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Submitted on 1 Jan 1972
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ACOUSTIC INTERFEROMETRY IN LIQUID HELIUM FILMS
C. Anderson, E. Sabisky
To cite this version:
C. Anderson, E. Sabisky. ACOUSTIC INTERFEROMETRY IN LIQUID HELIUM FILMS. Journal de Physique Colloques, 1972, 33 (C6), pp.C6-87-C6-90. �10.1051/jphyscol:1972620�. �jpa-00215138�
JOURNAL DE PHYSIQUE Colloque C6, suppltment au no 11-12, Tome 33, Novembre-Dtcembre 1972, page 87
ACOUSTIC
INTERFEROMETRY IN LIQUIDHELIUM FILMS
C . H. ANDERSON and E. S. SABISKY RCA Laboratories, Princeton, New Jersey, U. S. A
RCum6. - On a transform6 une mkthode de detection optique du paramagnetisme dans les solides en un spectromktre B phonons acoustiques pouvant travailler en balayage entre 10 et 300 GHz avec une resolution spectrale de 80 MHz et une grande sensibilitk.
La detection d'ondes acoustiques stationnaires dans des films d'h6iium liquide adsorbks sur es surfaces de clivage de CaFz dopks par 0,02 mole pour cent de thulium divalent cornrne ions paramagnktiques, est l'une des applications de cette technique. En effet, ceci a permis I'utilisation aux longueurs d'ondes acoustiques comprises entre 40 et 120
A
(20-60 GHz) d'un interferomktre de Fabry-Perot acoustique a balayage, dans lYh6lium liquide. La determination prkcise des epais- seurs de films ainsi obtenue a fourni les meilleures mesures du potentiel de van der Waals. En utilisant le systkme en tant qu'interferomktre de Fabry-Perot modulk, on a pu mesurer la dis- persion de la vitesse de phase du son dans I'h6lium liquide et montrer qu'elle est positive B 1,36 OK.De plus on a pu mesurer le dkphasage a I'interface cristal h6lium.
Abstract. - Optical detection of paramagnetism in solids has been developed into an acoustic phonon spectrometer capable of scanning from 10 to over 300 GHz with an 80 MHz spectral resolution and high sensitivity. One application of this technique has been the detection of simple acoustic standing waves across films of liquid helium adsorbed on cleaved surfaces of CaF2 containing divaient thulium as the paramagnetic ion. In effect this has allowed a scanning acoustic Fabry-Perot interferometer to be operated at acoustic wavelengths of 40-120
A
in liquid helium.The precise determination of the film thickness this provides has produced the best measurements of the van der Waals potential. Operating the system as a modulated Fabry-Perot has enabled the dispersion of the phase velocity of sound in liquid helium to be measured and shown to be positive. In addition it has been possible to measure the phase shift introduced at the crystal- liquid helium interface.
Optical detection of paramagnetism in the tradition of Kastler has been extended to form the basis of a new acoustic phonon spectrometry capable of operating in the frequency range of 10 to over 300 GHz [I]. The spins of a paramagnetic impurity in a dielectric crystal a t very low temperatures can dynamically interact with the host lattice through the emission and absorption of resonant acoustic pho- nons. Therefore the effective temperature of the spin system is determined by the temperature of the resonant phonons ; classically one would say it is determined by the power density of the resonant acoustic radiation. Using a magneto-optical property associated with the paramagnetic ions it is possible to optically detect the spin temperature with great precision and hence the temperature of the resonant phonon modes. Optical detection unlike conventional EPR absorption operates freely at any Larmor frequency as determined by an externally applied magnetic field. Thus the spin impurities form a sensitive tuneable monochromatic detector of incohe- rent acoustic radiation which can be readily addressed optically.
One successful application of this concept has been the operation of an acoustic Fabry-Perot interferome- ter in liquid helium at acoustic wavelengths of 40-
120 A, corresponding to 60 to 20 GHz respectively 121.
This acoustic interferometer, operating at wave- lengths only an order of magnitude larger than the size of helium atoms, has been used in the usual way to measure lengths, dispersion in the velocity of sound and phase shifts. Each measurement providing new conclusive results on very interesting problems in physics.
The interferometer consists of a film of liquid helium on a cleaved crystal of CaF, having a surface with large areas which are atomically flat. The van der Waals force holds the liquid helium on the crystal so that the free helium surface, which forms the other end of the Fabry-Perot, is also essentially atomically flat. To a very good approximation the boundary conditions on the acoustic waves in the film are given by assuming the hard crystal surface is incom- pressible and the free helium surface can sustain no stresses. The resonant standing wave modes across the film therefore have a node at the crystal surface and an anti-node at the free surface. Because the velocity of sound in liquid helium is an order of magnitude smaller than in the crystal, refraction at the interface couples the acoustic modes at all angles in the crystal with essentially only the simple standing waves in the film.
