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Microwave ultrasonics-some physical aspects
Microwave
ultrasonics-some
physical
aspects
FL W. B. STEPHENS* Discussions concerning Debye’s theory of vibratory motion in solids and Brillouin’s method for detecting the Debye waves are presented. Mention is made of the fundamental research now being carried out into microwave ultrasound for use in
In the investigation of the propagation of sound in a medium, some type of generator and detector is necessary and normally a device known as a transducer is used for both purposes. The function of the transmitter transducer is effectively to add a high intensity ‘monochromatic’ wave, ie a wave of a single frequency,to the multitude of thermally excited waves that were considered by Debye in his theory of the specific heats of solids. As an initial approximation, the elastic properties of a solid may be considered to arise from an assemblage of point masses connected together by ideal massless springs. The masses and the springs respectively represent the atoms and their electrical binding forces and such a system will support a very wide spectrum of possible elastic waves, extending in wavelength from the dimensions of the specimen to inter-atomic distances, ie distances of the order of angstrom units. In terms of frequency they will range from effectively zero to about 101*Hz but at frequencies greater than 1012Hz the wavelength becomes ccmparable with the dimensions of the atomic structure and the wave propagation becomes dispersive, ie the wave velocity is a function of frequency, or a plot of o against l/x is no longer linear (Fig 1). In an actual solid the ‘springs’ are not exactly linear in response so that a single defined frequency wave will degrade through the irreversible flow of energy into all the other frequency waves. Since the density (N) of elastic wave modes increases rapidly with frequency (Fig 2), the decay time of the impressed wave will decrease the higher its frequency. The frequency distribution curve has been obtained experimentally by studying the inelastic incoherent scattering of slow neutrons from a solid and by the study of diffuse X-ray reflections, and these both give fairly good agreement with theory. Additionally, the point masses of our ideal system are actually less definable in a solid due to the clouds of outer electrons which can interact in various ways with the system of elastic waves, leading to such phenomena as rotary polartsation of an elastic wave, and anomalous dispersion etc. The vibratory motion of the ions in a crystal lattice about their equilibrium positions may be interpreted in terms of a transverse and longitudinal wave system. For frequencies less than about 101aHz the adjacent ions move in the same sense and this motion is referred to as an acoustical mode (Fig 3). For slightly higher frequencies the oppositely charged ions on adjacent lattice sites may be displaced in opposite senses giving rise to optical modes. Many physical properties of a solid such as electrical and thermal conductivities are controlled through interaction with these thermal (lattice) phonons. The X-ray diffraction patterns from liquids were found by Stewart to be similar to those from powdered crystals or crystallites which led to the suggestion that a liquid consisted of quasicrystallites, having no discontinuous boundaries. * Physics Department,Imperial Road, London SW7
College, Prince Consort
high speed data processing devices. The paper serves as an introduction to the article ‘Microsound components, circuits, and their applications’ presented elsewhere in this issue of ULTRASONICS by E. Stern.
