C 54 i 24 Cr O k e V Figure 2.2: The decay scheme of 5 4 M n . 5 4 M n has a magnetic moment of p = 3.2819 ± 0.0013/Jiv and a quadrupole moment of Q = +0.33 ± 0.03 b. 2.5785 h O k e V 56 25 3" 1.16% Mn 14.6% 27.9% 56 .3% fit" * ^ ^ ^ ^ V cy -v », 2 + c y c - ^ <V~ , /y ^ $ 3369.74 k e V 2959.923 k e V 2657.562 k e V 846.776 k e V O k e V 56 26 Fe Figure 2.3: The simplified decay scheme of 5 6 M n . The mixing ratios for the 2523 keV, 2113 keV and 1811 keV transitions are, respectively, 6 = 0.25 + 0.15, 5 = 0.27 + 0.03 and 5 = -0.18 ± 0.01. 5 6 M n has a magnetic moment of /J = 3.2266 ± 0.0002/i^. Chapter 2. General introduction to NO and its NMR related techniques 10 5/2" 3/2" 1/2" i> A 7/2" 271.79 d O k e V 57 27 Co E C 136.4745 keV 99.8% 14.413 keV O k e V 57 26 Fe Figure 2.4: The simplified decay scheme of 5 7 C o . The mixing ratio for the 122 keV transition is 5 = 0.120 ± 0.001. 5 7 C o has a magnetic moment of /z = 4.720 ± 0.010/ZAT and a quadrupole moment of Q = +0.52 ± 0.09 b. 5 + 5.2714 y 60 27 Co 99.925% O k e V A + St St ©I 2505.765 k e V or 1332.516 k e V O k e V 60 28 Ni Figure 2.5: The simplified decay scheme of 6 0 C o . fx = 3.799 ± 0.008/iAr and a quadrupole moment of Q 6 0 C o has a magnetic moment of = +0.44 ± 0.05 b. Chapter 2. General introduction to NO and its NMR related techniques 11 Isotope 7 energy (keV) u2 A2 uA A4 5 4 M n 834.848 0.828079 -0.597614 0.417855 -1.06904 5 6 Mn 846.771 0.638722 -0.597614 0.125907 -1.06904 S 7 C o 122.061 0.874818 0.141692 0.580288 0.01001 136.474 0.874818 -0.534522 0.580288 -0.61721 6 0 C o 1173.237 0.939374 -0.447702 0.797724 -0.30438 1332.501 0.703732 -0.597614 0.227128 -1.06904 Table 2.1: The nuclear orientation coefficients Uk and Ak for various isotopes. The angular distributions can also be expressed as functions of the level populations, W(8) = Y,am(8)Pm (2.13) m The am satisfy ] C m a m ( # ) = 2/o + 1> and, of course, YimPm = 1- This expression of the angular distribution is particularly useful for N M R when different level populations are equalized or inverted. The change in W{9) can then be easily calculated. 2.3 Nuclear Magnetic Resonance (NMR) We shall consider an isolated spin in a magnetic field or electric field gradient along the z-axis. A resonance occurs when there are transitions between an initial and final nuclear substate. The simplest way to induce transitions between magnetic substates is to apply an oscillating magnetic field perpendicular to the applied field, B0, or the electric field gradient. The oscillating field has a frequency v = u/2ir. Energy conservation leads immediately to the resonance condition u = Ef — E{. The case of an applied, constant magnetic field, B0, will be considered in some detail, although many of the results also apply to an electric field gradient. The total magnetic field is given by B = B0uz + 6(cos ut ux + sin ut uy) (2-14) Chapter 2. General introduction to NO and its NMR related techniques 12 Isotope h 7 energy (keV) m a m (0° ) a m (90°) 5 4 Mn 3 834.848 ± 3 0 5/4 ± 2 5/3 5/4 ± 1 4/3 3/4 0 1 1/2 5 6 M n 3 846.771 ± 3 0.36296 1.24319 ± 2 1.20088 1.07533 ± 1 1.30187 0.82395 0 1.26858 0.71505 5 7 C o 7/2 122.061 ± 7 / 2 1.19398 0.96142 ± 5 / 2 1.01844 0.99636 ± 3 / 2 0.91687 1.01654 ± 1 / 2 0.87071 1.02569 136.474 ± 7 / 2 0 5/4 ± 5 / 2 10/7 5/4 ± 3 / 2 10/7 25/28 ± 1 / 2 8/7 17/28 6 0 C o 5 1173.237 ± 5 0 5/4 and ± 4 1 5/4 1332.501 ± 3 4/3 13/12 ± 2 4/3 7/8 ± 1 26/21 5/7 ± 0 25/21 55/84 Table 2.2: The am(6) coefficients for various isotopes for 6 = 0°,90°. Note that both 7 decays of 6 0 C o have the same coefficients. Although the Uk and Ak coefficients are different, the product UkAk is the same. For pure transitions, the results are exact and are given as fractions. When transitions of mixed multipolarities are involved, the accuracy of the values are limited by the knowledge of the mixing ratio, 5. A l l terms are calculated assuming that Qk = 1. Chapter 2. General introduction to NO and its NMR related techniques 13 If we have an ensemble of spins, with TVj spins in the initial state and Nf in the final state, then dN--£- = NfQfi-NiQif (2.15) where Qif is the transition rate between initial and final states induced by the oscillating magnetic field. The transition rate increases when the magnitude of the oscillating field is increased. In the case that Qif = Qfi — Q and taking n = Ni — Nj then ^ = -2Qn (2.16) and n = ra(0) exp(-2Qt) (2.17) The effect of the transitions is to equalize the spin populations in a time on the order of 1/Q. O f course, the initial level populations are given by the Boltzmann factor, and relax with a characteristic time T i , due to spin-lattice relaxation processes. For a spin 1/2 system n = n0 + Aexp(-^ (2.18) where n 0 is the thermal population difference. The overall rate equation due to resonance and thermal relaxation is dn „ _ no — n ,„,„> — = - 2 Q n + ^ - — (2.19) dt 1\ In a steady state, n 0 n (2.20) 1 + 2QTX v ' If Ti <§C 1/Q then n ~ n 0 , but if 7\ » 1/Q then n —> 0 and the level populations become approximately equal. Therefore, when performing N M R O N , systems with long T i are desirable (and this is generally the case because the temperature is very low) Chapter 2. General introduction to NO and its NMR related techniques 14 because it is easier to equalize the level populations and destroy the 7-ray anisotropy. However, if Xi is too long, the initial cooling of the nuclear spins is rendered difficult. The absorbed power is simply P = n(Ef - Ei)Q = n0(Ef - Et) ± + ^ (2.21) The maximum absorbed power with respect to Q (when the oscillating magnetic field becomes large) is n0(Ef - Ei)/2T1. For N M R - T D N O , it should be noted that when Q » 1/Ti the system is saturated and the transition rate due to N M R cannot be increased, even if the amplitude of the oscillating magnetic field is increased. Therefore, for this technique to be useful, systems with short Ti are desirable because the resonance is being used to warm the lattice. 2.3.1 Classical treatment The rate of change of the angular momentum, 7, is given by the torque ft = H x B (2.22) where B is the sum of the static an oscillating magnetic fields, as defined in equation 2.14 and since the magnetic moment is jl = j^I, we have ft = fl x (lNB) (2.23) We can change coordinates from the fixed reference frame (fix), of unit vectors uz, ux and iiy, to the reference frame rotating at angular velocity u = uuz (rot), of unit vectors ez = uz, ex = cos utux +sin ujtuy, ey = cos ut uy — sin utux. Since the time derivative in the rotating reference frame is given by 3 , = 3 , - a x ( 2 - 2 4 ) / rot / fix Chapter 2. General introduction to NO and its NMR related techniques 15 then dp p x (-yNB + u)= 7^/2 x Beff (2.25) / rot —* —* where Bejf = (B0 + UJ/^N)Uz + bex. If we choose Q = — j^Bo then f ) = /Zx(6e x) (2.26) / rot The solution for the z component of the magnetic moment is pz = pcos^^bt). Therefore, in the rotating reference frame, the magnetic moment precesses in the y-z plane when the resonance condition to = —y^Bo is achieved. The z-component of the magnetic moment oscillates between ± / j with frequency j^b/2n. 2.3.2 Q u a n t u m m e c h a n i c a l t r e a t m e n t The Hamil tonian for the system is H = -n0IZ + -^(I+b'(t) + rb+{t)) (2.27) where 7 ± = IX ± ily, b±(t) = bx(t) ± = bexp(±iut) and i l 0 = 1NB0. We can transform to the rotating reference frame with the rotation operator R = exp (iutF) (2.28) If 77 = RHR) and | ^ >= > then Schrodinger's equation can be writ ten as i ^ l Z = (H-iRRi)\y > (2.29) dt and, since iRR) = LJIZ, ((Oo + w)/ z + ^ ( b + e - i u , t r + fe-eiw*/+)) | * > > = ffr> , . . \ T Z , 7Nfu „-i»t] dt = - Wo+uj)r + yNbIx) |* > = -ft-/|*> = - 7 j v 5 e / / • / ! * . > (2-30) Chapter 2. General introduction to NO and its NMR related techniques 16 where ft = (ft 0 + ui)uz + ^Nbux = 7ATB 6//. This is exactly the same effective field as found in the classical treatment. Clearly, the eigenvalues of this system are given by Em = | f t |m (2.31) and the eigenvectors are the usual eigenvectors of P, but rotated an angle 9, given by tan(0) = towards the z-axis. In other words, the eigenvectors are given by |tfT O >=e-mv\m> (2.32) For NMR, we are interested in the rate of transitions between different eigenstates of the unperturbed Hamiltonian, under the influence of the oscillating magnetic field. If, at time t = 0, the system is in a state |* > (0) = | ^ > (0) = \m >, then |tf > (t) = > (t) = fit exp (-zfi • It) | m > (2.33) We can calculate < \iz >= 7^ < P > when u = —fto and find < iiz >= l N < m\eiiNbIHIze-i™bIxt\m > (2.34) Since < m\P\m >= 0, and using jr,Nbl*tlze-iyNbI*t = jy s i n ( 7 i v f a ) + J* C0S(lNbt) (2.35) Then, if m = I, we have < \xz >- jNI cosijubt) (2.36) In other words, the magnetic moment precesses between ± 7 ^ / = ± / i at a frequency jnfb/2ir. This is exactly the result of the classical case. Chapter 2. General introduction to NO and its NMR related techniques 17 The probability of finding the system in a state |n > at time t is given by P(t) = |<n|#>(rj) | 2 = | < n\R) exp {-iCl • It) \m > |2 = | < n\ exp (-ift* • It) \m > | 2 (2.37) In the case 7 = 1/2, P(t) is easy to calculate, although for NO involving 7 radiation there is no anisotropy when 7 = 1/2. Using the Pauli matrices, a/2, as a representation of 7 and exp (iau • a) = COS(OJ) 4- iu • <rsin(o;) (2.38) we have the probability of transition after a time t, from the |+ > state to the |— > state, given by „ , , sin2(|Q|t/2), , l 2 , n n n . P(t) = l7Ar&l ( 3 9 ) The approximate solution for general 7, needed for NO, is the same, with |7Arfr|2 replaced with 4|W/ j | 2 where Wfi is the matrix element connecting final and initial states [7]. In pulsed NMR, the pulse time can be calculated to give ir or 7r/2 rotations of the spin based on equation 2.36 or 2.39. In this case, it is clear that b should be as large as possible. In C W NMR, in the limit where 7; is large but t\jNb\ <C 1, we can use the relationship U m e s i n ^ ) and, with t = 1/e, P(t) ~ 27r\Wfi\2S(Q0 - w)t (2.41) The transition rate, Qji = dP(t)/dt, is Qfi~2ir\Wfi\28(n0-u) (2.42) which is exactly the result predicted by Fermi's Golden Rule. Note that Qji = Qij, a condition necessary to show that C W NMR will equalize the populations of two levels. Chapter 2. General introduction to NO and its NMR related techniques 18 2.4 Spin-lattice relaxation We will now consider in more detail the interactions with the lattice which allow the nuclear moments to achieve thermal equilibrium with the lattice. In many cases, the —# —* dominant mechanism orienting the nuclear spins is the hyperfine interaction, H = I -A-S where I is the nuclear spin, S is the electron spin and A is the hyperfine coupling tensor. The electronic spins have time dependent fluctuations away from their equilibrium values. It is these dynamic fluctuations, equivalent to a fluctuating magnetic field, that provide the mechanism for the nuclear spins to relax to the lattice temperature. In order to calculate the transition rate, it is convenient to write S(t) = So + SS(t) where So is the static electronic spin and SS(t) is the small time dependent fluctuation of the electronic spin. We can then write the hyperfine interaction in the form HHF = I-A-S(t) = I-A-S0 + I-A-5S{t) = HHF(0) + H'(t) (2.43) and find the transition rate using time dependent perturbation theory in H'(t). We define Nm as the number of nuclei with F = m. For N M R O N , the interaction between the radioactive nuclear spins is very weak because they are so dilute. In this case, the relaxation of the nuclear ensemble back to thermal equilibrium is governed by dNm ^ — N m + i Q m + i , m — Nm(Qm,m+l + Qm,m-l) + Nm-lQm-\,m (2.44) where Qm,n is the transition rate between a level m and n. We have assumed only transi-tions between adjacent levels because of angular momentum conservation. At equilibrium Chapter 2. General introduction to NO and its NMR related techniques 19 with a thermal bath (the lattice) at a temperature TL, we know that i V m + i -AE = exp( — - ) (2.45) Nm ^kBTL where AE = Em+l - Em, and therefore Qm,m+\ = Qm+i,mexp(f^-). The nuclear spin-lattice relaxation time, Tx, is defined by 1 Qm+l,m ~r~ Qm,m+1 T\ ~ 7(7 + 1) - m ( m + l ) (1 + exp(A7;/A; BT L)) 7(7 + 1) - m ( m + l ) (2.46) and therefore 1 7(7 + 1) - m ( m + l ) Q m ' m + 1 ~ Ti l + exp(AE/kBTL) _ 1 7 ( 7 + l ) - m ( m + l ) Vm+l.m - T l ! + e x p ( _ A j B A b T l ) Qm,m+i can also be calculated using Fermi's Golden Rule. If we assume that the electronic spin system has a probability exp(—En/kbTL)/Zs of being in a state n of energy En and the nuclear spin is in a state F = m of energy Em. The transition rate is Qm,m+i = 2 7 r £ | <m + l,ri\H'(t)\m,n> \H{En>A--En^AE)^^^- (2.48) n,n> AS The only terms of 77' (t) that contribute are those involving 7 + . Assuming for simplicity that the hyperfine coupling, A, is isotropic then Qm,m+i = nA2(I{I + 1) - m(m + 1)) x £ I < n'|<JS-(t)|n > \H(En, -En + AE)^t^Mll (2.49) / ZQ n,n' J Since - En + A E ) = ^ d t e - ^ ' - ^ + ^ S and = eiHtSS-(0)e-iHt we have Qm,m+i = (/(/ + 1) - m(m + 1))— / d t e - i A £ f < 8S+(t)SS-(0) > (2.50) Z J—oo Chapter 2. General introduction to NO and its NMR related techniques 20 We can calculate Q m + i t m in the same fashion and it can be shown that Q m + i , m = Qm,m+i exP(fcf^")) a s expected. Comparison with equation 2.47 gives a value for l / 7 i of 1 42 /.oo - = — / dte~iAEt < 5S+(t)6S-(0) > (1 + exp(AE/kBTL)) (2.51) 11 Z J-oo where the brackets <> indicate a thermal average, ie < O >= Tr[e~0HO]/Z. This relates the relaxation time to the Green's function < 6S+(t)SS~(0) > and is the same expression given by [8, 9]. For magnetic systems, the relaxation time can, in principle, be calculated at low tem-peratures using spin wave theory. Since S0 from equation 2.43 is simply Suz, 8S+ = S+ \/25a and 5S~ = S~ ~ \/2ScJ, where a and a) are bosonic annihilation and creation operators. The relaxation time can be calculated if the electronic magnon spec-trum is known. In particular, it is clear that the relaxation time will become shorter if the electronic magnon energy gap becomes small. If it is possible to modify the size of the gap, then 7\ can be changed. This effect will be discussed further when dealing with magnon cooling in MnCl2-4H 20. 2.5 RF enhancement When NMR is performed on a magnetic material, the oscillating field b is enhanced. This makes it easier to perform NMR because the strength of the oscillating field at the nucleus can be much larger than the applied oscillating field. The effective field orienting an electronic spin is given by Beff = BA + B0 (2.52) where BA and B0 are the anisotropy and applied fields respectively. When the R F field is applied, in general perpendicular to the orienting field, the total field on the electronic Chapter 2. General introduction to NO and its NMR related techniques 21 spin is Bor = Bef! + b (2.53) If the electronic magnetic moment follows the orienting field adiabatically, the hyper-fine field Bhf = AS/JN also follows the direction of the field B o r . The total magnitude of the field at the nucleus precessing at frequency cu in the x-y plane is given by the sum —• —* of b and the perpendicular component of B^f, therefore hot = b + B h f - ^ -Dor = » ( l + | k ) (2-54) Since it is often the case that Bhf/Beff » 1, there can be significant enhancement ofthe applied R F field. This effect is shown schematically in figure 2.6. Chapter 2. General introduction to NO and its NMR related techniques 22 Figure 2.6: A diagram of the enhancement effect. The perpendicular component of Bhj is added to the applied R F field b when performing N M R on magnetic materials. Chapter 3 Radiofrequency Circuits for Nuclear Magnetic Resonance When performing NMR, the radiofrequency (RF) magnetic field is usually provided by a coil, although in some very high frequency applications a microwave cavity is used to provide high frequency fields. In the experiments considered here, the oscillating magnetic field was provided by a coil because the frequencies used were always less than 1 GHz. For C W NMR, the strength of the magnetic field is not always the most important consideration. For long relaxation times, applied oscillating fields of 1 0 - 7 T can be sufficient to saturate the resonance, due to the enhancement of the R F field, so that good impedance matching of the resonant circuit to the transmission line is not always necessary. The NMR experiments presented here sometimes involved sweeping the frequency of the R F field over ranges greater than 100 MHz. This type of measurement requires an RF coil, tuning circuit and transmission lines that can provide a constant intensity of magnetic field over the entire sweep range. If the intensity of the field changes dramat-ically, for example due to cable or circuit resonances, it is possible to miss observing resonances due to an insufficient magnetic field at the resonant frequency, or to detect false resonances due to the increased R F heating near a cable or circuit resonance. In this chapter a quantitative analysis of circuits for N M R O N and N M R - T D N O ex-periments is presented. A model that includes the effects of an arbitrary number of transmission lines and an R F generator is developed to calculate the power spectrum of 23 Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 24 the circuits. 3.1 General considerations 3.1.1 Ideal case A n electromagnetic wave travelling down a transmission line terminated by an impedance, Z, is reflected with an amplitude ratio r = , where RQ is the characteristic impedance of the tranmission line. The voltage and current in the transmission line, for a wave of frequency v = tu/2n, are given by v _ U q fj(wt-k(x-L)) + rei(wt+k(x-L))^ i = — (eHv>t-Kx-L)) _ rei(wt+k(x-L))\ (3.55) Ro ^  ' where L is the length of the cable. The power dissipated by the impedance is given by P = v [ R(i)9t(v) Jo ( l - | r | 2 ) (3.56) 2R0 where 3?(i) and 3?(u) are the real parts of i and v respectively. Since the usual coil set-up is much smaller than the wavelength of the electromagnetic field, we can calculate the magnetic field produced using D C formulae. In our experi-ments, a single turn coil was used in order to keep the inductance as low as possible. In this case, the field at the centre of the coil is given by B = W (3.57) 2r c where r c is the radius of the coil. For a typical geometry, rc ~ 0.5 cm and oscillating fields of 10~7 T can be produced by currents on the order of 1 mA. Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 25 3.1.2 A more complete model At frequencies greater than 100 MHz, the equations for the voltage and current are more complicated because v0 in equations 3.55 and 5.38 depends not only on the generator voltage, but also on the precise setup of transmission lines between the generator and the circuit. Stray capacitances and mismatched impedances between the different sections of coaxial cable can have important effects. For example, at a frequency of 500 MHz, a capacitance of 1 pF has an impedance of approximately 300 Q. We shall consider a Figure 3.7: The typical configuration for an NMR coil. The frequency generator, of output impedance Ro drives n lengths of coaxial cable. Each cable, j, has a length Lj and characteristic impedance Rj. At the connection between cable j and j'< + 1 is an impedance to ground, Zj. This impedance is generally due to capacitive coupling in the connectors, and is typically on the order of 1 pF. Zn is the impedance of the NMR coil. typical situation (see Fig 3.7) involving a frequency generator of characteristic output impedance Ro, with an impedance Z0 to ground driving n sections of coaxial cable, each with characteristic impedance Rj, length Lj connected to the next cable (j 4- 1), and an impedance to ground, Zj. Zn is the impedance of the coil circuit. The voltage and current in the jth. section is given by Vj = voj (eW-kix-ij)) + r.ei(M(x-ij))^ {. = 1^1 ^(wt-kix-lj)) _ r,ei(wt+k(x-lj))^ Rj (3.58) Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 26 where lj is position of the end of each section of cable (ie Lj — lj — The generator voltage is v9 = vogelwt. Applying Kirchoff's laws to each cable connection, we find that Mn M12 Mj ( V* ) (3.59) where Mn e*jLj (1 + Qt± + Ri~l Zj-i Mii = -e-lK^> [ 1 - + R 'j'-i M, 1 .• 21 Rj Rj Zj—i Rj-i _ Rj-i \ R3 Z 3 - l ) R M22 = (l + ^ --=r-Z y stj ^j—l Applying Kirchoff's laws to the generator leads to Mn M11 \ Mn M22 ) i V voir 1 j = Mi 1 vm ^ (3.60) (3.61) where Mn and M i 2 have the same definitions as in equations 3.60, and $ is arbitrary and depends on the choice of M 2 i and M 2 2 . Since $ is not needed, M2i and M 2 2 can be chosen to have the same definitions as in equations 3.60. Finally, we can write / (v \ = MiM2...Mn If G = MiM2...Mn, then VOn = V0g Gn + rnGi2 (3.62) (3.63) Since rn = f"Tp", we can easily calculate the power spectrum, p - ^( i -k -P) 2Rn\G n + rnG 12 (1 - |r.|») (3.64) Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 27 A good coil design is one that provides a frequency response that is relatively in-sensitive to small changes in the parameters Zj, Rj and Lj. A coil not satisfying this requirement will present serious experimental difficulties, even if the power spectrum predicted in equation 3.56 appears satisfactory. These considerations are extremely im-portant because of the relatively high R F frequencies used (100 to 700 MHz). 3.2 Tuned L C circuits We shall first consider several tuned L C circuits and their usefulness in NMR at fre-quencies of 100 MHz to 1 GHz. The inductance of the coil can be calculated from its dimensions [10]. For a single turn, approximately 1 cm in diameter, of 1 mm thick magnet coil, L ~ 10~8 H. For an inductance of this magnitude, the series configuration (see Figure 3.8) is not desirable because any inductance in the leads of the circuit is of similar magnitude to the inductance of the coil itself. The inductance adds to the inductance of the coil thereby reducing the resonant frequency and making external tuning very difficult. L R C Figure 3.8: The series configuration. The resistor, R, may be only the resistance of the coil itself. The parallel configuration (see Figure 3.9) allows the coil itself to be isolated. This is useful for measuring the exact inductance of the coil. However, it is not possible to control the quality factor, Q, of the resonance without introducing a resistor. A useful configuration is a combination of the series and parallel circuits (see Figure Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 28 L R OOQQOO C2 Figure 3.9: The parallel configuration. The resistor, R, may be only the resistance of the coil itself. 