Method for calculating the induced voltage in a vibrating sample magnetometerdetection coil systemXiaonong Xu, Aimin Sun, Xin Jin, Hongchang Fan, and Xixian Yao Citation: Review of Scientific Instruments 67, 3914 (1996); doi: 10.1063/1.1147292 View online: http://dx.doi.org/10.1063/1.1147292 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/67/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A high sensitivity, wide dynamic range magnetometer designed on a xylophone resonator Appl. Phys. Lett. 69, 2755 (1996); 10.1063/1.117665 Shock wave and detonation wave response of selected HMX based research explosives with HTPB bindersystems AIP Conf. Proc. 309, 1413 (1994); 10.1063/1.46245 Computeraided analysis of precision oscillating systems with wideband fluctuations in circuit parameters AIP Conf. Proc. 285, 501 (1993); 10.1063/1.44646 Design of a detection coil system for a biaxial vibrating sample magnetometer and some applications Rev. Sci. Instrum. 64, 1918 (1993); 10.1063/1.1143977 An ultrasensitive fiberoptic magnetometer AIP Conf. Proc. 146, 676 (1986); 10.1063/1.35824

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Method for calculating the induced voltage in a vibrating samplemagnetometer detection coil system

Xiaonong XuPhysics Department and Center of Advanced Studies in Science and Technology of Microstructures,Nanjing University, Nanjing 210093, People’s Republic of China

Aimin SunPhysics Department, Nanjing University, Nanjing 210093, People’s Republic of China,and Physics Department, Northwest Normal University, Lanzhou 730070, People’s Republic of China

Xin Jin, Hongchang Fan, and Xixian YaoPhysics Department and Center of Advanced Studies in Science and Technology of Microstructures,Nanjing University, Nanjing 210093, People’s Republic of China

~Received 2 January 1996; accepted for publication 10 July 1996!

A numerical method for obtaining the vibrating sample magnetometer~VSM! sensitivity functionsdue to a uniformly magnetized sample in the shape of ellipsoid, cuboid or cylinder, and thencalculating the induced voltage in a VSM detection coil systems is presented. The induced voltage,which depends on the equilibrium position and amplitude of the sample in three-dimensional space,and depends on the geometric parameters of both sample and detection coils, is calculated from theviewpoint of magnetic charges. This numerical method is more accurate than others, which usedipole moment approximation without taking into consideration the sample shape and size. By usingthis method, the induced voltage, for which the shape and size effects of the sample must be takeninto account, can be calculated if there is knowledge of geometric parameters of the sample andcoils. The formulas for calculating the correction coefficient are also given for samples whose shapeerrors should be considered. Our results provide rigorous theoretical guidance for the exhaustivestudy of the VSM working principle, for accurate measurement of the sample magnetic moment,and for optimum design of configuration, position, and orientation of VSM pickup coils. ©1996American Institute of Physics.@S0034-6748~96!01410-4#

I. INTRODUCTION

A vibrating sample magnetometer~VSM! is a powerfulinstrument widely used in magnetic measurements. An opti-mized VSM detection coil system should provide maximumsignal, and induced voltage should be insensitive to the po-sition of the sample. Therefore, the induced voltage in thecoil system is the most important parameter for judging thequality of the detection coil system. A method to investigatethe correlation between the induced voltage and the time-varying magnetic field set up by the magnetic sample, andthen to calculate the induced voltage in the coil system isneeded, in order to optimize the design of detection coil sys-tem.

Since VSM was first reported by Foner in 1959,1 manypapers on VSM magnetometry have been published. How-ever, among them very few were dedicated to the calculationof induced voltage of a real coil system. Bragg and Seehra2

calculated the induced voltage of a detection system consist-ing of single-turn circular coils, but they made too manysimplifications. Pacyna and Ruebenbauer3,4 calculated the in-duced voltage of a spatially extended multiturn coil by re-ducing the problem of calculating a vectorial geometric fac-tor, which characterizes the dependence of the induced signalupon the geometry of the detection coil system. Althoughthey made a lot of physical approximations, their calculatingformulas were still very complicated. References 2, 3, and 4,used a point magnetic moment approximation,5 but a com-

parison between theoretical and experimental results was notreported. Their calculating methods cannot be used to calcu-late the induced voltage in a real detection system consistingof several multilayer coils, to say nothing of studying theeffects of shape and size of samples on the measured signalvalues.

