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QUANTUM TURBULENCE :
DECAY OF GRID TURBULENCE IN A DISSIPATIONLESS FLUID
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
S2007 Shu-Ch.! i. Liu
To my home country, Taiwan, R.O.C. To my mentor, Professor Gary G. Ihas. To my
parents. To my two sisters and my brother. To my grandparents.
I am thankful for all the help I have received in my Ph.D. program, in writing this
dissertation and in learning about this research subject. First of all, I would like to
thank my mentor, Professor Gary G. Ihas, who has led me to the amazing field of low
temperature experimental physics. Front hini, I have learned a lot of physics and I was
able to carry out my accomplishments. I really enjoy the beauty of quantum turbulence
science. Second, I would like to thank my parents, who have given me a lot of support,
emotionally, financially, and spiritually, and a lot of encouragement before and during
the pursuit of my Ph.D. Third, I would like to thank the nienters of our research group,
former and current, and the people in the machine and electronic shops. Because of them,
I was able to stand on those giants' shoulders and look higher and further. Fourth, I
would like to thank the whole Gainesville coninunity, which has given me much.
TABLE OF CONTENTS
ACK(NOWLEDGMENTS ......... . .. .. 4
LIST OF TABLES ......... ... . 8
LIST OF FIGURES ......... .. . 9
ABSTRACT ......... ..... . 12
1 INTRODUCTION TO QITANTITA TITRBITLENCE ... .. .. 14
1.1 Introduction ......... . .. .. 14
1.1.1 Basic Properties of He II . . .. .. 14
1.1.2 Two-Fluid Model, and Landau's Two-Fluid Equations .. .. .. 16
1.1.3 Quantization of Vortices and the Critical Velocities .. .. .. .. 18
1.1.4 K~elvin Waves ....... .. .. 20
1.2 Introduction of Towed-Grid Turbulence Experiments .. .. .. 20
1.3 Proposed Towed-Grid Superfluid Turbulence Experiment .. .. .. 21
1.4 High Resolution, Fast Responding Milikelvin Therntoneters .. .. .. 25
1.4.1 Neutron Transmutation Doped Gernianiunt Bolonieters .. .. .. 25
1.4.2 Miniature Ge Film Resistance Therntoneters .. .. .. 27
2 SHIELDED SITPERCONDITCTING LINEAR MOTOR .. . :34
2.1 Introduction ... .. . .. ... .. :34
2.2 Models of the Shielded and Unshielded Superconducting Motor .. .. :35
2.3 Simulation Results and Discussions ...... .. :39
2.3.1 Simulation All ll-k- ........ ... :39
2.3.2 Critical Magnetic Fields for Nichium ... .. .. 41
2.3.3 Required Voltage Input for the Solenoid .. .. .. .. 4:3
2.4 Unshielded Motor Testing Experiments .. .. .. .. 45
2.4.1 Capacitance Bridge and Lock-in Amplifier for Monitoring Armature
Motion ............. .. ...... 46
2.4.2 The 555 Oscillator for Monitoring Armature Motion .. .. .. .. 47
2.4.3 The Q-nieter for Monitoring Armature Motion .. .. .. .. .. 49
2..4Te Aic Bridge circuits for Monitoring the Motion of the Armature 5
2.5 Improved Design of the Armature and the Test Cell .. .. .. 57
2.6 Conclusion ......... .. . 59
:3 CONSTRUCTION OF SITPERCONDITCTING SHIELDED LINEAR MOTOR
AND EXPERIMENTAL CELL ......... ... .. 84
:3.1 Construction of Superconducting Shield .... .. . 84
:3.1.1 Electroplating Theory and Electrolyte Recipe .. .. .. .. 84
:3.1.2 Properties of C'I. !! I- ....... .I .. .. .. .. 85
:220.127.116.11 Lead ........... ...... ..... 85
:18.104.22.168 Methanesulfomic acid (j!HA) . . 86
:22.214.171.124 Lead carbonate . ..... .. .. 86
:126.96.36.199 Lab protective equip ..... .. . 86
:3.1.3 Lead Electroplating Procedures and Results .. . .. 86
:188.8.131.52 Procedure steps . ..... .. .. 86
:184.108.40.206 Results ......... ... .. 88
:220.127.116.11 Troubleshooting . . ... .. 88
:3.2 Testing the Experimental Cell at Liquid Helium Temperature .. .. .. 90
:3.2.1 Simulation Results ....... ... .. 90
:3.2.2 Experimental Testing Results ..... .. . 92
:3.2.3 Viscous Drag and Impedance Forces Discussion .. .. .. .. .. 92
:3.2.4 Heat Dissipation Discussion ..... .. .. 9:3
:3.3 Leak Tight Electrical Feedthrough Design .... .. .. 94
:3.3.1 Electrical Feedthru Construction .... ... .. 95
:3.3.2 Thernxistor Circuit Board Construction .. .. .. .. 96
:3.3.3 Conclusion ......... . .. 97
4 THERMISTORS ELECTRONICS . ...... .. 111
4.1 Introduction ......... . .. .. 111
4.2 The AC Bridge Circuit Analysis . .... .. .. 111
4.3 Resolution Measurement at Room Temperature ... .. . .. 11:3
5 EXPERIMENTAL PROCEDURE, DATA, AND ANALYSIS .. . 118
5.1 Introduction ......... . .. 118
5.2 Experimental Results ......... .. .. 118
5.2.1 Thernxistor Calibration . .... .. 119
5.2.2 Background Heating C'I. I 1: at 624 n1K .. .. .. .. 119
5.2.3 Performing QT Experiments at 520 n1K .. .. .. .. 120
5.3 Exploring the K~elvin Waves in the Energy Spectra ... .. . .. 121
6 CONCLUSIONS AND FITTIRE WORK( .... .. .. 127
6.1 Conclusions ........ . .. 127
6.2 Future Work ........ . .. 127
A DERIVATION AND NITMERICAL ANALYSIS OF MAGNETIC FIELD AND
FORCE ........ .... ......._ .. 129
B SOME C CODE ........ .. .. 1:31
C LABVIEW PROGRAM SHOTS ....... ... .. 140
D FEYN1\AN'S SPECULATIONS ABOUT THE NATURE OF TITRBITLENCE
IN THE SITPERFLITID ........_ .. 144
REFERENCES ..........._ ..........._. 145
BIOGRAPHICAL SK(ETCH ......... .. .. 148
LIST OF TABLES
2-1 Optimal parameters for the motor design. .... ... . 40
2-2 Parameters for the unshielded test motor. .... ... . 45
3-1 Parameters of superconductor shielded superconducting linear motor system. 91
3-2 Forces on the armature when moving up at 1 m/s (Download is positive). .. 94
4-1 Parameters of the AC bridge affecting the sensitivity and the power dissipation. 114
4-2 Sensitivity test of the AC bridge. ........ ... .. 115
LIST OF FIGURES
1-1 Viscosity of liquid heliuni-4. ......... .. 29
1-2 Superfluid and normal fluid densities as a function of temperature. .. .. .. 29
1-3 Photographs of stable vortex lines in rotating He II. ... .. .. :30
1-4 Damping on a sphere oscillating in liquid helium. .... .. :30
1-5 Donnelly-Glaherson instability of a quantized vortex line. ... .. .. :31
1-6 Energy spectrum for homogeneous and isotropic turbulence. .. .. .. :31
1-7 Electrical conduction mechanisms in semiconductors. .. .. .. :32
1-8 Calibration plots for three test thernxistors developed for calorintetry. .. .. :32
1-9 Design for the therntoneter test cell. . ...... .. 3:3
2-1 Grid turbulence in a classical fuid. . ..... .. 60
2-2 Superconducting motor model. ......... ... .. 60
2-3 Armature motion in unshielded and superconducting shielded solenoid. .. .. 61
2-4 Armature motion inside superconducting shielded solenoid (linear acceleration). 61
2-5 Armature motion inside superconducting shielded solenoid (square acceleration). 62
2-6 IUpper critical field versus temperature for niohium. ... ... .. 62
2-7 Voltage input to the superconducting shielded solenoid. ... .. .. 6:3
2-8 Simulated armature motion for the superconducting shielded solenoid.. .. .. 64
2-9 Driving voltage. . .. .... .. 65
2-10 circuit of the switch box. ..........6
2-11 1\achine drawings and photos of the motor system. ... .. .. 67
2-12 Experimental apparatus for unshielded motor. .... ... .. 68
2-1:3 The GR 1616 Capacitance Bridge ....... ... .. 69
2-14 Circuitry of the experimental apparatus for unshielded motor testing experiments
using the 555 oscillator circuit and LahView counter program to monitor the
motion of the armature. ......... . .. 69
2-15 Testing circuit of Q-nleter. ......... . 70
2-16 Simple AC bridge circuit.
2-17 Measuring the capacitance of the capacitive position sensor with the GR 1616
capacitance bridge and the lock-in amplifier.
2-18 Circuitry and setup for testing the superconducting motor at 4.2 K(.
2-19 Theoretical calculation of the magnetic force and RT sensor calibration curve.
2-20 Motion of the armature of the superconductingf motor.
2-21 Simulated armature motion with the input pulse sent to the sol
2-22 Motion of the armature of the superconductingf motor. ....
enoid. .. .. 75
. . 79
umbers.. .. 80
. . 82
Motion of the armature of the superconducting motor (I). ..
Motion of the armature of the superconducting motor (II). ..
Motion of the armature of the superconducting motor (III). .
Modified superconducting motor system. .....
Machine drawing of the grid and the corresponding Reynolds n
Calibration curve of the position sensor at 4.2 K(. ....
Electronics circuits for the superconducting motor system. ..
Motion of the armature of the superconducting motor. ....
Machinery drawings for the cell cap. .....
Machinery drawings for the cell body. ......
Electrode for the cell cap. .....
Electrode for the cell body. .....
Procedure steps for the lead plating for the cell cap. .....
Procedure steps for the lead plating for the cell body. .....
Masterpiece of the lead coated cell cap. .....
Masterpiece of the lead coated cell body. .....
Towed-Grid experiment cell. .....
Simulation for superconductor shielded superconducting linear motor system. .. 107
Motion of the armature of the shielded superconducting motor. .. .. .. .. 108
:3-12 External forces on the armature. ........ ... .. 109
:3-13 Leak tight electrical feedthru design for our cryogenic cell. .. .. .. .. 110
4-1 The AC bridge circuit simulation. ........ ... .. 116
4-2 AC bridge circuit for thernxistors resistance measurement. .. .. .. .. 117
4-3 Room temperature thernxistor resolution curve. .. ... .. 117
5-1 Thernxistors calibration curves. ....... ... .. 12:3
5-2 1\otion of armature and thernlistor response at vacuum around 600 niK. .. 124
5-3 Quantum turbulence at 520 n1K. ........ ... .. 125
5-4 Fitting the enthalpy of liquid helium as a function of time. .. .. .. .. 126
C-1 LahView programs for calculation of current, magnetic field, magnetic force and
armature motion. .. ... . .. 141
C-2 LahView programs to look for optimal parameters. .. .. .. 142
C-:3 LahView programs for data acquisition and data analysis. .. .. .. .. 14:3
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
QUANTUM TURBULENCE :
DECAY OF GRID TURBULENCE IN A DISSIPATIONLESS FLUID
Shu-Ch.! i. Liu
C'I I!1-: Gary G. Ihas
We produced grid turbulence in liquid helium at 520 mK( to compare with classical
experiments and theories. Above T = 1 K(, with viscosity present, it has been shown that
grid turbulence is equivalent to homogeneous isotropic turbulence in a classical fluid. We
seek to investigate the nature of grid turbulence when viscosity is zero. Specifically, in the
absence of viscosity in a quantum fluid, through what path does the turbulence decay?
To produce grid turbulence, an actuator was designed and built that can accelerate and
decelerate the grid rapidly in a short distance (~ 1 mm), and achieve glide speeds of up
to 1 m/s. To avoid Joule and eddy current heating of the liquid helium, a magnetically
shielded superconducting linear motor was built. The grid is attached to the end of a very
light insulating armature rod which has two hollow cylindrical niobium cans fixed to it
about 26 mm apart. This part of the rod is inside a superconducting solenoid which, when
driven with the properly shaped current pulse, produces a magnetic field resulting in the
Detailed computer simulations guided the motor design. The simulation and motor
control programs were written in LabView with an embedded C compiler. Using the
simulator, various designs of solenoid (with and without shielding) and armature were
investigated. We compared the simulation and the experimental results in which complex
current pulse shapes were required to produce the desired motion.
The motor and grid (1 mm square hole array with '71I' transparency) were mounted
in a copper cell containing a pool of liquid helium cooled to 520 mK( by dilution
refrigeration. We measured the decay of the turbulence produced, after one 28 mm stroke
of the grid, using calorimetry. Doped germanium thermometers less than 300 micrometer
diameter immersed in the turbulent liquid helium allowed fast calorimetric measurements
limited by the electronic time constant of 1 ms. The decay of turbulence was detected by
the rate of temperature rise in the isolated cell after the grid was pulled. Recent theory
-II- -_ -r ;the decay occurs through a K~elvin-wave cascade on the vortex lines which couples
the initially large turbulent eddies to the short wavelength phonon spectrum of the liquid,
yielding a characteristic rate of temperature rise. Initial measurements support the K~elvin
wave cascade theory.
INTRODUCTION TO QUANTUM TURBULENCE
Quantum turbulence is a very interesting and amazing field in physics. Richard
Feynman said that turbulence is the most important unsolved problem of classical physics.
At temperatures below 2.17 K( (the A- transition temperature), the turbulence produced in
superfluid helium II demonstrates quantization of vortex lines (i.e., quantized circulation).
Therefore, this turbulence is called quantum turbulence. The idea of quantized circulation
in superfluid helium was proposed by Onsager and Feynman in 1955  and discovered
experimentally by Hall and Vinen a year later .
Superfluid quantum turbulence produced by towed-grid experiments in liquid 4He
at very low temperatures is predicted to decay, not through viscosity, as in a classical
fluid, but by phonon radiation when the energy flows into the smaller length scales in a
K~elvin-wave cascade. We propose a new calorimetric technique to probe such a decay
mechanism of superfluid grid turbulence at extremely low temperatures, wi 520 mK(,
while the normal fluid density is only 8.6 ppm .
1.1.1 Basic Properties of He II
Helium is the only known element that remains liquid at extremely low temperatures,
even down to absolute zero, under saturated vapor pressure. Its remarkable superfluid
properties have been a drawn to scientists and they have begun studying its other
hydrodynamic properties as well.
