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NMR-ON studies of europium vanadate and hcp cobalt Willetts, Martin 1995

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NMR-ON STUDIES OF EUROPIUM VANADATE AND HCP COBALT B y Mar t i n Wil le t ts B . Sc. (Physics) University of Birmingham, 1993 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F . • M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A July 1995 © Mar t i n Wil le t ts , 1995 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 The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: 4L <2A Abstract The quadrupole interaction on E u 3 + in E11VO4 was investigated using low temperature nuclear orientation. The interaction strength, P, on the 1 5 2 E u nuclei was measured as 1±4 MHz, with a total cooling time of 24 hours. This low value is attributed to either the high symmetry of the europium lattice postion, or a very long (greater than one day) spin lattice relaxation time. The N M R frequency of 5 9 C o was measured at low temperature, using the first obser-vation of thermometric N M R in a conductor. The motive was to measure the frequency pulling in cobalt metal, and the resonance was observed to be 218.4(4) MHz at 70mK, which is 1.5 MHz below the 1 K value. The frequency pulling effect on the nuclear resonance is due to the Suhl-Nakamura interaction between the nuclear spins. i i Table of Contents Abstract ii Table of Contents iii List of Tables ( . vi List of Figures vii Acknowledgements ix 1 Theory 1 1.1 Introduction 1 1.1.1 Europium Vanadate . . 1. 1.1.2 Cobalt . . . . . . . . . . . . . . . . . 1 1.2 Introduction to N M R - O N 2 1.3 Probability Distribution from Oriented Nuclei 6 1.3.1 Physical Background . . . . 6 1.3.2. Nuclear Orientation Parameters . 9 1.3.3 DirectionalDistribution Coefficients . . 10 1.3.4 Deofientation Coefficients 10 1.3.5 Final Considerations 12 1.4 The Hyperfine Interaction 13 1.4.1 Ferromagnetic Cobalt 14 1.4.2 Europium Vanadate . . . 15 i i i 1.5 N M R on Ordered Nuclei . . • 16 1.6 Frequency Pulling Effect . 18 1.6.1 Nuclear-Nuclear Interactions • 18 1.6.2 Suhl-Nakamura Interaction 19 1.6.3 Nuclear Spin Wave Dispersion Relation . . . . 20 2 Experimental 25 2.1 Experimental Setup 25 2.1.1 The Cryostat 25 2.1.2 Supplying R F Power 26 2.1.3 Gamma Ray Detection 27 2.1.4 Miscellaneous • 28 2.1.5 Analysis. . . . '• 29 2.2 Sample Preparation 29 2.2.1 E u V 0 4 . • • • • • • •• ' 29 2.2.2 Cobalt . 29 3 Europium Vanadate 32 . 3.1 Results ; . . . .... • 33 3.2 Discussion 37 4 Cobalt 39 4.1 Frequency Pulling . . . . . . . . . . . • 39 4.2 The Thermometric Technique . 42 4.3 Samples • . 42 4.4 Results 44 4.5 Discussion . : 49 iv Bibliography A Radio-Frequency Tank Circuit B Mathematica Programs List of Tables 1.1 Nuclear electric quadrupole and magnetic dipole moments for the Co and . Eu nuclei studied 3 1.2 Nuclear orientation parameters for relevant decays of 6 0 C o , 1 5 2 E u and 1 5 4 E u . 13 2.1 Abundance, half life, thermal neutron capture cross section and neutron capture/decay products for the isotopes of europium studied 30 3.1 Gamma ray anisotropy for cobalt thermometer and europium, during E u V 0 4 cool down 37 4.1 Peak positions of nuclear resonances as measured in single crystal hep 6 0 CoCo . 47 4.2 Frequency pulling, measured and calculated, for hep Co 49 vi List of Figures 1.1 Nuclear sublevels for: a) 6 0 C o in a magnetic field and b) 1 5 2 E u in an E F G , both along the z-axis 4 ' 1.2 Calculated polar plot of normalised 7-ray intensity, W(6), for hep 6 0 C o C o at T=5, 15, and 100 mK. . . . . . . . 5 1.3 Simplified 6 °Co decay scheme. 6 1.4 Calculated orientation paramenters, Bk, for a) 6 0 CoCo (hep), versus tem-perature; b) 1 5 2 E u / 1 5 4 E u versus quadrupole interaction, P , at 13 mK . . 11 1:5 Calculated W(6) versus T for 6 0 C o C o at d = 0 and d = 90°. . . . . . . . 14 1.6 Calculated nuclear spin wave frequency verses wave number . . . . . . . 24 2.1 Arrangement of Co foil and rf coils on cold finger . 26 3.1 Simplified nuclear decay scheme for 1 5 2 E u . . 33 3.2 Simplified nuclear decay scheme for 1 5 4 E u 34 3.3 Gamma ray spectrum from E U V O 4 crystal and cobalt thermometer as measured by 4000s count on germanium detector . 34 3.4 Measured 7-ray intensity, W(0), for cobalt 7-rays and 1086 and 1408 keV 7-rays from E U V O 4 , during cooling of E U V O 4 crystal 35 3.5 Calculated W(0) for 1 5 2 E u and 1 5 4 E u , versus the quadrupole interaction, P, at 13mK 36 4.1 Frequency pulling for 5 9 CoCo , as predicted from the Suhl-Nakamura in-teraction 41 vii 4.2 Schematic representation of the important heat flow processes in N M R - O N 43 4.3 The best frequency sweep, showing counts versus frequency. This is a sum of five runs under the same conditions. . 45 4.4 The 6 0 C o N M R - O N resonance, showing counts versus frequency. This is a sum of three runs under the same conditions 46 4.5 Detail of Figure 1.5 used to calculate temperature of cobalt crystal during rf sweeps, corrected for solid angle subtended by detector . 48 A . l The tank circuit used in these experiments 54 vi i i Acknowledgements This thesis, and the experiences I have had in Canada, have been made possible by the generosity and support of many people. I would like to thank in particular: Brian Turrell, for his supervision, support, hospitality and patience; Andrjez, Gail and Zuneira for help and encouragement in the lab; Dr. Don Chapman, and Prof. B. Bleaney for providing samples; Lore Hoffmann for her friedliness and support; my family and friends at home, without whose support I would never have made it to B.C. ; Chris for keeping me in touch with events back home; everyone who came out to support me at the Graduate Student Society; James Pond for friendship, humour, and reviving my outdoors instinct; and all those people, far too many to mention, who made my experience in B C as fun and rewarding as it has been. In particular: Annie, Dave, Adam, Christian, Chris and Sara. ix Chapter 1 Theory 1.1 Introduction This thesis describes investigations made using low temperature nuclear orientation and nuclear magnetic resonance on ordered nuclei on EUVO4 and hexagonal close packed (hep) cobalt. 1.1.1 Europium Vanadate Europium vanadate is a Van Vleck paramagnet, with the zircon crystal structure. The conditions at E u 3 + defect sites have recently been investigated in some depth (see Chap-ter 3.) For reasons of crystal symmetry, the techniques employed in these experiments (which are based on optical hole-burning) are unable to investigate the E u 3 + lattice po-sitions. The aim of the investigations into.EuV.O4 was to measure the. strength of the quadrupole interaction on the europium nuclei at the Eu 3 + l a t t i c e sites. 1.1.2 Cobalt During investigations into the NO characteristics of 8 9 m Y in.hcp cobalt, the Munich group observed an anomalous line shape at approximately 217MHz in the yttrium nuclear magnetic resonance[l]. It has been suggested (Turrell[2]) that this was due to the nuclear resonance of the cobalt host being frequency-pulled by nuclear spin-spin interactions down to the yttrium resonance. The line shape then resulted from the heating effect due to 1 Chapter 1. Theory 2 absorption of rf power by the cobalt spins in the same manner as is observed in the thermometric technique pioneered by Kotlicki and Turrell[3]. The low-temperature resonance (T < 1 K) of 5 9 C o has not been measured before. Also the thermometric technique first used by Kotlicki and Turrell has not been applied to conductors. For these reasons, the measurement of the low temperature N M R line in stable cobalt is an interesting goal to pursue. This chapter describes the background theory required to understand the experiments and their results. Chapter 2 will explain the experimental setup. The results for europium vanadate are presented in Chapter 3 and those for cobalt in Chapter 4. In this chapter, Section 1.2 introduces the physical principles behind nuclear orientation (NO) and nuclear magnetic resonance on oriented nuclei (NMR-ON), section 1.3 covers the theory of NO in more detail, section 1.4 discusses hyperfine interactions, and Section 1.5 covers N M R . Frequency pulling—the effect investigated in cobalt—is introduced in Section 1.6. 