7
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972620
C6-88 C. H. ANDERSON A N D E. S. SABISKY
The CaF2 crystal contains .02 mole percent of the paramagnetic ion divalent thulium which has a simple spin
3
ground state and relatively strong circular dichroism properties in the visible spectrum.Detailed studies have shown that for temperatures below
-
10 OK and Larmor frequencies above 10 GHz these spins interact primarily with the reso- nant acoustic waves [31.
In these experiments the freshly cleaved samples are mounted in a copper holder at the end of a length of waveguide with small windows to permit the monitor light to pass through the crystal. The interior of the system is sealed against a surrounding liquid helium bath, but helium gas can be introduced or withdrawn through the waveguide and a separate tube allows pressure measurements in the copper holder to be made.
An overall picture of the somewhat complex dyna- mic situation of the experiment can be obtained through figure 1. The spins are tuned using an external magnetic field onto resonance with the microwave radiation sent down the waveguide, a higher spin temperature being detected optically at resonance.
MICROWAVE
-
SOURCE (v
OPTICAL MCD MONITOR
FIG. 1. - Schematic of experiment.
The microwave power level is then adjusted until the spins are about 30 % hotter than the bath tempe- rature. The heat in the spin system is lost to the host lattice by radiation of acoustic waves at the same frequency, warming up the resonant modes. This extra acoustic power in the resonant modes also feedsback on the spins and contributes to their net temperature. The magnitude of this excess power density is determined by the balance between the rate the spins radiate energy and the rate this energy escapes by simply leaving the crystal. In the presence of the liquid helium film most of the energy is reflected back at the crystal-liquid interface because the acous- tic mis-match is very large. However, if the film is slowly thinned by pumping off the helium then each time the thickness becomes a resonant length, i. e.
an odd multiple of a quarter wavelength, the excess acoustic energy readily leaves the crystal and the spins are observed to become cooler. This is shown in figure 2 for several frequencies, where the absissa
represents the film thickness in some monotonic non- linear fashion.
By simply counting the number of resonance peaks observed as the film grows from zero thickness it is possible to calculate the film thickness on each resonance using the formula
SATURATED (130g,
I .iM
T I M E
-
FIG. 2. - Interferometric resonances in liquid helium films.
where N is the number of wavelengths in the resonant mode. The wavelength A can be calculated using the velocity of sound in bulk liquid helium c, and the microwave frequency v. This formula leaves out corrections of a few percent due to dispersion in the velocity of sound and phase shifts due to the boundary conditions not being the simple ones mentioned above.
The thickness of the film is determined by a balance between the chemical potential of the gas in the vicinity of the crystal and the van der Waals potential due to the substrate acting on the helium atoms in the surface of the helium film. This is readily seen by considering a container with bulk liquid helium in its bottom and the liquid film on the walls, figure 3.
The surface of any liquid in equilibrium forms an equipotential surface so the potential energy of a helium atom in the bulk liquid surface equals that of a helium atom in the surface of the film. Therefore the gravitational potential energy gained by an atom in the film must equal that which it has lost falling into the van der Waals potential due to the substrate.
To be more exact, it is the van der Waals potential of the substrate relative to a substrate of liquid helium ; the helium atom in the film and that in the bulk liquid have the same helium neighbors down
ACOUSTIC INTERFEROMETRY IN LIQUID HELIUM FILMS C6-89 to a depth d. The chemical potential of the helium
gas at the height h is also in equilibrium with the surface of the film so we have the expressions
v(d) = mgh = k~ log (P,(T)/P) , (2)
FIG. 3. - Potentials in a container of liquid helium.
d F I L M T H I C K N E S S (i)
where V(d) is the van der Waals potential of a helium atom a distance d from the substrate and mgh is the gravitational potential energy of an atom a height h above the bulk liquid surface. In the expres- sion for the chemical potential of the gas k is the
Boltzmann constant, T the temperature, Po(T) is the vapor pressure of liquid helium at the tempera- ture T and P is the pressure in the vicinity of the crystal.
By simultaneously measuring the chemical poten- tial through either T and P or h with the detection of the resonance peaks it is possible to plot out V(d) with great precision, figure 4. These results are far superior to any previous measurements of the van der Waals potential simply because the acoustic interferometric measurement of the film thickness is far more precise than any other method can provide.
The most useful aspect of this measurement is that it confirms in great detail a computer program developed by biophysicists to calculate the van der Waals potential according to a formula developed by Lifshitz ; the solid line in figure 4 [4]. Thus our results afford them greater confidence in their compu- tation of this very important force between the mem- branes of biological cells.
The dispersion and phase shift measurements are made by precisely determining the center frequencies of two or more resonant modes at a given film thick- ness, which is aided by modulating the film thickness a few Angstroms about its equilibrium value [5].
