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ac Susceptibility Measurements inHigh-Tc Superconductors

Experiment ACS

University of Florida — Department of PhysicsPHY4803L — Advanced Physics Laboratory

Objective

You will learn how to use an alternating cur-rent susceptometer to study the magnetiza-tion induced in various magnetic materials inresponse to the alternating magnetic field in-side a driven solenoid. In particular, a high-Tc superconductor is studied as a function oftemperature near the superconducting transi-tion. A two-phase lock-in amplifier is used tomeasure the amplitude of the sample magne-tization and its phase relative to that of thedriving magnetic field.

References

M. Nikolo, “Superconductivity: A guideto alternating current susceptibility mea-surements and alternating current suscep-tometer design,” Am. J. Phys. 63, 65(1995).

W. Lin, L.E. Wenger, J.T. Chen, E.M. Logo-thetis, and R.E. Soltice, “Nonlinear mag-netic response of the complex AC suscep-tibility in the YBa2Cu3O7 superconduc-tors,” Physica C 172, 233-241 (1990).

C. Kittel, Introduction to Solid State Physics,6th ed. (Wiley, 1986). See Chapter 12and Appendix K.

N. W. Ashcroft and N. D. Mermin, SolidState Physics, (Holt, Rinehart and Win-ston, 1976). Chapter 34 gives intro-ductory descriptions of superconductiv-ity and lists references for more advancedreading.

W. E. Pickett et al., “Fermi Surfaces, FermiLiquids, and High-Temperature Super-conductors,” Science 255, 46 (1992).

M. Tinkham, Introduction to Superconductiv-ity (McGraw-Hill, 1996).

Introduction

The equipment list for this experiment in-cludes a vacuum pump, liquid nitrogen(LN2) Dewar, ac susceptometer, function gen-erator, two-phase lock-in amplifier, silicondiode temperature sensor, temperature con-troller, computer data acquisition system, andYBa2Cu3O7 (YBCO) superconductor. The-oretical considerations include superconduc-tivity, magnetization, susceptibility, Faraday’slaw of induction and ac circuit analysis. Manyexperimental and theoretical aspects will onlybe treated at a level necessary to understandtheir application to this experiment. The in-terested student is encouraged to explore the

ACS 1

ACS 2 Advanced Physics Laboratory

references, textbooks, and other material tofill in the gaps.

High-Tc superconductor

Superconductivity was discovered in 1911 byH. Kammerlingh Onnes in Leiden. Afterdecades of detailed experiments, a thoroughunderstanding of “conventional” superconduc-tors became available in 1956 with the ad-vent of the Bardeen-Cooper-Schrieffer (BCS)theory. There are now hundreds of knowncompounds which exhibit this remarkable phe-nomenon. The field had a major upheaval in1986, when Bednorz and Muller at the IBMResearch Laboratory in Zurich discovered su-perconductivity in La2−xSrxCuO4. The dis-covery has led to a new class of oxide super-conductors with significantly higher transitiontemperatures Tc. These “high-Tc oxide super-conductors” are now a subject of intense ex-perimental and theoretical study, as there areexperimental indications that they differ fromconventional superconductors. They also pro-vide a convenient demonstration of supercon-ductivity in the laboratory, requiring only LN2

as the cryogen.

Superconductors possess a unique collectionof physical behaviors: zero electrical resis-tance, the Meissner effect (perfect dc diamag-netism), an energy gap ∆ in their electronicexcitation spectrum, the quantization of mag-netic flux, and the Josephson effect. This ex-periment studies their magnetic response, inparticular their ac diamagnetism at low fre-quency (hω ∆).

Any conductor will partially shield its in-terior from changing magnetic fields throughthe generation of eddy currents. Because ofresistive losses, the shielding is incomplete. Ina perfect conductor (zero resistance material)the shielding becomes perfect; the magneticfield cannot change in a perfect conductor.

Figure 1: The Meissner effect in a supercon-ducting sphere cooled in a dc magnetic field.Upon passing below the transition temperature,the field is expelled from the material (right).

Superconductors are perfect conductors, butthey shield their interiors from dc magneticfields as well. Not only dB/dt but also B iszero inside a superconductor. The expulsionof static magnetic fields from the interior ofsuperconductors is called the Meissner effectand will not be considered in this experiment.However, it should be pointed out that theMeissner effect is not a consequence of zeroresistance. Figure 1 (left) shows a supercon-ducting sphere in a dc magnetic field abovethe transition temperature. The material is inthe normal state and the field completely pen-etrates it. The sample is then cooled into thesuperconducting state and spontaneously gen-erates eddy currents completely expelling themagnetic field. If the superconducting transi-tion is simply a transition to a perfectly con-ducting state (where dB/dt = 0), the mag-netic field could not change on cooling throughthe transition, and would be trapped in thematerial. Superconductors are more than per-fect conductors!