1.2 Introduction to NMR-ON Nuclei with non-zero spin have a magnetic dipole moment and those with spin greater than one half have an electric quadrupole moment. (The nuclear magnetic moments and quadrupole moments of 5 9 C o , 6 0 C o , 1 5 2 E u and 1 5 4 E u are shown in Table 1.1.) Interactions between the magnetic moment and a magnetic field, and/or the electric quadrupole moment and an electric field gradient (EFG), will lead to a discrete set of energy sublevels of the nucleus, the different orientations of the spin relative to the magnetic field / E F G having different energies. A nucleus with magnetic dipole moment ft and quadrupole moment Qij in a magnetic field B and electric potential V has a classical energy E = fl-B + ^Qii d2V (1.1) Chapter 1. Theory 3 Isotope Spin Dipole Moment Quadrupole Moment . A % bn "Co 7/2 +4.627(9) +0.41(3) S?Co 5 +3.799(8) +0.44(5) e fEu 3 -1.95 +2.71 3 . +1.5 • +2.3, 3.9 Table 1.1: Nuclear electric quadrupole and magnetic dipole moments for the Co and Eu nuclei studied [4]. If the B and V2V are directed along the z-axis, i.e. there is axial symmetry, then the nuclei can be described by the orthonormal states \IIZ), and the Hamiltonian for the nuclei is: 1 H = IzAe + P[I2z- -7 (7 + 1)] with pB I P=3-eQ^ 1 4 ^ dx2 7 ( 2 7 - 1) (1.2) (1-3) (1.4) Usually one deals with a situation where the interaction is dominated by either the dipole or quadrupole interaction. The nuclear sub levels in these cases are shown in Figure 1.1 for 6 0 C o (7 = 5) in a magnetic field and 1 5 2 E u (7 = 3) in an E F G . Gamma ray emission from an individual nucleus is, in general, anisotropic—the prob-ability distribution for the 7-ray will be a function of the angle between the detector and the nuclear spin axis. If, therefore, one can create an ensemble of nuclei with net orientation (for example, an alignment of the nuclear spins in a given direction,) one would expect to observe an anisotropy in the emission of 7-rays. If an ensemble of nuclei is held in a field at a low enough temperature (of the order of the nuclear level splitting) then the lower energy states will be more, populated than the higher states, leading to a Chapter 1. Theory 4 - +5 ± 3 -+4 . f ( A , Figure 1.1: Nuclear sublevels for: a) 6 0 C o in a magnetic field and b) 1 5 2 E u in an E F G , both along the z-axis. net orientation of the nuclei. This is the method of low temperature nuclear orientation (LT-NO.) Figure 1.2 shows the angular distribution of the normalised 7-ray intensity from hep 6 0 C o C o at three temperatures. Note how, as the temperature approaches zero, the ra-diation intensity in the direction of the c-axis (along which the magnetisation lies, and about which the system possesses axial symmetry) tends to zero, while the intensity of the radiation perpendicular to the c-axis reaches a value of approximately 1.25 times the warm intensity. In nuclear magnetic resonance nuclei are excited between their magnetic sublevels by radiofrequency (rf) radiation, tuned to the energy difference between these levels. This excitation causes a change in the level populations, and will therefore cause a change in the observed anisotropy of the 7 radiation from the sample. In N M R - O N the nuclear resonance frequency is measured by observing this change in anisotropy. -3 -4 -5 ± 2 -± 1 -0-3P b) Chapter 1. Theory 5 c-axis T = I nn Figure 1.2: Calculated polar plot of normalised 7-ray intensity, W(9), for hep 6 0 C o C o at T=5, 15, and 100 mK. The vertical axis is parallel to the c-axis, and the magnetic field at the nucleus. Chapter 1. Theory 6 6 0 C o 5+ 4+ 7 1.13MeV 2+ 7 1.33MeV 6 0 N i 0+ Figure 1.3: Simplified 6 0 C o Decay Scheme. Numbers on right indicate the spin and parity of the nuclear state. Both 7 rays are pure quadrupole. The P particle has a maximum energy of 330 keV. The theory of nuclear orientation is well established, and for a more complete deriva-tion of the results discussed below the reader is referred to Low Temperature Nuclear Orientation[5], and in particular Chapter 2 by Krane[6]. The NO notation used here was first formally set up by Blin-Stoyle and Grace[7]. 1.3 Probability Distribution from Oriented Nuclei 1.3.1 Physical Background The emission of a 7-ray from a nucleus will typically be preceded by a 13 or a particle decay, or electron capture. This will lead to one or more intermediate nuclear states, from one of which the observed 7-ray is emitted. A typical simplified decay scheme (in this case 6 0 Co) is shown in Figure 1.3. We shall first look at the 7 decay of a parent nucleus, of known initial conditions, and later introduce the effects of intermediate radiations between the known state and the state from which the 7-ray is emitted. Consider a nucleus of spin 7t- in a magnetic field along the z-axis, which decays via a 7 Chapter 1. Theory 7 decay of pure multipole order 1h to a nucleus of spin If. (We shall not consider here the effect of a mixed-order decay.) The nucleus has 2/j +1 possible values of its z-component of spin, rrii, with occupation probabilities ami. Ignoring polarisation effects, the directional (by which we mean angular) probability distribution of the gamma ray from a known state, mj, is given by summing the directional distribution of each of the possible decays, weighted by their relative probabilities, as given by angular momentum coupling theory. For a 7-ray of known z-component of spin, M , the directional distribution is the modulus-square of the spherical harmonic Yjy(9, <f>). The angles 9 and 0 are the usual spherical coordinates, between the detector and the z-axis. The distribution from the state is therefore: Wm(9)= £ w(mi^mf)\YLM(9,(l>)\2, (1.5) mi—*m.f with rrii = rrif + M. The.initial state of the nucleus, | / ; ,Mj ) , can be written as a superposition of the product states \L, M)\If, rrif) of the 7-ray and the final nuclear state. Using the Wigner-3j notation: ' Ii If L \ f , (1-6) y m-i —mf —M J ' ' \.\Ii,mi)=^\If,mf)\L,M)(-l)Ii+m'(2Ii.+ ^ M with m-i-mf-M = 0. Therefore, if we start in the known state \Ii} m-i) the w(mi -> mf) are simply the squares of the coefficients of the \If, rrif)\L, M): \ 2 (1.7) w(mi —>• rrif) — (27, + 1) m-i —rrif —M J (For those who are unfamiliar with the Wigner 3-j notation, they are related to the Clebsch-Gordan coefficients by: + 1 C(j1muj2m2; 3 -m) (1.8) Chapter 1. Theory 8 with \ji - j2\ < j < ji + j2, and m x + m2 + m = 0.) Returning to the case of the, parent nucleus in an orienting field, the distribution is Whilst this formulation of the directional distribution gives a simple physical insight into the process of NO, it is very unwieldy to calculate, and gives no simple comparison between the anisotropy of one 7 decay and another.. (It is useful, however, where one is selectively exciting a particular nuclear sub-level,, as, for instance, in a quadrupole interaction, or a magnetic interaction perturbed by a quadrupole interaction. In this case, writing the anisotropy in terms of the occupation probabilities is the most straightforward analysis.) Further, if the decay is preceded (as is almost always the case) by one or more (usually unobserved) nuclear decays, the calculation of the am't. would be a very laborious process, since the population distributions within the daughter nucleus would have to be calculated from the distributions of the parent, a m o , using angular momentum coupling, which gives: found by summing the weighted contribution from each parent state: (1.9) 8 = l/kBT.. (1.11) 2 (1.12) Chapter 1. Theory 9 1.3.2 Nuclear Orientation Parameters The oriented nuclear state may. alternatively be described in terms of the statistical tensors[6]: p}&) = * E ( - i ) ( 7 < + m ' _ (IMplUm') , (1.13) y — rrC —m q x = (2x + l ) 1 / 2 , (1.14) which describe the initial (mixed) state of the nuclei in an ensemble. Here p is the density matrix describing the initial state of the nuclei, s P = ^.\hmi){Iim2\. ( 1 1 5 ) m i ,7712 The directional distribution of the radiation is now written as ^ ( ^ ^ ^ v ^ / i E p S ^ F . ^ 1 ^ ^ , ^ ) , (1.16) kq where the Fk are the directional distribution parameters, which will be described later. If the system possesses axial symmetry about an axis (typically a crystal axis, or the applied field) and we choose the z-axis in this direction, then (I, m\I, m') — 5m>mi and the density matrix p will be diagonal, i.e. (Iim\p\Iim') = am5m>m>. (1.17) (Note that the off-diagonal terms of the density matrix are usually negligible in compar-ison to the diagonal terms. If the system possessed exact axial symmetry the nuclear lattice would never relax to the temperature of the crystal, but would remain forever in its initial state.) Since —m' + m + q = 0 (from equation (1.13)) only terms with q = 0 will now appear in the sum,, and therefore " W(0)^J2BkFkPk(cos9). (1.18) k Chapter 1. Theory. 10 This is the usual expression used to calculate the directional distribution of 7-rays in NO, with Bk = iiPkQ(ii) =kiiYt(-iYi+m k ""TO* • (1.19) —mi —rrii 0 These parameters are plotted versus temperature for 6 0 C o C o , and versus the quadrupole interaction strength for 1 5 2 E u / 1 5 4 E u at 13 mK, in Figure 1.4. Conservation of parity means that the 7-ray distribution is invariant under space-reversal, and therefore the sum in (1.18) is only over k = even terms (including k = 0.) 1.3.3 Directional Distribution Coemcients The directional distribution coefficients are given by Krane as / Fk(LIfIi) = (-l)If+I>+1kL2Ii L L k li Ii If J where < 3l 32 3l3 (1.20) is a Wigner 6-j symbol, coupling the three angular momentum t 33 3 323 states, j1,32 and j3. The Fk have been tabulated (see, for example, Appendix 5 of Low Temperature Nu-clear Orientation.) 1.3.4 Deorientation Coemcients The orientation of the state from which the observed 7-ray is emitted is described by the orientation parameters Bk(Ii). In general this transition of the nucleus is preceded by one or. more decays, and it is the orientation of the parent nucleus Bk(I0), which is known, or which we are trying to measure. The reorientation of the nucleus due to these intermediate decays can be described by writing the Bk(Ii) as Bk(Ii) = Uk{IoIiL)Bh{I0).. (1.21) Chapter 1. Theory 11 0 10 20 30 T, mK 0 . 05 0. 5 10 15 20 25 30 P, MHz Figure 1.4: Calculated orientation paramenters, Bk, for a) 6 0 CoCo (hep), versus temper-ature; b) 1 5 2 E u / 1 5 4 E u versus quadrupole interaction, P, at 13 mK Chapter 1. Theory 12 The Uk factors are the deorientation coefficients and are calculated from angular mo-mentum coupling theory. The deorientation due to a decay between two states I\ and J 2 which carries angular momentum L is: u ^ h L ) = (-iy^+L+kijJ (1.22) h h k h h L Where the observed decay is preceded by more than one decay, the total deorientation is simply the product of all previous decays: Uk(hh) = Uk(IQhLx)Uk{hhL2)... Uk(InIiLn) (1.23) These coefficients have been calculated for a large number of decays and are tabulated in Appendix 5b of Low Temperature Nuclear Orientation. 