The superfluidity of helium-4 was discovered in 1939 by Allen, Misener, and K~apitza
[4, 5] while Osheroff, Richardson and Lee did not discover the superfluidity of helium-3
until 1971 . At 2.17 K(, under saturated vapor pressure, the curve of specific heat versus
temperature for 4He shows a dramatic spike, which looks like the Greek letter 'A'. This
is called a lambda transition, and corresponds to a second order phase transition. In
3He, the superfluid transition occurs at 0.9 mK( under saturated vapor pressure. Below
the transition temperature, 3He and 4He liquids show superfluid phases with low to zero
The thermal de Broglie wavelength of liquid helium is
Ar= 8.9A1 (1-1)
at 2.0 K(. This is comparable to or greater than the mean interparticle distance of a 3.6P4
for helium. So the de Brogflie wavelength of each atom is larger enough to overlap with its
nA~ = 2.1 x 1028 1773 x (8.9 x 10-ton)3 = 14.8 > 2.61 (1-2)
This is why liquid helium is called a quantum fluid. Because there are two protons and
two neutrons in the nucleus, there is an even number of nucleons (total nuclear spin=
0), and the quantum mechanical behaviors of 4He can he explained by Bose-Einstein
statistics. Atoms of "He obey Fermi-Dirac statistics because their nucleii contain two
protons but only one neutron, totaling an odd number of nucleons (total nuclear spin
=1/2). (The spins of the two electrons cancel out.) Hence, 4He is a hoson and 3He is a
The superfluid 4He, also called He II, has almost zero viscosity, while the normal
fluid of 4He above T He I, has much higher viscosity which can dissipate energy via
interactions with the walls of the container. The viscosity of liquid 4He measured by the
method of oscillating disc viscometer is shown in Fig. 1-1 . Two of the most famous
experiments demonstrating the superfluid properties of He II are the beaker experiment
and the fountain (thermo-mechanical) effect. If you put the bottom of an empty weaker
in the He II hath, a 20-30 nm thick 4He mobile film forms on the walls of the beaker,
and then liquid He II flows along the film from the hath into the beaker until the levels
are equal. If you then lift the beaker above the hath level, the liquid inside the beaker
will also flow along the film out of the walls of the vessel into the hath until the beaker
is empty. Now if you connect two vessels containing the same level of He II at the same
temperature, wi 2 K(, with one very narrow capillary which can only pass superfluid, and
then increase the temperature of one vessel by 10-3 K(, the level in the other will rise, by
9.6 cm. In addition, if you apply extra pressure on one vessel, then the fluid level will fall
because superfluid flows into the other vessel. These are both manifestations of superflow
1.1.2 Two-Fluid Model, and Landau's Two-Fluid Equations
Landau and Tisza proposed the Two-Fluid Model in 1941 [8-10] to explain the
various interesting phenomena that occurred in He II. In this model, the liquid helium II
is considered as a mixture of two interpenetrating fluids, called the normal component and
the superfluid component, with densities, p, and p,, respectively. Hence, the total density
of liquid He II  follows:
p = p, + p, a 0.145 (1-3)
g/cm3 (at Saturated vapor pressure).
The superfluid and normal fluid component densities, as a function of temperature
below Tx under the saturated vapor pressure, are shown in Fig. 1-2. The superfluid has
zero entropy (S, = 0) and zero viscosity (rl = 0), while the normal fluid exhibits viscosity
(r7;) and entropy (S,), equal to the entropy of all the liquid helium. Also, the superfluid is
considered to be irrotational:
V x v4 = 0, (1-4)
where v6 is the velocity of the superfluid.
Modifying Euler's equations for the classical (Euler) fluids based on the continuity
equation, and using the first and second laws of thermodynamics, and his own postulate
that the chemical potential (p) is the driving force for the superfluid, Landau derived the
two fluid equations for He II :
+ pil = 0 (1-5)
,+ psy, = (1-6)
+ = (1-8)
where the total mass flow is:
7 pil py,v + ps@~ (1-9)
and stress tensor is:
Pij = p6ij + pay,,iva,, + psy,4vs,,4, (1-10)
In Landau's two-fluid model, the elementary thermal excitations, phonons and rotons,
depending on the wave number, arise in the flow of helium II through a tube or capillary
at T / OK when the normal fluid component has the interactions with the walls causing
energy dissipation and viscous loss. Suppose an excitation is created with energy E and
momentum p due to the loss of energy from the tube (AE = E). Then Landau's relation
(v > E/p) gives the minimum velocity of flow
vc = E/p (1-11)
required to produce an excitation. However, actual critical flow velocities in experiments
are much smaller (~ mm/s) than Landau's prediction (60 m/s at the vapor pressure and
46 m/s at higher pressure ), due to the quantized vortices.
1.1.3 Quantization of Vortices and the Critical Velocities
W;lith the assumption =D Vp in the liquid He II, the superfluid velocity
obeys a K~elvin circulation theorem [13 :
-nv d e = 0. (1-12)
In addition, the K~elvin circulation theorem,
D, d A1 = 4, (V x i ) S (1-13)
(by Stokes' law) implies that the superfluid circulation
n = d (1-14)
stays constant, and if at t = 0 the superfluid vorticity = V x is zero everywhere it
will stay zero. If V x disappears everywhere, then the superfluid circulation is:
i = -de i (V x ) dS = 0. (1-15)
Assuming there is some circulation, there must he a gl region where either
V x / 0 or there is no superfluid. So we consider the singular region as a very thin
cylinder, called a superfluid vortex line or vortex core. According to Gauss' theorem:
f,(V 7)~dv = J 7i d S, and V V x 7 -- 0 for any 7, we have f,,(V Vx ~)d-r=
J,~ Vx dS = J d -e = 0 = m. So a vortex line cannot terminate in the fluid, but
must end at a boundary or close in on itself (a vortex ring). Since the vortex line is the
only source of vorticity in the fluid for V x = 0 everywhere except at the line, all path
integrals encircling the vortex line have identical circulations.
As soon as liquid helium II rotates or moves hevond a critical velocity, superfluid
vortex lines appear and demonstrate either an ordered array of vortex lines by steady
rotation or disordered vortex tangles for counterflow (due to heat flow). That the
superfluid circulation is quantized was postulated separately by Onsager and Feynman
in 1955, a = J v, .97 x 10-48 C112 8, Where n is an integetr. The radius of
the vortex core is about ao ~ 1P1 (atomic dimensions). The stable quantized vortex arrays
in rotating He II can be visualized as shown in Fig. 1-3 : stable vortex lines in rotating
He II in a cylindrical bucket of 2 mm diameter to depth 25 mm placed at the rotation axis
of a rotating dilution refrigerator at 100 mK(. >' 3He was added to provide damping which
maintained stability. The negative ions are trapped on vortex cores and are imaged on a
phosphor screen and recorded on cinefilm. All superfluid vortex lines align parallel to the
rotation axis with ordered arrays of areal density (or length of quantized vortex line per
unit volume) as given by the following equation:
(in lines/cm2), Where R is the constant angular velocity for the rotation. This can derived
as follows: the circulation around any circular path of radius r concentric with the axis of
rotation = J -, d e = J(V x v) dS = 2xrr20. And the total circulation = Kr2no0 m,
where no is the number of lines per unit area. Therefore, no = 20m/h = 20/s .
The turbulent state, described as a mass of vortex lines, usually has two critical
velocities signaling the onset of turbulence in superfluid and in the normal fluid separately,
increasing the total length of vorticity with the increasing relative velocity of the two
fluids. In the experiment on the damping of the rotation of a sphere oscillating in liquid
helium II as a function of the maximum amplitude (or velocity) of the oscillation, the
result is shown in Fig. 1-4 . At region A, the damping is constant, relating to the
constant normal viscosity rls. The two critical velocities occurred at the transition from
region A to B (which is the onset of turbulence in the superfluid component), and C
to D corresponding to the onset of turbulence in an ordinary classical liquid, where the
damping increases dramatically. In regions B, C, and D, both superfluid and normal fluid
are coupled and move together due to their mutual friction. The critical velocity of the
superfluid rises with reduction in diameter of the channel.
1.1.4 Kelvin Waves
Quantized vortex lines in helium II at high temperature can vibrate parallel to the
vortex lines under the influence of the normal fluid if they have large enough velocity.
This is called the Donnelly-Glaberson instability  (see Fig. 1-5). The plane of these
vibrations (called K~elvin waves) process about the center core, growing exponentially along
quantized vortex lines. The lengths of the vortex lines increase, eventually resulting in a
vortex tangle as the energy is transferred from the normal fluid to the superfluid.
Feynman -II---- -1h I1 in 1955 that vortices approaching each other very closely undergo
reconnections. The K~elvin waves can he generated by vortex reconnections, leaving kinks
on the vortex lines, regarded as superpositions of K~elvin waves, leading to the continuous
generation of K~elvin waves with a wide range of wave numbers.
1.2 Introduction of Towed-Grid Turbulence Experiments
In classical grid turbulence, the eddy motion can range from large length scales,
(as large as the mesh of the grid or the size of the channel), to small length scales (or
larger wave numbers than the inverse of the vortex line spacing, -e-1). In the later case,
the Reynolds number R, = ~ 1 (where U is the characteristic velocity and n is the
kinematic viscosity) and energy dissipation because of viscosity occurs. However, if the
Reynolds number R, =-~ ~ inril oc 1 (in an "inertial regime") so that the viscosity can
he ignored, then the energy will flow in a cascade from large scales to smaller scales, as
described by the K~olmogorov spectrum :
E /) C2/3k_-'l" (1-17
where E(k)dk is the energy per unit mass for spatial wave numbers in the range dk. The
function E(k) has dimensions [L3/T2](L: length, T: time). C ~ 1.5 (the K~olmogorov
constant, w~hic~h is dimensionless), and E idk (that is the average rate of kinetic
energy transfer per unit mass flowing down the cascade, dissipated by viscosity at high
wave number, k > -e-1). The dimensions of E are [(L/T)2/T] = [L2 T3]; the dimensions
of k are [L-l]. As -II_t---- -1.. by the energy spectrum shown in Fig. 1-6 , the largest
eddies have the most turbulent energy, and decay most slowly, which determines the
energy dissipation rate (E). In the steady state, energy flows from these largest eddies to
the smallest eddies.
The superfluid grid turbulence experiments in helium II above 1 K(  can also be
described by the K~olmogorov spectrum in the inertial regime on length scales larger than
the spacing between vortex lines (e). In this case, the energy dissipation by viscosity
occurs on the length scale of order -e (the spacing between vortex lines) due to the
significant amount of normal fluid and the mutual friction between the superfluid and
the normal fluid.
1.3 Proposed Towed-Grid Superfluid Turbulence Experiment
We seek to verify or eliminate the K~elvin-wave cascade as the mechanism for
the decay of quantum turbulence produced by a towed-grid in liquid 4He at very low
temperatures. At these temperatures, the component of the normal fluid is negligible, so
mutual friction may be neglected. measurements above 1 K(  show that homogeneous
isotropic turbulence is produced behind a towed grid moving at the order 1 m/s.
Computer simulations  predict at zero temperature the K~elvin waves on intersecting
vortex lines produce the equivalent of the viscous regime in a classical fluid. This has yet
to be confirmed by experiments.
It is ell----- -r1 11 [16, 25] that energy flows to the smallest scale by a K~elvin wave
cascade on the vortices, leading to a K~elvin-wave energy spectrum for the wave number k
of Kelvin waves greater than the inverse vortex spacing -e-1. Kelvin waves do not have any
damping at very low temperatures until the wave numbers k (> k2 = 2 x 10s m-l) [16, 25]
become much greater than -e-1. It has been predicted  that at 0.46 K(, energy dissipates
by phonon radiation; however this has not been confirmed experimentally. The results of
the simulations demonstrate the continuous energy flow within the K~elvin waves towards
highest wave numbers at which the phonon emission dissipates the energy, and the steady
state (a K~elvin-wave cascade) with balanced energy input and dissipation.
The corresponding K~elvin wave spectrum, cut off by dissipation at k ~ k2, iS proposed
to be: [25, 26]
E(k) = Aps2 -1, 18
where E(k)dk is the energy per unit mass and unit length of vortex associated with K~elvin
waves with wave numbers in the range dk, p is the density of the helium, and A ~ 2. The
rate at which energy flows into the K~elvin wave cascade per unit mass of helium is given
E" (X12/2x)K3L (1-19)
where Lo = e-2 is the length of the smoothed vortex line per unit volume, and X2 ~ 0.3.
The energy contained in the "K~elvin wave cascade" per unit mass of helium is given by
1 K2 e
E =-E,AL =AL Inb( ) a ri2AL, (120)
p 4xr fo
where aL = ALoln(k2e), E, is the energy per unit length of vortex line, and (o is the
vortex core parameter. The cut-off wave number k2 is giVen by the formula 
k2e = ( 3/4, _21)
where c is the speed of sound in helium.