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ac Susceptibility Measurements inHigh-Tc Superconductors ACS 3

Susceptibility

A magnetic material is described by its magne-tization M(r), or magnetic dipole moment perunit volume at points r throughout its volume.Recall that the SI unit of magnetic dipole mo-ment is A·m2 and thus the SI unit of magne-tization is A/m.

Determining the magnetic field associatedwith an arbitrary magnetization distribution,M(r), can be difficult. However, if thematerial is in the shape of a long cylinder(solenoidal) and the magnetization is constantand points along the cylinder axis, the situa-tion simplifies.

Before considering this special case, recallthat for a long solenoid carrying a current I,the magnetic field outside the solenoid is zeroand inside B = µ0nI where n is the solenoid’snumber of turns per unit length. The fielddepends only on the product nI which can becalled the current density and has SI units ofA/m, the same as M.

The microscopic model for bulk magnetiza-tion is that it arises from the contributions of agreat many atomic or molecular current loops.In the interior of the material, the current fromadjacent loops are in opposite directions andcancel for a constant magnetization. On thematerial’s boundary, however, the loop cur-rents are unopposed, and lead to a surface cur-rent density (current per unit length) M× n,where n is the surface normal.

For a constant axial magnetization Mthroughout a long, cylindrically-shaped sam-ple, the surface current density is constant andequal to M . This current circulates aroundthe cylinder like the current in a solenoid, andthe magnetic field is the same as that of asolenoid with nI = M , i.e., inside the ma-terial Bin = µ0M (a constant), and outsideBout = 0.

If such a sample is placed in a uniform mag-

netic field Ba oriented parallel to the sampleaxis, the net magnetic field is the superposi-tion of the applied field and the field due to themagnetization. If the external field is written

Ba = µ0Ha (1)

(which defines the magnetic intensity Ha) themagnetic field outside the cylinder walls (butnot just outside the ends of the cylinder) isjust the applied filed Bout = µ0Ha and thefield inside the cylinder becomes

B = µ0(Ha +M) (2)

For some materials the magnetization iszero in the absence of an applied field and onlybecomes non-zero as a response to the appliedfield. For linear response materials, the mag-netization will be aligned with (or against) thefield and proportional to it. That is, we canwrite

M = χHa (3)

which defines the material’s susceptibility χ.Equation 2 can then be expressed

B = µ0(1 + χ)Ha (4)

For paramagnetic materials, χ > 0 and themagnetic field is enhanced in the presence ofthe material. For diamagnetic materials, χ <0 and the field inside the material is reduced.

Non-linear materials in dc magnetic fieldscan show saturation effects, hysteresis, andmagnetization containing terms proportionalto higher powers in the applied field.

If the applied magnetic field is not constantbut is instead oscillating sinusoidally along thesample axis at a frequency ω, i.e.,

Ha = H0 cos ωt (5)

the situation (for a linear magnetic material)is only slightly more complicated than that fora dc field. The general form for a material’s

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magnetization in an ac magnetic field is alsosinusoidal at the frequency of the applied field,but it can be shifted in phase relative to theapplied field. Relative to the applied field asgiven by Eq. 5, the general form for the mag-netization can be expressed

M = H0(χ′ cos ωt+ χ′′ sin ωt) (6)

Phasor representations are very useful forworking with ac quantities. Recall phasors asrotating vectors in the complex plane. Euler’sidentity in the form

eiωt = cosωt+ i sinωt (7)

illustrates the rotational behavior of a pha-sor of unit length with oscillating real (cosωt)and imaginary (sinωt) components. Also re-call that the real physical quantity associatedwith a phasor is the projection of the phasoron the real axis and is obtained by taking thereal part (abbreviated <) of the phasor quan-tity.

An applied ac field of the form of Eq. 5would be expressed

Ha = <H0e

iωt

(8)

A (linear) material’s susceptibility can then bespecified as a complex constant

χ = χ′ − iχ′′ (9)

The magnetization can then be expressed asthe real part of a phasor constructed as theproduct of the now complex susceptibility andthe phasor for the applied field

M = <χH0e

iωt

(10)

which is the ac equivalent of Eq. 3 for dc fields.

Exercise 1 Show how Eq. 8 is consistent withEq. 5. Show how Eq. 10 with Eqs. 9 and 8 isconsistent with Eq. 6.

For some materials, χ′′ is nearly zero. Ifχ′′ = 0, the magnetization will be perfectlyin phase with Ha for a paramagnetic material(χ′ > 0) and will be 180 out of phase fora diamagnetic material (χ′ < 0). With non-zero values of χ′′, the magnetization given byEq. 6 will be neither perfectly in phase or outof phase with the applied field.

