GLTRASZGNICALLY ENWJCEU FRANSINONS OF SQEMM EVUQEE 3N S‘Ofi‘éifiv‘: 6%LGREE-E Thesis gar We Degree 0? M. S. MICHEGAN STé'EE UNWERSETY Gordon Lee Jendragiak 1957 INIHIHIHIllllHlUHHIllllHllllHHlHHHUIIHHW 3 1293 016875 ULTRASONICALLY INDUCED TRANSITIONS OE SODIUM NUCLEI IN SODIUM CHLORIDE by Gordon Lee Jendrasiak A THESIS Submitted to the College of Science and Arts of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Physics 1957 5/012/5 7 i [/02 (f, / ULTRASONICALLY INDUCED TRANSITIONS OE SODIUM NUCLEI IN SODIUM CHLORIDE by Gordon Lee Jendrasiak ABSTRACT The attenuation, by ultrasonic energy, of the free nuclear induction signals from sodium nuclei, in a crystal of sodium chloride, has been studied. The attenuation was observed with the ultrasonic energy at both theisnizi:2 and Arn=itl nuclear transition frequencies. The attenuation was plotted as a function of the square of the voltage applied to the quartz crystal at the 4rn=122 nuclear transition frequency. .A calorimetric method of measuring the ultrasonic power deliv— ered to the sodium chloride crystal was studied and a measure- Inent of the power was made. Using the values of the ultrasonic attenuation and the power delivered to the sodium chloride CI‘ystal, a value of 6', the shielding constant for the sodium Tulcleus, was obtained and found to be 1.7. This value was COmpared with values for 6' found by other investigators. ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to Dr. Walter H. Tanttila for his help and encouragement dur- ing the progress of this work. He wishes to thank Dr. Harry Bendler and Messrs. Mack Breazeale, William Lester, Walter Mayer, and George Smith for their help and suggestions. The author also thanks the National Science Foundation for their support of this work. ii INTRODUCTION THEORY APPARATUS AND EXPERIMENTAL TECHNIQUES RESULTS Calorimetric Data Attenuation Data CONCLUSION REFERENCES TABLE OF CONTENTS iii Page 15 25 25 28 33 36 Table I II LIST OF TABLES Calorimetric Data .Attenuation Data iv Page 25 28 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure \fl \OCO-\]0\ 10 ll LIST OF FIGURES Block Diagram of Apparatus Bloch Head Diode Peak-Reading Voltmeter "Beat Phenomenon" Page 10 1U- 16 19 21 23 26 29 INTRODUCTION In a crystal of sodium chloride the nuclear spin-lattice relaxation of a sodium nucleus comes about by the interaction of the time varying electric field gradient, at the sodium nucleus, with the electric quadrupole moment of the sodium nucleus. In order to calculate the electric field gradient, J. Van Kranendonk1 assumed a model in which the sodium nucleus interacted with point charges placed at the positions of the six nearest neighbors. Using this model, one can calculate the interaction and thus the nuclear spin-lattice relaxation time. The electrons surrounding the sodium nucleus, however, may modify the electric field due to these point charges in such a way that the effective point charge is q = ETe rather than e. e is the ionic charge. 5 , then, is a measure of the extent of the shielding and antishielding of the nucleus due to these electrons. The electric field gradients in crystals can be modu- lated by distortions of the crystal lattice. One way of causing these distortions is by the application of ultrasonic energy to the crystal. Work with sodium chloride, utilizing this idea, has been done by Proctor and Robinson2 and more recently by Olen Kraus.3 This is the type of work which will be discussed in this thesis. These workers made measurements of 3' by studying the ultrasonic attenuation of the free nuclear induction signals from sodium nuclei. By using the ratio of the attenuated signal to the unattenuated signal, along with the ultrasonic energy density in the crystal, one can arrive at a value for K. A value for 3' is given in this report and is compared with the values found by Proctor and Robinson and later Kraus. The signal ratio, spoken of above, is a function of the ultrasonic energy density in the sodium chloride crystal. This energy density can be obtained by using the ultrasonic