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Spurious ringing in pulse NMR
JOURNAL
OF
MAGNETIC
33,
RESONANCE,
199-203
(1979)
Spurious Ringing in Pulse NMR* EIICHI Los Alamos
Scientific
Laboratory,
FUKUSHIMA
University
of California,
Los Alamos,
New
Mexico
87545
AND
S. B. W. Department
of Physics,
San Diego
State
ROEDER University,
San Diego,
California
92182
Received March 28, 1978 The theory of electromagnetic generation of ultrasonic standing waves in metals is reviewed and applied to the problem of spurious ringing detected in pulse NMR. The theory predicts that the spurious signal is proportional to the transmitter pulse amplitude end the square of the static field intensity. It also depends on some material properties and geometrical factors. A table of pertinent properties for various metals is presented showing that aluminum is one of the worst materials. Experiments confirm that replacement of aluminum with other metals in the probe attenuates the spurious signal and that the ringing of the coil is attenuated by a judicious choice of wire size.
Recently, we attempted a pulsed NMR experiment on a solid sample at a combination of lower Larmor frequency, higher magnetic field, and larger coil size than we were accumstomed to. Somewhat to our dismay, we encountered the well-known coil “disease” (I-3) as a large-amplitude ringing which persisted for about 2 msec and completely obliterated the desired signal. Although we had encountered this problem before, we had never carried out a systematic study of the effect. This report describes our results and some recommendations on how to avoid the spurious ringing. The source of our spurious signal is electromagnetic generation of ultrasonic standing waves in metals. The induced radiofrequency current within the skin depth of the metal interacts with the lattice in a static magnetic field through the Lorentz force and the coherent ultrasonic wave, at the frequency of the incident radiation, propagates into the metal to set up a standing wave. A reciprocal mechanism, then, converts the acoustic energy, in the presence of the static field, to an oscillating magnetic field which is picked up by the coil as a spurious signal. The first application of this effect known to us was for studies of internal friction in brass rods by Zener and co-workers in 1939 (4). Recently, there has been renewed interest in this phenomenon especially in its application to nondestructive testing of materials (5). * Research performed under the auspices of the U.S. Department of Energy. 0022~2364/79/010199-05$02.00/0 199
Copyright Q 1979 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
200
FUKUSHIMA
AND ROEDER
There has also been an attempt to detect nuclear magnetic/acoustic resonance from the bulk metal with this effect (6). We now consider the various parts of an NMR apparatus which can ring to give rise to the spurious signal in the coil. In this discussion, it is assumed that all parts of the probe have been physically stabilized so that the system is free from lower-frequency microphonics. The possibility of the probe body and walls ringing was recognized by Speight et al. (3). Another possibility is the coil itself, as discussed by Aksenov et al. (I ) and Clark (2). Because the primary transmitter current flows in the receiver coil of a single coil system, no induction is necessary and the current in the coil interacts with the field to produce acoustic waves in the coil wires. The third possibility is the sample itself if it is metallic. Choosing a suitable probe material and shielding the probe from the coil should attenuate the contribution from the probe body. The contribution from the coil itself is harder to deal with. Clearly, shielding will not work, and there is a greater constraint on the selection of coil material compared to that for the body because the signal-to-noise ratio is proportional to the qualityfactor of the coil. Fortunately, the spurious signal is maximized with the generation of acoustic standing waves at or near the Larmor frequency so it can be minimized by a judicious selection of wire size. There is even less choice when it comes to the sample. Since the material itself cannot be changed, the only recourse is to change the sample configuration. Powder is the form least susceptible to acoustic ringing. For a single crystal, a stack of thin sections will not only minimize the ringing but will also increase the signal size by increasing the surface area exposed to the radiofrequency. The electromagnetic generation of sound waves was reviewed by Wallace (5u), who gave the amplitude of the generated acoustic wave for ql<< 1, where 1 is the electron mean free path and q = w/v, is the acoustic wavenumber, as lu] = (B/47rmv,w)B1/(1
+p2)1’2.