Since a sample has a finite volume, if the distance be-tween the sample and pickup coils are not much larger thanthe dimensions of the sample, the point magnetic momentmodel may introduce pronounced error. Guy and Howarth,6

Zieba and Foner,7 and U. Asserlechneret al.8 considered theshape effect. Although sophisticated knowledge such asmagnetic multipoles and vector spherical harmonics were in-volved, they were not a method to calculate the induced volt-age in the real detection coil system. So far, none of thepublished papers give formulas to calculate the induced volt-age, in which the shape and position of both sample and coilsof a pickup system are considered.

In electromagnetism there are two methods for calculat-ing the magnetic field set up by a magnetic moment: one isbased on the molecular current viewpoint, the other is basedon the magnetic charge viewpoint.9,10 Identical results can beobtained by using these two methods. As a calculatingmethod, the magnetic charge viewpoint is much more trans-parent than the molecular current viewpoint, since magneticcharges are analogous to electric charges, many formulas inelectrostatics can be taken over, although there are no ‘‘mag-netic charges’’ in the physical world. In this article, we as-sume an isotropic and homogeneous magnetic sample being

3914 Rev. Sci. Instrum. 67 (11), November 1996 0034-6748/96/67(11)/3914/10/$10.00 © 1996 American Institute of Physics This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

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uniformly magnetized. This assumption is equivalent to thecondition that the sample is in the shape of an ellipsoid, orthe sample is of any other shape but in magnetic saturation.In fact, for nonellipsoidal samples with small magnetic sus-ceptibility, their magnetization is uniform to a high degree.The ferromagnetic and ferrimagnetic sample may be magne-tized uniformly in a sufficiently high field.7 We adopt themagnetic charge viewpoint to obtain the vector sensitivityfunctions of the sample, then to calculate the induced voltagein the VSM detection coil system. This method relates theinduced voltage to the shape, size, position, and vibratingamplitude of the sample, and to the geometric parametersand positions of the pickup coils. Therefore with our method,some new information about the induced voltage in the VSMdetection system is acquired. The formulas for calculatingthe correction coefficient to eliminate shape and size errorsare reported.

II. DERIVATION OF SENSITIVITY FUNCTIONINVOLVING SHAPE AND SIZE EFFECTS

The principle of VSM is based on Faraday’s law, ac-cording to which a voltage is induced through a detectioncoil by a time-varying magnetic flux. To define the sensitiv-ity function precisely, consider Fig. 1, which shows an arbi-trary ellipsoidal sample immersed in a uniform applied mag-netic field HW . The Cartesian coordinate systemO82X8Y8Z8 vibrates with the sample, and its originO8 islocated at the center of the sample, the sample is symmetricalabout its own center.O2XYZ is a static Cartesian system,andO is the symmetry center of the coil system. The equi-librium position of the vibrating sample inO-XYZsystem is(x0 ,y0 ,z0). Suppose a single turn pickup coil centers on anarbitrary pointP(x,y,z). Let i, j , k be unit vectors alongX,Y,Z axes, respectively;n5cosa i1cosbj1cosg k, theaxial unit vector of the coil;a, b, andg, direction angles ofn; and v the displacement vector of the vibrating sample.There are two vibrating modes of samples, one isv'Hini jcorresponding to coils mounted on the poles of a conven-tional iron cored magnet as shown in Fig. 2~a!, the other isviHinik, as shown in Fig. 2~b!, corresponding to coils in-side a superconducting solenoid. These two types of arrange-ment are the most commonly used. Usually a sample is in theshape of a cuboid, cylinder, or an ellipsoid. In this article, the

calculating formulas for the voltage induced in the coil sys-tem by an ellipsoidal sample are discussed in detail for thecase shown in Fig. 2~a!. For the case shown in Fig. 2~b!, onlyresulting formulas are presented.