power put into the crystal, the crystal volume, and a quantity called the phonon relaxation time. Unfortunately, the values of the phonon relaxation time, found in the literature, are, in gen- eral, rather unreliable. Proctor and Robinson calculated the power put into their sodium chloride crystal by using the equivalent resistance of a loaded quartz plate. The equivalent resistance was obtained by measuring the Q of the loaded quartz using a Q-meter. O. Kraus in his work, which is as yet unpublished, ob- tained this ultrasonic power by measuring the impedance of the quartz crystal loaded by the sodium chloride crystal. He used a high-frequency bridge for this measurement. This thesis discusses a calorimetric method of measuring the ultrasonic power put into the sodium chloride crystal. A thermocouple arrangement was used to measure the increase in the sodium chloride crystal temperature, as a function of the time the crystal was receiving the ultrasonic energy. From this measurement, an estimate is made of the power delivered to the sodium chloride crystal. The measurements of the ultrasonic attenuation of the free nuclear induction signal, from sodium nuclei, made by Proctor, Robinson, and Kraus, were made using pulsed ultrasonic energy. The work of this thesis was done with the ultrasonic energy applied continuously to the sodium chloride crystal. Other than this, the equipment and techniques were, in essence, the same as those used by Kraus. Finally, the work of Proctor and Robinson as well as that of Kraus involved ultrasonic frequencies equal to those necessary for AIn=2t2 nuclear transitions, m being the magnetic quantum number of the sodium nucleus. The work described in this thesis involved not only these ultrasonic frequencies but also those necessary f0r45n1=izl nuclear transitions. THEORY Consider the sodium nuclei in a crystal of sodium chlo- ride, which is placed in a steady magnetic field, Ho' The sodium nuclei have spin number 1, equal to g and the energy levels for these nuclei are shown in figure 1. m_-_3_ - 2 m_-_1_ ‘ _ 2 m- i.._____..____ . — 2 Ag m: 2 K Figure l m is the magnetic quantum number and takes values I, (I—l), ('1-2), ----- ' -(I—1), -1. If the crystal is in thermal equilibrium with its sur- roundings, the sodium nuclei assume a Boltzman distribution among the various energy levels. This distribution is given by E (l) N(m) = C exp. - K? , where N(m) is the number of nuclei in the m th level, C is a constant, Em is the energy of the m th level, k is Boltzmanis constant, and T is the absolute temperature of the NaCl crystal. The energy difference between adjacent energy levels is given by‘? Ho wherezy is the maximum measurable component of It the magnetic moment of the Na nucleus along the direction of HO. Thus, for a sodium nucleus to make a transition from one energy level to the next, it must absorb or emit a quantum of energy equal to é¥k>. The frequency ’g'of electromagnetic energy which may induce these transitions is given by the ex- pression hflg==é%fl9. h is Plank's constant. The selection rule governing these transitions is 4 m== i 1. From quantum mechanical considerations, the probability of electromagnetically inducing a nuclear transition from an energy state corresponding to m to a state corresponding to m + l, is equal to the probability of inducing, electromagnet- ically, a nuclear transition from the m + 1 state to the m state. If the numbers of nuclei in each energy level were equal, the average number of upward transitions would equal the average number of downward transitions. Since, however, the NaCl crystal is in thermal equilibrium, from equation (1) the nuclear population of an energy level exceeds that of the next higher level and there is a net transfer of nuclei from the lower to the higher energy states when the nuclei are sub- jected to electromagnetic radiation of frequency'flg. The nuclear population difference, at thermal equilib- rium, between two adjacent energy levels is obtained asfollows: e1; The energy of a given level is Um=—g MomHO , where Mo=-——. M is the mass of the proton and c the velocity of light in ,u lul ference, for Na’ nuclei, between the mz-g and m=é levels. Let us first consider the population dif- vacuum. 