Quartz Crystal Tuning Fork in Super uid Helium
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Quartz Crystal Tuning Fork in Superfluid Helium
Experiment TFH
University of Florida — Department of Physics
PHY4803L — Advanced Physics Laboratory
Objective
D W
A quartz crystal tuning fork, designed for
sharp resonant oscillations when operated in
vacuum at room temperature, is immersed in
liquid helium instead. The tuning fork behav-
ior is affected by the surrounding fluid and L electrodes
varies as the helium temperature is lowered
through the superfluid transition. A “suck
stick” cryostat is used to get liquid helium
to temperatures ranging from 1.6 to 4.2 K.
The tuning fork’s frequency and transient re-
sponses are measured in that environment and
compared with predictions based on simple
models for the tuning fork and liquid helium.
Figure 1: Rough geometry of our quartz crys-
References tal tuning fork. The electrode wires and the
1. B.N. Engel, G. G. Ihas, E. D. Adams and vacuum canister are not shown. For our tun-
C. Fombarle, Insert for rapidly producing ing fork: L = 2.809 mm, W = 0.127 mm and
temperatures between 300 and 1 K in a D = 0.325 mm. Electrode shape and place-
helium storage dewar, Rev. Sci. Instr. ment are crudely illustrated in the figure and
55, 1489-1491 (1984). not representative of an actual device.
2. Russell J. Donnelly and Carlo F. Be- Temp. Physics, 146, 537-562 (2007).
ranghi, The observed properties of liquid
helium at the saturated vapor pressure, J.
Phys. and Chem. Ref. Data, 27, 1217- Introduction
1274 (1998).
Mechanically, quartz crystal tuning forks are
3. R. Blaauwgeers, et. al., Quartz tuning highly tuned resonators with low damping.
fork: thermometer, pressure- and vis- They are shaped like the normal tuning forks
cometer for helium liquids, J. of Low used for checking pitch in musical instruments,
TFH 1TFH 2 Advanced Physics Laboratory
but are miniaturized and operate at ultrasonic ical temperature near 2.2 K. This transition
frequencies. The ones used in this experiment toward a state with zero viscosity causes sig-
are about 3 mm long and have a nominal fre- nificant changes in the tuning fork behavior.
quency of 32768 Hz. See Fig. 1.
Electrically, the tuning fork is a two- Phasor notation and relations
terminal device, having thin film electrodes
Phasors are complex representations of si-
on each tine with leads for external connec-
nusoidally oscillating quantities and tremen-
tions. Quartz’s piezoelectric properties are ex-
dously useful for the kinds of analyses needed
ploited in construction so that tuning fork vi-
in this experiment.
brations create alternating charges on the two
In this write-up, sinusoidally varying quan-
electrodes. The same physics ensures that an
tities will be typeset using traditional math
applied voltage of one polarity or the other
fonts, e.g., a voltage v might be expressed
squeezes the tines together or forces them
apart. v = V cos(ωt + δ) (1)
Figure 2 shows two ways to characterize
where V is the amplitude, δ is the phase con-
tuning fork behavior. The left graph shows the
stant, ω is the angular frequency and t is the
steady-state oscillation amplitude of a driven
time.
tuning fork as the drive frequency is slowly
The phasor associated with such a time de-
scanned over the resonance. Note the ex-
pendent quantity will be typeset in a bold-face
tremely narrow full width at half maximum
math font and is the complex number having
(FWHM); the amplitude rises and falls quickly
that amplitude and phase constant. For exam-
near the resonant frequency f0 . The right
ple, the phasor representing the source voltage
graph shows ring down behavior as the oscil-
above would be
lations in an undriven, but previously excited,
tuning fork exponentially damp away. v = V ejδ (2)
This data set is from a tuning fork still √
where j = −1.
sealed in its vacuum canister. The top of A phasor can also be represented using its
the canister is cut away in our apparatus to real and imaginary components.
expose the tuning fork to the surrounding
medium. When operated in a liquid or a gas, v = Vx + jVy (3)
the medium’s viscosity and density strongly where Vx = ℜ {v}, and Vy = ℑ {v} are signed
affect the fork’s damping and resonant fre- scalars and the functions ℜ {z} and ℑ {z} take
quency. You will study this dependence with the real and imaginary parts of a complex
the tuning fork immersed in gaseous and liquid number z.
helium. Euler’s equation
The apparatus used to create the bath
of low-temperature liquid helium is called a ejθ = cos θ + j sin θ (4)
“suck stick” and will allow the liquid to be provides the key relationship between the two
brought to temperatures as low as 1.6 K. Liq- representations. For the voltage example, it
uid helium has a temperature of 4.2 K at gives
atmospheric pressure, but becomes colder as
the pressure above it is reduced by a vacuum Vx = V cos δ (5)
pump. It becomes a superfluid below the crit- Vy = V sin δ
March 11, 2015Quartz Crystal Tuning Fork in Superfluid Helium TFH 3
Figure 2: Left: Typical resonance response of a tuning fork in vacuum as the drive frequency
is scanned. Right: The decaying oscillations of the tuning fork with the drive turned off. (The
natural frequency is too high to see the individual oscillations.)
The Tuning Fork Model
Exercise 1 (a) Show that a phasor v = V ejδ The tuning fork is basically two parallel tines
and its associated time dependent quantity v = attached at their base to a bridge—all part
V cos(ωt + δ) satisfy of a single quartz crystal. There are many
v = ℜ{ve }jωt
(6) normal modes of oscillation for such a complex
structure. The lowest modes have each tine
(b) Use v = Vx + jVy in Eq. 6 to show that vibrating with a node at the bridge and an
v = Vx cos ωt − Vy sin ωt (7) antinode at the tip—the fundamental mode
for a single tine. For the low-loss mode that
(c) Use Eqs. 5 to show that Eq. 7 is consistent
our tuning forks operate in, the two tines move
with Eq. 1.
out of phase—approaching and receding from
As you will see throughout this experiment, one another on alternate halves of a cycle.
the simple idea of assigning an oscillating For small amplitude oscillations, the motion
quantity to the real part of a complex os- of all its parts are proportional to one another
cillation can turn cumbersome equations into and the description of the fork as a three di-
nearly trivial ones. Basically, phasors allow mensional solid can be reduced to a single co-
a general oscillation with an arbitrary phase ordinate, here taken as the position x of the
constant A cos(ωt + δ) = ℜ{Aej(ωt+δ) } = tip of one tine (along a line between the tips).
