Return-flux corrections for SQUID susceptometry

A temperature controlled sample in a SQUID magnetometer or susceptometer cannot fill the cross-section o f the gradiometer pickup due to the requirement o f a cryogenic insulating space between the sample and the superconducting pickup. Return-flux corrections beyond those accounted for by the ordinary demagnetizing factor are thus required. Such corrections as applicable to spheroidal samples are calculated herein, and some of their applications and consequences discussed.

Return-flux corrections for SQUID susceptometry M. Denhoff, S. Gygax and J.R. Long

The development of SQUID flux sensors has kindled new interest in mutual inductance techniques for measurement of the magnetic susceptibility X. Force methods (eg, Faraday balance) that sense the magnetization M in the gradient of a strong, steady field H have been traditionally preferred to mutual inductance susceptometry in applications requiring maximum sensitivity. High sensitivities with a mutual inductance technique is made difficult by the need to effectively null the empty coil flux, and the callbration is complicated by geometrical factors. Conventional mutual inductance techniques survive, however, because they are more economical for routine requirements and because they measure the complex differential susceptibility dM/dH and can be used in weak and swept fields with continuous temperature variation. Commercially available ~ SQUID sensors will detect flux changes as small as 10 -~ G cm 2. Thus there exists the attractive possibility of a single instrument that combines the flux sensitivity of a SQUID magnetometer with the induced moment resolution of a Faraday balance and the versatility of a mutual inductance method. Peculiar to SQUID detection, however, is the feature that the transformer loop, coupling flux to the SQUID, is superconducting. Thus, one has the option of measuring X = M / H using dc excitation as well as the complex ×* = dM/dH that results from conventional ac excitation. Due largely to the field and temperature dependences of the background produced by construction materials, however, it has been found. difficult2 to make a mutual inductance SQUID susceptometer competitive with a good Faraday balance. Recent s u c c e s s 3 toward this objective has stimulated our interest in solving some of the problems caused by the dependence of a mutual inductance method upon geometrical factors.

flux when empty. A sample placed in one of the coils then drives the sensor off null. To maximize the signal, the coil radius R, sample radius r and length l are usually chosen so that (R-Off "~-O. Then the net flux per gradiometer turn is given in cgs Gaussian units by the elementary expression = (1 - D/4rc) ~bm

(1)

where y

Cm =

2rr

(2)

~ 41rMpdp=41rUr2xHo(1 +Dx) -l 0

is the flux through the sample due to its magnetization only, D is the demagnetizing factor of the sample, and X is the internal susceptibility. This analytical form is valid only for sample shapes that result in uniform magnetization. The most useful shapes of this kind are ellipsoids of revolution, including cylinders and disks as limiting forms. The demagnetizing factors for these shapes are well known. We consider the case where the external field Ho is directed along the axis of revolution, and let m denote the ratio of the long axis to the short axis. The standard results a are as follows: Prolate spheroid: Dp

= 4 ; r ( m 2 - 1) -1

(3a) Im(m 2 - 1)-Y21n[m + (m 2 - 1) 'A] - 11

Oblate spheroid: Do = 47rm2(m 2 _ 1)-1

(3b)

Return-flux corrections

I1 -

A SQUID magnetometer usually employs a first order gradiometer as the flux transducer. The gradiometer consists of two uncoupled coils connected in series opposition (astatic), Fig. 1. The coils, each of radius R and n turns on a common axis with the primary field Ho, are trimmed to null M.D. and S.G. are from the Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1$6. J.R.L. is from the Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA. Paper received 22 December 1980.

( ~

(m 2 - 1) -'A arctan (m 2 - 1)'AI

Flux trensformer

SQUID ~

@ Sample

Fig. 1 Schematic diagram of a superconducting gradiometer. With a sample in one arm, the gradiometer becomes either a magnetometer or the secondary of a mutual inductance susceptometer

0011-2275/81/007400-03 $02.00 © 1981 IPC Business Press 400

CRYOGENICS. JULY 1981

When a SQUID flux sensor is employed, (1) is usually inadequate. This complication results from the requirement that an insulating space be interposed between the superconducting pickup coils and the sample at some temperature higher than the T c of the pickup. The thermal and magnetic properties of this insulating space lead to a variety of problems which have been addressed by several authors) These authors appear not to have addressed our present concern. That is, how to account for magnetization return flux through the insulating space. The solution to this straightforward question begins by placing the sample symmetrically in one of the gradiometer coils and calculating the axial magnetic field H = Ho + Hm outside the sample in the plane of the coil a distance p from the centre. The flux due to the applied field H0 is nulled by the gradiometer, so we are only concerned with the field Hm(P) due to the magnetization of the sample, which, by an extension of the same techniques s which lead to (3), we find to be the following:

l

Ho

---~y

x Fig. 2 The arm o f the gradiometer containing the spheroidal sample. Some notation used in the calculations is defined