It turns out that only positive values of χ′′

are physically possible (and thus that the mag-netization can only lag the applied field). Thatonly χ′′ > 0 is physically possible can be seenfrom the relationship between the sign of χ′′

and the direction of energy flow between thesample and the applied field. The power den-sity p (power per unit volume) absorbed in thesample is given by

p = −MdBa

dt(11)

Exercise 2 Show that averaged over a com-plete cycle,

pav =1

2µ0ωH

20 χ

′′ (12)

Only pav > 0 (χ′′ > 0) is energetically pos-sible. In this case, the sample absorbs energyfrom the applied field which goes into heatingthe sample. Were pav < 0 (χ′′ < 0), ratherthan absorbing energy, the sample would becontinually radiating energy — a violation ofthe second law of thermodynamics. Thus χ′′ isassociated with energy absorption within thematerial.

Non-linear magnetic materials in ac fieldsdevelop magnetization with higher harmoniccomponents in addition to the fundamentalexpressed by Eq. 6. Thus, non-linear materi-als produce non-sinusoidal magnetization. Wewill emphasize the linear behavior, but keepyour eye out for evidence of non-linear behav-ior.

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Solenoid in an ac field

Since the surface current of a uniformly mag-netized cylindrical sample is equivalent to asolenoidal current density nI = M , it willbe insightful to investigate the behavior of asolenoid in applied ac field. We start howeverwith the dc behavior.

Exercise 3 For a solenoid of radius R andlength D R carrying a current Is, the self-induced flux through the solenoid is defined byΦs = NBsA, where N = nD is the total num-ber of turns and A = πR2 is the cross-sectionalarea. Show that Φs can be expressed

Φs = LIs (13)

where L = µ0n2V , and V is the solenoid vol-

ume.

L is called the self inductance.The solenoid is placed with its axis along an

applied field Ba.

Exercise 4 Show that the flux through thesolenoid due to the applied field Φ = NBaAcan be written

Φa = LIa (14)

where the pseudo-current Ia is defined by Ia =Ba/µ0n.

Thus, if the solenoid carries a current Is, thenet flux through the solenoid can be expressed

Φ = L(Is + Ia) (15)

If the applied field Ba oscillates at a fre-quency ω, Φa will oscillate, and from Faraday’slaw of induction, an ac voltage

V = −dΦ/dt (16)

will develop across the open solenoid ends. Ifa resistive load R is placed in series with thesolenoid, an oscillating solenoid current Is =V/R will develop, which then must be takeninto account.

Exercise 5 Show that if Ba is given by Eq. 5(i.e., Ia = Ia0 cos ωt with Ia0 = H0/n), thenthe current Is in the solenoid will be

Is = Ia0(χ′ cos ωt+ χ′′ sin ωt) (17)

where

χ′ = − ω2L2

R2 + ω2L2(18)

χ′′ =RωL

R2 + ω2L2(19)

Hint: This problem is easy if you representIs, Ia, Φ, V as phasors. Alternatively, youcan assume the solution as given by Eq. 17and solve for χ′ and χ′′. Either way you willneed the fact that the solenoid current obeysV = IsR, with V given by Eq. 16 and Φ givenby Eq. 15.

We can now compare the solenoid behaviorwith that for a linear induced magnetizationin a solenoid-shaped sample. The magnetiza-tion that would produce the same field as thesolenoid is given by M = nIs. With Is as givenby Eq. 17, this produces a magnetization Mas given by Eq. 6.

Exercise 6 The instantaneous electricalpower delivered to the solenoid by the oscil-lating magnetic field is given by P = V Isand is oscillatory with a DC offset. (a) Showthat the DC component (average value over acomplete cycle) is given by

Pav =1

2ωLI2a0χ

′′ (20)

(b) Show that this is consistent with

Pav =1

2|Is|2R (21)

demonstrating that energy is going into jouleheating of the resistive component of thesolenoid. (c) Show that this calculation agrees

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with Eq. 12. Hints: Keep in mind that Eq. 12is for a power density not a power. Usingphasors the formula for the average power isPav = 1

2<[V I∗s ], where I∗s is the complex con-

jugate of Is.

Exercise 7 For a given frequency, as thetemperature T decreases, the resistance de-creases from a high value (R ωL) to a lowvalue (R ωL). Assume that most of thischange occurs within a region ∆T around Tc,where Tc is the transition temperature of thesuperconductor. Sketch the temperature de-pendencies of χ′(T ) and χ′′(T ). You shouldexpect to see these dependencies in the exper-iment that follows.

In a superconductor, B must be zero andthe effective magnetization must obey M =−Ha. Thus χ′ = −1 and χ′′ = 0 for a super-conductor. (What do Eqs. 18 and 19 give forχ′ and χ′′ for a solenoid with zero resistance?)However, during the transition to the super-conducting state the study of the magnetiza-tion expressed by Eq. 6 (and higher harmon-ics) can lead to insights into the mechanismsresponsible for the creation and destruction ofsuperconductivity.