111
Here, o is the angular frequency, B is the static magnetic field, B1 is the amplitude of the incident rf magnetic field, m and vs are the mass density and the acoustic shear velocity for the material, and p = (qS)2/2, where 6 = (2p/p~o)“~ is the classical skin depth. The theory of reciprocity, then, shows that the rf signal induced in the coil is proportional to the initial rf amplitude B1 and the conversion efficiency E, where E = kB2/[mv,(l
+ p2)],
PI
with k a constant. To compare different materials for their relative efficiencies, it is convenient to rewrite p2 in terms of p and v,, Using the definitions for & q, and 6, we can show that p2 = (4.rr2/p2) (p2/&J2
= 2.5 x 10’3(p2/v:)v2,
[31
where p = 47r x lo-’ Wb A * m, v = w/27r, p is the resistivity, and all units are in MKS. Table 1 lists the pertinent material parameters at room temperature for a number of metals which are likely to be used in NMR applications and the factors mv, and p2 (at a Larmor frequency of 5 MHz) which enter into Eq. [2]. According to Eq. [2], E can be minimized by choosing materials with a large mvs or p2. Neither m nor v, depends appreciably on the temperature, so the temperature dependence is mostly due to p2. For pure metals, the resistivity decreases with
SPURIOUS
RINGING TABLE
ROOM
(lo3 Aluminum Brass (70% Cu) Copper Gold Lead Platinum Silver Stainless (347) Tungsten Zinc
IN
PULSE
NMR
1
TEMPERATURE PARAMETERS~ OF METALS POSSIBLE NMR PROBE APPLICATIONS
kmg,m3) 2.7 8.5 8.9 19.7 11.4 21.4 10.4 8.0 19.3 7.1
201
WITH
2
VS (m/s4
(10-‘&m)
mv,/mv,(Al)
3040 2110 2270 1200 690 1730 1610 3100 2640 2440
2.7 6.2 1.7 2.4 20.7 10.6 1.6 73 5.7 5.9
I.,0 2.2 2.5 2.9 1.0 4.5 2.0 3.0 6.2 2.1
(at 5’MHz) 0.005 0.121 0.007 0.173 118 0.783 0.024 3.61 0.042 0.061
a The m, a,, and p values are taken from Handbook of Chemistry and Physics, 50th ed., pp. E-41, F-140, F-141, Chem. Rubber Co., Cleveland, Ohio, 1969; Metals Handbook (T. Lyman, ed.), 8th ed., Vol. 1, pp. 52, 56, Am. Sot. for Metals, Metals Park, Ohio, 1961.
temperature so that at liquid nitrogen temperature or below p2 CC1 and E = kB2/mv,. The best pure metal at low temperature in Table 1 is tungsten with a value of mv, approximately six times greater than that of aluminum. On the other hand, it is well known that the resisitivity ratio of a metal at high and low temperatures depends on the purity; the resistivity is governed by the impurities at low temperature when the electron mean free path is limited by the impurities. Therefore, alloys such as brass and stainless steel not only have moderate to large p2 at room temperature as shown in Table 1 but have the additional advantage of nonvanishing p2 at lower temperatures. For example, the resistivity of stainless steel (347) at room temperature is only 1.4 times that at 4 K (7) so p2 changes by less than a factor of 2 between 300 and 4 K. So far, we have ignored geometrical considerations. The works of Aksenov et al. (I) and Gordon (6), among others, show that the geometrical factors are important in our application because they govern the acoustic standing wave patterns which, in turn, affect the acoustic resonance frequencies and the widths (and, consequently, the amplitudes) of these resonances. The acoustic resonance frequency is proportional to the sound velocity and inversely proportional to the appropriate dimension. Broadening can take place due to distributions of sound velocities and physical dimensions, for example, due to differently oriented microcrystallites, and it is more prevalent in thin pieces of metal and also in metals whose surfaces are stressed or constrained. Another important geometrical consideration is the actual physical configuration of the coil and the pertinent metal surfaces which can affect the coupling. The dependence of the coupling on the physical separation between the coil and the metal was found to be exponential (9) so that once a certain separation is achieved (say of the order of the coil diameter to ensure a reasonable quality-factor), little is gained by further separation. It is, then, more sensible to shield the metal from the coil.
