Contents

List of Figures

    2.1  Phonon \mathaccent "017E\relax k-vectors with \voidb@x w_{\mathaccent "017E\relax k} < 5
    2.2  Distribution function over w for w < 0.1 ·10¹⁵   \voidb@x \voidb@x s^-1
    2.3  Distribution function over w for w < 10¹² \voidb@x \voidb@x s^-1
    2.4  Distribution function over w for w < 10⁹   \voidb@x \voidb@x s^-1
    2.5  Distribution function over q for w < 10¹² \voidb@x \voidb@x s^-1
    2.6  Kapitza resistance between gold heater and glass substrate as a function of T

    3.1  Pumped ⁴He cryostat
    3.2  Experimental cell
    3.3  Twin bars setup
    3.4  Setup for electropolishing the copper bar
    3.5  Electrical circuit for electropolishing
    3.6  Typical electropolishing I,V-diagram
    3.7  Heater on copper
    3.8  Excitation electronics
    3.9  Tunnel diode and filtering electronics
    3.10  Detection system
    3.11  Gas handling system

    4.1  Third sound signal on glass. \voidb@x n _{\voidb@x in} = 16.83 mmol
    4.2  Third sound signal on glass. \voidb@x n _{\voidb@x in} = 35.9 mmol
    4.3  Third sound signal on copper. \voidb@x n _{\voidb@x in} = 35.9 mmol
    4.4  Third sound speed c₃ on copper and on glass as a function of the amount of ⁴He in the cell
    4.5  Third sound speed c_3,Cu on copper as a function of ⁴He film thickness d_Cu
    4.6  Third sound attenuation a on glass as a function of ⁴He film thickness d
    4.7  Third sound attenuation a on copper as a function of ⁴He film thickness \voidb@x d_Cu
    4.8  Brisson's heater

    A.1  Bottom plate in experimental cell
    A.2  Left frame piece
    A.3  Right frame piece
    A.4  Copper resonator bar
    A.5  Glass resonator bar
    A.6  Mounting platform

Chapter 1
Third Sound

1.1  Introduction

Liquid ⁴He becomes a superfluid, i.e., it can flow without friction, below the l-temperature (2.18 K). Superfluid ⁴He can be described by the two-fluids model[1]. This model states that the liquid consists of two parts:

• A normal part that is viscous and carries entropy.
• A superfluid part that is not viscous and does not carry entropy.

In a superfluid ⁴He film, a surface wave called third sound can be excited. Only the superfluid part takes part in the surface wave, because the normal part is stuck to the wall as long as the viscous penetration depth is smaller than the film thickness. Third sound consists of both temperature and film thickness oscillations and can be excited by either a heat pulse or a strong electric field.

Third sound has been studied extensively, mostly on a glass substrate[7]. Subjects of study have been the propagation speed and the damping coefficient, as a function of film thickness and temperature.

To extend these experiments to ³He, the film has to be cooled below 0.93 mK, the ³He superfluid transition temperature. Because it is hard to cool glass to these temperatures, a copper substrate has to be used. Experiments with third sound on a copper substrate, using ⁴He, have been described previously [14]. In these experiments a copper cylinder was used which enabled resonant measuments; in previous experiments of the same group a flat glass surface was used so that only time-of-flight measurements could be performed.

The present work was undertaken to link previous experiments with a copper cylinder to those with glass. For this, an experimental cell that contains both a glass and a copper cylinder was built in order to measure properties, especially third sound speed and attenuation, of third sound on both substrates under the same circumstances.

The experiments described in this report involve the use of a thin gold heater to excite the third sound. Theoretical work on phonon modes and Kapitza resistances at low temperatures and with thin materials was performed as well (Chapter 2).

All symbols used in this paper are given in Appendix B.

1.2  Third sound speed

From the film equations of the two-fluids model the following equation for the third sound speed can be derived[1]:

1.3  Third sound attenuation

The results from this theory, for 4.17 kHz third sound and the infinite geometry, are presented in Figure 1.2 for a glass substrate and in Figure 1.3 for a copper substrate. These calculations were performed using a program written by Brouwer[5]. The minimums in these plots indicate the transition from the thin-film to the thick-film region, where the dominant damping mechanism shifts from thermal to acoustic waves.

