INTERACTION BETVJEEN THE SODIUM TOG LEI AND THE LATTICE OF SODIUM CHLORIDE By 01en Kraus A THESIS Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1957 ProQuest Number: 10008527 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 10008527 Published by ProQuest LLC (2016). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 481 06- 1346 ACKNOV/LSDGMENT I thank Dr. W.H. Tanttila for arousing my interest in the study of interactions between a nuclear spin sys­ tem and a crystalline lattice; for suggesting the problem which the thesis discusses; and for acquainting me with the techniques of pulsed nuclear induction. I am grate­ ful for his guidance. I obtained helpful suggestions from Mr. Mack Breazeale and Mr. Walter Mayer. Their knowledge of ex­ perimental ultrasonic methods was useful. I appreciate the cooperation of Mr. Charles Kingston and Mr. Richard Hoskins of the Physics Shop, and Mr. Marvin Doerfler of the Electronic Shop. They constructed apparatus on rather short notice. I thank Michigan State University for a Graduate Assistantship which gave me valuable teaching experience and enabled me to become better acquainted with physics. I am grateful for financial aid from fellowships provided by the Standard Oil Company of Indiana and the National Science Foundation. ABSTRACT The thesis discusses the interaction between the lattice vibrations and the sodium nuclei of a crystal of sodium chloride. One supposes that the quadrupolar inter­ action between the sodium nuclear quadrupole moment and the electric field gradient in the crystal provides the mechanism which makes possible an exchange of energy be­ tween the crystalline lattice and the nuclear spin sys­ tem. If the sodium chloride crystal is in a constant ex­ ternal magnetic field, the Zeeman levels of the sodium nuclei are split; and the nuclear-magnetic-resonance fre­ quency for the sodium nuclei is in the radiofrequency re­ gion. At room temperature, the intensity of such fre­ quencies in the spectrum of the lattice vibrations is negligible. Using ultrasonic energy, one can, however, excite the low-frequency vibrational modes of the lat­ tice.If ation, jy ' is the frequency of the ultrasonic then one chooses radi­ zy' in such a way that where is the magnetic quantum number for the so­ dium nuclei. Consequently, the ultrasonic energy, by means of the quadrupolar coupling between the lattice and nuclear spin system, produces sodium nuclei. = ± 2. transitions of the Experimentally, one subjects the sodium chloride crystal to a pulse of ultrasonic energy. Several hun­ dred milliseconds later, one applies a radiofrequency pulse of frequency u , with . radiofrequency energy produces sodium nuclei. Consequently, the transitions of the The amplitude of the nuclear induction signal which follows the radiofrequency pulse is a meas­ ure of the difference between the populations of adjacent Zeeman levels; hence, the amplitude of the induction sig­ nal permits one to study how effectively the ultrasonic energy produces nuclear transitions. This means one can check the hypothesis that the nuclear spin system and crystalline lattice are coupled through the quadrupolar interaction. One assumes a simplified model for the sodium chlor­ ide lattice; the six nearest neighbors of a sodium nu­ cleus are replaced by six equal point charges of magni­ tude X C 5 where electronic charge. V is a parameter, and time, TJ is the The experimental data enables one to compute a numerical value for Y=o.(o& , G X , and the result is Measurement of the spin-lattice relaxation for the sodium nuclei in sodium chloride is also part of the experimental work. is the result of the measurement. A value of"T^-TiÇ sec, TABLE OF CONTENTS Page Chapter I. Introduction ............. ............... Chapter II. Equipment ..................... .......... 9 The magnet ..... .................... 9 The induction head ...............10 The radiofrequency oscillator ...... 11 The puiser ........................ 12 The receiver ..................... 15 The ultrasonic o s c i l l a t o r 17 Chapter III Theory ................................. The Ultrasonic Transition Probability .. The electrostatic potential ....... The quadrupolar interaction energy . The presence of a sound wave ...... Transition probability per unit time The Nuclear Magnetization............ The differential equations ......... Steady state equations ......... The linear approximation ........... Solution of the linear approximation The macroscopic magnetization ..... Chapter IV. The Experimental Method ................ h6 Measurement of spin-lattice relaxation time, .................. 1+6 Preparation of the sodioim chloride crystal for ultrasonic studies .... 50 Measurement of ultrasonic effects .. 52 Chapter V. Experimental Results .................... 55 The spin-lattice relaxation time ... 55 The parameter X .................. 55 Chapter VI. Discussion ........................... 60 The spin-lattice relaxation time ... 60 The parameter Y ................... 60 Appendix. Theory of Pulsed Nuclear Induction .... 63 Bibliography............................................. 71 1 19 19 20 2h 30 32 35 35 39 % h2 hh LIST OF FIGURES Figure Page 1• Block Diagram of Apparatus 2. Induction H e a d .......... 3. Puiser h. Receiver 5. Ultrasonic Oscillator 6. Positions of Nucleus, N and C h a r g e , ^ * Before and After Displacement ..... 20a 7. Mutual Spin Flipping for % = ^ .......... 38a 8. Measurement of Relaxation Time ...... 55a 9. Equivalent Circuit for Loaded and Unloaded Quartz Transducer ............... 56a 10. Real Part of Impedance of Quartz .............. 56b 11. Real Part of Impedance of Quartz and Sodium Chloride ................................. 56c Experimental Ultrasonic Attenuation of Induction S i g n a l ........ 59a 12. ................ . 9a 10a ............ 12a ....... I5a ..... 1 3 . Theoretical Ultrasonic Attenuation of Induction Signal ....... 17a 59b CHAPTER I INTRODUCTION Waller^ gave the first theoretical treatment of elec­ tronic paramagnetic relaxation in crystals. The theory assumed that the time variation of the crystalline magne­ tic field at the site of an electronic spin is responsible for the coupling between the lattice and the spin system. A simple modification made the theory applicable to n u ­ clear paramagnetism; however, the theoretical relaxation times were much longer than the experimental values. In order to account for the nuclear spin-lattic relaxation times which were observed experimentally, Bloembergen 2 and Hatton and Rollin^ proposed that electronic paramagnetic impurities were responsible for the spin-lattice coupling. Indeed the presence of paramagnetic impurities explained the relatively short spin-lattice relaxation times; how­ ever, nuclear magnetic resonance in solids had been con­ fined largely to either protons or nuclei with spin I Since the nuclear quadrupole moment of nuclei with spin I< 1 is zero, no quadrupole effects were observed. Pound 4 presented the first discussion of nuclear electric quadrupole interactions in crystals and their effects on the nuclear resonance of atomic nuclei in solids. Pound's investigations showed that the inter­ action of a nuclear quadrupole moment with the