Susceptometer

The sample magnetization is studied using aconventional ac magnetic susceptometer. Thesusceptometer probe illustrated in Fig. 2 in-cludes a primary (excitation) solenoid coilwhich produces a near-uniform ac magneticfield when driven by an ac source. Inside theprimary coil is the secondary coil consisting oftwo pickup coils which are wound in oppositedirections and electrically connected in series.The pickup coils occupy opposite halves of theprimary coil volume. The sample is placed inone of the pickup coils (the sample coil) while

Figure 2: Arrangement of the primary coil (cut-away), the secondary coil, and the sample in thesusceptometer. Note the secondary coil consistsof the sample coil (bottom) and the counter-wound reference coil (top).

the other pickup coil (the reference coil) is leftempty.

The ac voltage V across the secondary coilis due to Faraday induction.

V = −dΦ

dt(22)

where Φ is the net flux through the entire sec-ondary (both the sample and reference coil)and includes contributions from both the ap-plied field and the field due to the samplemagnetization. The self-induced flux from thesecondary can be neglected because the sec-ondary is connected only to high impedance(10 MΩ) voltage metering equipment (oscillo-scope, lock-in) and thus carries negligible cur-rent.

Because the two pickup coils are wound inopposite directions and connected in series,

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the net secondary flux due to the applied fieldwill be small. It is not exactly zero because ofimperfections in probe construction, and thusthere is some small mismatch in the flux forthe sample and reference coils. Of course, thenet secondary flux due to the mismatch, Φmm,will be proportional to the primary field

Φmm = αmmBa (23)

where αmm is an effective mismatch area whichmay be positive or negative depending onwinding directions, and whether the sampleor reference coil has the larger flux. Conse-quently, there will be some secondary signalV = −dΦmm/dt (called the mismatch signal)even with no sample present. If the primaryfield is given by Eq. 5, then the mismatch sig-nal will be

V = Vmm sin ωt (24)

where Vmm = αmmµ0ωH0 gives the amplitudeof the voltage due to mismatch.

If a sample is present (in the sample coil),the magnetic field µ0M associated with the in-duced magnetization will unbalance the fluxesin the sample and reference coils and there willbe a large net flux Φm in the secondary. Wecould write

Φm = αmµ0M (25)

with the effective area αm expected to be muchlarger than αmm. Again, the sign of αm de-pends on the winding directions. If M is givenby Eq. 6, there will be an induced secondaryvoltage V = −dΦm/dt given by

V = Vm(χ′ sin ωt− χ′′ cos ωt) (26)

where Vm = αmµ0ωH0 sets the scale for thesecondary voltage due to the sample.

The net secondary voltage will include bothsources, Eqs. 24 and 26.

Figure 3: Equivalent circuit for the function gen-erator, susceptometer and lock-in.

Primary field, current

The primary field is, of course, directly pro-portional to the current Ip in the primary coil.

Ba = κIp (27)

Exercise 8 The proportionality constant κcan be estimated from the primary coil geom-etry. Estimate κ (in gauss/amp) based on theinfinite solenoid formula (B = µ0nI) and the(approximate) primary coil geometry of 5 cmlong, 350 turns, i.e., n = 70/cm. The fieldhas been measured with an axial Hall probe andagrees well (within a few percent) with this ap-proximation. The average diameter of the pri-mary coil is about 1.7 cm.

A small junction box allows a variable re-sistor to be inserted in the return path forthe primary. The voltage across the resistorappears at the BNC terminal labeled Moni-tor when an external resistor is placed acrossthe banana jacks. This voltage should be verynearly in phase with the current in the primaryand is used for the lock-in reference. The am-plitude of the voltage, together with the valueof the monitor resistance is used to calculatethe amplitude of the primary current.

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Lock-in detection

In our application, the two-phase lock-in am-plifier is used to measure both the amplitudeof the secondary voltage and its phase (howmuch it lags or leads) relative to the monitorvoltage. The lock-in takes the monitor volt-age as the reference input and it takes the sec-ondary voltage as the signal input. Althoughthe amplitude of the reference voltage is rela-tively unimportant, the manufacturer recom-mends its value be near 1 Vrms for optimumsensitivity.

The origin of time is arbitrary, so the refer-ence voltage can be expressed

Vr = Vr0 cos ωt (28)

Relative to this reference, an arbitrary signalvoltage (at the same frequency) can be ex-pressed

Vs = Vs0 cos(ωt− δ) (29)

where δ is the phase angle by which the signallags the reference. Equation 29 can also beexpressed

Vs = Vx cos ωt+ Vy sin ωt (30)

where

Vx = Vs0 cos δ (31)

Vy = Vs0 sin δ (32)

The lock-in provides direct measures of Vxand Vy. (The lock-in outputs are actuallyrms values with signs, i.e., Vx/

√2 and Vy/

√2.

