AN ULTRASONIC EXPERNENT ON SOD-WM CHLORIDE Thesis gor ”12 Degree o§ M. S. R’EECEEEQAR SETS UNE‘é'EESETY Donald Alfred Jennings 1958 C.I IIIIIIIII III I IIII III II II IIIIIIIIIIIIIIIIII IIIIIIIIII III 3 1293 0168 67 5860 II E L I B R A R Y Michigan State University IIII AN ULTRASONIC EXPERIMENT ON 301mm CHLORIDE by Donald Alfred Jennings A THESIS Submitted to the College of Science and Arte of Michigan State university of Agriculture and Applied Science in partial fulfillment of the requirements for the degree or MASTER OF SCIENCE Department of Physics 1958 ACKNOWLEDGMENT I wish to thank Dr. H. H. Tanttila for suggesting the experiment which this thesis discusses. 1 am grateful for his aid and suggestions concerning the techniques of pulsed nuclear induction. I wish to thankimessrs. Charles Kingston and Richard Hoskins of the machine snap and Mr. Ernest Brandt of the electronic shOp for their fabrication and repair of exper- imental ecuipment. I am.grateful for financial aid from Michigan State University in the form.of a Graduate Teaching Assistant- ship. I thank the National Science Foundation for making this experiment financially possible. ABSTRACT When a sodium.chloride crystal is placed in a magnet- ic field Ho, the sodiuminuclei, having a spin 3/2, find their energy level split into four sub-levels. The sep- aration, in frequency, between neighboring levels is 3'Hol'h where (fie the gyromagnetic ratio. Ultrasonic energy has been used to make Am::t2 transitions between these levels. These transitions tend to equalize the pop- ulations of the four levels. The effectiveness of the ul- trasonic energy in producing amztz transitions has been measured by using pulsed nuclear induction. This method determined the population difference between adjacent levels during the ultrasonic excitation. The effectiveness of the ultrasonic energy in causing transitions has been measured with longitudinal waves prep- agated in the [1061 , the [110] , and the [11]] crystal di- rections. The eXperimental results are compared with the- oretical predictions. The results indicate that the ionic quadrupole polarization is independent of the direction of the ultrasonic waves. TABLE OF C ONTENTS mmODUCflON OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOCOOO‘O. 1 WHYO..OOOOOOOOOOOOOOOO...0......COCOOOOOOOOOOOOOOO 3 EQUIPMENT AND EHERIMENTAL TECHNIQUES ................ 12 mERImTAL EMTS OOOOOOOOOOO0.00000000000000000000 15 DIS'CUSSIONO0OO0.0...0.0.0....OOOOOOOOOOOOOOOOOOOOO... 23 REFERENCES OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO... 2‘ INTRODUCTION The spin-lattice relaxation between the spin levels of the sodium.nucleus in a magnetic field is brought about by the interaction of the quadrupole moment of the nucleus and the time varying electric field gradient at the site of the nucleus. The time varying electric field gradient is generated by the thermal lattice vibrations which bring about relative motion between the ions in the crystal. One may modulate the time varying electric field gradient with ultrasonic energy of such frequency as to cause 4m:t or Amztz transitions, where m.is the magnetic quantum number. Proctor and Robinson1 were the first to produce am=23 transitions between the sodium.spin levels in sodium.chlor- ide. Later more work of this nature was done by Kraus and also Jendrasiaka. Krausz has written a theory for the ultrasonic tran- sition probability. He uses van Kranendonltzs'4 model of the lattice; a sodium atom surrounded by point charges, b’ e, situated at the neighboring lattice points. There has been some question as to whether the X appearing in van Kranend- Oaks" and Kraus'z model possesses tensor preperties. By comparing the effectiveness of the ultrasonic energy in Producing transitions in each of the three crystal direc- 2 tions one can, with the aid of Kraus' theory, establish, within experimental limits, three components of the 3’ tensor. Since the ultrasonic energy density in the expression for the ultrasonic transition probability derived by Kraus is difficult to measure,the ratios of the three components of the 5' tensor: to each other are determined rather than their absolute values. THEORY When the sodium.chloride crystal is placed in a mag- netic field the energy level of the sodium spin 3/2 nucleus is split into four Zeeman levels as shown below: n1: - 3/2 m:: - 1/2 “ m: +l/2 m:+3/2 FIGURE 1 m is the magnetic quantum number and may take any of the (21-1-1) values in the series I, I- l, I- 2, ..., -(I~l), -I. The nuclei, at room