electric field gradient produced by neighboring charges broaden­ ed and split the nuclear resonance absorption line. Of more importance for our discussion, however, is the fact that the quadrupolar Interaction could account for the spin-lattice relaxation times. The experimental evi­ dence indicated that the nuclear quadrupolar interaction was important in crystals with cubic symmetry. Pound's experimental work J. van Kranendonk After wrote a the­ ory for the effect of the nuclear quadrupolar interaction on the spin-lattice relaxation time. Two processes account for the spin-lattice relaxation of electrons or nuclei in solids; processes. direct and indirect The indirect process is a Raman scattering of phonons, and the direct process requires the annihilation or creation of a phonon accompanied by a transition of an electronic or nuclear spin between magnetic energy levels. Van Kranendonk is concerned with the indirect process. assumes that a phonon of initial energy hi/ is scattered He 3 into a final state with the energy The absolute value of the difference between Kl/ and hi/' is the energy of the nuclear spin transition. The nuclear quadrupolar interaction is the only mechanism which van Kranendonk considers responsible for the phonon scatter­ ing. If one assumes a Debye distribution for the lattice oscillators, then the theory predicts that the probability of a nuclear transition as a result of a thermal Raunan process is proportional to T^ where ”T ” is the absolute temperature of the lattice, at temperatures below 0.020 (0 is the Debye temperature). At higher temperatures the transition probability is proportional to T^(a - b/T^), where "a" and "b" are constants. Van Kranendonk applies his theory to a model in which a lattice of the sodium chloride type is replaced by a central nucleus surrounded octahedrally by six equal point charges. The magnitude of the point charges is a variable parameter of the model. In order to obtain spin-lattice relaxation times which agree with experimental values, van Kranendonk finds the point charges must be from 100 to 1000 times the electronic charge. It is difficult to make a comparison between theoretical and experimental results. One knows very little about the quadrupolar interaction. 4 As Sternheimer 6 T and others' have shown, the nuclear quad­ rupole moment induces a quadrupole moment in the electro­ nic shells associated with the nucleus. The induced m o ­ ment can reduce or enhance (shielding and antishielding effect) the quadrupole moment of a bare nucleus. The ac­ tual electric field gradient which a nucleus in a crystal­ line lattice experiences is also difficult to estimate. Watkins,^ for example, found that strains in an ionic crystal can produce an electric field gradient. Apparent­ ly, the strains cause a mixing of p- and d-type orbitals to the original s-type electronic wave functions of the ions. Two articles by C. A. Altshuler q, 10 tion of ultrasonic energy by spin systems. treat the absorp­ One generally supposes that the interaction of the electronic magnetic moments with the lattice vibrations of a solid provides a means for the relaxation of the electronic spins after their excitation by resonance absorption of electromagne­ tic energy. Altshuler considers the possibility of split­ ting the electronic magnetic energy levels by placing a paramagnetic salt in a constant magnetic field and then producing transitions between the energy levels by means 5 of ultrasonic energy at the resonance frequency. (Accord­ ing to Altshuler, the effect was also considered by Zavoisky.) Lattice vibrations modulate the crystalline electric field. Variation of the crystalline electric field affects the orbital motion of the electron, and the lattice vibrations are coupled to the electron spin through the spin-orbit interaction. In the second paper Altshuler discusses the modulation of internal crystalline magnetic fields as a second mechanism for the transfer of ultrasonic energy from the lattice to electrons or nuclei. Our discussion has mentioned the direct and indirect processes which account for the spin-lattice relaxation time of electrons and nuclei. Since transitions of elec­ tronic or nuclear spins correspond to frequencies which lie in the radiofrequency region, the direct process is negligible until one reaches liquid helium temperature. One must keep in mind, however, that the ultrasonic excita­ tion of an electronic or nuclear magnetic moment to a high­ er energy state represents a direct process; hence, the frequency of ultrasonic energy will equal the transition frequency. Altshuler's order-of-magnitude estimates indicate that the resonance absorption of ultrasound by nuclei in solids would be very small. He considers both dipole- dipole and quadrupole coupling of the nuclei to the lattice of the crystal. In liquids, according to Altshuler, conditions are quite favorable for the ab­ sorption of ultrasonic energy by nuclei; and he states that one should observe ultrasonic saturation of the nu­ clear spin system. Now the possibility of Indirectly observing the absorption of ultrasound arises. For exam­ ple, suppose one observes nuclear magnetic resonance ab­ sorption of electromagnetic energy. Then the application of an ultrasonic field of the correct frequency and the subsequent saturation of the nuclear spins should destroy the resonance absorption of electromagnetic energy. At the present time absorption of ultrasound by nuclear spins has been observed only by means of this indirect method. Proctor and Tanttila^^ made the first observation of the influence of ultrasonic energy on the nuclear spinlattice relaxation time. Ultrasonic vibrations whose fre­ quency equaled the frequency for a nuclear magnetic dipole transition between the pure quadrupolar energy levels of chlorine nuclei in sodium chlorate saturated the energy levels. By applying pulsed radiofrequency energy of the same frequency as the ultrasonic energy and observing the 7 induction signal, the experimenters were able to detect the degree of saturation of the energy levels. (The ap­ pearance of an induction signal as a result of the pre­ cession of a nuclear quadrupole moment in a non-uniform electric field is discussed in detail by Bloom, Hahn, and Herzog. 