ℜ{Aejδ ejωt } to be separated into a constant With a sinusoidally forced tuning fork, the
part Aejδ (the phasor) and an oscillating part equation of motion for this coordinate takes
ejωt . Euler’s equation guarantees everything the familiar driven harmonic oscillator form
works out. To see what Euler’s equation did
d2 x dx
in Ex. 1, use it on each of the three exponen- m 2 + b + kx = F cos(ωt + δ) (8)
tials in the equation ej(a+b) = eja ejb and then dt dt
equate the real and imaginary parts on each The effective driving force F cos(ωt + δ) arises
side. from a sinusoidal voltage across the tuning
March 11, 2015TFH 4 Advanced Physics Laboratory
fork electrodes. It is specified in Eq. 8 with state motion. If the oscillator is momentar-
an arbitrary amplitude F and phase constant ily perturbed from the steady state motion,
δ that will later be related to that voltage. it returns to it after some time interval during
In Eq. 8, m is the effective mass of one tine. which it executes non-steady state or transient
It depends on the fork geometry and is pre- motion.
dicted to be approximately The difference between the transient motion
and the steady state motion gradually decays
m = 0.243ρq V (9) to zero and is referred to as the transient re-
sponse.
where V = DW L is the leg volume and ρq is With all else fixed, as the drive frequency
the density of quartz. The effective spring con- varies, the steady state oscillation amplitude
stant k is proportional to the Young’s modulus and phase offset vary. This dependence is
Y , with the approximate relation: called the frequency response.
( )3 Equations for both the transient and fre-
Y D
k= W (10) quency response arise when finding general so-
4 L lutions to Eq. 11—a non-homogeneous differ-
The effective damping constant b arises from ential equation. The general solution is the
internal energy losses which are very low in sum of any particular solution xp satisfying
pure quartz. In actual devices, additional en- that equation plus the general solution xh sat-
ergy loss mechanism arise, for example, from isfying the corresponding homogeneous equa-
the attached electrodes and from tuning fork tion:
d2 x dx
interactions with its environment. Manu- 2
+γ + ω02 x = 0 (14)
facturing variations among similar forks are dt dt
larger in this parameter than in the other two. This is the differential equation for an un-
Dividing through by m gets Eq. 8 into the driven tuning fork and has solutions xh that
another common form take on the familiar form of exponentially
damped oscillations. These decaying oscilla-
d2 x dx F tions are the transient response. Steady state
+γ + ω02 x = cos(ωt + δ) (11)
dt 2 dt m motion will satisfy Eq. 11 and will be the par-
ticular solution xp .
where ω0 is the natural frequency
√
k The homogeneous solution
ω0 = (12)
m Solutions to Eq. 14 are readily derived assum-
ing xh is the real part of a complex solution
and γ is the damping coefficient { }
xh = ℜ aejωt (15)
b
γ= (13)
m where a and ω are complex constants to be
determined by substituting Eq. 15 as a trial
After a time, the coordinate x satisfying
solution into Eq. 14. The resulting equation is
Eq. 11 settles into oscillations at the drive
just the real part of the complex equation
frequency that have a fixed amplitude and a { }
fixed phase offset from the driving force. This d d
+ γ + ω0 aejωt = 0
2
(16)
long-time, settled motion is called the steady dt2 dt
March 11, 2015Quartz Crystal Tuning Fork in Superfluid Helium TFH 5
Taking the derivatives and then dividing The steady state solution
through by −aejωt leaves the characteristic
The steady state solution is constant ampli-
equation
tude oscillations at the drive frequency. These
ω 2 − jωγ − ω02 = 0 (17) solutions can be expressed
Considering only the underdamped case ap-
xp = A cos(ωt + δp ) (22)
propriate for our tuning forks (for which ω0 >
γ/2), the characteristic equation is satisfied by or { }
two values for ω xp = ℜ xp ejωt (23)
√
jγ γ2 where xp = Aejδp is the phasor associated with
ω= ± ω02 − (18) those oscillations.
2 4
With the driving force related to its phasor
The general transient solution is then a lin- f = F ejδ by
ear superposition of the two trial solutions— { }
F cos(ωt + δ) = ℜ f ejωt (24)
one for each value of ω—with arbitrary coeffi-
cients. Defining the free oscillation frequency Eq. 11 with xp as a trial solution is just the
√ real part of the equation
γ 2 { }
ω0′ = ω02 − (19) d2 d f
4
2
+ γ + ω02 xp ejωt = ejωt (25)
dt dt m
this superposition gives the general solution
The derivatives now act only on the oscilla-
for xh as
tory factor ejωt giving
{ ( ′ ′
)}
xh = ℜ e−γt/2 a1 ejω0 t + a2 e−jω0 t (20) { } f jωt
−ω 2 + jωγ + ω02 xp ejωt = e (26)
m
Equation 19 shows that damping pulls the free
oscillation frequency somewhat below the nat- and after canceling ejωt shows that the solu-
ural resonance frequency ω0 . However, for the tion for the position phasor xp is simply
small damping γ ≪ ω0 associated with our f /m
tuning fork, ω0′ is indistinguishable from ω0 . xp =
−ω 2 + jωγ + ω02
(27)
Using Euler’s equation and trigonometric
identities, it is not hard to show that this so- A lot of physics is tied up in Eq. 27. For ex-
lution can also be expressed in the equivalent ample, from Eq. 23, the steady state solution
form is
{ }
xh = Ae−γt/2 cos(ω0′ t + δh ) (21) f /m
xp = ℜ · ejωt (28)
−ω + jωγ + ω02
2
This form is more readily recognizable as ex-
ponentially damped oscillations.
Transient motion in undriven systems oc- Exercise 2 (a) Show that the oscillation am-
curs only after an external excitation provides plitude is given by
a non-zero initial displacement and/or veloc-
ity; the values for A and δh would be chosen F/m
A= √ (29)
to meet those initial conditions. (ω 2 − ω02 )2 + ω 2 γ 2
March 11, 2015TFH 6 Advanced Physics Laboratory
Hint: Use A2 = xp x∗p where x∗p is the com- k and the resonant frequency f0 . Tuning forks
plex conjugate of Eq. 27. (b) Show that the are made from crystal quartz, not fused quartz.
phase difference between the displacement and Also quartz has two different Young’s mod-
the driving force is given by uli, depending on whether the strain/strain is
( ) along the z-axis of the crystal or perpendicular
−1 −γω to it. It turns out our fork’s stress/strain will
δp − δ = tan (30)
ω0 − ω 2
2
be perpendicular. (b) Show that for γQuartz Crystal Tuning Fork in Superfluid Helium TFH 7
voltage v across it. Zf
p = iv =
dx
= κ v (34) C R L
dt
Equations 33 and 34 are only consistent if the
effective driving force is associated with the Cp
source voltage according to
κ
f= v (35)
2
Figure 3: The equivalent circuit of the quartz
Substituting Eqs. 31 (and its first and sec-
crystal tuning fork.
ond derivatives) and Eq. 35 into Eq. 11 gives
m d2 q b dq k κ described by Eq. 40 and (2) a parallel capaci-
+ + q = V cos(ωt + δ) (36)
κ dt 2 κ dt κ 2 tance Cp from the electrodes and connections
(called the electrical arm).
where V cos(ωt + δ) is now the voltage across Recall that steady state behavior in a.c. cir-
the tuning fork. With the following associa- cuits can be determined from extensions of
tions: Kirchhoff’s rules for d.c. circuits. The d.c.