This modulation turned out to be the most difficult part of the experiment since the helium film thickness is an extremely stable quantity, but we shall not
FIG. 4. - Measured and theoretical van der Waals potential,
discuss this point here. The film thickness with these corrections is given by
where 6(v) is the dispersion correction to the phase velocity and do(v) is the phase shift correction due to the boundaries. The precise frequency measurements of the two modes are used to obtain a small length Ad defined by
This length is related to the dispersion and phase.
shift by the equation
where d, is the mean acoustic length as given by eq. 1 for the two modes. By measuring Ad for different film thicknesses it is possible to separate the dispersion and phase shift contributions. While this procedure gives only relative values for these correction terms it has the great advantage of being very sensitive.
The measurements obtained clearly show that the- phase velocity increases by as much as 3 % and is linear over the frequency range of 20 to 60 GHz, in rather good agreement with more indirect results.
obtained by analysing neutron scattering data [6].
The linear behavior probably does not continue indefinitely to lower frequencies [7]. The under- standing of this anomalous positive dispersion in liquid helium is a problem of very high current interest.
The phase shift results are shown in figure 5 with]
C6-90 C. H. ANDERSON AND E. S. SABISKY
FIG. 5. - Frequency dependance of the phase shift introduced been discussed here are results'obtained by basically at the crystal liquid helium interface. operating a classical acoustic Fabry-Perot interfero- meter at extremely short wavelengths in liquid helium.
6 o
"4
-
5.0--
a<
4.00 t 5 3.0- I V)
w 2 . 0 -
V) a
I a 1.0-
The measurements have proven useful in providing an absolute scale obtained by forcing the acoustic ultrafine data for a variety of both old and new and van der Waals length scales to have a common problems in physics. Finally it should be noted that origin. This very small quantity and its frequency this is only one aspect of a new technique in acoustics dependance gives for the first time precise data on which has many other promising applications.
References
Cl] ANDERSON C. H. and SABISKY E. S., Chapter 1, Physical [4] RICHMOND P. and NINHAM B. W., J. LOW Temp.
Acoustics, vol. 8, Mason and Thurston Eds., Phys. 5 (1971) 177.
Academic Press (1971). [5] ANDERSON C. H. and SABISKY E. S., Phys. Rev. Lett.
28 (1972) 80.
f21 ANDERSON C. H. and SABI~KY E. S., Phys. Rev. Lett. [61 Mo~INARl A. and R E ~ ~ E T., Phys. Rev. L ~26 ~ ~ .
24 (1970) 1049. (1971) 1531.
131 SABISKY E. S. and ANDERSON C. H., Phys. Rev. B 1 [7] ROACH P. R., ABRAHAM B. M., KETTERSON J. B. and
(1970) 2028. KUCHNIR M., Phys. Rev. Lett. 29 (1972).
Oo 20 40 60 so liquid helium are effected by boundaries.
FREQUENCY ( G H z ) In conclusion we wish to point out that what has
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DISCUSSION
the dynamics of liquid helium next to a substrate.
One immediate conclusion that can be drawn from this result is that this reglon cannot act as an efficient acoustic impedance matching element on atomically flat surfaces. This had been a possible explanation for the high heat conductance observed between solids and liquid helium, which is an old problem of liquid helium called the Kapitza resistance. The dispersion in the phase shift is very suggestive of the existence of a resonant mode at slightly higher fre- quencies, which could be related to the interesting question of how the elementary excitations of bulk
M. PAPOULAR. - 1) Regarding anormalous dis- persion of sound, is it immediately obvious that you are measuring a bulk property ?
2 ) In isotopic mixtures, 4He segregates near the walls. Do you think your technique is well adapted to studying that phenomenon ?
C. H. ANDERSON. - 1) Yes, we believe so. The dispersion was effectively measured for film thicknesses ranging from 30-160 A and found to be independent of film thickness. Our van der Waals potential measurements assure us that except for the first 5-10 A next to the substrate the density of the liquid helium in the film is identical to that of bulk liquid.
The phase shift measurements, figure 5, give us infor- mation about the dynamics in this 5-10 region, which indeed are quite different from the bulk pro- perties.
2) There is no simple answer to this question since we do not even understand the phase shift results for pure 4He. There is no doubt that a very interesting series of experiments on 4He-3He mixtures can be carried out with this technique. The reso- nances have been observed with normal liquid helium films.
JOFFRIN. - What is the order of magnitude of the relaxation time of the phonon population you can measure.
C. H. ANDERSON. - Because the coupling between the spins and the phonons must be very weak in order for the spins not to cause strong scattering of the phonons, the response of the spins to changes in the phonon population is very slow. Thus we cannot measure this quantity directly. However in our chapter in Physical Acoustics vol. 8 we describe a series of experiments which enable us to obtain a measure of this quantity. We find for example in a crystal 1-2 mm in cross-sectional dimension, life- times at 20-40 GHz of the order of 50 micro- seconds and a clear trend of a decreasing lifetime with increasing frequency. For frequencies above 120 GHz we have evidence that the transmission coefficient into the surrounding liquid helium bath is of the order of 50-100 %.
P. J. KING. - DO YOU have any way of determining the polarization of the phonons under observation ? C. H. ANDERSON. - TO first order a simple spin