12 ) The experimental results are compared with a theory proposed by Chang. IR Chang's theory specifically treats the pure quadrupole resonance of chlorine in sodium chlorate. Proctor and Tanttila find good agreement between experiment and theory for the temperature dependence of the indirect process; however, there is poor agreement between experiment and theory for the dependence of the directprocess relaxation time on the energy density of the ultra­ sound. In a later experiment Proctor and Robinson l4 studied the effect of ultrasound on the relaxation time of sodium nuclei in sodium chloride. The experimenters made use of the amplitude of the induction signal, which was produced by the processing nuclear magnetic moments in a steady magnetic field, to sample the population difference be­ tween the magnetic energy levels of a sodium nucleus. The ultrasonic vibrations produced nuclear spin transitions 8 which corresponded t o = ± 2 where "m” is the magnetic quantum number of the nuclear spin. At the same time, the radiofrequency field produced transitions which cor­ responded toA/vn«±i, On the basis of a classical model, Robinson computes the electric field gradient which a sodium nucleus experiences. (The ultrasonic waves re­ move the cubic symmetry of the crystalline electric field.) He compares the calcubted gradient with the value obtained from his experimental data. Agreement between the two values is good, and Robinson suggests that van Kranendonk*s theory is unnecessary. The data do not tell whether the dipolar or quadrupolar interaction is the more important in determining the spin-lattice relaxation time, but it suggests that the quadrupolar interaction is more important. This thesis reports the results of an experiment which is a modification of Proctor and Robinson’s experi­ ment. Proctor and Robinson used a roughened sodium chlo­ ride crystal for a sample and assumed that the ultrasonic phonons were scattered isotropically by the irregularities in the end of the sample. The experiment reported by this thesis concerns unidirectional standing ultrasonic waves in the sodium chloride crystal. CHAPTER II EQUIPMENT A block diagram of the apparatus appears in Pig. 1, and a discussion of the individual components of the ap­ paratus follows. The magnet An electromagnet supplied a constant magnetic field. Storage batteries served as a source of current for the magnet, and three batteries in series (96 volts) provided the most stable operating conditions. In order to obtain the sodium resonance with a radiofrequency field of five megacycles/sec., the magnet windings required a current of 6.8 amperes. Three rheostats made possible variation of the current through the windings. A "fine" adjustment of the field was obtained by a rheostat which consisted of flat nichrome wire wound on a ceramic cylinder. A screw of the proper pitch allowed an electrical contact to move along the nichrome wire as one rotated the screw. The magnet windings and the rheostats were water-cooled. 9 Oscilloscope Q. (/> Puiser RF Oscillator Ultrasonic Oscillator Receiver CO o "O CO Diode Voltmeter CO U_ Magnet Pole Face induction Head Fig. I Block Diagram of Apparatus 10 The pole pieces of the magnet were eight inches in diameter and unpolished* The experiments were perform­ ed with the pole pieces separated 1-1/8 inches. No at­ tempt was made to measure either the magnitude of the magnetic field in the gap or the homogeneity of the field. Unfortunately, the magnetic field did not remain constant for a period of time longer than one minute, usually less; hence, the drift of the magnetic field was a constant source of trouble. Although the storage bat­ teries had a capacity of 400 ampere-hours, the provision of a steady magnetic field by current from them was inade­ quate for the experiment* The induction head Fig. 2 is a drawing of the induction head. The trans­ mitter coils A and B are orthogonal to the receiver coil R. Each transmitter coll contains three turns of No* l6 copper wire, and the receiver coil contains l6 turns of No, l8 copper wire. Since a pulse of intense radiofrequency energy is ap­ plied to the transmitter coils, one must take measures to prevent saturation of the radiofrequency receiver because of coupling between the receiver coil and the transmitter 10a CO *o CO "w o 3 < " o o rO in ro I —v w ( | i >3 in Q > to o CÜ m ro CTi i l HH m CP mcj <1 LT) ^A/sAA/V-VVVVVV^Ii lO CD in ro ro + 13 vibrator VI, one changed the repetition rate. A double- pole, double-throw switch made it possible to insert two different combinations of resistance and capacitance; and the potentiometer R1 allowed a continuous variation of the repetition rate over the range determined by the resistance-capacitance combination. The circuit supplied repetition rates from approximately 0.80 pulses/sec. to 0.085 pulses per second. For the relaxation time measurements the plate of V3 was connected capacitively to the grid of V7. Tube V7 amplified the pulse, and one obtained a positive pulse whose amplitude was approximately 120 volts at the output of the puiser. Application of the positive pulse to the grid of the 6l 6 in the radiofrequency oscillator caused the tube, which was normally biased beyond cutoff, to con­ duct; hence, the pulse gated the oscillator. Potentiome­ ter R2 allowed one to vary the width of the pulse. The ultrasonic studies required two pulses in suc­ cession; first a pulse approximately four seconds long and then approximately 300 milliseconds later a pulse of the order of 60 microseconds long. Both the duration of the pulses and their spacing were variable. The four- second pulse required long time constants in the circuits 14 of V3 and V4. Two microfarad capacitors supplied suffic­ ient capacitance. Variation of potentiometer R2 changed the duration of the first pulse, and variation of R3 chang­ ed the spacing between the first and second pulses. tiometer r4 Poten­ determined the width of the second pulse. By means of a relay the first pulse gated an oscilla­ tor whose output drove a quartz transducer. The winding of the relay coil replaced the resistor marked plate circuit of V3. in the One pole of the relay was connected to a source of approximately 400 volts dc; and the other pole of the relay was connected to ground. The moving contact of the relay was connected to the screen grid of the 807 tube in the ultrasonic oscillator (see Pig. 5). Application of the positive voltage to the screen grid gated the oscillator on, and connection of the screen grid to ground turned the oscillator off. In this manner the first pulse controlled the ultrasonic oscillator. Tube V6 supplied the second pulse. fied and inverted the output from V6, Tube V7 ampli­ As a result, one obtained from V7 a positive pulse with an amplitude of approximately 120 volts. The positive pulse gated the radiofrequency oscillator just as it did for the relaxa- 15 tion time measurements. During the ultrasonic experi­ ments, the repetition rate was constant at O .085 pulses/ sec, A regulated power supply provided the plate volta­ ges for the tubes in the puiser circuits. The receiver The receiver supplied the most difficult electronic problems. The experiments required the detection of a nuclear induction signal which followed an intense pulse of radiofrequency energy. Since it was impossible to avoid pickup of the radiofrequency pulse by the receiver coil, the possibility of receiver saturation was always present. However, saturation of the receiver for more than several microseconds after the radiofrequency pulse made it impos­ sible to detect nuclear induction in solids. Detection of the sodium induction signal demanded that the receiver have considerable gain with a good signal-to-noise ratio. In general one can say that the experiments required a high-gain receiver with a band pass of several megacycles. Pig. 4 is a circuit diagram of the receiver. It has four stages and a detector. Each stage is single-tuned by means of a slug-tuned coil. The 6a c 7 has large transcon­ ductance; and hence, provides a good mean stage gain for a 15a cr tro 16 band pass of several megacycles. Adjustment of the re­ ceiver for the best