By measuring the rise in temperature of the helium after creating turbulence with
a high resolution thermometer, we can probe the turbulence decay as a function of time
since the temperature change corresponding to the decay of a random vortex tangle is
proportional to the change in the vortex line density. Therefore, it is possible to explore
the existence of a K~olmogforov spectrum on large length scales, a K~elvin wave cascade on
small length scales, and the dissipation mechanism. The enthalpy of the helium is given by
H(T) = 747T4 _122)
(in J/m3). If a Small amount of turbulence energy (AE) is released as thermal energy in
helium at temperature To, the enthalpy values of the temperature change would be
T4 TO4 = 1.34 x 10-3AE (1-23)
(in K4), Which can be monitored as a function of time. The grid turbulence is created by
drawing a grid through the helium by a superconducting linear motor at a constant speed
as fast as 1 m/s for 10 mm for approximately 10 ms, which is faster than the decay speed
of the turbulence (a few hundred milliseconds). Any energy dissipation in the helium from
moving the grid must be much smaller than the released thermal energy from the decay
of the superfluid turbulence. Suppose the square cross section of the channel is d x d and
a very dense vortex tangle as well as quasi-classical turbulence (on a larger scale than the
mean vortex line spacing e) are produced by the towed-grid. The energy can transfer to
either larger length scales (on the scale of d) until becoming saturated, or to smaller length
scales (less than e) where energy dissipation occurs, leading to the K~olmogorov energy
spectrum. The time required to build this spectrum should be less than the turnover time:
re = ,(1-24)
where d= eddy size, and u(d)= characteristic velocity relating to this eddy size, defined by
u(d) = CE2/3k-2/3. (1-25)
From the clues in the previous experiments above 1 K(, it is surmised that a K~olmogforov
spectrum joins smoothly to the "quantum v. I ~1, n
u() = .(1-26)
which is characteristic of motion on the scale e. So 
u(d) = p1/2 1/3 _127)
where p ~ 0.25. The total turbulent energy per unit mass, most of which comes from the
largest eddies of size d, is given by
Eclass = 32() __ 2a 2/3s. (1-28)
2 2 P 2x-
As -e increases, turbulence decays and energy dissipates via the classical K~olmogorov
cascade as described by the above equation. At time t > -r, total energy decay with time
as described by the K~olmogorov spectrum can be thought of as the energy flow rate with
time  ,
E = 27C3 2( t0 -3, (1-29)
where to ~ 9d, which varies with the towed-grid speeds or initial turbulent intensities. So:
Class = -C3 2( t0 -2. 30)
Comparing Equations 1-28 and 1-30, the time dependence of e is given by
= (t to)3/4 1 )
33/4 (2xT)1/4 C9/8 1/2
Another characteristic time for the K~elvin-wave spectrum (Equation 1-18) which
describes the fully developed K~elvin-wave cascade is given by
E 2; A 2
-r ( ) I(k2). (1-32)
Energy flows into the K~elvin-wave cascade at the rate E = E", and eventually dissipates by
phonon radiation. The energy per unit mass contained in the K~elvin-wave cascade is given
E = Ais2e-2 2 k~)* (33)
The total turbulent energy per unit mass is
E = Eclass + E = C3 2" t0 -2 + Ais2e-21ke) 2 )*)
Since -e increases with time, the decay of E will be dominated by the decay of Eclass for
small but by the decay of E for large For t > to, Eclass is proportional to t-2, While
E is proportional to t-3/2, Which will be expressed by the observed rate-of-change of
1.4 High Resolution, Fast Responding Milikelvin Thermometers
Thermometers must meet the following requirements to be used in our quantum
1. Operating temperature: 20 mK to 1 K( (dilution refrigerator temperatures).
2. Sensitive to temperature change: 6T ~ 10-3K, or 6T/T ~ 0.05 10-3
3. Short response time: 6t ~ 10-3s. The turbulence energy decays within a few hundred
milliseconds. In order to have good time resolution in the data, it is necessary that
the thermistors respond within 1 ms.
4. Small mass, small heat capacity, and good thermal conductivity.
So far we have found two excellent candidates to fulfill the above requirements which
can be used in our calorimetric technique: Neutron Transmutation Doped Germanium
Bolometers and Miniature Ge Film Resistance Thermometers. We have used the later in
1.4.1 Neutron Transmutation Doped Germanium Bolometers
The basic electrical conduction mechanisms in semiconductors can be classified into
four categories: (The diagram is shown as Fig. 1-7 )
1. Thermal generation of electrons and holes across the bandgap, which is negligible at
low temperature since kT
2. Generation of free charge carriers by ionization of shallow donors, which is negligible
at low temperature since kT
3. C'!I. ge carrier movement from one impurity to the next in heavily doped semiconductors
(metal-insulator transition) also called banding mechanism.
4. In heavily doped and compensated semiconductors, the compensating or minority
impurities create a lot of ill s .int il,y impurities which remain ionized down to absolute
zero. The charge carrier hops from an occupied 1!! lint ~l~y impurity site to an empty
site, which is the working principle for the low temperature bolometer. It is also called
The material we use for our high-resolution dilution refrigerator thermometer is
neutron transmutation doped (NTD) germanium [18, 19], which applies the fourth
electrical conduction mechanism as discussed above.
A NTD Ge has NVA 1!! I li~ ~ry Shallow acceptor impurities and NDo minOrity Shallow
donor impurities (NVA > ND). At very low temperatures (kT
binding energy of the electrons to the acceptors), and in the dark, (NVA 1VD) acceptors
have an electron vacancy and are neutral while NVD acceptors capture electrons from
The transmutation of stable germanium isotopes via the capture of thermal neutrons
is accomplished by the following procedure:
1. A single ultra-pure germanium < li -r I1 is grown in a hydrogen atmosphere (1
atm) from a melt contained in a pyrolytic carbon-coated quartz crucible using
the Czochralski technique.
2. Six 2 mm thick slices of 36 mm diameter are cut, lapped and chemically etched.
3. Irradiated with thermal neutrons, doses 7.5 x 1016 ~ 1.88 x 10lscm-2
4. Ater en hlf lves f ne(T1/2 11.2d), the samples are annealed at 400 oC for 6
hours in a pure argon atmosphere (1 atm) to remove irradiation damage.
Some papers demonstrate that, even at dilution refrigerator temperatures, the NTD
Ge thermometers still have sufficient sensitivity. For example: at 25 mK(, 6T/T ~ 4.8 x 10-6,
and response time < 20 ms for thermometers as small as 1mm x 1mm x 0.25mm .
Some very good circuitry for the NTD Ge bolometers has been developed [20, 21].
1.4.2 Miniature Ge Film Resistance Thermometers
Microsensors based on Ge film on semi-insulating GaAs substrates [22-24] have
already been developed and tested in our group. The typical size for the sensitive
element is 300 pm in diameter, and the mounted gold leads are 50 pm in diameter.
The conduction mechanism is variable range hopping. Fig. 1-8 shows the absolute
resistance values and the sensitivity (R ) over the temperature range o intret,,t for
three test thermistors against a ruthenium oxide calibrated thermometer. We used the
LR-110 picowatt AC resistance bridge as the detection circuit, and calibrated three of our
available thermometers. The LR-110 bridge can measure resistances between 10 R and
1.2 M R with high resolution (better than 0.1 .) and good accuracy (0.05'~ 0.25' .).
The calibration curves demonstrating the performance and the characteristics of the
three test thermometers are quite different due to the variations in doping and heat
treatment during manufacturing. Sensing powers were less than 10-13 WittS. TWO Of
the thermometers are not ideal as we can see from the curves. For thermistor 1, the
resistance ~ 7 or 8 kR under 100 mK( and was nearly constant below 38 mK( with
pretty stable sensitivity within the measured temperature range. For thermistor 2, the
resistance goes to infinity at low temperatures and only becomes measurable above about
49 mK(. Thermistor 2 demonstrates dramatic sensitivity change over a wide range of
temperatures. For thermistor 3, the resistance is 16 kR at 88.7 mK( and 526 kR at 24.8
mK(. The sensitivity ranges from 1.7 M~egR/K to 140.3 M~egR/K between 50 mK( and
20 mK(. Its response time is less than 0.001 s. This makes thermistor 3 a good candidate.
During turbulence decay, the resistance of thermometer 3 is expected to change from 4 kR
to 1.75 kR, and the change from 3 kR to approximately 2 kR representing the K~elvin wave
decay at 520 mK(, which appears in our final experimental results.
The machinery drawing for the thermometer test cell is shown in Fig. 1-9. The
thermometer test cell is used to simultaneously test the sensitivity and the resolution
of two thermometers (miniature Ge film resistance thermometers) separated by a small
distance. By applying current to the heater located next to these two thermistors, we
can also measure how fast heat is transported in the superfluid or measure how fast the
thermometers respond to heating.
Figure 1-1: Viscosity of liquid 4He measured in an oscillating disc viscometer .
56 % P
0 2.0 Th
Figure 1-2: Superfluid and normal fluid densities (p,z and p,)
below lambda transition under the saturated vapor pressure.
as a function of temperature
Figure 1-3: Photographs of stable vortex lines in rotating He II in a cylindrical bucket of 2
mm diameter to depth 25 mm placed at the rotation axis of a rotating dilution refrigerator
at 100 mK(.
1 2 3 4
Figure 1-4: Damping on a sphere oscillating in liquid helium at 2.149 K( with a period of
18.5 s .
Figure 1-5: Donnelly-Glaberson instability of a
component of the normal fluid velocity parallel
quantized vortex line occurs if the
to the vortex line exceeds a critical value
k (Airbinry U~n rdr
Figure 1-6: Energy spectrum for homogeneous and isotropic turbulence . (k,:
K~olmogorov wavenumber, where viscous dissipation becomes significant; kc 2Z, d: the
size of th~e con~ta~iner; ke(t) 7, -e,(t): eddy length? scale) .
YR L~PIIICE B~NC]
Test Thermistor 1
-200 Illistor 3
'Test Tiermistor 2
10 100 101
Figure 1-7. Electrical conduction mechanisms in semiconductors .
Figure 1-8: Calibration plots for three test thermistors developed for calorimetry. (a)
Resistance versus temperature. (b) Sensitivity versus temperature.
,l-ier, silver powder,
(b) tl-roucgh on 1.3120* ECD
(2) 4-40 toppFed the ougjh Top V/Eewn
( jack screws) on 2 21?0' BCD
00 6685' capi ary he e 00.125' clear hole through on
-thrmagn~~ on 0,750' BC 0,75C' BCD C(t:li fit -to par-t #6)
Ut~~ 6-32 tappeob alepth7 0.12t5"
~ r --- n000' BCD
DO :100 thermometer hle 2 -8 42pe, eph 012'
dept: a7:.i', n I loi~i:n on 0 700" BCD under surfaee
Figure 1-9: Design for the thermometer test cell. (a) Overview. (b) Top view of cap.
SHIELDED SITPERCONDITCTING LINEAR 1\1TOR
Grid turbulence in the classical fluid produces homogeneous isotropic turbulence (e.g.,
Fig. 2-1 ), which is the simplest case among the complex nonlinear dynamics systems.
In order to compare with this classical case, we intend to produce homogeneous isotropic
quantum turbulence (HIQT) in liquid helium II below 1 K(.
In order to produce HIQT, we have designed and built the shielded superconducting
linear motor for our towed-grid turbulence experiments. First of all, we build a model,
shown in Fig. 2-2 [28, 29]: a single superconducting solenoid motor with an armature
moving through its center. This light and hollow insulating armature is constructed of
:3 phenolic tubes separated by two hollow cylindrical niohium cans placed some distance
apart, with the turbulence-producing grid attached to one end. The requirements and
advantages of our motor system are as follows:
1. The superconducting shield can avoid eddy current heating in the cell walls.
2. The superconducting solenoid can avoid Joule heating front the solenoid.
:3. With an appropriate current pulse, the grid can he efficiently accelerated and
decelerated front 0 to 1.0 nt/s within 1 nin. And the grid can he driven at a nearly
constant speed 1 nt/s for 10 nin, producing homogeneous isotropic turbulence within
In the resulting grid motion in our simulator, the grid would be accelerated front 0
to 1 nt/s in 1 nin. Then it travels at almost constant speed, 1 nt/s, for 10 nin. Then
the grid would rapidly decelerate to cease within 1 nin when the third pulse is applied.
Then we put our simulation results into practice. We build our test cell guided by the
simulation. In our test cell, we have one superconducting solenoid driving the armature to
move through its center, with a grid attached at the end. This light insulating armature
is constructed of :3 phenolic tubes separated by two hollow cylindrical niohium cans
placed exactly 26 nin apart, with the grid attached to one end. A conducting section
on the armature, composed of one of the Nb cylinders and silver paint coating part of
the phenolic rod, is inside a closely fitting capacitor made of two semi-cylindrical copper
sheets. This capacitor, coupled to a bridge circuit, measures the armature position.
Our unshielded test motor system has been tested very successfully. We apply the
pulses to the superconducting motor to drive the armature inside the solenoid: two sine
pulses with DC level in between, followed by small DC level for 250 ms. In the resulting
velocity versus time curve, we already can accelerate the armature to 1.1 m/s. The
velocity remains zero for about 250 ms when the armature is held on the top. In the
velocity versus position curve, we can see that the armature moves at almost constant
velocity 1 m/s + 0.1 m/s for at least 8 mm.
We also improve this design in our superconductor shielded superconducting
motor system, discussed in the next chapter. One of the important task is to build
the superconductor shield on the interior of the cell. The details will also be discussed in
the next chapter.
In chapter four, we would like to discuss about the accomplishment of the followings:
* Run the superconducting shielded linear motor system at 520 mK( in the dilution
* Explore the energy spectra and the decay mechanism of homogeneous isotropic
quantum turbulence at 520 mK(.
2.2 Models of the Shielded and Unshielded Superconducting Motor
The motor is shown schematically in Fig. 2-2, made of three coaxial parts: one
superconducting shield with the radius b and length h, one superconducting solenoid with
the inner radius r and length 1, and a light insulating rod with two niohium cylinders
attached and separated by distance AS. The grid is attached to the end of the rod and
hence is pushed by it.
Suppose the diameter of the superconducting wire for the solenoid is d, and the
total number of turns, NV. We can estimate approximately the self-inductance of such a
4 NBA N1oNVx22r2
L 4x,2 x 10-10 x (2-1)
I I ll 1
(in H. ,:, a and all the lengths are in mm), where # is the total magnetic flux flowing
across the solenoid, I is the current flowingf through the superconducting wire of the
solenoid, B is the approximate magnetic field at the center of the solenoid, A is the
average cross section area of the solenoid, and r is the average radius of the solenoid:
F = r +(2-2)
The z component of the magnetic field at the position (p, 8, z) near the solenoid is
derived as the following by generalizing the problem in Griffiths :
PolR~ 27 pCOSeCOs@ painesin~
B,(l : -) C~4~ol4x [P2 + 2 _( 0 XO2 2pRcosHcos~ 2pRainesin@]3/2
Bz(p, 8, z) = s(p, 8, z)I, (2-4)
2n 1 1 2m 1
R = r+( )d, zo = + ( )d. (2-5)
2 2 2
For the magnetic field distribution inside the solenoid enclosed by the superconducting
shield, we cite Eq. 12 in Sumner :
41-oN1Vr S(r lo(kp) kl
Bz (p, z)=-Sxkb ik) sin( 2)cos(kz) = x(p, z)I, (2-6)
Sl(kst) = li(ks)Kl(kt) Kl(ks)II(kt), (2-7)
II and K1 are the Bessel functions of the first and the second kind, and
(m +1/2) x
k = (2-8)
with m zero or a positive integer.
Suppose the niobium is a perfect diamagnet, then the magnetic force experienced by
the niobium cylinder #1 with the center of mass at z = ze along the axis of the solenoid
(the center of the solenoid is defined as z = 0 position):
/zi+11/2 2 o
ma z) [771 z ,(p, 0, z)]z= z ,[M z p ,z,-d e. (2-9)
Since M~ = -~i for a perfect diamagnet , the magnetic force is proportional to the
gradient of the magnetic field square:
zi Izi+11/2 r2~ yrol ao
0 zi-11/2 0 0
where m, M, I-o, rol, 11 are the magnetic moment, the magnetization, the permeability
at vacuum, the outer radius and the length of Nb cylinder #1, respectively. For the
purpose of computer simulations, the practical formula for the numerical analysis and the
corresponding C code are in the Appendix A and B, respectively.