2m voltages and currents are replaced with their
L = (37) phasor counterparts and resistance is replaced
κ2
2b with impedance: jωL for an inductor, 1/jωC
R = 2 (38) for a capacitor, and R for a resistor.
κ
1 2k For example, the impedance of the mechani-
= (39)
C κ 2 cal arm, with the resistor, capacitor and induc-
tor in series is just the sum of each elements’
Eq. 36 becomes impedance: R + 1/jωC + jωL. Additionally,
d2 q dq 1 the admittance (inverse of impedance) of two
L 2 + R + q = V cos(ωt + δ) (40) parallel branches add—giving an overall ad-
dt dt C
mittance for the tuning fork
This should be recognizable as the equation for 1 1
a harmonically driven series RLC circuit with = jωCp + (41)
Zf R + 1/jωC + jωL
L, R, and C thus construed as mechanically-
associated inductance, resistance, and capaci- The current phasor in a circuit branch is
tance. the voltage phasor across that branch divided
Modeling the tuning fork mechanical prop- by the branch’s impedance. Consequently, the
erties as a series RLC circuit leads to the elec- electrical arm carries a current ip = v s · jωCp
trical model of a quartz tuning fork as two with a relatively weak ω dependence over the
arms in parallel as shown in Fig. 3. The equiv- small frequency range explored in a resonance
alent circuit has (1) a series RLC arm (called scan. The mechanical arm carries a cur-
the mechanical arm) arising from the piezo- rent im = v s /(R + 1/jωC + jωL) and has
electric/mechanical properties of the fork and a sharp resonant behavior. It peaks at a value
March 11, 2015TFH 8 Advanced Physics Laboratory
im = v s /R when 1/ωC = ωL, i.e., when penetration layer of thickness
ω 2 = 1/LC = ω02 . √
2η
λ= (43)
Exercise 4 The resonant behavior is more ρω
readily obvious if the resistance is scaled from
the mechanical arm admittance. Show that where η is the medium’s viscosity and ρ is its
this leads to the equivalent expression density.
As long as the penetration depth and the
1 1 1 overall vibration amplitude are small com-
= jωCp + · (42)
Zf R 1 + j(ω − ω0 )/ωγ
2 2 pared to the fork dimensions, the surround-
ing medium can be treated as producing an
This equivalent parameterization, in terms of additional force with a “drag” component pro-
R, ω0 , γ, and Cp is better suited for compari- portional to and directed opposite the velocity
son with experimental data. and a “mass enhancement” component pro-
portional to and directed opposite the accel-
Exercise 5 A typical tuning fork operating eration. The additional force Fm due to the
in vacuum, might be determined to have a medium can be expressed
mechanical resistance R ≈ 18 k Ω, a damp-
ing constant γ ≈ 1.6/s, and a natural fre- dx d2 x
Fm = −b∗ − m∗ 2 (44)
quency f0 = ω0 /2π ≈ 33 kHz. (a) Find the dt dt
mechanical inductance L and capacitance C. Both of these forces are calculable from hy-
(b) A determination of the tuning fork con- drodynamic equations for the flow field around
stant κ would require some measure of the tip the oscillating tines, which predict that the
displacement—something we cannot get with drag constant is given by
our apparatus. However, κ can be estimated √
based on the given values and an estimate of ∗ ρηω
b = CS (45)
the effective mass. Estimate κ and then deter- 2
mine, for a typical current amplitude of 1 µA and the mass enhancement is given by
what the corresponding maximum tip displace-
ment, velocity, and acceleration would be. m∗ = βρV + BρSλ (46)
Effects of a viscous medium where S = 2(D + W )L is the surface area
of a tine and C, β and B are all geometry-
Equation 11 is the equation of motion for a dependent factors of order unity. The first
tuning fork in vacuum and must be modi- term in the mass enhancement arises from the
fied if the tuning fork is operated in a gas or potential flow and does not depend on the
liquid. The vibrating tines cause motion in medium’s viscosity, i.e., it must be included
the medium which in turn creates additional even for a superfluid. The second term arises
forces on the tuning fork. Where the medium from the boundary layer entrained with the
is in direct contact the fork, its motion is en- fork motion and, via the penetration depth,
trained with that of the fork. Far from the fork depends on the viscosity and thus goes to zero
and other bounding surfaces, the fluid veloc- for a superfluid.
ity field can be expressed as the gradient of Adding the additional force (Eq. 44) to the
a potential. The two behaviors merge over a right side of Eq. 11 and then bringing it over
March 11, 2015Quartz Crystal Tuning Fork in Superfluid Helium TFH 9
to the left side gives
0.14
2
x′d dx
m 2 + b′ 0.12 superfluid total
+ kx = F cos(ωt + δf ) (47)
density (g/cm )
3
dt dt 0.1
0.08
where 0.06
′ ∗ normal fluid
m =m+m (48) 0.04
0.02
and 0
b′ = b + b∗ (49) 0 1 2 3 4 5
temperature (K)
Thus, the form of the solution remains largely
the same, but the resonance frequency and
width change. The increased mass decreases Figure 4: The two-fluid model of liquid he-
the resonance frequency and the increased lium. The graph shows the density of the
damping broadens the resonance. Variations normal fluid, the superfluid, and their sum.
of b∗ , m∗ and λ with ω can be ignored as there Below 1 K, helium is virtually all superfluid.
would be only very small variations over the Above Tλ it is all normal fluid. From refer-
narrow range of a resonance. Thus ω will be ence 2.
replaced with ω0 in those equations and b∗ and
m∗ will be treated as constants over the range
of a resonance scan. Figure 5 shows the viscosity of liquid he-
lium as a function of temperature. The normal
fluid has viscosity while the superfluid does
Liquid helium model not. Consequently, the superfluid component
contributes only via the βρV mass enhance-
Helium makes a transition to a superfluid state
ment term. The normal fluid contributes to
at Tλ = 2.1768 K where the specific heat ca-
both terms of the mass enhancement and to
pacity has a discontinuity in the shape of the
the additional damping. Thus, the two-fluid
Greek letter λ. Above this temperature, liq-
uid helium is a normal fluid with a density
around 0.14 g/cm3 (about 1/7th that of wa-
ter) and a viscosity around 3.3 × 10−6 Pa·s
(about 1/300th that of water).