3.10). The best use of this circuit is to fix C2 and allow C\ to be variable. C2 should be chosen to give the desired Q of the circuit, then the variable C\ can be used to tune the resonance to the desired frequency. The final adjustments to the resonant frequency should be made when the coil is mounted. Since the stray capacitance to ground is not negligible at the frequencies of interest (~ 500 MHz), there will be some shift, normally on the order of 10 MHz or less, when the copper shielding is placed around the sample in the dilution refrigerator. Finally, it is important to take into account the shifts in frequency as the circuit is cooled. Between room temperature and 77 K, the resonant frequency can shift as much as 40 MHz, but below 77 K, there is little change. The coil should, therefore, be tuned to the correct frequency in liquid nitrogen. L R Cl C2 Figure 3.10: Series-parallel configuration Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 29 Figure 3.11: The power spectrum for a series-parallel coil connected to an ideal trans-mission line, calculated from equation 3.56. The coil has L = 7 • 10 - 9 H, C2 = 33 pF, Ci = 16 pF and R = 1 Q. The generator voltage is v0g = 1 V 400 500 600 700 Frequency (MHz) 800 Figure 3.12: The power spectrum (calculated from equation 3.64) for the same coil as figure 3.11, but connected to a generator and a series of 4 transmission lines as in figure 3.7. The parameters are Lx = 10 m, L2 = 1.5 m, L3 = 0.10 m, L 4 = 0.30 m, Z0 = Zi = Z2 = Z3 = 1 pF, R0 = R1 = 50 Q, R2 = 53 O, R3 = 55 0 and R4 = 40 Q. Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 30 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 Frequency (MHz) Figure 3.13: The power spectrum for the same coil as figure 3.11, calculated from equation 3.64. The parameters are the same as figure 3.12 except for Z2. Here, Z2 = 5 pF. A simulation of a typical power spectrum for the series-parallel configuration is shown in figures 3.11, 3.12 and 3.13. Although no resistor is placed in series with the coil, the resistance of R = 1 fl was used. This value is higher than the DC resistance of the coil (typically less than 0.1 fl), but the skin effect as well as the losses due to the R F radiation at higher frequencies increase the resistance. The value of 1 fl is consistent with the observed quality factor, Q. The resistor is the only component of the coil that can dissipate power, and P is proportional to the square of the current through the resistor. The current through the resistor and the inductor (the coil) is the same and proportional to the magnetic field. Therefore, the power dissipated in the coil is proportional to the square of the magnetic field. It is clear that the transmission lines have a large effect on the power spectrum of the coil. In one possible case (see figure 3.13), the resonance is split into two peaks only because of a 5 pF capacitance to ground between two transmission lines leading to the coil. The series-parallel coil, with C\ = 16 pF and C2 = 33 pF, is a good design because the resonant frequency is not significantly shifted by the presence of the transmission Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 31 lines. When tuning coils at frequencies greater than 100 MHz, it is useful to measure the amplitude of the voltage at the input to the coil. This can be achieved with a fast oscilloscope, or using a detector which converts the high frequency signal to a D C voltage (a Pasternack 8000-50 is a good example of such a detector). Although the DC output voltage of the detector is not precisely a linear function of the input voltage amplitude, it does allow the general shape of the spectrum to be measured and the coil to be accurately tuned. The variation with frequency of the Pasternack 8000-50 is less than 20% from 100 to 600 MHz, for a given input voltage. The use of a sweep generator allows the voltage spectrum to be plotted directly on an oscilloscope. Since the voltage amplitude at the circuit is given by \vn\ — |i>on(l + fn)\, the voltage spectrum can be compared to simulations of a perfect transmission line being used to drive a series parallel coil (see figure 3.14). The signal actually observed resembles that in figure 3.15, simulated with the same parameters as the power spectrum in figure 3.12. 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 Frequency (MHz) Figure 3.14: The voltage amplitude across the coil as a function of frequency (calculated from equation 3.55) corresponding to the power spectrum in figure 3.11 The tuned L C circuit is useful when the magnitude of the oscillating magnetic field Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 32 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 Frequency (MHz) Figure 3.15: The voltage across the coil as a function of frequency (calculated from equation 3.63) corresponding to the power spectrum in figure 3.12 is an important factor. However, it has two disadvantages. Firstly, the amplitude of the magnetic field away from resonance can be too small to perform NMR, thereby reducing the frequency range over which the coil is useful. This is an important factor to consider when searching for resonances. Secondly, the amplitude of the magnetic field varies greatly with frequency near resonance and heating of the the cold finger due to increased R F power can lead to the observation of false NMR resonances. In many cases this problem can be avoided by tuning the coil resonance slightly off the NMR resonant frequency, such that the NMR frequency is situated in a nearly linear part of the coil resonance. The effect of non resonant heating can then be subtracted from the power spectrum. This is possible when the NMR resonances are narrow compared to the the coil resonance. Of course, the coil resonance cannot be infinitely broadened without sacrificing magnetic field intensity. Another solution is to compare NMR sweeps with and without frequency modulation. If the NMR lines are inhomogeneously broadened, the effects of non resonant R F heating on the cold finger can be subtracted. If the magnetic field strength is an important consideration, the NMR resonances Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 33 are homogeneously broadened, and the frequency needs to be swept over large region (comparable or larger than the width of the coil resonance), then the above solutions to non-resonant heating will not work effectively. One solution is to use feedback stabilization of the R F power. One example of this method uses a resistor on the cold finger that responds to direct heating by R F radiation. If the response time is short compared to the thermal coupling to the cold finger, the R F power can be accurately stabilized by keeping the value of the resistor constant. The method can be unsatisfactory for N M R - T D N O because the resonance is detected thermometrically so that the stabilization based on the temperature of the cold finger can work to negate the effect of the NMR. While this may not be an issue in specific cases, careful consideration of the time constants and relative thermal couplings of the thermometer and the sample are necessary. Another possibility is to stabilize with feedback from a small pickup coil. In principle, it is possible to maintain a constant magnetic field at all frequencies. In practice, this is only true over a certain region near the resonant frequency. The size of this region depends on the width of the R F coil resonance, the power of the R F generator, and, most importantly, the coupling between the transmission cables of main R F coil and the pick-up coil inside the cryostat. At frequencies far from resonance, the power to the main coil is often sufficiently high that the signal from the pickup coil is due mainly to coupling in the transmission lines and not in the R F coils themselves. With a careful choice of quality factor for our resonance, stable fields over regions larger than 100 MHz have been achieved with this technique. The feedback loop can be seen in figure 3.16. As we shall discuss in Chapter 4, this feedback stabilization technique is the best way of providing the R F field for N M R - T D N O measurements on manganese acetate tetrahydrate (MnAc). The pick-up coil should be significantly smaller than the main coil to minimize mutual inductance effects on the latter. The circuit diagram for the pick-up coil used is shown Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 34 in Figure 3.17. The resistors, R[ and R^ were both chosen to be 100 fl in order that the total impedance would be matched to the transmission lines. The parallel configuration of the two resistors was chosen to remove any effects of the stray capacitance, C, on the circuit. Since C < 1 pF it is necessary that R2' <C 1/wC for frequencies less than 1 GHz. The pickup coil has a flat response in the frequency regions of interest (less than 700 MHz) because its resonant frequency is much greater than 1 GHz. coil HP voltmeter Pasternack 8000-50 detector 20 dB amplifier Figure 3.16: The feedback loop which allows the R F power to be stabilized. The non-linearity of the pick-up circuit with magnetic field intensity is unimportant when sweeping frequency because the feedback loop works to keep its response constant. However, when modifying the R F power, one needs to keep a reference frequency at which the generator voltage can be measured. Then the output of the Pasternack 8000-50 detector can be measured and used as the new setpoint for the feedback loop. In this manner, the pick-up circuit set-point can be related to the magnetic field intensity. The generator voltage amplitude as a function of frequency during a feedback sta-bilized frequency sweep is shown in figure 3.18. The circuit is in the series-parallel configuration with component values similar to those used for the simulation in figure 3.11. The frequency is swept through the upper half of the resonance. Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 35 L' RI' c R 2 ' Figure 3.17: The circuit used for the pickup coil. The same circuit, with different com-ponent values, is used for the untuned main coil. 450 -, : , 400 A 50 A 0 -I 1 1 1 1 1 1 1 480 500 520 540 560 580 600 620 Frequency (MHz) Figure 3.18: The generator voltage, v9, as a function of frequency for a stabilized sweep near the resonance of a series parallel circuit similar to that of figure 3.10. Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 36 3.3 Untuned circuits When performing C W N M R O N to search for resonances of a dilute isotope grown or implanted into a crystal host, it is often necessary to sweep the frequency over large ranges, but the intensity of the field can be as low as 1 0 - 7 T if the relaxation times are long. In this good solution is to use untuned R F coils. The main coil design is identical to the pickup coil, shown in figure 3.17, except that R[ 3> coL'. If V ~ 10" 8 H, then LJL' ~ 60 fl at a frequency of 1 GHz. If we choose RI' = 300 fl, then, coL' <C R[. We can, therefore, neglect the effect of the coil inductance. R'2 should be chosen such that the total impedance matches the characteristic impedance of the transmission line. If Rn = 50 fl, then R^ = 60 fl. Since C ~ 1 pF, R'2 < 1/uC for frequencies up to 1 GHz. Our circuit is then approximately the same as two resistors in parallel. In the ideal case, the resistor R'2 should not be needed. However, as we have seen in the development of the more realistic model, there are reflections off the generator itself as well as the connections between coaxial lines leading down into the dilution refrigerator (DR). It is best, therefore to match impedances to reduce effects of standing waves as much as possible. Finally, there is the issue of how much power can reasonably be dissipated in the DR. In our setup, the coil was mounted on the 1 K shield. The heat dissipated in the resistors was removed through the 1 K pot. If the voltage at the input has a maximum value of 1 V ,(peak to peak) then the heat input to the D R is approximately 10 mW. In fact, a generator voltage of 0.4 V was sufficient to saturate the resonance. If the refrigerator is capable of handling a heat load of 1 to 10 mW, then this coil design is extremely practical. At T R I U M F , the LTNO DR has been equipped with an untuned coil design, and a pick up coil to stabilize the power if necessary. In this case, the coil is mounted on the magnet, at 4.2 K , and heat dissipated in the resistors is not a problem. Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 37 A simulation of the power spectrum for our circuit is given in figure 3.19 for the ideal case, and figure 3.20 for the more realistic model. The true quantity of interest is the power dissipated in R[. This quantity, Ptot - l " 0 n ^ + r n ) l 2 , is proportional to the square of the current in the coil and is shown in figures 3.21 and 3.22. A resistor mounted 1 0 9 . 9 9 9 8 S. 9 . 9 9 9 6 o 9 . 9 9 9 2 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0 8 0 0 F r e q u e n c y ( M H z ) Figure 3.19: The power dissipated in the total circuit (see figure 3.17) as a function of frequency for L' = 7 • 1 0 - 9 H , R\ = 300 ft, R'2 — 60 ft, calculated from equation 3.56. This is the ideal case of a single 50 ft transmission line leading to the circuit. on the cold finger was directly heated by the R F field at low temperatures (less than 100 mK). The resistance decreases with increasing temperature and therefore with R F power. A plot of this resistor value is shown in figure 3.23. Although it is difficult to relate the value of the resistor exactly to the R F field, it is possible to see the changes in R F field that are due to standing waves in the coaxial transmission lines in the cryostat. Although it was not necessary for the experiments discussed in this thesis, it would be possible to stabilize an R F coil by keeping the value of this resistor constant. Indeed, the immediate response of this resistor to the R F field make this a promising possibility for feedback stabilization in the future, particularly with tuned R F coils. However, we found that stabilization was unnecessary with our untuned coils because the relative variation in R F power over the frequency ranges of interest was small. There was enough power Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 38 200 300 400 500 600 700 800 F r e q u e n c y (MHz) Figure 3.20: The power dissipated in the total circuit as a function of frequency including the effects of the transmission lines and the R F generator, calculated from equation 3.64. The values in the transmission line are the same as in figure 3.12. Note that the most rapid oscillations are due to standing wave resonances in the 10 m line leading to the cryostat. The slower oscillations are due to the 1.5 m line inside the cryostat. The slowest oscillations are due to the 0.3 m line from the 1 K pot to the coil. The oscillations due to the 0.1 m stainless steel line from the 1 K pot to the 4.2 K He bath are not visible in this frequency range. 1 . 665 1 . 6625 I 1 . 6 6 1 . 6575 u <D IS 1 . 6 5 5 0 1 . 6525 1 . 6 5 1 . 6475 200 300 400 500 600 700 800 F r e q u e n c y (MHz) Figure 3.21: The power dissipated in R[, Ptot - l " 0 n ^ r " ) | 2 , as a function of frequency. This quantity is proportional to the square of the current in the coil. Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 39 200 300 400 500 600 700 800 Frequency (MHz) Figure 3.22: The power dissipated in R[, Ptot - M ^ ^ , as a function of frequency including the effects of the transmission lines and the R F generator. to saturate a resonance, but never so much that R F heating of the cold finger was a problem. Finally, to illustrate the danger of trusting ideal simulations, figures 3.24 and 3.25, show the results of a simulation of a single coil connected to ground. The resistance of the coil is assumed to be 1 ft, as before. In the ideal case, it appears that the variation of power (and therefore current and magnetic field) with frequency is small. It would seem possible, therefore, to either stabilize this field, or assume it is a constant over small frequency ranges. However, in the more realistic model, it is clear that there is a large relative variation in power (and magnetic field). Furthermore, this power spectrum is extremely unstable with respect to small changes in the model parameters. In practice, it is very difficult to ensure that enough R F power is reaching the sample, and feedback stabilization is difficult because of the high generator power required relative to the currents produced in the coil. Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 40 180 230 Frequency (MHz) Figure 3.23: The value of a cold finger resistor which responds directly to R F heating as a function of frequency. Figure 3.24: The power spectrum ofthe coil alone in the ideal model (R[ — 1 ft, i?^ = oo), calculated from equation 3.56. Chapter 3. Radiofrequency Circuits for Nuclear Magnetic Resonance 41 Figure 3.25: The power spectrum of the coil alone including the effects of the transmission lines and the R F generator (R[ = 1 Q,, R'2 = oo), calculated from equation 3.64. Chapter 4 A study of frequency pulling effects in M n ( C O O C H 3 ) 2 - 4 H 2 0 4.1 Introduction There has been recent interest in the spin dynamics of 2-dimensional spin systems [11] and the study of nuclear spin interactions ("frequency pulling") [12]. In a magnetic system, there is a relatively strong coupling between nuclear spins due to the Suhl-Nakamura interaction that involves the virtual emission and reabsorption of an electronic magnon [13, 14, 3]. At low enough temperatures, the nuclear spins collectively interact with the electron spins causing a shift in the NMR frequency. Also there are excitations of the nuclear spin system (nuclear magnons). We have chosen to investigate these effects in the quasi-2-dimensional ferromagnet manganese acetate tetrahydrate, Mn(COOCH 3 ) 2 -4H 2 0 (MnAc), which exhibits relatively large frequency pulling, by studying the abundant 5 5 M n nuclear spins and very dilute 5 4 M n radioactive spins which were doped in the sample during growth. This system has been studied previously by the techniques of NO, NMRON and N M R - T D N O [15] and a preliminary measurement of the frequency pulling showed a relatively large effect. MnAc has a crystallographic layered (a-b) plane structure [16]. The manganese ions in a given layer occupy two different sites (site 1 and site 2) and are arranged in triplet groups consisting of one M n l ion and two Mn2 ions which are internally coupled by 120° oxygen and acetate linkages. Each triplet is coupled to four other triplets within a layer by acetate linkages and there are no strong bonds between layers. 42 Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 43 The magnetic properties of MnAc have been studied extensively [16, 17, 18, 19, 20, 21, 22]. Each M n + + ion has S = 5/2 and g ~ 2.00 [22, 23]. Magnetically, the oxygen linkages within a triplet provide a strong superexchange that is antiferromagnetic with an exchange constant JAF,tripi = 48 K [18]. The superexchange between triplets in a layer is ferromagnetic and much weaker than the intra-triplet antiferromagnetic coupling [17], and the transition to the ordered state occurs at T = 3.19 K. The interlayer coupling (~ 1 mK) is very weak and antiferromagnetic accounting for the high degree of two dimensionality. The crystal structure and spin orientations of the Mn ions are shown in figure 4.26. The easy axis is the a-axis, the second easy axis is the c*-axis (perpendicular to the a-b plane) and the hard direction is the b-axis. In order to force the magnetization along the c*-axis (or b-axis) it is necessary to apply a field BCA (or BbA) along the c*-axis (or b-axis). These fields are called the anisotropy fields and in MnAc they are BCA = 0.135 T and B\ = 0.86 T [20]. At very low temperatures, a field B0 = 0.6 mT applied along the easy axis causes a transition from an antiferromagnetic ordering of the ferromagnetic layers to a mixed phase consisting of domains of ferromagnetic and antiferromagnetic ordering of the planes, and at B0 = 14 mT the planes are completely ordered ferromagnetically [15]. The strong antiferromagnetic coupling between the spins in a triplet allows us to treat the latter as an effective single spin with S = 5/2, ferromagnetically coupled to its neighbours in the ab-plane. We also ignore the interlayer interaction and treat the system as purely 2-dimensional. With these approximations all triplets are equivalent in terms of their magnetic interactions. Figure 4.26: The MnAc crystal lattice and magnetization. The easy axis of magnetization is the a-axis. Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 45 4.1.1 The 5 4 M n impurities A simplified spin Hamiltonian for both the 5 4 M n l and 5 4 Mn2 ions in an applied field B0 is H = Hx + H2 (4.65) Hx = gfiB(BE±B0)Sz + D sl - \s(s + 1) (4.66) H2 = AI-S (4.67) Here H\ and H2 represent the electronic and hyperfine interactions respectively. BE rep-resents the effective exchange field on a given electronic spin, calculated from mean field theory. The D term represents the crystal field interaction and is related to an anisotropy field by D = QIIBBAI'(25). The term Al • S represents the hyperfine interaction. The + and - signs are taken for the Mn2 and Mnl nuclear spins respectively. We have ignored the electric quadrupole interaction and dipole terms which are very small [15]. This Hamiltonian is clearly oversimplified because we have assumed uniaxial anisotropy. The H2 term, representing the hyperfine interaction, can be written as H2 = HHF + H'HF where HHF = ASZP H'HF = | ( S + / - + S - / + ) (4.68) The effect of H'HF can be treated with perturbation theory. The zeroth order states of the lowest energy multiplet are characterized by the states \SZ = —5/2, Iz = m >. The first order correction to the eigenvectors gives a small admixture of the \SZ = —3/2, Iz = m — 1 > states in the lowest energy multiplets. The energy difference between adjacent m and m + 1 levels in B0 = 0, to second order in H'HF I S AEm,m+1 = -SA + Pm (4.69) Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 46 Note that the second order effects lead to an unequal splitting of the lowest energy substates. The "pseudoquadrupolar" term P is given by P = JTVTH ( 4 7 ° ) gfjtBBE + 4D 4.1.2 The host 5 5 M n The abundant 5 5 Mn spins interact via the Suhl-Nakamura interaction [13, 14] that in-volves the virtual emission of an electronic magnon by one spin and its absorption by another. The Suhl-Nakamura interaction between spins i and j separated by distance r^ -can be represented by HSN = \Y. Unit IT (4-71) 2 M with U* = - — TeMlkUj)) (4.72) where UJk is the electronic magnon spectrum. In the case of a simple cubic, three dimen-sional lattice, U l j = -^--exY>(-7f) (4.73) Here a is the lattice spacing and b0 ~ (QIIBBE/Eg)a where Eg is the magnon energy gap. Typically b0 ~ 10, so that range of the interaction is ~ 10a. Actually, the mechanism for the pseudoquadrupolar interaction involves the virtual emission/absorption process by a single spin (Uu term). The Suhl-Nakamura interaction has a number of effects for abundant nuclei. First the coupling is usually relatively strong causing fast spin-spin relaxation, i.e. short T2, and this results in a significant homogeneous line broadening which is usually 5VSN > 1MHz and dominates other contributions. At very low temperatures (T < 1 K), when there is a significant nuclear magnetization < P >, the strong Suhl-Nakamura coupling causes Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 47 the N M R frequency to be pulled down from the value u0 = AS/h, to a value u0 — Av. Frequency pulling has been discussed e.g. [2, 3], and for the special case I = 1/2 at very low temperatures — = A < 1 * > (4.74) There is a band of nuclear magnons, i.e. excitations of the coupled abundant nuclear spins, the spectrum of which covers the range u0 — A < v < u0. These have been indirectly observed by their effect of enhancing the NSLR of the 5 4 M n 2 compared to 5 4 M n l because the frequency of the former is closer to the nuclear magnon band [15]. 