Prolate spheroid: Hm(P) = -47rxHo(1 +DpX) -1 m(m 2

-

-

Discussion and applications

1) -3/2

I ] n [ ( Y p + 1)/(yp -- l ) ] 1/2 _ yp-1]

(4a)

yp =- [(p/r)2(m 2 - - 1 ) -1 + 1] 1D

Oblate spheroid: H m (p) = -41rxHo (1 + Do X)-I m 2(m 2 _ 1)-3/2

[yo-1 -arctan yo-ll Y0 -

(4b)

f(e, R/r) = 2er/3R

[(P/r) 2 mE( m2 -- 1)-1 _ 1 ] 1/2

The magnetic induction B is, of course, solenoidal. We thus note, Fig. 2, that the net flux through the loop of radius R is equal and opposite to the flux returning outside the loop. This simplifies the flux calculations allowing us to write the net flux ¢ per turn as ¢ = -27r ( - Hm(p)pd p R

Integration by parts then leads to a generalization of (1) to the form ¢ = f ( e , R / r ) Cm

When R / r = 1, (5) and (6) reduce to (1). The quantity f = ¢/¢m then becomes the familiar (1 - D/4rr). The cryogenic requirements of SQUID detection usually results in R/r ~ 2. The dependence of f upon R/r is depicted in Fig. 3 for several decades of the sample aspect ratio e. Straight lines in Fig. 3 illustrate for comparison the useful limiting case of a dipole at the origin normalized to the same far field as the real sample. The dipole approximation for the parallel field configuration has the very simple form

(5)

(7)

and is good for nearly spherical (e ~ 1) or small (R/r >> 1 ) samples. The dipole approximation is particularly convenient when the axis of revolution of the spheroid is normal to the applied field. This configuration includes the important limiting cases of the disk in a parallel field and the cylinder in a perpendicular field. Although the exact flux integration in this perpendicular field configuration must be done numerically, the dipole approximation to the net flux per pickup turn yields the uncomplicated result ¢ = 2rrXHoR-1 V(1 +Dx) -1

(8)

with the 'flux fill factor' fgiven as follows: Prolate spheroid: f ( m , R/r) = m ( m 2 -- 1) -1/2

(6a) I(1 -Y~o) In [Cvpo + 1)/O'po - 1)] 1/2 +Ypo]

where V is the sample volume and D is the demagnetizing factor appropriate 4 to the field configuration. Very recently, R.A. Klemm ~ has shown that a general ellipsoid with a general permeability tensor in an arbitrarily directed uniform field still possesses uniform magnetization. His calculations can be used to find ¢ of such samples by numerical integration. Our results are useful for prediction and calibration of the response of a SQUID susceptometer. It is often desirable to predict or calibrate in terms of a standard sample. The relative flux ¢1/¢2 for two samples of equal radius but different shapes and susceptibilities is

Ypo = [(R/r) 2 (m 2 -- 1) -1 + 1] 1/2 e=---m

Oblate spheroid: f ( 1 / m , R/r) = (rn 2 - 1) -1/2

¢1/¢2 = f l X l ( 1

I(1 +Y~o) arctan y ~ -Yoo I Yoo = [(R/r)2 m2(m 2 -- 1 ) - ' - 1] 1/2 e= l/m

CRYOGENICS

. JULY

1981

(6b)

+D2x2)/f2x2(1 +DIX])

(9)

For specific material types, (9) simplifies. Some cases of special interest are these: two soft ferromagnets, where ¢1/¢2 = flD2/]'2 D1 ; two superconductors, in their Meissner state, where ¢1/¢2 = fl (4n - D2)/f2 (41r - D 1);

401

and materials with × ~ 1, where ¢1/¢2 = fl Xl/f2 )(.2. As a numerical example we consider sample 1 to be a cylinder with el= 10 and sample 2 to be a disk with e2 = 0.1. We set R/r = 2 for both samples and find by application of (3) and (6) that the respective ratios are 1000 for the ferromagnets, 3.6 for the superconductors and 25 Xl/X2 for the small × samples. When normalized by volume, each of these ratios is reduced by a factor 100. Thus, cylindrical samples are advantageous in ferromagnetic studies, as expected, and disks are best for a given volume of superconductor.