9 = O‘~ 2 2 g floHO Then, N(Z) exp. 2 kT 1 )4 H N(,—) ex . i 9 ° 0 3 p 2 kT = exp. g floHo = exp _2_ MHO RT 3 k '3 2/119 ‘ 1+ 3 k at room temperature and Ho'g ASOO gauss. Therefore, N(%) l _ l 2.HHo . .fl . 41(5) _ ME) 3—“ . We Will. call this difference do: the nuclear population difference at thermal equilibrium. Carry- ing out the above calculation for the nuclear population difference between the m = % and1n==-% levels and again for the population difference between the m = --é— and m = --3- levels, one finds that the Aio's are the same between any two adjacent energy levels. The method of inducing transitions of the sodium nuclei, reported upon in this thesis, is the pulsed nuclear induction method. In its simplest form, the method consists of applying a pulse of R.F. energy to the NaCl crystal and then observing a time decaying free nuclear induction signal from the sodium nuclei. Figure 2 is an illustration of this method. The free nuclear induction signal, after amplification and detection, is observed on the screen of an oscilloscope. The amplitude A0 of this signal is proportional to the popula- tion differences, between adjacent energy levels of sodium nuclei, that existed at the beginning of the R.F. pulse. xr————-R.F. Pulse Free Nuclear Induction Signal i A0 I —-%1 I<%——60 /4 Sec. Figure 2 Once the sodium nuclei have arrived in a higher energy state, they remain there for a finite length of time T called 1’ the "spin-lattice" relaxation time. Their return to the lower energy state is governed by the following expression: (2) A =AO(l-exp.-—%l) .A is the instantaneous population difference at a time t fol— lowing the cessation of the electromagnetic radiation. The spin—lattice relaxation time is the time necessary for all but 5 of the equilibrium population difference of the nuclei to reach the lower energy state from the higher state. The sodium nuclei have a spin number of 2 and have a 2 ELLe cmg. In the sodium chloride quadrupokrmoment of O.l)1I->[ k 60/.( sec. sec. r- __ Ultrasonics A, A ‘ . _J R.F. PUISB R.F. PUIS€—/ Time—+h Figure 3 When the ultrasonics is applied to the NaCl crystal in the (100) direction and at the lims=i.l nuclear transition frequency, we find that there should be no attenuation of the nuclear induction signal. This is due to the fact that the perturbation matrix elements used in the probability expres- sions for 45m==i 1 nuclear transitions are zero. Thus the probability for43rn=fiil nuclear transitions is also zero and no attenuation of the free nuclear induction signal is expected. The ultrasonic energy density, in the Nacl crystal, is given by 10 Figure A r~ ? The photograph o. Figure A shows the following se- quence of events: At the top of the photograph is shown a free nuclear induction signal, as it appears on an oscillo- scope screen, from sodium nuclei in a crystal of NaCl; in the center is shown the attenuation of the free nuclear induction signal caused by the application of ultrasonic energy at the ¢3m==ig2 nuclear transition frequency; at the bottom is shown the free nuclear induction signal with the ultrasonics again turned off. ll Figure A (3) 5- P Here, P is the ultrasonic power delivered to the NaCl crystal, V’ is the crystal volume, and Tp is the phonon relaxation time. Tp is the average time that it takes for the energy of a number of 10.2 MC. phonons to be lost to phonons of other frequencies by phonon-phonon collisions. Following Kraus, we have: (Li) Ao=1+§wrl A—- 3 where W is the ultrasonic transition probability per second. (5) W: 27 e1+ o.2 21282 Kepfwl lohe a6 ,o(w) is the line density function andgw/ofw) C/W = l. a is the lattice constant for NaCl, B is the amplitude of the ultrasonic vibrations in the NaCl crystal, K is the wave num— ber, e is the electronic charge, and Q is the quadrupole moment of the sodium nucleus. (6 ' ) 82 = --—-—2 where/O is the density of NaCl, /Ow andu/is 277 times the ultrasonic frequency. Then, using equation (3), we obtain (7) B2= 2PT2 V’/ow2 but (8) P = c><.V2 where O( is a proportionality factor and V is the R.M.S. voltage applied to the quartz crystal. We obtain this 0( by measuring, calorimetrically, the ultra- sonic power delivered to the NaCl crystal and the voltage across the quartz crystal. Putting these measurements into equation (8) allows °< to be obtained. 