In addition to the magnetic force, the gravity force E;',, is also considered, therefore,
according to the second Newton's law, the net force experienced by the whole system with
mass M.1 ... produce the acceleration, a: E;ma,, + E;,,,=M ...Eo
The system includes the insulating rod, the two niobium cylinder cans and the grid.
Because Fma,(ze) cm I2, We CR1 Write Fmayxi) ma f,(i 2. Suppose the second niobium
cylinder is some distance, AS, below the first one, then the current required to reach the
objective acceleration would be:
(a +9.8)i1 M.,
I = (2-12)
f mas ( z ) +f m,,( z A S)
(unit in Amp~ere).
By applying a properly shaped current pulse to the solenoid, the magnetic interaction
with the niobium cylinders produce the required motive forces, with the upper cylinder
accounting for the initial acceleration and the lower cylinder the deceleration of the grid.
In addition, we have several options for the mathematical form of the relationship
between the destined terminal velocity v (in m/s) and the traveling distance S t(in mm)
during the period of time of the acceleration a (in m/s2), Such as linear, square and
sine function. For each small traveling distance, dz (mm), the accumulated velocity,
acceleration and time at the ith increment, I (m/s), ai (m/s2), and ti (ms):
Linear acceleration: v oc S1
S= s dz (m/s) (2-13)
as=(i-1)v)2dz x 103 (l2) (2-14)
ti = ti_l + ( ) x 103 (ms) (2-15)
Square acceleration: vU oc S,
= S(idz)2 (m/s) (2-16)
as= 2i _2 + i-)()dz3 x 103 (l2) (2-17)
ti = ti-1 + (2i 1) ( aS) x 103 (ms) (2-18)
Sine function acceleration: v oc sin(S1)
=vain(i- ) (m/s) (2-19)
ai = -sin2(2 si2[(i 1) ]} x 103 (l2) (2-20)
2dz 2 S1 2 S1
is 41 si(2 -sin[(i 1) ]} x 103 (ms) (2-21)
a 2 S1 2 S
For the deceleration, we simply have the linear mathematical form. Suppose the
traveling distance would be S3 ill mm). FOr each small traveling distance, dz (mm), the
resultant velocity, acceleration and time at the jth displacement interval, vj (m/s), aj
(m/s2), and tj (ms):
vy = v j S3dz (m/s) (2-22)
as = (j )()2dz x 103 -x 103 (l2) (2-23)
tj = tj_, (aS3 x 103 (ms) (2-24)
2.3 Simulation Results and Discussions
2.3.1 Simulation Analysis
In order to find the best motor design, we have developed a LabView simulator
program, in which the various design parameters may be varied while maintaining the
required motion of the grid. Optimization in general yielded the lowest voltage and
current applied to the solenoid, hence minimizing the magnetic field produced. This
process yielded the optimized parameters of Table 2-1 (units of lengths in mm).
Fig. 2-3(a) shows the required current versus time curves for the unshielded and
superconducting shielded solenoid with a sine function acceleration (velocity is the sine
function of the niobium position). The curves show three peaks: the first and the third
peaks are to accelerate and decelerate the niobium cylinders. The middle peak is due to
the almost balanced magnetic forces on the two niobium cylinders at that position since
each niobium cylinder is almost equidistant from the ends of the solenoid. We can run the
simulator removing the middle peak (o- symbols) in Fig. 2-3(a), and see that the velocity
versus position curve in Fig. 2-3(b) (again 0-symbols) is virtually unaffected. The droop
in velocity after z = 10 mm is unavoidable since all forces, including gravity, are directed
downward for the rest of the stroke.
Table 2-1: Optimal parameters for the motor design.
Parameter Description Value
d the diameter of the superconducting wire 0.2
NV the number of turns of the solenoid 1500
r the inner radius of the solenoid 5.0
1 the length of the solenoid 10.0
zo the initial position of Nb cylinder #1 3.5
rol the outer radius of Nb cylinder #1 4.0
To2 the outer radius of Nb cylinder #2 4.0
11 the length of Nb cylinder #1 9.0
12 the length of Nb cylinder #2 11.0
as the distance between the center of masses of two Nb cylinders 21.0
& the length of the superconducting shield 50.0
b the radius of the superconducting shield 20.0
Ml. .,.. the mass of the whole system 2.0g
(the armature, including the insulating rod,
two Nb cans and a grid)
Note that the superconducting shield requires a slightly higher current (0.14 A
greater) to produce the same motion. The effect is small because the shield is significantly
larger than the solenoid.
In the velocity versus position curves in Fig. 2-3(b), we see that the motion is as
expected. The niobium cylinders travel at almost constant speed, 1 m/s, for 10 mm, but
start to slow down when the middle current peak occurs, which is quite reasonable. After
the first niobium passes z = 10 mm, the second niobium is closer to the solenoid and
experiences a stronger magnetic force in the opposite direction, resulting in the slight
deceleration. Therefore, applying the third pulse produces the desired deceleration to
rapidly stop the grid. The evaluation results prove that our superconducting linear motor
is a very feasible design.
For comparison, the required current versus time curves for the superconducting
shielded solenoid with a linear and square function accelerations and the corresponding
velocity versus position curves are shown in Fig. 2-4, Fig. 2-5. It requires a slightly
higher current, 0.149 A and 0.8 A greater, for the superconducting shielded solenoid to
produce the linear and square acceleration motion. Therefore, sine function acceleration is
a more efficient way that takes the least current among the three options.
The shots of all the simulation programs are in Appendix C.
2.3.2 Critical Magnetic Fields for Niobium
Niobium (Nb) belongs to transition metal (group IIIB). It has atomic number 41,
atomic weight 92.9 g/mol, density 8.58 g/cm3, melting point 2741 K( and remain solid
at room temperature. Niobium is type II superconductor, which forms a vortex stable
state mixed with normal and superconducting regions over the range between the lower
and upper critical magnetic field strengths, H,1 and Hc2, for partial penetration of the
magnetic flux, called the incomplete Meissner effect. Below the lower critical magnetic
field, it exhibits the same phenomenon as type I superconductor to expel the whole
magnetic flux, i.e. complete Meissner effect.
The theoretical estimates of the lower and upper critical magnetic fields for the type
II superconductors are :
Hc2 2 (2-26)
where @o is the superconducting flux quantum, called a flux~oid or flux~on:
Go = 2xh~c/2e T2 2.0678 x 10- (2-27)
(in Gauss -cm2). A and ( are the penetration depth and the coherence length, respectively.
For example, the niobium has the penetration depth A at absolute zero estimated from
the measurements to be 470 P1 , and the superconducting coherence length ( is 11 nm
. So H, a 2980 Gass Hc 54397 G~1S, H ~ causs. The penetration depth and the coherence
length are actually temperature dependent.
Experimentally, H,1 and Hc2 ValueS have been measured to be dependent on the
purity and the residual resistivity ratios [RRR = R30 4 2N Of Il-bu [35,- 36], the field
orientation , and the temperature [35-38]. R300 iS the resistance measured at room
temperature.I~ R4N2 is the normal state electrical resistivity measured at 4.2 K( in a field
of 0.6 T or higher after the magnetization data measurement. The higher purity the
niobium, the higher the RRR values, and the lower the Hc2 ValueS. For example, in Fig.
2-6 , at the temperature T s 1.5 K(, the upper critical field Hc2 m 12.8 K( Gauss, 7.3K(
Gauss and 3.5 K( Gauss for RRR of the niobium samples = 3.1, 8.8 and 505. For the
temperature dependence, the data has shown the thermodynamic critical field, He, with
the form He(T)= 1993[1 (g)2] (in Gauss) for RRR = 16300 , the lower critical field,
HeI(T) = 1735[1 (g)2.13] (in Gauss) for RRR = 1400 , anld th~e upper critical field
Hc2 = 4100- (in Gauss) for RRiR 300 .
In addition, the critical surface field, Hc3(T) = 1.695Hc2(T), has been predicted by
Saint-James and de Gennes  and measured [36, 38] from the onset of zero resistivity
and the AC susceptibility, which is also temperature, purity, RRR value and surface of
the samples dependent. Between Hc2 and Hc3, Superconductivity and surface supercurrent
appear in the form of a surface sheath with a thickness about ((T) on surfaces parallel to
the applied magnetic field.
Experimentally what we would do is to make thin hollow niobium cylinders with
end caps. Below the surface critical field, the enhanced -IIn! II -1,- I11, magfnetization
makes the thin surface sheath still superconducting while the bulk is in normal state. By
making the cylinder hollow we would enlarge the total volume of niobium to exclude more
magnetic flux with much smaller mass and reducing the total mass of the whole system
make the required driving magnetic force much less. Even when the field is beyond the
upper critical magnetic field and below the critical surface field, the surface sheath can
still form a quasi-perfect diamagnet.
2.3.3 Required Voltage Input for the Solenoid
If the resistance and the inductance of the solenoid are the only elements in the
circuit, and at the cryogenic temperature of 20 mK( the total resistance of the whole circuit
is R (Ohm), then the required voltage to supply the circuit would be:
V(t) = L + I(t)R (2-28)
In numerical analysis, we use:
[I(t) I(t dt)]
V(t) = Lx+ I(t)R (2-29)
(in volts), where dt is the infinitesimal time interval between the time (t-dt) and t when
the current flowing through the solenoid are I(t-dt) and I(t), respectively. The LabView
simulation program calculates the required voltage input to the superconducting shielded
solenoid with sine, linear, and square function acceleration as shown in Fig. 2-7. Since the
current for the sine function acceleration has smooth transition with respect to time, it has
the better performance than the other two, i.e. lower required voltage. Still the voltage is
too huge to have the practical application in the laboratory.
In order to solve this problem, we need an additional capacitor with capacitance
C (p-F) in the circuit to form an LRC circuit. In an LRC circuit under the sinusoidally
driven voltage, V = Voeiet, the K~irchhoff rule requires that the sum of the changes in
potential around the circuit must be zero, so
dl Q dl 1
V(t) = L + IR + L + IR + Id = ii Voeiwt, (2-30)
dt C dtC
where Q(Coulomb) is the total charge on the capacitor. The solution for the current would
I = Ioei(wt+ P, (2-31)
R29g2 (1- Lu2 @2+ (wL 1/wC)2~
1 LCw2 1 wL
p tn-() = tan- ( )(2-33)
RwC RwC R
At resonance, for w =iO = Wo 7= e"" .
We can calculate the required voltage input to the solenoid from the calculated
dl(t) 1 Pt
V(t) = L + I(t)R + I(t)dt (2-34)
dt C o
In the way of numerical analysis:
[I(t) I(t dt)] t
V(t) = L x dt + I(t)R + N (),(2-35)
ti = -( )(2-36)
Another alternative way of sending the pulses to the solenoid is to input the perfect
sinusoidal shape current pulse modified according to the previous calculated current
profile, which would still give the expected motion for the armature of the motor, as
shown in Fig. 2-8.
Suppose we supply the current I(t) = Iosinwt. If we only consider the solenoid with
inductance, L, and the resistance for the whole circuit, R, then the required input voltage
V(t) = L +I(t)R = L7,,... ..!: + IoRaincut. (2-37)
For Io = 2.2 A, R = 0.1 R, w = 1200 mad/s, L = 0.056 H, the same parameters as those
in Fig. 2-8, then the required voltage input to the solenoid versus time would be like Fig.
2-9. Some voltage pulses as large as 150 V seem to be too large to be applicable.
Experimentally, we might need C = 11.2 pF to tune the circuit on resonance. At
resonancei iWe. 11"' = o te = ~eime = ~. During the constant current period
for 10 ms, if the R = 0.1 R, L = 0.056 H, and we cut off the voltage, exclude and ground
the capacitor, then the current flowing through the solenoid would decay very negligibly:
I = Ioe-Rt/L o -Rt/L = -1.786t. It WOuld take 560 ms to decay to I = Ioe-l = 0.368Io-
The key is to find out the appropriate micro-electronics analog switches (IRF 7311, PVI
1050N) which can be controlled by the TTL signals from the computer to include or
exclude and discharge the capacitor simultaneously with the input pulse for the solenoid.
The schematic drawing for such a specially designed circuit is shown in Fig. 2-10. The
detailed circuitry for the whole experimental setup will be introduced very shortly.
2.4 Unshielded Motor Testing Experiments
Now it is time to put our simulation results into practice. Here comes the unshielded
motor testing mission. The dimension parameters of our unshielded motor system are
listed in Table 2-2. With those parameters, we run the LabView simulation programs to
find out the required current profile for the sine function acceleration, and then we modify
this current profile into two perfect sinusoidal pulses along with the constant DC current
in between. Fig. 2-11 shows the machine drawings and the photos for our unshielded
motor test cell. As we can see from the drawings that we build the capacitive position
sensor in order to monitor the motion of the armature.
Table 2-2: Parameters for the unshielded test motor. Refer to Table 2-1.
M. ... .2.0
2.4.1 Capacitance Bridge and Lock-in Amplifier for Monitoring Armature
The primary circuitry and the experimental setup for the motor test cell are shown
in Fig. 2-12. We send the negative pulse from the computer (analog output 1: DAC 0)
through our current amplifier. The current amplifier amplifies and inverts the pulse. Then
the pulse is fed into the switch box, where the switches can be controlled by the TTL
signals from the computer (analog output 2: DAC 1). Eventually the pulse is sent to
the solenoid in the cryostat at helium temperature, 4.2 K(. We also use the computer to
monitor the pulse sent to the solenoid (analog input 2: ACH 2), and simultaneously the
capacitive position sensor is measuring the position of the armature (analog input 1: ACH
The sensor is connected to the lock-in amplifier and capacitance bridge (GR1616).
We set the driving rms voltage 1 V, frequency 1.01 kHz from the reference sine waveform
output of the lock-in amplifier, and the time constant 100 ms. At room temperature the
capacitance bridge reads the capacitance of the capacitive position sensor 3.32 pF (the
magnitude, R, of the output on the di pl w~ of the lock-in amplifier reads minimum 39
p-V) when the armature is at rest and 3.00 pF (the magnitude, R, of the output on the
display of the lock-in amplifier reads minimum 108 pV) when the armature moves all
the way up hitting the brass plate 12 mm above in the air. By setting the sensitivity
of the lock-in amplifier 1 mV and the standard capacitance of the capacitance bridge
3.32 pF, the channel 1 output of the lock-in amplifier reads 0.44 V (R = 44.7 pV) as the
armature sitting at rest, while as the armature moves up 12 mm the channel 1 output
reads 2.54 V (R = 255 pV). With such significant voltage change, 2.10 V, which can be
easily recorded by the LabView data acquisition program, we can convert the voltage
shift into the position, or even the velocity of the armature motion. When cooling down
to liquid helium temperature, 4.2 K(, some conditions might change. We need to change
some of the settings of the lock-in amplifier to record the voltage shift from channel 1
output in a timely manner: the driving rms voltage 1 V, frequency :3.01 kHz from the
reference sine waveform output; time constant 1 ms (18 dB); sensitivity 1 mV. For the
capacitance bridge, we set the standard capacitance to be :3.00 pF while the channel 1
output of the lock-in amplifier reads 0.55:3 V (R = 0.28 mV) as the suck-stick probe is in
the liquid helium dewar and the armature is surely sitting all the way down. As long as
we adjust the standard capacitance to :3.30 pF, then the channel 1 output reads 2.38 V.