k (a f requency)
&
Figs 1 and 2 Wave propagation in linear chain of point ma8ses separated by a distance a
0
12 3 IWave number k [lOqcm]
Fig 3 The modes of vibration in an ionic crystal. represents the linear range
Line OL
This theory has received modifications but the concept of a short range order over 5 to 10 atomic diameters is still retained and is expressed by a radial distribution function. Debye specified that the thermal vibrations in a solid can be resolved into transverse and longitudinal elastic waves propagated in all directions, but in a liquid the former would be very highly attenuated at microwave frequencies. Brillouin was the first to suggest a method by which these Debye waves could be detected and their velocities determined. Brillouin suggested that the local density fluctuations arising from thermal agitation give rise to changes in the dielectric constant (and hence refractive index) of the medium, so scattering of an incident beam of light will occur. Constructive interference of the monochromatic light (wavelength X, in vacua and frequency Y), will lead to the scat tered light being mainly concentrated in a direction making an angle 0 with the incident beam to satisfy the Bragg condition, &o = 5Al sin (o/2), where hl is the wavelength of the ULTRASONICS October 1969
225
sound in the liquid and p is the optical refractive index. Since the density fluctuations and therefore their spatial Fourier components vary with time, the light scattered in any given direction is not monochromatic but has a spectral distribution. For a simple liquid the time variation of density will be in three modes, one non-propagating and two propagating. The former gives rise to the peak in the scattered intensity which is known as the Rayleigh component at the incident frequency, while the propagating modes yield the two Brillouin components, the Doppler-shifted peaks of frequencies w = wo f Aw, where Aw = 2(v/c)pw, (sin6/2). c is the velocity of light and v is the velocity of sound (which may be a function of frequency). It has been shown by Landau and Placzek that the ratio of the intensity (I,) of the Rayleigh component to the sum of the intensities (21s) of the Brillouin components is given by 1,/21, = (Al), where y is the ratio of the principal specific heats of the fluid. Recent developments due to the advent of laser techniques have made it possible to measure the widths and line shapes of the Rayleigh and Brillouin components leading to a revived interest in theoretical calculations of the light distribution. The theory has also been extended to relaxing liquids where the existence of internal degrees of freedom leads to the presence of other modes. At very low frequencies the interaction phenomena in simple mechanical systems obey classical laws but at high frequencies ie at kilomegaherz, quantum theory considerations are generally invoked, thus energy interchange will be in terms of quanta and expressed in phonons, the acoustic counterpart to photons in light. One method of generating coherent and monochromatic elastic waves at microwave frequencies is by extending the use of electromechanical transducers, eg quartz, to these higher frequencies using the technique of converting electromagnetic radiation into ultrasonic waves of the same frequency.‘At frequencies of the order of a few kMHz the sound wavelength is about 19-4cm in a solid and since it is difficult to establish electric fields with a significant variation over such short distances the only effective source becomes a surface excitation, as described in the article by E. Stern in this issue of ULTRASONICS. The other type of procedure is to cause various excited electronic systems to emit phonons by stimulated emission, which is the preferred technique at frequencies greater than 10irHz. A brief mention will be made of the types of fundamental research in microwave ultrasonics, which have their application as mentioned in Dr Stern’s paper. Hypersonic frequency mechanical waves in single crystal quartz are heavily damped at ordinary temperatures and the absorption is approximately independent of temperatures above 80°K. At lower temperatures the attenuation decreases rapidly with a lowering of temperature until 10°K is reached, when it becomes constant again, leaving a residual or intrinsic damping characteristic of the perfect crystal itself.
In the temperature dependent region the attenuation of an injected sound wave is attributed to discrete collisions between the low energy phonons (of the sound) and the thermal (lattice) phonons. This collision concept is only useful if wirt > 1 where w1 is the frequency of a soundwave phonon and rt is the lifetime of a thermal phonon. For single crystal natural quartz, rt has been estimated from the approximate expression K = .1!,Cvii2rt, where K is the thermal conductivity, C, the specific heat, and g the average phonon velocity. For a IOOOMHzsound wave, wlrt is of the order of unity at 40°K. Investigations concerning the propagation of these high frequency coherent sound waves are of great value in the understanding of transport phenomena such as thermal conduction in insulating materials. Recent investigations in microwave resonance phenomena have revealed the fact that at room temperature yttrium iron garnet (YIG), ruby (AzO,), rutile (TiO,) and magnesium oxide (MgO) all show extremely low attenuation. Furthermore, in YIG it was possible to propagate magnetic waves whose velocities can be varied by an external magnetic field. By obvious analogy these waves are referred to as magnons. The longer magnetic waves are termed magnetostatic and the shorter modes are the spin waves. The latter evolve from the exchange forces (essentially electrostatic) between adjacent atoms or ions in a crystal lattice and so will be influenced by any change in the interatomic distance, thus giving rise to elastic strains in the medium. The idea of spin waves is due to Bloch who considered them as waves of deflection of electron spin away from the ordered orientations which characterise a magnetic material. The deviated waves spin about their equilibrium positions with a precessing phase which has a periodic variation throughout the crystal. The distance between planes of equal phase is the spin wavelength so that the ground state energy level is equivalent to an infinite wavelength. In the intermediate region between the phonon and spin wave dispersion curves there is an admixed form known as magnetoelastic waves as shown in Fig 4. As with phonons, microwave phonons can excite transitions between the energy levels of a paramagnetic ion when these are split by a suitable static magnetic field. The energy transfer from the phonons to the electron spins results in the phonon induced electron spin resonance which is easier to observe than with the conventional ESR when the spin lattice broadening of the line width shrouds the induced resonance. Other aspects such as phonon interactions in semiconductors have received consideration in previous issues of this journal but the foregoing should suffice to indicate the stimulation given to solid state research by microwave ultrasonics.