2 2 szx'V v”/ow2 and substituting this value of B2 into equation (5), we (9) 33 = obtain ll 2 2 2 (IO) fi2=1+ 272 e O 6 2Tpo fiblfijgb By solving this equation, a value for 4X can be obtained. ILL _ L 42_ [GSignai Timer r Time enera or 0" f Delay 0? Bloch 4 Head Ultrasonic Case Transmitter Pulser ,“ Quartz R.F. Oscillator Magnet Pole Face (Field out of Page) I ....... M NaCl Sample I Receiver A ——L—- C110 Figure 5 Block Diagram of Apparatus APPARATUS AND EXPERIMENTAL TECHNIQUES A block diagram of the apparatus, used to study the ultrasonic attenuation of free nuclear induction signals, is given in Figure 5. The sample crystal of NaCl was in the form of a cylin- der, about l-%é inches long and é-inch in diameter. It was obtained from the Harshaw Chemical Company and was the same one used by Kraus. For the part of the experiment where the frequency of the ultrasonic energy used was the.AIn= i.2 nu- clear transition frequency, the NaCl crystal was placed in a "Bloch Head," a photograph of which is shown in Figure 6. The Bloch Head case was made of aluminum. The shield was removed for this part of the experiment. The Bloch Head was then placed between the poles of an electromagnet. The orientation of the system was such that the axis of the receiving coil was perpendicular to the axis of the transmitter coil. Both of these axes were per- pendicular to the direction of the steady magnetic field, H0. The pole pieces of the magnet were seven inches in di- ameter. lfiuzmagnetic field strength used was about uSOO gauss and the magnet operated at a current of about 3.u amperes. The R.F. energy, necessary for the Aim 2 i 1 nuclear transitions, was applied to the NaCl crystal by the transmitter coil shown in Figure 5. The frequency of the R.F. energy was 15 16 Screw for.Adjusting Coupling Between the Transmitting and Receiving Coils Outlet for Receiving Coil Input for Transmitting Aluminum Coil Shield NaCl Sample Input for Ultrasonics Figure 6 Bloch Head 17 approximately 5.1 NC. The R.F. energy was applied in pulses, about 60 Alsec. in length. The ultrasonic energy was introduced into the NaCl crystal as follows: The NaCl crystal was sealed to an X-cut quartz plate with Duco cement. The ultrasonic energy was applied to the NaCl crystal along its (100) direction. The ultrasonics was kept on continuously, during the time the attenuation measurements were being made, except for a short period of time when the R.F. energy was applied. This time interval was the period neCessary for the following sequence of events to take place: The timer activated the time delay circuit which in turn operated the pulser causing the R.F. oscillator to give off a 60‘p¢sec. pulse of energy; the timer then turned the ultrasonics back on. The time interval be- tween applications of the R.F. energy was approximately seven seconds. The free nuclear induction signal was generated in the receiver coil which was part of the tank circuit coupled to the first stage of an R.F. amplifier. This amplifier was an AN/APS radar receiver, modified to amplify at 5.1 NC. A variable grid bias was provided for the input stage. This bias regulated the gain of the receiver. The amplified signal was then detected and applied to a Tektronix, Type 531, oscil- loscope. The height of the amplified free nuclear induction signal, appearing on the oscilloscope, was then estimated using the grid on the screen of the oscilloscope. .Although 18 this estimation is not subject to great precision, it is ac— curate enough for an experiment of the type performed. The voltage applied to the quartz was obtained by using a General Radio Standard Signal Generator, Type