Therefore, we could expect some dramatic voltage increase around 1.8 V to be detected if
the armature moves all the way up to hit the brass plate.
It turns out that we do not need the computer controlled switch box, as the gray
region in the schematic, because current amplifier filters out the back emfs from solenoid.
The distorted pulse with spikes sent to the solenoid will be produced without both current
amplifier and the switch box, as mentioned previously. We turned up the current sent
to the solenoid little by little by turning up the gain of the current amplifier gradually.
Eventually, when the peak current reached the expected 2.86 A, we heard the sound from
the dewar like the armature hitting the brass plate 12 mm above it. What a success! Very
disappointedly, the capacitance bridge cannot respond fast enough to have any observable
change. Later on we realized from some experimental tests that the capacitance bridge
takes at least 100 ms to respond. It could be the inductance of the ratio transformer
slowing down the response time. A better method to measure quantitatively the position
of the armature of the superconducting motor system is necessary. So we have 555
oscillator circuit built and connect to the sensor in parallel and the LahView program with
the counter function counting the frequency change with time is developed as the second
attempt for the alternative measurement technique.
2.4.2 The 555 Oscillator for Monitoring Armature Motion
The improved circuitry and the experimental setup for the motor test cell is shown
in Fig. 2-14. Now instead of using the capacitance bridge and the lock-in amplifier,
we have the 555 oscillator circuit box, which can oscillate and output TTL pulses with
any destined frequency, which is determined by the resistances of the resistors in the
circuit and the capacitance of the capacitors in the circuit along with the capacitive
position sensor connected in parallel. The output frequency from 555 oscillator is inversely
proportional to the total capacitance of the whole circuit, and we would like the frequency
measurement in the LahView counter program to have the time resolution as small as
possible, 11- 1 ms; therefore, we set the gate width 0.5 ms. With the gate width 0.5 ms,
the uncertainty of the rising or falling edges counting during this short period of time
would be + 1, which means the frequency calculation would have the uncertainty +
2000 Hz. Hence, in order to have higher frequency resolution, we tune the 555 oscillator
as high as possible, ;?-- ~ 1 1\Hz. The capacitance change of the capacitive position
sensor for the armature at rest and moving up for 12 mm is roughly about 0.5 pF,
way too small if compared with all the other cables in the circuit, 11- 200 p-FF, which
gives the capacitance change the same as the frequency change about 0.25 So the
frequency change would be expected to be 2.5 kHz, not too much above our frequency
measurement uncertainty. We also have tried to decrease the gap between the sensor
and the niohium can to around half, but it still doesn't improve the capacitance change
too much! Since it takes time for the SuhVIs in the LahView to execute, about 1 ms for
reading one frequency measurement, therefore about 1.5 ms for one frequency data point,
and the frequency uncertainty is so high, we consider to run the experiments for dozens
to one hundred times and then do data averaging analysis. The signal-to-noise ratio is
proportional to the square root of the number of data sets for averaging. By averaging
100 runs of data sets, we can increase the signal-to-noise ratio by ten times, reducing the
time resolution down to 0.05 ms, and the frequency resolution to 200 Hz. We have the
LahView program developed to do continuous data acquisition and the averis?~ine-: however,
averaging one hundred times of experimental data is too much burden for our tiny, fragile,
and resistless armature. So we had to give up this way of measurement method. The
testing results for our unshielded motor test cell to monitor the motion of the armature
using the 555 oscillator circuit and the LabView counter program for the frequency
measurement show no frequency change with time within the frequency uncertainty.
2.4.3 The Q-meter for Monitoring Armature Motion
We also spent a week to build the Q-meter, i.e. resonant RLC circuits driven by a
sinusoidal voltage, like in Fig. 2-15 (a). Our theoretical simulation programs predict this
method feasible, for the capacitance change accompanying with noticeable current change,
which gives some noticeable voltage change across the resistor, capacitor or the inductor,
especially for higher Q (quality factor). Since the current in an RLC circuit in series is
I(t) =cos(wt + cp), (2-38)
tany = (2-39)
RwC R '
the maximum of the current versus frequency graph is at w = w0 = 1/& onreoane
The higher the Q, the sharper the peak will be due to the full width at half maximum
(FWHM) of I v.s. w curve, wo/Q, where Q = woL/R. In order to have more sensitive
measurement, we need to design the circuit with higher Q.
The testing circuit is shown in Fig. 2-15 (b). We tried to tune the frequency of the
reference output (sine wave) from lock-in amplifier to get the maximum voltage across any
one element of the circuit, which should fall at resonance of the RLC circuit. Therefore,
with a little capacitance change from the capacitive position sensor, the circuit will be off
resonance, then we are supposed to see the dramatic current decrease, therefore, dramatic
voltage change. Unfortunately, we didn't observe any noticeable voltage change (all less
than 2 .~) either across the resistor, capacitor or the inductor. We also exchanged the
position of the resistor and inductor, or rearrange the circuit to make the capacitors and
inductor in parallel. But none of the above worked well to show any large enough voltage
shift. Hence, we need to give up this approach.
2.4.4 The AC Bridge Circuits for Monitoring the Motion of the Armature
Eventually we find the excellent way to monitor the motion of the armature, the AC
We have used the capacitance bridge along with the lock-in amplifier to monitor the
motion of the armature before by measuring the capacitance change of the capacitive
position sensor. The output voltage shift from lock-in amplifier when moving the armature
up and down could be as large as 2 V or more, which gives enough sensitivity. The only
problem is to overcome the slow response time, 100 ms, of the ratio transformer. Fig. 2-16
shows one of the simple AC bridge circuits we built. As what we had expected, the AC
bridge circuit can respond as fast as less than a few microseconds without any delay. The
followingfs are the detailed descriptions about the experimental apparatus and setups.
The capacitive position sensor is composed of two copper semi-cylindrical sheets along
with the second niobium can. In the magnetic field, the slits of the copper cylindrical
sensor could prevent the eddy current and therefore the heat dissipation from being
produced. We use the stronger superconducting wire for the leads of the sensor and the
more flexible insulated fiber tube to protect the leads. Epoxy is used to glue the copper
sensor to the capacitor frame made of phenolic. Fig. 2-17 shows the circuit connection to
measure the capacitance of the capacitive sensor with the capacitance bridge. At room
temperature the capacitance bridge reads the capacitance of the capacitive position sensor
2.11 pF (the magnitude, R, of the output on the display of the lock-in amplifier reads
minimum 138.9 pV) when the armature is sitting all the way down, and 1.59 pF (the
magnitude, R, of the output on the di pl w~ of the lock-in amplifier reads minimum 153.4
p-V) when the armature moves all the way up hitting the brass plate 12 mm above in the
air. The capacitance difference is as small as 0.52 pF.
Now we connect the position sensor to the AC bridge circuit box. The circuit diagram
is shown in Fig. 2-18 (a). Instead of using the new digital lock-in amplifier (SR830), we
use the antique analog lock-in amplifier (PAR119) to get rid of the digitized problem
of repetitive ripples with periods about :3 ms superimposed to the signals. Here are the
settings for the analog lock-in amplifier: signal channel- frequency 92.25 kHz, mode
handpass; sensitivity 500 pV. We tune the variable capacitor to about :3.5 pF, and the
channel 1 output from the lock-in amplifier becomes 0.116 V when the armature sits all
the way down.
We also carefully calibrate and zero the offset of the current amplifier to within a few
nano-amperes. After adjusting the brass plate above the top of armature to be 21.8 mm,
the armature is expected to have enough space to move up and pass the second net zero
magnetic force position to be held there with some DC level. Fig. 2-19 (a) demonstrates
the theoretical calculation of the magnetic force per unit of current square experienced
by the armature inside the superconducting solenoid as a function of the position, z,
defined as the vertical distance between the center of the first niohium can and the middle
of the solenoid. Our plan is to start applying the acceleration pulse at xo = 6.4:3 mm,
around the peak position of the positive magnetic force zone, then the velocity of the
armature carries through the first net zero magnetic force position at about z = 12.5 mm;
afterwards it would experience some negative magnetic force until we apply the second
pulse for deceleration at the peak, z = 18.5 mm. If our second brake pulse didn't totally
stop the armature, then the residual velocity would be able to bring the armature further
up until it pass the second net zero magnetic force position, z = 26 mm. After this point,
the armature would experience positive magnetic force again and some small DC level
pulse would be enough to cancel out the downward gravity force and the armature would
be able to float there still for however long we need.
The room temperature calibration curve for the conversion of the voltage output from
channel 1 of the lock-in amplifier into the position, z, of the armature, precisely measured
by the depth micrometer gauge, is demonstrated in Fig. 2-19 (b). The probe was set to
be vertical to be as close to the situations of the actual experiments as possible and the
variable capacitor was tuned to about :3.8 pF when calibrated. The dip for the first ~
3 mm has two possible sources. One is due to the deviation of the second niobium can
from the central vertical axis of the sensor, therefore, resulting in more or less smaller
readings. Usually the armature will be brought to the center by the magnetic force as soon
as the pulses are applied to the solenoid. The other reason is due to the longer sensor of
the copper sheets (about 17 mm) than the length of the second niobium can (15 mm) by
around 2 mm. The capacitance of a coaxial cylindrical capacitor is
where an inner cylindrical conductor of length L and radius a is surrounded by an outer
cylindrical conductor of the same length and radius b, and Eo is the permittivity of
free space. Hence, the capacitive position sensor is expected to have the capacitance
proportional to the overlapping length between the second niobium can and the copper
sensor. During z = 0 and 2 mm, the overlapping length is about the same, 15 mm, so the
capacitance is supposed to have no change. However, due to the edge or boundary effect,
we see the dip occurs at z = 0 ~ 3 mm in the calibration curve and in the data we took
for the armature motion. In the data on~ ll-k- the conversion of signals into positions will
be modified accordingly.
In order to monitor the actual pulses sent to the superconducting solenoid, we
measure the voltage drop across the 0.1 R resistor in series with the solenoid. The 0.1 R
resistor box is connected to the output of the current amplifier (BOP) directly via banana
connectors and the other side is connected to the superconducting solenoid via the 12
pin connector on the top of the probe directly. The circuit diagram is shown in Fig. 2-18
(b). Due to the almost purely inductive (zero resistance) motor circuit, the pulses fed to
the solenoid are somewhat distorted and as the gain of the current amplifier turns up to
exceed 3 A of current output, the spikes with the opposite polarity appear, resulting from
the back emf due to the fast changing current. Therefore, we put cross diodes across the
solenoid in parallel to prevent this occurred.
Let's look at some of the data and testing results. For instance, in Fig. 2-20 the
motion of the armature of the superconducting motor is clearly examined. The pulse
profile sent to the solenoid is distorted due to the inductive behavior of the superconductor
solenoid. As the gain of the current amplifier is turned up, the higher the gain the more
the current would be fed to the solenoid. Here the first current pulse peak height reaches
around -1.8 A, then decays a little bit, followed hv the second peak as much as -2.369 A
then decaying to zero gradually. The current amplifier not only amplifies the pulses, but
also inverts the pulses output; therefore, we have the pulse profile with negative polarity.
The capacitive position sensor responds to the motion of the armature as the armature
moves up until out of the reach of the sensor, then drops back down by gravity force as
a function of time. The armature accelerates to 0.6 m/s at z = 5 mm within about 15
ms (average estimate acceleration 40 m/s2), then decelerates to zero at z = 15 mm in
40 ms (average estimate deceleration -15 m/s2). Afterwards, due to the gravity force
it drops back down and reaches the velocity as fast as 0.4 m/s at z = 4 mm within 70
ms (average estimate deceleration -5.7 m/s2). The armature probably moves further up
during the time 60 ms and 100 ms, but our sensor could not measure any changes because
the armature at that moment is already out of the detectable range. Fig. 2-20 (d) is the
original current profile sending to the current amplifier from the analog output DAC 0 of
our LahView data acquisition program.
If we compare with the theoretical prediction from our simulator with the same
distorted current profile as in Fig. 2-20 (a), then the predicted armature motion is like in
Fig. 2-21. The armature is supposed to reach the velocity as fast as more than 1.0 m/s,
but it is also predicted to move as far as approximately 10.5 mm only. The difference
between the theoretical prediction and the actual motion is still under exploration.
Now if we get rid of the second brake pulse and increase the DC level following the
first acceleration pulse, how would it affect the armature motion?
If we apply the acceleration pulse followed by 1.5 V DC pulse for 112.5 ms, Fig.
2-22 (d), then the current pulse profile in the solenoid actually looks like a smooth
transition from zero to some saturated DC level and the capacitive position sensor
responds correspondingly by oscillating around the equilibrium position about 0.9 V
and damping with time, Fig. 2-22 (a). From Fig. 2-22 (a) and (b), the armature seems
to be accelerated to almost 1.0 m/s within 10 ms and 4.5 mm, then it slows down until
stops at the highest point, about 6.8 mm, and drops back down to as fast as about -0.47
m/s, as far as the position about 4.3 mm. Afterwards, it oscillates back and forth around
the equilibrium position about 5.2 mm, like a harmonic oscillator. Due to the viscosity
and impedance of the liquid helium and the friction on the hearings contacting with the
armature, the oscillation motion also damps in 80 ms, then the armature -r li- stationary.
As soon as the DC level is turned off, the armature drops back down to the original
If we compare with the magnetic force versus position graph in Fig. 2-19 (a), this
kind of oscillation motion is quite understandable. We apply the acceleration pulse at the
maximum peak of the magnetic force zone, z ~ 6.5 mm. As soon the armature travels
passing through the zero magnetic force position, z = 12.5 mm, it experiences negfative
magnetic force, decelerating and pushing it back down. And then it experiences positive
magnetic force again helow 12.5 mm position, which will push it up. Such a cycle repeats
until it damps, then it will float still at the position where the gravity force and the
magnetic force all cancel out, resulting in zero net forces.