4.1.3 The effect of the Suhl-Nakamura interaction on the 5 4 M n impurities The dilute 5 4 M n spins are also coupled to their neighbours via the Suhl-Nakamura inter-action. The effect of this interaction on the energy levels of the lowest order multiplet can be calculated using perturbation theory. The 5 4 M n are sufficiently dilute that we can treat them as a single impurity (at R = 0) in a lattice of 5 5 M n spins. The Hamiltonian can be written as H = HQ + HSN = AS^I' + WUijI+Ii (4-75) j i,3 We can calculate the correction in energy to a level \m, $ > where m = Ifi is the state of the 5 4 M n spin and $ is an arbitrary state of the ensemble of 5 5 M n spins. The first order energy correction comes only from the self-interaction term. Since I+I~ = / ( / + 1) - (Iz)2 + P, the correction to the energy is simply given by 6EW = \U« ( / ( / + 1) - m 2 + m) (4.76) This first order correction is another way of calculating the pseudoquadrupolar interac-tion. Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 48 The hyperfine coupling constants for 5 4 M n and 5 5 M n are different and the second order correction can be calculated using non-degenerate perturbation theory: 6E(2) _ I < m,$\HSN\m',& > 12 ( /56( /55 + 1) - m'^mr - 1)) (754(/54 + 1) + m(m + 1)) \U0j\2 \ S(A54 - A55) _ (/ 5 5(/ 5 5 + 1) - m'jjm'j + 1)) (75 4(/5 4 + 1) + m(m - 1)) \U0j\2 \ S(A55 - A 5 4 ) [ ' ' where ^454 and AB5 are the hyperfine coupling constants for the 5 4 Mn and 5 5 Mn spins re-spectively. 754 = 3 and / 5 5 = 5/2 are the nuclear spins of the 5 4 M n and 5 5 M n respectively. We could find a mean field energy correction by replacing ml- with its thermodynamic average. However, it is clear that S(A54A55)2 ( 4 7 G ) 4iV(A55 - A 5 4 ) ^ LO\ The ratio of the second order correction to the first order correction is then SEW/SB^ „ ^ K U \ (4.79) ' 2(A 5 5 - A54) Zk.Uk Without calculating the electronic magnon spectrum, it is clear that £l/w*«l/wbX;iM (4.80) k k and ^ 5 4 ^ 5 5 2(AB5 - A54)LU0 < 0.1 (4.81) Since the first order energy corrections are on the order of 1 MHz, the second order corrections are much less than 100 kHz and can be ignored. The only significant effect Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 49 of the Suhl-Nakamura interaction on the 5 4 M n spins is the self-interaction (or pseudo-quadrupolar term). 4.1.4 Using NMR-TDNO to study the frequency pulling In order to study the spin dynamics we employ the techniques of N M R O N and NMR-T D N O . The magnetic crystal is doped with radioactive 5 4 M n (a few /iCi) during its growth from a saturated solution. The crystal is mounted on the "cold finger" connected to the mixing chamber of a dilution refrigerator and cooled to a temperature 10 mK < T < 100 mK. The nuclear spins are then oriented by the hyperfine interaction and a measure of this orientation is obtained from the angular distribution of 7-rays at 0° and 90°. W(0) and W(90) can be expressed according to equation 2.13, with am(9) from table 2.2, such that W(0) = ^(p2 + P- 2) + ^(pi+P-i)+Po (4.82) W(90) = ^(p3+P-3) + ^{p2+P-2) + ^(pi+P-i) + \po (4.83) with pm given by the Boltzmann factor, exp(f^) , "=T^m (4-84) Having achieved a significant 7-ray anisotropy by cooling the sample to a low tem-perature T ~ A/kBT, NMRON can be performed by applying an R F field and sweeping the frequency. For the dilute 5 4 M n spins, the resonant frequencies ^ m,m+i = L\Em,m+\/h corresponding to m —> m + 1 transitions, can be selectively observed. The dilute 5 4 M n spins were investigated by N M R O N and gave < 5 4 A S > i /h = 437.0 ± 0.2 MHz for the Mnl site, < 5 4 AS >2 /h = 480.5 ± 0.2 MHz for the Mn2 site, and P/h = 1.2 ± 0.1 MHz [15]. Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 50 The abundant host spins were investigated using N M R - T D N O . The beauty of com-bining these methods is that the effects of the Suhl-Nakamura interaction can be observed directly by comparing the NMR signals of the abundant 5 5 M n spins, which feel the inter-action, and the 5 4 M n that do not. Thus in a 1 cm 3 crystal of 5 4 M n - M n C l 2 - 4 H 2 0 with 5 fj,Ci activity, the separation of the 5 4 Mn atoms is ~ 1000a which is much larger than the range of the interaction. Thus the line width is ~ 35 kHz because there is no broadening from the Suhl-Nakamura interaction, whereas the line width of the 5 5 M n resonance is several MHz and there is frequency pulling. In MnAc values of (Av/v) (Mnl) = 0.05 and (Av/v) (Mn2) = 0.06 were observed at T ~ 40 mK [15]. The expected values of 0.09 and 0.10 for the Mnl and Mn2 spins respectively, calculated from equation 4.74, were significantly different from the measured values. It was noted that these effects would be investigated further, and this is the purpose of the present study. 4.2 Experimental Results 4.2.1 Sample preparation Seed crystals of approximately 1 mm in length were grown from a saturated solution of MnAc by evaporation at room temperature. Unfortunately, larger crystals could not be grown by evaporation because the MnAc rapidly oxidized with the oxygen in the air, creating a brown oxide powder throughout the solution which prevented further growth. It was not possible to increase the acidity of the solution to prevent this oxidation. To prevent oxidation, a seed crystal was placed in a groove cut in a piece of teflon, approximately 5 mm in width, as seen in figure 4.27. The goniometric data of Groth [24] was used to orient the crystal. The groove was covered with a microscope slide, sealed to the teflon with vacuum grease. Saturated MnAc solution was prepared and doped with approximately 100 /xCi of 5 4 M n . The solution was filtered to remove all the Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 51 Figure 4.27: The orientation of the seed crystal in the groove cut in teflon and cov-ered with a microscope slide in preparation for further growth with saturated solution containing some 5 4 M n . Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 52 oxide powder and pipetted into the groove containing the seed crystal. The solution wet the microscope slide well (but not the teflon) and would run along the glass until the groove was completely filled with solution. Both ends of the groove were then sealed with vacuum grease to prevent any contact with the air. Throughout the whole process, the solution, pipette, filter paper, teflon, microscope slide and vacuum grease were maintained at a temperature of 30° C in a feedback stabi-lized thermal bath controlled by a PC. Once the saturated solution surrounded the seed crystal, the temperature of the bath was lowered slowly over a period of 4 days from 30° C to 23°. It was not possible to grow a non-radioactive coating on the crystal and it had to be handled carefully. The final crystal had the orientation of the seed crystal, and an activity of approximately 1 fid. The orientation of the crystal was verified by X-ray diffraction. The crystal was covered with Apiezon N grease to protect it from the air, and was attached with unwaxed dental floss to the copper cold finger of a SHE dilution refrigerator, as shown in figure 4.28. A 6 0 C o in Fe thermometer was only mounted when fridge diagnostics needed to be performed because its higher energy 7-rays would otherwise contaminate the spectrum of the 5 4 M n . Hydrated crystals deteriorate at reduced pressure at T > 240 K so the samples were precooled under an atmosphere of air. 4.2.2 Data acquisition The N M R of the stable 5 5 M n spins was observed utilizing the technique of N M R - T D N O [1] in which the radioactive 5 4 M n are used as a temperature probe. The spectra were obtained with two Nal detectors and four single channel analysers. The data was acquired from the single channel analyser output by a P C , which also controlled the R F power and frequency, and monitored the value of thermal resistors inside the cryostat. Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 53 vacuum can detector Figure 4.28: The experimental setup inside the SHE dilution refrigerator. The Nal or Ge detectors are outside the cryostat, at room temperature. Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 54 In order for N M R - T D N O to work well, the spin-lattice relaxation time, 7\, should be short compared to the dwell time of the R F frequency step and the time constant for the cooling of the sample to the cold finger. In zero field, the spin lattice relaxation times are ~ 10 _ 1 s [15], and the 7-ray anisotropy of the 5 4 M n spins provides an accurate measure of the temperature of the 5 5 M n spins. The isotropic warm counts necessary to normalize the counting rates were obtained at T ~ IK. In the previous experiment[15], only one measurement of the 5 5 M n lines was made. The frequency step size was 1 MHz and the dwell time was 200 s. The R F power was stabilized by monitoring the temperature of the cold finger, but this has two disadvan-tages: first, the time constant of the feed-back loop was relatively long; second, the cold finger must undergo some warming at resonance, but the feed-back loop works to negate the effect. This stabilization procedure works well for N M R O N of the dilute radioactive nuclei because there is essentially no warming, but is not well suited to the N M R - T D N O technique. In the present experiment, the frequency was swept in 500 kHz steps with a dwell time of 100 s. The R F field was provided by the coil shown in figure 3.10. The coil inductance was L ~ 7 • 1 0 - 9 H, the resistance was from the coil itself, the capacitor in parallel with the coil was C2 = 33 pF and the capacitor in series with the coil was variable with C i = 2 — 20 pF. The series capacitor was used to tune the coil to the desired frequency. The main coil had a broad resonance centered approximately at 530 MHz at 100 mK. The R F power was monitored by a smaller, broad-band pick-up coil mounted behind the main coil, as shown in figure 3.17. The resistor values were R[ = R'2 = 100 ft. C was only due to capacitive coupling between the coil and ground and was < 1 pF. A feedback loop, as shown in figure 3.16 allowed the R F power to be stabilized over the entire range of the frequency sweeps (> 100MHz). This eliminated the effects of frequency dependent R F heating. Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 55 In the previous experiment the 5 5 Mn lines were observed in zero field at only one temperature and no structure was observed. Also, no measurements were performed above 600MHz. In this experiment, the improved detection method allowed the resolution of the structure of the lines. Several spectra were obtained corresponding to different 5 5 M n nuclear spin temperatures and these showed four peaks rather than the expected two. Two upward frequency sweeps can be seen in figure 4.29 In the thermometric method of detecting the resonances, sweeps upward and down-ward in frequency yield different line shapes in part due to the frequency pulling effect itself. Figure 4.30 shows three spectra at different temperatures in zero applied field, for a downward frequency sweep. The same spectra were observed in applied fields up to 110 mT. In Figure 4.30 the frequency pulling effect can be clearly observed. 4.2.3 Data analysis In order to estimate the line positions as a function of temperature, the peaks were fitted by assuming a Gaussian line-shape and an exponential relaxation to the temperature of the dilution refrigerator. The change in anisotropy at frequency u is then given by where the first term corresponds to the four resonance lines, and the second term is the relaxation to the base anisotropy, Wo(0). (f>o is the modulation, r is the relaxation time, and the Ai, a>j and CTJ are the amplitudes, positions and widths of the four resonance lines. The starting anisotropy was taken to be the average of the first five data points. W(0) was then calculated by integrating equation 4.85 with respect to t and a \ 2 w a s calculated and minimized with MINUIT. The free parameters were Ai, Ui, Oi, r and Wb(0). Wo(0) was not necessarily equal to the starting anisotropy because turning on the R F power (4.85) Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 56 1.00 , 0 I . ' • '— 505.0 555.0 605.0 Frequency (MHz) Figure 4.29: Two frequency spectra showing four resonance lines in zero applied field. The frequency was swept upwards. Although the temperature varies throughout the sweep, the lower line is in general warmer than the upper line and consequently experiences less frequency pulling. The unpulled resonant frequencies are 551.8 MHz for the M n l spins and 606.7 MHz for the Mn2 spins. Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 57 ) I , , i , i i i i i i , i — i — i — i — i — i — i 500 525 550 575 600 625 F r e q u e n c y ( M H z ) Figure 4.30: A comparison of 3 spectra. Although the temperature of the sample varies throughout the frequency sweep (from about 30 mK to over 100 mK), in gen-eral Ta > Tb > Tc and the frequency pulling can be clearly seen. The frequency was swept downwards. Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 58 did affect the base temperature of the crystal through non-resonant warming of the cold finger. An example of a fit can be seen in figure 4.31. Although the relaxation processes, 1.05 - r — : 1 0.75 -I 1 1 1 1 r — 1 500 525 550 575 600 625 Frequency (MHz) Figure 4.31: An example of the power spectrum with a best fit line. The frequency was swept downwards. At the start of the sweep the temperature is approximately 35 mK. When W(0) approaches 1, the temperature is over 100 mK. and the power absorption are more complicated than the assumptions of the model, the goal was to find a position of the lines as a function of temperature and this is clearly achieved. The width of the lines (<7j) are used as the error in line position and the temperature was measured by averaging ten counts immediately before the start of the each peak. The position of the lines as a function of temperature are plotted in figure 4.32. In order to fit the data exactly, a theoretical model of the entire power spectrum Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 59 40 50 60 70 Temperature (mK) Figure 4.32: The position of each of the four resonant lines as a function of temperature. as a function of temperature would be necessary. However, the Gaussian fit gives a very reasonable estimate, within the given error, of the position of the line as a function of temperature. 4.3 Conclusion The simple model of the Suhl-Nakamura interaction for the case 1 = 1/2 with uniaxial anisotropy and a single ion per lattice site is clearly insufficient to explain the observed spectra. To the best knowledge of the author, the Suhl-Nakamura interaction with I > 1/2 has not been considered. When I > 1/2 the self interaction term (Uu) cannot be neglected. When there is non-uniaxial anisotropy, the electronic magnon gap, which determines the overall amplitude of the frequency pulling needs to be calculated by a Bogliubov transformation. Finally, when there is a triplet of ions at each lattice site, the Chapter 4. A study of frequency pulling effects in Mn(COOCH3)2-4H20 60 interactions between the spins cannot be ignored, and will modify the frequency pulling. In the following chapter, a model is developed which takes all of these important effects into account. Chapter 5 The general Suhl-Nakamura interaction: a basis of spins, I > 1/2 and non-uniaxial anisotropy 5.1 Introduction As mentioned in the previous chapter, to the best of our knowledge, the Suhl-Nakamura interaction for 7 > 1/2 has not been rigorously dealt with theoretically. Therefore we have developed a theoretical model for this case in which the self-interaction term is important. The case of non-uniaxial crystal field anisotropy is also considered and it is shown to have an extremely important effect on the frequency pulling. The frequency pulling is calculated for a triplet of spins and compared to the ex-perimental data. Furthermore, the T = 0 spectrum of excitations is calculated by spin wave theory. The effects of the M n l spins on Mn2 spins and vice-versa is found to be important. A number of mechanisms which could explain the four observed lines are considered. The linewidth is calculated in the case of a single spin per lattice site down to T = 0 and compared to a high temperature expansion previously calculated. The results agree well at the temperatures where the high temperature expansion is valid. However, our calculation also includes the effects of the self-interaction on the linewidth, and is valid as T ->• 0. The exact power spectrum involving 1 to 4 spins is calculated numerically at a variety of temperatures and classical simulations of a crystal of 15x15 spins are performed. The 61 Chapter 5. The general Suhl-Nakamura interaction 62 results qualitatively confirm the predictions of our model. 5.2 The electronic magnon spectrum The strong coupling between the three electronic spins forming a triplet allows us to treat the latter as an effective S — 5/2 single spin, ferromagnetically coupled to its neighbours in the a-b plane. We can also ignore the magnetic interactions between different layers and treat the system as a 2-D ferromagnet. With these approximations, all triplets in the a-b plane are equivalent in terms of their magnetic interactions. The magnetic anisotropy in this system is not uniaxial. The easy axis is the a-axis, BA = 0.14 T and BCA = 0.86 T) [20]. The z-direction is chosen along the a-axis, and the x and y directions are chosen to be the b and c* axes respectively. Two of the sites within the triplet of spins are equivalent and are denoted as Mn2, while the other site is Mnl . The electronic magnon gap can be calculated from the electronic spin Hamiltonian A standard Holstein-Primakoff transformation can be used, (Sz = S — o^a, S+ ~ V2Sa, S~~ ~ \f2Sa)) and is expected to give good results because the spin is large (5* = 5/2) and the temperature is low compared to the coupling between electronic spins (the Curie temperature is ~ 3 K while we are working at T < 100 mK. Working in momentum space, the Hamiltonian can be written as H = JF £ Si-Sj + Y.dVB <nn> i (5.86) H-EQ = Y1 (^ o(£)4afc + MB-£-{%a-k + 4a-fc) (5.88) Chapter 5. The general Suhl-Nakamura interaction 63 where EQ is the ground state energy, fto(fc) = S\J\ in 5> o s ( £ • Si) + gixB{BX + B0), (5.89) BA = (BA ± BcA)/2, n is the number of nearest neighbours (4 in the case of MnAc) and 5i is the position of the zth nearest neighbour. The reciprocal lattice can be divided into two sections which will be referred to as ±k, and the Hamiltonian can be written in the following matrix form M = «„(+*) W . B - A \ ( 5 9 1 ) y gnBBA Clo(-k) J A Bogliubov transformation can be performed by diagonalizing the matrix M in the metric It is necessary to diagonalize in this metric to preserve the proper commutation relation of the new fields, which should themselves be bosonic creation and annihilation operators. The unit vectors of our space are ak and a L f e , and we define the scalar product between two unit vectors, a and (3 to be a • (3 = [a*, /?]. This definition of the scalar product gives our space the metric 77. If we diagonalize M with 77 as the metric, the eigenvectors are orthogonal in that metric, and the new operators will obey all the correct commutation relations. The eigenvectors can be found by solving (5.90) where (5.92) M $ i = £iTI$i. (5.93) Chapter 5. The general Suhl-Nakamura interaction 64 Since rj2 = 1, we can solve ?)M$ = £ $ . The eigenvectors corresponding to positive eigenvalues can be normed such that $-r/$i = 1, the negative eigenvalues such that ^rj^j = — 1, and $\r}$j = 0 for any i ^ j. If B is the matrix where each column is an eigenvector, then the orthogonality relations can be written as B]r]B = n (5.94) and B^MB is diagonal, and all the terms are positive, since equation 5.93 can be rewritten as MjkBkm = Emr}jqBom (5.95) and, multiplying by B\ we have BijMjkBkm = EmBJjrijqBqm (5.96) For a more complete discussion of diagonalization of bosonic and fermionic annihilation and creation operators Hamiltonians, see [25]. If we pose I ak \ V a - k J = Btp where ' e^1 0 N v 0 ei(t>2 j (5.97) (5.98) then H is diagonal in terms of the operators ak and a-k. The phases of the fields a±k and a±k are arbitrary. Using equation 5.94, we have = (p^nB^n V ^ J (5.99) Chapter 5. The general Suhl-Nakamura interaction 65 The Bogliubov transformation leads to the following electronic magnon spectrum and an electronic magnon gap of 0.35 T. With a convenient choice of the phases 4>\ and <j>2, the new fields are where ( «k ^ V « - * J \ ~Vk uk dk uk -Vk -ukvk = 2 , 2 uk + vk UL. — VL Q0(k) +u>(k) y/2cj(k)(to(k) + Q0(k)) ^2u(k)(cj(k) + Q0(k)) -gfiBB~A 2u(k) fto(fc) u(k) 1 (5.100) (5.101) (5.102) Typical values of the parameters give uk = 1.105 and vk = —0.470 when k —> 0. Plots of Uk and \vk\ over the first Brillouin zone are shown in figures 5.33 and 5.34. 