1.3.5 Final Considerations We should also consider here the reorientation of the nuclear spins whilst in the excited state of the daughter nucleus, and the solid angle correction of the detectors. In most decay schemes used in NO and N M R - O N experiments, the lifetime of the state Ii is much shorter than any reorientation times and so this effect can be ignored. The effect of the solid, angle subtended by the detectors can be described by a param-eter, Qk(0). In most experiments, the solid angle is small and the Qk(0) can be treated as constant and close to unity. Now we have as a final expression for the directional distribution of the radiation: W{9) = £ BkUkFkQkPk(cos6), (1.24) k even with k < 2L, 21. We have described the 7 radiation from an oriented ensemble of nuclei with known decay scheme in terms of constant parameters which can be read from tables or simply Chapter 1. Theory 13 Parent 7 Energy Ref. 7 Type Proportion u2 F2 F4 nucleus keV 6 0 C o 1173 1 E2 100% 0.939 . 0.798 -0.448 -0.304 1332 E2 100% : 0,704 0.227 -0.598 -1.069 1 5 2 E u 1408 1 E l . 21% 0.828 N / A -0.418 N / A 779 2 E l 13% 0.750* N / A 0.346 N / A 1085f 3 E2 10% 0.817* 0.382 -0.529 -0.816 1 5 4 E u 1274 1 E l ." 35% 0.828 N / A . -0,418 N / A Table 1.2: Nuclear orientation parameters for relevant decays of 6 0 C o , 1 5 2 E u and 1 5 4 E u [5]. Ref. refers to graph number in Figure 3.5; nuclear decay data from [8] and [9] *Angular momentum of beta decay is mixed; I have taken J = l . f includes. 15% contribution from 1089 keV decay via Gd. calculated, and one simple orientation coefficient which depends only on the interaction between the nucleus and its environment, and is independent of other nuclear consider-ations. In the experimental procedure W{6) can be compared at a low temperature to the normalisation value taken at a "high" temperature ( I K ) . The nuclear orientation parameters Uk and Fk, relevant to the decays of 6 0 C o , 1 5 2 E u and 1 5 4 E u used in this thesis, are shown in Table 1.2. Figure 1.5 shows the relative intensities of the 7 radiation, W(0), versus T, at 6 =.0 and 0 = TT /2 for 6 0 C o C o . j 1.4 The Hyperfine Interaction Nuclear magnetic moments are small—on the order of one thousandth of the magnetic moment of the electron. The magnitude of the interaction energy between the nucleus and an applied field is the product of the magnetic moment and the applied field. Since any system is disordered unless its thermal energy becomes comparable to, or less than, the energy differences which characterise the system, high fields and low temperatures Chapter 1. Theory 14 1.2 " 1 0.8 0.4 0.2 •° 2 5 10. 20. "SO. T, mK Figure 1.5: Calculated W{0) versus T for 6 0 C o C o at 0 = 0 and 0 = 90°. are needed to produce a significant nuclear orientation. For example, a 6 0 C o nucleus in a field of 20 Tesla has an energy splitting between adjacent sub-levels equivalent to 6 mK. It is possible, but experimentally difficult, to achieve measurable nuclear orientation using the combination of a high laboratory field and very low temperature in this way. This technique is called, appropriately, "brute force NO." 1.4.1 Ferromagnetic Cobalt In 3-d magnetic materials, the unpaired electrons in the 3-d shell are strongly polarised— this is the source of the magnetism of the sample. These electrons partially polarise the electrons in the other atomic orbitals, and of particular importance are the s-orbitals. The wavefunctions of these electrons are non zero at the nucleus, and even a small spin polarisation of their spins can lead to a very strong hyperfine interaction. This interaction, called the contact term can be represented by an effective magnetic field B^ acting on the nuclear moment. These fields can be large (e.g. the hyperfine field in iron is 33 T) and significant nuclear orientation can be achieved by cooling, to temperatures Chapter 1. Theory 15 obtainable with a helium dilution refrigerator (12mK in our experiments.) The wavefunction of an electron in the nth s-shell can be written as the sum of the spin-up and spin-down wavefunctions, ip}l + ip^, and therefore Bc = ^9»BEM\2-M2]Sn 1 (1.25) —* where Bc is the field at the nucleus, g the Lande-g factor and /J,B is the Bohr magneton. Sn is the net spin of the nth s-shell electrons. We assume a point nucleus, and therefore can take the value of the electron wavefunction at the origin in calculating the hyperfine —* —* —* interaction. Sn is proportional to the atomic spin, S, in a 3-d magnet, and S is parallel to the magnetisation, M. We therefore write the field at the nucleus as with ^ = yf^EM 2 - |^| 2 ]^. (1.27) 1.4.2 Europium Vanadate Previous investigations into the HFI on E u 3 + (see Chapter 3) indicate that the dominant interaction with.the nucleus is between the nuclear quadrupole moment and an electric field gradient (EFG) at the nucleus. Unlike the magnetic interaction, the quadrupole interaction is insensitive to the sign of the ^-component of the nuclear spin, and the splitting between levels is not equal (see Figure 1.1.) The strength of the quadrupole interaction is described by the parameter P (equa-tion (1.4)), which is usually given in MHz. •Chapter 1. Theory 16 1.5 NMR on Ordered Nuclei It is sufficient here to'consider the rf coil as simply providing quanta of electromagnetic energy (in packets TIUJ) to excite the nuclei between their levels. The consequent change in the level populations is observable as a change in gamma ray anisotropy. One can also consider the applied rf field as rotating (tilting) the nuclear moment in a classical sense. In this treatment, resonance occurs when the rf frequency matches the precession frequency of the spins in the magnetic field which causes the sub-level splitting. If the field at the nucleus is B, then i l = 7 n B , ' ' (1.28) where 7 n is the gyromagnetic ratio of the nucleus: j2n = jnhl. (1.29) For a thorough treatment.of nuclear magnetic resonance, the reader is referred to one of the standard texts, e.g. the book by Slichter[10]. Nuclear orientation will occur when the nuclei are cooled to temperatures low enough that the nuclear spins are significantly ordered. It is observed by studying dilute ra-dioactive nuclei in a cold host crystal. The nuclei relax to the temperature of the crystal because of spin-lattice interactions, which determine the spin-lattice relaxation (SLR) time, T\. (The existence of a nuclear-spin temperature itself depends on a strong nuclear spin-spin interaction, characterised by a time T2, which maintains a Boltzmann distribu-tion of nuclear sub-levels.) Observation of N M R - O N requires a SLR time which is short enough that the nuclear spins relax to the temperature, of the host, but long enough that the low power levels that can be applied at low temperatures (limited by rf heating and the cooling power of the cryostat) can significantly change the state of the nuclear system. Chapter 1. Theory 17 In conventional N M R - O N , one observes the resonance of dilute radioactive nuclei. Nuclear spin-spin interactions are. very small in this case, and the inherent resonance linewidth is correspondingly narrow (typically 10 kHz in a ferromagnetic host). This ho-mogeneous broadening gives the response of each nuclear spin packet to applied radiation. The resonance is generally broadened much further by variations in the environment at the atomic site. These variations (due to defects, impurities, demagnetising effects etc.) produce an inhomogeneous broadening of the resonance, typically of about 1 MHz . This means mean that there are ~ 105 independent wavepackets within the sample, or that an rf field at a precise frequency will only be on resonance with 1 0 - 5 of the nuclei. The rf radiation can be applied over a range of spin packets by modulating the frequency of the radiation, typically over a range 1/2 to 1/10 of the resonance width. This reduces the magnetic field applied to each nucleus, and would make the observation of N M R - O N difficult without the effect of magnetic enhancement, which results in an rf-field at the nucleus much larger than the applied field. The enhancement results from the electronic magnetisation of the sample being aligned along the easy-axis, or an easy-plane, of the crystal. The electronic magnetisation follows an rf field applied perpendicular to the easy axis, resulting in a field on the nucleus which is enhanced by the hyperfine interaction. The enhancement factor is -B0 being any applied static field, parallel to the easy axis. The field felt by the nucleus is r]Bi, where Bt is the rf field applied to the sample. Here we have represented the magnetocrystalline energy which causes the magnetisation to lie along the easy axis by an anisotropy field, BA, acting on the electronic magnetisation. Chapter 1. Theory 18 1.6 Frequency Pulling Effect •> ' * The dominant term in the N M R frequency of the host nuclei in a 3-d ferromagnetic ma-terial is the hyperfine interaction between the atomic electrons and the nuclear magnetic moment! This is due mainly to core polarisation of the s-wave electrons, interacting with the nucleus through the contact term (the overlap of the electron wavefunctions with the nucleus). The N M R frequency determined by this interaction is perturbed by weaker interactions between the nuclei, which change the sub-level splittings. 