202
FUKUSHIMA
AND
ROEDER
With these known characteristics, the electromagnetic generation of ultrasound in an NMR probe can be minimized. Equations [l] and [2] dictate small static and radiofrequency magnetic fields. That is a high price to pay for an NMR experiment, and other means should be considered first. The ringing in the probe walls can be suppressed by shielding the coil from the walls (3), but it is easier to simply choose a better material in the first place and then consider shielding if still necessary. For the probe walls, the data of Table 1 and associated discussions strongly suggest stainless steel as the ideal material at both high and low temperatures. There are other suitable materials, but aluminum, which is used extensively for NMR probes at room temperature, is one of the worst materials. Copper is nearly as good as any other material for a high quality-factor coil. Gold is slightly better but probably not enough to make it worthwhile except in special cases. The most effective remedial procedure for a coil is to first make certain that the wire surface is constrained, for example by being wound on a form, and then to change the wire, most probably to a smaller size. We encountered the “disease” with an aluminum probe which fits into a 4.5-cm magnet gap to do NMR at 6 MHz in a field of 20 kG. The single coil was made of No. 18 copper magnet wire on a plastic form, and it had a diameter of 19 mm and a length of 30 mm. The pulse had a circularly polarized radiofrequency magnetic field amplitude of about 35 G. In this configuration, the ringing was so bad that no NMR signals were observed at all in solid samples. Figure 1 shows the field dependence of the spurious ringing amplitude which clearly obeys the B” dependence of Eq. [2]. Presuming that the acoustic modes were in the aluminum probe walls, we replaced them sequentially with iron-cobalt alloy (i.e., the probe walls were simply removed to expose the coil to the pole faces in two directions), brass, and lead. As expected from Table 1, the best of these materials was lead, with which the dead-time was shortened by a factor of 5 but was still objectionable for any but the longest T2 samples. We next modified the coil by rewinding a same size coil using a smaller diameter wire (No. 22). The improvement was substantial and the net dead-time with a
FIG. 1. The spurious ringing amplitude in the NMR system described in the text as a function of the static magnetic field intensity.
SPURIOUS
RINGING
IN
PULSE
NMR
203
16-psec pulse became 27 psec from the center of the pulse. Since the original dead-time was about 2 msec, this is a two orders of magnitude improvement. Further improvements are, no doubt, possible even with the same probe geometry by screening the coil from the walls and using other wire sizes. It should be possible to optimize the parameters to achieve freedom from the spurious ringing discussed here, at least to the extent of many pulse NMR experiments which have been performed without any such problems. The most likely conditions for electromagnetic generation of ultrasound in NMR, to our knowledge, were those involved in the detection of 19’Au by Narath (8). The resonance was successfully observed at a Larmor frequency of 4.2 MHz, static field of 60 kG, sample temperature of 1.2 K, and a rotating rf field amplitude of 200 G. In summary, the choice of materials and configuration for a pulse NMR probe to be used at Larmor frequencies below 10 MHz in fields exceeding 20 kG at room temperature or below should be made with some care in order to avoid spurious ringing due to electromagnetic generation of acoustic standing waves in the probe walls, the coil, and the sample. Simply choosing some material other than aluminum for the probe body, preferably stainless steel, should be a first step in suppressing the “disease.” Further improvements can be made by proper shielding, a careful coil design, and, for a metallic sample, proper sample configuration. ACKNOWLEDGMENTS We would like to thank J. L. Smith of Los Alamos and A. G. Beattie of Sandia Laboratories for pointing out the relevant references in the field of electromagnetic generation of ultrasound in metals. D. E. Armstrong of Los Alamos executed the drawing. Note added in proof: A recent article on the same topic is by M. L. Buess and G. L. Petersen, Rev. Sci. Znstrum. 49,115l (1978). We thank the authors for communicating their results to us prior to publication.