5.3 The Suhl-Nakamura interaction between nuclear spins The Suhl-Nakamura interaction can be calculated for the MnAc crystal, with the as-sumption that the electronic spins in a triplet behave as a single, S = 5/2 spin coupled to each of the three nuclear spins. If we introduce the operator (5.103) Chapter 5. The general Suhl-Nakamura interaction 66 Figure 5.33: The value uk over the first Brillouin zone. Note that uk is only different from unity near k = 0. The graph is calculated with J/kB = 0.456 K, BCA = 0.14 T and BbA = 0.86 T. Chapter 5. The general Suhl-Nakamura interaction 67 a Figure 5.34: The value \vk\ over the first Brillouin zone. Note that vk is only different from zero near fc = 0. The graph is calculated with J/kB = 0.456 K, BCA = 0.14 T and BbA = 0.86 T. Chapter 5. The general Suhl-Nakamura interaction 68 where A i and A 2 are the effective hyperfine coupling tensors of the sites 1 and 2 and the indices a and b refer to the two equivalent Mn2 sites, then the hyperfine Hamiltonian can be expressed as HHF = E A , • s = E (A;S; + V+s- + AJS})) (5.104) where A * = AJ ± iAj . We can use the Holstein-Primakoff transformation, introduced above, to rewrite the electronic spins in terms of annihilation and creation operators. Ignoring constant terms, and working in momentum space, the hyperfine Hamiltonian becomes HHF = SEA)4 E ^ t ^ k ~ k ' ^ + W ^ E { ^ U ^ + A+afce**) (5.105) Finally, we can introduce the operators ak and a\_k. The Hamiltonian H = H0 + H' becomes Jfo = 5EA^  + E ^ 4 ^ ( 5 - 1 0 6 ) H' = - - ^ E A* (ufcttfc'atafci + i^al^a:-*; + ufcufc'(4aiLfc, + uk,vkak>a-k) e%{k~k')Ri +]f^T, (Aj(«*al + »fca_fc)c-^ + A+(u*a* + t;*alfc)e**) (5.107) If we consider i7' as a perturbation of H0, we can calculate the correction to the energy for a state | ^ >= \ipo<f> > where \ipn,ku-,kn > i s the electronic state with n magnons of wave vector ki,...,kn, and \<f> > is an arbitrary nuclear state. Clearly, there is no first order correction to the energy. To second order, only terms with one or two electronic magnons contribute. Since uk E<j> AE(2) J2 1 < ^O,fc0l#1^n,fc1,-,fc> > I 2 4>',n,kl kn Wfci + ... + WfcB + E# - E^ Chapter 5. The general Suhl-Nakamura interaction 69 e-i(fci+A;2)(Rj-H i;) - Tn £ i Ki U *a + uk2vkl)2 < 0|A||0' >< < £ ' | A ^ > -^TjF £ ( < 0 | A - | ^ X 0 ' | A J , | ( / > > ^ f c + < 0|At|0' >< 0 ' | A ^ > «* + < 0 | A - | ^ > < ^|A+|0 > w 2 + < 0|At|0' X </>'|A+|c/> > ukvk) (5.108) Using 1^ ' >< <f>'\ = 1) this expression is equivalent to the diagonal matrix elements of an effective nuclear Hamiltonian Heff = \ E { u t f i W + A+AJ) + UfrA+Aj. + UtfAjA} + 2 ^ , A * A ' , ) (5.109) with _. + + _ s r w ^ - ^ r „ 1 v ( u k l v k 2 + u k 2 v k i y e - ^ + k ^ - R ^ U n ' = —)v2 £ , ( 5 - 1 1 0 ) 5.3.1 The pseudoquadrupolar interaction We can first consider the pseudoquadrupolar interaction, for 5 4 M n . In this case, there is no coupling between nuclear spins because the spins are so dilute. Using 7 +7" = 7(7 + 1) — (7 2) 2 + P, and assuming an isotropic hyperfine interaction A, the Hamiltonian for each spin (1 or 2) reduces to 77 = AI' + *((l(I + l)-(I')*)(Ut + Uu-+) + Izm--Uti-+) +t / i [ + (7 + 7 + + 7-7") + 2UZU{P)2) (5.111) Chapter 5. The general Suhl-Nakamura interaction 70 Using equations 5.102 and 5.110, all the terms can be calculated numerically. Since k = xbi + yb2 where x = n/Ni and y = m/N2 and 6j • = 27T(% it follows that -J- 01,^-02)= dxdyf(2nx,2ny) (5.112) iV - .7-1/2 i-1/2 Considering equations 5.102, it is clear that L^}- » U£~ » C/j[+ and C / ^ - > Uu because uk 3> vk except for a small region near k = 0. This conclusion is supported by the numerical results because, for J = 0A56K, B0 = 0, BCA = 0.14T, BbA = 0.86T, we find U£~/h = -3.4 • l O ^ M H z - 1 [/-+/7i = -8.9 • l O ^ M H z - 1 U*+/h = 4.2 • l O ^ M H z - 1 Ul/h = -2.2 • l O ^ M H z - 1 (5.113) Consequently, equation 5.111 reduces to H = AI' + £ (/(/ + 1) - (Iz)2 + I') U+~ (5.114) The difference in energy between levels m and m + 1 is given by hvm,m+i = A - mA2U£~ (5.115) We can compare to values measured for 5 4 M n [15] where, in Bo = 0.2 T , the value of -(A\A)2Uu~/h for the Mn2 site is found to be 1.2 MHz. The mean field Curie tempera-ture for a Heisenberg Hamiltonian is given by [26] kBT™* = \S(S + 1)J (5.116) Our magnetic system is a Hamiltonian with nearest neighbour coupling (equivalent to a square lattice model since J is the same for all 4 nearest neighbours). From the exact Chapter 5. The general Suhl-Nakamura interaction 71 2 dimensional square lattice result for an Ising, nearest neighbour model, we know that Tc/T™f = 0.567 [27]. Since Tc = 3.19K for MnAc, we can predict that J/kB ~ 0.456 K. Using B% = 0.14 T, BbA = 0.86 T, B0 = 0.2 T and J/kB = 0.456 K, we find that — {Abi)2U^~ = 1.1 MHz, which compares very well with the experimental result of 1.2 MHz. 5.3.2 The interaction for the host spins In the case of the abundant 5 5 M n spins, the interaction between neighbouring spins cannot be ignored. The equations of motion for •, where a denotes the M n l or Mn2 site (a = 1,2), are given by djf dt si = i[H,gJ] (5.117) -i-—-This leads to -Aal'a,i E W ' + ^ + ) A i " - 2 ^ U j T + ( / * t i + I)/", (5.118) j and -AJU + U;+)Aj - 2AlU++(I^t - 1)/+ (5.119) j We can linearize these equations by replacing I* • with its thermodynamic average < J* >. The thermodynamic average of an operator O is defined as Tr le-^O] < ° > = -TTJe^V ( 5 ' 1 2 0 ) where /3 = 1 /kbT. This approximation leaves an indeterminacy in the equations because < J* > is a c-number and the commutation relations with I^j are lost. However, as we Chapter 5. The general Suhl-Nakamura interaction 72 shall see, the terms are small enough to be neglected. If we write the equations in Fourier space, using 1_ 1 • . 3 V — 3 1 4 = E e * ^ (5-121) VNj 3 then equations 5.118 and 5.119 become _A = (sAa + AlUt+2AlU?l(<Iza>-l) + -4AaUz<Az>)llk and +2AaS <Iza> ^AZk + SAa <Iza> <±^A+ wk wk -2AlU£+(< I'a > +l)I-.k (5.122) i*LgL = ^SAa — A2tU^+ + 2A2aUu(< Iza > +1) H—4AaUz < Az >) I~k +2AaS <Pa> ™±Atk + SAa <I*a> ^ ^ A k wk wk -2AlUtt+{< Iza > - l ) /+_ f c (5-123) where The numerical results shown in equation 5.113 show that we can ignore all the terms involving ,U^+ and Ufr. Finally, the term involving Ujj" only leads to a renormal-ization of the unpulled frequency. Since A2aU^~ ~ 2 MHz in B0 — 0 and SA\ ~ 600 Chapter 5. The general Suhl-Nakamura interaction 73 MHz, we will ignore this term as well. Finally, we have —i d_ dt ( I + \ 1k T+ a,k T+ 1b,k a,—k ^a.k = Gk{T) T+ 1b,k l-k a,—k where the indices a and b refer to the two Mn2 sites, Gk(T) = «12 «12 <5ll #12 &12 «21 ^2 + 0 2 «22 <$21 #22 «21 a 2 2 OL2 + 02 $21 $22 822 -<$12 -^12 -Oil ~ 01 - « 1 2 -Oi\2 — ^ 21 - 5 2 2 -<$22 - « 2 1 - a 2 - 02 -OL22 — ^ 21 -822 - « 2 1 -OL22 -a2 - 0 (5.125) (5.126) and ai(T) = AlS + A\S<Tl>'^±rt Wk a2(T) = A2S + A22S <IZ>^-+VK AiA2S < I{ > 4 + vk a 2 1 (T) = A ^ S K I ^ ^ wk 4 + vl wk a22{T) = A'2S<IZ2> 6n(T) = 2A\S < I{ > S22(T) = 2A\S<IZ2> wk UkVk wk ukvk wk Chapter 5. The general Suhl-Nakamura interaction 74 812(T) = 2A1A2S<Izl 82i{T) = 2A1A2S<I2Z> A CO 4A 1 f / 2 (A 1 < /* > +2A2 <T2>) AA2Uz{Ai < I* > +2^2 < Ii >) (5.127) We expect that Ujj~ S> Uz, and numerical results show that Uz/h = 1.0 • 10 - 1 0 MHz We will therefore ignore terms involving Uz, namely /3i(T) and P2(T). Finally, the power spectrum can be calculated In the case of a single spin per lattice site, one of the terms in equation 5.128 is zero due to energy conservation. For MnAc, both terms contribute because the effective hyperfine field is in opposite directions for the M n l and -Mn2 sites. The resonant lines in the power spectrum will correspond to the eigenvalues of the matrix G. There are three excitation modes. The first corresponds to an unpulled optical mode where the 2 Mn2 sites precess in opposition of phase and there is no precession of the Mnl spins. From equation 5.128 and 5.129 it is clear that this mode should not be absorbed. Indeed, there is no net precession of the magnetization and this mode should not couple to the precessing magnetic field [28]. Although the other two modes involve precession of all three spins it is worthwhile to note that the spins of the Mn2 sites precess exactly in phase. Also, the higher frequency mode corresponds almost entirely to precession of the Mn2 spins, while the lower frequency mode corresponds mostly to P(u,k) = Jdte^ (< Tfc-(i)T+(0) > + < T+(i)T*(0) >) (5.128) where T is the sum of the spins at each of the three lattice sites, Tt(t) = ikV) + i£k{t) + *& (5.129) Chapter 5. The general Suhl-Nakamura interaction 75 motion of the M n l spin. This admixture between Mnl and Mn2 spins decreases with increasing separation of the unpulled frequencies. In principle it is possible to integrate equation 5.125 by assuming that dP/dt = 0, without making the approximation /? =< P >. Indeed, this is justified in the sense that < dP/dt >>C< d / ± /d t >. It is then possible to calculate the first moment of the power spectrum. The first moment gives the resonant lines already calculated. 5.4 The frequency pulling at T = 0 calculated with spin wave theory The low temperature limit of the theory can be calculated using spin wave theory. We can make following Holstein-Primakoff transformation, // = I-c]c3 If ~ y/21cj IJ ~ V2Ic) Jli = -1 + afa /+• ^ V2Ia] I-d ~ V2Iaj l'bd = -l + b% ir. ~ y/2Ibj (5.130) where lj refers to the M n l sites, and Iaj and Ibj refer to the 2 Mn2 sites. The Hamiltonian can then be written in terms of the annihilation and creation operators introduced in equations 5.130. If we then work in momentum space, ^ = 4 r 7 £ e i f c ^ ( 5 - 1 3 1 ) Chapter 5. The general Suhl-Nakamura interaction 76 where d, is any of the annihilation operators, a,-,bj or Cj, then the Hamiltonian can be expressed as H = £ ( 4 °-* c-fc 44 ) AH- V 7 a-k bk (5.132) where Qk = ai - Pi -Oi\2 -Ot\2 -Su -S12 S\2 0121 &2+ P2 OL22 S21 S22 S22 Oi21 OL22 OL2 + P2 S21 S22 S22 Sn ~S\2 Sn -ai - Pi -OS12 -OL\2 S21 S22 S22 &21 OL2 + P2 OC22 S21 S22 S22 OS21 OL22 0-2 + P: (5.133) V and o>\, a2, cui2> «2i> 1^2, 2^1, 2^2> Pi and P2 are defined in equations 5.127 but here are taken at T = 0. In order to diagonalize Qk and find the spectra of the excitations, it is necessary to diagonalize in the metric n in order to preserve the correct commutation relations between the bosonic annihilation and creation operators, where - 1 0 0 0 0 0 \ 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 (5.134) Chapter 5. The general Suhl-Nakamura interaction 77 As discussed earlier, it is necessary to find the eigenspectrum of T]Qk in the usual sense. However, r\Qk = Gk(0) and the spectra calculated by both techniques are identical at T=0. 5.5 Linewidth The calculation of the second moment is quite complicated because of the three spins. We shall return to the case of a single spin in order to understand the behaviour of the linewidth as a function of temperature. The Suhl-Nakamura Hamiltonian is H = HQ + HX = ASX // + ( 5 - 1 3 5 ) i ij The power spectrum is given by Pk(u) = Jdt < l£(t)Ik(0) > e x p M ) (5.136) where Ik are defined in equation 5.121. The thermodynamic average is defined as This expression of the power spectrum implies that A < 0 and we have chosen our axis of quantization such that < i | >—>• +1 as T —> 0. Because AS » ^ Uij we can, to an excellent approximation, calculate the thermo-dynamic averages with H = H0. Following Van Vleck's method of moments [29], we can calculate the nth moment of the power spectrum, defined as < w» > = I f ^ A M (5.138) Chapter 5. The general Suhl-Nakamura interaction 78 Using integration by parts, the first and second moments can be expressed as <UJ> = -i *+ * (5-139) < 2k Lk > 2 < d2l£/dt2 Ik > . . < OJ2 > = ^ r - (5.140) Since dl£/dt = i[H,l£], < U } > = < > (5-141) <u2> = _<B^SK> (5.142) The width of the resonance, (Aa;)2 =< cu2 > — < u> >2 can then be calculated. Since we are interested in the uniform mode (k = 0), we will restrict ourselves to that case from here on. The first moment can be evaluated from >/N[H, It=0] = ASY,Im-E Uimltlm (5-143) m im When calculating the thermodynamic averages, we have used H = H0 and therefore < IfImlJ >=< IflZ >< Im > $tj when i / m. Of course, when i = m we can, to a good approximation at low temperatures, use < IfImI~ >=< IflJ >< Im ~ 1 > With these approximations, we find < u> >= AS + Ujj -Y,Uim<Iz > (5.144) i where m is arbitrary but fixed. This is the expected result because J2Uim = - — (5.145) i uo and this gives the same frequency pulling as equation 4.74. The second moment has been calculated by an expansion in 1/T [4], and the linewidth is found to be (Aw) = ( A u U (l - ^^L) (5-146) Chapter 5. The general Suhl-Nakamura interaction 79 where ( ACJQQ) is the high temperature linewidth and Sco = £ i Uim < F > is the frequency pulling. However, the result of Pincus has two problems for our application. Firstly, it is only valid when AS <§C kBT which is certainly not the case at the temperatures of interest to us. Secondly, it is calculated assuming that there is no self interaction term (Uu — 0). The second moment can be calculated from VN[H, [H, 4t0]] = A2S2 E If - 2AS £ UimItFm - hlyUUI?^'> (5-147) j im where there are no restrictions on the indices i, j, n and m. The second moment can be evaluated using H = HQ to evaluate thermodynamic averages and by assuming that <itUK> < It I' >< (If - I)2 > (5.148) The final result for the linewidth is "3 < (F)2 > <F> 1(1 + 1) (AcoY + - < F >2 < J Z > 2 _ < F > , /(/+!) <{F)2> + -Ul < F >2 (5.149) The linear terms in < F > appear to indicate that the linewidth depends on the choice of quantization axis. However, the expression of the power spectrum in equation 5.136 changes if the quantization axis changes. If z —> —z, the Green's function used to calculate the power spectrum is < l~(t)l£(0) >. If the linewidth is recalculated in this case we find, of course, that the linewidth in independent of the choice of quantization axis. It is important to remember that in equation 5.149 the linear terms are positive. The linear term in the result of Pincus (equation 5.146) is expressed as an absolute value to avoid this confusion. Chapter 5. The general Suhl-Nakamura interaction 80 Note that, at high temperature < F >= 0 and < (F)2 >= . Therefore ( A ^ o o ) 2 = lYliU2jI(I + 1). This is precisely the result of van Vleck [29] calculated at the high temperature limit and is also the high temperature limit used by Pincus [4]. Since the result of Pincus is only valid for Uu = 0, we can compare the two predictions in this case (see figure 5.35). The results agree well when kBT ^> AS when the high temperature expansion is valid, but they deviate markedly at low temperatures when our approximation should be valid. T e m p e r a t u r e (mK) Figure 5.35: The mean square width for Uu = 0, £ i Uij = 14.1 MHz, and = 6.3 MHz. The dashed line is the prediction of Pincus. It is also useful to consider the case where only the self interaction term is present. In this case, (Aco)2 = Ul (< (F)2 >-<F > 2) (5.150) This is exactly the result expected for a pseudoquadrupolar splitting. At T = 0 only the Chapter 5. The general Suhl-Nakamura interaction Figure 5.36: The mean square width for Uu = 1.9 M H z , E/y = 14.1 M H z , ~Zi = 6.3 M H z . Chapter 5. The general Suhl-Nakamura interaction 82 lowest energy level is occupied and the line is infinitely narrow. At T = co, the resonance is made up of six individual transitions of equal amplitude, with a spacing Uu between each. The linewidth for such a resonance is clearly on the order of Uul. The result when Uu is not zero is shown in figure 5.36. The temperature dependence of the linewidth is very surprising, precisely in the region of interest for our experiments. These results indicate that the temperature dependence of the linewidth in the more complicated case involving triplets of nuclear spins may be unusual precisely in the re-gion of interest for our data. Indeed, some of the unusual temperature dependence of the linewidths in the experimental data may indicate that these effects are present. Un-fortunately, the second moment has not been calculated in the case of the triplet nuclear spins because it is difficult to find which terms contribute to which of the four lines. In order for a second moment calculation to be meaningful only terms contributing to a single resonance should be included. 5.6 Calculating the exact power spectrum for 1 to 4 spins In order to explore the change in linewidth with temperature, and the possibility that the splitting of the pulled Mn2 line could be lost in the linearization procedure, we have calculated numerically the exact power spectrum for up to four spins with 1=5/2, including nearest neighbour interactions. The simplified Hamiltonian used, for a single spin has the form where < ij > denotes a sum over nearest neighbours. In the simulations, a/h = 500 MHz, b/h = 2 MHz and c/h = 1 MHz. The power spectrum is calculated by diagonalizing the Hamiltonian and using equation 5.128. The results for one to four spins at ksT/h = 500 MHz can be seen in figure 5.37, and for four spins at different temperatures in figure 5.38. (5.151) Chapter 5. The general Suhl-Nakamura interaction 83 These numerical results indicate that the pseudoquadrupolar structure is smoothed out as the number of interacting spins increases. If there are N spins, the matrix to diagonalize is of dimension nxn where n = (2I + 1)N. Using the fact that the Hamiltonian conserves the total angular momentum ([H, £ i / / ] = 0) allows the matrix to be reduced to a block diagonal form. However, it is still very difficult to treat numerically the case of a real crystal lattice with longer range interactions. 1 1 1 1 1 1 1 spin | —t-^^, ^ — i i 1 i . 1 i 2 spins rt i —"1 1 L i 1 i h-== \ i i i i i 3 spins i 1 j i | i 4 spins A ^ — " i . i 1 480 490 500 510 490 500 510 520 Frequency (MHz) Frequency (MHz) Figure 5.37: The exact power spectrum for one to four spins at kBT/h = 500 MHz or T = 24 mK, including pseudoquadrupolar interactions of 2 MHz and nearest neighbour interactions of 1 MHz Chapter 5. The general Suhl-Nakamura interaction 84 1 1 1 1 ' 1 T=100MHz | i — i ^ — i 1 i 1 i • i T=500MHz |\ 1 — i I i 1 i u i 1 1—=i^ - • | i i i T = 1 0 0 0 ^ ^ 1 | 1 1 l | i T=1500MHz f\ — - i , i , r , 480 490 500 510 490 500 510 520 Frequency (MHz) Frequency (MHz) Figure 5.38: The exact power spectrum for four spins at different temperatures, including pseudoquadrupolar interactions of 2 MHz and nearest neighbour interactions of 1 MHz. The temperatures (T = hu/kb) are: 4.8 mK, 24 mK, 48 mK and 72 mK. Chapter 5. The general Suhl-Nakamura interaction 85 5.7 Classical model and simulations In order to perform numerical simulations involving more spins, it is simpler to work in a classical system. We shall first consider the classical theoretical predictions of a crystal with a single nucleus per lattice site. We will then compare these predictions to some classical simulations. We show that, qualitatively, these simulations confirm the picture we have developed from the quantum mechanical treatment of the frequency pulling. The Hamiltonian can be written as i ij 1 = Y/\ASu2 + -Y,Uij(I*ux + lVuy) -Ii (5.152) i The equation of motion of each spin is, in the reference frame rotating at angular velocity to, dh dt (5.153) where Bejj = Bi + uj/yu. The equations of motion can be linearized by replacing if with its thermodynamic average, < F > in all quadratic terms. The result for the uniform mode (k = 0 ) , in the rotating reference frame, is ( U{AS + tv-\Y.iUij<P>) N bIS ~ IS (AS + to - \ Zt Ui3 < F >) (5.154) where b is the amplitude of the oscillating magnetic field. The resonance condition is clearly UJ = -(AS-1-J2 Uij < Iz (5.155) dh dt Chapter 5. The general Suhl-Nakamura interaction 86 In other words, the frequency pulling effect predicted classically is the same as the effect predicted quantum mechanically in equation 5.144 except for the factor of 1/2 in the second term of equationclassicalpulling. Classical simulations were performed with an array of 15 by 15 nuclear spins. The spins were first thermalized without including the interaction terms. The interactions and the oscillating magnetic field were then added, and the system was evolved for a short period of time, At. The change in P was measured for each spin, and the absorbed power is then A ~ P - ^ = A * ) - / ; ( * = o) 3 oc < / ; - / / > (5.156) where the average <> is taken over all the spins in the lattice, If = Pit — At) and If = P(t = 0). At was chosen in order to remain in the limit of C W NMR. The centre frequency of the resonance was obtained by plotting the P as a function of frequency. The result of several simulations at the same temperature (10 mK) with longer and longer range interactions is shown in figure 5.39. The results of several simulations at different temperatures is shown in figure 5.40. The agreement with the theory is reasonable, even with a short range interaction. The discrepancy may be partly due to errors associated with the strength of the oscillating magnetic field. In order to perform the simulations in a reasonable time, the oscillating field was stronger by five orders of magnitude than it is experimentally. The discrepancy between the theory and the simulations was reduced when the oscillating field was reduced. A simulation involving an array of 12 by 12 triplets of spins was also performed, and the results can be seen in figure 5.41. Only interactions within a triplet of spins were included in the calculation. The quantity plotted is < If — If > where the average is over the Mnl spins or one of the Mn2 spins. It appears there are only two resonances. Chapter 5. The general Suhl-Nakamura interaction 87 14 12 10 £ 8 3 6 0 y = 2.046x , ' , ' V y = 1 -8656x - 0.0084 0 1 2 3 4 5 6 7 (1/2)U(k=0) (MHz) Figure 5.39: The frequency pulling for a variety of interactions at T = 10 mK. Each data point is calculated by increasing the range of Uij. The first point corresponds to only a self interaction of 2 MHz. A nearest neighbour interaction of 1 MHz was added, then a second nearest neighbour interaction of 0.5 MHz and finally a third nearest neighbour interaction of 0.2 MHz. The solid line is the best fit to the data, and the dashed line is the prediction of equation 5.155 because at T = 10 mK, < T >= 2.046. Chapter 5. The general Suhl-Nakamura interaction 88 7 0 -| , , , , 1 0 0.5 1 1.5 2 2.5 <lz> Figure 5.40: The frequency pulling for a variety of temperatures (10, 20, 50, 100 mK) with a self interaction of 2 MHz and a nearest neighbour interaction of 1 MHz. The solid line is the best fit to the data, and the dashed line is the prediction of equation 5.155. Chapter 5. The general Suhl-Nakamura interaction 89 The first corresponds mainly to precession of the M n l spins, while the higher frequency line corresponds mainly to precession of the Mn2 spins. In order to observe the unpulled optical mode, as discussed in section 5.3.2, the quantity < \Ij — I*\ > was calculated for the Mn2 spins (see figure 5.42). The increased self interaction was of 10 MHz and the temperature of 1 mK was used in order to separate the pulled and unpulled modes. The result can be seen in figure 5.42. Both the unpulled optical mode and the frequency pulled mode are visible. It was verified that the Mn2 spins precess in opposition of phase in the unpulled mode, and in phase in the pulled mode. Therefore, in the unpulled mode, the power absorbed by one of the spins is given up by the other Mn2 spin and there is no net power absorption. It should be noted that the frequency pulled Mn2 mode involves some precession of the Mnl spin. It was also found that the Mnl mode contained some precession ofthe Mn2 spins. However, the unpulled optical mode involves no precession of the M n l spins. The unpulled optical mode has no net precession of magnetization to couple to the oscillating RF field and the optical mode is not visible in the plot of the power spectrum. These results confirm qualitatively the picture obtained from quantum mechanical calculations in section 5.3.2. Unfortunately, due to numerical limitations, it was not possible to include sufficiently long range interactions to compare the simulations to experimental data. 5.8 Comparing the theoretical model to the experimental results The data and models are plotted together in figure 5.43. The data were obtained from spectra such as that shown in figure 4.31, as explained in section 4.2.3. The theoretical line is a fit using B\ and B°A as the only free parameters. The fit had to be performed in several steps because the power spectrum has four lines rather than the expected two. In Chapter 5. The general Suhl-Nakamura interaction 90 0.2 0.15 A -0.15 -I , , . 