1.6.1 Nuclear-Nuclear Interactions The direct interaction between nuclei is given simply by the energy of two interacting dipoles, and has a maximum value of ^dipole - (1.31) For a typical atomic seperation r ~ l A , and a nuclear magnetic moment of /x n this is on the order of 1kHz. This interaction is clearly negligible between non-nearest neighbours. Rudermann and Kittelfl l] showed that there is an indirect interaction between the nuclei through a nuclear-driven polarisation of the conduction electrons. This interaction has an energy on the order of AC/EF, where EF is the Fermi energy of the sample, and Ac is the energy of the interaction between the nucleus and the conduction electrons (which will be due to the contact term if the conduction electrons are s-wave.) This is typically equivalent to a frequency shift of a few kHz. Chapter 1. Theory 19 1.6.2 Suhl-Nakamura Interaction As discussed in Section 1.4, the hyperfine interaction on the nucleus of a 3-d ferromagnet can generally be described by an isotropic interaction with a Hamiltonian: Uhfi = -AS-T. (1.32) This Hamiltonian is correct only to first order in A, since it neglects all possible couplings —* —* between I and S. In the case of dilute nuclei in a ferromagnet, the correction to (1.32) due to interactions between I and S acting on neighbouring nuclei is negligible, since the coupling A is much smaller than the exchange interaction between the electrons, J . (Since the electrons are very strongly coupled, any perturbation due to a nuclear spin has a negligible effect on the interaction energy with neighbouring nuclei.) For instance, if one tilts the spin, Ij, of the nucleus at lattice site j, by an angle 9, then the relative perturbation of the electronic states will be of order (AfJ)IjO There is a perturbation to this Hamiltonian due to the mixing of atomic states with different values of S coupled by the nuclear spin. This interaction causes a quadrupole-like change in the energies of the sub-levels, but is negligible in cobalt. Where the nuclei are dense, however, (as is the case for the 5 9 C o in 6 0 CoCo) , the —* —* interaction between nuclei through the coupling of T and S can be significant because a perturbation to the electron states is carried across the order of J/hwe atomic neighbours by the exchange interaction. The interaction between dense nuclei and the electron spins is now of order (A/J)Iz(J/huje) = AIz/hue, where Iz = (I?) is the mean value of 7| over the ensemble of. nuclei. This interaction leads to a frequency pulling—a change of the N M R frequency at temperatures where the nuclear magnetisation (Is) becomes appreciable. The frequency pulling effect can be derived from the macroscopic equations for the Chapter 1. Theory 20 coupled nuclear and electron magnetisation or by considering the interaction between the nuclei due to the Suhl-Nakamura interaction (see, for instance, de Gennes et al [12]). Here we take the second approach, which shows that the frequency pulling is due to the Suhl-Nakamura interaction. This interaction was first derived by Suhl[13] for a ferromagnet and Nakamura[14] for an anti-ferromagnet, and is due to the presence of electron spin wave states within a magnetically ordered lattice of electron spins. Spin waves (also called magnons) are the elementary collective excitation of the magnetised electronic spin system in ferro-and antiferromagnetic materials, and are the source of the observed reduction of the magnetism of ferromagnets with increasing T. The excitation energy of these magnons is much greater than the sub-level splitting of the nuclei, and so the nuclei cannot di-rectly excite an electron spin wave state, but they can interact via the absorption and emission oi virtual magnons. The Suhl-Nakamura interaction is much weaker than the hyperfine interaction, but is long range (typically about thirty atomic spacings), and its effect becomes appreciable at low temperatures. Here, where the nuclear magnetisation becomes significant (the expectation value of the z-component of the spin.greater than 1%), the correlation of. nuclear spins caused by the interaction leads to the nuclear ex-citations taking the form of nuclear spin waves. The clearly defined dispersion relation for nuclear spin waves changes the N M R frequency by an amount proportional to the nuclear magnetisation. 1.6.3 Nuclear Spin Wave Dispersion Relation The Hamiltonian for an electron spin wave with wave vector k is written in terms of the magnon creation and annihilation operators, ai and respectively: Hmagnon{k) — ^ f c 0 ^ 0 ^ - (1.33) Chapter 1. Theory 21 We wish to consider the Hamiltonian of the combined nuclear-spin/electron-spin-wave system, which can be written as a sum of two terms: H = nQ + W. (1.34) The dominant term is simply the independent energies of the nucleus and the electron spin waves: ' ' 3 k and %' contains the interaction between the nuclei and the spin waves, which will be treated as a perturbation to %Q. We have included the hyperfine interaction by treating J the nucleus as being in an effective field B + (AS/jnh); j is summed over the positions of the N nuclei in the lattice,, k over the wave vectors of the spin waves. Consider an eigenstate of %Q in which the nuclear state, \If, Jf • • • '!%)•> 1S arbitrary, and the electrons are in their ground state |0), i.e. there are no spin waves. This is clearly a physical state at low temperatures since the nuclear level splitting is much less than the electron spin wave energy. (The k = 0 magnon has energy hue] huie = HBBA ^> AS—see below) Turov and Petrov[15] show that there is no first order correction to the eigenvalues of Tio due to but that there is a second order correction to the energy, and this can be represented by the Hamiltonian 'rieff = \Y.UjjlIp^ ' . • . _ (1.36) with : <742 - - -k The If and Ij are usual ladder operators, acting on the nucleus at site j: n = (i-38) IJ = I*-iP]. (1.39) Chapter 1. Theory 22 Equation (1.36) is the Suhl-Nakamura interaction, derived by Suhl by summing oyer all possible virtual spin-wave exchanges in a cubic ferromagnet. The interaction term only operates on the nuclear states, and so we can split the Hamiltonian (to second order in %') into independent terms for the nuclear and electronic states: •  "' U = Hn + He, (1.40) with Ue = Y,^tH 0-41) k • , .. . .. ' • and nn = -(AS + 7nhB)^I* + Wujflll];. (1.42) • j • . j,f The ladder operators, 7+ and 17, raise and lower the z-component of the spin of the nucleus j. According to Turov and, Petrov, the N nuclei in the ensemble under consideration are identical particles, and therefore the state vector of the ensemble is given by.the Slater determinant constructed from the individual nuclear states |7|). An operator acting on one nucleus therefore acts on the whole ensemble—it can be said-to belong to the whole crystal. There is, therefore, an equivalent representation of the operators, given by their Fourier transforms into the /c-space of the nuclear Bravais lattice: h-^Ee-^Ii- • .. (1.44) These operators represent the elementary excitations of the nuclear spin system. In other words: they are the nuclear spin wave creation and annihilation operators. The eigenfrequency spectrum of these operators will therefore give the frequency spectrum of possible excitations of the nuclear lattice. The~NMR frequency'will be given by the Chapter 1. Theory 23 eigenfrequency of the k = 0 wave, since this is the wave representing coherent excitations of the nuclear spins. We shall therefore solve for the eigenfrequency spectrum. In the Heisenberg representation, we can write the equation of motion for it as dlt i f i[(AS + jnhB)lf + £ U^Ip*] (1.45) (1.46) If. we substitute for Iff], the second term contains the Fourier transform (FT) of If, and simplifies to: (1.47) Substituting the inverse F T of (1.43) for If. in the first term of (1.46), and using the fact that the .it are independent we can remove the summation over A;..-If we also linearise the equation by replacing 7J by its thermodynamic mean value, Iz = (7J), then dt = i AS + inhB+SA2lz huJk 'I (1.48) By inspection this leads to an eigenfrequency spectrum for the it of k ft- - ^ k ~ h hu)k. + 1nB. (1.49) The replacement of the operator I? with its thermodynamic mean value is easy to justify in the very low temperature case, where w 7J. At higher temperatures this is not true, and this linearization requires justification. De Gennes et a/.[12] show that even at moderately low T (with ~ 1% nuclear alignment), there are still well defined nuclear spin waves (characterised by a clear position in £-space) and therefore an eigenfrequency spectrum, and thus this substitution is still valid. Thus, through the interactions between the nuclear spins, via the weak but long range Suhl Nakamura interaction, there exists within a cold, concentrated nuclear lattice Chapter 1. Theory 24 Wave Number Figure 1.6: Magnitude of the nuclear spin wave frequency verses wave number, calculated from Equation (1.49) The straight line is the N M R frequency without the Suhl-Nakamura interaction, f2[j. the possibility of nuclear excitations in the form of nuclear spin waves. The dispersion relation for these waves is shown schematically in Figure 1.6. In this graph Q,0 is the value of at k = 0, and OQ is the N M R frequency without the Suhl-Nakamura interaction. Chapter 2 Experimental 2.1 Experimental Setup 2.1.1 The Cryostat To obtain the low temperatures (T ~50 mK) necessary to observe nuclear orientation, a helium dilution refrigerator (made by S.H.E.) was used for all experiments. At 100 mK the refrigerator had a cooling power of about 60 pW, which varied roughly as T2 below this temperature. The base temperature of the fridge, for a sample with no radioactive self heating, was about 12 mK. The dilution fridge is a standard tool for low temperature physics, and therefore will not be described here. The interested reader is referred to Experimental Principles and Methods Below 1 Kby Lounasmaa[16]. The cryostat required 35 litres of helium to cool and fill, and boiled-off about fifteen litres of helium per day. Once cooled, experiments were run for durations varying between two days and two weeks. Samples were mounted on the end of the copper cold finger, in the centre of a su-perconducting solenoid. This was mounted outside the vacuum can, and could provide a vertical field of up to 2 Tesla. The E U V O 4 crystal was wrapped in copper foil and clamped to the cold finger with two small copper strips. The cobalt samples were sol-dered directly to an extension on the cold finger. The arrangement df the cobalt foil and the rf coils used is shown in Figure 2.1. . 