lOOl-A” driving a Healthkit, Model DX—35 transmitter which, in turn, was at— tached to the quartz by means of aluminum plating. The voltage applied to the quartz was measured by means of a diode, peak-reading voltmeter. The circuit for this voltmeter is shown in Figure 7. In order to make observations, the following procedures were followed: After the sample was placed in the magnetic field, no measurements were taken for about a 30 minute period of time. During this time the magnetic field stabilized and the NaCl crystal came into thermal equilibrium with its sur- roundings. .At the end of this 30 minute period, with the ultrasonic energy off, the height of the free nuclear induc- tion signal was observed on the oscilloscope. Then the ultra- sonics was turned on and the height of the signal was again observed along with the voltage on the quartz. Finally, the ultrasonics was turned off and after waiting for equilibrium to return, the height of the signal was again observed to see if it was the same height as that prior to the turning on of the ultrasonics. This procedure served to provide verifica- tion that the attenuation of the signal was due to the ultra- sonic energy and not to some other influence, such as the drifting of the magnetic field. N Quartzl 50K 6AL5 —‘I 0.1;4f . D.C.Voltmeter Figure 7 Diode Peak-Reading Voltmeter 20 Lastly, care was taken to see that the steady magnetic field was made as homogeneous as possible. Before this precau- tion was taken, a "beat" in the free nuclear induction signal was observed. This beat was barely observable without the ultrasonics on but became quite pronounced when the ultrasonics was turned on. When the magnetic field was made quite homo- geneous, however, the beat was not observed. This beat phe- nomenon is shown in Figure 8. The procedure followed, when the ultrasonic energy was applied at thezam = i lnuclear transition frequency, was the same as that for thezsm = iEBnuclear transition frequency, ex- cept that the Bloch Head was used as it is shown in Figure 5. The metal shielding was put in to prevent stray R.F. fields, arising at the quartz, from inducing nuclear transitions. Without this precaution, these stray fields might have caused nuclear transitions since their frequency was the45m = i 1 frequency. The selection rules forbid nuclear transitions, where4Am = i£2,being induced electromagnetically and thus the shield was unnecessary when the ultrasonic frequency was that of the¢3m = i.2nuclear transition frequency. .An addi- tional small NaCl crystal was sealed to the end of the original lVaCl crystal with Duco cement. This additional crystal was necessary, inasmuch as the original NaCl crystal was not long enough to pass through the receiving coil with the shield in the Bloch Head. 21 Figure 8 "Beat Phenomenon" The sequence of events in Figure 8 is as follows: At the top of the photograph is shown what.was observed on the screen of the oscilloscope when the ultrasonic energy had been applied to the NaCl crystal, as shown in Figure 3, at thezsm = iz2nuclear transition frequency. Rather than a single nuclear induction signal following the second R.F. pulse of Figure 3, two signals are observed. The second ex- posure from the top of Figure 8 shows what is observed on the screen with the ultrasonics off and the R.F. applied as in Figure 2. Note that a second nuclear induction signal is barely visible to the right of the now more prominent first induction signal. The third and fourth exposures of Figure 8 again show the "beat phenomenon" with the ultrasonics on; the fourth exposure again shows the free nuclear induction signal with the ultrasonics off. 22 Figure 8 The procedure for the calorimetric part of the experi- Inent is as follows: The NaCl crystal, sealed to the quartz, ‘was taken out of the "Bloch Head." One junction of a c0pper- constantan thermocouple was fastened to the outside of the NaCl crystal by means of wire wrapped around the junction and NaCl crystal. Duco cement was then placed over the contact point of the thermocouple and NaCl