response to a nuclear induction sig­ nal determined the final tuning. Variable capacitor Cl together with the receiver coil formed a tuned input circuit for the receiver. Only the first stage was gain controlled, and the control was accomplished by varying the negative grid bias of the first stage. The circuit diagram shows two crystal diodes con­ nected to the grid lead of the first stage; their purpose was to prevent excessive saturation of the receiver. The diodes with proper bias bypassed to ground all positive and negative signals whose amplitudes exceeded seven micro­ volts . Consequently, the diodes reduced any pick up from the radiofrequency pulse; but they allowed the nuclear in­ duction signal to pass freely. The receiver was completely shielded, and it was con­ nected to an external ground through a heavy lead. A regu­ lated power supply provided the plate and screen voltages. The receiver performed satisfactorily and adequately for the experiments. One could obtain an induction signal from protons in glycerine with a signal-to-noise ratio of forty to one; and the signal-to-noise ratio for the indue- 17 tion signal from the sodium nuclei in sodium chloride was approximately 10:1, when the signals were displayed on a Tektronix type 531 oscilloscope. The ultrasonic oscillator A Hartley-connected oscillator supplied the driving voltage for the ultrasonic transducer, and Fig. 5 shows a circuit diagram for the ultrasonic oscillator. The re­ sistor in the screen-grid circuit made the oscillator res­ pond sharply to the gating pulse. A description of the gating device and the method of gating appears in the dis­ cussion of the puiser. In order to obtain a large output voltage, the pickup coil was wound concentrically about the inductor in the tank circuit. The output from the oscillator was connect­ ed through a coaxial cable to the terminals of the quartz transducer. A variable plate supply voltage for the 807 tube permitted one to change the output voltage of the os­ cillator. In turn, a Variac connected across the primary of the power transformer for the plate power supply made possible variation of the plate voltage for the oscillator As a result the peak voltage, measured with a diode volt­ meter, across the terminals of the transducer was continu­ ously variable from 10 to 55 volts. 17a CD O o N y § g > m LftûfiJ CD o 5 'Ô (/) O o 'c o 8 O LO CT» CO '^^W\— |i 5 Hi' O D 18 The diode voltmeter used to measure the transducer voltages consisted of a 6AL5 diode and a capacitor. A Simpson meter connected across the capacitor indicated peak voltage. The voltmeter was inserted in the coaxial line which led from the oscillator to the transducer by means of a coaxial "T" connector. The "T" was located approximately six inches from the induction head. CHAPTER III THEORY The Ultrasonic Transition Probability Consider a single crystal of sodium chloride placed in a constant magnetic field, We are interested in the interaction of the magnetic moment of a sodium nucleus with and the interaction of the sodium nuclear quadrupole moment with an electric field gradient produc­ ed by charges surrounding the sodium nucleus. We suppose that the Zeeman interaction is much stronger than the quadrupolar interaction; hence, we treat the quadrupolar interaction as a perturbation of the Zeeman energy. A sound wave in the crystal modulates the quadrupolar interaction, and the quadrupolar interaction supplies a coupling between a sodium nucleus and the crystal lattice. We choose a Cartesian coordinate system which is fixed in the crystal and whose z-axis coincides with the direction of the magnetic field, H^. What is the probability per unit time that the sound wave will produce a transition of the sodium nucleus between its Zeeman energy levels? 19 20 The electrostatic potential Let N be a sodium nucleus and charge. N by S be an external Suppose the sound wave displaces the nucleus, by S ^ ( s e e Pig. 6 ). and the charge Here is the nuclear charge density at a position from the center of mass of the nucleus. We assume that after a displacement of the nucleus the nuclear charge distribution remains unchanged with respect to the nu­ clear center of mass. static potential at We expand and ^ After displacement the electro­ due to the charge as a function of is andCo^ (S^; theterm which arisesfrom the nuclear quadrupole mo ­ ment is ^ Where Pg (Cd%, the Legendre polynomial of order two. ) is Now we have We write the Cartesian components of the vectors in Fig. 6 as follows ; Rt"'\ ( x T = C x f ) = C x r ). 1,1,3) H=.,i,3) 20a 2 CD CD CP o cr cd" CJ) o _c Ü ID c o COr CT D 02 o c 0) E CD CJ _D a. g Q o CO c g H— *cô “O o Û_ 10 Lu < c o 21 Co) S = U T ^ ‘T 1,i) Then or U where summation over the indices i and j is implied• From eq. (2) _ t y J y (f) (4%) c* Ajt c| . , Cr^"*0' The quadrupolar term of the electrostatic potential is now Summation over indices j and m gives Next we separate the coordinates % and we write the quantity in square brackets as i. ©•> 22 we find that Cr‘-’)V"'*] \ x[3X;xT-(mV'^''], % = y A c s i . l 3 x T ’x ^ ’- ( r ‘’">)V‘‘‘*] X %[ s x T x T - . (:) Each of the square brackets may be considered as the components of a tensor. The potential \/^ becomes where A and B are second-rank tensors. The multiplication indicated by the double dot we define as the product ob­ tained by multiplying the component B^*j^ of B by the compon­ ent A^j0 of A. Let us consider the tensor A; Aiif ^ - (r The tensor is symmetric; and there are five independent components, since it has zero trace. Each component is a homogeneous quadratic function of x, y, and z. For conveni­ ence we shall choose for a new basis the five linearly inde­ pendent functions where spherical harmonics of degree two. are the unnormalized Then the five indepen­ dent components, which are sufficient to define the tensor, become in Cartesian coordinates 23 Ao= \ (.3y - V') = y' CC.T».®) - )r (i*) 3 ^ U ± - i ^ ) = y^Pj'Cc<,»o>e"X y " y ^ ‘ = y'‘t^* Cc1 5& X “ *.y > ) ± ^ ( » Ï V î«>)]‘. We consider only processes which produce transitions of the central (the sodium) nucleus between nuclear magne­ tic energy levels determined by the quantum numbers ^ ±2. (the + z transitions). and Here the quantum number labels the allowed values of the nuclear spin operator I^. Experimentally, the angular frequency co of the sound wave I”*#1 to , must satisfy the condition We must calculate the matrix elements ( X Ana \ \ X Anrv± The only components of the nuclear quadrupole moment whose matrix elements connect the nuclear states m n and the Q±2. 5 hence, from eqs. (14) and (l6) Cl/vy>\H'\X 6 % * - ^ ttxf» j " ) X L X 0« U ^ l l ^X- /vv\l . ^ - C x f ’-J Ï ’ +( o . In) We shall use van Kranendonk’s model of a sodium chloride lattice and suppose that a sodium nucleus is surrounded by six equal point charges. We also assume that the axes of the Cartesian coordinate system coin­ cide with the cubic axes of the crystal. Then dinates of the six charges (the sodium nucleus is at the origin) are; the coor­ 29 o l-l 0(--| ^0^,0,o') U ,, (X T ' X T ' x 7 ’) = Co.