The sensitivity function indicates the ability of a magne-tized sample to produce an induced voltage in a detectioncoil, it shows the spatial distribution of the ratio of the in-duced voltage in a single turn infinitesimal coil to its area. Inthe case of Fig. 2~a!, the magnetizationM , magnetic momentm, and volume of the sample areM j , mj , andVM , respec-tively, m5M•VM . If x05y05z050, and the samplevibrates with an amplitude A as OO85Ztk5A sin(vt)k, the vector sensitivity functionG(x,y,z)5 Gi(x,y,z) i1Gj (x,y,z) j1Gk(x,y,z)k is only dependenton the vibrating sample itself, and is independent of the size,axial direction, and other geometric parameters of a detectioncoil. A voltage is induced in every turn of a multilayer coilwith turn numberN, the induced voltage in the entire coil is

V5(i51

N ESi

G•ndS5(i51

N ESi

~Gi cosa1Gj cosb

1Gk cosg!dS, ~1!

whereSi is the area enclosed by theith turn.Using the point magnetic dipole approximation,

Bowden11 studied the gradient function¹@H(r )#, and intro-duced the sensitivity function. For the situation shown in Fig.2~a!, he found

5Gi0~x,y,z!5C0

5xyz

~Ax21y21z2!7

Gj0~x,y,z!5C0

z~2x214y22z2!

~Ax21y21z2!7

Gk0~x,y,z!5C0

y~2x22y214z2!

~Ax21y21z2!7

, ~2!

where a constantC0 } (m•A) but the author did not furtherstudy the shape effect and the induced voltage on the basis ofthese equations. We think that if the shape and size effectsare considered, for a uniformly magnetized sample,Gi , j ,k8 (x,y,z) 5 C08***V8Gi , j ,k

0 (x 2 x8, y 2 y8, z2z8)dx83 dy8dz8, with C08 a constant, anddV85dx8dy8dz8 an el-

FIG. 1. The sample centered atO8 with the definite shape and a detectioncoil centered atP in CartesiansO-XYZandO82X8Y8Z8. The coordinatesof O8 in O-XYZ system are (x0 ,y0 ,z01zt), zt5A sinvt.

FIG. 2. Two types of detection coil system in common use.~a! v'Hi j , ~b!viHik.

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ement of volume of the sample. This expression is a tripleintegral. It seems very difficult to be simplified, even if thesamples, such as an ellipsoid, a cuboid and a cylinder, havecentral symmetry.

From the viewpoint of magnetic charges, we can takeadvantage of the symmetry of the sample in the derivation ofsensitivity functionGi , j ,k(x,y,z). Thus the triple integral isreduced to a double integral. Consider Fig. 2~a!, wherev'Hini j . For a uniformly magnetized sample, magneticcharges are induced only on the surface of the sample asshown in Fig. 1. We assumen8 is the unit normal vector tothe surface at P8, which is expressed asn85 cosa8i1cosb8j1cosg8k, wherea8, b8, g8 are directionangles of n8. Surface magnetic charge density iss85m0M•n85m0M cosb8, and the magnetic charge onsurface elementdS8 is ~in SI units!

dQm8 5s8dS85m0M ~dS8 cosb8!5m0Mdx8 dz8.

We use 1dQm8 and 2dQm8 to designate the magneticcharges on surface elements centered atP8(x8,y8,z8)andP8(x8,2y8,z8), respectively. PointP(x,y,z) is at a dis-tance r6 from 6dQm8 where ur6u5uOP2OO82O8P8u,OP5xi6yj1zk, OO85x0i1y0j1(z01zt)k, and O8P85x8i6y8j1z8k. Whenx05y05z050, the static magneticpotential produced by6dQm8 at P is

dU6561

4pm0

dQm8

r6, $r65@~x2x8!21~y7y8!2

1~z2z82Zt!2#

12%.

From the above two equations, the magnetic field set up bythe dipole is given bydH52,(dU11dU2)5 dHi i1 dHj j1dHkk, (,5(]/]x) i1(]/]y) j1]/]zk). Therefore,the magnetic field set up by the entire sample can be ex-pressed as

Hi5M

4pE E ~x2x8!S 1r13 2

1

r23 Ddx8 dz8, ~3!

Hj5M

4pE E S y2y8

r13 2

y1y8

r23 Ddx8 dz8, ~4!

Hk5M

4pE E ~z2z82Zt!S 1r13 2

1

r23 Ddx8 dz8. ~5!

In what follows we shall discussGj (x,y,z) only, whileGi(x,y,z) andGk(x,y,z) can be obtained in the same man-ner. The integrand in Eq.~4! is

I j~x,y,z,x8,y8,z8,t !5y2y8

r13 ~ t !