BIBLIOGRAPHY Taylor, R. G. F.and Pointer,A. J., ‘Microwave ultrasonics’, Contempory Physics, Vol 10 (1969) p 159
Longitudinal lattice
Transverse
Jacobsen, E. H., ‘Microwave ultrasonics and the study of the solid state’, International Congress on Acoustics, Copenhagen (1962) paper 115
lattice
Stevens, K. W. H., ‘Microwave ultrasonics’, 5th International Congress on Acoustics, Liege (1965) paper 363 Ziman, J. M., ‘Electrons and phonons’, Oxford University Press, England (1960) Maris, H. J., ‘On the mean free path of low energy phonons in single crystal quartz’, Philosophical Magazine, Vol 9 (1964) p 901
-I
Wave
number
(k)
Fig 4 Disperbion curves for spin waves and phonons. The dotted lines indicate the strong coupling between elastic and magnetic waves
226
ULTRASONICS October 1969
Maris,H. J., ‘The absorption of sound waves in perfect dielectric crystals’, Philosophical Magazine, Vol 12 (1965) P 89 Strauss, W. J., ‘Spin wave experiments’, Applied Physics, Vol 36 (1965) p 118 Fletcher, P. C. and Kittel, C., ‘Magnetostatic and spin waves’, Physical Review, Vol 110 (1958) p 836
ultrasonics-some
physical
aspects
FL W. B. STEPHENS* Discussions concerning Debye’s theory of vibratory motion in solids and Brillouin’s method for detecting the Debye waves are presented. Mention is made of the fundamental research now being carried out into microwave ultrasound for use in
In the investigation of the propagation of sound in a medium, some type of generator and detector is necessary and normally a device known as a transducer is used for both purposes. The function of the transmitter transducer is effectively to add a high intensity ‘monochromatic’ wave, ie a wave of a single frequency,to the multitude of thermally excited waves that were considered by Debye in his theory of the specific heats of solids. As an initial approximation, the elastic properties of a solid may be considered to arise from an assemblage of point masses connected together by ideal massless springs. The masses and the springs respectively represent the atoms and their electrical binding forces and such a system will support a very wide spectrum of possible elastic waves, extending in wavelength from the dimensions of the specimen to inter-atomic distances, ie distances of the order of angstrom units. In terms of frequency they will range from effectively zero to about 101*Hz but at frequencies greater than 1012Hz the wavelength becomes ccmparable with the dimensions of the atomic structure and the wave propagation becomes dispersive, ie the wave velocity is a function of frequency, or a plot of o against l/x is no longer linear (Fig 1). In an actual solid the ‘springs’ are not exactly linear in response so that a single defined frequency wave will degrade through the irreversible flow of energy into all the other frequency waves. Since the density (N) of elastic wave modes increases rapidly with frequency (Fig 2), the decay time of the impressed wave will decrease the higher its frequency. The frequency distribution curve has been obtained experimentally by studying the inelastic incoherent scattering of slow neutrons from a solid and by the study of diffuse X-ray reflections, and these both give fairly good agreement with theory. Additionally, the point masses of our ideal system are actually less definable in a solid due to the clouds of outer electrons which can interact in various ways with the system of elastic waves, leading to such phenomena as rotary polartsation of an elastic wave, and anomalous dispersion etc. The vibratory motion of the ions in a crystal lattice about their equilibrium positions may be interpreted in terms of a transverse and longitudinal wave system. For frequencies less than about 101aHz the adjacent ions move in the same sense and this motion is referred to as an acoustical mode (Fig 3). For slightly higher frequencies the oppositely charged ions on adjacent lattice sites may be displaced in opposite senses giving rise to optical modes. Many physical properties of a solid such as electrical and thermal conductivities are controlled through interaction with these thermal (lattice) phonons. The X-ray diffraction patterns from liquids were found by Stewart to be similar to those from powdered crystals or crystallites which led to the suggestion that a liquid consisted of quasicrystallites, having no discontinuous boundaries. * Physics Department,Imperial Road, London SW7
College, Prince Consort
high speed data processing devices. The paper serves as an introduction to the article ‘Microsound components, circuits, and their applications’ presented elsewhere in this issue of ULTRASONICS by E. Stern.