The two-fluid model is used for tempera-
tures below Tλ where the liquid helium be-
haves as if it were a mixture of a normal fluid
and a superfluid with the proportion of each
a function of temperature. The size of the
two fractions is illustrated in Fig. 4 where the
solid line gives the superfluid density ρs and
the dashed line gives the density ρn of the nor-
mal component. The total density ρ is the sum Figure 5: The viscosity of liquid helium. The
of the two two-fluid model attributes the viscosity to the
ρ = ρn + ρs (50) normal component only. From reference 2.
March 11, 2015TFH 10 Advanced Physics Laboratory
model gives To see how the damping parameter γ =
√ b /m′ should vary, first note that m′ /m =
′
ρn ηω
b∗ = CS (51) (ω00 /ω0 )2 giving γ(ω00 /ω0 )2 = b′ /m = (b +
2
b∗ )/m = γ0 + b∗ /m. Thus, if we define
and √
2ηρn ( )2
m∗ = βρV + BS (52) ω00
ω G=γ − γ0 (58)
ω0
Temperature dependence it is then b∗ /m and thus predicted from Eq. 51
Measurements and fits of the transient re- √
ρn ηω0 CS
sponse and of the frequency response will be G= (59)
described shortly that will provide values for 2 m
R, ω0 , γ and other parameters. These param-
eters will be obtained as the temperature of The Data Acquisition System
the liquid helium is varied.
The data acquisition computer for this ex-
Determining the temperature dependence of
periment is equipped with a National Instru-
ω0 and γ are the basic goals of the experiment.
ments PCI-GPIB+ IEEE-488 interface card
It turns out useful to compare these two pa-
used to communicate with a function gen-
rameters against their vacuum values, which
erator and a dual phase lock-in amplifier.
values will now be given an additional 0 sub-
These two instruments are used to deter-
script
k mine the tuning fork’s impedance as the drive
2
ω00 = (53) frequency is varied through the resonance.
m
and The computer also has a National Instru-
b ments PCI-MIO-16E-4 multifunction data ac-
γ0 = (54)
m quisition (DAQ) card for transient response
The square of the ratio of the resonance fre- measurements and for temperature measure-
quency in vacuum to that in the media then ments using a low-temperature thermometer
gives: installed in the cryostat. The important fea-
( )2 tures of these components and their use in the
ω00 m′
= tuning fork measurement circuit are presented
ω0 m
m∗ in this section.
= 1+
m
(55) Function generator and circuit
It is recommended that the experimental re- Figure 6 is a schematic of the circuit for mea-
sults for the resonance frequencies be plotted suring the tuning fork behavior.
as the function The function generator is the Stanford Re-
F = (ω00 /ω0 ) − 1
2
(56) search Systems DS340 with features similar to
others, e.g., an adjustable frequency and am-
which is then m∗ /m and thus predicted from plitude and a circuit model consisting of an
Eq. 52 ideal voltage source and a 50 Ω series resis-
√
βρV BS 2ηρn tance. The output is labeled FUNC OUT over
F= + (57) the bnc connector on the front panel.
m m ω0
March 11, 2015Quartz Crystal Tuning Fork in Superfluid Helium TFH 11
transimpedance ADC
function amplifier ACH1
generator quartz crystal 10 kΩ lock-in
tuning fork amplifier
50 Ω out reed relay
- signal
+
v0 Rs Cs Cd
sync
reference
Figure 6: Circuit schematic for measuring the tuning fork impedance.
The function generator output is only avail- A second common function generator out-
able after its 50 Ω output impedance. At this put is the sync signal. In the DS340 it is
point in the circuit, the voltage would depend a square wave synchronized with the voltage
on the current, which in turn depends on the source described above. It is labeled SYNC
load impedance and thus cannot be specified OUT over its bnc connector. The sync sig-
ahead of time. The function generator output nal’s rising edges have a fixed phase differ-
is specified irrespective of the load at the (in- ence with the positive-going zero-crossings of
accessible) source point labeled v0 in Fig. 6. v0 and will be used as a reference for deter-
This voltage can be expressed mining the phase of any voltage measured by
the lock-in. The ability to measure the phase
v0 = V0 cos(ωt + δ0 ) (60)
of the current in the tuning fork relative to
or equivalently as the phasor the source voltage is required to determine the
tuning fork impedance.
v 0 = V0 ejδ0 (61)
The minimum amplitude from the function
where δ0 is relative to the sync signal (de- generator is generally too big for directly driv-
scribed next). ing the tuning fork. To get the smaller exci-
The voltage waveform before the 50 Ω resis- tation voltages required, a shunt resistor Rs
tor, while inaccessible, could be measured by of either 0.5 Ω or 5.6 Ω is placed across its
using a high impedance probe with no other output as shown in the figure. According to
load attached. Be sure the DS340 is set in the Thévinin’s theorem, the shunt resistor reduces
High-Z mode so that V0 is shown on the func- the output impedance to the parallel combina-
tion generator’s display. The display will be tion
low by a factor of two if the DS340 is set to Rs · 50 Ω
Rs′ = (62)
50 Ω mode. Look for the High-Z/50 Ω indica- Rs + 50 Ω
tor under the output BNC connector and look
for the units indicator on the right side of the which is just a bit below Rs . The shunt resis-
front panel. The amplitude can be set or read tor also reduces the output voltage to
as peak-to-peak
√ values (2V0 ) or as rms values
V0 / 2 v ′s = ϵs v 0 (63)
March 11, 2015TFH 12 Advanced Physics Laboratory
where the reduction factor ϵs is given by The previous exercise should have demon-
strated that because of the low output
Rs impedance of the source, the coax capacitance
ϵs = (64)
Rs + 50 Ω Cs should have no bearing on the measure-
and is around 0.1 for Rs = 5.6 Ω and around ments.
0.01 for Rs = 0.5 Ω. The coax from the other side of the tun-
Coaxial cables connect to and from the tun- ing fork connects to the virtual ground input
ing fork. Coax is normally modeled as a trans- of the transimpedance amplifier (current-to-
mission line, but for the relatively low frequen- voltage converter). Because of the near-zero
cies involved in this experiment, the simpler input impedance of this amplifier, the coax ca-
model of the coax as a lumped capacitance to pacitance Cd on this side can also be neglected.
ground is appropriate. Approximately 2 m of The 10 kΩ transimpedance then gives the am-
LakeShore type SS cryogenic coax cable (ca- plifier’s output as
pacitance about 174 pf/m) connect each tine
of the tuning fork at the bottom of the cryo- v d = −i · 10 kΩ (67)
stat to the two feedthroughs at the top. About
2 m of RG58 coax cable (capacitance about where i is the current in the circuit and is
80 pf/m) connect the function generator to one given by
vs
feedthrough and a similar cable connects the i= (68)
Z s + Zf
other feedthrough to the transimpedance am-
plifier. Thus, Cs ≈ 500 pf on the source side Because it is negligible compared to Zf , the
of the circuit and Cd ≈ 500 pf on the detector source impedance Zs can be dropped from the
side. analysis. The factor of -1 in Eq. 67 arises from
the inverting behavior of the op amp in the
Exercise 6 According to Thévinin’s theorem, transimpedance amplifier.