2y1y8

r23 ~ t !

, ~6!

which is a periodic function, and thus can be decomposedinto a Fourier series:

I j~ t !5bj ,01 (K51

`

@bj ,K cos~Kvt !1aj ,K sin~Kvt !#,

~7!

where

bj ,05v

2pE2p/v

p/v

I j~ t !dt,

bj ,K5v

pE2p/v

p/v

I j~ t !cos~Kvt !dt,

aj ,K5v

pE2p/v

p/v

I j~ t !sin~Kvt !dt.

By calculating bj ,K , aj ,K in the whole space round thesample, we know that,aj ,2 ,aj ,4 , andaj ,K.0(K>6), bj ,1,bj ,3 , and bj ,K.0(K>5), and ubj ,0u@uaj ,1u@ubj ,2u@uaj ,3u@ubj ,4u@uaj ,5u. For instance, we consider the magnetic fieldat P~0, 0.015 m, 0.008 m!, which is set up by a pair ofmagnetic charges7dQ centered atP8(0,60.005 m, 0! andvibrating with the amplitudeA50.001 m, then fromthe above equation we getI j (t)5bj ,01aj ,1 sin(vt)1 bj ,3 cos(2vt)1aj,3 sin(3vt)1bj,4(4vt)1aj,5sin(5vt)1d(t),with bj ,052781.9294, aj ,15592.8353, bj ,25221.5108,aj ,350.1552,bj ,4520.0518,aj ,5520.0025, and Maxud~t!u,0.0013. Each harmonic will produce an induced voltage inthe pickup coils, and the induced voltage must be detected bya lock-in amplifier. The resonant frequency of the passbandfilter of the lock-in amplifier is set atv/(2p), the vibratingfrequency of the sample, which is usually round 1.5 times or2.5 times of the frequency of electric power, so that high-order harmonics of induced voltage cannot pass through thefilter. Therefore only the fundamentalaj ,1 sin(vt) of I j (t) iseffective, other items can be neglected

Hj~ t !5M

4pE E aj ,1~x,y,z,x8,y8,z8!sin~vt !dx8 dy8.

For a single coil, if its axial direction isni j , the inducedvoltage is

«~ t !52m0

]~**HjdS!

]t

5F ESS 2

m0Mv

4p Ex8Ez8aj ,1dx8 dz8D dSGcos~vt !,

and the measured induced voltage is the effective value of«(t), that is

V51

A2 F ESS 2

m0Mv

4p Ex8Ez8aj ,1dx8dz8D dSG

5ESGj~x,y,z!dS,

Gj~x,y,z!5CE E @2aj ,1~x,y,z,x8,y8,z8!#dx8 dz8

5CE E F jdx8 dz8, ~8!

F j@x,y,z,x8,y8~x8,z8!,z8#52aj ,1@x,y,z,x8,y8~x8,z8!,z8#,~9!

whereC5A2m0Mv/(8p).Now we study the relation betweenaj ,1 and I j .

By Eq. ~7!, we obtain( l50` aj ,2l11(21)l5 @ I j (t)u t5p/(2v)

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2I j(t)ut52p/(2v)#/25aj ,12aj ,31aj ,5 , since aj ,K50when K>4, and (K51

` bj ,K cos(Kvt)uvt5(p/2)

2(K51` bj ,K cos(Kvt)uvt52(p/2)50. Also the relations of

aj ,1 , aj ,3 , uaj ,5u, aj ,3 /aj ,1 , uaj ,5 /aj ,1u to the amplitudeA arecalculated and shown in Fig. 3 forP(0,0.015 m, 0.008 m!and P8(0,60.005 m,0!. It shows that if A<3 mm,uaj ,5 /aj ,1u<231024, aj ,5 can be neglected, ifA<1 mm,thenaj ,3 /aj ,1<231024, aj ,3 can also be neglected, thus un-der conditions of small amplitude,aj ,1 can be expressed as

aj ,151

2~ I j~ t !u t5p/~2v!2I j~ t !u t52p/~2v!!. ~10!

By Eqs. ~6! and ~10! we can obtain the expression ofaj ,1 ,then substitutingaj1 into Eqs.~9! and ~8! yieldsGj (x,y,z).Gi ,k(x,y,z) can be received in the same manner.