25 Chapter 2. Experimental 26 Cold Finger (to mixing chamber) IK Shield— Sample V Rf Coils-Figure 2.1: Arrangement of Co foiliand rf coils on cold finger 2.1.2 Supplying RF Power R F power was supplied to the sample from a tuned L-C tank circuit mounted on the 1 K shield. Care was required in setting up to avoid a thermal touch between the coil and the cold finger/sample. Because of the large cooling power of the 1 K cooling plate, commercial teflon coated copper coaxial cable could be used to transfer the rf power through the cryostat. The width of the 6 0 C o C o resonance is dominated by inhomogeneous broadening, and it is necessary to frequency modulate the rf signal to observe the resonance. The signal generator used had an internal control that produced a frequency modulation of depth 100kHz; broader modulation could be obtained by external programming. Removing the frequency modulation from the signal gives a very straightforward method for dis-tinguishing between genuine N M R - O N peaks, and false peaks which can result from rf heating of the sample due to resonances in the coil/tank circuit/transmission line system. We also wished to observe the resonance of the 5 9 C o nuclei (using the thermometric technique, which will be discussed in Chapter 4.) In the case of concentrated nuclear Chapter 2. Experimental 27 spins, the broadening of the resonance peak is dominated by nuclear spin-spin interactions and the nuclei respond to a wide range of frequencies. With this homogeneous broadening it is not necessary to modulate the rf frequency. However, it is important that there be( no resonances in the tank circuit close to the N M R frequency. For this reason the coil was tuned with a two capacitor circuit (mounted at the coil) to give a known resonance frequency and to match the 50S1 impedance of the coaxial cable connected to it. The coil was tuned with the resonance midway between the 5 9 C o and 6 0 C o N M R frequencies. The tank circuit was built for each experiment, and is described in Appendix A . A pick-up coil was included in the circuit, matched to the coaxial cable using a bOQ. metal oxide resistor, but was not required during the experiments. Two coil configurations were utilised: one was a Helmholtz pair on either side of the thin sample; the other consisted of a two turn ellipsoidal solenoid arranged with the sample along the axis. The latter gave the best resonance height/non resonant heating ratio, but the former was often necessary due to the geometric arrangement of the sample. The axis of the rf coil was horizontal. 2.1.3 Gamma Ray Detection. . For the experiments on cobalt, the gamma rays were counted using a pair of Nal(Tl) scintillation counters, each of which had a diameter of 4 inches and a length 4 inches. These were placed perpendicular to each other outside the cryostat, with one on a vertical axis through the sample. To reduce the effect of the applied magnetic field on the counters, the photomultiplier tubes (PMTs) were seperated from the crystals by a 60 cm plexiglass light pipe. The PMTs were stabilised against temperature drifts, and the effect of an applied field, by using a temperature stabilised diode to provide a reference pulse height[17]. The signal was passed through a stabilising amplifier, gated for the 7-ray photoelectric peak, and then converted to a square pulse. The pulses were counted Chapter 2. Experimental 28 on a PG running commercial acquisition software, which also controlled the rf frequency and collected data on the temperature of the cold finger. The 1 5 2 E u and 1 5 4 E u decays involve many 7-rays closely spaced in energy (see Fig-ure 3.3,) and in order to make quantitative measurements it was necessary to resolve these transitions. To achieve this resolution a 52 mm diameter x 45 mm single crystal germanium detector was used. Stabilisation was not necessary, but counting efficiency was very low. The raw spectrum was accumulated on a multichannel analyser program, and count rates determined from the areas of several 7-peaks. 2.1.4 Miscellaneous During the course of experiments, the temperature of the cold finger was monitored using a carbon resistance thermometer. The resistance was measured by a four wire resistance bridge, connected to the computer by optical cables, to prevent electrical noise affecting the bridge. For magnetic field sweeps, the coil current could be monitored with a digital voltmeter. The rf power was provided by a Marconi signal generator. Fractional frequency error was less than 1 0 - 6 . A frequency modulation of 100kHz depth was used, with a scan frequency of 1 kHz. Power was transferred to the top of the cryostat by 50S1 coaxial cable, where it could be amplified by a 30dB rf amplifier if needed, before being passed into the cryostat. For the experiments on E u V 0 4 , a 6 0 CoCo thermometer was used. This was a single crystal of hep Co, approximately 10mm x 1mm 2, with an activity of 1/iCi. By measuring the anisotropy of the 7-rays from this crystal we could calculate the temperature of the cold finger. Chapter 2. Experimental 29 2.1.5 Analysis. A l l collected data were stored on a P C . The 7-ray counts were taken while sweeping the frequency up and down, and filed separately. The results were manipulated and graphed on a spreadsheet. Where needed, curve fits were performed on a separate program. In general,, it was best to view the counts for each detector individually, looking for a change in count rate. This should be of opposite sign for the two counters, as can been seen from Equation (1.24), in which Pk(0) = 1 and Pk{n/2) = - 1 / 2 . 2.2 Sample Preparation 2.2.1 EuV0 4 This was a single crystal of E u ( S m ) V 0 4 with 1.5% Sm and 98.5% Eu. It was donated to us by Prof. B.. Bleaney, at the Clarendon Laboratory, Oxford, England. The sample was irradiated in the thermal neutron facility at T R I U M F (B.C., Canada) for approximately two hours in a neutron flux density of 1010.—>• 10 1 1 neutrons c m - 2 s _ 1 . The activity of the 1 5 2 E u was 5/iCi. Table 2.1 shows the natural abundances, thermal neutron cross sections, and lifetimes for the europium isotopes considered here. From these data we find that the activity of the 1 5 4 E u nuclei is approximately 4% of the activity of the i 5 2 E u . 2.2.2 Cobalt Three samples of cobalt were tested. The first.was a l cm x 1mm 2 single crystal of hep cobalt, donated to us by the Munich group, for the purposes of investigating the frequency pulling effect. This was irradiated at T R I U M F to give an activity of 1/^Ci, and was not annealed before use. This crystal was mounted to the cryostat in two configurations, first clamped directly to the cold finger, and secondly attatched to the cold, finger by a 2cm Chapter 2. Experimental 30 Isotope Natural Half Neutron Products number abundance life cross section 151 48% stable 5900bn 1 5 2 E u 2800bn 1 5 2 M E ] J 153 52% stable 320bn 1 5 4 E u 152 12y 78% 1 5 2 S m ; 28% 1 5 2 G d 152m 9.3h 23% 1 5 2 S m ; 77% 1 5 2 G d 154 16y 100% 1 5 4 G d Table 2.1: Abundance, half life, thermal neutron capture cross section and neutron cap-ture/decay products for the isotopes of europium studied [18][19]. piece of 22 gauge copper wire. In the latter configuration, the sample was attatched to a wooden support with G.E. varnish to provide mechanical stability. The second sample was a 5//m thick single crystal (hep) donated, to us by Dr. D. Chapman of the Australian Defense Force Academy, University of New South Wales, Australia. This sample had been spark cut from a single crystal rod, mechanically pol-ished, irradiated and finally cut and electropolished, and was sent to us soldered to a copper finger. The c-axis of -the crystal was in the plane of the foil. Its activity was approximately lOO/uCi. Given time constraints, it was not possible to prepare a third single crystal of hep cobalt. As it is not necessary to use a single crystal to investigate the frequency pulling effect, a 4//m foil of 99.9% cobalt foil was purchased from Goodfellows. This was cut to a suitable size (1/2 cm by 1cm), and irradiated at T R I U M F for a day. The foil was annealed for eighteen hours at 650°C in a hydrogen atmosphere. Cobalt undergoes a slow phase transformation from hep to fee between 350 and 400°C, and so the crystal; was cooled rapidly (within a few minutes) to prevent regions of hep structure growing. Chapter 2: Experimental 31 The foil was soldered at both ends to a suitably shaped piece of O F H C (oxygen free, high conductivity) copper foil (0.25mm