With a change of mode, the lock-in can alsobe made to provide Vs0/

√2 and δ.)

Assuming that the monitor voltage (lock-inreference) is perfectly in-phase with the pri-mary field and that the secondary voltage (sig-nal) is given by the sum of Eqs. 24 and 26, thelock-in outputs are expected to be

Vx = −Vmχ′′ (33)

Vy = Vmm + Vmχ′ (34)

Note, however, that the lock-in has a ref-erence phase offset adjustment which allowsone to introduce an arbitrary, constant phaseshift between the reference and signal voltages.The phase offset is often adjusted when theremay be an apparatus-dependent phase shift inthe reference and/or the signal, say due to ca-ble lengths or stray impedances. To correctfor the unknown phase shift, one needs somecondition that can be realized experimentallywhere the relative phase between the referenceand signal is known. If such a condition can beachieved, the lock-in phase adjustment can beset to produce the known phase difference, andas the experimental conditions change to thosewhere the phase relationship is unknown, theresults can be interpreted based on this set-ting.

For example, it is quite reasonable to ex-pect small phase shifts in either or both oursignal and reference voltages. Furthermore, asthey may be affected by circuit impedance,such apparatus-dependent phase shifts maychange with temperature. In this experiment,if the reference and signal voltage have noapparatus-dependent phase shifts (or if bothhave the same shift), there should be no (in-phase) x-signal when the sample is perfectlysuperconducting or when the sample is non-magnetic since in both cases χ′′ = 0. Thiscondition can be used to calibrate the phaseat room temperature, just above, and just be-low the superconducting transition. Simplyadjust and record the phase setting needed toproduce no x-output.

Temperature Measurements

The temperature of the superconducting sam-ple is measured using a silicon diode as a ther-mometer. If you do the low temperature trans-port experiment, you will measure the current-voltage characteristics of such a diode. Here,

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Figure 4: Calibration curve for our diode witha 100 µA excitation current.

you will use the forward voltage to sense thetemperature. The current I in the diode fol-lows

I = I0(eeV/ηkT − 1

), (35)

where e is the charge of the electron, V is theapplied bias voltage, k is Boltzmann’s con-stant, T is the absolute temperature, and ηis an “ideality factor.” The ideality factor de-pends on the details of the diode fabricationand typically varies between 1 and 2. Finally,I0 is called the reverse saturation current be-cause I = −I0 when the diode is reverse biased(V < 0) sufficiently that eeV/ηkT << 1. Equa-tion 35 can be solved for the temperature:

T =eV

ηkln(I + I0I0

). (36)

Generally, I0 I, allowing Eq. 36 to be sim-plified. The Lakeshore temperature controllersends a constant current (100 µA) through thediode and measures the voltage V . A calibra-tion (see Fig. 4) is build into the controller andthe temperature appears on the front panel

and is read by the LabVIEW program for thisexperiment.

Safety Considerations

Handling of cryogenic fluids requires cautionsince serious frost bite may result from mis-handling. In particular, never get cryogenicfluids in eyes or allow the fluid to remain incontact with your skin. DO NOT PLAYAROUND WITH CRYOGENIC FLU-IDS. Operating valves and other metal partsthat are or have been in contact with cryo-genic fluids requires wearing special protectivegloves. Otherwise, it is best to handle cryogencontainers and valves with bare hands. Thereason is that most gloves absorb cryogenicfluids which would then remain in contactwith the skin too long, possibly causing frostbite. On the other hand, fluids spilled overbare skins will rapidly escape and/or evapo-rate, reducing the risk of frostbite. If you spillthe liquid on your feet, quickly remove bothshoes and socks to avoid freezing your toes.The leather and the fabric of your footwearhave most likely soaked up the cold liquid, al-though they appear dry. The consequence ofnot following this advice can be serious andextremely painful.

The vacuum chamber for the apparatus willbe cooled to LN2 temperatures in a glass de-war. Read and observe the procedures forevacuating, cooling, and warming the cham-ber. Failure to follow proper precautions canlead to explosions and flying shards of metaland dewar glass.

The health effects of the superconductingcompound YBCO used in this lab have notbeen studied in detail. Therefore, you areadvised to wash your hands after handlingthe sample. As was discussed in the lecture,NEVER eat in the lab and always wash yourhands after handling any samples.

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General Procedures

Do not perform any of the procedures in thissection until instructed to do so in the mainProcedure section. At that time please rereadand follow the instructions given here.

The susceptometer is mounted inside a vac-uum chamber to decrease heat transfer rates.The superconducting sample is thermally con-nected to the vacuum chamber by a small cop-per braid and will cool down reasonably slowlywhen the chamber is evacuated and cooled toLN2 temperatures.