k (a f requency)
&
Figs 1 and 2 Wave propagation in linear chain of point ma8ses separated by a distance a
0
12 3 IWave number k [lOqcm]
Fig 3 The modes of vibration in an ionic crystal. represents the linear range
Line OL
This theory has received modifications but the concept of a short range order over 5 to 10 atomic diameters is still retained and is expressed by a radial distribution function. Debye specified that the thermal vibrations in a solid can be resolved into transverse and longitudinal elastic waves propagated in all directions, but in a liquid the former would be very highly attenuated at microwave frequencies. Brillouin was the first to suggest a method by which these Debye waves could be detected and their velocities determined. Brillouin suggested that the local density fluctuations arising from thermal agitation give rise to changes in the dielectric constant (and hence refractive index) of the medium, so scattering of an incident beam of light will occur. Constructive interference of the monochromatic light (wavelength X, in vacua and frequency Y), will lead to the scat tered light being mainly concentrated in a direction making an angle 0 with the incident beam to satisfy the Bragg condition, &o = 5Al sin (o/2), where hl is the wavelength of the ULTRASONICS October 1969
225
sound in the liquid and p is the optical refractive index. Since the density fluctuations and therefore their spatial Fourier components vary with time, the light scattered in any given direction is not monochromatic but has a spectral distribution. For a simple liquid the time variation of density will be in three modes, one non-propagating and two propagating. The former gives rise to the peak in the scattered intensity which is known as the Rayleigh component at the incident frequency, while the propagating modes yield the two Brillouin components, the Doppler-shifted peaks of frequencies w = wo f Aw, where Aw = 2(v/c)pw, (sin6/2). c is the velocity of light and v is the velocity of sound (which may be a function of frequency). It has been shown by Landau and Placzek that the ratio of the intensity (I,) of the Rayleigh component to the sum of the intensities (21s) of the Brillouin components is given by 1,/21, = (Al), where y is the ratio of the principal specific heats of the fluid. Recent developments due to the advent of laser techniques have made it possible to measure the widths and line shapes of the Rayleigh and Brillouin components leading to a revived interest in theoretical calculations of the light distribution. The theory has also been extended to relaxing liquids where the existence of internal degrees of freedom leads to the presence of other modes. At very low frequencies the interaction phenomena in simple mechanical systems obey classical laws but at high frequencies ie at kilomegaherz, quantum theory considerations are generally invoked, thus energy interchange will be in terms of quanta and expressed in phonons, the acoustic counterpart to photons in light. One method of generating coherent and monochromatic elastic waves at microwave frequencies is by extending the use of electromechanical transducers, eg quartz, to these higher frequencies using the technique of converting electromagnetic radiation into ultrasonic waves of the same frequency.‘At frequencies of the order of a few kMHz the sound wavelength is about 19-4cm in a solid and since it is difficult to establish electric fields with a significant variation over such short distances the only effective source becomes a surface excitation, as described in the article by E. Stern in this issue of ULTRASONICS. The other type of procedure is to cause various excited electronic systems to emit phonons by stimulated emission, which is the preferred technique at frequencies greater than 10irHz. A brief mention will be made of the types of fundamental research in microwave ultrasonics, which have their application as mentioned in Dr Stern’s paper. Hypersonic frequency mechanical waves in single crystal quartz are heavily damped at ordinary temperatures and the absorption is approximately independent of temperatures above 80°K. At lower temperatures the attenuation decreases rapidly with a lowering of temperature until 10°K is reached, when it becomes constant again, leaving a residual or intrinsic damping characteristic of the perfect crystal itself.
