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IMversity Mcrafihns International
8603031
Lorenson, Claude Pierre
DYNAMICAL PROPERTIES OF SUPERFLUID TURBULENCE
The Ohio State University Ph.D. 1985
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University Microfilms International
DYNAMICAL PROPERTIES OF SUPERFLUID TURBULENCE
DISSERTATION
Presented in Partial Fulfillment of the Requirement for
the Degree of Doctor of Philosophy in the Graduate
School of the Ohio State University
By
Claude Pierre Lorenson, B.S., M.S.
The Ohio State University
1985
Reading Committee: Approved by
James T. Tough i— C. David Andereck ' _ Advisor William F. Saam Department of Physics
Supported by the National Science Foundation, Low-Temperature
Physics, Grant No. DMR 8218052 ACKNOWLEDGEMENTS
I would sincerely like to thank my advisor Dr. J. T. Tough. I
realize that I pushed his patience to the limit throughout this
research effort. The completion of this dissertation is a tribute
to his creativity and patience. Every time we were stuck with a
problem, Dr. Tough could always come up with a suggestion, "something
to try" so that we could keep moving. I used quite a few "nicknames"
for Dr. Tough throughout the years ... yet the best one is advisor, never has his advice let me down.
One person who helped me a lot through this work is my friend
and co-worker Donald Griswold. Without his help this work would not
be completed by now. I thank him for writing most of the computer programs and also for helping in the construction of the apparatus and in the data taking. His presence in the lab made the day to day work enjoyable.
I would like to thank Dr. V. U. Nayak not only for teaching me how to run the experimental apparatus but also for stressing the importance of quality in a project. Because of/my association with him I learned a lot about the basic problems of experimental low-temperature physics and how to solve them.
I am grateful for all the support of the technical staff at
O.S.U. during my stay. Special thanks to Bob Merritt for keeping our helium supply constant and to Bob Kindler for the great work he has done on our experimental probe.
I greatly appreciate the love and support of my parents. They were always willing to help me throughout my career. I could never repay all the sacrifices made by them to help me.
I thank all my friends, especially Alain Gauthier, to help making my stay in Columbus so enjoyable. Finally I would like to thank my best friend, my wife Pandora for all the love, support and understanding she has given me. This work was supported by National
Science Foundation-Low Temperature Physics-Grant #DMR 8218052. VITA
April 29, 1957 Born: Jonquiere, P. Quebec Canada
1980 B.S.: Universite Laval Quebec City, Canada
1980-1982 Graduate Teaching Assistant, The Ohio State University, Columbus, Ohio
1982-1985 Graduate Research Assistant, The Ohio State University Columbus, Ohio
1982 M.S.: The Ohio State University Columbus, Ohio
Field of Study: Condensed Matter Physics TABLE OF CONTENTS
ACKNOWLEDGEMENTS ...... ii
VITA ...... iv
LIST OF TABLES ...... vii
LIST OF FIGURES ...... viii
CHAPTER PAGE
1. INTRODUCTON...... 1
1.1 Introduction ...... 1 1.2 Early Theories and Tests ...... 1 1.3 Turbulent Thermal Counterflow ...... 3 1.4 Theories and Models ...... 7 1.5 Previous Experiments and Motivation for This Research ...... 12
2. APPARATUS ...... 16
2.1 Introduction ...... 16 2.2 Design Consideration ...... 16 2.3 Probe Overview ...... 17 2.4 Vacuum C a n ...... 20 2.5 Helium Reservoir ...... 20 2.6 Flow Tube Assembly ...... 25 2.7 Counterflow C e l l ...... 26 2.8 Superleaks ...... 28 2.9 Thermometers and Heaters ...... 29 2.10 Pressure Transducer ...... 32 2.11 Feedthroughs and Electrical Wiring ...... 40 2.12 Cryostat and Vacuum Systems ...... 44
3. EXPERIMENTAL PROCEDURES ...... 49
3.1 Introduction ...... 49 3.2 Temperature Measurements ...... 49 3.3 Temperature Regulation ...... 53 3.4 Chemical Potential Measurements ...... 58 3.5 Counterflow Experiments ...... 61 3.6 Spectral Analysis ...... 63
v 3.7 Relative P o w e r ...... 66 3.8 Time Constant Experiment ...... 67
4. EXPERIMENTAL RESULTS ...... 73
4.1 Introduction ...... 73 4.2 Steady State Data in Turbulent Counterflow . . 73 4.3 Power Spectra Measurements ...... 82 4.4 Relative Power Measurements ...... 96 4.5 Time Constant Measurements ...... 103 4.6 Amplitude Probability Distribution ...... 106
5. DISCUSSION ...... 113
5.1 Introduction ...... 113 5.2 Observations of Fluctuations ...... 113 5.3 Amplitude of the Fluctuations and Relaxation Time at the TI/TII Transition ...... 117 5.4 Future Work ...... 120
APPENDICES
A. Construction of Transducer ...... 122
A.l Introduction ...... 122 A.2 Body Assembly ...... 122 A.3 Membrane Assembly ...... 125 A.4 Closing the Transducer ...... 128 A.5 Leak Testing ...... 129
B. Sensitivity T e s t ...... 131
B.l Introduction ...... 131 B.2 Physical Analysis ...... 131 B . 3 Resonance in the Transducer ...... 144
C.l Computer Programs ...... 144
BIBLIOGRAPHY ...... 165
vi LIST OF TABLES
TABLE PAGE
1. Values of dR/dT and R for all thermometers at 1.6 ° K ...... 31
2. Description of Figure 1 7 ...... 47
vii LIST OF FIGURES
FIGURE PAGE
1. Schematic diagram of a counterflow apparatus .... 4
2. Cubic dependence of the dissipation ...... 6
3. Crossing vortex lines reconnecting in the Schwarz m o d e l ...... 11
4. Vortex line density graph v.s heat current ...... 13
5. Schematic diagram of experimental probe ...... 18
6. Vacuum can ...... 21
7. Detailed diagram of Helium reservoir ...... 23
8. Thermometer m o u n t ...... 24
9. Counterflow Cell ...... 27
10. Detection of A p ...... 33
11. Detailed diagram of transducer platform and mounting ...... 35
12. Schematic diagram of capillaries on alluminum grill ...... 36
13. Pressure Transducer ...... 38
14. Plate and Spacer Assembly ...... 39
15. "Normal" Feedthrough ...... 41
16. "Superconducting" Feedthrough ...... 43
17. Schematic diagram of Cryostat ...... 46
18. DC-4 Terminals resistance measurement circuit . . . 50
19. AC-Bridge for resistance measurement ...... 52
viii Phase-Splitter used for exciting the bridge . . . . 54
Feedback circuit for temperature regulation . . . . 56
Block diagram of the temperature regulating feedback circuit ...... 57
Capacitance Measurement Circuit ...... 59
Set-up for Counterflow Experiments ...... 62
Experimental set-up for determination of Power Spectrum ...... 65
Experimental set-up for relative power measurements ...... 68
Experimental set-up for time constant measurements ...... 69
Typical digitized response signal ...... 71
Exponential fit to the response data ...... 72
AR vs. a ...... 75
AC vs. Q ...... 76
L^d v s . Q ...... 77 o High resoltuion data in the TI/TII transition . . . 80
Time varying signal of our background ...... 83
Power spectrum of the background signal ...... 84
Time varying signal as function of heat current . . 86
Power spectrum in TI Region ...... 87 i Power spectrum in TI/TII Region ...... 88
Power spectrum in TII Region ...... 89
Log-log graph for 3 heat currents in TI Region . . . 91
Log-log graph for heat currents in TI/TII Region . . 93
Log-log graph for heat currents in TII Region . . . 95 ix 43. Relative power at 1 Hz as a function of heat current ...... 99
44. Relative power at .63 Hz as a function of the heat current ...... 100
45. Relative power at .1 Hz as a function of the heat current ...... 101
46. Exponential response time as a function of the heat current ...... 104
47. Schematic sketches for bistable and a continuous transition ...... 107
48. Probability distribuion at 88 yW, a Gaussian fit is also included ...... 109
49. Probability distributon at 92 y W ...... 110
50. Probability distribution at 94 y W ...... ill
51. Baratron station used for leak testing and gluing the membrane ...... 124
52. Assembly Procedure for the transducer ...... 126
53. Geometry used to derive the capacitance ...... 133
54. 1c vs. P at room temperature ...... 136
55. Circuit to deflect the m e m b r a n e ...... 139
56. Dependence on the voltage of the capacitance for Q = 0 ...... 140
57. Ac vs. P at 1.6°K using voltage and temperature t e c h n i q u e ...... 141
58. Power spectrum at 0 yW for old transducer ...... 143
x CHAPTER 1
INTRODUCTION
1.1 Introduction 1 2 Since the discovery of superfluid helium (Hell) in 1938 ’ much progress experimentally and theoretically has been made in the under standing of this quantum fluid. Many of the flow properties of superfluid helium are related to the existence of quantized vortex lines. This hydrodynamic structure is unique to Hell and the turbulent flow of this fluid has been modeled as a tangled mass of vortex lines. One type of flow that has been well studied throughout the years is thermal counterflow and that is the flow under investi gation in this dissertation. In this chapter models and experimental results used in our understanding of turbulence in Hell will be reviewed.
1.2 Early Theories and Tests
The experiments done in the beginning of the "superfluid" era seem to give contradictory results. Experiments designed to measure 3 the resistance to flow would give a viscosity of zero . Experiments designed to measure the drag of a body moving through liquid helium 4 4 5 6 would give a value close to that of He gas ’ ’ . These results are explained by the basic model that describes the properties of
1 7 8 superfluid helium. This model which is due to London , Tisza 9 and Landau , considers the liquid to be composed of two interpenetra
ting fluids, each with its own density and its own velocity field.
One fluid has a density of pn and a viscosity n and behaves just
like an ordinary viscous fluid: this is the "normal fluid". The other
is called "superfluid". It flows without friction, carries no entropy
and has a density pg . The total density of the fluid is p-p^p^. The
assumption of different velocity fields implies two separate equations 9 for the two fluids. Using thermodynamic arguments, Landau derived
the dynamical equations for the two fluids as (neglecting effects
due to "second viscosities"):
i.i
2 n -» -* nV v - — VP - p sVT 1.2 n p s
In these equations P is the pressure at r, T the temperature at r, n
is the viscosity of the normal fluid, s the specific entropy of the
fluid, v and v are the velocities of the superfluid and the normal s n fluid at r. These equations describe the flow of Hell at low velocities very well. From this model, Tisza8 also predicted the existence of temperature waves, commonly called second sound. The superfluid can oscillate against the normal fluid, 180 degrees out of phase, giving rise to local temperature fluctuations. Tisza predicted from the model that such a local disturbance in the temperature could be propagated as a wave and derived the second sound velocity from the two fluid equations. Peshkov*^ discovered these temperature waves with the same velocity prescribed by the model. Despite the success of the model in predicting these waves the equations fail to explain the behavior of the dissipation measured in the fluid when it flows through a narrow tube at high velocities. In this situation the dissipation measured is much larger than what the two fluid model predicts. In this case the flow is no longer laminar but turbulent. This is what we call superfluid turbulence^.
1.3 Turbulent Thermal Counterflow
The easiest way to establish this turbulent state is to produce a heat flow through the helium and generate what is known as thermal counterflow. Figure 1 shows a schematic representation of a counter- flow apparatus. When we apply a heat current Q at one end of a sample of Hell contained in a channel, the helium supports the heat flow by the counterflow of the normal and superfluid component. The heat is carried away by the normal fluid component with average velocity Vn=*Q/(/osTA) and to conserve mass the superfluid moves toward the heat source with the average velocity V =-(p V )/p , since s n n s
Psvs+PnVn=0 in a steady state. The average relative velocity of the normal fluid with respect to the superfluid v=vn~vs gives for thermal counterflow V=Q/(p sTA). The relative velocity is proportional to Q. s Information about the flow can be obtained by several ways: the relative increase of T, P or p, the attenuation of second sound, or the transmission of ions through the flow. Below a critical heat 4
He Bath atT,P,/i
T + A T A P + A P T + Q rvwwvi
Heater Figure 1
Schematic diagram of a counterflow apparatus current Qc « the iPlow is laminar and the dissipation comes only from
the normal fluid viscosity. The pressure difference obeys the
Poiseuille formula for the laminar flow of the normal fluid. The
chemical potential difference is zero which agrees with the perfect
dissipationless flow of the superfluid. From the equations of motion
1.1, 1.2 we can find the laminar temperature difference flp Fn8,Vn AT, = — = --- — , H: length, d: diameter where F is a factor that lam p .2 s /osd depends on the geometry of the channel. This result is very well
established experimentally and provides another test of the model.
An excess dissipation due to superfluid turbulence is observed above
a certain heat current Qc> The pressure difference in this state is not very different than in the laminar region, but the temperature difference and the chemical potential difference increase dramati- . 3 cally by an amount approximately proportional to Q (see Figure 2), data from Reference 21 and 30). It is this extra dissipation that 12 lead Gorter and Mellink to add an extra force to the two fluid equations. They called this force the mutual friction force Fsn-
Because of this force, the independent motion of pg and p^ breaks down and their interaction through Fgn produces the extra dissi- 13 pation. Feynmann in 1955 proposed that the extra dissipative force has a microscopic origin in the scattering of the normal fluid excitations from a random distribution of quantized vortex lines.
This process gives rise to the large AT and Ap in the "non-linear" regime. Superfluid turbulence is the state of flow of Hell filled with a random vortex line density. A quantized vortex line consists AT (mK) m | - 0 2 0 4 ’ S CP 7 5 6 2 0 3 iue 2 Figure Cubic dependence of the dissipation the of dependence Cubic 0 0 0 0 0 0 0 70 60 50 40 30 20 10 T =I.60K 50 W(mW/cm2) Q (/x-watts) 100 150 6 7
of a cylindrical core of atomic size around which the superfluid
circulates with a velocity inversely proportional to the distance to
the core. The circulation around a vortex is quantized in units of
h/m^. In the next section I will discussed the theories and models
used in superfluid turbulence.
1.4 Theories and Models
The physics of superfluid turbulence involves making the connec
tion between the microscopic distribution of quantized vortex lines
and the macroscopic mutual friction force that is determined by this
distribution. To make this connection we need to establish a way to 12 treat Fgn in the two fluid model. Gorter and Mellink proposed the
addition of an average force F to the V and -F to the V sn n sn s 12 equation. They also introduce the "mutual friction approximation"
that makes the two fluid equations easier to handle. Most experi ments probe the properties of superfluid turbulence under a steady
state and the measured values of AP and AT are spatially averaged
so the two fluid equations become
- (VP) - p s
— <\7P> + p s
< >: time and spatial average of the axial component. Adding 1.3 and
1.4 we get
4 2 4 (VP) = nv
8.F qn AT = | fi.< VT> | = AT. + — — 1.6 lam p s
,■> , fi,F <3t) ,- sr i.7
Vinen2-^ in 1956 made the connection between Fsn and a vortex line density based upon his results of second sound attenuation in rotating Hell. He proposed that the superfluid turbulent state is described by a random distribution of lines that is homogeneous, isotropic, and drifting with the superfluid. He found1**
B p p K F = — f - 12- L V 1.8 sn 3p o
L q : steady state line density (length of line/unit volume), V: relative velocity, k: quantum of circulation, B: coefficient depending on temperature. This is the principal connection between experiments and the determination of the line density. Since Fgn is a force due to a scattering process it is proportional to the density of scatterers (i>Q). to the relative velocity between the scatterers and the excitations of the normal fluid, and also to the friction
16 17 coefficient which is the Bp p k/(3p) term. Vinen ’ also proposed s n a phenomenological model to account for the vortex line density Lq in a counterflow with relative velocity V. He assumes that the time rate
of change of the line density is due to two competing processes, one
of line creation, one of line destruction. The line density will
increase due to line stretching. He models this process on the growth
of a vortex ring and finds
dL X1B O/O _ = -J- (— )L3/2 V 1.9 dt p 2 p
where and B are determined experimentally. The line destruction
term is modeled on the mutual annihilation of two parallel and
oppositely polarized (opposite circulation) vortex lines when they
get close to an average interline spacing 9,=1/L. Vinen finds
dL X2K 2 dt 2ir~ L i-10
where X2 is determined experimentally. The time rate of change of
the line density is equal to
dL _ dL dL 1.11 dt ~ dt dt
by equating this to zero Vinen finds the steady state line density as
2 2 Lo = Y V 1.12
irB/o^x^ where y = — ---- . Different modifications of the Vinen model exist 2 and are discussed in references.