The outer can of the vacuum chamber easilyrolls off a lab table. If it falls on the flooror otherwise becomes dented, the vacuum sealwill be ruined. When not in use store thecan (seal-side in) in the wooden blockprovided for this purpose. Furthermore,keep the block away from table edges.

Probe Evacuation

1. Close the purge and pump valve and turnon the vacuum pump.

2. Wipe the brass contacting surfaces witha clean soft towel and coat both surfaceswith a thin layer of silicon grease.

3. Open the pump valve to the vacuumchamber, gently rotating the can witha slight pushing against the sealing sur-faces. As a vacuum is achieved, the ro-tation will become difficult and can bestopped. Continue pumping while youmake measurements.

Probe Cooling

1. Fill the dewar about 3/4 full with LN2.(LN2 can be obtained from the cryogenicsshop on the basement floor.)

2. Lower the can — slowly enough to avoidrapid boiling of the LN2 — until it is com-pletely submerged.

3. The pressure should fall as the chambercools. If it rises, LN2 may be seepinginto the chamber — proceed to the probewarming and check with the instructor.

Probe Warming

1. Raise the chamber completely out of thedewar. Continue pumping.

2. The chamber can be warmed by lower-ing it completely into the popcorn popperand running the popper on a Variac (ad-justable ac voltage supply). While pump-ing, warm the chamber with the Variac ata fairly low setting. When the tempera-ture gets slightly above room temperatureturn off the heating. Lower the ac voltageas necessary to make sure that the cham-ber never gets so hot that you can nothold your finger on it. Never walk awayfrom the chamber with the heatingon. You could overheat and destroy thesusceptometer. Do not open a cold cham-ber to air; water will condense on the sus-ceptometer making it difficult to evacu-ate later. As a precaution always arrangethings so that if, during the warming cy-cle, the vacuum can should slide off thechamber, it will not fall to the floor andpossibly become dented. If you use thepopcorn popper and position it and theprobe properly, the can would safely fallinto the popper.

3. Again, make certain that the vacuum canwill not fall on the floor as you next let airinto the chamber. Either lower the can soit is within a centimeter of the table orplace a block under the can leaving no

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more that a centimeter of space betweenthem.

4. While holding the can in your hand, closethe pump valve and open the vent valve.You should hear the air rushing into theprobe which should then be at atmo-spheric pressure. So be careful. The cancould easily fall off on its own at thispoint. If the can does not fall off, hold thetop brass sealing surface not the sup-port tubing and gently twist and pullthe can straight off. If the can does notcome off easily, it may still be under vac-uum. Check that the pump valve is closedand the vent valve is open. Place thecan in its holder.

5. Turn off the pump and open the pumpvalve.

Procedure

1. Keep the susceptometer at room temper-ature and open to the air for now. The su-perconducting sample is mounted arounda copper rod on a copper flange which isheld inside the susceptometer by a singlescrew at the bottom of the flange. Thesilicon diode used as a thermometer is sol-dered to the bottom flange. It connects tothe wires running up the tube via a twopin connector. Unplug it. The braid sol-dered to the copper flange is heat sunk tothe large conical brass flange by a shortscrew and washer. Remove this screw aswell as the one holding up the supercon-ducting sample. The sample should nowslide out easily. Take care not to lose any-thing and store all parts in the plasticcontainer. You will first test the opera-tion of the susceptometer with no sampleso the only signal will be from coil mis-match.

2. Using an ohmmeter, measure the re-sistances of the primary (approximately3 Ω) and the secondary coil (approxi-mately 60 Ω). Check the electrical isola-tion of the two coils from each other andfrom the coil supports. Report problemsto the instructor.

3. Disconnect the function generator fromthe primary coil and measure its outputdirectly on an oscilloscope. Set the ampli-tude to 10 Vpp and observe that it is in-dependent of frequency. (Make sure thatload-impedance selection has been set toHIGH-Z instead of 50 Ω and leave it atthis setting throughout the experiment.)

4. Reconnect the function generator to theprimary circuit and monitor the primarycurrent (monitor voltage) and secondaryvoltage with the oscilloscope. Trigger onthe primary current. Don’t try to mon-itor the primary coil voltage sinceneither end of the coil is grounded.Keep the generator amplitude (at the10 Vpp setting) and monitor resistance(about 10 Ω will work well) fixed duringthe measurements in the next step.

5. Draw and analyze a circuit model includ-ing: the function generator (ideal sourcein series with a 50 Ω resistance), primarycoil (ideal inductor in series with a re-sistor as measured previously), and themonitor resistance. Measure the primarycurrent and secondary voltage as a func-tion of frequency from 100 Hz to 200 kHzin a 1, 2, 5 sequence. Above 100 or200 kHz stray circuit capacitances not in-cluded in the model can lead to unpre-dicted resonances in the primary current.