1.4  Film thickness

There are several ways to determine the helium film thickness on the walls of a container. The determination can be performed with the use of one of the following parameters:

• The vapour pressure p_v
• The number of ⁴He atoms in the cell N
• The third sound speed c₃.

1.4.1  The vapour pressure p_v

1.4.2  The number of ⁴He atoms in the cell N

1.4.3  The third sound speed c₃

Chapter 2
Phonon modes in thin films recalculated

2.1  Introduction

Therefore an alternative for the standard (Debye) phonon distribution function was developed and tested by Monte Carlo simulation. Furthermore the Kapitza resistance between the film and another material was recalculated using the new phonon distribution function.

2.2  A new phonon distribution function

2.2.1  Crystal dimensions

All calculations will be performed assuming the film is a solid of dimensions V = L_x×L_y×L_z with a cubic lattice with lattice parameter a. There are N = N_x×N_y×N_z atoms in the crystal. The speed of sound in the solid is c. Only longitudinal phonons will be take into account. This treatment can easily be generalized to other phonon polarizations.

The phonon (reciprocal lattice) vectors take the form [k\vec] = (k_x,k_y,k_z) with

Assuming L_z << L_y << L_x it follows that the values of k_x and k_y are much more closely spaced than the values of k_z. Therefore k_x and k_y will be assumed to be continuous, while k_z will be considered discrete.

2.2.2  The density of states over w

Because the circular frequency of a phonon mode [k\vec] equals

As an example, in figure 2.1, the sphere depicts

The number of modes with circular frequency between w and w+dw at the fixed value of k_z now equals

This equation is tested using Monte Carlo techniques (section 2.3). Note that for large values of w equation 2.4 becomes equal to equation 2.2.

2.2.3  Angular dependence of phonon density

The values of [k\vec] do not have a uniform angular distribution because of the discreteness of the values of k_z. Therefore the phonon density of states is not only a function of w (depending on the absolute value of [k\vec] only), but also a function of the phonon directions.

The density of states in every k_x,k_y-plane is [(L_x L_y)/((2 p)²)] (there is one phonon state in every area [(2 p)/(L_x)] ·[(2 p)/(L_y)]). The Dirac d-function must be used because the k_x,k_y-planes lie a distance [(2 p)/(L_z)] apart. The phonon density of states as a function of [k\vec] is

This differential function is transformed into the new coordinates w,q,f with

If the discreteness of the values of k_z is neglected, the density of states as a function of [k\vec] becomes

2.2.4  The density of states over w recalculated as a check

The density of states over w is recalculated to check equation 2.7 by integrating it with respect to q and f.

2.2.5  Taking into account discreteness of k_y

If it is assumed that L_z << L_y << L_x the discreteness of k_y can be taken into account, just like the discreteness of k_z in the previous sections. The phonon density of states as a function of [k\vec] now becomes

In the rest of this chapter, the discreteness of k_ywill not be taken into account. Only the discreteness of k_zwill be taken into account because in the experiments described in Chapters 3 and 4 the effects of the discreteness of k_y are too small to play a role of importance. (With L_y = 0.1 mm, the discrete steps in k_y are sized [(2 p)/(L_y)] = 63  m^-1, which corresponds to (with c = 3390  m s^-1) an energy of E = (^h/_2p) c k_y = 2.3·10^-29 J. This is small compared to the thermal energy at T = 1 K, E = k_B T = 1.4 ·10^-23  J).

2.2.6  The density of states over q

The density of states over q (i.e. the number of phonon states with q between q and q+ dq) is calculated by integrating g(w,q,f) with respect to w and f. Only values of [k\vec] with w_[k\vec] smaller than a given W will be taken into account, because the modes will be occupied with a Bose-Einstein distribution, which doesn't allow a noticable number of phonons above a certain energy. The Bose-Einstein distribution will be used in section 2.4 to calculate the equilibrium occupation of the phonon state at a given temperature and hence the heat flux. In the following treatment the case q = p/2, phonons with k_z = 0 will be excluded. This case will be examined at the end of this section.