thick), which was then clamped to the cold finger. A galium alloy was used as a solder—its low melting point (30°C) ensuring that the foil was not heated during soldering, which could cause the growth of hep regions. The activity of the sample was approximately 3/xCi. r Chapter 3 Europium Vanadate The first ( 7 F X ) excited state of E u 3 + is just 500 K above the ground ( 7 F 0 ) state. Elliott[20] examined the hyperfine interaction in these ions, and predicted that mixing between these two low lying states should lead to a reduction in the nuclear Zeeman splittings. This reduction should be relatively insensitive to the host material. Shelby and Macfarlane[21], and Erickson and Sharma[22] have measured the effective nuclear magnetic moment of the Eu nuclei using optical hole burning on E u 3 + dilute in YAIO3. They report a nuclear magnetic moment shielded by about 80%. The rare earth vanadates have been studied by the Oxford group of Bleaney, Leask and co-workers. One material studied, europium vanadate, has a tetragonal crystalline structure of the zircon type with space group D\9h (14i/amd) (Stubican[23]), and is a Van Vleck paramagnet. Optical hole burning and optically detected nuclear quadrupole resonance experiments on EuV04[24][25] have concentrated on measuring the quadrupole interaction at defect sites and explaining the apparently high (approximately thirty).num-ber of defect lines. These optical hole burning techniques use the 7F0 —> 5D0 excitation, which is forbidden at lattice sites in zero applied magnetic field by symmetry consider-ations. N M R - O N is not restricted by this consideration, and measuring the resonance frequency would give the quadrupole interaction on the 1 5 2 E u and 1 5 4 E u nuclei at the lattice positions, and hence the intrinsic electric field gradients in the crystal could be calculated. Judd, Lovejoy and Shirley[26] made the first NO measurements on europium, using 32 Chapter 3. Europium Vanadate 33 :72% 6 3 0 - 2 8 % 2- E l 3+ 2+ 2+ 0+ 1530i 1234* E2 1085; 152 Sm 122) E l 1123 2+ 344 0+ 152 Gd Figure 3.1: Simplified nuclear decay scheme of 1 5 2 E u , showing decays relevant to this experiment, and nuclear level spin/parity assignments. A l l energy levels are in keV[9]. neodymium ethylsulphate as a host. . The crystal studied here was Sm(1.5%)Eu(98.5%)V04, supplied to us by the Oxford group. The ionic sizes of the Sm arid Eu ions are very similar, so the former should not perturb the Eu nuclear parameters. However, the magnetic Sm ions could reduce the Eu spin-lattice relaxation time, T i , significantly. 3.1 Results Figures 3.1 and 3.2 show the parts of the europium decay schemes relevant to this discus-sion. The nuclear orientation parameters for these decays of 1 5 2 E u and 1 5 4 E u are listed in Table 1.2. The 7-ray spectrum measured for the sample and the cobalt thermometer is shown in Figure 3.3. This is one of the spectra used to calculate peak areas. It can be seen that the 7 peaks are well distinguished by the detector. Chapter 3. Europium Vanadate 34 1 5 4 R n 63 a n E l 1397 i2+ 123 0+ 154 Gd Figure 3.2: Simplified decay scheme oP 5 4 Eu[8] , details as for Figure 3.1. 4 o 3 o o 0 250 '500 .750 1000 1250 1500 E n e r g y , k e y Figure 3.3: Gamma ray spectrum from E u V 0 4 crystal and cobalt thermometer as mea-sured by 4000s count on germanium detector Chapter 3. Europium Vanadate. 35 ,1 .1 •-, 1 . 05 -• 1 -to +J a 0 . 95 --3 o 0 u • 9. " TS -CD 0. 85 -cq •H 0 .8 -fd e • i-i o 0. 75. -2; • 0 .7 -a . 65 •-0 .6 -miy T eta 4 •4-•rtj •* C o b a l t . x ?.u:1086 • . Eu:1408 20 •> . 40 . .. 6C . . SO 100 '•• 120 '-; Time,,, seconds (thousands) '• 140 Figure 3.4: Measured 7-ray intensity, W(0),ior cobalt 7-rays and 1086 and 1408 keV 7-rays from E U V O 4 , during cooling of E u V 0 4 crystal The crystal was cooled for about 24 hrs. Figure 3.4 shows the counts (versus time) for the 6 0 C o thermometer and two Eu peaks (both from 1 5 2 Eu. j - Each point shows the counts in a live time of 4000s on the detector. The graphs were compiled from the peak-search results of the multi-channel analyser program being used. The temperature of the sample can be found from the value of W{0) for the cobalt thermometer. From Table 3.1, the value of W(0) is. 74.1±.8%. This corresponds to a temperature of 13.15±.3 mK. The TIFT at the Eu nucleus is predominantly quadrupolar, and has a Hamiltonian ' , :L % = 1) V ^ ( J * - J*)].' , (3.1) Chapter 3. Europium Vanadate 36 ..• 1.06 1.04 1.02 ; i 0.98 . - , . •'• 0.9.6 . 0 2 4 6, 8 10 12 14 • ;.. P, MHz Figure 3.5: Calculated W(0), for 1 5 2 E u and 1 5 4 E u , versus the quadrupole interaction, P, at 13mK. See Table 1.2 and Figure 3.1 for curve identity. T] is a function of the axial asymmetry at the ion position, and is zero for the intrinsic sites; P is the parameter (with dimensions of frequency) describing the interaction, given by Equation (1.4). Figure 1.4 shows the values of B2 and BA versus P for 1 5 2 E u and 1 5 4 E u at a temperature of 13.2 mK; Figure 3.5 shows the expected anisotropy along the c-axis, at 13.2 mK, versus the quadrupole interaction strength, P. From this graph it is possible to convert a measured anisotropy, W , into a quadrupole interaction, P . Table 3.1 lists, for the europium 7-rays with good, counting statistics and known orientation parameters, values of W(0) — 1 in two counting periods after the fridge had cooled down. These results were compiled from the data in Figure 3.4 by summing eight points, starting at t = 0, 40,000 and 90,000 seconds. The first value gives the warm (> 1 K) normalisation count for each peaks; The quadrupole interaction strength is derived from Figure 3.4. The errors on the values, of W(0) — 1 are one standard deviation (i.e. s/N) on the statistical error. The error on the.weighted mean was calculated from the standard deviation of the set of values, and is much greater than the error implied by the Chapter 3. Europium Vanadate 37 Isotope peak Peak Warm Cold values, 1 Cold Values 2 keV Ref. Counts, 103 W(0) - 1 P(MHz) W(0) - 1 P(MHz) 6 0 C o 1173 55 -0.249(13) -0.257(13) 1333 50 -0.267(9) -0.262(9) total 105 -0.258(8) -0.259(8) 1 5 2 E U 344 4 162 +0.013(9) +2.6(1.8) +0.021(9) +4.2(1.8) 780 2 49 -0.055(20) +17(6) -0.020(20) +6(6) 1086 3 36 -0.089(44) -16(8) -0.075(44) -13(8). 1408 1 54 -0.001(8) -0.2(1.8) -0.006(8) -1.4(1.8) 1 5 4 E u 1275 1 15 -0.046(27) -11(6) -0.029(27) -7(6) Weighted Mean: 2(14) 1(8) Table 3.1: Gamma-ray counts at different times during europium vanadate cool down. • "Peak Ref." refers to curve in Figure 3.5 counting statistics. The results for the 1 5 4 E u peak are not included in the mean because of the uncertainty in the value of the nuclear quadrupole moment (see Table 1.1), which makes normalisation between the two nuclei difficult. Approximately 50% of the decays in the 344 keV peak are preceded by the 780 keV 7 decays shown in Figure 3.2. Another 25% are preceded only by an L — 1 mixed beta decay. If we ignore the effect of the remaining 25% of the decays and assume that the beta decay is dominated by the J = 1 decay, then we can calculate a value U2F2 ~ —0.4. This is the value which has been used to calculate the value of P . 3.2 Discussion Taking the weighted mean of all the results for 1 5 2 E u gives a quadrupole interaction of 1.4+4 Mhz. The error is calculated from the distribution of the results. There are two possible explanations for such a small interaction: Chapter 3. Europium Vanadate 38 1. The electric field gradient at the nucleus is indeed small. This could be a result of the tetragonal symmetry of the E u 3 + lattice position in EuVCu. This symmetry is relatively higher than for the defect sites studied in the hole burning experiments, which reported values for P ~ 40 MHz. 2. There is a sizeable quadrupole interaction at the nucleus, but the spin-lattice re-laxation time,-Ti, is too long for NO to be observed in the time for which the experiment was run. This would mean that 7\ ~ 1 day, which is certainly possible. If this conjecture is correct, it means that the 1.5% Sm content was not effective in reducing Tx appreciably. In this case the experiment might be repeated with a crystal containing a greater amount of samarium (which could be provided by the Oxford group.) Chapter 4 Cobalt 4.1 Frequency Pulling Section 1.6 introduced the idea that there exists within a cold ferromagnetic conductor elementary excitations of the nuclei in the form of nuclear spin waves. These spin waves —# are localised in fc-space and have a defined dispersion relation, due to the Suhl-Nakamura interaction between the nuclear spins. The Suhl-Nakamura interaction results from the exchange of virtual magnons between nuclear spins. The frequency spectrum of the nuclear spin waves is given by: AS •n«.