C.Q. 1 Plot primary current vs. frequencyand use this data with the model to deter-mine the primary’s inductance. Compare with

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a calculated inductance based on the geome-try given previously. Plot the ratio of the sec-ondary’s voltage to the primary current as afunction of frequency and explain its behavior.

Hopefully, you have verified in the previousstep that the amplitude of the secondary volt-age for a given primary current is proportionalto the frequency. Thus, higher frequencies willgive larger signals. However, the impedanceof the primary also increases with frequencyand therefore the maximum attainable pri-mary current and primary field decrease. Fora frequency around 1 kHz, the secondary sig-nals are generally large enough to be easilymeasured and the maximum primary field canbe made large enough to observe the effectsof large amplitude applied fields on the super-conducting transition.

6. Set the function generator frequency to1 kHz.

7. Maintaining the oscilloscope connections,connect the lock-in — monitor voltage tothe reference input, secondary to the sig-nal input. Turn off all lock-in input fil-tering and output offsets. Set the out-put time constant to 1 sec. When makingmeasurements make sure the lock-in sen-sitivity is set to a reasonable value; toolow and the readings will lose accuracy,too high and the input amplifiers will sat-urate.

8. The monitor resistance and function gen-erator amplitude should be simultane-ously adjusted to produce the desired cur-rent amplitude in the primary while alsoproducing about a 1 Vrms monitor volt-age. Adjust the function generator am-plitude and/or monitor resistance to getan ac magnetic field amplitude of a few

gauss. Do not trust the nominal resis-tance values of the Ohm-ranger, particu-larly for small values. If the directly mea-sured resistance value is unstable or muchlarger than the nominal one, flip up anddown all the switches several times untilthe problem disappears.

9. Adjust the reference phase setting of thelock-in to zero. With just the mismatchsignal, the primary current and secondaryvoltage should be 90 out of phase. Thus,there should be nearly no signal on the x-or in-phase output and a maximum signal(plus or minus) on the y- or quadraturesignal. Record how the x- and y-outputschange as you vary the lock-in’s referencephase control and explain your results.Determine and record the reference phasesetting which zeros the x-output. Howwell can this be done? Does it depend onthe primary field amplitude?

A ferrite core inductor is used for this partof the experiment. The core is cylindricallyshaped and made from ferrite—a ceramic ma-terial having a large χ′ and a small χ′′. Theinductor is made by winding a solenoid aroundthe core. The leads of the solenoid can be leftopen so there is no inductor current or theycan be shorted to see the effects of non-zeroinductor current when it is placed in an acmagnetic field.

10. Keep the inductor open-circuit. Watchthe secondary voltage on the oscilloscopeand on the lock-in outputs as the core isjust barely inserted into the sample coil.Record whether the extra flux from theferrite core creates a signal which addsto or subtracts from the mismatch sig-nal. What does this behavior imply aboutthe mismatch (no sample) signal? Withno sample, does the sample coil (lower)

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or the reference coil (upper) have largerflux? What does this behavior implyabout the sign of Vm in Eqs. 33 and 34?Gradually insert it farther into the sus-ceptometer watching as the amplitude ofthe secondary voltage goes through sev-eral extrema. Describe how the oscillo-scope trace (amplitude and phase) andlock-in outputs change during this step.and explain the observations in terms ofthe flux changes going on in the sampleand reference coils.

11. Since the inductor was open circuit inthe last step, there was no current in it.Consequently, there is only very low re-sistive energy losses due to the small χ′′

of the core. Watch how things changewhen the leads are shorted. Starting withthe inductor open circuit and inserted tothe first maximum, describe how the sec-ondary signal changes. Relative to theprimary current (monitor voltage), doesit advance (lag) or retreat (lead)? Doesthe amplitude increase or decrease? Also,describe the changes in the lock-in out-puts. Explain how these observations areconsistent with the expected changes in χ′

and χ′′ when the inductor windings beginconducting.

12. Is there any position of the inductorwhere the secondary amplitude goes tozero? Explain why your answer dependson whether the inductor is open circuit orshorted.

13. Remove the inductor and gradually insertthe iron nail into the sample coil. De-scribe how the lock-in signals change. Ex-plain and compare with the prior obser-vations when inserting the inductor (bothopen circuited and shorted). How do yourobservations show that for the nail mate-

rial both χ′ and χ′′ are positive. Whatproperties of the nail are responsible?[Hint: What is the nail made from? Doesit behave like the open circuit or shortedinductor?] Do you think the ferrite corematerial is a good electrical conductor?Why or why not?