temperature, assume a Boltzman distribution and the number in each level is given by; Rm = c exp (-Bn/KT) where 3m is the energy of the (m) level, K is Boltzman‘s (sonatant, c is a constant and T is the absolute temperature Of the sodium chloride crystal. we are interested in the pepulation difference, 4° , between two adjacent levels. The energy of a given level 1E3; Em : - 8 PoHom where no: edi/ZMc and M is the mass of the proton and c the velocity of light and g zp/pol. Let us find A. between 111 =2 3/2 and m = 1/2, Ng/a __ C eXp (g pofia3/KT2) fly; 0 exp (6 polio/KR) = exp (spam/m) but g :p/p¢I and here I t: 3/2 expanding and we see that (113/2: N71} 2 N71 2m./3KT If this calculation is extended to the other levels we find that the pOpulation difference between any two adjacent levels is equal to Nyazpadsx'r. The energy difference between two adjacent levels is; Ra"... El/a “-3 (“SNoHO3/21 "' (“’3 None/2) “SI-leHe The selection rule for governing magnetic dipole transitions between levels is 11m;zl. Therefore a quantum of energy can excite transitions between the energy levels if it has the same magnitude as the level Spacing; her' :: g/poho whereiwé is the frequency or the electromagnetic radiation supplying the quanta of energy. if a magnetic dipOle)u is placed in a magnetic field no the dipole precesses about the direction of the applied field. The rate of precession is the well Known Larmor angular frequency; ()3 :: [H0211 where J is the gyromagnetic ratio of the dipole. This same effect takes place with the nuclear magnetic moment #1 . Since 3: [ll/I h we see that this frequency is Just; A6 : 2p a./an -.- gp.H./ 1: ln general the nuclear magnetic moments are precess- ing at random phase with respect to each other. lf a 11.1". magnetic field H, is applied perpendicular to 110 the assem— blxage of u's can be made to rotate in phase and by placing a (>011 around the sample with the 0011's axis perpendicular tCi both H, and Ho a small veltage will be induced in the coil. The magnitude of the induced voltage will be direct- ly Proportional to A , the population difference of the levels. There would be 0 v0ltage if the levels were equal in population. This method of detecting pOpulation differ- ences is called Pulsed Nuclear Inductions. In short the method is this: The sample is placed inside a detecting coil whose axis is perpendicular to the axis of the h, - transmitting c011 and then this combination inserted into Ho such that the three components; the detecting coil, 3,, and h, are mutually perpendicular. The R.F. magnetic field is turned on for a short time and the voltage induced by the precessing nuclear spins is detected, amplified and observed on an oscilloscOpe. We are concerned with the mechanisms by which the pep- ulation of the levels are equalized. There are three. {1) The R.F. pulse causing 4m: 9: 1 transitions, \2) spin-flip process, and (3) ultrasonic energy causing Am =t2 transi- ‘tions. The frequency of the R.F. pulse is equal to the freq- tiency difference between adjacent Zeemsn levels. Since the lJower energy levels are more populated than the upper levels there will be a net upward transfer of nuclei as a result 01' the R.F. excitation. Once a nucleus reaches a higher state it will remain there a finite length of time. Math- ematically this can be written; A -.: AJ l-exp- t/T.) 'here A is the instantaneous pOpulation difference at a time t following the R.F. pulse. T, is the spin lattice relaxation time and is the time required for all but l/e of the nuclei to return to equilibrium. 4° is the thermal equilibrium pOpulation difference. The spin-flip goes about in this manner: A nucleus in the m. 3/2 state may go to the m l/2 state with e si- multaneous transition of any one of the following, -3/2-4’ -l/2, -l/2 -’l/2, and l/2 -93/2. Since this process takes place via a magnetic dipolar interaction between spin systems and doesn't depend on the lattice vibrations it con- serves energy of the spin system. There is therefore no gain in the macroscopic magnetization but merely a redis- tribution of that magnetization in the spin system. Ultrasonic energy applied to the NaCl crystal at a frequency corresponding to the frequency difference between levels separated by 2 in the magnetic quantum number causes 'transitions between these levels. These transitions come about by means of the interaction of the quadrupole moment taf’the sodium.nucleus and the time varying electric field gradient at the site of the nucleus. The electric field gradient is modulated