Cs can also be modeled as part of the When the low source and detector
source. Show that adding a parallel capac- impedances are neglected, the final re-
itance to ground (a) changes the Thévinin sult for the phasor at the amplifier output
source impedance from Rs′ to is
104 Ω
Rs′ v d = −ϵs v 0 (69)
Zs = (65) Zf
1 + jωτs
and (b) changes the Thevinin source voltage to The Lock-in Amplifier
1 The transimpedance amplifier output is con-
v s = v ′s · (66) nected to the lock-in input for measurement of
1 + jωτs
v d . Consult the user manual for detailed in-
′
where τs = Rs Cs is an effective source time formation on our Stanford Research Systems
constant. (c) Evaluate the time constant for SR830 lock-in amplifier. Here we only need to
the Rs = 5.6 Ω shunt. (d) Noting that 1 + x ≈ appreciate that it analyzes an oscillating volt-
ex for xQuartz Crystal Tuning Fork in Superfluid Helium TFH 13
and returns two signed quantities Vx and Vy
given by
Vx = V cos ϕ
Vy = V sin ϕ (71)
Vx is called the in-phase component and Vy is
called the quadrature component. They are
given by the lock-in as rms values. The lock-
in can also provide the amplitude V (again,
an rms value) and the phase constant ϕ. Note
that Vx and Vy are just the real and imagi-
nary parts of the phasor v = V ejϕ . In effect, Figure 7: The manufacturer calibration for
the lock-in can be considered to provide the our Cernox solid state thermometer for tem-
phasor associated with its input. peratures from 1.4 to 100 K and crudely ex-
The lock-in determines the phase constant tended above this range as shown.
ϕ relative to the primary reference signal con-
nected to its reference input—in our case,
and Eq. 69 for the measured lock-in phasor
the sync signal. The lock-in adds a user-
becomes
adjustable offset to the phase of the primary 104 Ω
reference and uses that phase to create two v d = ϵs V0 (74)
Zf
secondary sinusoidal reference signals at the
primary frequency—one for each of the Vx and By making δ0 = 180◦ , only the amplitude
Vy output circuits—that are 90◦ out of phase V0 of the function generator source voltage
with one another. Each secondary reference is now appears and the overall negative sign
multiplied with the input signal, scaled, and from the transimpedance amplifier inversion
time averaged to generate the Vx and Vy out- is gone. A measured lock-in phase of ϕ = 0
puts. Unwanted noise in the signal that is not for v d (Vx > 0, Vy = 0) would then imply
at the reference frequency is largely filtered that v d is a positive real quantity, that the
out while the signal at the reference frequency circuit current is in-phase with v0 , and that
remains. Zf is a positive real quantity (resistive) with
The phase offset adjustment will be used to no imaginary (capacitive or inductive) compo-
take into account the phase offset between the nent. A measured lock-in phase of 90◦ for v d
sync signal and the source waveform v0 . This (Vx = 0, Vy > 0) would imply that v d is a
phase offset will be measured and 180◦ will be positive imaginary quantity, that the current
added to it before it is applied via the lock- leads the source voltage by 90◦ , and that the
in phase offset adjustment. This makes the impedance Zf has a negative imaginary part
phase constant δ0 = π in the source voltage and no real part, i.e., it is capacitive.
(Eq. 60) so it now becomes
v0 = V0 cos(ωt + π) = −V0 cos ωt (72) Thermometry
i.e., its phasor becomes A Cernox solid state thermometer is posi-
tioned next to the tuning fork. Its resistance
v 0 = −V0 (73) Rth near room temperature is about 60 Ω and
March 11, 2015TFH 14 Advanced Physics Laboratory
measured with the ADC on the DAQ card in
1.00 ΜΩ synchrony with the output waveform vdac .
The lock-in technique used to determine the
vdac
v amplitude of the resistor voltage Vadc is simi-
R th adc
lar to that of the SR830. The computer mul-
DAC
ADC tiplies the measured vadc waveform by a sine
and cosine waveform of unit amplitude and at
the exact frequency of the vdac waveform and
then the computer averages the result for each
Figure 8: The circuit for measurements on the product over many periods as specified by a
Cernox solid state thermometer. user selected time interval. The square root
of the sum of the squares of the sine and co-
goes to 145 kΩ at 1.2 K. The calibration sine components (properly normalized) gives
curve provided by the manufacturer is for tem- the amplitude Vadc of the ADC waveform at
peratures from 1.4 K to 100 K and is shown the frequency of the DAC waveform while av-
in Fig. 7 along with a crude extension above eraging away most of the noise.
100 K where the accuracy is expected to be Because the 19 Hz frequency is so low, cable
poor. Below 10 K the accuracy is expected to and other capacitance have virtually no effect
be around 0.5 mK degrading to around 2 mK and d.c. equations can be used to relate the
for the maximum calibrated temperature of voltage amplitudes and resistances involved.
100 K. The measured thermometer resistance Rth is
The Cernox thermometer is a four-wire re- then given by
sistor with two leads for supplying an exci-
Vdac
tation current and two leads for measuring Rth = Rcal (75)
Vdac − Vadc
the voltage generated by the current. It is
placed in a voltage divider circuit as shown in The data acquisition programs that report
Fig. 8 with an Rcal = 1.00 MΩ 1% series resis- temperature use this formula along with the
tor. The current through the series resistors thermometer manufacturer’s calibration for-
is driven by a sinusoidal voltage generated by mula (see the auxiliary material for the de-
a 12-bit digital-to-analog converter (DAC) on tails) to convert resistance to temperature.
the DAQ card installed in the computer. The Because the amplitudes of both the drive
sinusoidal voltage waveform across the ther- waveform and the signal waveform are deter-
mometer is measured by a 12-bit analog-to- mined relative to the same internal reference
digital converter (ADC) also on the DAQ card. voltage on the DAQ card, inaccuracy in this
The Cernox thermometer is a delicate sen- reference value plays no role on the ratio used
sor that must never be driven by voltages to determine the thermometer’s resistance.
large enough to cause power dissipation above
2 mW. The 1.00 MΩ series resistor should pre-
vent any possibility of an overdrive situation.