REFERENCES 1. S. I. AKSENOV, B. P. VIKIN, AND K. V. VLADIMIRSKII, Zh. Eks. Teor. Fiz. 28,762 (1955) [English transl.: Son Phys.-JETP 1,609 (1955)]. 2. W. G. CLARK, Rev. Sci. Znstrum. 35,316 (1964). 3. P. A. SPEIGHT, K. R. JEFFREY, AND J. A. COURTNEY, J. Phys. E 7,801 (1974). 4. R. H. RANDALL, F. C. ROSE, AND C. ZENER, Phys. Rev. 56,343 (1939); C. ZENER AND R. H. RANDALL, Am. Inst. Mining Metallurg. Eng. 137, 41 (1940). 5. W. D. WALLACE, Znt. J. Nondestructive Test 2,309 (1971). (b) R. B. THOMPSON, IEEE Trans. Sonics Ultrasonics SU-20,340 (1973). 6. R. A. GORDON, “Electromagnetically Excited Acoustic Standing-Wave Resonances in Metals,” Ph.D. thesis, Brown University, 1972 [University Microfilms, Ann Arbor, Mich. No. 73-22751. 7. “Mechanical and Physical Properties of the Austenitic Chromium-Nickel Stainless Steels at Subzero Temperatures,” The International Nickel Company, Inc., New York, 1963, referenced in T. D. MOORE, “Structural Alloys Handbook,” Vol. 2, Belfour Stulen, Traverse City, Mich., 1974. 8. A. NARATH, Phys. Rev. 163,232 (1967).
OF
MAGNETIC
33,
RESONANCE,
199-203
(1979)
Spurious Ringing in Pulse NMR* EIICHI Los Alamos
Scientific
Laboratory,
FUKUSHIMA
University
of California,
Los Alamos,
New
Mexico
87545
AND
S. B. W. Department
of Physics,
San Diego
State
ROEDER University,
San Diego,
California
92182
Received March 28, 1978 The theory of electromagnetic generation of ultrasonic standing waves in metals is reviewed and applied to the problem of spurious ringing detected in pulse NMR. The theory predicts that the spurious signal is proportional to the transmitter pulse amplitude end the square of the static field intensity. It also depends on some material properties and geometrical factors. A table of pertinent properties for various metals is presented showing that aluminum is one of the worst materials. Experiments confirm that replacement of aluminum with other metals in the probe attenuates the spurious signal and that the ringing of the coil is attenuated by a judicious choice of wire size.