crystal. This point of contact was made approximately midway between the ends of the NaCl crystal, Then the thermocouple-NaCl crystal combination was put in the circuit of Figure 9. Reference Junction R x/r— . l 4?,rNaCl Junction /\ z R2 . Galvanometer l.S\I. Figure 9 Rl-ng was approximately 1000 JL . The reference junction was placed in ice. Ultrasonic energy was then continuously transmitted to the NaCl. At specified intervals of time, readings of the voltage across the quartz and the galvano- meter deflection were observed. Using the sensitivity of the galvanometer and its resistance, the voltage necessary for the deflection, at each observation, was calculated. 21+ Using the value of 39 $4 volts//°(3 for a copper-con- stantan thermocouple, as given by Michels in "Electrical Measurements and Their Applications," the temperature rise of the NaCl crystal in each time interval, between the obser- vations, was calculated. One difficulty with this procedure was experienced: After taking about two sets of temperature rise measurements, each set taking about ten minutes, the Duco cement holding the NaCl to the quartz apparently began to crack. .Although this cracking did not manifest itself in any apparent loss of holding power of the cement, it did show up in sudden jumps of the temperature of the NaCl crystal. It appears that the heat generated at the quartz-NaCl inter- face, by the ultrasonics, for extended periods of time, is enough to make the Duco cement quite brittle. This results in cracks appearing in the cement layer. Further work must be done on finding a more suitable agent to hold the Nacl crystal to the quartz. The calorimetric measurements, used for the power calculation, were not very precise; however, great precision in the measurement of the ultrasonic power is not needed since the phonon relaxation time is not known with any great accuracy. RESULTS Calorimetric Data TABLE 1 Setting of Signal Generator -- 10.2 M C. Average Temperature Average Time (minutes) Rise (CO) Voltage (volts) 0 _- -- 1 0.89 Ell 2 2.6) Sb, 3 u-) 5H )4 bit 5h 5 8.0 Su 6 9.5 ELL 7 10.7 EU- 8 11.5 su 9 12.5 Ell 10 13.1 SD 15 15.3 SM 25 C o in C Temperature Rise 15.0 0 O 1‘ TWT’T 28 x Experimental Point 1 l S 10 Time in Minutes Figure 10 27 From the data of Table l, we find that the average temperature rise per minute, for the first 10 minutes, is 1.3 Co. The graph of Figure 10 shows the rise in temperature for the NaCl crystal plotted as.a function of the time the ultrasonics was applied to the NaCl crystal. The time period from 10 to 15 minutes was not considered in obtaining the slope since it was felt that the temperature rise for the first 10 minutes would be representative of that during the attenuation measurements. One can see, however, from the temperature rise during the 10 to 15 minute time period that the graph of temperature rise versus time would begin to level off if it were plotted for times greater than 10 minutes. The ultrasonic power delivered to the NaCl crystal is obtained as follows: al The specific heat of NaCl is 0.2011 ——C——5'E— (Ref. 5) gn.— The density of NaCl is 2.185 9127 (Ref. 5') cm. The volume of the NaCl crystal is 5.93 cm3. Then, C81. 1 O P = O.2Oh.————3—- x 5.93 cm.3 x 2.165 92;- x 2—3LE;. g"!- C cm.3 60 sec. = 0.057 —:—§—:— = 238 milliwatts Figure 11 shows a representative graph of 52_ plotted A against the square of the voltage applied across the quartz. The voltage obtained with the diode voltmeter was divided by 28 Attenuation Data TABLE 11 Length of Time Ultrasonics was Kept on Between R.F. Pulses was Approximately 7 Seconds Square of Voltage Applied Ao Across Quartz (R.M.S. volts) Er- 5,000 (approximately) 21 8,820 (approximately) in 3,505 10 2,670 0.7 1,750 5.0 1,180 3.8 895 3.0 238 1.9 an 1.3 29 x Experimental Point T T d A _2 A.u 10.0 _ fl 1 0 TT [ l T I I TI l [ 0 1000 2000 3000 u000 5000 Square of Voltage Applied Across Quartz Figure 11 30 Arz-to obtain the R.M.S. voltage since the diode voltmeter is a peak-voltage-reading instrument. At voltages higher than about 65 R.M.S. volts, the