14. Launch the High Tc data acquisition pro-gram. This program reads the lock-involtages Vx and Vy as well as the sili-con diode temperature via the Lakeshorecontroller. However, you must make sureboth instruments are set up properly, andon the correct scales. The program makesthese three readings at a user-definedrate. After a number N of scans are col-lected, new data overwrites older data.You can sent N in the data acquisitionprogram. N = 4096 is the default butyou can increase or decrease this number.

To save the data to a text file that can eas-ily be read by a spreadsheet, use the SaveData button on the program front panel.Do not use the File|Save or File|SaveAs menu items which are for saving theprogram, not the data. Besides the threevoltages, the data saved includes the lock-in settings and the data collection rate.Record all other information, e.g., mon-itor resistance and voltage, in your logbook.

15. Plug the two pin connector coming fromthe silicon diode into its connector. Checkthat the Lakeshore controller is read-ing “room temperature,” typically 293–295 K. If it is not, or shows an error, re-verse the two pin connection.

16. Secure the YBCO sample in the probewith a single nylon screw. Cool to LN2

temperature as per the previous instruc-tions while recording data. It is impor-

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tant to monitor the lock-in voltages anddiode temperature when cooling down.The lock-in voltages initially are quitesmall but become much larger when theYBa2Cu3O7 sample is superconducting.It takes an hour or so to cool the probefrom room temperature to LN2 tempera-ture, so take data spaced about 5 secondsapart. Save the data. Graph and analyzethe temperature versus time with respectto a thermal model.

17. While the sample is at LN2 temperature:(a) Measure the dc resistance of the pri-mary and secondary coils and explain theresults. (b) Look at the secondary sig-nal on the scope and increase the pri-mary current until the signal becomesnon-sinusoidal. How big does the appliedfield have to be to see this effect? Atthese higher fields, is the sample still su-perconducting? (Partially? Totally? Notat all?)

18. You should realize by now that even atLN2 temperatures, the sample may not betotally superconducting at high appliedfields. With your sample at LN2 temper-ature, and with the primary field suffi-ciently low (so that the sample is super-conducting), zero the Vx-signal with thereference phase adjustment and recordthe setting. Be careful. If you take theapplied field too low, the lock-in mightbe measuring noise rather than a super-conducting signal. Determine how big thenoise is and whether it can be a problem.

19. Set the primary current to produce afield amplitude of about 0.5 G. Raise thechamber out of the LN2 (partially or to-tally, depending on how fast you wouldlike the temperature to change). Takedata as the sample warms up through

the superconducting transition tempera-ture. You may want to take data pointsmore often now; the runs will be shorter.The fastest setting is 1 second per point;one second is about the minimum timethis program needs to read the lock-in.(There is probably a way to do this fasterif needed.)

20. While the sample is around 100 K orso: (a) Measure the dc resistance of theprimary and secondary coils and explainyour results. (b) Look at the secondarysignal on the oscilloscope and increase theprimary current as much as you did in thestep 17. Interpret your results. (c) Deter-mine and record the reference phase set-ting which produces no Vx-signal. Resetthe reference phase to the prior value fromStep 18.

21. Make several complete thermal cycles(from LN2 temperature to about 100 K).

22. Repeat the measurements as you vary theamplitude of the primary magnetic fieldfrom about 0.01 G to 10 G.

CHECKPOINT: Data from severalthermal cycles for at least one primarycurrent setting should be recorded.

Comprehension Questions

2. Discuss the change in the dc resistance asthe coils are cooled to LN2 temperature.Can resistance changes affect the phase ofthe monitor signal relative to the currentin the primary? If so, how?

3. Compare and discuss the various refer-ence phase settings that produce no in-phase components under the three recom-mended conditions. Do there appear tobe temperature dependence phase shifts?

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4. As the inductor was shorted, how did thex-signal change? How is the direction ofthe change consistent with expectations.

5. Explain how your measurements demon-strate that YBCO is diamagnetic at LN2

temperature?

6. Use your recordings to plot the ac suscep-tibility of the sample as a smooth functionof temperature. What is the transitiontemperature and how sharp is the transi-tion? Do you observe different tempera-tures for the transition when cooling andheating through the transition? Is thisreal physics or an experimental problem?Discuss the issue of controlling and de-termining sample temperature. Estimatethe transition temperature and its uncer-tainty from your data and describe howyou chose them.

7. Explain how your measurements demon-strate that χ′′ for the superconductor dur-ing the transition to the superconductingstate is positive. Where is the associatedenergy going?

8. Discuss the reproducibility of your tran-sition temperatures. Discuss the depen-dence (or lack of dependence) of the sus-ceptibility and the transition temperatureon the amplitude of the primary field. Doyou see evidence of two transitions? If so,do they have the same dependence on pri-mary field amplitude? Keeping in mindthat the sample is polycrystalline, whymight there be more than one supercon-ducting transition?

April 19, 2018

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