Recently, we attempted a pulsed NMR experiment on a solid sample at a combination of lower Larmor frequency, higher magnetic field, and larger coil size than we were accumstomed to. Somewhat to our dismay, we encountered the well-known coil “disease” (I-3) as a large-amplitude ringing which persisted for about 2 msec and completely obliterated the desired signal. Although we had encountered this problem before, we had never carried out a systematic study of the effect. This report describes our results and some recommendations on how to avoid the spurious ringing. The source of our spurious signal is electromagnetic generation of ultrasonic standing waves in metals. The induced radiofrequency current within the skin depth of the metal interacts with the lattice in a static magnetic field through the Lorentz force and the coherent ultrasonic wave, at the frequency of the incident radiation, propagates into the metal to set up a standing wave. A reciprocal mechanism, then, converts the acoustic energy, in the presence of the static field, to an oscillating magnetic field which is picked up by the coil as a spurious signal. The first application of this effect known to us was for studies of internal friction in brass rods by Zener and co-workers in 1939 (4). Recently, there has been renewed interest in this phenomenon especially in its application to nondestructive testing of materials (5). * Research performed under the auspices of the U.S. Department of Energy. 0022~2364/79/010199-05$02.00/0 199
Copyright Q 1979 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
200
FUKUSHIMA
AND ROEDER
There has also been an attempt to detect nuclear magnetic/acoustic resonance from the bulk metal with this effect (6). We now consider the various parts of an NMR apparatus which can ring to give rise to the spurious signal in the coil. In this discussion, it is assumed that all parts of the probe have been physically stabilized so that the system is free from lower-frequency microphonics. The possibility of the probe body and walls ringing was recognized by Speight et al. (3). Another possibility is the coil itself, as discussed by Aksenov et al. (I ) and Clark (2). Because the primary transmitter current flows in the receiver coil of a single coil system, no induction is necessary and the current in the coil interacts with the field to produce acoustic waves in the coil wires. The third possibility is the sample itself if it is metallic. Choosing a suitable probe material and shielding the probe from the coil should attenuate the contribution from the probe body. The contribution from the coil itself is harder to deal with. Clearly, shielding will not work, and there is a greater constraint on the selection of coil material compared to that for the body because the signal-to-noise ratio is proportional to the qualityfactor of the coil. Fortunately, the spurious signal is maximized with the generation of acoustic standing waves at or near the Larmor frequency so it can be minimized by a judicious selection of wire size. There is even less choice when it comes to the sample. Since the material itself cannot be changed, the only recourse is to change the sample configuration. Powder is the form least susceptible to acoustic ringing. For a single crystal, a stack of thin sections will not only minimize the ringing but will also increase the signal size by increasing the surface area exposed to the radiofrequency. The electromagnetic generation of sound waves was reviewed by Wallace (5u), who gave the amplitude of the generated acoustic wave for ql<< 1, where 1 is the electron mean free path and q = w/v, is the acoustic wavenumber, as lu] = (B/47rmv,w)B1/(1
+p2)1’2.
111
Here, o is the angular frequency, B is the static magnetic field, B1 is the amplitude of the incident rf magnetic field, m and vs are the mass density and the acoustic shear velocity for the material, and p = (qS)2/2, where 6 = (2p/p~o)“~ is the classical skin depth. The theory of reciprocity, then, shows that the rf signal induced in the coil is proportional to the initial rf amplitude B1 and the conversion efficiency E, where E = kB2/[mv,(l
+ p2)],
PI
with k a constant. To compare different materials for their relative efficiencies, it is convenient to rewrite p2 in terms of p and v,, Using the definitions for & q, and 6, we can show that p2 = (4.rr2/p2) (p2/&J2
= 2.5 x 10’3(p2/v:)v2,
[31
where p = 47r x lo-’ Wb A * m, v = w/27r, p is the resistivity, and all units are in MKS. Table 1 lists the pertinent material parameters at room temperature for a number of metals which are likely to be used in NMR applications and the factors mv, and p2 (at a Larmor frequency of 5 MHz) which enter into Eq. [2]. According to Eq. [2], E can be minimized by choosing materials with a large mvs or p2. Neither m nor v, depends appreciably on the temperature, so the temperature dependence is mostly due to p2. For pure metals, the resistivity decreases with
SPURIOUS
RINGING TABLE
ROOM
(lo3 Aluminum Brass (70% Cu) Copper Gold Lead Platinum Silver Stainless (347) Tungsten Zinc
IN
PULSE
NMR
1
TEMPERATURE PARAMETERS~ OF METALS POSSIBLE NMR PROBE APPLICATIONS
kmg,m3) 2.7 8.5 8.9 19.7 11.4 21.4 10.4 8.0 19.3 7.1
201
WITH
2
VS (m/s4
(10-‘&m)
mv,/mv,(Al)
3040 2110 2270 1200 690 1730 1610 3100 2640 2440
2.7 6.2 1.7 2.4 20.7 10.6 1.6 73 5.7 5.9
I.,0 2.2 2.5 2.9 1.0 4.5 2.0 3.0 6.2 2.1
(at 5’MHz) 0.005 0.121 0.007 0.173 118 0.783 0.024 3.61 0.042 0.061
a The m, a,, and p values are taken from Handbook of Chemistry and Physics, 50th ed., pp. E-41, F-140, F-141, Chem. Rubber Co., Cleveland, Ohio, 1969; Metals Handbook (T. Lyman, ed.), 8th ed., Vol. 1, pp. 52, 56, Am. Sot. for Metals, Metals Park, Ohio, 1961.