1 560 580 600 620 640 Frequency (MHz) Figure 5.41: The quantity < Ij — If >, which is proportional to the power spectrum, as a function of frequency for triplets of spins. The self interaction term which allows interactions between the three spins within a triplet is 2 MHz. The unpulled frequencies are 573.8 MHz and 631.7 MHz. The temperature is 10 mK. The black line is the average change of the the Mnl spin, while the grey line is the average change of one of the Mn2 spins. Chapter 5. The general Suhl-Nakamura interaction 91 0.14 615 620 625 630 635 640 645 Frequency (MHz) Figure 5.42: The quantity < \If - I-\ > as a function of frequency for triplets of spins. The self interaction term which allows interactions between the three spins within a triplet is 10 MHz. The unpulled frequency is 631.7 MHz. The temperature is 1 mK. The black line corresponds to the M n l spin, while the grey line corresponds to one of the Mn2 spins. Chapter 5. The general Suhl-Nakamura interaction 92 620 20 40 60 80 100 Temperature (mK) Figure 5.43: The spectrum of the zero momentum nuclear spin excitations as a function of temperature and the experimental data. The best fit values are B\ = 0.27 ± 0.03 T and B°A = 0.81 ± 0.05 T. Chapter 5. The general Suhl-Nakamura interaction 93 each of the x2 fits described below, four data points were used. A reasonable fit should yield xLn - 4-resonance (corresponding mainly to precession of the M n l spins). A x2 fit w a s first performed on only the lowest frequency line (corresponding es-sentially to the precession of the Mnl spins). This is the obvious choice because the resonance corresponding to Mn2 spins precessing in phase is split into two lines. A graph of 1/x2 is shown in figure 5.44. The inverse of x2 is shown because it is easier to see a maximum, rather than a minimum. This plot is interesting because it shows that only one of the two parameters can be determined from this fit. This is essentially because the errors on the low frequency line are relatively large. The slope of the line is therefore difficult to determine from the data. What can be determined is the overall magnitude Chapter 5. The general Suhl-Nakamura interaction 94 of the frequency pulling effect which depends almost entirely on the smallest of the two anisotropy fields. It is impossible to determine from this model whether the smallest field corresponds to the b or the c directions. However, we know from the results of Cowen et al. [20] that BA is the smaller anisotropy field. Our fit then yields BbA = 0.27 ± 0.03 T. In order to determine the value of BCA we need to fit the model to the data from the higher frequency lines corresponding essentially to precession of the Mn2 spins. There are two choices of lines. A x2 fit was performed on both of the possible resonances independently. The free parameter was B°A while BA was fixed at 0.27 T. The plots of X2 vs BCA are shown for both the lower and higher frequency Mn2 lines in figures 5.45 and 5.46 respectively. Clearly, the lower frequency Mn2 line gives an excellent fit (figure 5.45) with BCA = 0.81 ± 0 . 0 5 while the higher frequency line (figure 5.46) cannot be fitted reasonably. The unpulled optical mode is not frequency pulled and does not have any dependence on the anisotropy fields. The conclusion is that M n l resonance, and the lower of the frequency pulled Mn2 resonances are described well by our model, with B\ = 0 . 2 7 ± 0 . 0 3 T and BCA = 0.81 ± 0 . 0 5 T. The higher of the frequency pulled Mn2 lines is an excitation not described by our model. Furthermore, our model correctly predicts the existence of an unpulled optical mode involving precession of the Mn2 spins, but does not explain how this mode is excited by an R F field. The values of the anisotropy fields previously measured at 1.2 K are BA = 0.14 T and B\ = 0.86 T [20]. While our results for BCA are entirely consistent with this result, B\ is significantly different. There are several explanations for this discrepancy. The first is that the anisotropy fields depend on temperature, and were only previously measured down to 1.2 K. The trend for BA is to increase with decreasing temperature. Cowen et al. did not include the trend for BCA. Another source of discrepancy is relating Chapter 5. The general Suhl-Nakamura interaction 95 Figure 5.45: x2 as a function of B% for the lower of the 2 resonances corresponding mainly to precession of the Mn2 spins. BbA was fixed at 0.27 T. The graph shows the best fit be B°A = 0.81 ± 0.05 T. Chapter 5. The general Suhl-Nakamura interaction 96 X2 2 0 0 : 1 0 0 : 0 i , , — , i — • — ' i i — i i i i — • i — i — i — i — J 0 2 4 6 8 1 0 B £ ( T ) Figure 5.46: x 2 a s a function of BA for the higher of the 2 resonances corresponding mainly to precession of the Mn2 spins. B\ was fixed at 0.27 T. There is no physical minimum for x2i indicating that the model does not describe the magnitude and temperature dependence of this resonance. Chapter 5. The general Suhl-Nakamura interaction 97 the parameter measured by Cowen et al. with the parameter of our model. Indeed, Cowen et al. measured the anisotropy fields by measuring the field necessary to force the magnetization into the b and c directions. Relating that field to our parameters requires the assumption that only quadratic terms in Si- and Sf are present in the original electron spin Hamiltonian (equation 5.86). Ignoring higher order terms is justified in our model because the < Sf >~ S, but is not justified when forcing the magnetization into the b and c directions. The relatively small errors reported in the values of the anisotropy fields are an indication of the sensitivity of the frequency pulling to the anisotropy fields. It is not possible to fit both the Mnl and Mn2 frequency pulled lines with a model involving only uniaxial anisotropy. A plot of the predictions with uniaxial anisotropy (BA = 0.35 T and B~A = 0) is shown in figure 5.47. Although the resonant lines measured experimentally are shifted downwards in fre-quency as a function of temperature, and the magnitude of the overall shifts agree well with the theory, there are several places where the model fails. The existence of four observed lines rather than two, the mechanism of absorption above 600 MHz, near the unpulled frequency, and the small temperature dependence of the unpulled frequency is not explained by our model. Our model predicts the existence of an unpulled optical mode, but also predicts that this mode should not couple to the precessing magnetic field. We believe that our model correctly predicts the existence of this mode, but the absorption mechanism is lost due to the approximations made. The linearization procedure consisting of replacing If with its thermodynamic average, < F > at each site in equations 5.118 and 5.119, and the artificial equivalence of the two Mn2 sites makes it impossible to predict an absorption mechanism for the optical mode. If we do not treat each triplet of electronic spins as a single, S=5/2 spin, then we must Chapter 5. The general Suhl-Nakamura interaction 98 Figure 5.47: The spectrum of the zero momentum nuclear spin excitations as a function of temperature for uniaxial anisotropy (B\ = 0.35 T and B~ = 0) and the data. Chapter 5. The general Suhl-Nakamura interaction 99 include 6 atoms per unit cell. There will then be 12 electronic magnon modes, and as many nuclear magnon modes. Of the nuclear magnon modes corresponding essentially to precession of the Mn2 spins, there will be 2 modes with all nuclear Mn2 spins in phase, 4 modes with Mn2 spins in opposition of phase within the same triplet, and 2 modes with Mn2 spins in phase within the same triplet but in opposition of phase with the other triplet in the same unit cell. It is important to note that a careful consideration of the crystal structure and the exchange and super-exchange bonds (see [16]) shows that each Mn2 spin is strongly coupled to two nearest neighbour triplets. The two nuclear Mn2 sites are then coupled differently to different electronic magnon excitations. Since the exact couplings between electronic spins within each triplet, and with nearest neighbour spins outside the triplet are not known, there are too many free parameters to make a full calculation worthwhile. However, these different couplings will clearly break down many of the symmetries in our model that prevent absorption of optical Mn2 line (presumably there will be some admixture of the Mnl nuclear spin). A full calculation should also predict another excitation mode not presently seen in our model. This mode would correspond to the observed resonance that could not be fitted with our model. There could also be a small temperature dependence of the optical line. Furthermore, there is the possibility of different anisotropy fields for each of the 6 ions. Crystal symmetries prevent different anisotropy fields for the two triplets in each unit cell, but it is very likely that the anisotropy fields for the M n l sites and the Mn2 sites are different. Again these parameters are not known. What has been measured previously [20] is the average anisotropy field over a triplet of ions. There is the possibility that some of the observed structure is due to domains. The magnetic behaviour of MnAc at low temperatures has been extensively studied by SQUID magnetometry [15]. At very low fields (JB < 14 mT), there exist mixed phases of ferromag-netic and antiferromagnetic alignment of the a-b planes. However, no domain structure Chapter 5. The general Suhl-Nakamura interaction 100 was observed at higher applied fields. In our experiment, the same line structure was observed at applied fields up to 110 mT, in two separate measurements. (Higher fields were not possible due to the increase in Ti.) Therefore, it seems unlikely that domains can account for the observed structure. The exact power spectrum calculated for four spins gives a reasonable qualitative picture. It seems improbable that the number of lines in our spectrum is due to an enhanced pseudoquadrupolar splitting. On the other hand, our numerical results indicate that the linewidth and lineshape are strongly dependent on temperature even if the individual transitions cannot be observed. This could partly explain the asymmetry of the peaks observed and may introduce some error into the estimation of the centre frequency of the line. In an effort to explain the fourth resonant line without calculating the electronic spectrum including all 6 ions per unit cell, we have applied the perturbation technique of Kubo and Tomita [8], where we regard a single ion Hamiltonian as unperturbed. In this approximation, it is necessary to identify the observed lines in the unperturbed Hamil-tonian and the corrections to the positions of the lines can then be calculated. However, we are unable to account for the observed spectra by including the pseudoquadrupolar lines in the unperturbed Hamiltonian. There is no energy scale of the order of 10 MHz in the basic Hamiltonian except the symmetry breaking effect between Mnl and Mn2. It is difficult, therefore, to account for four lines, separated by more than 10 MHz using this perturbative approach. The MnAc crystal has low symmetry so that the hyperfine coupling constant might have some anisotropy which is also indicated by the relatively strong anisotropy fields. However, we expect the anisotropy to be small since the magnetic moment of Mn++ is spin only. The effect of anisotropy is essentially to include I?lf and 171 J in the effective Hamiltonian. In other words it has the same effect as the anisotropy fields. Chapter 5. The general Suhl-Nakamura interaction 101 Since the hyperfine anisotropy should be small, we expect this to lead to small corrections. Demagnetizing effects can also yield If If and I~I~ terms, but the demagnetizing factor for the thin, planar shape is very small, and, in zero magnetic field, the antiferromagnetic coupling between different layers (in the a-b plane) ensures that no demagnetizing fields are present. Further, we observe no change in structure on applying a sufficiently large magnetic field to make the sample fully magnetic. The nuclear spin in MnAc is large (I = 5/2), and it it seems reasonable to expect that spin wave calculations using the Holstein-Primakof transformation (1/1 expansion) should yield good results at low temperatures. Spin wave theory does indeed agree with our prediction at low temperatures. However, to treat the effect of higher temperatures using spin wave theory, it would be necessary to introduce coupling between the bosons, making the calculation difficult. Since the T = 0 results agree with our predictions, however, we have some confidence that our model is accurate. 5.9 Conclusion We have measured the frequency pulling effect in MnAc. The temperature dependence of the frequency pulling could not be explained by a simple model assuming uniaxial anisotropy fields and no coupling between Mnl and Mn2 spins. Furthermore, the exper-imental data shows a spectrum involving four resonance lines, rather than the expected two. A model including coupling between all the nuclear spins and non-uniaxial anisotropy was developed. This model agrees well with the magnitude and temperature dependence of the frequency pulling and shows that it is highly sensitive to the strength of the anisotropy fields. Classical and quantum simulations support qualitatively the results of the calculations and a spin wave calculation, accurate at T = 0, agrees exactly with this Chapter 5. The general Suhl-Nakamura interaction 102 model at zero temperature. The absorption of the unpulled optical mode, and the existence of another resonance mode (presumably mainly involving precession of the Mn2 spins) is lost in the approxi-mation that each triplet of electronic spins behaves as a single, 5 = 5/2 spin. Although there are too many unknown parameters to make a full calculation worthwhile, it is clear that including the full electronic magnon spectrum will break many of the artificial sym-metries that prevent absorption of the unpulled optical line, and the existence of another frequency pulled Mn2 resonance. We believe, however, that the essential physics of the nuclear magnon excitations has been captured by the model. The model is also of interest in itself because it extends the theory of frequency pulling to cases with I > 1/2 where the self-interaction term is important and to cases where the crystalline anisotropy fields are not uniaxial. Chapter 6 First NO of a radioactive isotope implanted in insulators 6.1 Introduction The great majority of experiments exploiting the technique of on-line nuclear orientation (NO) of implanted isotopes have been motivated by nuclear or fundamental physics. All experiments to date have used a metallic magnetic material, usually an Fe foil, as the host. However, from the condensed matter point of view, there is a very wide range of magnetic insulators that would be interesting implantation targets and we decided to investigate the performance of these as hosts for on-line experiments. In this regard, we note that implantation of radioactive isotopes has been used to study the physics of semiconductors and insulating systems by the Mossbauer [30] and perturbed angular cor-relation (PAC) [31] techniques. Usually the experiments are performed off-line although there have been on-line measurements using PAC on, for example, high temperature su-perconductors [32]. For nuclear orientation, on-line measurements would open the door to many new experiments to investigate interesting magnetic structures. Although NO and NMR on oriented nuclei (NMRON) have many orders of magnitude more sensitivity than conventional NMR, it is often very difficult to obtain samples doped with a suitable isotope. This problem would be obviated if an isotope could be implanted in an on-line experiment. An insulating host may also, in specific cases, provide a larger hyperfine interaction than an Fe host. In these first experiments to investigate the efficacy of the method, we chose to implant 103 Chapter 6. First NO of a radioactive isotope implanted in insulators 104 5 6 M n into antiferromagnetic crystals of MnCl 2 -4H 2 0 and CoCl 2 -6H 2 0. MnCl 2 -4H 2 0 is antiferromagnetic below T = 1.6 K. The crystal structure is monoclinic with /? = 99.7° [33], and the magnetic easy axis (along which the magnetization is aligned in zero field) is close to the c-axis [34]. The magnetic properties have been studied by NO and NMRON using 5 4 M n as the radioactive probe [35, 36, 37, 38]. In fact, it was the first insulating ordered magnet in which NMRON was successfully performed [35]. CoCl 2 -6H 2 0 is also antiferromagnetic with a similar crystallographic structure, but with f3 = 122.3° [39], and an ordering temperature of 2.3 K [40]. The easy axis of magnetization is the c-axis. In this case, the magnetism is more complicated with some orbital contribution. Previous NO studies have shown that a significant orientation of 5 4 M n can be achieved by incorporating it as an impurity in the crystal [41]. The magnetism is close to "spin only" in these crystals and the hyperfine field is large with a value B^f ~ 60 T. Radioactive 5 6 M n has spin 7 = 3, magnetic moment [i = 3.227/i^, and a half-life ti/2 = 2.6 hr. It decays by f3~ emission to the daughter 5 6 Fe and the 7-ray observed in the experiment is the E2 transition from the 2 + 847 keV level to the 0+ ground state. Fig. 2.3 represents a simplified decay scheme in which only the f5~ decays with intensities greater than 1 % and the subsequent 7-rays feeding to the observed 847 keV transition are shown. The isotope 5 4 M n , with 1 — 3, magnetic moment fx = 3.282/ZAT, and a half-life ti/2 = 303 d, was also utilized in these experiments. In this case, the decay to the daughter 5 4 Cr is by electron capture to the 835 keV level and the observed 7-ray is the subsequent E2 transition to the 0+ ground state. The decay scheme is shown in Fig. 2.2. Chapter 6. First NO of a radioactive isotope implanted in insulators 105 6.2 Sample Preparation and Experimental Procedure Seed crystals were grown from saturated aqueous solutions of MnCl 2 -4H 2 0 and CoCl 2 -6H 2 0 The crystals were then grown from saturated solutions of the salts in a temperature con-trolled environment. The final crystals had an area of ~ 1 cm 2 and a thickness of ~ 0.2 cm. During growth, the MnCl 2 -4H 2 0 crystal was doped with the radioactive isotope 5 4 M n while the CoCl 2 -6H 2 0 crystal was doped with both 5 4 M n and 6 0 C o . These isotopes are long lived, have simple decay schemes and their 7-ray anisotropics were used both to ascertain the temperature of the crystals and to provide a comparison with the 7-ray anisotropy of the implanted isotopes. A special sample holder was designed to clamp the crystal to the cold finger using Apiezon N grease yet allow ~ 1 cm 2 of its face to be exposed to the ion beam. The cold finger was "top-loaded" into the Louvain-La-Neuve dilution refrigerator [42], but, because the hydrated crystals deteriorate under reduced pressure at temperatures T > 240 K, the top-loading equipment was modified to enable the crystals to be pre-cooled before insertion into the dilution refrigerator. A 5 7 Co-Fe foil was also soldered to the cold finger in order to monitor its temperature. A magnetic field, B0 was applied not only to magnetize the 5 7 Co-Fe foil, but also to reduce the nuclear spin-lattice relaxation time, T i , of the Mn spins and so increase their rate of cooling. The 5 6 M n ions were produced at the C Y C L O N E cyclotron in Louvain-La-Neuve. A 100 MeV, 2.5 mA deuteron beam was directed on to a 5 5 M n target foil mounted in an ion guide source. Reaction products recoiling out of the foil were thermalized in helium buffer gas at 200 mbar pressure inside the ion source. The positive ions leaving the source through the extraction hole were accelerated to 50 keV, mass separated and then transported to the dilution refrigerator where they were implanted at a rate ~ 103 s _ 1 into the crystal samples. Chapter 6. First NO of a radioactive isotope implanted in insulators 106 Germanium detectors were placed in various directions with respect to the crystalline axes to measure the intensities of the 835 keV 7-ray in the 5 4 M n decay (electron capture to 5 4 C r ) , the predominant 847 keV 7-ray in the 5 6 M n decay, the 1173 keV and 1332 keV 7-rays for 6 0 C o (in the 5 6 Mn-CoCl 2 -6H 2 0 experiment) and the thermometric 122 keV transition in the 5 7 C o decay. Pulser signals were also recorded in each detector to correct for electronic dead time and to check the stability of the data acquisition system. A magnetic field, B0, was applied in a horizontal direction, and two high efficiency detectors mounted in the horizontal plane, counted radiation emitted parallel (detector 1) and perpendicular (detector 2) to Bo- A third, lower efficiency detector monitored counts in the vertical direction, i.e. perpendicular to Bo-In order to analyse the Ge detector spectra, software was written that used a peak finding algorithm to track any changes in peak position due to gain changes. The peaks were then integrated and the background was subtracted using the average counts to the left and right of the peak. The pulser peaks were also integrated in this way, and the dead time corrections were made by dividing the areas of the peaks by the area of the pulser peaks. This software allowed the hundreds of spectrum files to be analysed easily. In the first experiment, the MnCl 2 -4H 2 0 crystal was oriented so that the crystalline c-axis, which is close to the magnetic easy axis, was horizontal. Thus the directions of the c-axis and the applied field, B0, were parallel within the angular uncertainty of aligning the crystal that was ± 4 ° . With this geometry, detectors 1 and 2 measured radiation emitted at angles 9X = 0° and 62 = 90° respectively. When 5 6 Mn was implanted into the sample, a problem was immediately encountered. The insulating crystals warmed up to T ~ 100 mK during the implantation process. This was not due to the deposition of energy by the implanted isotopes, but because of the exposure to thermal radiation down the side access. This problem was overcome because the half-life of 5 6 M n (2.6 hr) allowed the experiment to be done pseudo-on-line in cycles. The detectors collected data Chapter 6. First NO of a radioactive isotope implanted in insulators 107 at all times, and in each cycle the following operations were performed: • 5 6 M n was first implanted into the crystal for a time > Tx/2 SO that sufficient activity built up; • The side access was closed and the system cooled for a time > Ti/2; • The dilution refrigerator was warmed up to a temperature T > 100 mK to obtain "warm" counts for normalization. The cooling of the crystal is limited by the extraction of the heat capacity of the abun-dant nuclear spins ( 5 5 Mn or 5 9 Co). There are two "bottlenecks": the Kapitza boundary between the crystal and the copper cold finger, and the nuclear spin-lattice relaxation characterized by a time T\. In B0 = 0, the cooling is initially limited by the Kapitza resistance, but as the temperature falls, T\ increases exponentially [38] with an exponent proportional to (- Eg/ksT) where Eg is the magnon