The Vinen model and the different modifications of the basic
Vinen equation remained the guiding light in this field until 1978. A more realistic approach to the problem has been produced by 18 19 20 Schwarz ’ ’ at IBM. His model can be used to compute the
microscopic distribution of vortex lines. He writes an equation to
describe the instantaneous motion of a vortex singularity as a
function of the relative local velocity between the normal fluid and
the superfluid components using the localized self-induction approxi
mation. This approximation assumes that only the local radius of
curvature and the local velocity are important and neglects long
range non-local effects. This approximation is not valid when a line
whose motion is being described comes close to another one. Schwarz
postulates that in this case, the lines reconnect to each other.
Without this assumption the tangle would eventually be annihilated
at the channel walls. Figure 3 shows how the runaway growth is
offset by line reconnections which result in the creation of new
small rings that can in turn grow and reconnect. Schwarz numerically
simulates the time development of this description of a vortex line
system in a uniform velocity V for various initial conditions. In
the steady state he finds a homogeneous distribution, independent of
2 initial conditions, where the line density LQ is proportional to V
and agrees with Vinen's model. The mutual friction computed from
this model and the parameter y can be compared to experiment and 11 agree very well at large relative velocity or high Q. Both Vinen's and Schwarz's models predict equally well the behaviour of superfluid turbulence but Schwarz's theory does not contain any adjustable parameters and is based on microscopic considerations. Both theories 11
Figure 3
Crossing vortex lines reconnecting in the Schwarz model 12 make predictions about the steady state properties of the turbulence and almost all experiments have probed that aspect of the turbulence.
In the context of the Schwarz theory the vortex line density fluctuates about the steady state value and this can be detected as fluctuations in the chemical potential difference. In the Schwarz’ picture the line density fluctuations are due to random line crossing events and to the motion of the lines driven by the normal fluid.
1.5 Previous Experiments and Motivation for this Research
Many types of flow11, different flow tube materials11 and geometry11 have been used to probe the properties of superfluid turbulence. The most popular flow isthermal counterflow. In smaller channels (=100 pm) the probes of choice are measurements of 21,22,23 , „ /iB1 22,24,25,26,27,28 the pressure (P) , temperature (T) and 29 30 31 chemical potential (p) differences ’ ’ . The material of choice is glass because it has proven difficult to reproduce data obtained 2 in metal flow tubes. In the larger diameter tube (area =cm ) the differences in P,T,y are too small to be measured so the probe of 24 30 32 33 34 choice is second sound attenuation * > • > or ^on 35 36 37 transmission ’ ’ . Most experiments done previously have studied the steady state properties of the dissipation due to superfluid turbulence. Figure 4 shows a typical line density plot obtained from steady state chemical potential difference data in thermal counter- flow. We can distinguish a laminar region where the line density is zero. At a certain heat current (or relative velocity) there is a 13
L0: Length of Vortex Line / Unit Volume d = Flow Tube Diameter L'o2d: Dimensionless Quantity
50 i— '— r i— r
40 - TI
30 "O CM T I - O 20
1 0
Laminar III 0 1 0 20 40 60 80 100 120 140
Q (/jlvj) Figure 4
Vortex line density graph vs. heat current 14
discontinuous (this will be apparent in Chapter 4) transition to a
state TI of larger dissipation due to the onset of superfluid turbu
lence. At a larger heat current there is a continuous transition to
a state XII of yet larger dissipation and increased turbulence. It
is this XII state that appears to be homogeneous and described by the
Schwarz theory.
Xhe purpose of this work is to gain information about the
dynamics of the turbulent states. We want to measure fluctuations of
the line density (our dynamical variable). In a classical turbulent
flow the dynamical variable exhibits a noisy or chaotic time
dependence. Xhe transition to turbulence can be rich in details.
Even if the average dissipation in the flow shows no structure, a
study of the fluctuations in the dissipation can reveal different
flow states of complex character. Xhe main focus of this experiment
is to study the transition from XI to the homogeneous state XII.
Xhe first group to report noise from vortex line turbulence was
38 39 Moss et a l . in 1975. Since then Donnelly et al. also reported
an observation of fluctuations in turbulent counterflow. Xhe inter
pretation of these experiments is made difficult by the lack of
characterization of the turbulent state being studied. Xhese experi ments were performed in very large channels using second sound and 40 negative ions. Recently Smith et al. have refined the ions measurements and established a high frequency asymptotic behavior of
1/f in the power spectral density of the fluctuations of a negative
ion current. In the large tube used in that experiment only the XII turbulent state is present and the fluctuations are only representative of that state. Our interest is to use a small diameter channel and try to observe fluctuations in the chemical potential and the temperature differences. Experiments performed in small diameter channels before never observed any fluctuations because of their long length (=10 cm). We thought that by using a short tube (1 cm) we would have a better chance of observing the fluctuations because they would not be averaged over as long a length of tube. In a small channel we are able to link the noise power with the line density in the two turbulent states and also we are able to study the dynamical features of the transition between the two states. This transition remains somewhat of a mystery and the different experiments described in this dissertation provide new data to understand better the nature of this transition. CHAPTER 2
APPARATUS
2.1 Introduction
In this chapter I will describe the experimental apparatus used to perform the different experiments describe in this work. I will start with design considerations and a probe overview. In the beginning of this project the design was performed by a postdoctoral research associate Dr. V. U. Nayak and a graduate student Don
Griswold. The problems faced in the building of the probe will be explained in each section dealing with the proper probe subsystem.
2.2 Design Considerations
Previous experiments that were designed to look for fluctua tions in turbulent counterflow were based on second sound attenuation 38 39 40 or negative ion ’ ’ techniques limiting those experiments to large tubes where only one turbulent state exists. We wanted to look for fluctuations in a small glass tube where two turbulent states exist and we could also look for dynamical properties in the transition between the two states. Using a small (130 pm) tube meant measuring temperature differences and chemical potential differences fluctuations. Previous measurements using long but small diameter glass tubes never saw any fluctuations in those quantities,
16 probably because they were averaged over the whole length of the tube. To rectify this problem we had to design an experimental probe which could accommodate a short (1 cm) flow tube. We wanted to have a "modular" design for the flow tube holder so that we could easily change to a longer flow tube and re-do the experiment to test length effects and do some local measurements of the temperature differ ences. We thought at the beginning of this project that temperature regulation would be a key to successfully detect fluctuations. To minimize heat leak we went to a "sub-pot" design, where the main helium reservoir is isolated from the 4 K helium bath by a vacuum can. All leads going down to the actual experimental area had to be thermally anchored and the materials used in the reservoir area had to be poor thermal conductors so that the heat leak to the reservoir / was kept to a minimum (545 yW). We also thought that external noise would be a key in this search for fluctuations so we wanted to use coaxial cables for all leads going down the cryostat. We also needed provision for a "superconducting feedthrough" to isolate the leads going to the S.Q.U.I.D. from external pick-up.
2.3 Probe Overview
A schematic diagram of the entire experimental probe is shown in
Figure (5). The probe consists of a helium reservoir to which the flow tube and the measuring devices are connected, enclosed in a vacuum can. This assembly is suspended by stainless steel tubing.
The tubing serves as pumping lines for the helium reservoir and for Feedthroughs 18
Helium Reservoir Vacuum Can Pumping Line Pumping Line Superleak Tight Valve Stem- S.Q.U.I.D. Liquid Helium Stand - Of f- Transfer Port
Support Rod Radiation Baffle
Valve Stem Housing
Normal Superconducting Feedthrough- Feedthrough Pumping Line Offset
Helium Reservoir- -Vacuum Can
Flow Tube Counter Flow Cell Transducer Inlet Super leak- Thermal Pressure Connection I I'ti Transducer Supporting Plate Figure 5
Schematic diagram of experimental probe 19
the vacuum can. The probe includes all standard cryogenic techniques
like radiation baffles spaced over the whole length of the probe and
pumping line offsets at the ends of the lines. There is also a
superleak tight valve that connects the bath to the helium reservoir.
Provision for this valve stem had to be made on the baffles. A
S.Q.U.I.D. probe is needed for some measurements and again a routing
for that probe had to be included in the baffles. The flange that
separates the bath space and the vacuum space ("lower flange")
contains two feedthroughs, one to pass the superconducting wires to
the S.Q.U.I.D. and the isolated wires to the pressure transducer, the other one to pass through all the other leads needed in the experi mental area. The feedthroughs to bring the wires from the room to the bath space consists of two brass standoffs with an inner lip to 41 hold a "macor" ceramic disk. The "macor" disks are pierced each with 36 holes (27 mils in diameter) and a 25 mils diameter copper rod
(1" long) is inserted in each hole. The seal is made by pouring 42 Stycast (Emerson and Cumming 2850 FT) in the brass standoff. An infrared lamp was used to cure the stycast. The copper rods are used to solder the coaxial cables used in the experiments.
The superleak tight valve used to fill the helium reservoir con sists of a needle valve that can be opened from the room using a long stem. Turning the stem lifts the "needle" from its seat letting the liquid from the bath fill the reservoir. Every other component on the probe will be discussed separately in a section of this chapter. 20
2.4 Vacuum Can
The vacuum can is assembled from three pieces of machined brass.
The body of the can consists, of a cylinder of brass (21" long, 2.825" in diameter) to which a disk of brass is hard soldered to make the bottom (see Fig. 6). The top collar was machined to produce an
0-ring surface between the vacuum can and the lower flange of the probe. The collar was hard soldered to the body. The 0-ring was
43 made from .0625" diameter indium wire . The 0-ring surfaces were cleaned with methanol and freon. The 0-ring itself was cleaned in an ultra-sonic cleaner in freon. After everything was cleaned care was taken not to touch any surfaces with fingers so that grease would not be present when the 0-ring was pressed. A superconducting shield consisting of a 5 mils thick lead shield was soldered on the surface of the can using alien salt flux and "soft solder" (lead-tin solder).
This provides a shield against external field to protect the
S.Q.U.I.D. measuring device inside.
2.5 Helium Reservoir
This reservoir consists of a body made out of OFHC copper in cylindrical shape of 7.125" long and 2.5" in diameter, this is big enough to contain .6 liter of liquid. On the middle of the reservoir there is a thermal anchor which consists of a OFHC copper block with
76 slots milled into it. In each slot a 45 mils copper pin was glued and the wires which were used in the measurements were soft soldered on one end. On the other end the coaxial cables coming from the 21
0 - Ring Surface
Top Collar
Threaded Hole
Brass Cylinder
Brass Disk
Figure 6
Vacuum can 22 normal "feedthrough" were soft soldered. With this thermal anchor we could eliminate thermal E.M.F. between coaxial cables and measur ing wires and also we could stabilize the heat leak from the bath
(throughout the coaxial cables) to a fixed value. One of the stain less steel supporting rods of the probe is used as a pumping line for the reservoir. The use of stainless steel minimizes the heat leak from the 4.2 K bath to the cold region of the experiment. To the bottom of the reservoir is "hard soldered" (silver alloy solder) a piece of copper which consists of a disk with a stand-off pipe to which an 0-ring surface is hard soldered (see Figure 7). A matching piece of this 0-ring surface is machined out of OFHC copper and gold plated to prevent oxidation. The purpose of this arrangement is to make flow tube removal and assembly easier. On a separate jig we can assemble and glue the flow tube on the bottom piece. When the protective shield on the flow tube is cured (see Section 2.6) we can attach the flow tube assembly to the reservoir by means of an indium
0-ring (.031" in diameter). The bottom of the reservoir has a hole of .5" diameter to which a brass 0-ring surface is hard soldered.
This is matched by a mating 0-ring surface to which two thermometers 44 are attached (see Figure 8) on a block of AGOT graphite . In this way the thermometers used for temperature regulation are immersed in the liquid. The bottom flange is also drilled to accomodate a large
(75 mils) capillary to connect the reservoir to the bottom chamber of the pressure transducer. To the bottom of the reservoir is also hard soldered three long (24 cm) OFHC copper rods of .125" in diameter. 23 | To Pumping Line
Helium
Thermal Anchor for Wires
Stand-Off Thermometers P i p e - " \ Holder
Regulating Heater Hard Solder s >
O-Ring Surfaces Piece to Attach Flow Tube Copper Rod (Thermal Anchor Hole for Hole for of Transducer) Vespel Rod Flow Tube
Hard Solder Support Platform
Figure 7
Detailed diagram of Helium reservoir 24
Helium Reservoir
" 7 - r ? y - F 7 .
Hard Stand-Off Pipe Solder
0-Ring Surface AWAV
-Germanium Carbon Thermometer Glass Thermometer Themometers - Wires Feedthrough
Stycast 1 1 2850 Ft E23: Copper Seal : Brass : Agot Graphite Side View of Mount and Thermometer
Figure 8
Thermometer mount 25
These rods are needed to support a holder for the pressure
transducer. The diameter of the rods make them also a good thermal
link from the support to the reservoir. This is an important
feature for chemical potential detection as will be seen later.
2.6 Flow Tube Assembly
The flow tube consists of a 130 pm diameter glass tube cut to
a 1 cm length using a diamond blade saw. The tube is cleaned in
methanol then freon in an ultra-sonic cleaner. A tungsten wire of 3
mils diameter is used to free the tube of all foreign matter. We
first glued the flow tube to the bottom flange of the reservoir after
it had been wrapped with an impregnated (with Stycast 2850 FT) paper
towel. Using the bottom flange of the flow tube assembly as a
support, three vespel SP22 rods are screwed in the flange. The
counterflow cell was then glued at the bottom of the flow tube being
supported by the vespel rods. Vespel SP22 was chosen because of its r very low thermal conductivity (10 pW/cmK) since we want the helium
in the flow tube to be the main thermal link between the cell and the reservoir. The rods were .125" in diameter and to avoid very difficult machining in threading this flexible material we had to tape the rods tightly using teflon tape and mark the threads on the tape with a nut. This technique has proven to be easy and reliable.
After the shield around the flow tube and the glue of the counterflow cell had cured we pressed that assembly with an 0-ring on the reservoir bottom flange. Only then we could test for blockage of the tube using a leak detector and looking for a leak from an opening in
the cell to the reservoir. The main problem in this assembly is to
avoid violent "movements'* when the 0-ring is being pressed since we
broke a few flow tubes before we got something sturdy on the cell.
2.7 Counterflow Cell
The counterflow cell, Fig. (9), is made of OFHC copper and its 3 helium volume is very low (.057 cm ). This is necessary to insure
that the cell body would respond quickly to temperature changes in
the flow tube. Since the cell contains two thermometers for steady
state temperature measurements we want the cell to quickly follow the
changes in the flow tube since we are measuring the temperature of
the copper, not the helium. The time constant of the cell to temper
ature changes in the tube is 7 msec. The counterflow heater was
wound of Evanohm wire which has a resistance of 200 ohm/foot. The
resistance of the heater is 997.28 ohm. After the heater has been 47 wound it was coated with GE varnish to make a compact coil. The
coil was encapsulated in a copper vessel and that was glued (using
Stycast 2850 FT) into a hole in the cell which under experimental
condition is filled with helium. This insures good heat transfer
from the heater to the liquid. The cell is connected to the top of the pressure transducer by a 24 mils copper-nickel capillary which is soft soldered into the small helium reservoir of the cell. The cell also contains a copper clamp to which a superconducting thin film bolometer can be attached. It is now used to thermally anchor one T
27
Reservoir
Carbon Glass Thermometer Flow Tube GE Thermometer Counterflow Heater Leads | Vespel Stycast 2850Ft Support Rod Seals To Reservoir
Counterf ^Pressure Heater Transducer Top Capillary Copper Outlet (0032") Vessel
Hard Solder
Copper Clamp
Figure 9
Counterflow cell end of a gold-iron thermocouple to the cell. Since the main thermal
link between the cell and the reservoir is the helium in the flow
tube, care has to be taken when cooling the system down. Even if
the reservoir is filled with helium at 4.2 K, the cell can remain at
77 K. There is advantage in filling the vacuum can with exchange 3 gas (typically He) and letting the cell temperature equilibrate to
the temperature of the reservoir before cooling the system below the
X-point. The three vespel rods used to support the cell are screwed
in the cell but that thermal link is negligible compared to the
helium in the flow tube.