= a 1 + AL hujr. (4.1) and this spectrum is plotted in Figure 1.6. . Nuclear magnetic resonance is the excitation of nuclei between their magnetic sub-states. Classically the phenomenon occurs when the frequency of the applied rf radiation matches the Larmour precession frequency of the nuclei in the magnetic field producing their level splitting. The latter case corresponds to a coherent rotation of the nuclei about an axis in a frame of reference rotating about the z-axis at the frequency of the applied radiation. In the case that there are clearly defined nuclear spin waves, the nuclear res-onance resonance is the excitation of the k = 0 mode. In- this case we can calculate the frequency of the "wave" from equation 4.1 with k = 0. This gives us a nuclear frequency of magnitude: 0 = AS_ h 1 + z huj0 (4.2) 39 Chapter 4. Cobalt 40 The value of u>o is given by: ^%BA = u;e, (4.3) where BA is the magnetocrystalline anisotropy field. The nuclear frequency can be writ-ten O = O 0 + A O (4.4) where O 0 is the unpulled nuclear resonance frequency:. I A S On = h + 7 n £ (4.5) and A O is the frequency pulling: A O AS Ah (4.6) In the usual case where the nuclear frequency is dominated by the HFI, and the ap-plied field, B, represents a small correction to this, the frequency pulling can be well approximated by AT, (4.7) A O ~ ~ OQ or A O On _Op_ ' Su)P (4.8) At the low temperatures at which this, effect is significant, O0, ue and: S are constant, and so Iz contains all the temperature dependence of the frequency pulling. The magnetocrystalline anisotropy energy in hep cobalt may be approximated by E = K0 + Kx sin2(0) + K2 sin 4 (6) + (4.9) where 0 is the angle between the magnetisation of the crystal and the c-axis (the easy axis). This anisotropy energy can be represented by an effective field (directed along Chapter 4. Cobalt 41 o 20 40 60 80 100 T> mK Figure 4.1: Frequency pulling for 5 9 C o C o , as predicted from the Suhl-Nakamura interac-tion. the easy axis) acting on the electron spins. This anisotropy'field, S^, has a magnitude 1K\jBs, where B$ is the magnetic induction due to the spontaneous magnetisation, MS, i.e. Bs = ^KHQMS-In hep cobalt the spontaneous magnetisation is 1.48 A m - 1 at low temperatures, which corresponds to an induction.of 1.85T, and ifi=0.75 MJm - 3 [27]. This gives an anisotropy field of 8.1x105 A m - 1 , or 1.0 T. We therefore have an electron frequency The magnetic moment per atom is, 1.72//B, which implies an atomic spin of 0.85/i, and therefore: . ue = 27T28GHZ = 1.8 • l O 1 ^ - 1 (4.10) AO 9.2 • 10~3IZ O0 (4.11) The frequency pulling, as predicted by these values, is shown in Figure 4.1 Chapter 4. Cobalt 42 4.2 The Thermometric Technique In nuclear magnetic resonance power is absorbed by the nuclear spins. The energy deposited will be transferred to the host material through spin-lattice interactions. If the rf field is applied at the resonance frequency of the nuclei of the host material, then the power absorbed by the nuclei may be comparable to the rf absorption in the material. In this case the temperature of the sample will be raised, and the N M R of the stable nuclei can be detected by the change in the 7-ray anisotropy of the radioactive dopant. This thermometric technique of measuring N M R was first used by Kotlicki and Turrell on 5 4 Mn 5 5 MnCl2 .4H 2 Q[3l . It is dependent on suitable SLR times to transfer power from the abundant host spins to the dilute radioactive spins via the lattice. This technique is considerably more difficult to apply in conductors because there is eddy current heating in the surface of the sample. This thesis reports the first measurement of thermometric N M R in a conductor. Figure 4.2 shows schematically the important heat flow processes. 4.3 Samples The samples have been described in Section 2.2.2. Two samples were tested on which no thermometric N M R was observed. They were a 10mm x 1mm 2 hep single crystal, and a 4/xm thick polycrystalline fee foil. The sample which gave positive results was the 5 ixm-thick hep single crystal from Australia. There is a fundamental restriction on the measurement of the thermometric technique: the ratio of the power absorbed by the nuclei to the rf heating in the skin depth of the sample. This means that one cannot improve oh a poor signal by. simply increasing the rf power, and that the measurement should be insensitive to sample thickness. The thin samples were used so that the 6 0 C o resonance could also be observed—allowing Chapter 4. Cobalt 43 Figure 4.2: Schematic representation of the important heat flow processes, in N M R - O N Chapter '4. Cobalt 44 calculation of the unpulled 5 9 C o resonance. It is believed that the resonance was only detected in the one sample for two reasons: 1. Rf efficiency: the only configuration in which this crystal did give observable res-onances was with a two turn solenoid applying rf parallel to the crystal surface. With rf applied perpendicular to the surface no resonance was found. 2. Statistics: the observed peak was small compared to the change in anisotropy due to the non-resonant heating of the sample. This means that the high count rates given by this crystal were essential in observing resonance. (There was typically a change of 0.5% in the 7-ray intensity at resonance, compared with a 6% change due to the non-resonant heating.) It should be noted that a highly active sample does produce radioactive self heating, due mainly to the absorption of beta particles. A sample of 6 0 C o with an activity of 100 /j,Ci has a radioactive self heating of 65nW. However, using the formula for dE/dx given by Krane[28], and by integrating over the foil on a spreadsheet, it is estimated that ~ 10% of the energy of the (3 particles was deposited in the crystal. This corresponds to a self heating of ~ 7nW, and a temperature rise of approximately l m K , assuming that the Q= T2 relationship for the cooling power of the fridge is valid at base temperature, and that the base temperature can be taken as a constant correction. . The base temperature of the dilution fridge was never reached with either of the thin films (and self-heating was certainly not a problem in the fee foil). The reason for this problem is not understood at this time. 4.4 Results Figure 4.3 shows the best thermometric N M R spectrum obtained for the 100 fxCi crystal. It is the sum of five sweeps (in the downward direction), with a dwell time at each Chapter 4. Cobalt 45 0.99 $ 0.988 •p § 0.986 o o'. TS CD »0.984 14 2 0.982 0 . 98 215 -+- + 216 217 218 ' 219 . 220 Frequency, MHz 221 222 Figure 4.3: The best frequency sweep, showing counts versus frequency. This is a sum of five runs under the same conditions. frequency of 100 seconds, and a step size of 100 kHz. \ The run clearly shows the 5 9 C o peak at a frequency of 218.91(6) MHz, with a width of 1.0(1) MHz. Figure 4.4 shows the 6 0 C o resonance, measured on three runs sweeping the frequency down. The dwell times and step sizes are as for Figure 4.3 Table 4.1 shows the measured resonances,, both for 6 0 C o and 5 9 C o , with their re-spective anisotropics. The frequency was swept up and down; sweeps with the same conditions were taken consecutively, and added to give the results shown. The peaks were fitted assuming a Gaussian peak shape on a straight line background. The results from sweep (a) should be treated with some caution, as it only contains a single sweep, and is subject to a large amount of noise and poor statistics. The unpulled 5 9 C o resonance can be obtained by scaling the measured 6 0 C o resonance Chapter 4. Cobalt 0 . 9 8 .% 0 . 9 7 5 — PS 0 . 9 6 -I 1 1 t- H 1 1 1 1—: 1 1 1 2 2 1 2 4 1 2 6 1 2 8 1 3 0 1 3 2 F r e q u e n c y , MHz Figure 4.4: The 6 0 C o N M R - O N resonance, showing counts versus frequency. This sum of three runs under the same conditions. Chapter 4. Cobalt 47 Ref: field . rf counts W(0). sweep Resonance (MHz) T dBm 106 direction position width 6 0 C o N M R - O N resonance: 0.3 9 1.5 0.968 up 126.38(3) 0.24(4) down 126.36(3) .38(5) 0.15. 7 4.2 0.969 up 126.49(5) 0.26(7) down 1.26.52(4) .16(6) 0.3 12 1.5 0.971 up 126.36(3) 0.17(5) 5 9 Co thermometric resonance: a 0.3 9 0.4 . 0.979 up 217.98(8) 0.7 down 218.37(14) 1.2 b 0.3 9 0.8 0.975 up 218.23(4) 0.46(5) down 217.97(10) c 0.15 ,9 2.3 0.984 up 218.77(3) 0.67(4) down 218.91(6) 1.0(1) . d 0.3 12 : 1.1 0.982 up 218.73(5) 0.80(7) Table 4.1: Peak positions of nuclear resonances as measured in single crystal hep 6 0 CoCo . 'Ref.' gives entry in Table 4.2. See the text for a full description K Chapter 4. Cobalt 48 0 . 9 8 8 B 0 . 9 8 6 3 0 . 984 0 . 982 60 65 70 7 5 80 T , mK Figure 4.5: Detail of Figure 1.5 used to calculate temperature of cobalt crystal during rf sweeps, corrected for solid angle subtended by detector. by the ratio of the nuclear ^-factors (1.740(5), see Table 1.1.) The measured 6 0 C o reso-nance frequency was 126.42(2) MHz. This can be compared with the value of 126.8(1) MHz observed by Hagn and Zech[29]. The small difference is probably due to different sample geometry. The unpulled frequency is then calculated to be 220.0(6). MHz. The N M R frequency of hep cobalt has been measured by Kawakami et. a/.[30], and they report a value of 219.9MHz at a domain wall edge, which agrees with our result. The external field, which is applied perpendicular to the c-axis of the crystal, has an effect ~ 0.01% on the frequency, and has been neglected. Figure 4.5 shows the anisotropy, W(0), for 6 0 CoCo, for the range of values of interest here. From this we can read off the temperature, of the crystal as measured by the anisotropy, and calculate the expected frequency pulling. The calculated values are shown in Table 4.2, and it should be noted that the appropriate temperature was given by the anisotropy at the top of the resonance. At these temperatures the variation in W(0) with temperature is small—an uncertainty of ±.001 in W(Q) gives an error of ±2.5 Chapter 4. Cobalt 49 Ref. Sweep Direction Temperature mK Frequency Pulling (MHz) Measured Calculated b d a c up down up down up down up 71 64 64 62 71 77 74 2.0 1.5 .1-7 2.0 .1.1 1.0 1.2 1.6 1.7 1.6 1.8 1.6 1.4 1.5 Table 4.2: Frequency pulling, measured and calculated, for hep Co. 'Ref.' gives entry in mK. This error in the temperature gives an uncertainty of approximately ±0.2 MHz in the calculated frequency pulling. The measured values of the frequency pulling are also shown in Table 4.2, and can be compared with the calculated values. 