Also note that the preceding treatment is only valid for q ¹ [(p)/2], because then cosq = 0. Of course there are a lot of possible modes with q exactly equal to [(p)/2], since this corresponds with the k_z = 0-plane. The number of modes with q = [(p)/2], N_{w < W; q = [(p)/2]; 0 £ f £ 2 p}, can be calculated as follows:

Equations 2.14 and 2.15 will also be checked with a Monte Carlo-simulation in section 2.3.

If the classical phonon distribution function of Equation 2.9 is used, calculation of the density of states over q gives

2.3  Monte Carlo simulations

The idea of using Monte Carlo techniques to check our calculations was first conceived after finding Equation 2.4 and its unexpected discontinuities. It was decided to generate a large number of phonon [k\vec]-vectors of the form [k\vec] = (k_x,k_y,k_z) with k_i = [(2 p)/(L_i)] n_i and to choose a random integer number in the range -[(N_i)/2] ... [(N_i)/2] for each n_i (see section 2.2.1). With a large number of such random [k\vec]-vectors, a good approximation of the phonon distribution function g(w,q,f) will be generated. Next, all these [k\vec]-vectors can be classified with respect to their values of w, q and f to generate, for instance, the density of states over w.

In figures 2.2 to 2.4, g(w) from these Monte Carlo simulations is plotted against w. The material properties have been chosen to resemble a gold strip of dimensions 30 mm × 0.1 mm × 100 nm with c = 3390  m s^-1 and a = 0.257  nm. The three figures plot the same function but on a different w-scale: Figure 2.2 for w < 10¹⁴  s^-1, Figure 2.3 for w < 10¹²  s^-1 and Figure 2.4 for w < 10⁹  s^-1. In Figure 2.2, it is demonstrated that, for w >> c[(2 p)/(L_z)] but w < c [(p)/a] Monte Carlo simulation gives the same results for g(w) as the classical result of equation 2.2. For w > c [(p)/a] the finite crystal dimensions that the classical approximation does not take into account causes a breakdown of g(w). In Figure 2.3 the discrete steps of Equation 2.4 can be seen both in the analytical results and in the Monte Carlo simulation. In Figure 2.4, on an even smaller scale, the discontinuities of Equation 2.12 are shown to comply with the Monte Carlo simulation.

In figure 2.5, the distribution function over theta g_W(q) is plotted. An upper limit W = 1 ·10¹²  s^-1 was used. The figure shows the correspondence between Equation 2.14 and the Monte Carlo simulation. The number of modes with q = p/2 also agrees with the analytical result from Equation 2.15.

2.4  Kapitza resistance

2.4.1  Heat flux

The expression for the heat flux from a thin heater of dimensions V = L_x ×L_y ×L_z, with the interface to the substrate in the z-direction, is (from [12])

2.4.2  Kapitza resistance

Because the heat flux in the classical limit is proportional to A T⁴, Equation 2.19 can be rewritten for the heat flux from the heater to the substrate to

In this figure, the fraction a is assumed to be (from [12])

2.5  Discussion

2.5.1  Experimental support

Mester et.al. [13] describe experiments on heat transport between solids and superfluid ⁴He films. In these experiments the heat flux is dissipated by evaporation of ⁴He atoms. They find a giant effective Kapitza resistance (3 orders of magnitude greater than for bulk helium), but no dependence of the Kapitza resistance on film thickness. This indicates that the phonon distribution as described in this chapter is not responsible for the giant Kapitza resistance found by Mester since the Kapitza resistance depends on the heat flux (Equation 2.23) which in turn depends on film thickness L_z (Equation 2.18).

Eggenkamp [8] found an increasing thermal resistance between a gold heater film and a ⁴He film for temperatures below 0.60 K. Here evaporation and third sound are the dissipation mechanisms. Eggenkamp found a thickness dependence of the heat transport, although only 2.5 a.l. and 1.9 a.l. films were studied. However, Eggenkamps experiments describe heat transport from a gold heater film to a superfluid ⁴He film. The theory presented in this chapter is able to describe the phonon distribution in the heater, but it is not able to describe the heat transfer from the heater to the ⁴He film.