2.8 Superleaks 48 The superleaks were made of Vycor glass which is cut to a
length of 1.5 cm using a diamond saw. The Vycor glass has voids of
the order of 60 K. Because of those small pores only the superfluid
can pass through. Since the normal fluid is being blocked the main
heat conductor across the superleak is the Vycor glass itself. The
length of the superleak is determined by physical considerations and
to keep the heat flow negligible compared to the helium in the tube.
Since Vycor is porous in all directions a coat of Stycast 2850 FT is put on the side of the glass. Small brass caps of the same diameter of the superleak are glued to the end to permit soldering of the superleak to the capillaries going to the cell and to the pressure transducer. At the beginning we had two superleaks to insure no differential pressure on the transducer during warm-up. We went to a 29 one superleak design when the reservoir side superleak started to give us trouble with leaks. It is still possible to warm-up without breaking the membrane if it is done very slowly and the reservoir keeps being pumped.
2.9 Thermometers and Heaters
The thermometers used for the temperature regulation are immersed in the helium reservoir. The thermometer part of the feed back loop to the temperature regulator is a carbon glass thermometer,
49 model CGR-1 500, from hake Shore Cryotronics, Inc. Its resistance at 1.6 K is 128.730 Kohm with a dR/dT of -505.38 Kohm/K. The other thermometer immersed in the liquid is a germanium thermometer, model
CR 500, from Lake Shore. Its resistance at 1.6 K is 3.0006 Kohm with i dR/dT of -8.016 Kohm/K. The purpose of this thermometer is to be our temperature standard when we are running. The first thing we do after a cool down is to calibrate all thermometers on the probe against the vapor pressure of helium. To make sure everything is at the same temperature we leave the valve to the reservoir open. The 50 vapor pressure of helium is measured with a MKS Baratron 200 pressure gauge. It was found that the calibration of the germanium thermometer remains basically the same, in the region of interest, after every cool-down, warm-up cycle so it gives us a good tempera ture standard. A carbon glass thermometer is used for the tempera ture regulation because of its greater sensitivity. We have mounted a carbon glass thermometer on the outside of the reservoir just in case the one inside would fail during a run. The problem in having the regulating thermometer outside is an offset in temperature between the copper of the reservoir and the helium inside. The outside thermometer is therefore used only in an emergency. The counterflow cell has one carbon glass and one germanium thermometer glued to its surface using GE varnish (thin coat). A thin .5 mil mylar sheet is put on the cell to eliminate electrical shorts between the thermometer and the cell. Day after day the germanium thermo meter has the same resistance and it is therefore the preferred thermometer for cell temperature measurement. The carbon glass thermometer was added with the hope to provide enough sensitivity to observe temperature fluctuations but it was inadequate. We also added a germanium thermometer on the body of the pressure transducer so that we can monitor temperature changes of the transducer. When we calibrate the thermometers we need to be able to monitor the bath temperature since we are regulating the temperature of the bath. A germanium thermometer is used for this purpose. Table 1 shows the values of resistance and of dR/dT for all thermometers.
The heaters we are using for the temperature regulation are noninductively wound on two of the three copper rods that connect the reservoir to the pressure transducer. The heaters are located as close as possible to the reservoir and they are glued in place with
GE varnish. The Evanohm wire used to wind the heaters has a resis tance of 199.7 ohm/foot. Two heaters were made and installed to insure against failure of one. They each have a resistance of TABLE 1
Values of dR/dT and R for all thermometers at 1.6°K
Thermometers dR/dT (Kfi/°K) R(KQ)
Bath Germanium - 5.960 4.1996
Reservoir Germanium - 3.968 3.0453
Reservoir Carbon Glass (in) -554.900 128.375
Cell Germanium - 8.013 5.171
Cell Carbon Glass -499.366 122.07 .
Reservoir Carbon Glass (out) - 67.493 18.898
Transducer Germanium - 5.262 3.786 1000 ohm. The heater in the bath is mounted on the bottom of the
vacuum can far away from the bath thermometer. That heater is also wound of Evanohm wire, the resistance is 1037 ohm. The heater is
"packed" inside a teflon tube for compactness. After a few warm-up,
cool-down cycle when we had troubles with leaks, we decided to add a
power resistor (1 watt, 1000 ohm) to aid in boiling the helium out
of the cryostat. We also wound a heater around the hole where the
liquid enters the reservoir when the superleak tight valve is open.
This is a precaution against frozen air during cool-down, we flush
the valve with helium gas before cooling but if air is still trapped we would be unable to open the valve. The heater wound of Evanohm
(R=1000 ohm) would hopefully help us get rid of the block.
2.10 Pressure Transducer
The pressure transducer is used in conjunction with the cell
side superleak to form a chemical potential gradiometer. Since the
superleak acts as a chemical potential short, the change in Avi at the
cell is the same as the top chamber of the pressure transducer (see
Fig. 10). It is important for the helium on both sides of the pressure transducer to be at the same temperature. By making the
temperature difference across the capillaries be zero, the chemical potential measurement is converted into a pressure measurement. This can be seen from the Gibbs-Duhem relation Ay=AP/p-SAT. If AT=0 then
Ajj«AP. To insure that this temperature difference be zero we first used a heat exchanger that was put between the two capillaries, the 33
Helium Reservoir at
T , P , ^
LJ I_____ T+AT Lt+Au P+AP
Super leak (Chemical Potential Short)
Pressure Transducer at T so A fjL a SP' Platform at
Figure 10
Detection of Ap one from the reservoir and the one from the cell. It consists of a
long copper-nickel capillary wrapped in a coil, the inside of it was
filled with the helium of the large reservoir capillary and the
smaller cell side capillary was running through the helium of the
reservoir side equilibrating the temperature of both capillaries.
This worked well except that wrapping 50 cm of capillary with small
(=20 mils) outlet capillaries at the output of the exchanger gave us
a lot of troubles with leaks. We decided to make do without this
exchanger, since the body of the pressure transducer is OFHC copper,
thermal equilibrium on both sides of the membrane of the transducer
is very quick. The trick is to anchor well enough the transducer to
the reservoir. The three copper rods coming from the reservoir are hard soldered to a platform that provides us another anchor at the
reservoir temperature (see Fig. 11). We mount the transducer to that platform by bolting another plate on the bottom with threaded rods. 51 The transducer is firmly bolted in place with Apiezon N-grease used on both platforms to insure better thermal contact. To help even further this temperature equilizing process, the capillary coming from the reservoir and the cell side capillary are positioned against each other on an aluminum grill which is attached between two of the copper rods (see Fig. 12).
We are left to design a device to measure pressure. We are using a capacitive pressure gauge where a change in pressure is detected by a change in capacitance. The transducer is designed with two chambers: the upper one is connected to the cell and the lower 35
Copper Rods to Reservoir at T
Hard Solder Hard Solder wwwww Apiezon N-Grease Platform atT Pressure Transducer (Copper) ^Threaded Apiezon Rod (Brass) I N-Grease
'* x '■ V V V V V"
Bolted-On Platform at T Figure 11
Detailed diagram of transducer platform and mounting To j 0 Counterf low Cell Reservoir Copper Rod V/ A to Reservoir X
Aluminum Grill 2 o o -*-»0 o-«o °l 0| ,o ol jo 0 |o Superleak o. io o| lo 01 lo 9 0 * 0 o i i o
o,°! |o!° 0 | ,o ol IO ol lo io °! . Ol o ol Jo O jo IO C4-*40 Z
To Transducer To Transducer Bottom Top
Transducer Platform—
Figure 12
Schematic diagram of capillaries on aluminum grill one to the reservoir. The two chambers are separated by a flexible
52 membrane made of 1 mils gold-plated mylar . Stycast (Emerson- 42 Cuming 1266) is used to make this superleak tight seal. The body of the transducer is made of OFHC copper. The dimensions of the pressure transducer and the different parts are shown on Figures 13 3 and 14. The volume of helium in each chamber is .3 cm . The sensitivity of the device depends on the gap between the measuring plate and the membrane. The bottom side of our device is the most sensitive side because the gap between the bottom plate and the membrane is less than one mil. The gap can be calculated from the value of the capacitance when the membrane is in its equilibrium position. Assuming parallel plate configuration C=A(e^c0)/(d^cQ+de^)
A: area of the plate, cQ : permittivity of vacuum, e : permittivity of mylar, d^: thickness of mylar, d: gap dimension (see Fig. 13). When heat is applied to the counterflow cell the chemical potential decreases at the cell therefore the pressure in the bottom chamber
t increases and the membrane moves toward the top plate. The capaci tance read between the membrane and the bottom plate decreases. The plates, machined of copper, are glued inside a "macor" spacer which are in turn glued in the body of the transducer using Stycast 1266.
"Macor" is a machinable ceramic, it is a good electrical insulator so the plates do not short with the body of the transducer and its thermal contraction is low so we do not change the spacing much by the cool-down. After the membrane is glued, gold facing up, to the bottom rim, we glue a mylar sheet- to the top plate to make sure that 38
Macor" Spacer
0.035 Capillary (Cu-Ni) Wire Feedthrough Soft Solder 1.375 ^ T o p Plate Wire Feedthrough ^
0-Ring Surfaces Gold Plated Mylar Gap d Bottom Plate (Cu)
A "Macor" Spacer 1.1875" Wire Feedthrough S, r0.035nCapillary (Cu-Ni) Stycast 1266 Soft Solder 0.875
Figure 13
Pressure Transducer 0.188 1/16 r3/!6
0.562 0.750
0.500
Copper,Transducer Plate
0.500
0.830
"Macor" Transducer Spacer
Figure 14
Plate and Spacer Assembly the membrane never shorts to the plate. The .5 mil mylar is glued i in place using GE varnish. A lead (3 mils copper) is soldered to 53 the membrane using indium and Indalloy U 5 flux . The superleak
tight joint between the top and the bottom is made with a .030"
indium wire 0-ring. Brass screws are used in pressing the 0-ring to match the thermal contraction of the copper body. The leads to the
plate are soldered using soft solder. Three superleak tight feed- 54 throughs are made for the lead, using a teflon-lined capillary as 55 a support coming out of the body. Fiberglass braid (#28) is
slipped over the capillaries and Stycast 1266 is coated over the
braid. The inlet capillaries to the two chambers are soft soldered
in place and then reinforced with fiberglass braid painted with
Stycast 1266. Construction of the device and the calibration will be discuss in detail in Appendix A and B.
2.11 Feedthroughs and Electrical Wiring
There are two feedthroughs on the lower flange and two on the upper one (room temperature to 4.2 K). The two on the top flange were discussed in Section 2.3. One feedthrough ("normal" feed- through) brings the coaxial cables from the bath (58 of them) to the experimental region. The other feedthrough ("superconducting" feed- through) brings the superconducting wires of the S.Q.U.I.D. and the three coaxial wires used for the pressure transducer from the experi mental area to the bath flange. The normal feedthrough design can be seen in Figure 15. The first step in assembly is to hard solder the 41
Copper Pins
Stycast 2850 Ft
0\*.0.0-.
m ' M ' 0-V-*
2A
“ Macor" Disk 4 9 /7 7 7 /A Soft Solder Brass Cup
Hard Solder 0 .5 0 0 Stainless Steel Stand-Off Pipe
Stainless Steel Co-axial Cables
To Bottom Flange (Soft Solder to Bushing)
Figure 15
"Normal" Feedthrough brass cup to the stainless steel stand-off. The long stand-off permits easier soldering to the flange since it is possible to
thermally anchor the stand-off while soft soldering it to the flange.
A "Macor" disk drilled with 58 holes (41 mils in diameter) is then put on the lip of the brass cup. Every hole is provided with a copper pin (40 mils) to which a stainless coaxial cable is soldered.
The seal is made by pouring Stycast 2850 in a teflon cup which is
slipped on the brass piece. When the Stycast has cured the teflon cup is cut away and the wires from the bath are solder to the pins.
The superconducting feedthrough is of similar design (see Fig. 16).
To shield the wires from external fields, every wire used on the feedthrough (22 mils copper) are enclosed in a 63 mils copper-nickel capillary which is tinned with lead-tin solder. The capillaries are soft soldered to the brass cup after it has been hard soldered to the stainless steel stand-off. To make sure that the copper wires are not shorted to the capillaries, a teflon sleeve is slid over the wires. The feedthrough is then soft soldered to the flange and
Stycast is poured into the brass cup. The wires used for the
S.Q.U.I.D. and the pressure transducer are then solder to the copper wires.
The wires used for the S.Q.U.I.D. are Niobium-Titanium super- 56 conducting wires except for the part going in the feedthrough.
All wires running from the room temperature flange to the normal feedthrough on the bottom flange are stainless steel coaxial cables of model S-l from Lake Shore Cryotronics. The inner conductor and 43
Stycast 2 8 5 0 Ft
z m m Ofa-ZVl7 0 . 5 0 0 9 s m
7 ’Stay-B rite" Solder
Brass Cup Hard Solder
Stainless I I Steel S ta n d -O ff 0.063 Cu-Ni Capillary Tinned with Soft Solder
0.022"Copper Wires
To Brass Bushing on Bottom Flange
Figure 16
"Superconducting" Feedthrough the braided shield are made of stainless steel. All the cables were
57 cut and pre-tinned using Lancol flux and soft solder before being
put on the probe. All the shields were soldered at the ends of the
cables to the cryostat. From the normal feedthrough to the thermal anchor the same type of cable is used and all shields are connected
at one end of the probe. To make the connection from the thermal anchor to the diverse thermometers and heaters standard 8 mils copper wire is used. These are the only wires which are not shielded. The connection of the wires from the cryostat to the equipment racks is made with the help of a "break-out" box. All the pins of the outside feedthroughs are soldered to coaxial cables which are in turn soldered to a 19-pins Amphenol connector^8 (male). We need A of those connectors for our 72 wires. The A connectors go to the
"break-out" box where all the wires are internally routed to female
6-pins connectors, one for each thermometer and heater. The leads for the pressure transducer go to 3 BNC outputs. All the cables from the "break-out" box are connected to the racks using coaxial cables braided together to reduce interference. The end of the cables are connected to male 6- pins Amphenol connectors and these are used to make the connection with the instruments.
2.12 Cryostat and Vacuum Systems 4 The cryostat used for this work is a Helium cryostat.
Temperatures below the \-point are achieved by pumping on the liquid. Helium is contained in a glass dewar which is isolated from 45
a liquid nitrogen space by a vacuum jacket. On cool-down from room
temperature that vacuum space is filled with exchange gas (nitrogen
gas). The vacuum can is also filled with exchange gas during cool
down. The dewars are mounted on a metal frame which is vibration
isolated from the room.
A schematic diagram of the pumping system and the associated
plumbing system is shown on Figure 17 and Table 2. The bath space
59 and the helium reservoir can be pumped by a Stokes Microvac pump
The reservoir can also be pumped by a diffusion pump when we are
warming up. The vacuum can is pumped by a diffusion pump. To make
sure that back pressure does not reach the cold region of the
experiment in case of electrical failure we always close off this
system overnight. The dewar wall vacuum space is now constantly
being pumped by a water cooled diffusion pump. This system is
protected against electrical failure by a solenoid valve that
automatically shuts the system off when the power goes off. The
reservoir is filled with helium everyday and we pump on the liquid
to make sure that the transducer is always full of liquid. There is
a connection between the bath to the MKS Baratron pressure measuring
system so that we can read the vapor pressure of the helium
in the bath. The other side of the Baratron is constantly being 4 3 pumped by a diffusion pump system. Gas supply lines allow He, He
and nitrogen gas to be routed to the various parts of the probe as needed. The cryostat has the capacity to be vibration isolated since
it is mounted on coil springs. The frame of the cryostat has two > 4
7 7 7
Gas Lines A ir Cooled Water Cooled Valve 7 / 7 Helium Reservoir — 8 i i i Helium Bath Vacuum Can 77^7777777777777777777777777 Bath Reference Baratron Pressure Gauge To Stokes Pump
Schematic diagram of cryostat 47
TABLE 2
Description of Figure 17
1: Water cooled diffusion pump
2: Air cooled diffusion pump
3: Bellows (copper)
4: Diffusion pump oil trap
5: Solenoid valve
6: Mechanical pump
7: Gas entry valve
8: Dewar wall boxes filled with sand (=^2000 pounds). When we want to isolate the cryostat we try to disconnect all pumping lines and the recovery system because those connections bring mechanical vibrations to the dewar. When the cryostat is not vibrationally isolated it is supported by wooden blocks. CHAPTER 3
EXPERIMENTAL PROCEDURES
3.1 Introduction
The investigation of the dynamical properties of superfluid turbu lence involved several types of measurements. Fluctuations in the chemical potential were analyzed to determine the probability amplitude distribution and the power spectrum. The noise power at fixed frequency and the system response time were also measured as a function of the heat current. To characterize the properties of the dissipation it was also necessary to do some steady state ("DC") measurements.