-a,o) OC “ 3 Q^%-3 ( X ': ; x % x T ) - - co,o,c') (X T ' X T : x T > i o , o , _ . ^ Here "a" is the equilibrium lattice constant. Then for ^ - ^ , substitution of eq. (20 ) into eq. (19 ) gives ) L ~ " i r t ^ 1=0 We multiply eq. (21) by ^ •A»/ , the k-oomponent of the relative displacement, and sum over the index k; J»-* •*^‘r O If we retain only linear terms, we find C • v zco.— =1» ^ o ) ’G ^ o i ;J E - ^ • B«' Similarly, we find the linear terms for all the — — The results are : AO BA 30 . ^ c _ o o «3 " C23C) R!, The complex conjugate of any of eqs. (23a, b, c) equals the linear terms for the expansion of the corresponding -2zi • The presence of a sound wave Consider a unidirectional standing sound wave propa­ gated in the direction of the x-axis. The displacement of a lattice point is S = AcvCJxX-i^O,.u,Jt A ' Where is the amplitude of the sound wave, and trary phase factor. is an arbi­ For the central nuofeus we find - Aca«-J C.cra.uj'ifc C2.4) while for the charge g, S, =- s) Since the product «Aa.-_ 5 ^- - o i T = % r = s r = « s T = t r = s T " = - Substitution of the relative displacements In eqs. (23a, b, c) yields — AXct.,r*.;.^Xc*«w:JC Uoo.) CL^-0 R ‘ ~ "Ar A«Wcx Ok? UoV.') LU 8«. _6^_ C ”ioc) nX Ri R-l' Next we expand eq. (17a): Q w) - ^ \ ” l gW - c g o) \CX'-V/yv\+»>tI-/wx)C3^ I §|“ j ^ „t-0 Let 2«i" ^ ^ f») W) V for ail ex , where Y Is a constant and G is the electronic charge. Substitution of eqs. (30a, b, c) in the expanded form of eq. (l?a) determines the matrix element C ^ a v \ \ H lXAfv\*^z.^* 32 (Xory\jH'|X/W\*t-x)= - X X ^ C ^ ‘V/Yn'V\l(E"^wV) L 0 ^ c^^uuZt Y C 3 \) , then from eq. (17b) we If we recall that find the matrix element \W \ X ^ - X ^ = •“ %1 \/CE"^vv\-Vi)^X-V/vv\-l) Y ^ ^ (^X-^4^ %)(X 4^/vv^ ^"IT A ^ » - qa »v x 5 With I * , eqs. (31) and (32) become CX/>r\\ V\*\X ArVV-VO ~ ^^Arv\ \V\*\X/VV\—z) — = (.33) Transition probability per unit time Since the matrix elements of the perturbation are time dependent, we must use time-dependent perturbation theory to compute the probability per unit time of a transition. We expand the state ^ tem in terms of the eigenstates system. Let in the expansion. of the perturbed sys­ of the unperturbed be the coefficient of the eigenstate Then first-order perturbation theory shows that (see Schiff, Quantum Mechanics, p. 195)* 33 X(k). - CuJjt- ^ iO - --fr where w~- h - % ■: . - , is time independent and If o the first term of eq, (34) is negli­ gible compared with the second; and 2k“1^ — LU If the system is initially in the statenr^ and the perturba­ tion is turned on at time the probability of finding the system in the state JL , after a time is I lag ) Since the nuclear resonance line has a finite width, we express the probability that a nucleus has a Larmor precessional frequency between Lu and c u + j u u by the normalized line-shape function g Then the probability that the system makes a transition in the time X from the state to one of the states ^ is — GO The function “x is approximately a J — function, and g C w ) is zero outside a definite frequency 34 interval; hence, we write eq, (36 ) as The transition probability per unit time is We average over the frequency spread of the resonance line and find the average transition probability per unit V W H W i v ^ where fz/ is the line width in frequency. We have considered the case for o , This means that the perturbed system absorbs energy from the radiation field. When ((xA 4- uu energy by the perturbed system. )^o , we have emission of In this case the second term of eq. (34) is negligible in comparison with the first term. Calculation of XA^^yields the same result as eq. (37). If we substitute the value of the matrix element from eq. (33) into eq. (37), we find the probability per unit time for a A w = ± 2 . transition induced by ultrasonic energy. The result is 35 w - _L_ Finally, when we average 4* over all values of the arbitrary phase factor «T , we find The Nuclear Magnetization We shall compute the magnitude of the macroscopic magnetization of the sodium nuclear spin system and find the effect of ultrasonic energy on the magnetization. We consider the spin system in a steady magnetic field and in the presence of ultrasonic energy; however, there is no radiofrequency field present. First, we write the dif­ ferential equations which determine the populations of the nuclear spin states as a function of time. Solution of the equations for steady state conditions yield the number of sodium nuclei in each spin state, and the macroscopic nuclear magnetization follows immediately. The differential equations In order to write the differential equations we must take account of all processes whereby a sodium nucleus can make a transition from one spin level to another. Since 36 ultrasonic energy is present, all^/w\ = t Z , where w is the magnetic quantum number, transitions are allowed. The probability "W" of such a transition has been found in a previous section, and eq. (38 ) is the expression for "W", The spin system of the sodium nuclei is in thermal contact with the sodium chloride lattice. Transitions of the sodium nuclei between spin states strive to bring the spin system and the crystalline lattice into thermal equi­ librium. For the thermal processes both =± z transitions are allowed. be the probability of a % If we let and and and "Wg" - -Z transition, re­ spectively, the probabilities for the corresponding downward transitions ^ -v % and A . % are KT and As an approximation we expand the expo­ nentials and write and where N Lk' v^T/ Vs/ ^ J a l U ' ) ^ for for = transitions transitions, Wv. The expansion of the exponentials is valid, since Viv^cXT except at very low temperatures. 37 There is a third process which is responsible for re­ arranging nuclear spins among the spin states. Mutual spin-spin flipping of the sodium nuclei is the third pro­ cess, and its effect on the populations of the spin states was suggested by Abragam and Proctor^^. Each sodium nu­ cleus experiences a local magnetic field because of neigh­ boring sodium nuclei. We may suppose that the local magne­ tic field consists of two components. The first, a static component, arises from the dipolar interaction of the so­ dium nuclear moments. ting field. The second component is an oscilla­ Since the magnetic moment of each sodium nu­ cleus is processing in the constant external magnetic field, a neighboring sodium nucleus experiences an oscillatory field whose frequency is the Larmor processional frequency of a sodium nucleus in the external magnetic field. As a result, the oscillating field from nucleus "a" can flip the spin of nucleus "b". While the spins flip there is an ex­ change of energy between nuclei "a" and "b". The process re­ quires conservation of the energy of the spin system, and for mutual spin-spin flipping no energy leaks from the nu­ clear spin system to the crystalline lattice. Spin-spin flipping takes place only between identical nuclear species. 