If we apply the acceleration pulse followed by 2.0 V DC pulse for 112.5 ms, Fig. 2-22
(d), then the current pulse profile in the solenoid actually looks like a rapid transition
from zero to the same saturated DC level and the capacitive position sensor responds
correspondingly by oscillating around the equilibrium position about 0.85 V and damping
quickly with time, Fig. 2-22 (a). From Fig. 2-22 (a) and (b), the armature seems to
be accelerated to almost 0.9 m/s within 10 ms and about 1.3 mm, then it keeps almost
constant speed until at the position about 4.7 mm it slows down to zero and stops at the
highest point, about 6.3 mm. Afterwards, it drops back down to as fast as about -0.3 m/s,
as far as the position about 4.6 mm. It oscillates back and forth around the equilibrium
position about 5 mm, like a harmonic oscillator. Due to the viscosity and impedance
of the liquid helium and the friction on the hearings contacting with the armature, the
oscillation motion damps very soon in :30 ms, then the armature cr li- stationary. As soon
as the DC level is turned off, the armature drops back down to the original position.
With higher DC level, the armature can he accelerated to higher velocity followed
by more rapidly damped oscillation afterwards because the actual current pulse in the
solenoid is ramped faster up to the saturated current of BOP. Since we really need to
have the armature travel with constant speed for at least 10 mm, this result with our
current design cannot satisfy us. So the armature must he modified. With the distance
between the centers of the two niohium cans about :35 mm, we could just simply apply one
sine wave pulse followed by a DC pulse, then we are able to accelerate the armature and
hold it around 10 mm position for as long as we need. The drawback of this design is the
unavoidable oscillation which might more or less affect the turbulence just produced and
the DC pulse to hold the armature must he at least a few amperes.
Now if we reverse the polarity of the input pulse going to the current amplifier,
how would it affect the armature motion? We apply the original current pulse to the
superconducting solenoid, Fig. 2-24 (d), then the measured pulse profile is positive and
the inductive effects are not so significant, Fig. 2-24 (a). The actual pulse peaks saturate
up to 7.5 A, more than what we have expected. The armature is accelerated to about 0.7
m/s within about 10 ms and 1.5 mm and up to 0.85 m/s at 4.2 mm position and at the
time 15 ms. The armature goes up as high as 8.7 mm, and then drops back down with
the velocity as high as 0.2 m/s. The armature seems to get stumbled for a little bit while
at about 4.8 mm position, where is estimated to be the zero net magnetic forces position.
We figure out that the DC level pulse as high as 2 A could have decelerated the armature
rapidly beyond the position z = 5 nin before the brake pulse. That is the reason why the
armature could not have gone farther and reached the expected height, 12 nin.
We lengthen the DC level to 25 nis while keeping the rest of the original current
profile the same, Fig. 2-25 (d), then the measured pulse profile fed to the superconducting
solenoid is positive and the inductive effects are not so significant as well, Fig. 2-24 (a).
The actual pulse peaks saturate up to 7.5 and 10.5 A, while we do keep the gain of the
current amplifier the same as in the previous case. The armature is accelerated to about
0.72 nt/s within about 10 nis and 1.2 nin and up to 0.9 nt/s at 4 nin position and at
the time 12.5 ms. It seems to get stumbled and slow down between 4 and 6 nin position,
but speeds up to 0.95 nt/s afterwards. The armature goes up as high as 15.2 nin, beyond
which the armature is actually out of the detectable range of the capacitive position
sensor. It shows zero velocity during the time 30 nis and 110 nis, when the armature is
either out of the reach of the sensor or it is floating above the second zero net magnetic
position, z = 26 nin in Fig. 2-19 (a), or 19.5 nin equivalently in Fig. 2-25 (c). It then
drops back down with the pretty constant velocity, 0.15 nt/s, all the way down. The DC
level pulse as high as 2A is long enough not to decelerate the armature in time, so it can
move up beyond the detectable position 15.2 nin or higher before the brake pulse applies.
This provides us one of the important successful experiences of how we could achieve our
goal. The armature is proposed to be accelerated, travel for about 10 nin with almost
constant velocity, he decelerated then pass the second point of the zero net magnetic force
position and then he held above it for however long we need. The drawback of this design
is that we don't know if we are goingf to make the armature hit the top of the cell. The
DC pulse level should be carefully adjusted to be just as much as what we need and the
whole trajectory of the motion should be monitored. In order to be able to monitor the
whole trajectory of the motion of the armature, the capacitive position sensor needs to be
lengthened to appropriate length without modifying our current armature. Also we should
lower down the sensor for a few nin (2 ~ 3nin) to avoid the dip problem.
2.5 Improved Design of the Armature and the Test Cell
In order to be able to monitor the whole trajectory of the motion of the armature,
and solve the dip problem at the initial positions, we actually lengthen the copper sensor
to 27 mm and move it down by 7 mm. Conducive silver paint is also applied to the
armature for total conductive length 31 mm. A conducting section on the armature,
composed of one of the Nb cylinders and silver paint coating part of the phenolic rod, is
inside a closely fitting capacitor made of two semi-cylindrical copper sheets (see Fig. 2-26
(a) ). This capacitor, coupled to a bridge circuit, measures the armature position. The
total mass of armature is now 2.60 g. The grid, made of 0.125 mm thick spring steel with
1 mm square holes and 70 transparency, is attached to the lower end of the armature
(see Fig. 2-26 (b), (c) and Fig. 2-27 (a)). The corresponding Reynolds number for the
different velocities of the grid motion is shown Fig. 2-27 (b). Let's find out the Reynolds
number of our motor system in the liquid helium-4 bath at 4.2 K(. Reynolds number is
defined as the ratio of inertial force and viscosity force, or the velocity scale multiplied by
the length scale, divided by the kinetic viscosity:
U/L inertial force
V v .o~lu force
At motor testing temperature, 4.2 K( in liquid helium, grid Reynolds number:
U x 1.016 x 10-3m
Re 3.927 x 104 x U (2-42)
2.587 x 10-sm2 s-1
(in m/s). If we consider the size of the mesh of the grid as the length scale, velocity of the
grid motion as the velocity scale, then we have Reynolds number ranging from 4000 to
40,000 when the grid velocity varies from 0.1 to 1 m/s, above the onset of turbulence.
The room temperature calibration curve shows the monotonic increase in voltage as
moving up the armature without any dip. The entire assembly is mounted in a helium
test cell and cooled in a transport dewar cryostat . The armature position sensor was
calibrated at 4.2 K( using a micrometer drive mounted at room temperature. Fig. 2-28 
shows the calibration curve at liquid helium temperature, 4.2 K(.
In the electronics, we have two independent circuitry for the superconducting solenoid
and for the capacitive position sensor (Fig. 2-29). We drive with the properly shaped
current pulse profile determined by simulation, then amplify the current via the bipolar
power supply. The input pulse to the solenoid is monitored by measuring the voltage drop
across the 0.1 ohm resistor shunt. Two cross diodes are to protect the solenoid from the
spikes due to fast changing current. For the position sensor, we use the AC bridge circuit.
Two arms are 100 ohm resistors, and the other two are variable capacitor and the position
sensor. While the armature moves, the capacitance would change, giving different voltage
response. We convert the voltage into the position of the armature with our calibration
curve performed at 4.2 K(.
The experimental results are shown in Fig. 2-30 . We apply the pulse profile: two
sinusoidal shape pulses followed by -0.1 V DC level for 250 ms (Fig. 2-30(e)). The solenoid
is protected by two crossed silicon diodes which have cut-in voltages of 4 V and 18 V at
4.2 K(. Therefore, reversing the polarity of the solenoid current produces an .I-i-mmetry
in response (Fig. 2-30(a) and Fig. 2-30(b)). In Fig. 2-30(c) and (d), with the current
pulse like Fig. 2-30(a) the armature is seen to accelerate to 0.8 m/s within 40 ms over
a distance of 9 mm (estimated average acceleration 20 m/s2). Then it undergoes a near
constant deceleration to zero velocity within 33 ms in another 12 mm (estimated average
deceleration 24 m/s2). It WaS held at the position of 22 mm for about 250 ms until the
current was turned off, it then dropped under gravity, as fast as 0.48 m/s in 75 ms over
10 mm position (estimated average acceleration 6.4 m/s2). As seen in Fig. 2-30(c) and
(d), with the current pulse in Fig. 2-30(b) the armature accelerates to 1.09 m/s within
20 ms over 9 mm (estimated average acceleration 55 m/s2). Then it slows down to zero
velocity within 38 ms over the remaining 12 mm of stroke (estimated average deceleration
29 m/s2). It WaS held at 22 mm for about 250 ms until the current was turned off, then
dropped under gravity as fast as 0.45 m/s in 49 ms over a movement of 12 mm (estimated
average acceleration 9.2 m/.s2). At the position between 4 and 14 mm, the velocity of the
armature was between 0.8 and 1.09 m/s, constant within about 15
We have written the code in LahView to calculate the current and voltage required
to supply the solenoid in the magnetically shielded superconducting linear motor
system to produce the homogeneous isotropic turbulence for our low temperature
quantum turbulence experiments. This kind of magnetic driven motor shielded by the
superconductor can avoid the heating due to eddy current on the wall resulting from the
changing magnetic field from the surroundings, the Joule heating from the solenoid with
current and the friction during the process when the grid is towed through the liquid
helium at 520 mK(. We also find the optimal parameters for our design demanding the
least voltage, current or magnetic field. The evaluation results from the velocity versus
position curves demonstrate that the grid can he efficiently accelerated to 1 m/s over
1 mm, then proceed with that about constant speed for 10 mm and slow down quickly
within 1 mm, which is a very reasonable and feasible design. We plan to implement our
successfully developed superconducting linear motor to our quantum turbulence studies.
Figure 2-2: Superconducting motor model with grid, thernxistors and superconducting
Figure 2-1: Grid turbulence in a classical fluid .
0 2 4 6 8 lb1012
Position of Nb #1 (mm)
114 ~ 16
Figure 2-3: Armature motion in unshielded (pink curves) and superconducting shielded
(black curves) solenoid under the sine function acceleration: (a) I(t) curve; (b) v(z) curve.
(o: modified data without the central peak in the current profile) 
4 6 8 10 12 14
Position of Nb #1 (mm)
Figure 2-4: Armature motion inside the superconducting shielded solenoid under the linear
function acceleration: (a) I(t) curve; (b) v(z) curve.
14 16 18
0 2 4 6 8 10 12 1416 1820 2224 26 28
4 6 8 10 12 14
Position of Nb #1 (mm)
Figure 2-5: Armature motion inside the superconducting shielded solenoid under the
square function acceleration: (a) I(t) curve; (b) v(z) curve.
10 N 4 T '8T"
Figure 2-6: Upper critical field versus temperature for niobium samples with different
1)(b i 0
4 6 8 10 12
Position, z (mm)
2 4 6810 12 1'4 1'6
2.0 -2.0 -
-* Magnetic field at r-rol, z=0
1.5 Magneticforce1. -
1.0 -i 1.0-
-0. -. -05
S-1.0 -I -1.0
tR-1.5- -F -1.5-
0,, 2 4 6 8 10 12 14 16 4 0
Z ~Time (ms) E
*Magnetic field at r=rol, z=0
6b 81 1'2
Position, z (mm)
Figure 2-8: (a) Sinusoidal current input to the superconducting shielded solenoid with
the dimensions as stated in Table 1.; (b) the corresponding velocity versus position of
the shaft motion graphs; (c) the corresponding magnetic field, and magnetic force versus
position of the shaft motion graphs. 
Figure 2-11: Machine drawings and photos of the motor system: (a) machine drawing ;
(b) photo of the test cell; (c) photo of the test cell installed at the end of the suck-stick
Figure 2-15: Testing circuit of Q-meter.
Figure 2-16: Simple AC bridge circuit.
(a) r7111CUUy IU~reft- rlll fi I
Sne Out ~LVrms-5V
Variable I Capacitive
capacitor /I / position
I I sensor
Analog lock-in Amp
100X 1 100
ground Probe, Crost~at at 4.2K
,Analog output Irpoolpolar output R-0.1ohm I r=1 ohlm
Oxnputer DAC O input Ftueaapy
Lab~eur -- llferaal 4
ground rr 56~dutg
Analog input ACHO ACH8 lni
Oxnputer = sm
Figure 2-18: Circuitry and setup for testing the superconducting motor at liquid helium
temperature, 4.2 K(. (a) monitoring the motion of the armature with the capacitance
position sensor; (b) monitoring the pulse sent to the superconducting solenoid.
*Current pulse profile
(a) Capacitive position sensor response
200 points adjacent normal averaging
J I I I I 10.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
0.00 0.05 0.10 0.15 0.20
200 points adjacent normal averaging
0 2 4 6 1 12 1'4
10 12 14 16 18 20
Position, z (mm)
Figure 2-20: Motion of the armature of the superconducting motor. (a) Pulse profile sent
to the solenoid and capacitive position sensor response as a function of time. (b) Velocity
of the armature as a function of time. (c) Velocity versus position of the armature. (d)
Original current profile from analog output DAC 0 of LabView. 
3 10 12 14 11
Position of Nb #1 (mm)
Magnetic Field at r=rol, z=0
Magnetic Field at r=rol, z=0
-0 4 -
8 1 12 14
Position of Nb #1 (mm)
10 20 30 40 50 60
Figure 2-21: (a) Distorted current profile due to the almost purely inductive behavior of
the solenoid (small resistance ~ 10) in the superconducting shielded solenoid with the
dimensions as stated in Table 2.; (b) corresponding velocity versus position of the shaft
motion graphs; (c) corresponding magnetic field, and magnetic force versus position of
the shaft motion graphs; (d) corresponding magnetic field, and magnetic force versus time
F4 -0.4 -
100 points adjacent normal averaging
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
o n ,,
(a) -- Capacitive position sensor response
One pulse, DC level=2V for 112.5ms
1.-.in the original pulse profile
Actual pulse peak height= ~ -0.55V
a 0.4 -
0.00 0.05 0.10 0.15 0.20
100 points for adjacent normal averaging
-0.4 -( '
Position, z (mm)
20 40 60 80
Figure 2-23: Motion of the armature of the superconducting motor. (a) Pulse profile sent
to the solenoid and capacitive position sensor response as a function of time. (b) Velocity
of the armature as a function of time. (c) Velocity versus position of the armature. (d)
Original current profile from analog output DAC 0 of LabView. 
(a) Capacitive position sensor response
DC level= 0.32V for 25ms
in the original pulse profile
3.0-1 Actual pulse peak height=
2.5-1 1 1.05V
0 00 0.05 0.10 0.15 0.20 0.25 0.30
(c) 1~100 points adjacent normal averaging
8 10 12 14 16 18 20 22
Position, z (mm)
(b)1.0100 points adjacent normal averaging
a .0.8 -
0.00 0.05 0.10 0.15 0.20
5 10 1'5
20 2'5 30
Figure 2-25: Motion of the armature of the superconducting motor. (a) Pulse profile sent
to the solenoid and capacitive position sensor response as a function of time. (b) Velocity
of the armature as a function of time. (c) Velocity versus position of the armature. (d)
Original current profile from analog output DAC 0 of LabView. 