2.5.2  Suggestions for improvement

Chapter 3
Experimental set-up

3.1  Cryogenics

Third sound experiments with ⁴He have to be done at temperatures around or below 1.3 K, because otherwise third sound is strongly attenuated due to the ⁴He vapour. To reach 1.3 K, a pumped ⁴He cryostat is used (Figure 3.1). It consists of a set of stainless steel vessels placed inside each other. The outermost is vacuum, the one inside it is filled with liquid nitrogen at 77 K. Inside the nitrogen vessel there is a vacuum vessel again and the innermost vessel is filled with liquid helium. The vapour in this vessel is pumped away to reach a minimum temperature of about 1.19 K. The experimental cell is hanging in this helium bath.

3.2  The cell

The copper cell made by Pinkse [14] was used in the experiments. It has enough space for a (10.8 ±0.6) m² copper sinter, also made by Pinkse. The sinter is needed to stabilize the ⁴He film thickness inside the cell and thermally anchor the film to the cell. The cell also features a feedthrough for electrical wiring of which 19 connections are still usable.

A new interior for the cell was designed, capable of holding both a copper and a glass resonator bar at the same time (see Figure 3.2 and detail on front page). The following parts were made for this interior:

• A copper bottom plate, which is screwed to the top flange of the cell (Appendix, figure A.1).
• A copper resonator bar (Appendix, figure A.4).
• A glass resonator bar (Appendix, figure A.5).
• Two frames that together hold both bars (Appendix, figure A.2 and A.3).
• A mounting platform that can be screwed to the bottom plate (Appendix, figure A.6). Pinkse's copper sinter can be screwed to this platform.

The cell also features a sapphire capacitive pressure gauge, hanging in the free space below the sinter, and four resistor thermometers. One thermometer is fitted to the frame, two are glued to both bars and one is hanging in the helium bath.

3.3  The bar geometry

Both bars have a diameter of 10 mm and a length of 30 mm. They are suspended above the (silver-plated copper) bottom plate at distance of about 15 mm for the copper bar and about 32 mm for the glass bar. Both bars feature a mechanism for third sound excitation and third sound detection. Third sound is excited by a heater (H1 and H2 in Figure 3.3) and detected using a detector capacitance (D1 and D2). The excitation and detection mechanism are described in sections 3.5 and 3.6. A third sound wave, excited at one of the heaters, would travel both up and down. The wave that starts moving down would reach the detector after running along a quarter of the bar circumference. After running completely around the bar, it would reach the detector again after 5/4 of the bar circumference, and again at 9/4, 13/4, etc. The wave that starts moving up would reach the detector after 3/4 of the bar circumference, and again at 7/4, 11/4, etc. In this way a signal can be expected at the detector after 1/4, 3/4, 5/4, 7/4, etc. times the time needed to complete one loop around the bar.

3.4  Electrolytic polishing of copper

The copper resonator bar needed to be polished in order to have a clean third sound signal. If the surface is rough, the circumference of the cylinder will vary from place to place. This will cause a difference in time needed by a third sound pulse to travel around the cylinder, resulting in a distorted signal. Mechanically polishing copper is very hard, because the material is soft. In this case it is even harder because the item to be polished is a copper cylinder . The solution chosen was to use a technique called electrolytic polishing[17]. Electrolytic polishing (or electropolishing ) provides the following two effects on the surface to be polished:

• ``smoothing'' of the surface by elimination of the large-scale irregularities (above 1 mm in size)
• ``brightening'' by removal of the smaller irregularities (down to a around 10 nm).

The item to be polished (in our case the copper cylinder) forms the anode in an electrolytic cell (Figure 3.4). The item is first mechanically polished with sandpaper of successively finer grade (up to 4000). A solution of 70% orthophosphoric acid (H₃PO₄) in water is used as electrolyte.

The electric motor rotates the anode (the copper bar) with a frequency of about 1 Hz.

The electrical circuit used is presented in Figure 3.5.

Here R is a 1  W resistor used to measure the current running through the cell, V_R is the voltage across this resistor, V is the voltage across the cell and V_T = V_R + V is the voltage across the total circuit. By using an X,Y-recorder with V on the X input and V_R on the Y input it is possible to measure the I,V-diagram of the electrolytic cell. A typical I,V-diagram is presented in Figure 3.6.