temperature so that at liquid nitrogen temperature or below p2 CC1 and E = kB2/mv,. The best pure metal at low temperature in Table 1 is tungsten with a value of mv, approximately six times greater than that of aluminum. On the other hand, it is well known that the resisitivity ratio of a metal at high and low temperatures depends on the purity; the resistivity is governed by the impurities at low temperature when the electron mean free path is limited by the impurities. Therefore, alloys such as brass and stainless steel not only have moderate to large p2 at room temperature as shown in Table 1 but have the additional advantage of nonvanishing p2 at lower temperatures. For example, the resistivity of stainless steel (347) at room temperature is only 1.4 times that at 4 K (7) so p2 changes by less than a factor of 2 between 300 and 4 K. So far, we have ignored geometrical considerations. The works of Aksenov et al. (I) and Gordon (6), among others, show that the geometrical factors are important in our application because they govern the acoustic standing wave patterns which, in turn, affect the acoustic resonance frequencies and the widths (and, consequently, the amplitudes) of these resonances. The acoustic resonance frequency is proportional to the sound velocity and inversely proportional to the appropriate dimension. Broadening can take place due to distributions of sound velocities and physical dimensions, for example, due to differently oriented microcrystallites, and it is more prevalent in thin pieces of metal and also in metals whose surfaces are stressed or constrained. Another important geometrical consideration is the actual physical configuration of the coil and the pertinent metal surfaces which can affect the coupling. The dependence of the coupling on the physical separation between the coil and the metal was found to be exponential (9) so that once a certain separation is achieved (say of the order of the coil diameter to ensure a reasonable quality-factor), little is gained by further separation. It is, then, more sensible to shield the metal from the coil.
202
FUKUSHIMA
AND
ROEDER
With these known characteristics, the electromagnetic generation of ultrasound in an NMR probe can be minimized. Equations [l] and [2] dictate small static and radiofrequency magnetic fields. That is a high price to pay for an NMR experiment, and other means should be considered first. The ringing in the probe walls can be suppressed by shielding the coil from the walls (3), but it is easier to simply choose a better material in the first place and then consider shielding if still necessary. For the probe walls, the data of Table 1 and associated discussions strongly suggest stainless steel as the ideal material at both high and low temperatures. There are other suitable materials, but aluminum, which is used extensively for NMR probes at room temperature, is one of the worst materials. Copper is nearly as good as any other material for a high quality-factor coil. Gold is slightly better but probably not enough to make it worthwhile except in special cases. The most effective remedial procedure for a coil is to first make certain that the wire surface is constrained, for example by being wound on a form, and then to change the wire, most probably to a smaller size. We encountered the “disease” with an aluminum probe which fits into a 4.5-cm magnet gap to do NMR at 6 MHz in a field of 20 kG. The single coil was made of No. 18 copper magnet wire on a plastic form, and it had a diameter of 19 mm and a length of 30 mm. The pulse had a circularly polarized radiofrequency magnetic field amplitude of about 35 G. In this configuration, the ringing was so bad that no NMR signals were observed at all in solid samples. Figure 1 shows the field dependence of the spurious ringing amplitude which clearly obeys the B” dependence of Eq. [2]. Presuming that the acoustic modes were in the aluminum probe walls, we replaced them sequentially with iron-cobalt alloy (i.e., the probe walls were simply removed to expose the coil to the pole faces in two directions), brass, and lead. As expected from Table 1, the best of these materials was lead, with which the dead-time was shortened by a factor of 5 but was still objectionable for any but the longest T2 samples. We next modified the coil by rewinding a same size coil using a smaller diameter wire (No. 22). The improvement was substantial and the net dead-time with a
FIG. 1. The spurious ringing amplitude in the NMR system described in the text as a function of the static magnetic field intensity.