Figure 2-26: (a) New machine drawings of the modified motor system . (b) Picture of
the armature with the grid mounted at the end. (c) Picture of the test cell showing the
2-56 clear holes through
(smooth fit to pittar rodl)
0.0 0.2 0.4 0.6
Grid Velocity, U (m/s)
Figure 2-27: (a) Machine drawing for the grid. (b) Reynolds number versus grid velocity
at 4.2 K(.
11515 1 IT ------a
(b) 2.0 -
U 2.8 -
Position, z (mm)
0 2 4 6 8 10 12 14 16 18 20 22
Position, z (mm)
Figure 2-28: (a) The calibration of the armature position sensor (4.2 K(). Position sensor
bridge voltage output front lock-in amplifier versus the position of the armature . (b)
The capacitance of the capacitive position sensor versus the position of the armature,
measured by using the capacitance bridge, GR 1616.
I~o pols lbuor simply
Probe Cryostat at 42K
Figure 2-29: Electronics circuits for the superconducting motor system.
2.5- sensor response
Pulse profile (b)
-0.5-11 Actual current pulse peak
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.r
1.2 '- -Pulbs (bi.
0 2 4 6 8 10 12 14 16 18 20 22 2
POSitiOn, z (mm)
DC level -0 10V for 250 ms
10 ms delay
10 ms DC level 0.32V
2. ims Sine pulse with peak 2.86V1
2.5 ms Sine pulse with peak 2.86V
)0 0.05 0.10 0.15 0.20 0.25 0.30
Figure 2-30: Motion of the armature of the superconducting motor . (a) Current
through the solenoid and capacitive position sensor response as a function of time. (b)
Same as (a) with polarity reversed. (c) Velocity of the armature as a function of time.
(d) Velocity versus position of the armature. (e) Original current profile from analog
output DAC 0 of National Instruments board sent to K~epco BOP current amplifier driving
CONSTRUCTION OF SUPERCONDUCTING SHIELDED LINEAR MOTOR AND
3.1 Construction of Superconducting Shield
For our quantum turbulence research, we are building a shielded superconducting
linear motor to tow a grid through superfluid helium. Hence, one of our tasks is to
construct a superconducting shield on the interior of the experimental cell. To accomplish
this, we have decided to electroplate our cell with lead. Lead has the advantage over other
possible superconductors of a high critical magnetic field, 803 Gauss, and high critical
temperature, 7.2 K . Lead plating is also approachable and economic to be done in
the laboratory. We have decided to electroplate, rather than chemically deposit the lead,
because chemical vapor deposition (CVD) requires high temperatures and is difficult to
perform in the laboratory. For reference, lead metal has very high boiling point, about
2022 K, and melting point, as high as 600 K, using physical vapor deposition method
requiring vaporization of the metal element at very high temperatures, which is also not
applicable for our lab. Another superconducting metal, niobium, with critical temperature
9.5 K  has much higher melting point, 2740 K, and even higher boiling point, about
5017 K. Therefore, considering the options we decide to electroplate a lead magnetic
shielding enclosing the motor at very low temperature.
3.1.1 Electroplating Theory and Electrolyte Recipe
Basically, the idea of lead electroplating is to have lead deposited on the desired parts
while a pure lead electrode is dissolved in the electrolyte. The electrode is used to supply
and keep the lead concentration in the solution at a sufficient level. The process is as
Anode : Pb Pb2+ + 2e- (Oxidation) (3-1)
Cathode : Pb2+ + 2e- Pb(Reduction) (3-2)
We used methanesulfonic acid (ilr;A) as our electrolyte in lead plating. Recently,
(since the 1980's) MSA has been used in industry for electre~lpta filr_ rather than lead
fluoroborate (Pb(BF4)2), fluOroboric acid (HBF4) and boric acid (H3BO3) because it
is less corrosive and the produced effluent is easier to deal with. Besides, the organic
additives aid to produce a smooth, fine-grained deposit and to increase throwing power.
We got a nice recipe for the electrolyte solution makeup from Technic Inc.:
* Techni-Solder NF Acid (70 .~): 20 .~ or 200.0 mL in one liter of electrolyte.
Ingredients: 70 .~ or 10 M methane sulfonic acid (CH4SO3 -
* Techni-Solder 700 HS MakeUP: 2.5 or 25.0 mL in one liter of electrolyte.
Ingredient: two non-ionic surfactants, which can reduce the surface tension to
have the lead deposited more delicate and fine-grained.
* Lead carbonate (PbCO3): 15.0 g of lead or 19.3 g of lead carbonate in one liter of
electrolyte. Lead can be dissolved in MSA very slowly. It would be more efficient to
put lead carbonate into MSA resulting in lead (II) methanesulfonate solution, carbon
dioxide (CO2) and water (H20) produced.
* DI water: for balance and adding up to one liter of solution.
* The optimal current density would be 20 ASF (ampere per square feet) for deposit
1 p-m of lead per minute. Agitation is required to make the lead ions in the solution
Since the coherence length of lead is 0.083 pm , we would like to deposit 25 pm of
lead, which is more than enough. For our purposes, our cell is made of pure copper free of
oxygen, Fig. 3-1 and Fig. 3-2. For the cell cap and cell body, the interior surface areas are
about 68.3 cm2 and 78.9 cm2. We need 1.469 A and 1.695 A of current to deposit 1 pm of
lead per minute on them, respectively.
3.1.2 Properties of Chemicals
Lead has a wide range of applications in our daily life, like lead paint, solder joints on
PCB in electronics, car batteries, etc. However, over the past few decades, more and more
information has come ut about the hazards of lead and lead compounds and it has seen a
gradual decline in use. Lead accumulates in bone and body, primarily through inhalation
or ingestion of dust and fumes, resulting in headache, nausea, vomiting, abdominal
spasms, fatigue, sleep disturbances, weight loss, anemia, and pain in legs, arms, and joints;
an acute, short-term dose of lead could cause acute encephalopathy with seizures, coma,
kidney damage, anemia, and even death .
18.104.22.168 Methanesulfonic acid (MSA)
The appearance of methanesulfonic acid (CH4SO3) is ClarT, traUSparent liquid and it
has a faint sulfur oxide odor. Thermal decomposition of MSA may release sulfur dioxide
and sulfur trioxide. It is corrosive to skin and causes severe burns if contacted, swallowed
or inhaled. Therefore, it should be handled carefully. If released or spilled, MSA can be
neutralized by slow and careful applications of a solution of soda ash and water .
22.214.171.124 Lead carbonate
Lead carbonate (PbCO3), alSO known as cerussite, is a white
powder without any odor. It is harmful if inhaled or swallowed. Lead is a cumulative
poison and exposure even to small amounts can raise the body's content to toxic levels.
Risk of cancer depends on level and duration of exposure. Target organs: K~idney, central
nervous system, blood, reproductive system .
126.96.36.199 Lab protective equip
CI.- --1. s and face shield; lab coat and acid resistant apron; vent hood; rubber gloves
(vinyl gloves are the best, and latex gloves are acceptable); respirator or NIOSH/j\!SllA4
3.1.3 Lead Electroplating Procedures and Results
188.8.131.52 Procedure steps
Making new electrolyte solution:
* 100 mL of Techni- Solder NF Acid (70 ~) is added to about 200 mL of DI water in
* Add 12.5 mL of Techni- Solder 700 HS MakeUp to the beaker.
* Add 15 grams of lead (or 19.34 grams of PbCO3) to the solution. Note: this is double
amount of lead as that in the recipe. Now the solution becomes turbid and a lot
of exquisite white bubbles of CO2 float above the surface filling up the 1000 mL of
beaker. After a while, the solution becomes clear again.
* Add DI water to the beaker up to 500 mL of electrolyte solution.
The followingfs are the actual procedure steps of lead electroplatingf performed in the
1. Cleaning the cell with the Electronic shop's help. They use the commercial copper
cleaner and ultrasonic vibration equipment for cleaning the cell. Fig. 3-3 (b) and
Fig. 3-4 (b) show the copper interior of the cell cap and cell body, and the well-built
silicone wall, ready for the lead plating.
2. Sealing the holes with silicone [? ]and building about one inch high of silicone wall
leaning against the paper wall about a few mm extended outward along the edges of
the top rim of the cell cap or cell body (Fig. 3-3 (b) and Fig. 3-4 (b)).
3. Making the lead electrode for lead plating of the cell cap and cell body (the drawing
of the exact size shown in Fig. 3-3 (a) and Fig. 3-4 (a)). The lead electrode is well
trimmed and carefully adjusted to be about 2 mm clearance away from every cell
wall. With the aid of a magnifying mirror, an ohmmeter, and a round rubber pad
attached to the center of the turntable, we can do an almost perfect job, as shown in
Fig. 3-3 (c), (d) and (e) and Fig. 3-4 (c), (d) and (e). The cell cap is supported by
a rubber piston underneath to stand more stably. The cell body has gold plated at
the bottom. While rotating the turntable, we make sure the resistance between the
electrode anode and cell cathode is infinity.
4. The electrolyte is injected to the cell cap or cell body about half to one inch higher
than the rim enclosed by the silicone wall (see Fig. 3-5 (a) and Fig. 3-6 (a)). The
resistance between the two electrodes (the lead electrode as the anode and the cell as
the cathode) now is only a few ohms. The cell is rotated steadily and slowly on the
turntable of a broken microwave oven. Therefore, the lead anode can also agitate the
electrolyte to make the lead ions in the solution distribute very uniformly while lead
plating is performed. We start to supply the current of about 1.47 A (Fig. 3-5 (b))
for the cell cap and about 1.70 A (Fig. 3-6 (b)) for the cell body. The lead electrode
(anode) connects to the positive terminal of power supply, while the cell (cathode) is
in contact with the thin stainless steel piece connecting to the negative terminal of
the power supply. No white bubbles occur with high concentration of lead, 30 grams/
liter of lead in the electrolyte. The exterior part of the cell wall in contact with the
thin stainless steel piece, reduces to shiny pure copper from the oxidized copper (Fig.
3-5 (c)). Under the liquid surface, it is clear to see that the lead electrode is well
adjusted- 2 mm clearance away from every cell wall (Fig. 3-5 (d) and Fig. 3-6 (d)).
The dissolved lead precipitates at the bottom of the cell.
5. The power supply is turned on to supply the current for approximately 25 minutes.
The current fluctuates slightly, but overall very stable. The voltage output from
the power supply is 8.2 V when lead plating the cell body. Note: Do not touch the
electrodes during the electlll rfpiing. Otherwise, you will get burn because they are
very very hot due to high current flowing through them. The whole lead plating
process is done in the vent hood.
6. Rinse with ample DI water (about one gallon). Dry the cell with clean tissue and
flush with a high flow of nitrogen gas or cool wind from the heat gun immediately
7. The electrochemical equivalent for the reaction: Pb2+ + 2e- Pb is 3.86
g/(Almpere hour) . By lead plating on the cell body with current 1.70A for
27 minutes, the deposited lead mass is 2.95g. Since the density of lead is 11.34g/cm3,
the surface area of the cell body is 78.85cm2, the thinkness of the deposited lead
would be 33.0 pm.
Fig. 3-7 and Fig. 3-8 show the masterpieces after lead electroplating with shiny lead
of silver white color in the interiors, except that covered by the silicone. A close view of
the silver white interior of the cell cap shows the rims, the walls and the edges of the well
down to the cell bottom fully covered by the lead. Fig. 3-7 (c) and Fig. 3-8 (c) shows
the residue lead electrodes while the parts immersed under the liquid surface become
darker, dissolved to almost only half left.
1. If the lead anode were too small, then it might all get dissolved into the electrolyte,
resulting in zero current before the lead plating process is complete. Especially, the
dissolution of lead electrode happens at the interface between the electrolyte and air,
where oxygen is available. If the current were too small, like 90 mA, then it would
take 7 or more hours to achieve 25 p m of lead thickness, and it is possible that the
lead deposition process is much slower than lead dissolution into the electrolyte,
hence, the actual thickness of deposited lead could not be sure. Therefore we make
the lead electrode as large as 2 mm clearance away from the cell wall, so that we
could turn up the current up to the optima.
2. Deposited lead on the cell interior could be oxidized to yellow-colored and powder like
lead oxide (PbO) after drying out naturally without some special care, which could
be due to the residual acid or just water causing the oxidation. The lead oxide can
he easily wiped off to make the copper on the rim edge exposed. After sanding off
the oxide or sponge on the interior surface of the cell, the shiny metal lead appears.
Therefore, we know that oxide on the outer surface 1,o ;r can protect the inner
1.w-;r of lead metal to be further oxidized. To prevent this situation, we need to use
abundant DI water to clean thoroughly the acid residue and dry the wet immediately
:3. White and delicate bubbles show up on the surface of the electrolyte with lead
concentration only 15 g per liter of electrolyte when turning up to the operating
current during lead electroplatingf. The white bubbles on the surface of the solution
occurred at high current density is fine 1 When the current efficiency is not 100' .~
the hydrogen ions, competing with the lead ions, capture the electrons instead, then
the hydrogen gas would be produced, forming the bubbles. When the current is
high, as high as 1.70 A like the above process, and the current density is not the
same everywhere on the cell, the lead might he deposited more or less at different
positions. If the bubbles occur, then it could result in uniform thickness throughout
the whole cell walls. The sponge like, dark color lead forms rough surfaces at high
current density when the lead concentration in the electrolyte is too low. We solved
this problem by doubling the amount of the lead in the electrolyte.
4. "1\odern Electroplatingt by Schlesinger and Paunovic (2000) on page :366 :
lIs-u~llah!.- anodes cannot he used in lead plating electrolytes as lead dioxide, PbO2,
will form on the surface of the anodes. The purity of the soluble lead anodes used
determines the extent to which a film forms on the surface of the anodes." On page
:369 : "During the deposition of thick lead coatings (up to 200 pm) formation
of nodules or ,y ~.--IlI!-" can occur. This failure does not generally occur with a
freshly made-up solution, and when it does occur, it can in most cases he rectified
by a purification of the electrolyte with activated carbon. contamination of
the electrolyte by breakdown products of the organic additives. together with
a rapid decrease of the lead concentration in the electrolyte. The anodic current
efficiency was reduced, which caused the drop of the lead content in the hath. By the
passivation of the anode, lead dioxide was formed on the anode surface, which caused
a partial oxidation of the organic additives."