In this diagram a number of stages can be recognised and the procedure for electropolishing is as follows:

1. Start increasing V_T. This causes I to increase. In this stage the anode surface is etched. This results in impairment of the anode surface and should last as short a time as possible.
2. At a certain moment, a blue, translucent viscous layer starts to form on the anode surface. This layer probably consists of copper phosphates. It is this layer that controls the polishing process. The current I will drop because the viscous layer increases the cell resistance. V_T is now held constant. The cell resistance still increases, resulting in a drop of I, a decrease of V_R and consequently an increase of V.
3. When the viscous layer is completed, the current reaches equilibrium. Increasing V_T a little again reveals the ``polishing plateau'', where I does not depend on V. The viscous layer limits the current. The anode surface is now being polished. Total cell voltage is about 2.5 V. To check this condition, V_T is increased further.
4. If V_T is increased too far the polishing plateau ends. Gas evolution occurs, destroying the viscous layer and pitting the anode surface. The destruction of the viscous layer allows the current to rise again. The total voltage V_T is quickly decreased to avoid pitting.
5. After some time, the current I will decrease again slowly, probably because the viscous layer still grows a little. This decrease of I causes a corresponding rise of V, like at point 2.
6. Increasing V_T reveals the polishing plateau again. Find the end of the plateau to make sure the anode is still being polished and go back.
7. Repeat steps 5 and 6 for about 15 minutes. The item is removed from the electrolyte approximately every 5 minutes for washing with distilled water.

There are a number of practical points to be observed when electropolishing:

• It should be possible to insert and remove the item to be polished quickly from the electrolyte (for washing purposes).
• Either rotate the anode or stir or rotate the electrolyte. This allows fresh electrolyte to reach the anode. Do not rotate too fast, because this may destroy the viscous layer.
• Make sure the item is in contact with the electrolyte only when the required potential difference across the cell is present, since otherwise etching may occur. Use a test piece to determine the appropriate voltage.
• The relative position of the anode to the cathode should remain fixed during the treatment. If this position shows variations, the internal resistance of the cell will also variate, which will make it make difficult to maintain polishing conditions.
• At the cathode, gas evolution will occur. Take care this will not destroy the viscous layer at the anode.
• Remove the item from the electrolyte every 5 minutes, wash with distilled water.
• After polishing, quickly remove the item from the electrolyte and wash with distilled water and alcohol.
• Getting the ``feeling'' for electropolishing takes some time. First practice on some test pieces and see how they get severely pitted if the voltage gets too high.

The copper bar used in the cell was polished brightly overall, but still has a few pits. Further polishing was not thought to be a good idea because the diameter of the bar had already decreased from 10.0 mm to 9.6 mm as a result of repeated etching and polishing.

3.5  Excitation of third sound

By sending one half-sine current pulse through the heater, a third sound pulse can be excited.

In Figure 3.8, the excitation electronics are drawn. R_heater is measured using a four-point method. The current through the heater can be measured by measuring the voltage across R_shunt on the oscilloscope. If that voltage is V, then the current I through the heater equals

3.6  Detection of third sound

These filtering circuits separate the approx. 20 MHz RF signal from the oscillator from the voltage needed to bias the tunnel diode. The RF signal is first amplified with a 30 dB HF amplifier and then mixed down to 70 kHz with a synthesizer sine wave (Figure 3.10. Numbers in this figure are meant as example). This signal is then amplified with a 20 dB, 50-90 kHz amplifier. A frequency counter (HP 5300B) is used to keep track of the TD resonant frequency. Next, the frequency (corresponding to ⁴He film thickness) is ``translated'' into a voltage with a P.L.L. (Phase Locked Loop). This signal is amplified 50-200 times (with a Princeton 113 pre-amp) and filtered with a 25 kHz LP-filter to eliminate the 70 kHz (and harmonics) that come from the P.L.L. The resultant signal follows the ⁴He film thickness change in the detector after a pulse is sent through the heater. This signal is averaged up to 16384 times in a Nicolet 370 signal averager.