38 Fig. 7 illustrates spin-flipping for the magnetic energy levels of a sodium nucleus. Only ^ ^ 5 + % transitions are allowed, and the probabilities and for mutual spin-flipping are explained in the diagram. If we let be the instantaneous number of sodium nuclei in the energy state specified by the magnetic quan­ tum number , then the differential equations which des­ cribe the populations of energy levels as a function of time are: jh/J. r M. /lii. + /VLi.p) d A/-X Mj. W- r , „ M v (*+ 4 ^ * - A ^ ^ W x + yV6- /VL +A^F} ^ j _ A L ^ W + / V ^ - //-ty, - + A/L^w, C*4C) + A/^^^ • - 2 V . , , - W , A / . j ^ W - * - / V - t W, + + ^(|_W + w^ * 38a H E (y CJ ro (\J ro CJ \ + + OJ ro ro O II CL- I— I III .P OJ cn Ç CL CL OJ O I U_ .92 _o II 0_ $ II o _Q o A g c: CM Œ III O "CL CJ II (f) CO jf CM c Z3 o -< ■ O + O /I jQ CM ro cn + H E CM ro CM I CM CM ro + + _o II o 39 In these equations we have let Zl- . The terms in the square brackets express the spin-flipping processes, and they require a word of explanation. Consider the situation whereby a nuclear spin makes a transition from the state to - -L through mutual spin-flipping. Then from Fig. 7 we see that the spin-flipping involves proces­ ses b' and c^ . (We can ignore processes a, a' , and a'* , since they do not change the spin populations.) We shall restrict ourselves to process b* , and compute the change per unit time in the population number Ni. due to process b^ is proportional to . The change , the probability per unit time of the mutual spin flip it is proportional to the number of spins in the level and finally the process b' cannot occur if the state is not occupied. r^s-l^is r = t e ^ /y ^ where N is the total number of spins in the spin system. The factor Consequently, we w ^ i represents the probability that the level N occupied. Steady state equations For steady state conditions we equate the derivatives to zero, and we find the following equations: 40 4-ALt^$+6W^^ 4- N-i. (^W-V W, 4"Z =o (,39Cl) “*"A/-x^?-j^^- A/j^(^W4-2W,4 aw,^44. ^ W 4 W| 4 ^-2 - ^-a O- - A'-iïî ^ f 2/V-;tfî %+• + '^ ^ + A A ^ ( . W 4- w, )-A/_i ^ W 4 i W , 4-iAw, ) 44- V - ^ L^^4^W.)+ -+-3AW, -V w ) - o j) ^ In these four equations we have assumed that^ w, = w^. That is, the probabilities of a transition ^^vv\ = *\ ^/y a ^ - 2 because of thermal relaxation are equal. and We add a fifth equation which expresses the fact that the total number of spins must be constant: /VV+A/4 .4 ^ ^ a /-, a. C39e) - A/. ^ Simultaneous solution of any four of the five equations will yield the population numbers A/Xv»' The linear approximation We notice that the population numbers A ^ appear quadratically in eqs. (39a, b, c, d). To find a solution we shall approximate the quadratic equations by linear equations. 41 Consider, for example, the product ( A/% \ -1— . Ac^ X ' N cording to Ahragam and Proctor's suggestion, the mutual spin flips distribute the nuclear spins among the energy levels in such a way that the population numbers are determined by a Boltzmann distribution for a spin tempera­ ture , Therefore we write ■ f e We may treat all terms of the form t^f> similarly and find in general Using eq. (2), we find for eq. (la) -i5 + + 4j V-JL-)P^- -if 1 w,^ + ) 4- 4- V - ^ Q W 4 - w , 4 2.AW,) -=: o . We find similar expressions for eqs. (39a, b, c, d), and the linear approximation consists of the following equa­ tions : 42 - A / ^ ( P , 4 - P ^ 4 4'V s; 4 ' ^ W , ) 44- A/-JU 'V (^ 4 4 W % 4 4 C iW ,')4 ' P,4- 4 W 4- 4 W | 4» 9 T a W ,) — A/lj^P^^ o A / ^ l ^ ^ 4 - ^ 4 4 W , ^ - A - (me) - /V .^ C S f, 4-<^+ 4 W 4 t w ,4 - S-AW,) 4 - /V/-3^ (%P[4 P^4- 4 W ,4“ “ ^ O. f%4"4 W 4 4 NV*)4 A/Lj^(%f>4- P ^ 4W,^ — QP|4-P^4 9:w , 4 \i a W , 4 - 4 w ) ** O C4W) V s . 4- A4. 4- A L , 4 A /-X - A/ 'i We shall ^ C 4 ie ) ^ solve the system of equations formedby eqs. (4la, b, d, e). Solution of the linear approximation The determinant of the coefficients of the system of equations can be expemded to yield the following equation: Det. -1 ^ + C'W\w,-vs-w+ \î.«‘>'^,'>3 K X jj-SP; tH )(.w+ w , + - 4 (.w + \w,+6VJ,) - f* LSvv, + . As an approximation, we drop all terms which contain Ch i ) t as a factor and retain only terms which involve the proba- 43 bilities and quadratically. which contain Our neglecting terms or 9^ linearly is justified, since the spin flips take place in a time which is very short com­ pared with the time required for a thermal relaxation process, and W j W » Z.< . The approximations reduce the determinant to Using eq. (43) and the system of equations, eqs. (4la, b, d, e), we compute the population differences a. and ( - A/^^ . X * i. iV To find the population differences, we neglect terms which involve (a )^ but retain terms which contain ^ linearly. fkl \ For example, -A^CVôW,)Clol^4-t'W4ŸW| + lfc^W, — We also retain only terms which contain ally. Then ^ Is/ ^ - Since IT, s — w, A/ZlW, ÇS /y& H --------- P, or quadratic- ) C.Py4 >.Px )________ _ I (.44) / + 5 w, , we may express eq. (44) in terms of the spin-lattic relaxation time CWj.-A(L)ei^ L X x/ V / -hX 5 : t4s) w t ; 44 Similarly, we find the same result, eq. (45), for the popu­ lation differences (A(i. - /Ui.) X X and X ) ; con- sequently, we have CAIt- - Cw.- ^ The result, eq. (46), of the linear approximation pre­ dicts that in general the temperature of the spin system is higher than the lattice temperature. If the ultrasonic transition probability W is zero, the population differences equal ; but is the population difference for the spin system and the crystalline lattice in thermal equilib­ rium. The macroscopic magnetization The macroscopic magnetization of the nuclear spins is In the absence of ultrasonic radiation, w ■ =o ; and we find the nuclear magnetization Mo = % ^ ^ \ ^ ^ s ^ . cm) With the ultrasonic transition probability not zero, we have M ^ 3 — X y ---- i---- -t-J. / 4- X w t , ^ !----y 45 The ratio of eq. (47) and eq. I . Mo (48) yields (44) /-l-XvsJT, Eq. (49) predicts that for sufficiently large values of the ultrasonic transition probability W, we can reduce the nuclear magnetization to a small fraction of its equilibrium value M g * CHAPTER IV THE EXPERIMENTAL METHOD Measurement of the spln-lattlce relaxation time, Ti The spln-lattlce relaxation time, sometimes called the longitudinal relaxation time, was introduced phénoménologieally by Block 17 . Consider a system of nuclear spins, for example, a crystal, in a vanishingly small magnetic field. The nuclear Zeeman levels are degenerate. If we thrust the spin system into a strong magnetic field, the degeneracy of the Zeeman levels is removed, but the nuclear spins are nearly equally distributed among the Zeeman levels. After sufficient time, the Boltzmann law determines the distribu­ tion of the nuclear spins among the magnetic energy levels as the spin system and lattice approach thermal equilib­ rium. We may say that the initial temperature of the spin system in the magnetic field was greater than the lattice temperature, and its final temperature is the lattice tem­ perature. The spin lattice relaxation time T^f is a meas­ ure of the time required for the spin system to cool from a high temperature to the lattice temperature. 