Some general techniques applicable to dynamical and steady state experiments will be discussed first. The electronics and signal processing needed for each dynamical experiment being different, I will discuss them ,in a separate section.
3.2 Temperature Measurements
Since we are using resistance thermometers as our temperature standard when we are running, it is necessary to calibrate all thermo meters against the vapor pressure of helium. We used a DC-4 terminals technique using the circuit shown on Figure 18. By using a 4.2 volts mercury battery connected to various resistors we can change the measuring current between 1 milli-ampere to 100 nano-ampere (na).
49 S w itc h V ' B N C -| r -
------r ^Standard or VThermometer Switch y r E T — t Standard I „ » Thermometer S w itc h or Battery Test Current I,I"+ Switch External Source h :
oo c "j 'V B a tte ry 1 1 There is also provision to use an external pico-ampere source if the current needed is smaller then 100 na. The current is sent to a standard metallic film resistor^0 of known resistance and then to the thermometer of interest. By measuring the voltage across the standard, we know the current going through the thermometer and a measure of the voltage across the thermometer gives us the resistance of the thermometer from R^^V^./V *R . The polarity of the battery th th st st (and the source) and the measuring leads on the thermometer can be switched so that thermal EMF are effectively eliminated by averaging the values of the voltage read for every configurations. By choosing above and below the temperature of interest, a value of dR/dT is also determined. When the calibration is done all resistance measurements are made using an AC bridge measuring technique. The bridge is designed to performed a three terminals measurements on the thermometers (see Figure 19). R on that figure is a 1 Mega-ohm frequency independent s decade resistor box. The variable capacitor has a maximum value of 20 picofarad. For some thermometers that value is too small to balance the capacitive part of the signal. We then add the desired capacitance in parallel to the variable capacitor and use the variable one to fine tune the zeroing of the capacitive part. The value of Rq (equal arms bridge) is 500 ohm. Four bridges were built and they were all included in one box, "AC bridge box", built as an 61 M.S. project by Alex Childs. Each bridge is in a separate shielded compartment inside the instrument to insure minimal interaction between each circuit. The bridge is excited at about 1000 Hz by 52 Phase Lock-in S p litte r A m p lifie r TH Figure 19 AC-Bridge for resistance measurement 53 6 2 using the internal oscillator of a PAR-124 lock-in amplifier (LIA). We run the signal from the oscillator to a phase splitter (see Figure 20). By using a phase splitter, the leads for the off-balance signal are at a virtual ground reducing the capacitance to ground from these leads. The two signals from the bridge are sent to the pre-amplifier (PAR model 116) of the PAR 124 LIA. We use the in-phase signal from the LIA to balance the resistive part of the bridge (using the variable resistor R ) and the out of phase signal to balance the s capacitive reactance after we made sure that the two signals were orthogonal to each other. The amplitude of the exciting signal is chosen so that no self-heating is detected, usually that means peak to peak voltage of 400 milli-volt. Because of the number of bridges we are limited in the number of thermometers we can monitored simul taneously. Usually we only need to monitor the regulating thermo meter and the cell thermometer. The reservoir germanium thermometer is monitor using a commercially available AC bridge, (Linear Research model LR-110: picowatt AC resistance bridge). The temperature regulation, feedback loop will be discussed in the next section. 3.3 Temperature Regulation The thermometer used for temperature regulation is the carbon glass thermometer immersed in the helium reservoir. The regulator consists of a resistance bridge (described in the previous section) whose off-balance signal is sent to a LIA 124. The off-balance signal of the LIA is sent to a feedback circuit which controls the 54 10/xf, Tantalum IO/Lif, Tantalum > 100.37X1 f 100.37X2 I jjii, Tantalum II .OI3kX2 Input Ceramic LM 394 o I.OI46kX2 Output 5.026kX2 Output O o (1 -3 0 X 2 )^ 130.09X2 100.44X2* i 100.92X2 530.09X2 2.936kX2 VA 2 N 2 2 I9 Motorola 2.701 kX2? 10/if, Tantalum - I 5 V Figure 20 Phase-splitter used for exciting the b rid g e 55 heat input into the regulating heater on the reservoir (see Figure 21). The feedback circuit was designed by a postdoctoral research associate, Dr. V. U. Nayak and built as an M.S. project^ by Alex Childs. Since the heat leak into the reservoir and also the heat removed by the pump both fluctuate, it is necessary that the heat input sent to the heater also fluctuates. The off-balance signal of the regulator LIA, which represents the temperature fluctuations, is processed in three different ways. We summed the three resulting signals and then sent.it to the heater. Figure 22 shows a block diagram of the feedback circuit. The signal is first passed linearly to a summing amplifier. The signal is then integrated to correct for long term drift. At last the signal is differentiated to correct for rapid fluctuations. A DC balance is then incorporated after the three previous stages via another summing amplifier. The signal out of this summing amplifier is used to drive a current source connected to the regulating heater. The actual components used for each part 61 are described in Alex Child’s Master degree report . Each part can be switched in .and out of the feedback circuit and each part is also connected to potentiometers so that their magnitude can be controlled. The regulator was typically able to keep the bath temp erature within 20 yK of the desired temperature for extended time periods (5 to 6 hours). 56 Helium Reservoir Thermometer Feedback Resistance Circuit Phase Splitter PAR 124 Figure 21 Feedback circuit for temperature regulation Ln ||l ------CL Ik —VA 0DC (\ — 'Q -(2)—I1' Diffentiator-(2)—I1' - O Linear —It' - Q Integrator —li' DC Balance Summing Summing Amplifier 2 Amplifier Amplifier Amplifier I \ o o a r \ ikft ■AM, ■AM, ||l HH ^ ^ Ik f t , /^— r\—v\A< ||t s\ lk w - t O ? 1 0 k : Source Current Buffer Inverting Integrator Differentiator J-OUT Heater Regulating lnput:PURPLE Output: YELLOW Ground Leads1 BLACK Buffer } From LIA V,N C n Block diagram of the temperature regulating feedback circuit 58 3.4 Chemical Potential Measurements As mentioned in Section 2.10 when heat is applied to the counterflow heater, the chemical potential in the cell decreases creating a pressure change in the pressure transducer (look at Figure 10). This change is detected by measuring the change in capacitance between the membrane and the bottom plate of the pressure transducer. This change in capacitance is measured with a General Radio capacitance bridge model 1615-A. The change in capacitance consists in an average value (AC) and a fluctuating value 6C(t) which depends on time. The average value gives us the steady state change in chemical potential ("DC value") and the fluctuating component ("AC value") is the signal we are processing to gain dynamical information on the turbulent counterflow. Figure 23 shows a block diagram of the measuring circuit. The capacitance bridge is excited at 3.5 volts peak-to-peak by the internal oscillator of a LIA (either a PAR-124 or Ithaco 393 Dynatrac) which is used as a null detector for the bridge. The bridge is set in the three terminals mode eliminating stray capaci tance to ground. The capacitance bridge is balanced by adjusting the capacitance decades (in-phase component) and the resistance decades (out of phase component) to produce a null reading on the LIA. When a counterflow experiment is being done (AC vs. Q) we are interested only in the steady state change of the chemical potential so we can set the output signal filter to a relatively high time constant (125 to 400 millisecond) so that rapid fluctuations are Membrane Pressure ~ V Transducer Bottom Plate --- 1 I___ -1 I___ D etector Low I “ ~ 1 Standard! _ j General Radio 1615-A Capacitance Bridge I— Pre-Amp Input PAR 124 In tern al LIA O s c illa to r Figure 23 Capacitance Measurement Circuit filtered out making the nulling procedure easier. We can run the LIA at a sensitivity of 500 nanovolt to 1 microvolt for heat currents above and below the TI/TII transition. For heat current in the transition region, the fluctuations are so big that we usually lower the sensitivity by a factor of 10. We are able to resolve AC to 3 5 parts in 10 in the transition region, better in the other regions. To convert AC into Ap we need to know the change of capacitance for a change in pressure, AC/AP. When we are at room temperature we can easily do this measurement directly, connecting the two chambers of the transducer to the Baratron pressure gauge and putting different amounts of gas in the lower chamber. We monitor the change in capacitance for every pressure and we can find AC/AP. When we are cold we do not have a direct pressure measurement, we have to rely on a "deflection study" or on pressure computed from the temperature differences measured for different heat currents. By measuring the voltage across the membrane and the bottom plate needed to move the membrane back to its equilibrium value for different heat currents we can deduce AC/AP when the experiment is cold and running. Appendix B will discuss this procedure in more detail. Using thermo dynamic relations we can also convert the temperature difference measured across the flow tube into a pressure equivalent, giving us another way to check the calibration of the pressure transducer. By knowing AP (AC **AP/AC) we can get the chemical potential measure difference Ap=AP/p. Using the equations discussed in Chapter 1, Ap is easily converted into an equivalent homogeneous line density. 61 L = 3.1 o 8.Bp Kv n 3.5 Counterflow Experiments Counterflow experiments involve measuring the dissipation across the flow tube for different heat currents. In our case measuring the dissipation meant measuring the temperature difference across the flow tube using the germanium thermometer on the counterflow cell and the chemical potential difference using the set-up discussed in the previous section. Figure 24 is a schematic diagram of a typical counterflow experiment. The chemical potential detector is the more sensitive device for dissipation measurements. Even in the low turbulence region we can easily resolve changes in the capacitance of 3 femtofarads. The thermometer was not sensitive enough to resolve small temperature differences occurring for heat currents below 50 pW. To measure the power across the counterflow heater we sent the current first through a stable metal film "standard" resistor of 997 ohm. By measuring the voltage across that standard and across the heater we can get the heat current Q=V . . ,/R . _ j^V. standard standard heater The power supply used with the counterflow heater is a Hewlett-Packard model 6114A: Precision power supply. This power supply is very stable and also it allows us to change the heat current by a very small increment (=1 pW) which is needed when we want a detailed map of the line density. The first step in a counterflow experiment is to set a zero, meaning the value of the capacitance (CQ ) of the pressure transducer 62 Helium Reservoir at I.6K Flow Tube Superleak Power Supply Pressure Transducer GR Capacitance Bridge Lock - in Amplifier Figure 24 Set-up for counterflow experiments 63 and the value of the thermometer (R^) for zero heat current. For every heat current applied to the cell the new values of C and R are measured and AC,AR computed. Knowing the value of dR/dT for the thermometer enables us to convert AR into AT. We can plot AR vs. Q, AC vs. Q or convert those measurements into line density to study the behavior of the turbulence. Results on our flow tube will be discussed in the next chapter. 3.6 Spectral Analysis It is convenient to think about our data as a combination of a static (time independent) component and a dynamic or fluctuating 6 3 component : Ap= proportional to the off-balance voltage of the LIA. To gain spectral information on the data we have to analyze the fluctuating part. Our chemical potential detector is sensitive enough to detect fluctua tions equal to .5% of the steady state value decomposition of the data is obtained by performing a Fourier transform on a digitized time series and then computing the power spectrum P(f)=|X(f)|2, where X (f) is the Fourier transform6^ ’ The power spectrum is also related to the mean square value of V(t) 00 (the off-balance signal voltage of the LIA) by V^(t)=J P(f)df. The o first step in computing the power spectrum is to convert the analog fluctuating signal (time series) into a digitized sample (digitized time series). Since our power spectra are computed from a Fourier transform it is important to take precautions in choosing the sampling 64 66 rate to avoid an effect known as aliasing . The "sampling theorem" assures us that the sampled data represents the original data. If we are interested in gaining information in a frequency bandwidth B e (0 to f ), we need to make sure that we sample at a frequency (f )>2f and our digitized time series should contain no S 6 frequency dependent part for f>f0 in other words |X(f)|=0 for Since we specify the sampling frequency from the computer it is easy to make sure that f >2f . To make sure that |X(f)|=0 for f>f s e e we use an eight pole Butterworth low-pass filter (Frequency Device model 901F) at the output of the detector lock-in. We can choose the corner frequency of the filter according to the data range of interested. The output of the filter is then sent to the analog/ digital module of a Minc-11 computer through a voltage divider to make sure that the fluctuating voltage is within the range acceptable to the digitizer (±5.12 V). Figure 25 shows a schematic diagram of the set-up necessary to determine the power spectrum. All digitized time series are stored on disk. We have to average many time series (from 3 to 10) to get an acceptable linewidth on the power spectra. The data taking program and the power spectrum computation program are shown in Appendix C. The factor 1/f in the computation of the s power spectra is necessary to scale the data so that those power spectra taken in a region from 0 to f would give the same results as the power spectra taken from 0 to 3 f for the overlapping frequencies. 65 Helium Reservoir at 1.6 K Flow Tube Superleak Power Supply Pressure T ransducer GR Capacitance Bridge Lock-in Amplifier Butterworth Computer Filter A/D X-Y Figure 25 Recorder Experimental set-up for determination of Power Spectrum 3.7 Relative Power After gaining a lot of information from the power spectrum we needed to study the growth in the amplitude of the power for a given frequency as a function of the heat current. Since the linewidth on our spectra are too big to distinguish small effects for the heat currents close together we had to use a different approach to measure that growth. The time involved in taking many power spectra necessary to reduce the scatter in the data to gain information at one frequency would also have been much too long. Since the power at f is related to the voltage squared, we can find the power at fQ 65 by processing the SQUARE of the voltage . Exactly viTtT = /“W ) |h(f)2 |df 3.2 1 o where h(f) is the transfer function of the bandpass filter used to eliminate all frequencies components except for fQ . We used an Ithaco 4211 electronic filter in the bandpass mode. The filter has a roll-off of 18 dB/octave on both sides of the center frequency fQ . After the signal has been filtered we sent it to an analog squarer ("home" made) and then to an amplifier to make the signal in the volt range. We then digitized this signal obtaining 2048 points and we averaged all those points to get a number proportional to the power at f^. Assuming that P(f) is constant in the range Af then P(f)=P(fQ) 65 so that Equation (3.2) becomes 67 If we can approximate h(f) by a rectangular function of width Af=f/Q, -where Q is the quality factor of the filter, for a sharply peaked 00 2 38 signal at f_, then J |h(f)| df=fA/Q so V(t)=f "P(fA)/Q . So for U q U L) U a center frequency f V| (t) P(fQ ) = Q — p 3.4 o the average over V2 is the average of 2048 points digitized at the output of the analog squarer. We stored this number proportional to the power and then repeated the procedure for the same heat current about 10 times to get "acceptable" scatter in the data. Figure 26 shows a schematic diagram of the set-up used to perform this experiment. 