Therefore, we should use 100 .pure lead as the anode to prevent the lead dioxide
from forming on the surface of the electrode, causing anodic oxidation of the organic
additives on the anode surface, or the oxidation of the organic additives by the lead
dioxide, resulting in the roughness of lead deposition. Also if the electrolyte solution
1 Private communication with Chuhua Wan--_ from Technic, Inc.
is made for a while, it should be discarded following the hazardous waste disposal
rules. The solution should be made just before we attempt the lead pI rfl-,Hi and the
work should be done within two or three d we~ before the solution becomes old.
5. "Modern Electroplatingt by Schlesinger and Paunovic (2000) on page 373 :
"Discoloration of the lead deposit to a brown or black color occurs due to deposition
of copper onto the lead surface by a displacement reaction that can happen in the
electrolyte if the current is left switched off while the parts are still immersed, or in
the rinses if they are heavily contaminated with copper."
Therefore, after finishing the lead plating, we should dump out the electrolyte
immediately after the power supply is switched off. And rinse the parts with plenty
of DI water immediately.
3.2 Testing the Experimental Cell at Liquid Helium Temperature
Fig. 3-9 shows the machine drawing with the dimensions and the pictures of our
experimental cell. We have one superconducting solenoid driving the armature to move
through its center, with a grid attached at the end. This light insulating armature is
constructed of 3 phenolic tubes separated by two hollow cylindrical niobium cans placed
26 mm apart, with the turbulence-producing grid attached to one end. A conducting
section on the armature, composed of one of the Nb cylinders and silver paint coating part
of the phenolic rod, is inside a closely fitting capacitor made of two semi-cylindrical copper
sheets. This capacitor, coupled to a bridge circuit, measures the armature position. The
dimension parameters of our superconductor shielded superconducting linear motor system
are listed in Table 3-1. The electronics circuits are the same as described in chapter 2,
e.g. Fig. 2-29, except the lead resistance 0.7 R. The experiment cell is mounted to the
0.25 inch diameter probe and tested in a transport dewar cryostat  at liquid helium
3.2.1 Simulation Results
If we define the center of the solenoid as the position, z = 0 (see Fig. 3-10 (a)), then
the magnetic force distribution along the z-axis of the solenoid are calculated, as in Fig.
3-10 (b). To be more efficient, we would apply the acceleration pulse when the center of
mass of the niobium can #1 is located at z = 6.5 mm position, and deceleration pulse at z
Table 3-1: Parameters of superconductor shielded superconducting linear motor system.
Ml. ... 2.6
=18.5 mm position. After the armature passes the z = 26 mm position, we would apply
the holding pulse, just enough to balance out the gravity.
In the current profile from our simulation program (Fig. 3-10 (c)), the first pulse
produces sine function acceleration, and the third pulse deals with linear function
deceleration. The central peak is due to the almost balanced magnetic forces on the
two niobium cans at that position, where each is almost equidistant from the ends of
the solenoid. We can remove the central peak and the inertia will serve to carry through
it. It takes slightly more current, 0.147 A more, for the superconducting shielded motor
system than the unshielded one. In the resulting grid motion (Fig. 3-10 (d)), the grid
would have sine function acceleration from 0 to 1 m/s in 1 mm. Then it travels at almost
constant speed, 1 m/s, for 10 mm. In the mid-way, the grid starts to slow down at 12.5
mm position after the niobium can #2 becomes closer to the solenoid, meaning stronger
magnetic force in the opposite direction. Then the grid would rapidly decelerate to cease
within 1 mm when the third pulse is applied. The empty circles represent the grid motion
without the central peak in the current profile. Without the central peak in the current
profile, the grid motion is not significantly different.
3.2.2 Experimental Testing Results
Fig. 3-11 shows the experimental results. Considering the inductive behavior of
the solenoid, an inductor, the time constant of an LR circuit is L/R = 16.34mH/0.70
~ 23.3milli seconds. Even if we apply a square pulse, ;?i 1 V, it takes five times
the time constant (116.5 ms) to saturate to 1 V. Therefore, we need to take this into
account when we apply the pulses to the superconducting solenoid. We apply the
appropriate pulse profile: DC 2.89 V for 14 ms, followed by DC -0.3 V for 10 ms, then
2.5 ms delay, eventually the holding pulse -0.02 V for 100 ms (Fig. 3-11(d)). The solenoid
is again protected by the same two crossed silicon diodes as described in chapter 2. In
Fig. 3-11(b) and (c), the armature is seen to accelerate to 0.7 m/s within 6.5 ms over a
distance of 2.5 mm (estimated average acceleration 98 m/s2). Then it undergoes a near
constant deceleration to zero velocity within 66 ms in another 14 mm (estimated average
deceleration 17.5 m/s2). It WaS held above the position 28 mm for about 100 ms until the
current was turned off, it then dropped under gravity, as fast as 0.4 m/s in 60 ms over 12
mm position (estimated average acceleration 6.7 m/s2). At the position between 8 and 17
mm, the velocity of the armature was above 0.6 m/s and between 0.6 and 0.7 m/s.
3.2.3 Viscous Drag and Impedance Forces Discussion
In our theoretical calculation, we have tried to simplify the real physical situation by
only considering the magnetic force and gravitational acceleration force. However, inside
the fluid, even in our quantum fluid- less viscous liquid helium-4, there are still some other
significant forces, e.g. viscous dr I_ Buoi- maly force, impedance force due to the flow based
on Bernoulli law, exerted upon the armature and the grid. Now we are going to estimate
the sizes of these forces.
1. At 4.0 K(, dynamic viscosity p S;:I,,.:. I opoise 36 x 10-7Pascal second =
36 x 10-71N s/m2. (K~inematic viscosity: v p/~p, strokes 1 cm2 -1)[4
2. At saturated vapor pressure, at 4.20 K(, the liquid helium-4 has density,
p 0.1254075g/cm3 125.4075kg/m3 .
3. Reod ubr Rg Up/ lm/s x25 075iksm 2m3 = .4835 x 10 ( in meter).
So it must be turbulent since very small length scale .
4. Total length of armature (Fig. 3-12) L = 3.1875" = 8.10cm 8.10 x 10-2m. The
diameter of the armature D (1/4)" = 0.635cm = 0.635 x 10-2m
5. Rer UsipL/pL = (1/sx1 475g 3 1010 = 2.82 x 106. Skin friction coefficient:
Cf 0.455(logloRee)-2.58 = 3.7094 x 10-3. Drag = plf, xa 12405/3
(1m/s)2 x 3.7094 x 10-3 0.2326N/lm2
Viscous drag drag x surface area Drag x xD x L = 0.2326N/lm2 x xrx 0.635
x 10-2m x 8.10 x 10-2m 3.758 x 10-4NV .
6. Force exerted upon the upper side plates of the armature (Fig. 3-12 (a)) = Fext:
d 0.3" 0.762cm 7.62 x 10-3m; t0181 CTOSS SeCtiOUS A' x( )2+
,[( )2\ Do\l 2] x 2 ing 7.347 x 10-sm2
Fext d(M~omentum)/dt pA'v2 9.214 x 10-3NV .
7. Force exerted upon the bottom side plates of the armature (Fig. 3-12 (b)) F'me:
Bernoulli's equation: p p= -~i 62.70N/m2l ; Fit, pA' 4.607 x 10-31V
8. Buoi- oi-i force = pgV = 125.4075kg/m3 x 9.8m/s2 x 2.565 x 10-6 3
3.1524 x 10-31V
9. Gravity force = 2.60g = 0.02548N\1.
10. If the grid is mounted at the end of the armature, the total non-transparent surface
area of the grid, excluding the area underneath the armature, is A" 1.7765cm2
Force exerted upon the upper side of the grid pA"v2 22.28 x 10-31V; foTCe
exerted upon the bottom side of the grid pA" 11.14 x 10-3NV. So the total forces
exerted upon the grid is 33.42 x 10-31V
So the total net force (downward) when the armature travels upward at the velocity
of 1 m/s = 0.06994 N = 2.745 x Gravity force (Table 3-2). In our simulation program,
we only consider the gravity force and the magnetic force. As the armature moves at
higher velocity, the viscous drag force and impedance forces become significant, pretty
comparable to the gravity force.
3.2.4 Heat Dissipation Discussion
The Joule heating dissipation is avoided by using superconducting wires winding
the magnet. Building the superconductor shield enclosing the magnet is to prevent the
Table 3-2: Forces on the armature when moving up at 1 m/s (Download is positive).
Viscous drag 3.758 x 10-41
Fezt 9.214 x 10-31
F'z 4.607 x 10-31
Buonne1y force -3.1524 x 10-31
Gravity force 0.02548N\1
External forces on the grid 33.42 x 10-31
eddy current heat dissipation on the cell wall. As far as the AC loss (including hysteresis,
coupling and eddy-current losses), splice and mechanical losses  are concerned, we
estimate the resulting heating due to the varying magnetic field, which is not significant.
We use epoxy-resin impregnation to prevent frictional displacement without noticing
any deformation of the coil. We do not have any splice loss problem without using any
slices, we assume that the heating due to the above three factors could be neglected.
When we run the motor in the dilution refrigerator, we will run it without any liquid
helium and measure the background temperature fluctuation. Possible AC losses of the
superconducting solenoid due to the varying current can be measured by substracting
from the blank background without any current in the solenoid. And then we will run
the motor again with grid immersed in the liquid helium bath. We can then do the
subtraction to get the net temperature change simply due to the quantum turbulence
3.3 Leak Tight Electrical Feedthrough Design
The electrical feedthrough for the cryogenic cell has been designed and constructed.
We also check the leak tight of the feedthru pins imbedded in our experimental cell at the
liquid nitrogen temperature, 77 K(, with our homemade "R. II. i-n (Roving Experimental
Device Investigator) apparatus. Fig. 3-13 shows the design of our experimental cell for the
towed-grid studies of quantum turbulence experiments.
3.3.1 Electrical Feedthru Construction
One of our tasks for building our superconductor shielded superconducting linear
motor system is to construct the circuits, such as for the superconducting solenoid, the
position sensor and thernxistors leads. When designing these feedthroughs, we need to
concern ourselves with keeping the cell leak tight, especially when the wires come out
front the inside of the experimental cell to the exterior circuit connection. An extra
challenge occurs when different jointed materials cool down to the cryogenic temperatures,
the distinct thermal contractions could cause a crack. We have designed and built the
electrical feedthroughs and a novel way of mounting the wires through the cell walls.
Fig. :3-13 shows the whole experimental cell assembly for the towed-grid experiments
and the circuit board for the thernxistors with the electrical feedthru pins mounted.
The electrical feedthru pins are gold plated on all surfaces to benefit the best electrical
conductivity 2 and tight fit to the homemade polycarbonate plugs. Polycarbonate
material has very close thermal expansion coefficient to that of the copper, so the plugs
and the copper cell wall will have nmininial differential thermal contractions when cooling
down to nxilliiE~levin temperatures. We have our cell made of pure copper free of oxygen.
Each individual electrical feedthru pin is imbedded in each plug, and then inserted to each
individual hole on the cell wall. In order to have leak tight, we seal each feedthru with
epoxy. Epoxy is mixed with 10 g of Stycast 2850FT and about 0.65 ~ 0.75 g of Catalyst
24LV (proportion 100:7).
Before putting everything together and applying the epoxy for seal, the degrease and
cleaning process is very important:
2 Electrical feedthru connector pins: Mouser Electronics (Order Number 575-11:3164)
Manufacturer's (11.1141ax) Part Number: :310-1:3-164-41-001000. Each individual machined
pin is gold plated on all surfaces. Each strip contains 64 pins.
1. Clean the polycarbonate plugs and electrical feedthru pins by immersing them under
the liquid surface of methanol or ethanol in a ultrasonic vibration equipment for a
2. Clean the dirt or dust with the soft iron brush, rough sponge and sand paper on the
exterior surface of each feedthru hole on the cell wall.
:3. Spray the acetone to wet the Q-tip and then apply the Q-tip to clean the interior of
each feedthru hole on the cell wall.
4. Spray the acetone to wet the K~imwipes wipers and then apply the wipers to clean the
exterior surface of each feedthru hole.
Applying the epoxy to the joint surfaces between the copper cell wall and the
feedthru plug needs enough surface edges for the epoxy to adhere well, so we make the
polycarbonate plugs higher than the cell wall surface, sticking from outside and the
non-connection part of the every electrical feedthru pin is longer than every plug (Fig.
:3-13 (a)). It usually takes a day to dry out the epoxy; however, we can illuminate with
the infrared light to shorten the time to about a half d w-. Shinning this light also helps
strengthen the epoxy. We should be very careful when cooling down and warming up the
cell very slowly, avoid chemical touching or also putting large forces on the joints. In this
way, the seal can he used for thermal cycles for many times without any crack.
3.3.2 Thermistor Circuit Board Construction
We have doped germanium thermometers . They are less than :300 pm diameter
and will be immersed in the turbulent helium allow fast calorimetric measurements to be
made. Therefore, we have the thermistor circuit board specially made (Fig. :3-13 (b)). The
circuit board has radius 0.5 inch are on one side to match the interior are of the cell; on
the other side, an extension part is made for a 0-80 slotted flat screw to mount the board
to the bottom of the cell and also for the ease of our fingers handling. Two miniature
germanium film thermometers are mounted to the circuit board hv indium solder and
attached/glued to the board by the cryogenic grease. Three electrical feedthru pins are
mounted in parallel to the circuit board hv indium solder and the two thermistors are
soldered to those pins for electrical connections. This design allows the circuit board to
be independent (e.g. Fig. 3-13 (b): electrical feedthru pin # 1) and to slide into (e.g.
Fig. 3-13 (b): electrical feedthru pin # 1') another three counter electrical feedthru
pins imbedded in the cell wall (e.g. Fig. 3-13 (b): electrical feedthru pin # 2). At room
temperature, the resistances for the two thermometers measured from the electrical
feedthru pins outside the cell are 60.0 and 57.5 R, respectively.
We have leak checked the cell at room temperature and at liquid nitrogen temperature,
77 K(, with our homemade "R. II. i- instrument. This proves that our electrical feedthru
design is a successful design.
Outside the cell, we will plug the electrical feedthru pin # 2 with the electrical
feedthru pin # 3 connected to a miniature coaxial cable 3 Which on the other end
connects to a ultra miniature (Lepra/Con) connector plug 4 to be mounted to the dilution
We use epoxy to vacuum seal the electrical feedthru, tight fit to the polycarbonate
plugs, imbedded in the cell walls. We also built the circuit board to mount our thermistors.
This design promises the flexibility of mounting the experimental cell and is leak tight.
3 COTO COnductor: 30 AWG copper; stainless steel braided shield; insulator: PFA jacket
(Janis <.l~l Ilr .n).
4 ManufaCtuTOT: TyCO Electronics; distributor: March Electronics.