SPURIOUS
RINGING
IN
PULSE
NMR
203
16-psec pulse became 27 psec from the center of the pulse. Since the original dead-time was about 2 msec, this is a two orders of magnitude improvement. Further improvements are, no doubt, possible even with the same probe geometry by screening the coil from the walls and using other wire sizes. It should be possible to optimize the parameters to achieve freedom from the spurious ringing discussed here, at least to the extent of many pulse NMR experiments which have been performed without any such problems. The most likely conditions for electromagnetic generation of ultrasound in NMR, to our knowledge, were those involved in the detection of 19’Au by Narath (8). The resonance was successfully observed at a Larmor frequency of 4.2 MHz, static field of 60 kG, sample temperature of 1.2 K, and a rotating rf field amplitude of 200 G. In summary, the choice of materials and configuration for a pulse NMR probe to be used at Larmor frequencies below 10 MHz in fields exceeding 20 kG at room temperature or below should be made with some care in order to avoid spurious ringing due to electromagnetic generation of acoustic standing waves in the probe walls, the coil, and the sample. Simply choosing some material other than aluminum for the probe body, preferably stainless steel, should be a first step in suppressing the “disease.” Further improvements can be made by proper shielding, a careful coil design, and, for a metallic sample, proper sample configuration. ACKNOWLEDGMENTS We would like to thank J. L. Smith of Los Alamos and A. G. Beattie of Sandia Laboratories for pointing out the relevant references in the field of electromagnetic generation of ultrasound in metals. D. E. Armstrong of Los Alamos executed the drawing. Note added in proof: A recent article on the same topic is by M. L. Buess and G. L. Petersen, Rev. Sci. Znstrum. 49,115l (1978). We thank the authors for communicating their results to us prior to publication.
REFERENCES 1. S. I. AKSENOV, B. P. VIKIN, AND K. V. VLADIMIRSKII, Zh. Eks. Teor. Fiz. 28,762 (1955) [English transl.: Son Phys.-JETP 1,609 (1955)]. 2. W. G. CLARK, Rev. Sci. Znstrum. 35,316 (1964). 3. P. A. SPEIGHT, K. R. JEFFREY, AND J. A. COURTNEY, J. Phys. E 7,801 (1974). 4. R. H. RANDALL, F. C. ROSE, AND C. ZENER, Phys. Rev. 56,343 (1939); C. ZENER AND R. H. RANDALL, Am. Inst. Mining Metallurg. Eng. 137, 41 (1940). 5. W. D. WALLACE, Znt. J. Nondestructive Test 2,309 (1971). (b) R. B. THOMPSON, IEEE Trans. Sonics Ultrasonics SU-20,340 (1973). 6. R. A. GORDON, “Electromagnetically Excited Acoustic Standing-Wave Resonances in Metals,” Ph.D. thesis, Brown University, 1972 [University Microfilms, Ann Arbor, Mich. No. 73-22751. 7. “Mechanical and Physical Properties of the Austenitic Chromium-Nickel Stainless Steels at Subzero Temperatures,” The International Nickel Company, Inc., New York, 1963, referenced in T. D. MOORE, “Structural Alloys Handbook,” Vol. 2, Belfour Stulen, Traverse City, Mich., 1974. 8. A. NARATH, Phys. Rev. 163,232 (1967).