-~

\\\

~oo~\\\

0.1 -\ E

Caution must be exercised for the thin superconducting disk, so the internal field Ho/(1 -D/4zr) does not exceed the lower critical field. 7 In this respect, SQUID detection has a substantial advantage over conventional detection because adequate sensitivity with the SQUID can be obtained with quite low Ho, even in very small samples. Caution must also be exercised in applying a calibration procedure where flux signals in the normal small X state and superconducting state of the same sample are compared. In this case (9) becomes en/¢sc = (4rr - D) Xn, so the comparison is not straightforward unless the sample is a long cylinder oriented along the field so that D ~ 0.

-0

Our results are also useful for evaluating the sensitivity of a SQUID magnetometer or susceptometer. The flux eSQ actually coupled into a SQUID is only a fraction a, the flux transfer function s, of the flux ¢ we have been calculating. We will assume a typical value of a = 0.05 for our example below. For a spherical sample of volume V we thus have eSQ = 2rra V×Ho/R if X < 1. The smallest change (SX)min that can be detected is then determined by the flux noise 8¢SQ that results after a given choice of the other variables. Consistent with our earlier example, we take R/r = 2 for our spherical sample in a pickup loop of radius R = 0.5 cm. Then, using Ho = 100 G and ~ieSQ = 10-2¢o ~ 2 x 10 -9 G cm 2 , we find (6X)min = R~¢SQ/27rctVH o ~ 5 x 10 -1°.

R/r Fig. 3 The fill factor ¢/~m as a function of R/r for prolate (e > 1 ) and oblate (e < 1 ) spheroidal samples with their axes of revolution in the direction of the applied field. The net flux ~) is due to a sample of aspect ratio e = c/r placed in one arm of radius R of the gradiometer. The sample height along the field is 2c, its radius is • and q~m = 4~r2r2H0x (1 + DX )-1 is the magnetization flux in the sample

It is possible to do much better than this. 3 We have reduced 8¢SQ to the level of the intrinsic SQUID noise of approximately 3 x 10 -tt G cm 2 (Hz) -'/2 where H0 is restricted to fields lower than 10 G or to a low frequency excitation of about 1 G. Attempts by us and many others 2 to raise sensitivity by applying a larger dc Ho tend to result in a proportionate rise in 5esQ. Recent advances in SQUID susceptometry 3 are primarily due to measures that permit a substantial increase of rio without a proportionate noise increase. As for the dependence of the flux signal on the sample shape, size and its orientation within the pickup loop, the calculations presented here will give the desired information.

402

0.01

0.1

IO

IOO

References 1

2

3

4 5

6 7 8

SHE Corp, San Diego, CA; CTF Systems, Coquitlam, BC,

Canada Good, J.A., SQUID, Hahlbohm, H.D., Lubbig, H., Eds (w. de Gruyter, Berlin, 1977) 225; Cukauskas, E.J., Vincent, D.A., Deaver, B.S., Jr Rev $ci Instrum 45 (1974) 1 Dearer, B.S., Jr, Bucelot, JJ., Finley, JJ. Future trends in superconductive electronies,AIP Conference Proceedings 44 (1978) 58; Philo, J.S., Faitbank, W.M.,Rev Sci Instrum 48 (1977) 1529 Osbom, J~.PhysRev 67 (1945) 351;Stoner, E.C.PhilMag 36 7 (1945) 803 ; American Institute of Physics handbook, third edition (McGraw-Hill,New York, 1972) 5:247 Smythe, W.R. Static and dynamic electricity (McGraw-Hill, New York, 1950), chapter 5, 111 ; Hobson, E.W. Theory of spherical and ellipsoidal harmonics, Cambridge Univ Press, London, (1955) Kiemm, R.A., JLow Temp Phys 39 (1980) 589 Tinkham, M., Introduction to superconductivity (McGrawHill, New York, 1975) chapter 3-4, 88 Lounasmaa,O.V. Experimental principles and methods below 1 K, Academic Press, New York, (1974)

CRYOGENICS.

JULY

1981