Data acquisition and analysis
The a.c. drive waveform vdac is at 19 Hz and All data acquisition and analysis programs are
its Vdac = 10 V amplitude is set from a second in the Tuning Fork folder in the PHY4803L
DAC available on the DAQ card. The result- folder on the desktop. To use the frequency
ing waveform across the thermometer vadc is scanning program requires enabling the GPIB
March 11, 2015Quartz Crystal Tuning Fork in Superfluid Helium TFH 15
(IEEE-488) communications on the DS340. It Equations 74 and 42 give the lock-in phasor
must be enabled on the DS340 every time it
is powered up (shift then 1 key then up ar- ϵs V0 104 Ω
vd = · (76)
row). The SR830 powers on with the interface ( R )
already enabled. Furthermore, once the com- 1
jωRCp +
puter sends a command to the DS340 or to 1 + j(ω − ω02 )/ωγ
2
the SR830, the instrument goes into remote
command mode and disables the front panel Analysis of frequency scans over a resonance
controls. If the software leaves the instrument is performed by the Analyze Resonance pro-
in remote mode, you will have to manually gram.
return it to local (front panel) control mode— To take into account that the lock-in phase
shift then 3 key for the DS340, the Local but- adjustment may be off by a small angle ϕ, an
ton on the SR830 front panel. overall phase factor ejϕ should multiply the
prediction of Eq. 76. In addition, small off-
sets Cx and Cy are expected in the lock-in’s x-
and y-outputs arising from d.c. errors in their
amplifiers. Adding these two effects give the
final prediction for the output phasor from the
lock-in.
Aejϕ
vd = + (77)
Frequency scan and Analyze resonance 1 + j(ω 2 − ω02 )/ωγ
program
Cx + Dx ω + j(Cy + Dy ω)
where
Frequency scans are performed with the Fre- ϵs V0 104 Ω
quency scan program. In it you will find con- A = (78)
R
trols for the starting and ending frequency, the
frequency step size, the time to wait after each Dx = −ARCp sin ϕ (79)
frequency change before reading the lock-in,
Dy = ARCp cos ϕ (80)
and whether to do a forward scan, a reverse
scan, or both. The program displays the pre- To describe a few unique details associated
dicted time for the scan to complete after any with fits involving complex variables, it will be
change to these parameters. useful to distinguish the measured lock-in pha-
sor v m from the prediction v d of Eq. 77. The
The resulting data set is the in-phase Vx and measured data are the signed scalars for the in-
quadrature Vy values at each frequency. The phase Vmx and quadrature Vmy lock-in outputs
program also averages temperature readings as the frequency is varied in N steps through
during the wait at each frequency and so has the resonance. The corresponding predictions
a temperature for each frequency. are the real and imaginary parts of Eq. 77 for
each frequency.
When complete, the program saves the data Assuming both lock-in outputs have equal
to the file specified at launch time. uncertainties σv , the reduced chi-square χ2ν =
March 11, 2015TFH 16 Advanced Physics Laboratory
s2v /σv2 is proportional to the sample variance be random and should not show any system-
s2v taken as atic dependence on frequency.
The Save button on the front panel writes
N [
∑
1 a single row of data containing the Run #,
2
sv = (Vmx (ωi ) − Vdx (ωi )) 2
2N − M i=1 the average temperature for the run, its rms
]
+ (Vmy (ωi ) − Vdy (ωi ))2 (81) deviation over the run, the y-scale factor (typi-
cally 10−3 ) and all fitting parameters and their
where M is the number of fitting parameters. sample standard deviations. Supply a new file
While there are only N independent variables name for the first data set and you can repeat-
(scan points ωi ), there are two measurements edly save to it; the program will append one
(Vmx and Vmy ) for each of them and hence the new row each time. The file must not be open
number of degrees of freedom is 2N − M . in another program when you try to write a
The fit minimizes the sample variance using new row.
a standard nonlinear fitting algorithm avail-
able in LabVIEW and reports the resulting Acquire and analyze transient program
sample standard deviation sv . It also provides
graphs of the Vx - and Vy -deviations. Transient solutions or “ring downs” will be
The parameters ω0 and γ are both scaled measured and analyzed using the Acquire and
down by 2π and are labeled f0 and ∆f in the Analyze Transient program.
program. This is a program feature, not a bug. During ring downs, a computer-controlled
It is designed to make it easier to estimate reed relay quickly disconnects the function
and compare parameters with standard data generator and reconnects this point to ground
plots in which the independent variable is the as shown in Fig. 6. Any initial charge on the
frequency f rather than the angular frequency tuning fork’s parallel capacitance Cp will de-
ω. cay away on a time scale around Rd Cp (where
To decrease the covariance between the Rd is the transimpedance amplifier’s input
C and D parameters and improve the pro- impedance) that is quite short compared to
gram’s performance, both the x- and y-terms the decay time for the current in the me-
in Eq. 77 of the form C +Dω are replaced with chanical arm. The current through the tran-
the equivalent terms simpedance amplifier’s virtual ground input
should then be given by Eq. 32 with Eq. 21
C ′ + D(ω − ω0 ) = C + Dω (82) for x. The transimpedance amplifier output
voltage v will then be that current times the
Thus, for both the x- and y-terms, 104 Ω feedback resistance
C = C ′ − Dω0 (83) d { −γt/2 }
v = 104 Ω κ Ae cos(ω0′ t + δh ) (84)
dt
Thus C ′ is the offset voltage near resonance.
Resonance curve fits may fail if the initial Exercise 7 Show that Eq. 84 gives
guesses for the parameters, particularly f0 , are
not close enough to the correct values. Play v = Av e−γt/2 cos(ω0′ t + δv ) (85)
with them a bit before hitting the Do Fit but-
ton. Also keep an eye on the plots of the Vx - where
and Vy -deviations. For a good fit, these should Av = 104 Ω κ ω0 A (86)
March 11, 2015Quartz Crystal Tuning Fork in Superfluid Helium TFH 17
and In the data acquisition program’s Time
2ω ′
δv = δh + tan−1 0 (87) domain|Acquire tab you will find controls for
−γ setting the ADC sampling rate, the number of
Hint:
{ Show that Eq. }21 can be expressed xh = samples to acquire, and the ADC range. The
′
ℜ Aejδh e(−γ/2+jω0 )t , then show that the order ADC range should always be set to the most
of differentiation and taking the real part can sensitive 50 mV range; our tuning fork sig-
be exchanged and perform the calculation in nal should never go higher than about 20 mV.
that order. Our ADC runs at a top speed of 500,000 sam-
ple per second. Use this speed whenever pos-
This exercise shows that the voltage measured
sible and adjust the number of samples so that
in a free oscillation decay has the same fre-
an entire ring down is acquired and the signal
quency and decay constant as that of the dis-
has decayed well into the noise. Because tem-
placement oscillations.
perature monitoring also uses this ADC, it is
The output of the transimpedance amplifier
shut down during the short intervals needed
is wired to channel 1 of the DAQ board and
to record ring downs, but temperature read-
to a bnc connector on the top of the interface
ings are made just before and just after these
box for connection to the lock-in. The lock-in
transient measurements and are displayed on
is used to monitor vibration amplitudes when
the front panel.
you “ring up,” or excite, the tuning fork.