energy gap. In order to increase the cooling rate one can utilize the "magnon cooling" effect [37]. B0 is adjusted to a value very close to the "spin-flop" field, BAE ~ (2BEBA)1/2 ~ 0.7 T, where BA and BE are the anisotropy field and exchange field respectively. This causes Eg to be decreased, at least for one branch of the magnons, and Ti is much reduced. In the second experiment 5 6 M n was implanted into CoCl2-6H 20 and measurements made using the same pseudo-on-line procedure discussed above. The same 7-rays were counted as in the MnCl2-4H 20 experiment as well as the 1173 keV and 1332 keV tran-sitions in the 6 0 C o decay. In this case, because of the way in which the crystal grew, it was fitted into the sample holder with the a-axis vertical. The b-axis was horizontal and perpendicular to B0. Thus the angular directions of detectors 1 and 2 were 32° and 90° to the easy c-axis respectively assuming exact alignment of the crystal. Chapter 6. First NO of a radioactive isotope implanted in insulators 108 6.3 Results and Analysis The 5 4 M n nuclei sit at lattice sites in the crystal because the ions are incorporated into the crystals during their growth. The nuclei thus experience a unique magnetic hyperfine field, Bhf, and the normalized 7-ray intensity observed at angle 6 to the magnetization axis at temperature T is given by equation 2.5 which can be written as W(9)54 = 1 + £ BkUkAkQkPk(cosO) (6.157) k=2A The Ak and Uk coefficients can be found in table 2.1. The Qfc are corrections for the solid angle subtended by the detectors and are Q 2 = 0.820 and Q 4 = 0.490 for the geometry used. For implanted nuclei there is a possibility that not all of them go into lattice sites. In the case of metals (e.g. iron), this effect is usually taken into account by a simple, two site model which assumes that a fraction / of the ions experience the full hyperfine field Bh/ and a fraction (1 — /) feel zero field. Although the hydrated crystals used in these experiments have a more complicated crystalline structure, we assume the same model so that the normalized intensity 7-ray intensity for the 5 6 Mn nuclei is given by W(9)56 = 1 + / £ B'kU'kA'kQkPk(cos9) (6.158) fc=2,4 In the 5 6 M n decay scheme there are several preceding transitions feeding the observed 847 keV 7-ray and the NO parameters can be found in table 2.1. Note that, even if the assumptions of the two site model are inappropriate, the factor / is still a useful parameter to describe the effect of implantation because it does describe the reduction in the 7-ray anisotropy, i.e. Chapter 6. First NO of a radioactive isotope implanted in insulators 109 where W(9)5e is the observed intensity and W(9)5e,m is the maximum intensity that would be observed if all the 5 6 M n ions were at substitutional sites. In experiments on nuclei with a long half-life it is usual to measure W(0) and W(90), by comparing cold counts to warm counts measured in directions along the quantization axis (0°) and perpendicular to it (90°) respectively. The effect of the decay of a short lived isotope can be obviated by measuring the ratio e(0,90) = W(0)/W(90) and comparing this ratio cold and warm. 6.3.1 5 6 M n - M n C l 2 4 H 2 0 experiment Figure 6.48 shows the results obtained from four successful implantation/cooling/warming cycles performed in the 5 6 M n - M n C l 2 - 4 H 2 0 experiment. In each cycle counts were taken at intervals of 300 s. The values of W(0)/W(90) for 5 4 M n and 5 6 M n obtained from the cold counts were normalized against a weighted average from warm counts taken before and after the cooling to give the anisotropy e(0, 90). In each separate run counts were av-eraged over five consecutive 300 s intervals and then the grand average for e(0,90) taken. The lowest value of e(0,90) for 5 4 M n , about 0.87, corresponding to a temperature T ~ 60 mK, was obtained in ~ 3 hours. This temperature is a little higher than expected, but may have been due to extraneous heating of the crystal by thermal radiation from 4 K walls. It should be noted that even in off-line experiments with optimum heat shielding, an initial cooling of a MnCl 2 -4H 2 0 crystal from 100 mK to a low temperature, e.g. 30 mK, takes many hours [37]. The 5 4 M n data can be fitted by a curve calculated from equation 6.157 with an exponential dependence on time given by with the time constant n = 5500 s. This is an effective relaxation time for the cooling (6.160) Chapter 6. First NO of a radioactive isotope implanted in insulators 110 1 \ 0.96 o ON O 0.92 0.88 0.84 -\ 1 1 1 1 1 1 0 2500 5000 7500 10000 12500 15000 Time (s) Figure 6.48: Variation of the 7-ray anisotropy, e = W(0) / W(90), versus time for implanted 5 6 M n (•) and 5 4 M n (O) in MnCl 2 -4H 2 0 single crystal. The dashed curve through the 5 6 Mn data is calculated from the fit through the 5 4 Mn data that is a simple exponential decay described in equation 6.160, with / = 0.96. Chapter 6. First NO of a radioactive isotope implanted in insulators 111 which, as mentioned above, is limited by the Kapitza boundary and nuclear spin-lattice relaxation. A very rough estimate using T\ = R K C , where the Kapitza resistance RK, for area A, is denned by RKAT3 ~ 5 • 10 - 3 K 4 m 2 W _ 1 [43] and the thermal capacity C is given by C T 2 ~ 5 • 10 - 5 JK, gives a time constant of 3000 s at T ~ 60 mK for the crystal used, indicating that both bottlenecks are limiting the cooling. As well as the two bottlenecks, the spin-lattice relaxation involves the solution of a master equation that includes populations of the 21+1 magnetic substates and the transition rates between them [2]. Thus the simple exponential behaviour described by equation 6.160 is very oversimplified because the actual relaxation process is quite complicated. However, the objective was to find a model that described the variation of the 5 4 M n anisotropy with time in order to determine how the 5 6 M n anisotropy changed. The dashed curve through the 5 6 M n data was obtained by calculating the 5 6 M n anisotropy from the 5 4 M n curve (see figure 6.49) using equations 6.157 and 6.158 with a value / = 0.96,and assuming the same relaxation time, T\. The comparison of e(0, 90) for 5 4 M n and 5 6 M n is shown in figure 6.49. A least squares analysis of the data gives the solid line corresponding to / = 0.96io!o7- The dashed line is for / = 1. The value obtained for / indicates that the occupancy of lattice sites by the implanted ions is very high in the two site model. It should be noted that, even in metals, the two site model, which is generally used for the analysis of nuclear orientation data, is rather crude. Nuclei can end up in sub-stitutional lattice positions, in interstitial sites (and perhaps there are more than one of these), or in positions damaged by the implantation process. Thus the nuclei might feel the full hyperfine field, a reduced one, or none at all. The structure of MnCl2-4H 20 is more complex than a metal like iron so that there may be more non-substitutional sites. The 2 + M n ion has a half-filled 3d shell and "spin only" magnetism, but there are crystal field effects that reduce the hyperfine field from that expected for the free ion value [36]. Chapter 6. First NO of a radioactive isotope implanted in insulators 112 0.88 -, 0.96 0.91 0.86 [ W ( 0 ) / W ( 9 0 ) ] 5 4 Figure 6.49: The comparison of the 5 6 M n anisotropy W(0)/W(90) with that for 5 4 M n in MnCl 2 -4H 2 0 single crystal. The solid line is the best fit to the data and corresponds to a value for the lattice site occupancy / = 0.96lo:o7> i e - c l ° s e t o unity. The dashed line corresponds to a value for the lattice site occupancy / = 1. Chapter 6. First NO of a radioactive isotope implanted in insulators 113 Therefore it is likely that the hyperfine interaction for non-substitutional Mn ions will be different from, but possibly close to, the full value. The observed anisotropy might be due to a majority of implanted ions in substitutional sites feeling the full hyperfine field and a few experiencing zero effect (two site model) or to many ions having a field slightly different from the full value. Future experiments using NMR experiments on the oriented nuclei (NMRON) should clarify the situation. 6.3.2 5 6 M n - C o C l 2 6 H 2 0 experiment These experiments were performed in B0 = 0.6 T. Because of the alignment of the crystal, the magnetization is turned away from the easy axis towards the applied field as the latter is increased. We calculate, using the reported values for the exchange and anisotropy constants [44], that the electronic sublattice magnetizations are turned 4° towards the direction of B0 that is also the direction of detector 1. In this case, the angular position for the horizontal detector 1, which was at 32° to the magnetization axis in B0 = 0, would be 9i = 28° assuming exact alignment of the crystal. The orientation of detector 2 remained 02 = 90°. The angle 6i can actually be determined from the 5 4 Mn data. The plot of the normalized intensities of the 5 4 M n 7-rays measured in the two detectors is shown in figure 6.50 and the best fit to the data (dashed line) indicates that 6\ was actually 27° instead of the calculated 28°, but this difference is well within the uncertainty in aligning the crystal. For the data analysis we used the measured value 6\ = 27°. The average results for four successful implantation cycles are shown in figure 6.51. The final value of e(27,90) = W (27)/W(9Qi) corresponds to a lower crystal temperature of T = 40(2) mK. The dashed curve through the 5 4 M n is a fit using a double exponential function e(27,90) = 1 - 0.05 1 - exp ( - 0.15 1 - exp (6.161) Chapter 6. First NO of a radioactive isotope implanted in insulators 114 W ( 9 0 ° ) 5 4 Figure 6.50: The plot of the normalized intensities for 5 4 M n measured in the two detec-tors. Detector 2 is at an angle of 90° to the magnetization direction and therefore mea-sures a normalized intensity W2 = W(90). Because of the alignment of the CoCi2-6H20 crystal, detector 1 is at angle 6\ to the magnetization direction, and the best fit to the data (dashed line) shows that di = 27° so that Wl = W(27). Chapter 6. First NO of a radioactive isotope implanted in insulators 115 where r 2 and r 3 are 250 s and 4000 s respectively. This expression was chosen to fit the data, but the first term could be considered to represent the removal of 5 9 C o nuclear spin heat capacity (which is much smaller than for the 5 5 M n heat capacity in MnCl 2 -4H 2 0 because of the much smaller hyperfine field) while the second term is due to the nuclear spin-lattice relaxation (expected to be weaker for the same reason). The dashed curve through the 5 6 M n data is then calculated from the fit to the 5 4 M n data using equations 6.157 and 6.158 with a value for / = 0.53 estimated from the comparison of e(27,90) for 5 4 M n and 5 6 M n shown in figure 6.52. The best fit for the data shown in figure 6.52 is obtained using / = 0.53 ± 0.10. There are various explanations for the smaller 5 6 Mn 7-ray anisotropy relative to that for 5 4 M n in the CoCl 2 -6H 2 0 experiment. Because the 5 6 M n ions were implanted into a Co crystal host, it seems likely that fewer of them ended up in good lattice positions. The hyperfine field in interstitial sites may be less than in lattice sites. Also, the interstitial Mn spins may not have been aligned with the easy axis of the crystal. In each case, the average anisotropy of the 7-rays would be reduced. Incident thermal radiation could also cause a temperature gradient across the crystal that has quite a low thermal conductivity. The 5 6 M n are implanted only a few hundred A into the surface and, therefore, might be at a higher temperature than the 5 4 M n deep in the interior. However, this reduced effect in 5 6 M n should then also be observed in the MnCl 2 -4H 2 0 experiments. The quality of the surface might be a factor, and, indeed, the surface of the MnCl 2 -4H 2 0 was visually better than that of the CoCl 2 -6H 2 0 crystal so that in the latter there may have been a significant number of ions in damaged sites feeling a low or zero hyperfine field. Again, future experiments, measuring W(0) vs T down to lower temperatures and performing NMRON would provide more precise information concerning these effects. A very small value of e ~ 1% was measured for the 6 0 Co indicating that the hyperfine Chapter 6. First NO of a radioactive isotope implanted in insulators 116 0.96 % 0.92 El 0.88 0.84 0.8 s I \ * - - -0 2500 5000 7500 10000 12500 Time (s) Figure 6.51: Variation of 7-ray anisotropy, e = W(27)/W(90), versus time for implanted 5 6 M n (•) and 5 4 M n (O) in CoCl 2 -6H 2 0 single crystal. The dashed curve through the 5 6 Mn data is calculated from the fit through the 5 4 Mn data that is the double exponential decay described by equation 6.161, with / = 0.53. Chapter 6. First NO of a radioactive isotope implanted in insulators 117 0.84 8 0.88 H o (N 0.92 -0.96 1 0.96 0.92 0.88 0.84 0.8 [ W (27) / W (90) ] 5 4 Figure 6.52: The comparison of the 5 6 M n anisotropy with that for 5 4 M n in CoCl 2 -6H 2 0 single crystal for the data shown in figure 6.51. Because the last two points in fig. 6.51 have the same value of 0.811 for [W(27)/W(90)]54, the average of the two [W(27)/VF(90)]56 values is taken to give the last point in this figure. The best fit is given by the solid line and gives a value of the lattice site occupancy / = 0.53 ± 0.10. The dashed line corresponds to / = 1. Chapter 6. First NO of a radioactive isotope implanted in insulators 118 interaction in the cobalt atoms is quite small. 6.4 1 0 4 A g implantation A further experiment to to implant 1 0 4 A g into MnCl2-4H 20 doped with 5 4 M n was at-tempted, again at Louvain-La-Neuve. Unfortunately, due to experimental difficulties, only a short period of data could be obtained and in zero field. Without the benefit of a reduced Ti due to "magnon cooling" the crystal could not be cooled enough to see anisotropy in the 5 4 M n . It appeared, however, that there was a 2% anisotropy in one of the analysed 1 0 4 A g 7 lines. Subsequent careful analysis, using techniques that will be described in section 7.2, showed that the observed effect was due to a small drift in the position of the beamline. This could be determined by comparing areas of 7 rays which should increase with areas which should decrease in the same detector. Although the experiment is inconclusive, it is important to note that beamline drift can be an important source of error if the detectors are close enough to the sample. A future experiment using Ag as an impurity in MnCl2-4H 20 is planned. The isotope 1 1 0 m A g is suitable for growth into MnCl 2 -4H 2 0 because the half life is 250 days. U O m A g is also a suitable isotope for NO because = +3.6/xw, 1 = 6 and there are suitable 7-rays. Production of 1 1 0 m A g is currently underway at T P J U M F . 6.5 Conclusions Our experimental results demonstrate that radiation damage does not preclude significant 7-ray anisotropies being obtained after implantation into insulators. For 5 6 M n implanted into MnCl 2 -4H 2 0 an effect close to that for 5 4 M n was observed, corresponding to / = 0.96lo!o7 m t n e simple two site model. This value for / might also represent a large number of 5 6 Mn ions being in sites in which the hyperfine field is less but close to the Chapter 6. First NO of a radioactive isotope implanted in insulators 119 full substitutional value. It appears that the situation is similar to that in metallic hosts, in which there may be some radiation damage, but the final position of the implanted nucleus is not in the region of greatest damage. For 5 6 M n implanted into CoCl2-6H20, the observed 7-ray anisotropy was significantly less than that for 5 4 M n with / = 0.53 ± 0 . 1 0 . It is also shown that the long spin-lattice relaxation times that can occur in insulators does not prevent the nuclear orientation of implanted isotopes. These experiments have demonstrated that insulators can be used as hosts for on-line experiments, at least for implanted isotopes with half-lives > 1 hr. Potentially, a new and exciting area for condensed matter physics has been opened up. Interesting magnetic structures which are difficult to dope could be studied by nuclear methods by implantation of a suitable isotope. Chapter 7 Cobalt impurities in MnCl 2-4H 20 In order to use magnetic insulating crystals as hosts for the implantation of a variety of ions, it is important to understand the properties and effects of impurities. The implantation fractions measured in Chapter 6 of / — 0.96io!o7 f ° r 5 6 M n into MnCl 2 -4H 2 0 and / = 0.53 ± 0 . 1 0 for 5 6 M n into CoCl 2 -6H 2 0 indicate that the implanted ions find good lattice sites when they are the same element as the host crystal, but that a significant fraction do not when they are different. If NMR measurements are to be attempted on implanted isotopes it is important to understand the possible arrangements of the impurities in the lattice. In order to investigate the effects of impurities, a series of MnCl 2 - 4H 20 crystals were grown with Co impurities (see Table 7.3). 5 4 Mn was also included in the crystals for two reasons. Firstly, the temperature of the crystals could be monitored at all times. Secondly, the effects of the impurities on the NMR line of 5 4 M n was studied. crystal 5 4 M n (/iCi) 5 7 C o (//Ci) 6 0 C o (/iCi) 5 9 C o % A 1.4 3.3 0 0 B 0.6 0 3 0 C 2.5 3.2 1.6 3.96 Table 7.3: The activities of the 3 crystals grown to investigate the properties of impurities in MnCl 2-4H 20 120 Chapter 7. Cobalt impurities in MnCl2-4H20 121 7.1 Sample preparation and experimental details Seed crystals of approximately 2 mm in length were grown from saturated aqueous solu-tions of MnCl2-4H20 at room temperature. The seed crystals were then suspended in a saturated solution containing the radioactive isotopes and stable Co in a thermal bath maintained at a temperature of 30°C. As the water evaporated from the solution, the crystals grew to a size of 1-2 cm in length over a period of 12 to 24 hours. Finally, the crystals were suspended in a saturated solution of stable MnCl 2 -4H 2 0 in order to grow a layer of non-radioactive layer to protect from possible contamination of the apparatus by radioisotopes. The smooth, (100) face of the crystal was cleaned by rubbing on a damp piece of filter paper. The crystal was then mounted with the (100) face in contact with the cold finger, in the SHE dilution refrigerator, as shown in figure 4.28. Apiezon N grease was used to improve the thermal contact, and the crystal was tied down tightly with unwaxed dental floss. The easily identifiable c-axis was mounted vertically. The easy axis of magnetization is at an angle * = 7° to the c-axis (towards the a-axis). In zero field, therefore, we measured the angular distribution of 7-rays, W(7°), rather than W(0). Prior to mounting the crystal, the copper cold finger was annealed overnight at 800°C under an atmosphere of hydrogen. The 6 0 C o in Fe thermometer was only mounted with crystal A for refrigerator diagnostics purposes. The crystal was precooled to 240 K under an atmosphere of air because hydrated crystals deteriorate rapidly under reduced pressure at room temperature. The R F coil used in this experiment was an untuned, single turn shown in figure 3.17, with R\ = 60 fl and i? 2 = 60 fl. The capacitance, C", came from capacitance between the coil and ground and was less than 1 pF. No stabilization was used, although the cold finger resistor, which is extremely sensitive to direct R F heating was monitored at all Chapter 7. Cobalt impurities in MnCl2-4H20 122 times to ensure that R F power was being received by the sample. The typical voltage settings were |u 9 | = 400 mV. This corresponded to a current through the coil of ~ 1.3 mA and, using equation 3.57, the oscillating field was ~ 1.6 mG. This field is similar to employed in previous NMR experiments on MnCl 2 - 4H 20 [37]. Measurements were taken with either 2 Nal detectors at 7° and 90° to the easy axis, or with 1 Ge detector at 7° to the easy axis. In general, the high resolution Ge detector was used for NO measurements, while the higher efficiency Nal detectors were used for NMR. All results for 5 7 C o were obtained with the Ge detector since the 122 keV and 136 keV 7-rays cannot be resolved with the Nal detectors. 7.1.1 Stable Co Crystal C (see Table 7.3) was grown with a fraction of stable 5 9 C o . This fraction can be precisely determined because the crystal was also grown with 6 0 C o and 5 7 C o . A solution was prepared with 13.1586 g of MnCl 2 - 4H 20 and 0.8330 g of CoCl 2 -6H 2 0. The total number of Mn and Co ions in solution is then where J\f is Avagadro's constant. The fraction of Co ions in solution is = 5.356 • 10~2 (7.163) The fraction of Co atoms in the crystal is certainly less than the fraction of Co ions in the master solution. The fraction of Co ions that grow into the crystal can be calculated based on the activities of the radioactive isotopes in the crystal. The activity of the original master solution and the final crystal were counted for a 1000 s live time at 50 cm from the Ge detector. The background counts were subtracted Chapter 7. Cobalt impurities in MnCl2-4H20 123 and the counts from the 122 keV line of 5 7 C o and the 1173 keV and 1330 keV lines of 6 0 C o were added together to find an overall fraction of Co that grew into the crystal. The fraction of 5 4 M n was calculated from the 835 keV line. The results are shown in Table 7.4. Isotope fraction % 5 4 Mn 4.49 ± 0 . 