3.7  Gas handling system

Chapter 4
Experimental results

For each layer thickness, a large number (up to 16384) excitation pulses were given consecutively on both bars and the resulting detector signals were averaged. Due to interference between both detector circuits, it was not possible to measure on both bars simultaneously. The excitation energy per pulse calculated according to Section 3.5 amounted to 0.11  mJ on the glass bar and to 0.13  mJ on the copper bar.

4.1  Third sound signals

A typical third sound signal on glass (n_inol) is presented in Figure 4.1. On the horizontal axis is the time after sending the heater pulse. On the vertical axis is the signal as captured by the Nicolet signal averager (Section 3.6). This signal corresponds with the ⁴He layer thickness at the detector. The first peak seen in the signal is an artifact which originates from interference by the heater pulse. This peak is not seen at time 0 because of delaying effects of the detection electronics. The second peak is the first third sound pulse arriving at the detector. This pulse has travelled 1/4 of the bar circumference (Section 3.3). Up to eight third sound peaks can be seen in the signal. By measuring the mean time between two adjacent peaks the third sound speed can be calculated, using the circumference of the bar, L = pd = p·1.0  cm = 3.142  cm with d the diameter of the bar. The third sound speed on glass c_3,gl was in this case (13.2 ±0.8)  m s^-1. This corresponds to a film thickness of 8.5 atomic layers (using Equation 1.1 and Figure 1.1). 1 atomic layer or a.l. of ⁴He corresponds to 0.36 nm. On the glass bar, third sound was seen when at least 6.2 mmol ⁴He was admitted, with a corresponding measured third sound speed c_3,gl = (23.4 ±0.9)  m s^-1 and a calculated film thickness d = 5.6  a.l.

In Figure 4.2, the third sound signal on glass with n_inol ⁴He in the cell is presented. The third sound speed c_3,gl now equals (5.02 ±0.22)  m s^-1, corresponding to a film thickness of 16.4 a.l. In the plot again first an interference peak, then two clear third sound peaks. After the second peak reflections are seen (e.g. the positive peak at 7.5 ms), making the determination of further peaks problematical. These reflections are probably caused at the detector, where the small gap causes interaction between the ⁴He films on both electrodes. This interaction can cause a partial reflection of the third sound wave.

A typical third sound signal (at n_inol) on copper is presented in Figure 4.3. Here six third sound peaks can be distinguished. Third sound speed c_3,Cu is now (4.47 ±0.5)  m s^-1. Under the same conditions, the third sound speed on glass was measured (see above) and was c_3,gl = (5.02 ±0.22)  m s^-1. The ⁴He film thickness on copper d_Cu was calculated from c_3,gl using the method described in the next section and amounted to d_Cu = 21  a.l. On the copper bar, third sound was seen when at least 28.8 mmol ⁴He was admitted, with a corresponding third sound speed c_3,Cu(6.7 ±0.4)  m s^-1. The film thickness on copper, calculating using c_3,gl, was in that case d_Cu = 16.3  a.l.

4.2  Third sound speed

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4.2.1  Comparison of c_3,gl and c_3,Cu

4.2.2  Calculation of d_gl and d_Cu

4.2.3  c_3,Cu versus d_Cu

In Figure 4.5 the measured third sound speed on copper c_3,Cu is plotted against the film thickness on copper as calculated from the third sound speed on glass as described in Section 4.2.2 (diamonds). The theoretical relation (Equation 1.1, now for copper) is given in the figure as well (solid line). To calculate d_Cu from the measured c_3,gl, g_Cu = 5.92 ·10^-24 m⁵s^-2 (from Brisson[3]) and dead layer thickness D(T) = 1.91  a.l. (from Equation 1.5) were used. The theoretical values of c_3,Cu lie 20% higher than the measured values.