free nuclear induction signal was attenuated almost 100%, showing saturation of the energy levels for the sodium nuclei. At these voltages it became extremely difficult to detect the nuclear induction signal from the background noise on the oscilloscope. The Kg-meas- urements, at these voltages, were quite uncertain and the slope of the curve for these voltages would riaeverjlrapidly. Since the slope of the curve at these voltages could not be used in equation (11), these points were not plotted. The slope of the curve of Figure 11, used in the calculations, 1.2 500 volts2 Using equation (11) and the constants given below, a was value for 0 was obtained and found to be 1.7. Constants Velocity of sound 5cm in NaCl -- u.75 x 10 sec: (Ref. 8) ‘fi -- 1.05 x 10'27 erg-sec. (Ref. 7) e -— 8.8 x 10"10 statcoulombs (Ref. 7) Q -- 0.1 x 10-2A cm.2 (Ref. 8) Tp -- 0.36 x 10'3 sec. (Ref. 2) T1 -- 7.3 sec. (Ref. 9) w -- 27 x 107 :22: a —- 2.813 x 10"8 cm. (Ref. 2) /o(w) -- 2100 sec. 31 Observations were also made of the effect of ultrasonic energy on the free nuclear induction signal when the ultrason— ics was applied at thezgni = i lnuclear transition frequency, .A i.e., 5.1 14C. Several measurements of A2 were made with a A voltage across the quartz of 70 R.M.S. volts. The A2 ratio was found to be 3.0 in this case. The main source of uncertainty in the attenuation meas— urements came about in obtaining the height of the free nuclear induction signal on the oscilloscope screen. This uncertainty A. in the-—9 ratio is relatively small at low voltages across the .A quartz. At high voltages where the A values become quite small, the experimental uncertainty is quite sizable as shown by the error flags in Figure 11. Also, at high voltages across the quartz, it became exceedingly difficult to distinguish the at- tenuated nuclear induction signal from the background noise on the oscilloscope screen. The calorimetric measurements were uncertain mainly for two reasons: First, no account was taken of the heat radiated by the NaCl crystal to its surroundings. Secondly, it was not known exactly how much of the temperature rise in the NaCl crystal was due to the ultrasonic energy in the NaCl crystal itself and how much was due to the heat generated at the quartz- Duco cement-NaCl crystal junction. The heat generated at this junction could have been conducted into the NaCl crystal itself thus making the ultrasonic power value greater than it should be. In view of the fact that the phonon relaxation time is 32 not known to any great degree of accuracy, it is felt that the uncertainties in the temperature measurement caused by these factors are negligible in comparison to the uncertainty in the phonon relaxation time. As regards the "beat phenomenon" illustrated in Figure 8, the explanation is as follows: Since the R.F. pulse is very short in length (60 [4 sec.), it may consist of a wide spread of frequencies about 5.1 M C.; on the other hand, the ultrasonic energy is applied to the NaCl for a comparatively long length of time (7 sec.) and thus has a very narrow fre- quency spread. Since the magnetic field HO was inhomogeneous, there was a wide energy spread of the Zeeman levels for the sodium nuclei. The R.F. pulse also had a wide spread of fre- quencies and, therefore, the nuclear induction signal was derived from a group of nuclei that had a spread in preces- sional frequency corresponding to the inhomogeneity in HO. When the ultrasonics was applied, its narrow band of frequen- cies influenced only a portion of the sodium nuclei, which in the case illustrated in Figure 8 was in the middle of the spectrum of transition energies. The nuclear induction sig- nals, shown in Figure 8, arise from the remaining two groups of sodium nuclei, uninfluenced by the ultrasonics, precessing at slightly different frequencies and, therefore, causing a beat signal to appear on the oscilloscope screen. CONCLUSION Definite attenuation of the free nuclear induction signals, from the sodium nuclei in a NaCl crystal, was ob- served when ultrasonic energy was applied to the crystal at thezfi m = i£2nuclear transition frequency. The slope of the . Ao _ 2 . . . 