0 4 total 5 7 C o and 6 0 C o 3.32 ± 0.02 Table 7.4: The fraction of each isotope in solution which grows into the crystal. The ratio of 5 9 Co in the crystal is then —2°. = r = ——0.0536 = (3.96 ± 0.04) • 10 - 2 (7.164) Mn 4.4» Since each Mn ion has 6 nearest neighbours, the probability p(n) of an Mn ion having n nearest neighbour Co ions is p(n) = rn(l - r ) 6 - " C 6 n (7.165) if the Co ions grow into substitutional lattice positions. In this case, p(l) = 0.19, while p(2) = 0.02. If the Co ions sit at substitutional lattice sites, and have an effect on the hyperfine field at neighbouring Mn sites, we would expect to see a satellite line approximately 1/4 the intensity of the main 5 4 M n resonance line. 7.2 Data collection and analysis The data were collected on 2 PCs as shown in figure 7.53. Four single channel analysers (SCA) were used to monitor the 5 4 M n NMR resonances. A multichannel analyser (MCA) was used while searching for 5 7 C o and 6 0 C o resonances, and when collecting data for detailed NO measurements. Although both systems were employed in parallel, it was important to to set the window positions and widths on the SCAs several times because Chapter 7. Cobalt impurities in MnCl2-4H20 124 the input impedance of the SCA changes with each position and width setting. Once the desired windows were achieved, the stability of the gain was monitored on the M C A . PC 1 rf generator 4 SCAs rf coil A sample Amplifier Ge or Nal detectors preamp Figure 7.53: Data collection setup. Data from the M C A was typically collected over 1000 s live time intervals. The Ortec software supplied with the M C A could output a list of peak positions and intensities. Since hundreds of spectra were collected, a programme was written to read these files, extract the necessary peak information collect it into a single file. However, it was found that the Ortec software did not perform the best possible fitting, particularly for the low energy 5 7 Co 7-rays where background subtraction is extremely important. Therefore, software was written to analyse the spectrum data directly. The Chapter 7. Cobalt impurities in MnCl2-4H20 125 method employed was a x 2 fit, with an asymmetric Gaussian fitting function given by (s-p) x < p (7.166) k + mx + A exp k 4- mx + A exp [^ ~^ 2 ; x > p where k, m, v4, p, <7/ and or are parameters of the fit. The minimization of x 2 was performed with MINUIT, and the results of several hundred spectra could be fit in less than one minute. The areas of the lines were calculated using several techniques, the results of which were then compared: • Area = S/TTA^^ • Integration over a fixed window, or integration over a fixed width window centered at p. • Integration over a window centered at p with baseline subtraction using k and m • Integration over a window centered at p with linear subtraction based on the average counts in fixed windows to the left and right of the line. Although it is customary to fit Ge detector data with skewed Gaussians, it was found the fitting with asymmetric Gaussians gave comparable results while being simpler to implement numerically. The accuracy of the results were tested by comparing the various techniques to a series of warm count spectra (typically around 100). Although the average counts found by each technique were consistent with each other, it was found that the smallest variance was obtained with a calculation based on the Gaussian fit parameters in the case of Ge detector data. In this case, k and m are strongly correlated with each other (because the background is almost constant), but very weakly correlated with the Gaussian fit parameters A , p, oi and oT because the lines are extremely narrow [pi and oT are on the Chapter 7. Cobalt impurities in MnCl2-4H20 5000 4500 4000 3500 to 3000 3 2500 o <•> 2000 t • • 500 1000 1500 2000 2500 3000 3500 Channel 4000 Figure 7.54: The full Ge detector spectrum for crystal C. Chapter 7. Cobalt impurities in MnCl2-4H20 127 800 3690 3700 3710 3720 3730 3740 Channel Figure 7.55: The Ge spectrum for the 1330 keV 6 0 C o line. The line is the best fit to the data. Chapter 7. Cobalt impurities in MnCl2-4H20 128 order of 2 keV). Therefore, the area calculated from the Gaussian fit parameters gave excellent results. 1400 T 1 1200 H 700 900 1100 1300 1500 1700 1900 Channel Figure 7.56: The full spectrum for the Nal detector for crystal B. The gray line is the best fit. In the case of the Nal detectors the best results were obtained with a fixed-width window centered at p. The reason is that the Nal lines are extremely broad (on the order of 100 keV), and the parameters k, m, A and oi and or become more strongly correlated. Using only the Gaussian parameters to calculate area then introduces more errors. Since p is, in all cases, very weakly correlated with the other parameters and has a very small uncertainty (less than 0.1 keV), the fixed window centered at p was the best choice. A fixed window alone did not give good results because of the drift in gain over time of the Nal detector setup. The baseline was not subtracted from the Nal spectra since they Chapter 7. Cobalt impurities in MnCl2-4H20 129 were only being used in the search for NMR resonances. Background subtraction would have only folded errors in k and m into the results. 7.3 NO Results In order to make use of the "magnon cooling" [37] effect to cool the crystals, a field of 0.72 T was applied. This is the field at which the spin-flop transition of MnCl2-4H 20 occurs [37, 41] and 7\ is dramatically reduced. The precise field needed to induce the spin-flop transition in a given experiment depends on the exact orientation of the crystal. The initial cooling of the crystal was performed in a field B0 ~ 0.7 T , and the value of W(0) as a function of time is shown in figure 7.57. The current in the superconducting magnet was then increased slowly to determine the field at which the spin flop transition occurred. The transition can be seen in figure 7.58 where W(0) is plotted as a function of the current in the superconducting magnet. Currents of 6.35 A were subsequently used to provide the optimum magnetic field for cooling the crystal. The final results of NO for the three crystals are given in table 7.5. The data were obtained from warm and cold counts in zero applied field. The counting times ranged from 27000 s to 85000 s. The data were collected on the M C A every 1000 s. Each 1000 s of data were analysed using Ortec's software and, independently by fitting with asymmetric Gaussians using MINUIT as described in section 7.2. This was necessary to ensure that that there was no change in counts for the duration of each collection period. The total counts were then the sum of the results of each of the 1000 s fits. As a final verification, all the 1000s spectra for each run were added together and fit with asymmetric Gaussians. It was found that the anisotropy calculated by each of the three methods consistently agreed within 1 a statistical error for the 6 0 C o and 5 4 M n lines. The comparison of the 5 7 Co did not always agree so well due to the important contributions of the background Chapter 7. Cobalt impurities in MnCl2-4H20 130 1 0 20 40 60 80 100 120 140 Time (1000s) Figure 7.57: W(0) as a function of time, showing the initial cooling of the entire crystal. Chapter 7. Cobalt impurities in MnCl2-4H20 131 0.75 6.25 6.3 6.35 6.4 6.45 Magnet Current (A) 6.5 6.55 Figure 7.58: W(0) as a function of the current in the superconducting magnet. The spin-flop transition begins at approximately 6.35 A. The transition is broadened by do-main effects. Chapter 7. Cobalt impurities in MnCl2-4H20 132 at low energies. The background is due to 2 sources: Compton scattering from the higher energy 7-rays, and electrical noise. The Compton background changed when the anisotropy of the higher energy 7-rays increased - particularly the 5 4 Mn contribution which displayed the largest anisotropy. This is an issue when performing NO, although it is a less important concern when performing NMR on 5 7 C o because the Compton background does not change in that case. The contribution from low energy electrical noise is essentially impossible to control and predict. It varies with the time of day, and with the activities of other experimentalists in the building. Changes in the background did affect the measured fits. Careful comparisons with the three fitting techniques, and with skewed Gaussian fits, showed that the statistical errors were larger than predicted by the errors in fitting parameters. It was found that a realistic assessment of the systematic errors on the anisotropy for the NO results was approximately 1% for both the 122 keV and 136 keV 7-rays. This 1% error is used as to calculate the errors on the hyperfine field. Unfortunately, it meant that the 122 keV 7-ray of 5 7 C o was useless for calculating a hyperfine field, since the total expected effect at the temperatures achieved was approximately 4-1%. The base temperature of each crystal was calculated from the NO results of the 5 4 M n , the hyperfine field of which is known precisely. The anisotropy of the 5 4 M n was monitored over time throughout the measurements to ensure that this base temperature did not change. The hyperfine fields of 5 7 C o and 6 0 C o were calculated by comparing the measured anisotropy with that calculated from the NO parameters. The results of the 1170 keV and 1330 keV 7-rays were averaged to find the hyperfine field for 6 0 C o . The 136 keV 7-ray was used to determine the hyperfine field for 5 7 C o . In Crystal A, the hyperfine field is 15.5 ± 2 T for 5 7 C o . In crystal B, the hyperfine field is 29.1 ± 0.5 T for 6 0 C o . In crystal C, the hyperfine fields are 15.6 ± 2.7 T and 25.6 ± 0.4 T for 5 7 C o and 6 0 Co respectively. The base temperatures and hyperfine fields calculated for each Chapter 7. Cobalt impurities in MnCk-4H20 133 crystal isotope 7 energy (keV) W(7°) A 5 7 C o 122 0.999 ± 0 . 0 1 ( 0 . 0 0 1 ) 5 7 C o 136 0.956 ± 0 . 0 1 ( 0 . 0 0 3 ) 5 4 M n 835 0.743 ± 0.002 B 5 4 Mn 835 0.812 ± 0.004 6 0 C o 1170 0.944 ± 0.002 6 0 C o 1330 0.946 ± 0.002 C 5 7 C o 122 1.021 ± 0 . 0 1 ( 0 . 0 0 1 ) 5 7 C o 136 0.970 ± 0 . 0 1 ( 0 . 0 0 3 ) 5 4 M n 835 0.811 ± 0 . 0 0 1 6 0 C o 1170 0.955 ± 0.002 6 0 C o 1330 0.958 ± 0.002 Table 7.5: The nuclear orientation results for the 3 crystals The statistical errors are given from the fit parameters except for the 122 keV and 136 keV lines in which case a systematic error of 0.01 is assumed and the statistical errors from the fit parameters are given in parentheses. crystal Temp. (mK) isotope 7 energy (keV) Hyp. field (T) fiB/hl (MHz) A 36.0 ± 0 . 2 ^ 5 7 C o 136 15.5 ± 2.01^ 159 ± 20+3 B 44.4 ± 0.6^;{ 6 0 C o 6 0 C o 6 0 C o 1170 1330 avg(1170+1330) 29.3 ± 0.7-^i 28.8 ± OJtol 29.1 ± 0.b±°Q;l 170 ± 4 l 2 167 ± 4±l 169 ± 3+1 C 44.3 ± 0.2^;{ 5 7 C o 6 0 C o 6 0 C o 6 0 C o 136 1170 1330 avg(l 170+1330) 15.6 ± 2 . 7 ^ ; } 26.0 ± 0.6lg;^ 25.1 ± 0.6±gj 25.6 ± 0.4lg;2 160 ± 28+} 151 ± 3+2 145 ± 3±l 148 ± 2t\ Table 7.6: The base temperature and hyperfine fields for the 3 crystals. The first error given is statistical. The second is the systematic error due to possible misalignment of the crystal. A possible misalignment of 7° was used to calculate the systematic error. The error on the 136 keV line is purely systematic: the first error is from the 1% systematic error in the detectors (the statistical errors in the detector are negligible compared to the systematic error) and the second is from the misalignment of the crystal. Chapter 7. Cobalt impurities in MnCl2-4H20 134 observed transition (except the 122 keV line) in each crystal are shown in table 7.6, as well as the frequency (in MHz) at which an NMR resonance would be expected. The first error reported is statistical. The second is systematic and is calculated based on a possible 7° misalignment of the crystal. The discrepancy between the 5 7 C o hyperfine field and the 6 0 C o hyperfine fields is most surprising. There are several possible explanations but none are entirely satisfactory: • It is possible that the Co impurities do not grow substitutional^ into the lattice, but into a variety of lattice positions, and experience a variety of hyperfine fields. The relative occupancy of each different site could depend on the particular growth conditions. However, it is difficult to understand the results of crystal C with those of A and B, since crystal C shows different hyperfine fields for the different isotopes in the same crystal. • It is possible, but unlikely, that the occupancy of particular sites could be strongly affected by the different mass of each isotope. • It is possible, but unlikely, that the Co magnetization is not aligned with the Mn, and the anisotropy is being measured at some very different angle than 0°. • If strong electric field gradients were present, a strong quadrupole interaction might explain these results. However, the relatively weak quadrupole moments involved (Q = +0.44 ± 0.05 b for 6 0 C o and Q = +0.52 ± 0.09 b for 5 7 Co) make this an unlikely explanation. • Finally, it has been observed by NMR, that the same isotopes of Fr in Fe experience a different hyperfine fields [45] and these fields can vary by as much as a factor of two, but it seems highly unlikely that Co isotopes should experience similar effects. This discrepancy is certainly worthy of further study. Chapter 7. Cobalt impurities in MnCl2-4H20 135 7.4 NMR results 7.4.1 Determining RF power 1800 1750 1700 1650 (0 1600 c 1550 3 o o 1500 1450 1400 1350 1300 499 499.5 500 500.5 501 Frequency (MHz) 501.5 502 Figure 7.59: The number of counts at 0° as a function of frequency showing the 5 4 M n resonance in an applied field of 0.2 T. The data were collected using a Ge detector and a SCA. The generator voltage is 400 mV, the step size is 25 kHz and the R F modulation is 50 kHz. The R F system was first tested by observing the NMR resonance of the 5 4 M n . The resonant frequency is already known for this system [35, 36, 37]. A plot of the resonance is shown in figure 7.59. The resonance is split into two subresonances because the applied field affects the two sublattices differently. The observed transitions here are from the P = - 3 to P = -2 states. Chapter 7. Cobalt impurities in MnCl2-4H20 136 1600 1550 1500 -{ 1450 1400 1350^ 1300 0 200 400 600 Generator voltage (mV) 800 Figure 7.60: Determining the saturation point for the RF. The frequency was set to 499.7 MHz, the resonant frequency of one of the 5 4 M n lines with a modulation of 50 kHz. The generator voltage was increased in 200 mV until the resonance was saturated. In subsequent frequency sweeps, 400 mV was used as the peak to peak generator voltage. Chapter 7. Cobalt impurities in MnCl2-4H20 137 The optimum R F power to use without warming the fridge was then determined by increasing R F generator voltage while staying at the resonant frequency. Although the resonance appears to be nearly saturated at 200 mV, 400 mV was used for the subsequent sweeps to be certain that enough R F power was delivered to the sample at all frequencies. The anisotropy is plotted as a function of R F generator voltage in figure 7.60. 7.4.2 5 7 C o and 6 0 C o N M R Frequency sweeps to find the NMR resonances of 5 7 C o and 6 0 C o were performed based on the NO results of table 7.5. Since the NO results predict significantly different hyperfine fields for the two Co isotopes, it was expected that the result of an NMR resonance would resolve the issue completely. In the case of 5 7 C o in crystal A, the frequency was swept from 120 to 240 MHz, in 25 kHz steps with a dwell time of 10 s and an R F modulation of ± 5 0 kHz. Based on the NMR of the 5 4 M n , it was assumed that 10 s was sufficient to saturate the modulation. The spin-lattice relaxation time for 5 4 M n in the crystal has been measured to be between 106 and 107 s at lattice temperatures below 65 mK [38]. Above 65 mK the dominant relaxation process is a Raman process while below 65 mK it is a direct process. Although the orbital contribution to the spin of 5 7 Co may make the spin-lattice relaxation shorter for the direct process, it seemed reasonable to assume that the spin-lattice relaxation time for Co satisfied Tf° >^ 103 s. A resonance should have been visible in the data, despite the short dwell time because 1000 s was sufficient to obtain approximately 6000 net counts in the M C A for the 136 keV 7-line of 5 7 C o , (statistical errors of approximately 1%). The frequency spectrum should then contain low temperature counts until the resonance frequency, and then higher counts relaxing very slowly to the low temperature counts in a time sufficiently slow to achieve good statistics. Unfortunately, no significant change in anisotropy was observed throughout, the entire frequency spectrum. Chapter 7. Cobalt impurities in MnCl2-4H20 138 Although the frequency generator used had a maximum modulation of ± 1 0 0 kHz, a variety of different sweep patterns and modulations were attempted in an effort to find the NMR resonance. In particular, the frequencies 155 MHz to 205 MHz were split into 5 MHz blocks. The R F field was applied in 50 kHz steps at 100 kHz modulation with a 1 s dwell time. The entire 5 MHz block was covered every 100 s. The total dwell time for each block was 12100 s, which allowed significant statistics to be achieved for each frequency block. Resonances of a variety of widths (from 0 to more than 5 MHz) would be observable by this technique, even with relaxation times, Tf° as short as 1000 s. Careful analysis of the data, as explained in section 7.2, did not indicate the existence of a resonance because the 7-ray anisotropy did not change throughout the sweeps. For the 6 0 C o NMR, it was possible to use the much higher efficiency Nal detectors, and it was possible to cover a greater frequency range. In this case, the frequency was swept through 1 MHz blocks, with 50 kHz steps, 5 s dwell time and a modulation of 100 kHz. Each MHz block was counted for 1000 s, giving more than 130000 counts, or a relative statistical error of approximately 0.3%. A resonance with Tf" as short as 1000 s and a width of several MHz would be visible with this technique. The total frequency range covered was 70 to 363 MHz. Again, a careful analysis of the data showed no resonance anywhere in the spectrum. One explanation for not observing the NMR resonance is that some Co ions do not grow into the crystal lattice substitutionally. This seems quite possible because MnCl 2 -4H 2 0 contains four water molecules while CoCl 2 -6H 2 0 contains six. Conse-quently, the Co ions could be situated in a variety of interstitial lattice configurations and experience a wide range of different hyperfine fields. Resonances could also be suf-ficiently broadened as to be unobservable with our NMR techniques. It is also possible that Tp0 «C 1000 s, which would also make the resonance difficult to observe with our sweep technique. The NMR results on the spin-lattice relaxation of the 5 4 M n show that Chapter 7. Cobalt impurities in MnCl2-4H20 139 Co may have a short T i , and this could also explain why the resonance was not observed. 7.4.3 NMR results on 5 4 M n - the influence of Co impurities The NMR data on 5 4 M n was mainly collected on Nal detectors. The high efficiency of the detectors allowed for dwell times of 100 s. The data were fitted using a x 2 minimization. The model is similar to that used in previous NMR studies of 5 4 M n in MnCl2-4H 20 [36]. The changes in level populations due to spin-lattice relaxation are governed by equation 2.44. The NMR resonance is assumed to be Gaussian in shape and inhomogeneously broadened. The R F field is assumed to be sufficiently strong to completely saturate the resonance. If the NMR resonance involves the equalization of levels m and m + 1 then, at a frequency OJ with a modulation of ±</>o, the change in population of the mth sublevel is given by where Ag is the amplitude of the resonance. The levels must be completely equalized by integrating over the entire frequency range, therefore If the system is at thermal equilibrium before the resonance, then pm+\ = pm exp(—AE/kBTL) where TL is temperature of the lattice and AE is the difference in energy between the two levels and is therefore equal to the resonant frequency. However, the populations are not at equilibrium in a sweep through the pseudoquadrupole-split resonances when the sweep time is < T\. The change in level population are governed by 27 + 1 equations (one for each level), or 21 equations by using ZmPm = 1- The change in anisotropy can be calculated using (7.167) Pm+1 ~~ Pm 2a (7.168) Chapter 7. Cobalt impurities in MnCl2-4H20 140 equation 2.13, or -^r = ^am{9)-w ( 7 J 6 9 ) In the simplest case, the x2 fit then involves 4 parameters: a, T L , UJ0 and Ti where a is the width of the resonance, TL is the temperature of the lattice, UQ is the centre frequency of the resonance and 7\ is the spin-lattice relaxation time. In reality, the data were collected using SCAs with baseline contamination from the higher energy 6 0 Co 7-rays. It was therefore difficult to determine the exact anisotropy, but only the change in counts due to the resonance. This means that the lattice temperature, T L , which appears in the equation 2.44 cannot be determined precisely from the data. Therefore, TL was assumed to be a constant and equal to 44.3 mK, the base temperature determined by NO data. The possible error in TL can be estimated by the maximum change in base temperature anisotropy between the sweeps. This change is at most 1% and means that the TL = (44.3 ± 2) mK for all the fits. Furthermore, test fits where TL was included as a fit parameter did not significantly alter the values of the other parameters. In order to analyse the data, two more parameters are introduced: K was is the number of counts measured in the given dwell time (typically 100 s) before resonance, and B which is the factor of proportionality between the change in counts, dN(9)/dt, and the change in anisotropy. B is given by dN(9) _ BdW(9) dt ~ dt = B E U ^ (7-170) The fits used five parameters in the simplest case: a, u0, 7\, K and B. An example of the data and fits can be seen in figures 7.61 and 7.62. The resonances do not have satellite lines, as might be expected from Mn ions with nearest neighbour impurities. However, the linewidth is much broader than the pure MnCl2-4H 20 crystal. Kotlicki et al. report a H W H M of 35 kHz [36], which corresponds to o — 42 kHz, while Chapter 7. Cobalt impurities in MnCl2-4H20 141 39000 33000 -I 1 . , 1 1 494 496 498 500 502 504 Frequency (MHz) Figure 7.61: The number of counts at 0° as a function of frequency, showing the NMR resonance of 5 4 M n in zero field. The frequency was swept upwards. The frequency step is 4-100 kHz, the modulation is ± 1 0 0 kHz. The best fit parameters are K = 34170 ± 27, B = 257 ± 7, a = (254 ± 32) kHz, u0 = (500.27 ± 0.01) MHz and 7\ = (176 ± 68) • 103 s. Chapter 7. Cobalt impurities in MnCl2-4H20 142 the crystal with impurities has a a > 200 kHz. This increase in width is certainly due to the presence of impurities. The most accurate value of the spin lattice relaxation time, T i = (253 ± 18) • 103 s is obtained from figure 7.62 because it has the longest relaxation tail past the resonance. 