3.8 Time Constant Experiment In order to study the response time of the system around the TI/TII transition it is necessary to make very small changes in the heat current and then digitize the output voltage of the pressure transducer which is due to the resulting small chemical potential change. The circuit used in this experiment is shown in Figure 27. We had to use the power supply in the programmable mode in order to get the small changes needed (<5Q/Q<.4%>. The digital/analog module of the Minc-11 was programmed to change the heat current in small steps. The off-balance signal of the LIA resulting from the heat change is then digitized. We repeat the sequence 26 times for the same step in heat current to reduce the effects of fluctuations 68 Helium Reservoir at 1.6 K Flow Tube Super leak Power Supply Pressure Transducer GR Capacitance Bridge Lock-in Computer Amplifier A/D Band Pass Analog Filter Amplifier Squarer ITHACO 4211 Figure 26 Recorder Experimental set-up for relative power measurements 69 Helium Reservoir at I.6K Flow Tube Superleak Programmable Power Supply Pressure Transducer GR Capac itance BrieJge 1 Lock-in Amplifier Butterworth Filter Recorder Computer Computer Figure 27 " D/A A/D Experimental set-up for time constant measurements ("signal averaging"). The average signal is then stored on disk for analysis. Figure 28 shows a typical averaged response signal. An exponential function (Figure 29) is fitted to the data and the time constant is extracted from the fitted function. The programs used in this experiment are shown in Appendix C. The results from all the experiments discussed in this chapter will be presented next. 10.23 8.184 4.092 6.138 Time (Arbitrary Units) (Arbitrary Time 2.046 0 52.1571 79.4207 24.8935 -56.8972 <1 -29.6336 “2 -2.37004 oo no Typical digitized response signal ro 819.2 1024 Fit Data 409.8 614.4 Time (Arbitrary Units) (Arbitrary Time Data Fit 204.8 53.4233 25.3662 -2.69 0 8 -86.8620 <1 -58.8049 -2 -30.7479 'Figure 29 Exponential fit to the response data CHAPTER A EXPERIMENTAL RESULTS 4.1 Introduction In this chapter I will present in chronological order the different experiments we did in order to gain better understanding of the dynamical properties of superfluid turbulence. Some experiments were conceived in advance, others had to be designed when we were faced with a particular puzzle. All data presented was taken at a reservoir temperature of 1.6°K. 4.2 Steady State Data in Turbulent Counterflow The first set of data we took when we got the experiment running was to measure the steady state temperature differences and the chemical potential differences as a function of the heat current. In doing so we can check to see if the flow tube is obstructed or if there is an extra heat load on the cell. When those situations arise, instead of seeing three regions on the dissipation plots (either temperature differences or chemical potential differences), we see an abrupt change even for the smallest heat current: There is no laminar 22 67 state ’ . Because of pump failures, it happened a few times that the counterflow plot did not show the right behavior and we were forced to warm up at least to liquid nitrogen temperature to evacuate the vacuum can which can be full of gas vapour. We can also compare the 73 results we get in our tube with other small glass tube experiments (the AT and the Ap differences scale with the length of the tube 8.) to check the behavior of the system. At first, thermal counterflow gives us a means to check the operation of the reservoir, flow tube, cell assembly. All the results presented in this chapter were obtained at a reservoir temperature of 1.6°K. Results for the AR (same as AT) and AC (same as Ap) are shown on Figure 30 and Figure 31. We can distinguish a region of low dissipation on the chemical potential plot and then a region where the dissipation is approximately propor tional to the cube of Q. On the AT plot we should see a laminar temperature difference (linear in Q) at low heat currents but with a short flow tube those differences are so small that we cannot resolve them with our germanium thermometer. Figure 32 shows a plot of the vortex line density as a function of the heat current. This way of presenting the results is very convenient in reducing the dynamical range of the data. On one side of the graph the quantity /AC/Q is on ordinate and that is connected to LQd (dimensionless quantity since d is the diameter of the flow tube) on the other side. /AC/Q is really what we measure and to convert to line density we need to use the calibration number for the transducer. The abscissa is the heat current and the top scale shows the conversion of Q to relative velocity using the equations described in Chapter 1. In Figure 32 we can see the region of no dissipation and then a "discontinuous" jump to a region of lower line density. That transition from Lq = 0 to the TI region is hysteretic^®’^ as can be seen from some data points on the abscissa extending past the TI region. It was really R (ohm) AR 34.78 52.17 86.95 17.39 69.56 180 144 108 72 36 2.17 8.68 4.34 10.80 > 6.51 c CD % Figure 30 AR vs. AC (Picofarads) 4.3696 5.4620 2.1848 3.2772 1.0924 36 iue 31 Figure C s Q vs. AC Q (/i.W) Q 218144 108 72 76 V (cm /sec) 8 12 16 0.1 10 4 0 0.088 TI 30 > 0.066 "O CM O 0.044 0.022 10 0 0 56 84 112 140 6 (yxW) easy to create this situation in this short flow tube. We even could increase the heat current way up past the TI/TII transition and still detect no dissipation. A small mechanical disturbance on the cryostat would suffice to "create" the vortex lines and produce the correct dissipation. Another feature of the laminar to TI transition that we observed is "intermittency". We could see the chemical potential jump from its value in the TI region to zero suddenly. In a few rare occasions the chemical potential jumped back to the TI value without any "external" help. Most of the time a little tap on the cryostat was the only thing needed to bring back the chemical potential. If the temperature regulation was not particularly good on some days (AT>20vjK) we could associate this jump back to the laminar state at low heat currents with big fluctuations in the reservoir temperature. This phenomenon of lines being "flushed away" 68 suddenly has been seen in some earlier work in this field . It is puzzling to us to think of those lines being flushed down the tube and disappearing so quickly. Future work with this apparatus will try to measure the fluctuations of the chemical potential or of the temperature difference at very low heat currents just at the edge of the laminar/TI transition to see if we can gain more insight on this phenomenon. This will be a difficult experiment to perform because of the great instability of the transition. Another interesting feature on the line density graph (Figure 32) is the rather broad and continuous transition from the state of low line density TI to the state of homogeneous, higher line density 79 Til. Previous experimental data concerning this transition consist of the measurements of Qc2 as a function of temperature and tube 26 70 size ’ . From Figure 32 it is difficult to identify Q g- During all the time we took data at 1.6 K we could never detect any sign of hysteresis in the TI/TII transition. It would have been difficult to detect because of the dissipation already present from the first turbulent state. 5 The resolution (3 parts in 10 ) of our chemical potential detector allows a more detailed study of the TI/TII transition (box on Figure 32). Figure 33 shows the high resolution data taken in the TI/TII transition. The points a,b,c marked are going to be discussed in the following sections. The transition turned out to be more complex than anticipated at first. We can see from ~80 yW to 93 yW that the line density has a different dependence on the heat current than the state TI but is still "smooth" in Q. At abut 91 yW there is an abrupt change in the line density to reach an almost vertical slope and then the line density rounds off to settle in a smooth dependence in the heat current that is characteristic of the homo geneous state TII. The state TII is the one present in large channels and the steady state properties of that state have been well established experimentally and also have been compared with the Schwarz theory. It is the behavior of the dissipation in the transition regon which is less known. Evidence of the very steep change in L^d has been seen before in the work of Ladner*^ and also 22 25 68 in the work of Brewer and Edwards ’ ’ . Except for that steep V (cm /sec) \ iue 3 ih eouindt i ITI transition TI/TII in data resolution High 33 Figure > v o / d J P \ 3/1° 1 •o . 3 80 change, the TI/TII transition was believed to be "smooth". We were surprised to see all the structure in the transition and to "convince" us we went ahead and tried to reproduce the data. All the points taken were reproducible and we could change the heat current by a very small amount to make sure we did not miss any important features. At every heat current that we took data, we could reproduce the value of AC (over 100 points). From this high resolu tion data, it looks like two different transitions are taking place. At the first onset the line density changes from its value in TI to a value of higher slope. Then another phenomenon takes place that causes the line density to change its Q dependence again. Those two phenomena make the determination of Qc2 very ambiguous ... what transition defines a critical heat current? Previous measurements of * 26 Qc2 in tubes of comparable diameter for 1.6 K was ~88 yW . In Figure 33 nothing indicates a different behavior at 88 yW, it is just in the middle of the broad and continuous transition from TI to TII. There has been earlier effort in trying to understand fluid 26 71 instabilities in terms of a phase transition ’ . If we define an order parameter in our system as t\> = L%d(TII) - L%d(TI) 4.1 o o we can easily establish an e fwhere e=(Q-Qc)/Qc) dependence for i|i if we choose the "right" Qc2' This result is consistent with the Ginzburg-Landau picture of a continuous phase transition. We can actually define Qc2 this way, but because of the steepness and the 82 structure of the TI/TII transition we cannot get a good fit for \j> everywhere. We really have to think of the TI/TII transition as a broad, continuous transition with no particularly well-defined heat current. In the next section I will discuss our first measurements of the dynamical properties of the system. 4.3 Power Spectra Measurements When this experiment was in the design stage, it was considered to be a "search" for fluctuations in chemical potential and tempera ture differences. When it started running it became clear that we would not have any problems observing the fluctuations in the chemical potential differences. To this point we have evidence of fluctuations in the temperature differences but some further t research and development is necessary on the temperature sensor so that we can obtain a good data base on that phenomenon. The first analysis performed on the chemical potential fluctua tions was the determination of the power spectrum for different heat currents. In Section 3.6 I explained how the data was processed. Here I will present some relevant results. Figure 34 shows the time series taken when the heat current is zero, our background. This is the signal at the output of the Butterworth filter (input of the filter being the signal out of the chemical potential detector LIA) which narrows the bandwidth of interest from DC to 5 Hz. Figure 35 is a plot of the power spectrum of the time series computed from a 72 Fast Fourier transform analysis on the digitized data. To see if - O/zW (Lock-in Sensitivity 0./x3/iV) ie ayn sga o or background our of signal varying Time iue 34 Figure ■U h- CL) E 83 84 YiV • •* Frequency (Hz) Frequency CO C\J o i i i U ) d°'&on Figure 35 Power spectrum of the background signal the two turbulent states have different characteristics and to see if the transition from one state to the other changes these character istics, it is necessary to perform this analysis for many heat currents. Figure 36 shows the time series for different heat currents as we go from the TI region to the TII region. Figures 37 through 39 show the power spectra for three different heat currents, one in the TI region, one in the transition, one in the TII region. We can see a change in the frequency dependence of the power spectra for heat currents in the TI/TII region. Since the power at low frequencies in that region is so much higher than background noise, the low frequency rise of the spectrum is covered. This explains the shape of the spectrum in the TI/TII region. When the power decreases again when the state TII has evolved, the low frequency rise becomes evident again and the shape of the power spectrum for heat currents in the TII region is similar to the TI region. We did not know what to expect when we started the search for fluctuations. In the back of ours mind we were hoping to see maybe a preferred frequency or evidence of period doubling. Looking at all the power spectra, we observed that the fluctuations reveal them selves as broad-band noise with no sharp structure in the spectra. Even if the frequency dependence is featureless, the dependence on heat current is very interesting and will be studied next. Because of the bandwidth of the spectra, they are obtained by averaging between three to ten runs, it is difficult to observe small frequency effects that may be present. To remedy this situation it 86 $ T ime T si OA c s6dhoa Figure 36 Time varying signal as function of heat current 00 5.0 4.0 V* 2.0 3.0 Frequency (Hz) Frequency V • v: *, ■* W r*. ll 1.0 • -• • -• •••»- £.• •• I*■ ^ < , -V. , 4 0 / 0 2 2 4 6 0 o c n O CL T CD CO H- C TJ (TO £ (D 1 T3 0 spectrum in TI region 88 in •« | O' e r s •• U_ O • ••• •v (1) d°'6oi Figure 38 Power spectrum in TI/TII region 89 s • • • N X > * Oc CD 3 ct o I 01 U ) d ° '6 o n Figure 39 Power spectrum in TII region is convenient to look at many spectra on a log-log graph. When we plot the data we can average a group of thirty points around each frequency and therefore create a graph which is easier to read. By placing spectra obtained for different heat currents on the same graph we can also see slight changes in frequency dependence for the different regions of turbulence. Figure 40 shows a graph for three different heat currents in the TI region. Because of the low frequency rise in the background, the frequency range for low or high heat current is limited (TI region or well into the TII region). This low frequency rise is an artifact of the temperature regulation. Because of continuous temperature changes in the reservoir, there is a superfluid motion from the top part of the pressure transducer toward the counterflow cell. In a way the temperature changes make the cell act like a superfluid pump. Because of that motion, the membrane moves and creates a change in capacitance which is perceived as a change in chemical potential with a low frequency response. The low frequency behavior of the spectrum can be changed by changing the time constant of the regulator LIA. That permits us to extend the range of data above background to another half decade in frequency. We did that at many heat currents data to satisfy ourself that there was no other structure in our spectra. It became clear early in our research that the most interesting aspect of the fluc tuations was happening around the TI/TII transition where we are above background even at low frequencies. We therefore have concen trated on that part of the data. 91 o o o o _o : oo o in iri o n o o □ 1^ 00 < 0 D 1 0 <30 <1 O Loglo f(hz) Loglo 1 0 00 m m 1 U)d 0 l6 < n Figure 40 Log-log graph for 3 heat currents in TI region 92 In Figure 40 we can see that the amplitude of the fluctuations increases with heat current. As we get closer to the TI/TII transition (the spectrum at 75 pW) a small change in the frequency dependence occurs, the spectra start to exhibit some ’’roundness" due to the increase in low-frequency, noise. That trend continues as can be seen in Figure 41. The amplitude of the fluctuations is changing very rapidly in this region. At all heat currents from ~85 pW to 93 pW all the spectra have the same shape. The frequency dependence is hard to establish analytically because we have to use functions with four different parameters. We need two different relaxation times (t ) and two different exponents (n,nQ) for a function of the form P(f) = ------f n f no (-) + (--) t To To fit the data over the whole frequency range, we typically have to use relaxation times which are one order of magnitude apart. The exponents obtained are also not indicative of any known process. For 73 example one fit would have n=.07,T=.04,nQ=2.7,To=.4 . Previous 3 publications on noise from vortex line turbulence reported a 1/f 38 dependence (Moss 1975) and more recently an ion experiment reported 4 40 74 a 1/f dependence ’ at high frequencies (high compared to ~ 5 mHz). In the counterflow channels used in those two experiments only one turbulent state is present, the homogeneous state TII. The fluctuations are characteristic of that state only. Smith explains 93 <30 O m o <3 O O c \ j o in oo o J o 00 O') 00 <3 O Nl in CP <3 O o in oo ro in (i)d Figure 41 Log-log graph heat currents in TI/TII region 94 4 his 1/f dependence by considering the motion of a charge trapped on 40 a vortex line in the tangle as a massive particle in a viscous medium undergoing a biased (electric field) Brownian motion toward the ion collector. He shows that such an analysis produces the current frequency dependence in the power spectra. It is difficult to separate in his work the effect of ions moving from line to line to the collector or the effect of an ion trapped on a line and that line fluctuating toward the collector. Also Smith found an important 40 74 dependence of the power spectral density on vortex line density ’ The channel used' was 8 cm long by 3 cm in diameter ... we can see using the equations of Chapter 1 that the line density (Lq ) in our experiment is about 10,000 larger than Smith’s experiment because our flow tube is much smaller. It is difficult for all these reasons to / compare the results obtained from our spectra to previous measure ments in this field. Figure 42 shows two spectra for heat currents in the TII region (95 pW and 120 pW) . The spectrum at 88 pW is also on the graph to show how much the amplitude of the fluctuations drops when the state TII has evolved. For very high heat currents (Q>110 pW) the vortex line density noise is barely above background. Where the spectra in the TI region were frequency independent in the low frequency region, the spectra in the TII region have some frequency dependence at low frequencies before becoming flat around 1 Hz. It is important to note here that the amplitude of the fluctuations decreases with increasing Q in the TII region, the region of homogeneous turbulence. The number of lines is therefore not the dominating factor in this 95 in 0 0 0 0 o o 0 0 . 