by the application of ultrasound to the crystal. This interaction causes a pOpulation equal- ization of the levels. by the application of enough ultra- souxui one can essentially equalize the p0pulation of the levels, ICraus5, taking these three mechanisms into account, arrives at this equation; a (Ag/A) = l + a 'm tam/5 where A. is the amplitude of the free induction signal with no ultrasound, A is the amplitude at a certain ultrasonic energy density and "Eel is the ultrasonic transition prob- ability corresponding to the amplitude A. The subscript [0"] denotes a dependence on the crystal direction. a~ a. w 8" Q3 (can DC“) 33KB. L7] - 192 £1“ a" erf' where e is the electronic charge, Q is the quadrupole moms ant, B is the amplitude of the ultrasonic wave, K is the propagation constant, a is the lattice constant, aefis the ultrasonic line width in frequency and Dcéjis the contri- bution to the quadrupolar interaction energy due to the position of the Xe's and is equal to; (8‘)“ for a single 6 e where the X's are the coordinates of the 3 3e Now Ba: 81ty, F is the density of NaCl, and w is the ultrasonic (Fifi-(x; iXaI[-—1'§" (x,t 1x9) 12+ “Saki-15W] I1) 22/9 at)" where E is the ultrasonic energy den- aHauler frequency. The ultrasonic energy density is given by; E :. P'Tkl V‘ where P is the ultrasonic power delivered to the crystal, T? is the phonon relaxation time and V' is the volume of the sample. The power is; P N valz : k va/z where V'is the voltage across the quartz, Z is the imped- ance of the NaCl rod and equal to f°c[}]S. S is the area of contact between the quartz and NaCl, and cEQIis the speed of sound in NaCl. Therefore P : kVa/r’ccqs Substituting we have i a 1 1 a O Q XEQ‘JDLQj Kfi‘ '1'”:qu 35 fi‘a“ {Jr‘s (3ch v0 wot (A,/l)3k = 1+ Experimentally we can plot a a (A./A) : l + Kqu By comparision 3 a K e" I.)‘1 (m Din] K‘k rm] [“3 35 haa‘ 5;; ("S °t~] V' 00* 10 Then a 2 Kt“)- X?) D Ex] 0 Lb] T. [1qu (2\ Kit] Jr. b] DU] ° Lori T'U] The ratios that are compared are K [10% K too} K [111] '___‘———'J ———————- and . Kind" Kala Ktnol in order to detect the presence of any anisotropy in the 5m . 1}. manna N .0: maéaamd ho amedHn moonm gag r ”doom / _ .933 .a .m I Omahm_IIJ /\ snug“ |||+II 12 EQUIPMENT AND EXPERIMENTAL TECHNIQUES A block diagram of the equipment is shown in figure 2 and with the exception of the receiver is the same as used by Irene and Jendrasiak. The essential parts of the receiver are, a tuned input consisting of the detector coil and a variable capacitor in parallel followed by three slug tuned stages of n.F. amplification. The signal is then rectified by a diode with the large n.F. pulse cut off by an lN34 crystal diode as shown in Figure 3. From last RF one stage :1- To scOpe r-L 11:34 ‘L ,J_ a ‘1! ‘fi. 5 volt 3" Figure 3 {The large magnetic field Ho was made as homogeneous “5 Possible by using proton resonance. The method is as fOIlovvs: The proton resonance is detected with a marginal oscillator. Then the pole faces are adjusted such that the Proton resonance does not change appreciably in frequency 0'31‘ the area of the pole faces. IPhe sodium chloride crystal was 1 7/8 inches long and 13 1/2 inches in diameter. The ultrasound in the NaCl crystal was generated by a 10 lie X—cut quartz crystal coated with silver paint and glued with Duco cement to the end of the NaCl sample. A preliminary experiment showed that if the NaCl sample was used with the ends parallel to each other and perpendicular to its length the ultrasonic impedance would change by a factor of 4 over a range of 20 K07. When one end was made approximately 3° out of parallel with the other end the ultrasonic impedance was constant. The RJ'. pulse was at a frequency of 5.07 Mo and the frequency of the ultrasound was 10.14 up which was well within the tuning range of the quartz. Iith the preliminary arrangements taken care of and the equipment adjusted for optimum performance the measure- ments were taken as follows: The ultrasound was left on continuously while the measurements were being made. The 3.1". field, H, was puls- ed on for a period of cup sec. every 3 seconds. The ampl- itude of the free induction signal was observed for differ- ent voltages on the quartz crystal. an A5 signal, the 3181181 with no ultrasound present, was observed at the be- ginning and at the end of a series of readings to make sure that the attenuation was caused by the ultrasound. The measurement of T, for each crystal was made with the pulsed nuclear induction apparatus. To make this mea- Bur(fluent a 