46 In the Intro- 47 duction we mentioned several proposed mechanisms which bring the spin system and the crystalline lattice into thermal contact. For a collection of nuclear spins each with spin X * one can show that the approach of the spin system to thermal equilibrium with the lattice is described by the equation; ^ CWo-w)= e^ (5o) where = number of excess nuclei in the lower energy state at thermal equilibrium /V^= number of excess nuclei in the lower energy state at time t = . o If the nuclear s p i n X > ^ , it is not always possible to in­ troduce a single spin-lattice relaxation time. Also a sing­ le relaxation time may not be valid if more than one process is effective in bringing the spin system to thermal equilib­ rium with the lattice. Although for sodium X - ^ , the quan­ tity which we determine experimentally is a measure of the effectiveness of the relaxation mechanism or mechanisms; and we shall call it the spin-lattice relaxation time, T^. Pulse techniques provide a convenient means for measur­ ing . In order to understand the method, we regard the 48 net magnetization of the spin system as a macroscopic magnetic moment. Let M o he the magnitude of the macro­ scopic moment for conditions of thermal equilibrium. Con­ sider a sample of sodium chloride placed in the coil R of Fig. 2, Introduce a coordinate system with its x-axis co­ incident with the oscillator coils A and B; its y-axis co­ incident with the receiver coil R; and its positive z-axis coincident with the direction of the steady magnetic field . When the spin system and crystalline lattice are in thermal equilibrium, M© is parallel to VA© , We subject the sodium nuclei to a pulse of radiofrequency en­ ergy by connecting the coils A and B to a gated oscillator tuned to the proper frequency. If the oscillator is tuned to the Larmor frequency of precession of about VA© , the macroscopic moment M © will precess toward the xy-plane through a polar angle ^ : e= v-w, where gyromagnetic ratio for sodium intensity of the radiofrequency magnetic field durâtion of the pulse of radiofrequency energy. It is convenient to choose VA, and make e the pulse ^ in a way which will (a 90^ -pulse); consequently, immediately after M © lies in the xy-plane. field no longer present, M q With the radiofrequency precesses in the xy-plane and 49 induces a voltage in the receiver coil R, The magnitude of the induced voltage is proportional to M © • After amplifi­ cation one observes on an oscilloscope screen the nuclear induction signal which follows the pulse. If after application of a 90*^ -pulse we apply a second 90^ -pulse at a time t o 7 5 3 0 2 3 4 5 6 7 8 9 10 II 12 13 14 Time Interval Between Pulses (sec.) Fig.B Measurement of Relaxation Time. 56 the ultrasound in the sodium chloride, then ~ where 9 = density of NaCl w = angular frequency of the ultrasound We compute the energy density £ , by finding the power dissipated in the sodium chloride. The quartz transducer and the transducer loaded with the sodium chloride were re­ placed by equivalent electrical circuits^^"^ The equiv­ alent circuit together with the values of the electrical com­ ponents are shown in Fig. 9« The capacitance C © is the capacitance of the X-cut quartz, plated on opposite faces, and treated as a parallel plate capacitor. The frequency for operating the transducer was chosen from Fig. 10 and Fig. 11, and the value chosen was = 10.2 mc/sec. From Fig. 10 and the reactive part of the impedance of the quartz one can determine the effective resistance of the unloaded quartz. From Fig. 11 and the re­ active part of the impedance of the loaded quartz one can de ­ termine the effective resistance of the loaded quartz. The small inductance in series with resistance was necessary to maintain the capacitance constant. It is attributed to the difficulty in reading small changes in the reactive part % of an impedance from the radiofrequency bridge. If V is the rms voltage across the quartz, one sees 56a ■a T5 is the effective resistance of the sodium chloride, and R t is the total effective resistance of the loaded quartz. 9y Prom Fig. ZioXOJl Hence, ^ \/ vvncttv\^ ■X) - ^ OwC‘ - S X *0 owx ^ CL= wave length of the ultrasonic waves in sodium chloride equilibrium lattice spacing for sodium chloride From eq. (52) and eq. (53), we find where the density of the sodium chloride is ijle ^yv\/c c - The width of the nuclear resonance line is given where is the spin-spin relaxation time. Me estimate from the length of decay of the nuclear induction signal. Experimentally, we estimate 3 Hence, A \o Æfl-o"* . CST) Substitution of the numerical values for the constants, eq. (56 ), and eq. (57) into eq. (38 ) yields w = 13 9) C \o‘‘^) . C59) 59 Fig. 12 is an experimental plot of the ratio A//\o as a function of the square of the rms voltage across the transducer. Here A is the amplitude of the induction signal with ultrasonic attenuation, and is the signal amplitude in the absence of ultrasonic attenuation. (49) is the theoretical expression for If we arbitrarily set A / A o - 0.384. where With A/Ao* = % , then from eq. (49) we find Fig. 12 shows that when A y ^ o - o .384, the =- 800 volts. voltage V Eq. Hence, if we write is a proportionality factor, we find JU- X lo" ^ , eq. (49) becomes A ______ Ao L / -4- ^ O'iSXio Fig. 13 is a plot of the theoretical curve, eq. (59). Since 800 volts, we can use the ex­ vsJT, - \ and perimental value of mine the parameter V . = 7.15 sec. and eq. (58) to deter­ We find . 59a o c g> CO c _o %— o 3 “O c “O o Z3 CT CO o o 3 c 0) (/) E lO (D O CP Ü ‘c o (/) o c (U ro E cz OJ CL X LU g Li- LO O — Ow s|Du6!S ^ O P oipy apn^jiduuv 59b a c O' CO CM lO > CM 73 2 O 3 00 m cr LO 'U) c .9 o 3 T3 C E QJ o> E m .o o 3 C < “5 u CD ro o O) ro O' in O in — Ô S|du5!S P Ojioy 9pn;!|duuv CHAPTEJK VI DISCUSSION The. _spin-lattice relaxation time The experimental value value “Ti-'I.S = agrees with the found by Proctor and Robinson^^. The writer is unaware of other determinations of Tj for so­ dium nuclei in sodium chloride. The parameter Y The magnitude of IT- calculated in this thesis is one-half the value found by Proctor and Robinson^^. The value stated in reference (lU) is incorrect, and in a private communication to the writer Proctor and Robinson find charge model, a V • of On the basis of the six pointindicates that a sodium nu­ cleus sees an effective charge slightly less or, within the limits of accuracy of the experiment, equal to the charge of a single electron on each of its six nearest neighbors. One should not expect to find a V<^looo as suggested by van Kranendonk^, since his calculation treats only indirect processes; but the treatment in this thesis is concerned with direct processes. Comparison of Fig.12 and Fig .13 shows that the experimental results and theory 60 61 agree quite well 5 however, the experimental curve does not show sufficient attenuation for large voltages. The effect is due to inhomogeneities in the external magnetic field. It is difficult to determine the amplitude of the displacement of an ion in the presence of a sound wave. Both this experiment and Proctor and Robinson^^ use the phonon relaxation time and the ultrasonic energy density in the sodium chloride to find the displacement of an ion. The computation of V used the value of the phonon relaxation time measured by Proctor and Robinson for their experimental conditions. It is possible that a different experimental condition would yield a different phonon re­ laxation time. The bridge measurements indicated that energy losses in the quartz were small; hence, energy in­ put to the composite oscillator (quartz and sodium chlor­ ide combination) was