Gi , j ,k~x,y,z!

5CE2 f X8

f X8 E2 f Z8

f Z8Fi , j ,k@x,y,z,x8,y8~x8,z8!,z8#dx8 dz8,

~11!

where

Fi5~x2x8!$@~x2x8!21~y2y8!21~z2z81A!2#23/2

2@~x2x8!21~y1y8!21~z2z81A8!2#23/2

2@~x2x8!21~y2y8!21~z2z82A8!2#23/2

1@~x2x8!21~y1y8!21~z2z82A8!2#23/2%, ~12!

F j5~y2y8!$@~x2x8!21~y2y8!21~z2z81A!2#23/22@~x

2x8!21~y2y8!21~z2z82A!2#23/2%2~y1y8!$@~x

2x8!21~y1y8!21~z2z81A!2#23/22@~x2x8!2

1~y1y8!21~z2z82A8!2#23/2%, ~13!

Fk5~z2z81A!$@~x2x8!21~y2y8!21~z2z81A!2#23/2

2@~x2x8!21~y1y8!21~z2z81A!2#23/2%

1~z2z82A!$@~x2x8!21~y1y8!21~z2z82A!2#23/2

2@~x2x8!21~y2y8!21~z2z82A!2#23/2%. ~14!

For an ellipsoid with the axial lengthsa,b, andc, andai i,bi j , and cik, respectively, by making use of the surfaceequation of the ellipsoid, we get the expressions ofy8(x8,z8), f Z8 and f X8 in O82X8Y8Z8 system, which occurin the equations above

H y8~x8,z8! 5bA12~x8/a!22~z8/c!2

f Z8 5cA12~x8/a!2

f X8 5a

. ~15!

For a cuboid with three edges 2a,2b,and 2c, 2ai i, 2bi j ,and 2cik, respectively

H y8~x8,z8! 5b

f Z8 5c

f X8 5a

. ~16!

For a cylinder of length 2c and radiusa with its axial direc-tion parallel to theZ axis

H y8~x8,z8! 5Aa22x82

f Z8 5c

f X8 5a

. ~17!

Equation~11! possesses the same symmetric and asymmetricrelations asGi , j ,k

0 (x,y,z) in Eq. ~2!. Taking the advantage ofthe symmetry and asymmetry properties of Eqs.~2! or ~11!,we can obtain the values of the sensitivity function in wholespace only by calculatingGi , j ,k(x,y,z) in the range(x,y,z>0).

In the case shown in Fig. 2~b!, using the point magneticdipole approximation and Bowden’s method,11 we can cal-culate (]Hki /]z) and get

Gki0 5C0i

z~3x213y222z2!

~Ax21y21z2!7, ~18!

where a constantC0i } (m•A) . If the shape, size and vibrat-ing amplitude of the sample are considered, by steps similarto Eqs. ~3!–~9!, the sensitivity functionGki5Gkik is ob-tained as

Gki~x,y,z!

5CE2 f X8

f X8 E2 f Y8

f Y8Fki@x,y,z,x8,y8~x8, z8!,z8,A#dx8 dy8,

~19!

Fki5~z2z81A!@~x2x8!21~y2y8!21~z2z81A!2#23/2

2~z2z82A!@~x2x8!21~y2y8!21~z2z82A!2#23/2

2~z1z81A!@~x2x8!21~y2y8!21~z1z81A!2#23/2

1~z1z82A!@~x2x8!21~y2y8!2

1~z1z82A!2#23/2. ~20!

For an ellipsoid,

FIG. 3. Relation ofaj ,1 , aj ,3 , uaj ,5u, aj ,3 /aj ,1 , uaj ,5 /aj ,1u vs A.

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H z8~x8,y8! 5cA12~x8/a!22~y8/b!2

f Y8 5bA12~x8/a!2

f X8 5a

. ~21!

For a cuboid,

H z8~x8,y8! 5c

f Y8 5b

f X8 5a

. ~22!

For a cylinder,

H z8~x8,y8! 5c

f Y8 5Aa22x2

f X8 5a

. ~23!

The definitions ofa,b, andc in Eqs.~21!–~23! are the sameas in Eqs.~15!–~17!.