To see a ring down, the tuning fork must As you change the helium temperature, the
already be oscillating with appreciable ampli- resonance frequency and damping will vary.
tude. To get it oscillating, you will initiate a Manually adjust both the frequency and am-
ring up. The data acquisition program has a plitude on the function generator after a ring
toggle button that will send a signal to the re- up so it is running near the resonance fre-
lay to connect one side of the tuning fork to quency and the lock-in amplitude is around
the drive voltage for a ring up or connect it to 10 mV.
ground for a ring down. The Save button in the Acquire tab saves
The function generator frequency must be the ring down data to a file that can be read
set near the resonant frequency to get any ap- by a spreadsheet. The first two numbers in
preciable amplitude on a ring up. The lock-in the file are the before and after measured tem-
is needed for this step. With the relay in the peratures, then the y-scale factor (typically
ring up position, adjust the function genera- 10−3 ), then the time interval then the array
tor frequency for maximum amplitude on the of scaled ADC readings. This data file can
lock-in. But keep an eye on the lock-in ampli- also be reread by the program by hitting the
tude. If it goes above 10 mV lower the func- Read button.
tion generator drive voltage before continuing. Fitting is performed in the Time
If it goes above 100 mV, the tuning fork mo- domain|Analyze tab. The program has
tion is getting large enough for it to shatter. built in delays so that the ADC will start
Continue to adjust the frequency to get near acquiring readings about 50 ms before the
the resonance and the drive amplitude to get relay switches. The program will fit the data
a lock-in signal around 10 mV. You do not between the two cursors on the graph, so find
have to be right on resonance before initiating where the relay switched and set the starting
a ring down. You just need a lock-in ampli- cursor right after the oscillations begin to
tude around 10 mV. decay. Take a look in this region with an
March 11, 2015TFH 18 Advanced Physics Laboratory
expanded time scale so you can better see append one new row each time. The file must
the start of the decay. Generally, the ending not be open in another program when you try
cursor should be set so the fitting region to write a new row.
includes all of the freely decaying oscillations, The Acquire and Analyze Transients pro-
but does not include too many points after gram can also perform a Fourier transform of
the decay is complete. However, there is a the ring up or ring down and, for ring downs
limit of around 700,000 for the number of only, can perform fits to expectations for these
points LabVIEW will allow in this fit. If transforms. This kind of analysis greatly re-
the entire decay has more points than that, duces the number of points needed in the fit
use a smaller fitting region or lower the at the expense of the extra step to compute
acquisition rate. The rate is divided down the transform. The instructor can show you
from a 20 MHz clock and so a divisor of 40 these features if you are interested and an ad-
gives the recommended and maximum 500k dendum on the subject is on the web site.
samples per second rate. Other reasonable
rates to try are 400k (divisor of 50), 250k
(divisor of 80) 200k (divisor of 100). Keep in Apparatus
mind that at 200k samples per second there Figure 9 (at the end of the write-up) is a
are only about 6 measured data points on schematic drawing of all relevant cryogenic
each cycle of the 32 kHz oscillations. components. It is not to scale and does not
The Do Fit button then initiates a nonlin- include all gauges in the gas handling mani-
ear fit of the data between the cursors to the fold. Refer to it for valve and other component
form of Eq. 85 plus a constant to take into locations.
account any offset in the transimpedance am-
plifier and/or the ADC. The program assumes
The Suck Stick Cryostat
t = 0 at the starting cursor, and returns the
oscillation amplitude at that point. It also The suck stick is inserted into the neck of the
returns the resonant frequency and damping liquid helium dewar as shown in Fig. 9. It is
constant scaled by 2π, i.e., it returns f0′ = designed to hold a small volume of liquid he-
ω0′ /2π and ∆f = γ/2π. Check the graph of lium that can then be brought under vacuum
residuals to make sure the fit was successful. conditions. An insert inside the suck stick
If it was not, the starting guesses for the fit holds the tuning fork and thermometer at the
parameters may need to be closer to correct. bottom with wiring to electrical feedthroughs
The time scale must be expanded considerably at the top. The suck stick and insert comprise
to see the fit. the cryostat.
The Save button on this tabbed page writes The suck stick is an invention of low tem-
a single row of data containing the Run #, an perature researchers here at UF. The design
Excel time stamp giving the date and time principle is simple. Insert the stick in a liquid
right after the ring down, the temperature helium dewar, pump on the volume inside the
readings before and after the ring down, the stick and the pressure difference will suck liq-
y-scale factor, and all fitting parameters and uid helium from the dewar through the cap-
their sample standard deviations. Supply a illary and into the volume. The length and
new file name for the first set of results and diameter of the capillary are chosen to give a
you can repeatedly save to it; the program will mass flow conductance that is neither too high
March 11, 2015Quartz Crystal Tuning Fork in Superfluid Helium TFH 19
nor too low. The flow rate of liquid helium en- 100
90
tering the volume should be just about equal 80
to the evaporation rate from the thermal load 70
P (kPa)
60
on the volume. 50
40
If the flow rate is too low, the volume will 30
20
never fill. If it is too high, the volume will 10
overfill and the incoming liquid will be at 0
1 1.5 2 2.5 3 3.5 4
a higher temperature—closer to the ambient T (K)
4.2 K temperature in the dewar than the lower
temperature in the volume. When the con-
ductance is just right, the liquid flowing into Figure 10: Saturated vapor pressure of liquid
the volume will just make up for the amount helium. From reference 2.
of helium gas being pumped away. Moreover,
the capillary will have a temperature gradient point entering liquid begins to pool inside the
such that the incoming liquid helium will be volume. Above the pooling liquid is helium
at the temperature inside the volume. in the gaseous state. As the gas is pumped
Various low temperature techniques are away and the pressure above the liquid de-
used to keep the heat load low. Most impor- creases, the liquid cools further. The gas
tantly, there is a vacuum jacket around the reaches a steady state pressure that depends
volume to insulate it from the 4.2 K environ- on the pumping speed and the heat load. The
ment inside the dewar. A few torr of helium liquid will ultimately reach a steady state tem-
gas can be let into the volume to increase the perature for that pressure as determined by
heat conductance, but only when the appara- the temperature dependence of the conden-
tus is being cooled down from or warmed up sation and evaporation rates. The relation-
to room temperature. The helium gas must be ship between the equilibrium vapor pressure
pumped out of the jacket once the apparatus is and the liquid helium temperature is shown in
cold so as to insulate the experimental volume Fig. 10.
from the 4.2 K liquid in the dewar. Adding Thus the temperature of the liquid he-
helium gas to the vacuum jacket and then re- lium can be adjusted by changing the vac-
moving it does not save much time, and so uum pumping speed. The pumping speed is
we simply keep the vacuum jacket evacuated changed by partially opening or closing valves
throughout the experiment. 6 and 7 in the plumbing lines from the vacuum
There are radiation baffles along the inner pump to the experimental volume.
volume to minimize radiative energy barreling
down from the top of the suck stick where the
Pressure Meters
temperature is near ambient. In addition, the
materials used, such as stainless steel, polycar- The three main pressure meters all use differ-
bonate and phosphor-bronze wiring and coax ent units and none are the SI unit of pascal
are chosen for their low heat conductance or (Pa) for which 1 standard atmosphere (atm)
low heat capacity. is 101325 Pa.