0 0 LO 00 m o o o o 0 0 05 cvi on o □ < > □ ( oq, o o Loglo f(hz) Loglo CM CM ro 1 1 U)d °'&on Figure 42 Log-log graph for heat currents in TII region 96 region. The fluctuations could be averaged over a larger number of lines. The bandwidth where the noise generated by the vortex lines is higher than background noise is of the same range as what was 38 40 74 observed before ’ ’ . I have only presented data up to 5 Hz because at frequencies above that, the amplitude is just barely above background and also because of a resonance in the transducer which gives rise to a large peak in the spectrum. That is the reason why all the spectra seem to be converging toward the same point. They are on the rise of the resonance bump. Different mechanisms to change the position of that resonance will be discussed in Appendix B. For the first time, chemical potential fluctuations which are directly related to the vortex line density fluctuations were observed in turbulent thermal counterflow. The spectral analysis revealed two different behaviors for the fluctuations in the turbu lent region TI and TII. The frequency dependence of the fluctua- v tions is rather featureless but the dependence on heat current is spectacular. Figure 41 shows how much the amplitude changes drastically (3 orders of magnitude above background) around the TI/TII transition. In the next section a closer look at this heat current dependence will be presented. 4.4 Relative Power Measurements The most interesting phenomenon that we observed in the chemical potential fluctuations is the rapid growth and then the decay of the fluctuations around the TI/TII transition. This is reminiscent of order parameter fluctuations in a second order phase transition. There has been a lot of work done recently on the relationship between "dissipative structures" in a system far from thermal equili- 71 75 brium to phase transitions in equilibrium systems ’ . As reported 26 71 76 77 in Section 4.2 earlier workers ’ ’ ’ in this field have used the ideas of phase transitions to try to explain some of their data. Now that we have available some dynamical measurements on the TI/TII transition, can we use the theory of equilibrium phase transition to give some clues on the TI/TII transition? One thing we can do is to measure the divergence of the amplitude of the fluctuations with respect to an "external force". In our case, our microscopic variable is the chemical potential (or vortex line density) and the strength of the external force (X) is the value of the heat current. Now we are not interested in the spectral decomposition of the fluctuations but we are going to study the amplitude of those fluctuations at only one frequency. Details of the measurements were described in Section 3.7. The important assumption in this technique is that we can replace the transfer function of the filter used to narrow the signal to one frequency by a rectangular function. Another important point is that the power spectra have the same frequency dependence around the frequency of interest. In the region where we have performed this divergence test, all the power spectra have similar frequency behavior. 78 Swift and Hohenberg showed that the amplitude of the hydrodynamic fluctuations at a convective instability diverged as 1/e ( ) just like the "static susceptibility" diverges in a second order phase transition. It is with this analogy in mind that we performed this divergence experiment. In our case \=Q. The results for the relative power at 1 Hz are shown in Figure 43. The points a,b,c refer to Figures 33,41. Looking at Figure 43 it becomes evident that we are not observing a simple divergence, instead we see a rather complex behavior of the power amplitude. The rise and fall of the power on both sides of some critical heat current Q „ would c2 have a completely different exponent. It is easily seen from Figure 43 how the amplitude decays so rapidly when the second turbulent state has evolved. We can also distinguish a well resolved peak at 92.25 pW. We were surprised to see so much structure in the AMPLITUDE graph when the power spectra were rather featureless. To convince ourselves that this was not a "frequency effect" (i.e. the shape of the power spectrum) we repeated the experiment at .63 Hz and .1 Hz. Figure 44 and 45 show those results. The error bars on these figures represent the spread of the data for one heat current. As explained in Section 3.7 some "signal averaging" is needed to determine a proper value of the power. We did take some more data at 1 Hz than the other two frequencies, therefore the spread is bigger at .63 Hz and .1 Hz. Again in Figures 44 and 45 we can distinguish a well resolved peak at 92.25 pW. The reason that the smaller peak on the .1 Hz graph is as high as the other peak is easily seen from RELATIVE POWER AT I Hz 0 0 . 0 1 0.01 0.10 1.00 eaie oe a 1 z s fnto o te et current heat the of function a as Hz 1 at power Relative 43 Figure 8 0 5 95 0 9 85 80 78 6 6 (/zW) 99 RELATIVE POWER AT 0.63HZ 0 . 0 0 1 0.01 0.10 10.0 iue 44 Figure eaie oe a .3 z s fnto o te et current heat the of function a as Hz .63 at power Relative Q (fjM) 0789 9580 85 100 RELATIVE POWER AT O.IHZ 100.0 0.01 0.10 10.0 eaie oe a . H a a ucin f h ha current heat the of function a as Hz .1 at power Relative 78 iue 45 Figure 085 80 (i ) (/iW Q 0 9 95 101 102 Figure 41. At low frequency the amplitude of the fluctuations between 92 pW and 88 pW is not very different, the power spectra are "converging" to one point in that region. Again we can see how fast the power of the fluctuations decays when the turbulent state TII has evolved. Because of this rapid drop, the exponent of the diver gence would be quite different on both sides of some critical heat current. I want to emphasize that these graphs represent RELATIVE power at one frequency. To get the proper frequency dependence of the power from these graphs, the quality factor of the filter has to be taken into account and that differs with the center frequency. We have looked at the reproducibility of the structure in the amplitude of the power. We have taken data by increasing Q, decreasing Q, diminishing the interval between points and reproducing the measurements on different days. All these tests give us the same results. We find a well-resolved peak at 92.25 pW, a region of heat currents where the amplitude is at a maximum and different rise and "drop" of the amplitude on both side of that "plateau". One interesting aspect of this experiment is that we can associate the structure seen on the relative power graphs to the vortex line density graph obtained from steady state data. Point c on Figure 43 corresponds to the point where the slope of the line density is maximum on Figure 33. The points of maximum power (86-89 pW) do not correspond to anything in particular in the steady state data, it'is just in the middle of the broad transition. Point a on Figure 43 is there for reference, it is where the dissipation changes its 103 characteristic to give rise to the TI/TII transition. Despite a rather featureless power spectrum the amplitude of the fluctuations exhibits a complex dependence on the heat current. We can associate a small peak in power to the maximum slope in the line density graph. For the first time we can associate some of the dynamical features of the dissipation to its steady-state value. In the next section the response time of the system will be presented for the same heat currents as the one used in the study of the amplitude of the fluctuations. 4.5 Time Constant Measurements Previous experiments have shown a dynamical effect where the response time of the system increased near the TI/TII 26 70 79 transition ’ ’ . This effect, similar to "critical slowing down" in a second order phase transition, also has been predicted to occur 78 near a convective instability. Swift and Hohenberg showed in their work that the response time of the system near that hydro- dynamic transition should increase. In that context of second order phase transitions we set-up to measure the response time of our system near the TI/TII transition. The details of the measurements are given in Section 3.8. Figure 46 shows the result of our analysis. The symbol t is the exponential response time determined from an exponential fit to the data. Error bars are omitted in that graph for clarity. We are interested in the ratio of the response time for different heat (sec) 0.4 2.0 3.0 iue 46 Figure xoeta rsos tm a a ucin f h ha current heat the of function a as time response Exponential 3 5 0 5 97 95 90 85 83 6 (fj. ) W • • , V • •• ••• • 104 105 currents, not the absolute value of the response time. The response time is sharply peaked at 92.25 yW. The maximum value of the time constant occurs at the same heat current as the narrow peak in the power amplitude. That point c refers to Figure 33 and corresponds to the maximum slope in the steady state data. The peak in response time is very sharp. Within 2.5 yW the system exhibits a sharp rise and then a relaxation to a steady response time. When the system has evolved into the homogeneous state TII, the response time reaches a constant value. The value of t is larger in state TI than state TII. The different value of t before and after the transition also has been observed in the work 70 of Ladner et al. In Ladner's Ph.D. dissertation some data for t at 1.6 K are presented and the peak in the response time is observed at 92 yW, in excellent agreement with our data. Other workers have associated the second critical heat current 26 70 with the peak in response time ’ . From our steady state data we have seen that the determination of a critical heat current is quite arbitrary due to the structure of the transition. The maximum fluctuations occur just in the middle of this transition and the response time is rather constant in that region. We can associate a "critical slowing down" effect with the narrow peak in the amplitude of the fluctuations. It is like another phenomenon takes place in the system just before reaching the homogeneous turbulent state TII. For the first time we have associated some dynamical features of the TI/TII transition to the steady state vortex line density in 106 superfluid turbulence. We have borrowed a lot of ideas from the 80,81 field of non-linear dynamical systems . In that context 82 Dr. Moss of the University of Missouri, St. Louis suggested to us to also look at the amplitude probability distribution of the fluctuating chemical potential. Details about the results of that analysis will be described in the next section. 4 .6 Amplitude Probability Distribution The probability distribution function for a random signal describes the probability that the signal will assume a value within some defined range at any instant of time. In our case the signal is the amplitude of the fluctuating chemical potential. One reason we looked at the probability distribution functions was to see if we could detect a bimodal distribution. If the probability function has two peaks, that would indicate that there are two preferred values for the chemical potential (or vortex line density). Since hysteresis is associated with a first order (bistable) transition, for one value of the driving force (in our case the heat current) the average value of the measured parameter could have two different 75 values . There are some simple arguments that justify the growth of the fluctuations near a bistable transition. Is the TI/TII transition bistable but too narrow to be resolved? Figure 47 shows sketches of a measured parameter vs. driving force for a bistable transition and a continuous, second order like transition. We have never detected any hysteresis in our data in the TI/TII transition, ceai sece frabsal n otnos transition continuous a and bistable a for sketches Schematic Figure 47 Figure Measured Parameter Continuous Stable S - i B D r i v i n g F o r c eX D r i v i n g F o r c eX , 2 \ X, 107 108 including around the narrow peak in power. A look at the probability distribution was another test to see if this could be a bistable transition. We have recorded the digitized time varying signalof the chemical potential on disk. We now want to look at how many points there are in an interval of .1024 volt. Our digitizer has a range of -5.12 volts to +5.12 volts, the total range is therefore 10.24 volts. We took our digitized time series and looked for how many points were in the interval -5.12 V+N^“ .1024 where goes from 1 to 100. Every probability graph therefore contains 100 points. The vertical axis gives the probability that a fluctuation given on the horizontal axis is present in the noise. The abscissa unit is the voltage out of the chemical potential LIA. We studied many heat currents. Figures 48, 49, 50 show the probability distribution function for 88, 92 and 94 jjW. The crucial point in analyzing the time series is to make sure that the current in the counterflow heater does not change while the data is recorded. A slight variation in the heater power will change the value of the line density. That problem can be seen on Figure 49 where there seems to exist a small "satellite peak". Further measurements at that heat current eliminated that peak. A Gaussian 73 fit is included on Figure 48. All the heat currents studied showed a Gaussian distribution which is characteristic of a random process. The half-width of these functions is proportional to the 6 3 standard deviation of the chemical potential change. Since the Probability (# of Points) rbblt dsrbto t 8 W aGusa i i as included also is fit Gaussian a pW, 88 at distribution Probability 48 Figure 691 2.8 20.48 -20.48 J I ■ I 0 (J ■ If I ------B/j, ( A r b i t r a r y U n i t s ) 1 ______I ------Gaussian Fit - -- -- 109 110 - — 20.48 S/i (Arbitrary Units) (Arbitrary S/i Q = 92 u.W 92 = Q •20.48 •20.48 -12.288 -4.096 0 4.096 12.288 68.2 136.4 341.0 272.8 204.6 c o o (/) O O Q_ CL =H= M— JD -tl jQ Figure 49 Probability distribution at 92 yW 5.12 3.072 (Arbitrary Units) (Arbitrary S/i. -3.072 -3.072 -1.024 0 1.024 Q = 94 pW94 = Q 195.6 - 195.6 586.8 - 586.8 782.4 978.0 c co o O O CL CL ' o JQ H— -- -- 391.2- JQ C o ^3 H* n (D VJ' CTQ Probability distribution at 94 yW , 112 fluctuations are larger at 88 yW than 94 yW, the rms change of the chemical potential is larger, therefore the half-width should be larger. We have established that the fluctuations in the chemical potential is a random process. No bimodal distribution was found near the TI/TII transition or any heat currents studied. CHAPTER 5 DISCUSSION 5.1 Introduction In this chapter I will summarize and discuss the results obtained from this experiment. Some of the phenomena observed are qualitatively similar to experiments done in superfluid turbulence and in other systems. I will point out the comparisons and I will also talk about future work that can be done to improve our under standing of superfluid turbulence. 5.2 Observation of Fluctuations The main goal of this research was to look for fluctuations in the chemical potential difference and in the temperature difference in turbulent thermal counterflow. We have succeeded in observing fluctuations in the chemical potential difference. These fluctua tions are directly related to the vortex line density through Fsn- Not only have we observed these fluctuations but we also have discovered many new interesting effects associated with the fluctuations at the TI/TII transition. Up to this point we have evidence of fluctuations in the temperature differences but a careful analysis does not exist yet. 113 114 38 39 83 Only four other experiments ’ ’ ’ have observed fluctua tions associated with turbulent thermal counterflow. In all of these experiments the channels used were large and this restricted the study of the fluctuations to the TII region. By using a chemical potential gradiometer to study the fluctuations we are measuring a quantity which is directly related to the line density. Our gradio meter also let us study the steady state response of the dissipation so that we can characterize the turbulent region we are studying. We can map out the value of the line density for all heat currents used and we’ can associate the effects seen in the dynamical response of the line density to its steady state features. The other experi ments that have studied the fluctuations in the line density did not provide any information about the steady state of the dissipation. The first dynamical property I will compare is the power spectrum which is the most popular way to look at a fluctuating quantity. The problem we are faced with is the different probes used to obtain these spectra. Some spectra were obtained from the attenuation of second sound, some from ion measurements and some from a resonantly mounted pannel in a rectangular channel. The study of an ion current injected across a counterflow channel involves different processes than the study of the fluctuations of the chemical potential across the length of the channel. In the ion case, it is difficult to separate effects due to random ion trapping or release from a fluctuating line. These processes can reveal themselves through different power spectra. The group at University 115 38 of Missouri obtained spectra using second sound and negative ions in the SAME channel and obtained different spectral behavior. Because of this, comparing specific features of the power spectra obtained from different experiments is not very useful. One general characteristic observed from previous experiments is the lack of sharp frequency features in the spectra within a bandwidth of 5 mHz to 20 Hz. In our case the noise lies within .1 Hz to 5 Hz and we do not observe any sharp peaks in the spectra either. Contrary to some other experiments our spectra do not exhibit a well-defined roll-off frequency for the heat currents studied. There exists a prediction of this roll-off frequency based on the lineari- 16 39 84 zation of the Vinen equation ’ ’ . That roll-off frequency depends on the line density Lq and that fact has been verified by 74 the group at the University of Maine . For typical values of Lq in our experiment (-10,000 x higher than the University of Maine group) we are below that roll-off frequency. We are therefore studying the low frequency end of the spectra they obtained. It is not informative therefore to compare the high frequency behavior of all these spectra because of the different regime probed in each experiment. From the data obtained on the vortex line fluctuations it is becoming clear that the Vinen equation cannot produce all the dynamical features observed in the turbulent state Til. In the Schwarz picture the line density fluctuations are due to random line crossings and to the motion of the line driven by the normal fluid. Viewing superfluid turbulence as turbulence in the classical sense we would have expected to observe the transition to turbulence manifest itself through a succession of periodic to chaotic behaviour in the power spectra of the line density fluctuations85. Previous experiments never probed the properties of the fluctuations close to the critical heat fluxes. They all started their study of the fluctuations well into the turbulent state Til. Our experiment can probe the transition from the laminar to the turbulent state TI and from the state TI to the homogeneous turbulent state Til. Up to now the data obtained close to the first critical heat current indicate intermittency near the laminar to TI transition. Further refinements in the experiment will be necessary before we can determine the characteristics of the power spectra at this transition. On the other hand our experimental set-up enables us to study the fluctuations at the TI/TII transition very well. We have discovered that the line density fluctuations grow by two orders of magnitude around a narrow region of heat current. Unlike other systems where the transition between different states manifests itself by some frequency effects in the power spectra, the TI/TII transition is characterized by the growth of the amplitude of the fluctuations. For the first time we have observed fluctuations of the chemical potential in thermal counterflow and have characterized the fluctuations in the dissipation at the TI/TII transition. 