The dashes in Figure 4.5 give the theoretical relation for c_3,Cu as a function of d_Cu now using g_Cu = 4.3 ·10^-24  m⁵s^-2 and D(T) = 1.91  a.l. This line seems to agree with the measured values of c_3,Cu. Note that here d_Cu is calculated from c_3,gl using g_Cu = 5.92 ·10^-24  m⁵s^-2 while the theoretical values of c_3,Cu are calculated from d_Cu using g_Cu = 4.3 ·10^-24  m⁵s^-2. It would be better to use the same value of g_Cu for both calculations, but in that case it is not possible to get a good fit. Better results are obtained if the dead layer thickness D(T) (Section 1.2) is involved in the fitting procedure as well. The + marks in Figure 4.5 represent c_3,Cu as a function of d_Cu where g_Cu = 4.3 ·10^-24  m⁵s^-2 and D(T) = 5.66  a.l. are used. The theoretical relation with these values for g_Cu and D(T) is given as the dots&dashes. Now the measured values for the third sound speed on copper c_3,Cu agree within a few % with the theoretical calculations.

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4.3  Third sound attenuation

4.3.1  Calculation of the attenuation coefficient a

4.3.2  Attenuation on the glass bar

Picture Not Created.

4.3.3  Attenuation on the copper bar

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4.4  Discussion and conclusions

4.4.1  Third sound on copper

4.4.2  Comparison with Brisson's experiments

4.4.3  Attenuation coefficient on copper

4.4.4  Problem: Pressure determination

New cables were used for the leads of the pressure gauge, because the old cables caused capacitance shifts as a result of (unavoidable) vibrations. For both leads now a coaxial cable was used. Also a different, more stable capacitance bridge was used to determine the capacitance of the pressure gauge. The new bridge could measure capacitances of 50 pF with a sensitivity of 0.0001 pF, corresponding with a pressure of 0.007 mbar. The new bridge has not been used for the experiments described in this chapter.

4.4.5  Problem: Film thickness determination error

A possible explanation for the discrepancy in the value for the ⁴He film thickness calculated from the amount of ⁴He admitted to the cell and from the third sound speed on glass may be that most of the helium that was let into the cryostat may end up in the filling capillary in stead of in the cell, because of counterflow effects in the capillary. The inside of the filling capillary is covered by a superfluid helium film from the cell up to the helium level in the bath. Above the helium level the capillary gets warmer, reaching room temperature at the top flange. Around the superfluid transition (2.17 K) the helium film will get very thick, while the vapour pressure reaches 50 mbar. A superflow up in the film combined by a flow of gas down may trap a lot of helium. To be able to work around this problem, a low-temperature valve was fitted just above the cell. Using this valve the filling capillary can now be closed. By admitting helium to the cell at temperatures larger than 2.17 K and closing the valve during cooldown it was hoped to solve the problem. New experiments may show if this extra valve indeed solves the problem.

Appendix A
Drawings

This appendix contains the drawings A.1 to A.6 that were made, mostly for the parts of the cell.

Appendix B
Symbols

Symbol	Unit
pv	Pa	Vapour pressure
psv	Pa	Saturated vapour pressure
[Q\dot]	W	Heat flux through a surface
R	Jmol-1K-1	Ideal gas constant, R = 8.31411 Jmol-1K-1
R	KW-1	Thermal resistance
RK	K4m2W-1	Kapitza resistivity
S	m2	Surface area
T	K	Temperature
V	m3	Volume (V = Lx ·Ly ·Lz)
Vtot	m3	Room temperature volume for 4He admission
Z		Set of integer numbers
a(q)		fraction of phonons, coming in at an angle q, that cross the interface
a	m-1	Attenuation coefficient
b	a.l.	Retardation constant (b = 15 nm for glass)
g	m5s-2	Van der Waals constant
ggl	m5s-2	Van der Waals constant for glass (ggl = 2.6·10-24  m5s-2)
gCu	m5s-2	Van der Waals constant for copper
d(x)		Dirac delta function
d	m	4He film thickness
dgl	m	4He film thickness on glass
dCu	m	4He film thickness on copper
q,f		Angular coordinates of wave vector
l	m	Wavelength
( [(rs)/(r)])		Superfluid fraction in bulk 4He
( [(rs)/(r)])f		Superfluid fraction in a 4He film
w	s-1	Circular frequency
W	m2s-2	Van der Waals potential
Wgl	m2s-2	Van der Waals potential on glass
WCu	m2s-2	Van der Waals potential on copper
w[k\vec]	s-1	Circular frequency corresponding with wave vector [k\vec] (w[k\vec] = c | [k\vec]| )
ëx û	[x]	Largest integer smaller than or equal to x

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