90°-pulse was used. This is an R.F. pulse of 14 the shortest length such that when repeated in a time t such that t «T. the free nuclear induction signal, A, is O. A Ooipulse for the sodium nucleus is 140 )1 sec. long with 328 peak volts of R.F. with the transmitter coil used. We note that A : 50(1 ’ 61p 't/T.) ln (Ac-Also) = - t/r. where t is the time between pulses. Then from a plot of ln (Ao’A/Ao) against t on semi-log paper one can obtain '1‘. from the sIOpe of the line. The ultrasonic line widths were measured for each crystal direction to insure that they were invariant. These measurements were made by observing the amplitude of the signal A as a function of the ultrasonic frequency at con- stant ultrasonic voltage. The characteristic line width was then determined from a graphic plot of A vs. frequency. 15 EXPERIMENTAL RESULTS The measurement of T, for the three sodium chloride crystals were found to be; TIQOQ :: 8.1 t .5 seconds TODICJ :: 9.? 1" .5 " TIL—.111] :: 11.0 I .5 " Figures 4, 5, and 6 are examples of the type of curve from which T. was obtained. It was noted that not only did T, change as one changed the direction of the crystal but also it changed slightly as one rotated the crystal about its axis. Since the latter effect had only slight variations the average of the T, measurements for each crystal was taken. Measurement of the ultrasonic line width for the three crystals showed invariance with respect to the crystal dir- ection and a value of 5.30: .30 lie was obtained for the three directions. The speed of sound for the three directions of Noel '88 obtained from Kittele. They are c [100] :- 4.72 x 105 cm/sec ”:an : 4.65x lo5 " OBI-lg 2 ‘e37 x 10: ' calculation of the Die)" from equation (1) gives; D [100] 3 18.00, Dana: 12.70, and 1)ng = 14.12. Thea. values were calculated from the nearest and next 16 meow .a. so 92%ng « .eE 828% E mange Ego a... a e. n a w a o 1 _ . - - 1 ; com in u .9 mood Sex 5 83833qu a. l “moon .4 4nd 1? 63.9 so assumes: a .eE mnzoomm zH mam-HE Em go. «H cu m o v m 4 fl q J 4 3 one So u a. mafia Sea 5 soaaoaaaouon .9 a In OQI‘Q 4 seq 18 may .8 5ng a 62 228% E mange g SE. 0 r. a. n . N A A . _ — a . _ one at: H .9 m2“— .Soz as aoaaoeaaaeuoo a. 19 nearest ions. The theoretical ratios calculated from equation (2) are; a ____K£1°<fl 1. (LEM I. KLné) {[110] ——[——K 1oq : 1.10 ( 5ro2 )3; .05 K £111 2{[111] \ 9; m— "’ 1.50 ( ‘Ellfl ) + .08 K[11q 5&10] '- Figures 7, 8, and 9 show a plot or (Ac/A) against the square or the quartz voltage for each crystal direction. The slape of the line, K3], was found to he; Kama -.—_— 10.5/1000 t 20% I‘D-la :: 6.7 /1000 1’ 20% Ktllfl : 10.0/1000 i 2075 and the ratios of these K values are; K E _’ 1.56 + 026 K [110] i M — 1.03 Kflll'J LEA. "' 1049 Kind) H .20 H .26 20 Goa a Egg; 1. 6E B553 @453 2.5% ooon coon oooa _ A 82%.“: u a r mood..— Soz . . 4 r OH om mm Wags mo magmas: m 52 55:3 ”845$ $543 21 ooon ooom 8.2 a fi 4 1 fi 83st n... u r 4. 4. m5... Hooz G 1 n- is _ a OH om mm 22 Had a 1-? so azmsmmpmoms o .on @553 mambo» ~58 ooon ooom oooa q fl oooQoa H M T. .4. wom— 8% J... r' 4 I . I O OH on ma 23 DISCUSSION The experimental results show that 5 is isotropic within 20% since the ratios, 5’on ,_. ”[1003 .. Z‘Eniu = 1 5 £116] ”[111] - 2‘ Lina . This would indicate that K'is a scalar within the experi- mental accuracy. One possible source or error which was not taken into account was the phonon relaxation time, Tp, which was as- sumed the same in all directions. That the phonon relax- ation time is the same in all directions is not at all obvious. However, phonon relaxation times are not easy to measure and no attempt was made to measure them. A small error enters in when one pulses the R.F. at-a rate (3 sec.) faster than T.. TheAo is not the true AO' reading but since one is dealing in ratios, Ao/A, the errer is small and compensating. The results or this experiment support rather well the theory or the interaction or the sodium nuclei and the lat- tice of sodium chloride given by Kreus. l. 2. 3. 4. 5. 6. 7. 24 REFERENCES W. A. Proctor and w. G. Robinson, Phys. Rev. $93, 1344 {1956) O. Kraus, Thesis, \M.8.U.) G. L. Jendrasiak, Thesis, (M.S.U.) J. Van Kranendonk, Physics 39, 781 (1954) I. Bloom, 3. L. Hahn and B. Herzog, Phys. Rev. 21, 1699 (1955) O. Kraus and W. H. Tanttila, Phys. Rev. $93, (1958) D. A. Jennings and W. H. Tanttila, Acoust. Soc. Am §_0_, (1958) C. Kittel, (John wiley'& Sons, Inc., New York, 1953) I. H. Tanttila, Private communication 1|HIlll|1||1||||1||1111111111111111111HI 31293016587