117 5.3 Amplitude of the Fluctuations and Relaxation Time at the TI/TII Transition As we approach the TI/TII transition the amplitude of the fluctuations increases by two orders of magnitude for a 10% increase in the heat current. After reaching a plateau the amplitude of the fluctuations drops very quickly to a value smaller than in the TI region. There is evidence of this effect in the work from the University of Maine. It can be seen from Smith's data that as the heat current is increased further into the TII region, a drop in the amplitude of the low frequency noise was observed. A growth and decay of broad band noise similar to what we have observed has been seen near a hydrodynamic transition by Donnelly et al8^. More 87 recently Iansiti et a l . have studied the noise in a driven / Josephson junction in a highly nonlinear regime. They have observed the growth and decay of noise at a fixed frequency. This increase in the noise is observed at the transition between two Josephson steps on the I-V curve. This transition shows some hysterisis and they associated the noise at this transition with hopping between the two possible states. We searched for underlying hysterisis in the TI/TII transition by analyzing the probability distribution of the chemical potential fluctuations and found nothing but a Gaussian distribution for every heat current studied. Our study of the probability distri bution is limited to the frequency range used to obtain the power spectra data. Any information about a hopping frequency which would be outside that bandwidth could not be revealed in the probability 118 distribution because the frequency information would not be present in the data to start with. Despite the fact that the TI/TII transition is between two non- equilibrium states we thought we could model this transition with a continuous second order phase transition. Our steady state data reveals no hysterisis in the TI/TII transition suggesting a "second order" like behavior. We even had a prediction for the "critical" 78 exponent of the divergence of the noise in this transition . We did not observe a simple divergence b,ut instead a complex structure of the amplitude with heat current. Another problem we had with the analogy to phase transitions was our failure to find the "right" order parameter to fit our steady state data. Our high resolution steady state data shows detailed structure in the dissipation in the TI/TII region. It is not possible to think of this transition as a smooth bifurcation like the theories predict for the order parameter 75 77 as a function of the drive parameter ’ A strong point of our experiment is the ability to associate dynamical features of the turbulence with the steady state features of the dissipation. We have discovered a well-resolved peak in the amplitude of the fluctuations that we have associated with the very steep change in the steady state line density. We thought maybe that this peak could be due to hopping between two different states in that region but again a study of the amplitude distribution proved this idea wrong. As I mentioned earlier this does not rule out the possibility of a high hopping frequency. We can associate an 119 increase in the response time fo the system at this heat current. This effect, similar to "critical slowing down", also has been 70 observed in the work of Ladner at the same heat current for a tube of the same diameter as ours and at the same temperature. 88 Sancho et al. have reported a strong increase in the relaxation time in a non-linear dynamical system. Even in such a simple system there is evidence of a phase transition behavior. The 89 same group of workers also have observed the increase in the noise near a continuous transition in an analog simulation of a bistable system. To study the characteristics of these non-linear dynamical systems, the main technique used is to model the equation describing the system using electronic circuits. It looks like the TI/TII transition in superfluid turbulence shows a lot of the same characteristics. It could provide a real physical system to study these transitions. This experiment has unveiled a unique continuous transition where the fluctuations can be observed directly. A search in the literature reveals only a few direct observations of fluctuations 90 near a continuous transition. Brophy and Webb have observed the fluctuations of the polarization at the Curie temperature in a 91 crystal. Kim et al. have observed the fluctuations of the electrical conductivity at a critical point in a binary fluid mixture. Despite the increase of fluctuations observed in these systems neither of them shows the complex phenomena associated with the TI/TII transition. The study of critical phenomena is based on 120 the existence of these fluctuations. They are revealed indirectly through light scattering near a critical point. It is our hope that this work will give another system for the study of this phenomena in more detail. 5.4 Future Work The next step in this research is a study of the temperature dependence of the fluctuations. Changing the temperature is like varying another control parameter in this system. This will give us the opportunity to see how the structure observed in the fluctuations at 1.6 K depends on the fluid parameters such as the superfluid density. Another important step is the study of the temperature difference fluctuations. According to the steady state data this should reveal. the same kind of structure because the temperature difference is also probing the properties of the line density in the same manner as the chemical potential difference. Hopefully this is very difficult to do with the chemical potential detector because of its sensitivity to large temperature fluctuations in the regulated reservoir. We have observed intermittency at that transition and maybe the temperature fluctuations sensor will permit a detailed study of this interesting phenomenon. In a larger context we hope to be able to study the effect of 92 93 added noise on the transition ’ . Observation of postponement of 94 critical onsets have been reported in some analog simulations. 95 In the case of superfluid turbulence one experiment has studied 121 the effect of noise on the fluctuations but no study exists on the effect of added noise on the TI/TII transition. This will give another tool for the study of continuous transitions. APPENDIX A CONSTRUCTION OF TRANSDUCER A .1 Introduction Many tries were necessary before we found a procedure that would give us a reliable, sensitive, leak-free pressure transducer. I will describe the different procedures to prepare the body for assembly and also how to prepare the membrane. A.2 Body Assembly The first step in assembling the body is to prepare the upper and lower plate for gluing in the macor spacers. An eight mils copper wire has to be soldered on the flat of each plate. It is important to take care in not destroying the insulation on the wire. After the soldering is done the plates should be cleaned with methanol and freon. A very thin coat of Stycast 1266 is then applied to the shaft of the plate and the plate is pushed in position while the wire is being threaded out of the spacer through a hole drilled in it. The Stycast should cure for 24 hours. After the plate is glued on the spacer it is necessary to mill a small groove on the spacer to avoid cramming the wire in the space between the transducer chamber and the macor spacer. A "Dremel" tool is adequate for this job. 122 There are five copper-nickel capillaries that need to be soldered on the transducer. Three for the electrical feedthroughs and two for the helium inlets. It is best to solder the "feedthrough capillaries" first because it is less critical for those to have a perfect joint since the superleak tight joint is made with Stycast. A torch is needed to bring the transducer body to a high enough temperature so that the solder can wet the surface. Before soldering the inlet capillaries it is important to have the body in a very stable position with the capillaries already in place. As soon as the body is hot enough, the solder should be applied quickly and uniformly. The use of rosin core solder makes this process easier. It is important that the solder wets the capillaries and the body of the transducer, those solder joints have to be superleak tight. / After soldering, the two pieces of the transducer have to be cleaned in an ultra-sonic cleaner, first in methanol then in freon. The next step is to glue the plate-spacer assembly in the body of the transducer with Stycast 1266. The electrical wires from the plate should be threaded through a small teflon capillary before going through the feedthrough capillaries. This is an extra precaution in avoiding electrical shorts. The coat of Stycast 1266 applied to the macor spacer has to be very thin, otherwise a risk of plugging th inlet capillaries exists because the Stycast can run up these capillaries. To reduce that risk it is best to solder each inlet capillary to the gas system of the Baratron station (see Figure 51). By letting gas flow through the capillary while Stycast 124 Baratron Helium Gas Reservoir -Water Cooled Diffusion Pump ® = j To Cryostat To Transducer Stainless Steel Capillaries— " ® ValveMechanical Pump ® ValveMechanical Figure 51 Baratron station used for leak testing and for gluing the mem brane 125 is curing you reduce the risk of Stycast creeping up the capillary. Since some machining needs to be done on the glued assembly it is important to let the Stycast cure for 24 hours. When the plate-spacer assembly is in place in the body of the transducer it is necessary to mill down the bottom plate so that it is flush with the body to insure the smallest possible gap between the plate and the membrane. It is also necessary to mill down all surfaces and grooves to insure that the plates are parallel. All heights have to be measure precisely and enough space for the indium to flow (~3 mils) has to be allowed in the final machining. The top plate should be at 4 mils from the bottom plate to allow enough space for the membrane, the solder joint on the membrane, and for the mylar glued to the top plate. That leaves a gap of 1 mil between the membrane and the top plate. When all the machining is done, the membrane can be put on. A.3 Membrane Assembly The tension of the membrane is an important parameter in the sensitivity of the transducer. The different tests needed to determine the "right" tension will be explained in the next appendix. To get this tension it is necessary to use an aluminum rim of 210 gm to hold the mylar used for the membrane (see Figure 52). The gold- plated mylar should be put on a piece of plexiglass (previously cleaned with methanol) with the gold surface up (see Figure 52). The aluminum rim and the gold should be cleaned with freon using cotton Aluminum Rim Gold Plexiglass Aluminum Rim Mylar Aluminum Rim Transducer Bottom Part n 200gm Cylindrical "Press' ■Aluminum Rim -Transducer i Plate i Bottom Part Figure 52 Assembly procedure for the transducer tipped applicators. To glue the mylar to the rim, "5-minutes" 96 epoxy is used. After the rim is deposited on the mylar it is necessary to wait about 15 minutes for the quick setting epoxy to cure. The surface can be keep flat by using another plexiglass plate and a 500 gm weight as a press. Before applying the membrane on the transducer it is necessary to clean the bottom part of the transducer in freon in the ultra-sonic cleaner. The mylar side of the membrane also has to be cleaned with freon to get rid of finger grease. A 97 static brush is good to brush off dust and also to neutralize static charges on the membrane and the bottom plate. We found out that this procedure reduces the risk of the membrane sticking to the plate creating small peaks on the mylar because of dust. A VERY thin coat of Stycast 1266 has to be applied on the rim of the bottom part of the transducer. A thick coat would spill over from the rim to the plate and immobilize the membrane to the plate. The rim with the membrane glued to it is then deposited on the transducer body, the gold face up (Figure 52). It is important to keep the rim centered. To avoid forming creases on the membrane it is best to use a cylindrical press on the transducer body with a weight of 200 gm on top (Figure 52). When the Stycast is hard, we cut the membrane off the aluminum rim using a sharp cutter. It is necessary to cut the membrane following closely the contour of the transducer rim to avoid the gold of the membrane shorting to the top part of the transducer. When the membrane has been cut is should be cleaned with freon and a cotton tipped applicator. A 3 mils copper wire has to be 128 53 soldered on the gold of the membrane using Indalloy flux #3 and indium solder. It is important to keep the small tip soldering iron on low heat so that the mylar is not burned. It is best to practice before hand on a spare piece of mylar to determine the proper setting on the transformer used in conjunction with the soldering iron. The solder joint should be small and far (~5 mm) from the outer edge of the rim to reduce the risk of the indium shorting to the top part of the transducer. A piece of .5 mil clear mylar should be glued on top of the solder joint using thinned GE varnish so that shorts to the top plate are eliminated. The same kind of mylar should also be glued to the top plate of the transducer so that the membrane does not short. A.4 Closing the Transducer The joint between the top and bottom part of the transducer is made with a 31 mils indium O-ring. The O-ring is formed on a piece of brass which has the same dimension as the O-ring step of the transducer. The O-ring and the step should be cleaned with freon to remove all finger grease. The membrane wire has to be threaded care fully through the feedthrough while the top part of the transducer is being deposited on the bottom. To avoid crimping the electrical wire, a constant tension should be maintained on it until the trans ducer is ready to be screwed together. The transducer is closed and the O-ring pressed by using brass screws and nuts. To apply an even pressure on the O-ring the screws should be tightened diagonally and 129 turn by turn. To seal the electrical feedthroughs we slip a fiberglass sleeve on the capillary and use Stycast 1266 to wet the sleeve and seal the wire coming out of the capillary. A small teflon capillary is used inside the wire capillaries to diminish the risk of short. Another precaution to take is to remove the insulating material of the electrical wires around the areas where the joint is going to be made so that no leak can occur because of the insulation cracking. To reinforce the capillaries used for the helium inlets it is good to slip a fiberglass sleeve over them and wet it with Stycast' 2850 to make a strong joint. These capillaries will have to be twisted and moved around to solder them to the rest of the experimental area. I A.5 Leak Testing After the Stycast has cured on every feedthrough and inlet capillary, intensive leak testing on all these joints needs to be done. The top and bottom chamber are soldered to the Baratron station and that gas system can be connected to a leak detector. From outside it is possible to leak check all the joints. To test if the membrane is leak tight we put He gas in the bottom chamber and connect the top chamber to the leak detector. Before installing the transducer