quite effective in producing nuclear transitions ultrasonically. One may question the validi­ ty of the use of the radiofrequency bridge for measuring the losses in the sodium chloride. The sodium chloride crystal does not cover the entire face of the transducer; hence, the quartz is also radiating into the air. There­ fore, all of the energy losses indicated by the bridge were not dissipated in the sodium chloride. The equiva­ lent circuit which was used neglected losses in the so­ dium silicate-talc cement; however, these losses should be 62 small. In general the use of a phonon relaxation time and equivalent circuit to compute the displacement of an ion in the sodium chloride is unsatisfactory. It would be desirable to measure the ionic displacements directly, since the weakest point in this experiment was the compu­ tation of the displacement. The values of do not agree. has found Y found by different experimenters In his most recent experiment Proctor^i . Proctor's experiment employed a Q- meter measurement and the phonon relaxation time to deter­ mine the amplitude of the sound wave in the sodium chlor­ ide* Until experimental methods are refined and one can make reliable calculations of Y it is without point to speculate about the significance of the parameter V and the correctness of the six point-charge model of the so­ dium chloride lattice. APPENDIX Theory of Pulsed Nuclear Induction VJe consider a system of nuclear spins placed in a constant magnetic field H q . At time « o we apply a pulsed radiofrequency field of magnitude 2Hj and frequen­ cy cv . The time duration of the field is , and its direction is at right angles to the steady field -a» Hq . During the time the Hamiltonian for the system of nuclear spins is HLet the steady magnetic field Hq foe directed along the zaxis of a Cartesian coordinate system and the field directed along the x-axis. where V Then eq.(l) becomes is the gyromagnetic ratio of the nuclear species which we are considering, and are the operators for the z- and x- components, respectively, of the nu­ clear spin. For the total wave function ^ we must solve the Schrodinger equation Let the nuclei have a spin I = wave function V* : 63 . We expand the 6^ where Substitution of eqs,(2) and (h) into eq,(3) yields -in«,c^wS; e ~ “ “- ='^*éc-«)&.e'>-* If we use the relationship x *-*- (l + x ^ " Z ^ “h where = (I"y 4and X_= cs) we may write eq.(5) as = - If^ <^~ c . w:X - /w% The non-zero matrix elements of the operators g^p0 Q/W\ \ /WL+ (,/W^ \X^,\'VVN— %. X+ ((.) and X _ — \ / 4 v w > + 1) — \/CX4-/yv\^(^-/VwV*0 • Using these matrix elements and the orthonormality of the wave functions • C /vv%— XUy»• , we find from eq.(6) p. —X^Wvv,-7 € /W\—I -i-C Wvvt+7 +( / mï X | / w \ l r i ) C , 6 /VW-I-I where oj,-=V>A,. J ) X C “ l ) If we let and require that the spin system absorbs energy from the 65 radiofrequency field, then we have (Jjo- a v o ) = o ; and eq. (7) yields the following differentia ficients C. . equations for the coef- ; C%a'V C jl vJ3 - ^ CYL') Cÿ c ') j_ 1. û.,h In eq. (8) we replace by the operator O , and we aX have a set of four linear homogeneous equations for the If the equations have a non-trivial solution, the deter­ minant formed from the doefficients of the C/w^ must vanish: D O .w .f D o o o - auJj o O O D Expansion of eq. (9) yields with the roots O - "i Lu^ 3 ^ D =* T For llOQ.) C»ob> 0='X'^» we find from eq. (8) that OO Since we may write A^6 66 where A + is an arbitrary constant, eq,(ll) determines in terms of the constant A + . We treat eq. (8) similarly for each root of eq. (10). The complete solu­ tion of eq.(8) will express each as a linear combi­ nation of four terms: one term for each root of eq.(lO), We find, therefore, A.e" A.e " 6+ e" b. e " u,.) ;3w. ^ Cub) C - i = - W 6 C->=-A,e , e cuo e_e" o»d) \/e must determine the four arbitrary constants A+^ A-, B w , in eq.(12) from the condition of the spin system at time -o . Suppose the system has been in the steady magnetic field prior to application of the pulse long enough for the spins to reach thermal equilib­ rium. We are interested in the number of excess spins in each energy level relative to the level . The en­ ergies of the magnetic energy levels are At thermal equilibrium we assume a Boltzmann distribution of the spins among the energy levels. spins in each level are Then the excess 67 From e q . (13) we find that the probabilities for excess spins in the four levels are: p. vp- o ) P - v Ù i \ = i > P\= t Consequently, at time ^-o the coefficients <■“»> must satisfy the conditions I =O I C j L C o ' iI=^ ic-xcoir^-^ ic^coii^= t . (is) A solution of eq. (15) is c.^to^=o where = > and cT C x C o > ’=-;^e-'°‘ ^ C;^CiAare arbitrary phase factors. Using eq.(12) and eq.(l6), we obtain ReCAi-+A-4-B^'^B-')= J Cna> 'àv« %.(- A^+A-4r e ^ - B_ ^ - o %,l-Ax+A_+6^-6_) end) 68 Let R&. X A_? ^ ^ u ^ tr, Substitution of eq. (18) into eq. (17) yields the system of equations ; Lv-v vr - ^ Q ^ S X 'à I 77= “■ — z. 4- >jy Lk— o x vr = TP cer%,ol U3 vTi V3 """X “ -^-''JTvA.-bvJ3ur = - ^ cBif we replace "cosine" by "sine". arbitrary constants A ^ 1 i k 6^.= t C e ^ + ^ B _ = % - 1- 6 ""+ A_ ^ follow from eq. (20) Our solution for the ^and - & "s B- is: ) cii^) 3)= e'^b Uid) -h 69 For time X tion is , the solution of the Schrodinger equa­ ^ = ^ 0 L (b ^W\ 1 F The phase or q . ^ C _ U w ^ is the angle through which a spin in the state w \ processes in the time wave function for time ^w- • Finally, the becomes The magnitude at any time of the nuclear signal induced in a coil with its axis parallel to the x-axis is proportional to the x-component nuclear magnetic moment. -^x y oT the resultant Since V i ; , we must compute the expectation value of the nuclear spin operator . We have (V'*’/rj(/<^) = i C v ' V x ^ + x _ / V ' ) . Using the wave functions of eq.(22), we find that i<4') = vTI 1 * C^--e-^)+ X » ( ^ i . - O i _ i , ")(..;<■- - t u r ) ■*■ 1- % IrvtAvJ -HCiC a. _ xÏ4 . -\TTCt ^ C jx_Jh J where % (zi) 70 Also (V'*’/ X./1/.) = .XAw u cT C.X+ i.+\JTcicJ . ü s) I. ^ ^1» X X We can compute the quantities in the square brackets of eq. (24-) and eq. (25) by using eq. (21) and eq. (12). eq. (12) we must set .) (In Computation shows that any term in the expansion of eq.(2h) or eq.(25) which contains a function of the phase factors ^ , and cf as a factor vanishes when one averages the functions over all possible values of the phase factors. Consequently, one finds that CV'*I1^1 V ) - Then -e It follows from eq. (26a) and eq.(26b) that ^ ' y — ^^ /Q-^-vx ^uj " * ^ BIBLIOGRAPHE (1) I. Waller, Zeits. f. Physik 22» 370 (1932) (2 ) N. Bloembergen, Physica 1^, 386 (19^9) (3) B.V. Rollin and J. Hatton, Phys. Rev. 2!±5 3^6 (19*+8) (^) R.V. Pound, Phys. Rev. 22) 685 (1950) (5) J . van Kranendonk, Physica 20, 78 I (195^) (6 ) R. Sternheimer, Phys. Rev. 8J+, 2^-^ (I95 I) (7) H.M. Foley, R. Sternheimer, and D. Tycko, Phys. Rev. 22) 73^ (195 ^) (8) G.D. Watkins, Thesis, Harvard University (1952), Unpublished (9) G.A. Altshuler, JETP 28, 38 (1955) (10) G.A, Altshuler, JETP 2 8 , h9 (1955) (11) W.G. Proctor and W.H. Tanttila, Phys. Rev. 101, 1757 (1956) (12) M. Bloom, E.L. Hahn, and B. 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