When the liquid helium first enters the vol- The main vacuum meter for the experi-
ume, it quickly evaporates—cooling the con- mental volume is the Bourdon-type Matheson
tents until they are below 4.2 K, at which gauge which works off the pressure difference
March 11, 2015TFH 20 Advanced Physics Laboratory
inside and outside a spiral-shaped tube. It sure everything is working properly and well
reads in torr (1 atm is 760 torr) and can be understood. The following sections describe
calibrated with a two point procedure. First, one regimen that should help you fulfill these
meas
make a reading Patm with the inlet opened to goals. It starts with measurements that do not
the room. This reading should be about 760 require liquid helium.
torr and would be independent of the actual
room pressure. The actual atmospheric pres- 1. Check how the DS340 function generator
sure Patm can be obtained from the physics works. Set the DS340 for “High-Z” mode
department weather station web site where (Shift then 6 key). Set it for 10 kHz sine
it is labeled inHg (inches of mercury). The wave with an output amplitude of 0.2 V.
conversion factor is a 25.4 torr/inHg. Next, (Remember to set the amplitude in rms
pump the air out of the Matheson gauge un- volts. Check the indicator to be sure.) Si-
til the thermocouple gauge bottoms out. The multaneously look at the function genera-
true pressure P0 and the thermocouple reading tor waveform output and sync output on
should be well below 0.3 torr and, if so, P0 = 0 a two-channel oscilloscope. Trigger on the
should be an accurate approximation. Record rising edge of the sync. What is the rough
the Matheson reading P0meas at this pressure. phase difference of the waveform’s rising
The actual pressure P in terms of the gauge zero-crossing relative to the rising edge of
reading P meas is then the sync? Express it in degrees and note
whether it leads (occurs before the sync
Patm − P0 crossing) or lags (occurs after the cross-
P = (P meas − P0meas ) + P0 (88)
Patm − P0meas
meas
ing). Does the phase difference change as
you change the frequency to 1 or 100 kHz?
The thermocouple gauges read in torr up
to a maximum of 2 torr. The meter reading 2. Set the frequency to 33 kHz. This is near
may go above 2 torr at higher pressures, but the frequency needed to measure the tun-
these readings are very inaccurate and essen- ing fork response.
tially useless. A thermocouple gauge depends
on the thermal conductivity of the residual gas 3. Check out what the lock-in does. Set the
and reads differently for different gases at the lock-in time constant to 1 s with a 24
same pressure. It is calibrated for air, but db/octave slope. Set the sensitivity to
don’t try to make corrections for helium when 0.5 V with no line filters in and set the
recording readings. All values given in the in- Reserve to Normal. Set the input for A,
structions are raw readings. DC Coupling and Ground. Connect the
The diaphragm-type Magnehelic gauge on DS340 sync signal to the lock-in reference
the helium dewar reads the amount the dewar input and its output to the lock-in A in-
pressure is above atmospheric pressure and is put. Set the reference channel for rising
in inches of water (1 atm is about 407 inches edge and set the lock-in phase offset to
of water). 0◦ . Set the front panel displays to x and
y (Vx and Vy ). Record the Vx and Vy and
then change the display to R and θ (V
Initial observations
and ϕ) record these values. In particular,
There are a lot of ways to explore the appa- note the sign of the θ in comparison to
ratus and the physics of the tuning fork to be whether the input led or lagged the sync.
March 11, 2015Quartz Crystal Tuning Fork in Superfluid Helium TFH 21
4. Hit the Auto Phase button. This button resistor in place of the tuning fork in the
adjusts the phase offset to make the in- circuit diagram of Fig. 6. Predict the
put signal in phase with the reference (af- lock-in outputs, adjust the lock-in sen-
ter the reference has been shifted by the sitivity and record the results. Is the
phase offset). Record the new phase offset current in-phase with the drive voltage?
and values for x, y, R, θ. Should it be? Why?
5. Change the DS340 amplitude to 0.1 V 8. Hit the lock-in Auto Phase button, record
and record how long it takes the lock-in the new lock-in outputs and phase offset.
to settle to the new correct amplitude. How does the phase offset change? Why?
Set the lock-in time constant to 1 ms
and change the DS340 amplitude back to 9. Switch the Device Selector to the 220 pf
0.2 V to see how long it takes now to re- capacitor. Do not adjust the phase offset.
act to a quick change in amplitude. Set Predict the lock-in x and y outputs, ad-
the lock-in time constant back to 1 s. just the lock-in sensitivity and record the
results. Does the capacitor current lead
6. Check how the shunt resistor affects the or lag the drive voltage?
function generator output. The shunt re-
sistors are located in the interface box.
Frequency scans
Connect the output of the function gen-
erator to the bnc labeled Signal Input on 10. Switch the Device Selector to the tuning
the interface box. The function generator fork still sealed in its canister (labeled
output with the effect of the shunt resistor Vacuum on the Device Selector).
is then also available at the Input Signal
Monitor bnc on the interface box. Con- 11. Set the lock-in sensitivity to 20 mV and
nect it to the lock-in input. The Signal the time constant to 10 ms. The lock-
Input/Input Signal Monitor bncs are con- in time constant is being kept short to
nected to the rotary switch labeled Input isolate the effects of the tuning fork time
Signal Attenuation. In the ×1 position constant.
there is no shunt, ×0.1 puts in a 5.6 Ω
shunt, and ×0.01 puts in a 0.5 Ω shunt. 12. Set the DS340 drive amplitude to 1 V. Be
Set the function generator amplitude to sure the 0.5 Ω shunt (×0.01 on the Atten-
1 V. Predict the lock-in outputs for each uator switch) is connected so the actual
shunt position, adjust the lock-in sensi- drive level is now around 10 mV. Find
tivity and record the results. the resonance by manually adjusting the
DS340 frequency around 32760-32770 Hz,
7. Study how the transimpedance amplifier further homing in on the frequency where
works. Leave the function generator con- the lock-in R maximizes; the amplitude
nected to Signal Input, but move the lock- should be on the order of 5 mV on reso-
in input so it measures the output of the nance.
transimpedance amplifier—the bnc con-
nector labeled Output Signal Monitor. Ad- 13. Change the drive amplitude to 2 V and
just the Device Selector switch for the record how long it takes for the lock-in to
100 kΩ resistor. This will connect a 100 Ω settle at the new equilibrium value.
March 11, 2015You can also read