on the experimental probe it is good to place the base of the transducer in a liquid nitrogen bath and repeat the tests mentioned above. If a leak is seen at LN^ temperature it is sometimes possible to pinpoint the faulty joint and re-seal when the transducer is warm again. A Stycast seal can be etched away by soaking it in an epoxy strip bath. After the joint has been etched, a good cleaning in methanol using an ultra-sonic cleaner is necessary 98 to get rid of epoxy strip residues. Baking the part in a warm oven (~75 C) will help getting rid of those residues. Epoxy strip can destroy the membrane so it is necessary to bake and clean many times before rebuilding the transducer. After a transducer has passed all the room temperature leak checks the transducer can be checked for superleaks by immersing it in a Hell bath, each capillary connected to the leak transducer. A room temperature sensitivity measurement can be done after that by connecting the transducer to the Baratron system. Those procedures will be explained in the next appendix. APPENDIX B SENSITIVITY TEST B.l Introduction In this appendix I will describe the model used to describe the behavior of the membrane. When the first transducer was leak tight at helium temperature we discovered that the sensitivity was much less than what it was at room temperature. The transducer also had a preferred frequency in its frequency response giving rise to an undesirable peak in the power spectra. We therefore had the need to understand what role the tension of the membrane played in determin ing the sensitivity. The calibration procedure at room temperature and the deflection test at 1.6 K will also be described. B.2 Physical Analysis When we first rebuilt the pressure transducer after having a long fight with leaks, we did not realize how important a role the tension in the membrane played in determining the sensitivity at low temperature. It was suggested to look in a former student's Ph.D. 99 dissertation (J. Landau, 1969) to gain some insight on the physics of the stressed diaphragm transducer. We want to have a transducer that will give us reproductible results for the same experimental parameters day after day. 131 132 processes happening in the membrane material. To minimize hysterisis effect, the dependence of the membrane on elastic properties must be minimize. This is the context in which the tension will come into play. The hysterisis effects come from "wrinkling" of the membrane and this also reduces the sensitivity of the device. The proper tension necessary to eliminate these problems will be determined from a trial and error method. Figure 53 shows the geometry used in the following derivation. When the differential pressure is zero the capacitance is Ac, e C ’ rt rt' ~ B -1 lco C1 G]_: permittivity of vacuum, e0 : permittivity of helium, A: area of the plate, d: gap between the membrane and the plate, d^: thickness of mylar. This result assumes parallel plate configuration. Figure 53 also shows how the displacement of the membrane (X) depends on the geometry of the transducer. This displacement X is given from the so-called "soap bubble" model. In this model the restoring force on the membrane does not have an elastic origin. The total capacitance between the top plate and the membrane will be the series combination of the capacitance between the mylar on the top plate and the membrane and the capacitance between the mylar glued on the top plate and that plate (Ae^/d^. This series capacitance has two dielectrics. The expression for X assumes a spherical displacement for the membrane. Using an infinitesimal element of capacitance using cylindrical coordinates to find C* between the membrane and the mylar glued on the top plate Top P late R „ M ylar 2ZZZZ2ZZZ2Z22 id. Rr R M em brane Bottom Plate Transducer Body zzzzzzzzzzzzzzz 6, M em brane Equilibrium Position R, y = d - x PRo x = I 4 T RS P : Differential Pressure T : Tension on Membrane Figure 53 Geometry used to derive the capacitance 4Tire R o C“ = J dC an [1 -4 B.3 o P ire R2 o B . 4 d-PR2/4T o and the total capacitance is This is the ideal case where the restoring force does not depend on the elastic properties of mylar. We found out by varying the tension on the membrane we could go from a situation where AC/AP was linear to a regime where AC/AP had a plateau but high sensitivity at low pressures. Typical tension for this crossover is around 800 dynes/cm. For these smaller tensions we had very high sensitivity but a lot of problems with wrinkling and hysterisis. The theory is that at such low tension the membrane is governed by the elastic properties of mylar: this is the "elastic model". When we fit the calibration curve we got at room temperature for high tension that would give us the "linear regime" for large pressure range to the equation for C we needed tensions of at least 4,000 dynes/cm to fit the data. Since 135 the calculation of the tension we apply on the membrane using our "rim technique" is rather involved, we decided to find the right tension by a trial and error method. We would build capacitors with a different tension in each and do a calibration at room temperature. We are looking for a linear AC/AP curve and a sensitivity in the _3 order of 10 pf/u- We would cool the transducer to LN^ temperature and observe if the membrane would wrinkle. After that cool-down we would check the calibration again to see if we would get reproducible results. This is how we determined the weight of the rim we used to put tension on the membrane. Figure 54 shows the value of AC vs. P for the pressure trans ducer now working in the cryostat. The line through the data points _3 has a slope of 6.9**10 pf/u. This transducer was also checked for / superleaks by using a test probe in a small cryostat. The trans ducer is in a helium bath and both capillaries are connected to a leak detector. The helium level is monitored as the bath is pumped through the \-point. The calibration of figure was done after the transducer was warmed back up to room temperature. The value of AC/AP was similar to what we obtained before the cool down. While the experiment is running it is highly desirable to have a way of checking the calibration of the capacitor. The way our experimental probe is set-up it is not possible to do a direct pressure test. We have to do a voltage deflection test to check the linearity and calibration of the transducer. The idea is to DC-bias the bottom plate-membrane system. When some heat Q is applied to the Capacitance (Pico Farads) 64 62 63 65 67 66 0 40 0 80 00 20 40 60 1800 1600 1400 1200 1000 800 600 400 200 rsue mcos f ,/i) g H of (microns Pressure iC vs. P at room temperature room at P vs.iC 54 Figure 136 137 cell the pressure increases in the bottom chamber of the transducer and the membrane moves away from the bottom plate. We use a DC voltage to bring the membrane back to its equilibrium value. We can convert the voltage needed to restore equilibrium into a pressure and we then have a measure of AC vs. AP. Another way to determine the pressure difference between the top and bottom chamber of the transducer is to measure the temperature difference resulting from Q. From Equation 1.6 8.F AT = AT + — — B . 6 lam p s s The temperature difference due to the vortex lines is AT’ = AT - AT. lam From Chapter 1 we can calculate the laminar temperature difference AT = — --- B . 8 lam 4, ,2_ ira (ps) T > a: radius of tube, 8.: length of tube, n: viscosity of normal fluid. From the measured AT we calculate the part due to the vortex lines and we can convert to pressure using AP = psAT’ B. 9 What kind of dependence on the voltage do we expect the capacitance to follow? If we assume parallel plate configuration, the energy density U is d: gap, A: area of plate, C = eQA/D (first order approximation) so 2 2 2 F/A=P=1/2**C V /c A . We measure V,C and calculate the pressure using o this equation. Figure 55 shows the circuit100 used to bias the membrane with DC voltage. Figure 56 shows how the capacitance depends on V for Q=0. The straight line is plotted through V. Figure 57 shows AC vs. AP obtained with temperature and the voltage _3 technique. The slope obtained is l^S^lO pf/y which compares _ 3 with 6.9**10 pf/y at room temperature. The slopes were computed from a least-square fit method. Correlation coefficient for the data fit is very close to 1. Both techniques give the same slope. The y-intercept is very close to zero for the voltage method and negative for the AT method. The offset between the two curves can be traced to the different zeros when the DC power supply is included in the measuring circuit. Also ATlam never has been measured from the data, it was computed using Sl=l cm and d=130 ym but those parameters can be different therefore changing the value of AT' that comes into play in determining the pressure difference. The important thing we wanted to check from this test is the linearity of the transducer and also how much the calibration has changed from room temperature is 1.6 K. The calculations for the line density computed from the chemical potential were done using the calibration obtained at 1.6 K. Because of fear of breaking the membrane we have limited the deflection test to a pressure of ~200 y, which corresponds to a voltage of 45 volts across the plate to bring the transducer to 139 Capacitance Bridge Low High Membrane Bottom P late—— H.R6II4A Power Supply Figure 55 Circuit used to deflect the membrane AC (Femto Far rads) 200 100 120 140 180 160 40 60 80 20 eedneo tevlaeo h cpctnefrQ= 0 = Q for capacitance the of voltage the on Dependence Figure 56 Figure o AC vs. V AC o □ AC vs.V AC □ 100 otg Apid (Volts) Applied Voltage 2 6 0 .24 20 16 12 otg2 (Volts2) Voltage2 28 32 36 140 220 O O From A T □ From Voltage Pressure (microns) Pressure 80 100 120 140 160 180 200 20 40 60 AC vs. P at 1.6°K using voltage and temperature technique 142 equilibrium. At room temperature we did break a membrane for a voltage of 50 volts. We felt that 45 volts was a safe maximum voltage to use. Since we have established that the AT technique gives us the same results as the deflection study we can use it to study the linearity of the transducer for higher heat currents. As far as we can tell from these tests, the transducer is well-behaved up to 145 jjW but some non-linearity occurs after that. To study the dissipation above that heat current we will need to rely on tempera ture differences. B.3 Resonance in the Transducer The first transducer had a resonance at 2.4 Hz as can be seen in Figure 58. We found out how to excite this resonance using AC current in our counterflow heater and we could gain some insight on the behavior of the transducer. We took data with that transducer even though that "bump" was always on top of the "real" vortex noise. When we opened up the bandwidth we could also see other harmonics. We can think of the transducer as a vibrating membrane coupled to two Helmotz resonators which are the helium volumes on both sides of the membrane. The way to change this resonant frequency is to change the tension in the membrane. The higher the tension the higher the resonant frequency. Ideally we want that frequency to be larger than 6 Hz. Above that the vortex noise is at the same level as our back ground so a peak outside 6 Hz is not important. One of the benefits of our work to determine the right tension for sensitivity purpose was to move this resonance to 7.5 Hz where it does not show in our data. 143 • •• N X o c CD 1 3 CT CD C\J o U)d°'6o-| Figure 58 Power spectrum at 0 yw for old transducer APPENDIX C In this appendix, I have listed 6 computer programs used in this experiment. Here is a list of what they are used for. p. 145, record time series: Repeat p. 146-147, average, store, plot Power Spectra: PWSPT2 p. 148, record data for power amplitude: CONTPW p. 149-150, change the heat current and record data for time constant experiment: TAU p. 151-152, fit exponential to time constant data: TAUFET p. 153-154, find probability distribution of chemical potential data: PROB 4 10 L=1024 1 4 5 20 DIN VC 1124) 35 OPEN 'LP!' FOR OUTPUT AS FILE 41 40 PRINT 41.DAT! 45 CLOSE 1 50 PRINT 'ENTER START RUN4 , STOP RUN! . SAMPLING FREQ . HEAT' 40 INPUT S1.S2.F.H \ S=1 \ N=1 BO PRINT ’ YOU HAVE ENTERED RUM'S " S l * - ^ . IS THAT CORRECT (N OR Y)T* 90 INPUT 04 \ IF 040'Y' EO TO 50 95 FOR H1=S1 TO S2 94 N4=STR4( HI> 100 OPEN 'LP!' FOR OUTPUT AS FILE *1 110 IF H1=S1 THEN PRINT 41 \ PRINT ♦l.'RUHt'N4.CLK4,'FR£0'F.'HEAT'H \ PRINT 41.'A/D'.'DEVICE'.'FILENANE' \ PRINT 41 140 F$='SY11B1' +N4+'»DAT' 150 OPEN F4 FOR OUTPUT AS FILE 42 140 PRINT 41.S.'LOCK IN1'.F4 170 PRINT 42,L/N'.'F'.'H 190 CLOSE 1 200 PRINT N4,'COLLECTING DATA' 210 AIN!.V< ),L+100,1/F.S»N) 220 PRINT 'FINISHED COLLECTING DATA' 230 PRINT 'STORING DATA' 240 FDR J=0 TO L 260 PRINT 42.V( J+100> 280 NEXT J 310 CLOSE 2 315 GRAPH!'-H'...Vi30)) 314 PRINT CHR4I7) 320 NEXT HI 322 PRINT 'MORE DATA (D) OR STOP 590 X=FNXa> \ Y=FXY( Y) tco if leinstpvxdt ? then x=o 510 IF Ltfi-:STH(Y))>9 THEN Y=0 o20 .IS-- Sfi.iFA' iSTF.SI XM'l'iSTWt Y H‘ iPUi' ii: S0S:JE 530 o40 NEXT ii oSO r.S-'PUiSFOr' \ GOSUt 830 6 : 0 REN *' 670 PRINT -fj YOU UANT STORE THE POWER SPECTIW (N OR Y)?' \ INPUT OS i 4E0 IF (Kr'N' then 770 685 PRINT ' » » IMP DISK POHSPT « « ' 690 PRINT ' ENTER FILENAME FOR STOREP SPECTRA (AS IN SYIIFILENAHE.PSP)' \ INPUT N< 700 OPEN 'SYlI'FNSt'.PSP' FOR OUTPUT AS FILE #2 710 PRINT *2»LVF 720 FOR J=0 TO Ml 730 PRINT *2.Y(Ji 7/0 NEXT J ' '50 CLOSE 2 760 REP. 770 PRINT 'DO YOU UANT H3RE PLOTS (N OR Y)?' \ INPUT OS 700 MSPLAY.CLEAP r?c if os=t two; stop BOO FOR 2=0 TO 1024 \ Y(J)=0 \ (EXT J 305 HUO 810 bO TO 40 820 REP 83 0 SERIN KS,5 ) \ RETUR'N B40 PEP FNX(P?)-(PF-XO)/(Xl-XO)S10 550 PEF FMYl P9 >=( PS-YG )A Y l-Y O »10 840 REH 870 PS^SYr.Br+STPKJW'.MT' 330 PRINT PS 8 Rf) OPEN PS FOR INPUT AS FILE *2 900 INPUT *2.1,F,H \ Nl=L/2 <*)5 PRINT ' LENGTH - 'L m 'FREO - 'F m 'HEAT- 'H 910 FOR 1=0 TO L-l 920 INPL'T *2.X 925 RHI)=INTUOaO»X> 930 I» I)= 0 9<0 NEXT I 950 CLOSE 2 960 FFTi iL iR Z l ),Ia( )»P) 970 Gl=((2TQ)tS/(NlllOOO))t2/(HIF) 980 FOR 1=0 TO Nl-1 990 Y( I )=Y{ I )F( RZ( I )t2+II< I )12 )S01 1000 NEXT I 1010 RETURN 1020 ENP 6 REH AMPLIFIER F-BUTTERHORTH GAIN ----- 10 L=2048 20 DIH VI2148) 30 OPEN 'LP!' FOR OUTPUT AS FILE I I 35 PRINT llrDATI 37 CLOSE 1 40 PRINT 'NO OF RUNS f FREO f HEAT f PRE AHP F f SQUARING F r WFLIFIER F' 50 INPUT SIi FfH.B i CfA 70 S=1 \ K=1 \ S2=0 100 OPEN 'LP!' FOR OUTPUT AS FILE I I 105 PRINT *1f'FRE0='Ff'HEAT='H 106 PRINT I I f'PRE AHP='B f' SQUARING='Cf'AhP GAIN='A 110 PRINT I1 f'RUN4VTIHEVP0UERV>5 VOLTS' 180 E-B.'IO 190 FOR K=1 TO SI 200 PRINT 'COLLECTING BATA' 210 AIN (fV!)fL+100f1/FfS fN) 220 PRINT ' FINISHED COLLECTING DATA' 230 P=0 \ S6=0 240 FOR J=100 TO 2148 245 F-F+‘J< J1 250 IF A B S ( W J » / 5 THEN S6=S641 a60 Nlai J 265 f-P/((B12)*2049IC»Ai 275 PRINT 41 \ PRINT I 1fKfCLK$fPfS6 276 PRINT K fP fS6 325 PRINT CHR'K 7) 326 32=S2+P 330 NEXT K 340 PRINT CHR$( 7) \ PRINT Ct*«7) 345 -RIMT ' AVERAGE--'S2/S1 34c PRINT 41 f 'AVERAGE^ S2/S1 347 ClOSE 1 350 50 TO 40 340 END 10 riTM ok sm .A?) tnut.& v m 9i> 20 S=300 \ Tl=5000 30 S1-1023-S 40 DEF FNNSa X )=IHT( B004X-2048) 50 OPEN 'LP:' FOR OUTPUT AS FILE 12 60 PRINT *2»PAT» 70 PRINT 12,'RUN 1','TIH EVI OF RUNS', ' START HEAT VEND HEATVFREQVFILENAHE' 80 CLOSE 2 90 PRINT 'ENTER START VOLTAGE , STOP VOLTAGE' \ INPUT V1.V2 100 IF V1>1 THEN PRINT 'START VOLTAGE TOO HIGH' \ GO TO 90 110 IF V2>1 THEN PRINT 'STOP VOLTAGE TOO HIGH' \ GO TO 90 120 N1Z=FNNZ(V1) \ N2Z=n«KV2) 130 PRINT \ PRINT 'SET VOLTAGES' \ PRINT 140 AOUT(,N1Z) 150 PRINT t t m START VOLTAGE **» « ' \ PRINT \ INPUT 0* 160 A0UTUH2Z) 170 PRINT m n STOP VOLTAGE** * » ' \ PRINT \ I (PUT W 180 PRINT 'ARE THESE VOLTAGESOK?' \ INPUT W 190 IF OKO'Y' THEN GO TO 90 200 PRINT \ PRINT 'ENTER RUN NUKKR , NUMBER OF RUNS , START HEAT , END HEAT » FREB' 210 INPUT RliNlr01r82rF 220 OPEN 'LPT' FOR OUTPUT AS FILE #2 230 R1»=STR«R1) 240 F*=' SY1 IT' +Ttl*f' .DAT' 250 PRINT 42,Kl,CUtt,Ml,Ol,02,F»F* \ PRINT 260 aOSE 2 270 FOR J=1 TO N1 280 PRINT 'SHEEP 'Ji 290 A0UTUN1Z) 300 FOR K=0 TO T1 310 NEXT i; 320 AIN(»A1(),S,1/F,1,1) 330 A0UTI.N2Z) 340 AIN( »A2( ),S1,1/F,lrl) 350 PRINT ' FINISHED' 360 FDR H=0 TO S 370 A3(H)=A3(HHA1(H) 380 IffXT H 390 FOR H=0 TO SI 400 G=HfS+l 410 A3(G)=A3(G)FA2(H) 420 NEXT H 430 NEXT J 440 PRINT 7) \ PRINT CHR«7) ’ 450 GRAPH! m iA3(0)) 460 B1=0 \ B2=0 470 FtW J=0 TO S 480 Bl=A3(J)tSl 490 IffXT J 500 FOR J=S1 TO 1024 510 B2=A3( J HB2 520 NEXT J 530 Bl=Bl/(S+l) \ B2=B2/( S+2) 540 B=B1+(B2-B1)*»63212 545 C=INT<.1754FK5) 550 FOR J=S+C+1 TO Sl-1 560 IF A3UKB THEN IF A3U41»=B THEN T= 340 C2-X2 350 S9=SGN! C1-C2)+2 \ ON S9 GO TO 360>320i380 7:0 2=-P T(C):T(J1FD 37u 58-Cl \ C1=C2 \ C2=S6 3d0 lT-CTt! it ; • gosuf goo \ C3=X2 4ju PRii.T CHi-3GJARE-'C3i'HIFF='C3-C2t '0FFSET='T(0)»'TAU='T(l)f'AHP='T(2) 410 ?*=SGwr3-C2H2 \ ON S’ GO TO 420r440i440 S': CSC: \ 22=C3 '70 GO T0 390 *.:0 21-lit! i/; IF! C1-C2)/{C3-C2) )f.5) •: 4 2 a • )=d: 7» soft 2/< n» : c3-2»C2fci j )) 450 r(.:'-T( .M-P! \ KJ)=B!J)*C?/3 \ PRINT 'FINAL 0FFSET='T(0)f'TAU='T(l)f'AhP='T(2) 151 NET 3 \ GOSUP 580 452 ?=AF5(!X-X2)/X) \ IF P>.01 THEN X=X2 \ PRINT 'CH1-SQUARE='X2'ZDIFF='P \ GO TO 280 454 MNT(F»T(0)-S) 4 60 P 3--1 0 2 7 -F :-:S F C ) \ S3=5FC 470 GRAW•-HSP3 m YIS3)) <90 PRINT 44r'LD» V0LT='Sl/Nh'HIGH V0LT='S2/N1,'BASELINE DIFF='(S2-S1) *500 PRINT 44•'OFFSET='(T< 0 HNT( T( 0)) ) f ' TAU=' T( 1)»' SIC TAU='G<1) 504 print ♦4»'A*R>='T<2)»'CHI-S0UARE='X2 505 PRINT *4 \ print 44 510 CLOSE 4 r»>A DOTirr /unr>r t>*t* / t>\ prnn /rw» \ TMWrr ru 152 530 IF 0$='B' THE/,’ OPEN 'LPJ' FOR OUTPUT AS FILE 14 \ GO TO 80 550 imAY.CLEAR 540 STO° 570 REK — SUBROUTINE THAT CALCULATES THE 'CHI-SQUARE' AW THE FITTED FUNCTION 590 X2=0 590 FOR I=SFC TO 1023-B2 STEP 2 600 TO=I/F-T< 0) 610 Y( I )=T< 2 )*{ 1-EXP( -TO/T( 1)) )fSI 620 X2=X2«A 10430 f1*='PA' iSTRtiF«X 1. J. F. Allen and A. D. Misener, 1938, Nature, 141, 75. 2. P. 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