39000 -, 1 33000 -I . , , 1 1 494 496 498 500 502 504 Frequency (MHz) Figure 7.62: The number of counts at 0° as a function of frequency, showing the N M R resonance of 5 4 M n in zero field. The frequency was swept downwards. The frequency step is -50 kHz, the modulation is ±50 kHz. The best fit parameters are K = 34196 ± 2 7 , B = 260 ± 4, a = (205 ± 21) kHz, co0 = (500.36 ± 0.01) MHz and Tx = (253 ± 18) • 103 s. Upward and downward sweeps of the main resonance with smaller frequency steps can be seen in figures 7.63 and 7.64. The fit to the downward sweep (figure 7.64) appears reasonable, but the upward sweep (figure 7.63) is not fitted well at the start of the resonance. It seems clear that a simple Gaussian is not sufficient to fit the data. Chapter 7. Cobalt impurities in MnCl2-4H20 143 39000 38000 H 37000 -i 36000 -o o 35000 -34000 33000 499 499.5 500 500.5 501 501.5 502 502.5 Frequency (MHz) Figure 7.63: The number of counts at 0° as a function of frequency, showing the NMR resonance of 5 4 M n in zero field. The frequency was swept upwards. The frequency step is +5 kHz, the modulation is ± 5 kHz. The best fit parameters are K = 34685 ± 22, B = 266 ± 3, a = (196 ± 8) kHz, u0 = (500.21 ± 0.01) MHz and T x = (288 ± 9) • 103 s. Chapter 7. Cobalt impurities in MnCl2-4H20 144 39000 33000 H 1 1 1 1 500 500.2 500.4 500.6 500.8 Frequency (MHz) Figure 7.64: The number of counts at 0° as a function of frequency, showing the N M R resonance of 5 4 M n in zero field. The frequency was swept downwards. The frequency step is -2 kHz, the modulation is ± 5 kHz. The best fit parameters are K = 34311 ± 35, B = 241 ± 15, o = (168 ± 10) kHz, CJ0 = (500.42 ± 0.01) MHz and Tx = (488 ± 146) • 103 s. Chapter 7. Cobalt impurities in MnCl2-4H20 145 In order to fit the data more accurately, a more elaborate model was introduced. The assumption is that a fraction, r, of the Mn ions are sitting in sites perturbed by neighbouring Co impurities. These perturbed sites have a different resonant frequency and line width than the unperturbed sites. The relaxation time for the two sites was assumed to be the same. This model introduces too many free parameters into the fit. Therefore the upward and downward sweeps (in figures 7.63 and 7.64) were fitted simultaneously and Ti was fixed at the value of 253 • 103 s found from the fit to the data in figure 7.62. It was observed that the resonance appeared to be in a different position for the upward and downward sweeps. This could result from the frequency modulation being larger than indicated by the R F generator and a modulation parameter 0 O was introduced to take this effect into account. A subsequent measurement with a spectrum analyser at v ~ 500 MHz showed that the true center frequency is not the expected frequency to the precision indicated on the front panel of the generator. Furthermore, when the indicated modulation was less than 20 kHz, the spectrum had an approximately Gaussian shape with a F W H M of 20 kHz. It is therefore possible that the resonance was being excited at frequencies as much as 20 kHz from the centre frequency. However, the modulation parameter from the fits was found to be 84 ± 2 kHz, ie. four times greater. The statistical error on the resonant frequencies found from the fits are only approximately 10 kHz, but this discrepancy between upward and downward sweeps can be explained by a systematic error of ± 1 0 0 kHz in the long term frequency stability of the R F generator. In the final analysis there were 9 free parameters used to fit the data: K\ and K2, the counts prior to resonance for the upward and downward sweeps respectively; 0\ and <T2, the linewidths of the main and secondary resonances respectively; uj\ and u2, the centre frequencies ofthe main and secondary resonances respectively; B, as described by equation 7.170; r, the ratio of spins in perturbed sites; and (f>0, the modulation parameter Chapter 7. Cobalt impurities in MnCl2-4H20 146 or frequency shift between the upward and downward sweeps. The results of the fit can be seen in figures 7.65 and 7.66. There is definitely im-provement to the fit in the ini t ia l rise of the upward swept resonance (figure 7.65). The ratio of perturbed spins is r = 0.14 ± 0.04 and is consistent wi th the expected value of approximately 0.25 calculated for substitutional impurities wi th only nearest neighbour effects. This value of r is also consistent with a variety of interstitial lattice positions. Note that the effects of an impurity is to decrease the resonant frequency. The model, however, is not entirely satisfactory. In particular, the width of the main resonance, o\ = (155 ± 7) kHz, is s t i l l much larger than the width of 42 kHz previously measured for a crystal with no impurities [36]. The assumption that an impuri ty affects only the spins in its immediate vicinity is not valid, and the effects must be of longer range. The model is s t i l l useful because it shows that the resonance is not Gaussian in shape, nor even symmetric about the centre frequency. Thus the effect of the impurities is to shift the resonant frequency of the 5 4 M n downward, although the magnitude of this shift is relatively small . The N M R resonance in an applied field of 0.5 T can be seen in figure 7.67. The line is split into two subresonances corresponding to the spins from each of the two sublattices. The fit parameters were then K, B, o\, uj\, a2, OJ2, Tn and Ti2 where the index 1 and 2 refer to the lower frequency and higher frequency lines respectively. In this case, a more complicated model using nearest neighbour effects was not attempted because it introduces too many parameters, and T\ is not known and cannot be fixed. The fit is not entirely satisfactory, particularly in the final relaxation to the base counts. This is likely due to the assumption of Gaussian lineshape, and the fact that the modulation is larger than indicated by the generator. A larger modulation has the effect of preventing the spins from relaxing as early and the fit then overestimates the value of T x . It is clear Chapter 7. Cobalt impurities in MnCl2-4H20 147 39000 499 499.5 500 500.5 501 501.5 502 502.5 Frequency (MHz) Figure 7.65: The number of counts at 0° as a function of frequency, showing the NMR resonance of 5 4 M n in zero field. The frequency was swept upwards. The frequency step is +5 kHz and the modulation is ± 5 kHz. The best fit parameters are K\ = 34577 ± 31, K2 = 34432 ±31, B = 287 ± 4, ox = (155 ± 7) kHz, ux = (500.32 ± 0.01) MHz, o2 = (205 ± 58) kHz and co2 = (499.97 ± 0.06) MHz, r = 0.14 ± 0.04 and <f>0 = 84 ± 2 kHz. The thick line is the best fit. The thin line is the shape of the double Gaussian NMR resonance in arbitrary units. Chapter 7. Cobalt impurities in MnCl2-4H20 148 39000 i 33000 500 500.2 500.4 500.6 500.8 Frequency (MHz) Figure 7.66: The number of counts at 0° as a function of frequency, showing the NMR resonance of 5 4 M n in zero field. The frequency was swept downwards. The frequency step is -2 kHz and the modulation is ± 5 kHz. The best fit parameters are the same as in figure 7.65 because both curves were fit simultaneously with the same parameters. The thick line is the best fit to the data. The thin line is the shape of the double Gaussian NMR resonance in arbitrary units. Chapter 7. Cobalt impurities in MnCl2-4H20 149 though that the relaxation time for the higher frequency line, T 1 2 is significantly shorter than for the lower frequency line. The sublattice corresponding to the high frequency line is more strongly coupled to the electronic magnons with the lower energy gap. This can be seen by considering the orientation of electronic and nuclear magnetic moments of the two sublattices relative to the applied field. 36500 33000 -I 1 1 1 1 497 498 499 500 501 Frequency (MHz) Figure 7.67: The number of counts at 0° as a function of frequency, showing the NMR resonance of 5 4 M n in an applied field of 0.5 T. The frequency was swept upwards. The frequency step is +5 kHz and the modulation is ±5kHz. The The best fit parameters are K = 33838 ±14, B = 288 ± 13, ox = (216 ± 22) kHz, UJX = (498.25 ± 0.01) MHz, CT2 = (176 ± 13) kHz and u2 = (499.62 ± 0.01) MHz. An upward frequency sweep where all six transitions ( P —> P + 1) are observed is shown in figure 7.68. The fit was based on a model with six Gaussian transitions, of Chapter 7. Cobalt impurities in MnCl2-4H20 150 equal width a, centered at positions u>0 + MP where P goes from -3 to +2. There were 6 free parameters for the fit: K, B, a, w 0, T\ and M. The result of the fit is excellent. The zero field Hamiltonian for an electronic and nuclear spin in the crystal is H = gu.BBESz + D[(SZ)2 - ±S(S + 1)] + Al - S + P[(P)2 - + 1)| (7.171) where BE is the exchange field, D is the crystalline field anisotropy interaction strength, and P is the quadrupole interaction strength. The energy difference between two sublevels P = m and P = m 4-1 is given by [36] A £ m > m + i = -AS + P'm + P{2m + 1) (7.172) where P' = g(lB^^_iD is the pseudoquadrupole interaction strength. The quadrupole and pseudoquadrupolar splitting, M — P'+2P is found to be (3.130± 0.011) MHz. This agrees well with the value for a pure crystal of 3.1 MHz [36]. The value of u>o = — AS + P is found to be (509.68 ± 0.03) MHz, in agreement with the result for a pure crystal of 509.72 ± 0.25 MHz [36]. It is interesting that the value of the spin-lattice relaxation time measured is 7\ = (83 ± 4) • 103 s. This is significantly shorter than the value of (253 ± 18) • 103 s found in figure 7.62 where the majority of the fit involves the relaxation of the spins after a single resonance. Although the value of K = 33794 ± 19 indicates that the crystal may be as much as 2 mK colder in this case, that should make the value of 7\ longer. It is unclear to what this discrepancy may be due, but the value of Ti = (253 ± 18) • 103 s is certainly more accurate because the fit involves fewer free parameters, and the majority of the data only involves spin-lattice relaxation after a single resonance. It is also interesting to compare the value of 7\ measured with the Co impurities to the value measured in the pure crystal [38]. In the pure crystal, the dominant relaxation process below 65 mK is found to be a direct process, with values of Xi ~ 3 • 106 s. The Chapter 7. Cobalt impurities in MnCl2-4H20 151 presence of 3.96 % Co impurities reduces the value of Ti by an order of magnitude at 44.3 mK! Presumably the magnon spectrum is affected by the impurities and this effect is worthy of future theoretical study. 38000 -r 37500 -37000 -36500 -co 36000 § 35500 -o ° 35000 -34500 -34000 -33500 -33000 -485 495 505 515 525 Frequency (MHz) Figure 7.68: The number of counts at 0° as a function of frequency, showing all six transitions between magnetic substates in zero field. The frequency was swept upwards. The frequency step is +100 kHz and the modulation is ± 1 0 0 kHz. The best fit parameters are K = 33794 ±19, B = 264 ± 4, o = (269 ± 30) kHz, co0 = (509.68 ± 0.03) MHz, Ti = (83 ± 4) • 103 s and M = (3.130 ± 0.011) MHz. The changes is populations of the magnetic sublevels throughout the resonances is shown in figure 7.69. These are calculated from the data in figure 7.68. Each resonance equalizes the two levels which subsequently relax due to Tx processes. In figure 7.70, the level populations are plotted as they would be if no Ti processes were present. Chapter 7. Cobalt impurities in MnCl2-4H20 152 0.50 0.45 . 0.40 H </> £ 0.35 H 1 0.30 g - 0.25 -I - 0.20 CD 5 0.15 - I 0.10 0.05 0.00 P-3 P-2 P-1 Po Pi P2 P3 480 490 500 510 Frequency (MHz) 520 530 Figure 7.69: The change in level populations throughout the frequency sweep shown in figure 7.68 Chapter 7. Cobalt impurities in MnCl2-4H20 153 480 490 500 510 Frequency (MHz) 520 530 Figure 7.70: The change in level populations throughout the frequency sweep shown in figure 7.68 as they would be if no relaxation effects were present, i.e. if 7\ = oo. Chapter 7. Cobalt impurities in MnCl2-4H20 154 7.5 Conclusions The NO results on 6 0 Co and 5 7 Co predict quite different hyperfine fields for the two isotopes. In Crystal A, the hyperfine field is 15.5 ± 2 for 5 7 C o . In crystal B, the hyperfine field is 29.1 ± 0.5 T for 6 0 C o . In crystal C, the hyperfine fields are 15.6 ± 2.7 T and 25.6 ± 0.4 T for 5 7 C o and 6 0 C o respectively. This discrepancy is extremely surprising, and we do not presently have a reasonable explanation for it. An N M R resonance, which would allow us to measure the hyperfine field very precisely was not found. This indicates that the relaxation times are much less than 1000 s, or that the resonances are more than 10 MHz in width. It seems likely that many Co ions do not sit at Mn substitutional lattice positions. The NMR results on 5 4 M n also indicate that the Co impurities do not enter the lattice substitutionally. If substitutional replacement occurred, we would expect to see an NMR resonance with a narrow main line due to Mn ions with no nearest neighbour impurities and a H W H M on the order of 35 kHz [36], and a satellite line of approximately 1/4 the intensity due to Mn with one nearest neighbour impurity. Instead we see a possible two line structure, but with linewidths on the order of 150 kHz. The ratio of intensities of the two lines is 1/6. Although this could be due to substitutional replacements with long range effects on the neighbouring ions, it is more likely that the replacements are not substitutional and cause important imperfections in the lattice which have long range effects. We suspect that the different hydration of M n C l 2 • 4H 2 0 and C o C l 2 • 6H 2 0 makes substitutional replacements difficult during growth. This might indicate that the 5 4 M n grown into CoCl 2 • 6H 2 0 as described in Chapter 6 is not substitutional. It would be unreasonable to expect that the implanted 5 6 M n would then be substitutional. The site occupancy factor of / = 0.53 ± 0 . 1 0 might indicate that several interstitial sites are Chapter 7. Cobalt impurities in MnCl2-4H20 155 possible and some are favoured during growth, others during implantation. The position of the resonant frequency and the pseudoquadrupole interaction for 5 4 M n are not measurably affected by the presence of the impurities. The width of the resonance is increased significantly, and the shape is not Gaussian. The presence of the (3.96 ± 0.04)% impurities reduces the spin-lattice relaxation time, T\, by an order of magnitude. This reduction in Ti can be explained if the impurities serve as scattering centers for magnons. These scatterers would reduce the magnon lifetime and broaden their spectrum. Direct processes would be greatly enhanced by a broadening of the magnon spectrum, and since we expect these to be the principle spin-lattice relaxation mechanisms at T < 65 mK [38], Tx would be reduced. Interestingly, this explanation also provides some insight into our inability to find the NMR resonance for the Co isotopes. If the impurities scatter electronic magnons appreciably, these scattering processes will provide important spin-lattice relaxation mechanisms. The spin-lattice relaxation time could be much less than 1000 s, making the resonance undetectable with our search techniques. It would be necessary to use a generator with a much larger modulation ( ± 5 MHz would be sufficient) in order to observe the resonance. The results of this experiment have led us to try doping 5 4 Mn into FeCl2 • 4H 2 0. Since this crystal has the same hydration as M n C l 2 • 4H 2 0 it is hoped that substitutional replacement will occur. A 7% anisotropy has been observed, but the N M R resonance has not yet been found. This work is currently in progress. Chapter 8 Summary and conclusions Nuclear Orientation and its related NMR techniques can be used to study the properties of interesting magnetic systems. The NO gives information on the electronic environ-ment of the nucleus which provides the hyperfine field. NMR can be used to measure the hyperfine field more accurately, to study quadrupole and pseudoquadrupole split-tings and to probe collective nuclear excitations. The spin-lattice relaxation can also be measured in order to elucidate the coupling of the nuclear spin system to the lattice. In the magnetic insulators studied in this thesis, the pseudoquadrupole interactions, the spin-lattice relaxation and the collective nuclear interactions can be used to study the electronic magnon spectrum. When combined with an isotope implantation facility, the isotopes which can be used as-suitable probes is vastly increased because it is feasible to perform experiments with shorter lived isotopes online or pseudo-online. The collective nuclear excitations, which lead to a "frequency pulling" effect were studied in MnAc using NMR-TDNO. Feedback stabilization of the R F power is neces-sary for this type of experiment and a new, improved, feedback system was analysed and utilized. The MnAc system is quasi-2-dimensional, with strong anisotropy fields. It therefore provides a convenient system for studying 2-dimensional Heisenberg systems with anisotropy, and both the electronic and nuclear excitations, or spin waves, can be studied. The theory of nuclear collective excitations had previously only considered sys-tems with 1 = 1/2 where pseudoquadrupole interactions are non-existent. This theory has been extended here to include the pseudoquadrupole interaction and non-uniaxial 156 Chapter 8. Summary and conclusions 157 anisotropy fields. The agreement between theory and experiment is good, with discrep-ancies being likely due to simplifying assumptions when calculating the electronic spin wave spectrum. The system is complicated by the elaborate crystal structure where lattice sites are occupied by triplets of ions, in two inequivalent sites, M n l and Mn2. Future work in this area should test the theory in a system with I > 1/2, highly anisotropic crystalline fields, but a simpler crystal structure with a single ion per lattice site. This system could be either two or three dimensional, but would allow the predic-tions of the theory to be tested without the complications of complex electronic magnon excitations. The first NO experiments with implanted isotopes into insulators were performed at Louvain-La-Neuve, using MnCl 2 -4H 2 0 and CoCl 2 -6H 2 0 as hosts. These targets were doped with 5 4 M n during growth. It was found that when 5 6 M n was implanted into MnCl 2 -4H 2 0 the implantation fraction was consistent with unity, but in CoCl 2 -6H 2 0 it was not. In future experiments, NMR should be performed on these sytems to determine the precise hyperfine fields. This would clarify which sites were occupied by the implanted isotopes. Once the lattice positions of implanted isotopes can be determined, a variety of condensed matter physics, experiments could be performed. In some cases, the hyperfine fields in an insulator can be larger than in an Fe host, and nuclear physics experiments could also be performed in insulators. The studies of Co impurities in MnCl 2 -4H 2 0 showed that the hyperfine fields for 5 7 C o and 6 0 C o are significantly different although, in one experiment, both were grown into the same crystal. The reason for this discrepancy cannot be explained simply, and should be explored further. An NMR resonance for the Co was not observed, and this is likely due to a short spin-lattice and/or a large linewidth both of which makes the observation of a resonance difficult. The NMR of 5 4 Mn was measured in a crystal with ~ 4% Co in order to determine the Chapter 8. Summary and conclusions 158 effects of impurities. The line was found to be broadened and non-Gaussian in shape. The spin-lattice relaxation time was found to be an order of magnitude shorter than in the pure crystal. At the temperatures utilized in the experiment, the nuclear spin-lattice relaxation proceeds via a direct process. A possible reason for the reduction in Ti is that the impurities act as scattering centres for the electronic magnons thereby broadening the magnon spectrum and enhancing this direct process. This indicates that the impurities act as scattering centres for electronic magnons and that spin-lattice relaxation time of the impurities is short, thereby making the observation of an N M R resonance for the impurities difficult. Finally, it seems likely that the impurities grow into interstitial lattice positions. Further experiments should include a study of a system with the same hydration as the impurity. This is currently underway with a study of 5 4 M n in FeCl 2 -4H 2 0. Although 7% anisotropy has been observed, the NMR resonance has not yet been observed. It would also be of interest to study the effects of stable Fe in MnCl 2 -4H 2 0 and to compare to the effects of the Co. Furthermore, several measurements with different amounts of impurities should be performed to determine the spin-lattice relaxation time as a function of impurity concentration. If the spin-lattice relaxation time could be controlled by including impurities during growth, some systems with long 7\ could be studied more easily. Finally, if the impurity concentration were high enough, it should be possible to make X-ray diffraction measurements to find the positions of the impurities in the lattice. Bibliography [1] A. Kotlicki and B. Turrell, Phys. Rev. Lett. 56, 773 (1986). [2] P. De Gennes, P. Pincus, F. Hartmann-Boutron, and J. Winter, Phys. Rev. 129, 1105 (1963). [3] E. Turov and M . Petrov, Nuclear Magnetic Resonance in ferro and antiferromagnets (John Wiley & Sons, New York, 1972), see chap. 2. [4] P. Pincus, Phys. Rev. 131, 1530 (1963). [5] See e.g. K.S. Krane in Low Temperature Nuclear Orientation, ed. N.J. Stone and H. Postma (North Holland 1986) Ch. 2. [6] See e.g. R. B. Firestone, Table of Isotopes, 8th Edition, ed. V.S. Shirley (John Wiley 1996). [7] C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (John Wiley & Sons, New York, 1977), see page 1340. [8] R. Kubo and K. Tomita, J. Phys. Soc. Japan 9, 888 (1954). [9] T. Moriya, Prog. Theor. Phys. 16, 23 (1956). [10] F. Grover, Inductance Calculations (Dover publications, New York, 1973). [11] See, e.g.: D. Landau and M. Krech, J. Phys.: Condens. Matter 11, R179 (1999); T. Nagao and J. Igarashi, J. Phys. Soc. Japan 67, 1029 (1998). [12] See, e.g.: G. Seewald, E . Hagn, and E . Zech, Phys. Rev. Lett. 79, 2550 (1997); G. Seewald, E . Hagn, and E . Zech, Phys. Rev. Lett. 78, 5002 (1997); W.D. Hutchinson, M.J. Prandolini, S.J. Harker, D.H. Chaplin, G.J. Bowden and B. Bleaney, Hyp. Interact. 20/21, 215 (1999). [13] H. Suhl, Phys. Rev. 109, 606 (1958). [14] T. Nakamura, Prog. 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Mermin, Solid State Physics (Saunders College Pulishing, New York, 1976), see chap. 33. [27] M . Fisher, Repts. Prog. Phys. 30, 615 (1967). [28] S. Vonsovskii, Ferromagnetic Resonance (Israel Program for Scientific Translations, Jerusalem, 1964), see chap. 3. [29] J. V. Vleck, Phys. Rev. 74, 1168 (1948). [30] See e.g. A. Perez, G. Marest, B.D. Sawicka, J.A. Sawicki and T. Tyliszczak, Phys. Rev. B28 (1983) 1227; L. Niesen, Hyperfine Interactions 13 (1983) 65; G. Lan-gouche, Hyperfine Interactions 84 (1994) 279. [31] See e.g. J .C. Soares, Nucl. Instr. Meth. Phys. Res. B64 (1992) 215. [32] Y. Kawase, S. Uehara, and S. Nasu, Nucl. Instr. Meth. Phys. Res. B64, 329 (1994). [33] A. Zalkin, J. Forrester, and D. Templeton, Inorganic Chemistry 3, 529 (1964). [34] R. Altman, S. Spooner, D. Landau, and J. Rives, Phys. Rev. B 11, 458 (1975). [35] A. Kotlicki and B. Turrell, Hyperfine Interactions 11, 197 (1981). [36] A. Kotlicki, B. McLeod, M . Shott, and B. Turrell, Phys. Rev. B 29, 26 (1984). [37] A. Allsop, M . de Araujo, G. Bowden, R. Clark, and N. Stone, J. Phys. C: Solid State Phys. 17, 915 (1984). Bibliography 161 [38] M . Le Gros, A. Kotlicki, and B. G. Turrell, Hyp. Interact. 77, 203 (1993). [39] J. Mizuno and K. Ukei, J. Phys. Soc. Japan 14, 383 (1959). [40] J. McElearney, J. Forstat, and P. Bailey, Physical Review 181, 887 (1969). [41] B. Turrell, P. Johnston, and N. Stone, J. Phys. C5, L197 (1972). [42] D. Vandeplassche, L. Vanneste, H. Pattyn, J. Geenen, C. Nuytten, and E . van Walle, Nucl. Instr. Meth. 186, 211 (1981). [43] O. Lounasmaa, Experimental Principles and Methods Below IK (Academic Press, London, 1974). [44] I. Lowe and D. Whitson, Phys. Rev. B6, 3262 (1972). [45] B. Will, P. Herzog, R. Paulsen, J. Camps, P. D. Moor, P. Schuurmans, N. Severijns, A. V. Geert, L. Vanneste, I. Berkes, M. D. Jesus, M. Lindroos, and P. V. Duppen, Phys. Rev. B 57, 11527 (1998). 

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