In the temperature dependent region the attenuation of an injected sound wave is attributed to discrete collisions between the low energy phonons (of the sound) and the thermal (lattice) phonons. This collision concept is only useful if wirt > 1 where w1 is the frequency of a soundwave phonon and rt is the lifetime of a thermal phonon. For single crystal natural quartz, rt has been estimated from the approximate expression K = .1!,Cvii2rt, where K is the thermal conductivity, C, the specific heat, and g the average phonon velocity. For a IOOOMHzsound wave, wlrt is of the order of unity at 40°K. Investigations concerning the propagation of these high frequency coherent sound waves are of great value in the understanding of transport phenomena such as thermal conduction in insulating materials. Recent investigations in microwave resonance phenomena have revealed the fact that at room temperature yttrium iron garnet (YIG), ruby (AzO,), rutile (TiO,) and magnesium oxide (MgO) all show extremely low attenuation. Furthermore, in YIG it was possible to propagate magnetic waves whose velocities can be varied by an external magnetic field. By obvious analogy these waves are referred to as magnons. The longer magnetic waves are termed magnetostatic and the shorter modes are the spin waves. The latter evolve from the exchange forces (essentially electrostatic) between adjacent atoms or ions in a crystal lattice and so will be influenced by any change in the interatomic distance, thus giving rise to elastic strains in the medium. The idea of spin waves is due to Bloch who considered them as waves of deflection of electron spin away from the ordered orientations which characterise a magnetic material. The deviated waves spin about their equilibrium positions with a precessing phase which has a periodic variation throughout the crystal. The distance between planes of equal phase is the spin wavelength so that the ground state energy level is equivalent to an infinite wavelength. In the intermediate region between the phonon and spin wave dispersion curves there is an admixed form known as magnetoelastic waves as shown in Fig 4. As with phonons, microwave phonons can excite transitions between the energy levels of a paramagnetic ion when these are split by a suitable static magnetic field. The energy transfer from the phonons to the electron spins results in the phonon induced electron spin resonance which is easier to observe than with the conventional ESR when the spin lattice broadening of the line width shrouds the induced resonance. Other aspects such as phonon interactions in semiconductors have received consideration in previous issues of this journal but the foregoing should suffice to indicate the stimulation given to solid state research by microwave ultrasonics.
BIBLIOGRAPHY Taylor, R. G. F.and Pointer,A. J., ‘Microwave ultrasonics’, Contempory Physics, Vol 10 (1969) p 159
Longitudinal lattice
Transverse
Jacobsen, E. H., ‘Microwave ultrasonics and the study of the solid state’, International Congress on Acoustics, Copenhagen (1962) paper 115
lattice
Stevens, K. W. H., ‘Microwave ultrasonics’, 5th International Congress on Acoustics, Liege (1965) paper 363 Ziman, J. M., ‘Electrons and phonons’, Oxford University Press, England (1960) Maris, H. J., ‘On the mean free path of low energy phonons in single crystal quartz’, Philosophical Magazine, Vol 9 (1964) p 901
-I
Wave
number
(k)
Fig 4 Disperbion curves for spin waves and phonons. The dotted lines indicate the strong coupling between elastic and magnetic waves
226
ULTRASONICS October 1969
Maris,H. J., ‘The absorption of sound waves in perfect dielectric crystals’, Philosophical Magazine, Vol 12 (1965) P 89 Strauss, W. J., ‘Spin wave experiments’, Applied Physics, Vol 36 (1965) p 118 Fletcher, P. C. and Kittel, C., ‘Magnetostatic and spin waves’, Physical Review, Vol 110 (1958) p 836