4.5 Discussion The 5 9 C o resonance has been measured at approximately 70 mK. using thermometric N M R - O N . The resonance position varied between 217.97(10) and 218.91(6) MHz, with a mean value of 218.4 MHz, across a measured temperature range of 62—>77 mK (mean: 69 mK). This frequency is 1.6 MHz lower than the unpulled value of 220.0(6)MHz. This difference, which agrees with the calculated value, is attributed to frequency pulling via the Suhl-Nakamura interaction. This is also the first measurement of thermometric N M R on a conductor. This tech-nique has the potential to investigate any material where a suitable nuclear orientation dopant can be implanted (provided the ratio of nuclear absorption to rf heating is large enough). It is particularly useful when the objective is to compare N M R frequencies and Table 4.1. Chapter 4. Cobalt 50 lineshapes of abundant and dilute nuclear spins. • This experiment on cobalt could be improved in two ways. First, it would be ad-vantageous to obtain lower temperatures where higher anisotropies could be measured. Failure to reach lower temperatures may have been partly due to radioactive self heat-ing, but there appeared to be other cryogenic difficulties. The rf arrangement may have produced extra heating, and/or the dilution refrigerator, which is twenty years old, may have deteriorated. Secondly, a different radioactive nucleus which experiences a larger hyperfine field in cobalt could be used, either to replace the 6 0 C o or in conjunction with it. This would allow a more accurate determination of the frequency pulling and its variation with temperature. Bibliography [1] E. Hagn and E . Zech. Private Communication to B . G. Turrell. [2] B . G. Turrell. Private Communication. [3] A . Kotlicki and B. Turrell. Thermometric N M R of stable nuclei by low-temperature nuclear orientation. Physical Review Letters, 56:773, 1986. [4] P. Raghavan. Table of nuclear moments. Atomic and Nuclear Data Tables, 42:189, 1989. [5] N . J . Stone and H . Postma, editors. Low-Temperature Nuclear Orientation. North-Holland, Amsterdam, 1986. [6] Kenneth S. Krane. Nuclear orientation formalism. In Stone and Postma [5], chapter Two. [7] R. J. Blin-Stolye and M . A . Grace. Oriented nuclei. In Handbuch der Physik, volume 42, page 555. Springer Verlag, 1957. [8] R. G. Helmer. Nuclear data sheets update for a=154. Nuclear Data Sheets, 69:507, 1993. [9] L.K.Peker. Nuclear data sheets for a=l52. Nuclear Data Sheets, 58:93, .1989: [10] C. P. Slichter. Principles of Magnetic Resonance. Springer Verlag, Berlin, 3rd edition, 1990. [11] M . A . Ruderman and C. Kittel . Indirect exchange coupling of nuclear moments by conduction electrons. Physics Review, 96:99, 1954. [12] P. G. de Gennes, P. A . Pincus, F . Hartmann-Boutron, and J. M . Winter. Nu-clear magnetic resonance modes in magnetic material. I: Theory. Physical Review, 129:1105,1963... [13] H . Suhl. Effective nuclear spin interactions in ferromagnets. Physics Review, 109:606, 1958. [14] A. Nakamura. Indirect coupling of nuclear spins in antiferromagnet, with particular reference to M n F 2 at very low temperatures. Progress of Theoretical Physics, 20:542, 1958. 51 Bibliography 52 [15] E. A . Turov and M . P. Petrov. Nuclear Magnetic Resonance in Ferro- and Antifer-. romagnets, chapter Four. Halsted Press (John Wiley and Sons), New York, 1972. Translated by E . Harnik. [16]. O. V . Lounasmaa. Experimental Principles and Methods Below IK. Academic Press, London, 1974. [17] M . Bottcher, W. D. Brewer, and E . Klein. A simple method for the temperature stabilization of scintillation detectors. Nuclear Instruments and Methods, 206:177, 1983. . : ' [18] D. R. Lide, editor. CRC Handbook of Chemistry and Physics. C R C Press, Boca Raton, 74 edition, 1993. [19] C. Lederer and V.S. Shirley, editors. Table of Isotopes: John Wiley and Sons, New York, 1978. [20] R. J. Elliott. Theory of nuclear magnetic resonance in E u 3 + . Proceedings of the Physical Society of London, B., 70:119, 1957.. . . [21] R. M . Shelby and R. M . Macfarlane. Measurement of the anomalous nuclear mag-netic moment of trivalent europium. Physical Review Letters, 47:1172,1981. [22] L . E. Erickson and K . K . Sharma. Nuclear quadrupole resonance measurement of the anisotropic magnetic shielding and quadrupole coupling constants of 1 5 1 E u 3 + and 1 5 3 E u 3 + dilute in Y A 1 0 3 single crystal. Physical Review B, 24:3697, 1981. [23] V . S. Stubican and R. Roy. Relation of equilibrium phase transition pressure to ionic radii. Journal of Applied Physics, 34:1888, 1963. [24] R. L. Cone, P. C. Hansen, and M . J . M . Leask. Eu3+ optically detected nuclear quadrupole resonance in stoichiometric europium vanadate. Journal of the Optical •> Society of America B, 9:779, 1992. [25] R. L . Cone, R. L . Harley, and M . J. M . Leask. Nuclear quadrupole optical hole-burning in stoichiometric EuV04. Journal of Physics C, 17:3101, 1984. [26] B. R. Judd, C. A . Lovejoy, and D. A . Shirley. Anomalous quadrupole coupling in europium ethylsulfate. Physical Review, 128:1733, 1962. [2.7] M . B . Stearns. Magnetic properties of Fe, Co, Ni . In Landolt-Bdrnstein: Numerical Data and Functional Relationships in Science and Technology; New Series., volume III/19a, page 22. Springer-Verlag, 1982. [28] K . S. Krane. Introductory Nuclear Physics. John Wiley and Sons, new York, 1988. Bibliography 53 [29] E . Hagn and E. Zech. New method for determination of the sign and magnitude of the electric quadrupole splittings. II. The electric field gradient of Co in hep and fee Co. Physical Review B, 25:1529,1982. [30] M . Kawakami, T. Hihara, Y . Ko i , and T. Wakiyama. The 5 9 C o nuclear magnetic resonance in hexagonal cobalt. Journal of the Physical Society of Japan, 33:1591, 1972. Appendix A Radio-Frequency Tank Circuit Reliable formulae and guidelines for building impedance matched tank circuits are hard to find. This information will hopefully be usefull. Figure A . l shows the resonant circuit used in these experiments. The resistance, r, is the internal resistance of the inductor. The inductor was a coil wound from two to four turns of copper wire. The capacitors were surface-mount rf/microwave ceramic capacitors from A T C (American Technical Ceramics). These are known to work in helium temperatures. It is likely that any surface-mount ceramic capacitor will work. Energy losses in this circuit are dominated by radiated energy, and so the Q-value is measured rather than calculated. From a known Q-value, and inductance, L, the resonant frequency of the coil will be given by: 1 L(C-+C')' (A. l ) r 4 r SIMM © L GO C w Figure A . l : The tank circuit used in these experiments 54 Appendix A. Radio-Frequency Tank Circuit 55 The impedance near resonance is: QLOL (A.2) ~ (1 + C'/Cf Using these formulae and a few simple measurements it is comparatively easy to match the impedance of the coaxial cable accurately enough to avoid transmission line resonances. The experiments described here typically used a coil of two of four turns, of diameter 4 mm to 10 mm. The capacitors used were in the range 2 —>• 30 pF. To estimate the number of turns required for a given inductance, if a solenoid (with gaps between the turns equal to the diameter of the, wire) has n turns, length a and diameter b cm, then 2 2 • 35a + 406: v ; L is in //II. . Appendix B Mathematica Programs The calculated graphs in this thesis were produced using simple routines on mathematica. The main functions, and an example of their use, are shown below. Enter physical constants Units: mK, eV, T, MHz. boltzmann=8.62*10"-8 nuclearMagneton=3.lS+lCr-S plank=4.136*l(T-9 «Graphics' Graphics' Define the functions needed. bCoefficient[k,I,m,a]is the orientation coefficient of order. k, operating on m and a —lists of m values and their occupation probabilities. aMagnetic[delta,T,m] and aQuadrupolar[P,T,m] return occupation probabilities for m values in list m, for magnetic and quadrupole field respectively. I is the nuclear spin. Clear[bCoefficient,aMagnetic,aQuadrupolar] Module[{k,i,mlist,alist}, bCoefficient[k_,i_,mlist_,alist_]:= Apply[Plus, (2k+l)-.5 * (2i+l)"\5 . . •' * ((-l)~(i+#))& /@ mlist 56 Appendix B. Mathematica Programs 57 * (ThreeJSymbol[{i,-#},{i,#},{k,0}]& /fl mlist) * a l i s t -] ] Module[{delta,t,mlist}, aMagnetic [delta. ,t_,mlist_].: = Exp[# * delta/(boltzmann * t)]& /@ mlist ) / Apply[Plus, Exp[# * delta/(boltzmann * t)]& /fl mlist] ] Module[{p,t,mlist}, aQuadrupolar[p_,t_,mlist_]:= Exp[-#~2 * p *plank/(boltzmann * t)]& /@ mlist / Apply[Plus, Exp[-#"2 * p*plank/(boltzmann * t)]& /fl mlist] ] This routine plots the graph ofW versus t in Figure 1.5. The orientation parameters are supplied in lists, {{11%, U%}, {U^ U4}}. etc., (a) and (b) being the two gamma decays. spin=5 magMoment=3.799 > field=21.9 levelSplitting=magMoment*nuclearMagneton*field/spin u={{.9394,.7977},{.9394*.7491,.7977*.2847}} f={{-.4477,-.3044},{-.5976,-1.069}} mValues=Table[m,{ra,-spin,spin}] LogLinearPlot[ Appendix B. Mathematica Programs 58 Evaluate[ Table[ 1+ Apply[Plus, (ufMean *Table[bCoefficient[ k,spin,mValues,aMagnetic[levelSplitting,t,mValues] ] *LegendreP[k,Cos[theta]],{k,2,4,2} ] ,{theta.,.0,Pi/2,Pi/2} ] • ],{t,2,50}, PlotRange->{Automatic,{0,1.3}}, Axes0rigin->{1,1}, DisplayFunction->Identity Show [{'/., Graphics [lineLabels] }, Frame->True, FrameLabel->{"T, mK", "W"}, DisplayFunction->$DisplayFunction 

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