2 . line AT — 1 + 11V , where b 18 the coeff1c1ent of V in equa- tion (10), is about u.9° in the region V‘= o to v = 85 R.M.S. volts applied to the quartz. At higher voltages, the free nuclear induction signal is attenuated almost 100% showing ultrasonic saturation of the magnetic energy leVels of the sodium nuclei. With the ultrasonics at thezs m = i lnuclear transi- tion frequency and the voltage across the quartz about 70 R.M.S. volts, definite attenuation of the free nuclear in- duction signal was again observed. Since the theory predicts no attenuation of the nuclear induction signal, when the ultra- sonics is applied at theASIn = i lnuclear transition frequency and in the (100) direction of the crystal, it is felt that the ultrasonic waves generated in the NaCl crystal were not plane waves to which the theory is applicable. Apparently, the ultrasonic waves, in the course of passing through the NaCl crystal and being reflected from the free end of the crystal, are scattered at various angles from the cylindrical surface of the crystal. 33 3L1 The calorimetric method of measuring the ultrasonic power delivered to the NaCl crystal, as used in the work re- ported in this thesis, is a very simple and inexpensive way of measuring power. Although, as mentioned previously, it is not a high precision method, it could be made so with some rather simple refinements. Using the calorimetric method, a temperature rise of 1.3 CO per minute for the NaCl crystal was obtained when the ultrasonics was applied at 10.2 14C. The power delivered to the crystal was found to be 238 milliwatts. Tsz 5 value, as here calculated, is found to be 1.7. in Van Kranendonk's theory, the quadrupole moment of a sodium nucleus is considered to interact with the electric field produced by six equal point charges placed at the six neigh— boring lattice sites. The magnitude of each of these point charges is q = U’e. Thus, on the basis of Van Kranendonk's model and the X of 1.7, one can say that the sodium nucleus interacts with an electric field created by point charges which are 1.7 times the ionic charge e. it is interesting to note, at this point, that(D.Kraus has recently calculated a value for B'cmithe basis of his experimentation, and found it to be around 0.7. Kraus, how- ever, used a value for the phonon relaxation time which is more than twice as large as the value used in the work of this thesis. If the value for the phonon relaxation time used by Kraus is used with the data of this thesis, a value 35 of about 1.1 is found for 3' instead of 1.7. At the time Kraus took his data, however, the magnetic field, H drifted o’ quite badly. The data for this thesis was taken after this difficulty was eliminated. Thus it is felt that the value of 6' obtained by Kraus and the one given in this thesis are in quite close agreement. The value obtained for K , in NaCl, by Proctor and Robinson was 2.87 i 0.80 for the quadrupole case. The agree- ment between this value of 5 and 1.7 does not appear to be very good. The method of measuring 2f used by these investi- gators was, however, somewhat different than that reported in this thesis. In conclusion, it can be seen that before any value of U is obtained in which one can place confidence, a more precise measurement of the phonon relaxation time must be obtained. REFERENCES J. Van Kranendonk, Physica 20, 781 (l95u). . W. A. Proctor and W. G. Robinson, Phys. Rev. 10“, 13AM (1956). O. Kraus, thesis, (Michigan State University, unpublished). R. V. Pound, Phys. Rev. 72) 685 (1950). "Handbook of Chemistry and Physics," 36th ed., (Chemical Rubber Publishing Co., Cleveland, Ohio, 1958). J. K. Galt, Phys. Rev. 73, lu60 (l9u8). G. Shortley and D. Williams, 2nd ed., (Prentice-Hall, Inc., New York, 1955). "N—M-R Table," 3rd ed., (Varian Associates, Pal Alto, Calif., 1955). ‘ . W. H. Tanttila, (Private Communication). 36 HICHIGQN STATE UNIV. LIBRARIES llI||||)||I)l|||lllllllllllllllllll||||||l|l|l|ll|l||| 31293816875886