III. SPACE DISTRIBUTION OF SENSITIVITYFUNCTION Gi,j ,k (x ,y ,z)

Equation~11! indicates that the induced voltage of thepickup coils is directly proportional to the angular frequencyv. Figures 4, 5, and 6 show the distribution functionGi(x,y,z)/C on the x515 mm plane,Gj (x,y,z)/C andGk(x,y,z)/C on y515 mm plane, respectively,Gi , j ,k(x,y,z)/C are dimensionless functions. The sample is asphere or an ellipsoid with axial lengthsa 5 b 5 c 5 A 5 1mm. From Eqs.~12!, ~13!, and ~14!, it is easy to findsymmetric and asymmetric relations,Gi(2x,y,z)52 Gi(x,y,z), Gj (x,2y,z)5Gj (x,y,z) andGk(x,2y,z)5 Gk(x,y,z), thus beside the sample,x5615 mm are twoasymmetric planes forGi , y5615 mm are two asymmetricones forGj and symmetric ones forGk , so there are eightpeaks forGi , four for Gj , and two for Gk in three-dimensional space. The positions of peaks in these figuresindicate the coordinates of points where maximum signal, ormaximum ratio of signal to noise can be obtained. The num-ber of peaks determines the number of detection coils neededwhen the axial direction of coilni i,j or k, respectively. The

total voltage output of a detection system is the superpositionof signals of individual coils, we should connect in series thecoils located at points where the peaks have the same sign,and use the series opposition to connect the coils at pointswhere the peaks have opposite signs. For example, when theunit normal vector of coil surface ni i, G5Gi i,Gi(6x,y,z)/C have four positive peaks and four negativeones located on planes6x5constant. Thus there are eightpickup coils in the pickup system, among them four coilssituated in positions of positive peaks should be connected inseries, another four coils situated in positions of negativepeaks should also be connected in series, and then the seriesopposition connection is adopted for these two sets of coils.In this way we can obtain the largest system voltage which iseight times of the induced voltage in a single pickup coil.

In order to get the greatest absolute sensitivity, i.e., themaximum induced voltage, we usually increase the numberof turns of a coil. In this case the size of the coil will increaseif the diameter of enameled wire, with which the coil iswound, remains unchanged. In three-dimensional space, thecontour surface forGi50 consists of three planesx50,

FIG. 4. The curved surfaceGi(x,y,z)/C for fixed x 5 15 mm. FIG. 5. The curved surfaceGj (x,y,z)/C for fixed y 5 15 mm.

FIG. 6. The curved surfaceGk(x,y,z)/C for fixed y 5 15 mm.

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y50, andz50, and those forGj ,k50 are curved surfaces.On both sides of these planes and curved surfaces, the valuesof Gi , j ,k change signs, therefore if the pickup coils extendcross these planes and curved surfaces whenni i, ni j , andnik, respectively, positive and negative parts of the inducedvoltage in a pickup coil would cancel each other out.

In experiments the pickup coils are usually oriented insuch a way thatni j as shown in Fig. 2~a!, so we studied therelation between the peak valueuGj (x,y,z)uM /C and peakcoordinates (XM ,YM ,ZM), where uGj uM is the absolutevalue of the peak ofGj (x,y,z) for fixedYM value. From Fig.5 we know the positions of peaks are located atx 5 0 plane,hence XM50. Figure 7 shows the peak position ofuGj uM /C on the plane (x50,y > 0,z<0) when a differentYM value is given,ZM increases with the increasing ofYM . The angle between theY axis and the asymptotic line ofthe curveZM(YM) is about 21.2°.Gi , j ,k decreases rapidlywith the increase of the distance between the sample andP(x,y,z). There is an approximate expression for the peakvalues shown in Fig. 7, uGj uM /C5318.3YM

23.96 for(XM50,YM . 5 mm!. For a densely wound circularmultilayer coil with given geometric parametersW, R2 ,R1 , the total turn number of coilsN } r c

22 , r c is the radius ofenameled wire, the resistanceRm } N/(pr c

2) } r c24 so the ra-

tio of signal to thermal noiseK } N/ARm is a constant, whichonly depends on the sensitivity function. If a very smallpickup coil is centered at a point, whereuGj u has a maximumvalue anddGj /dx5dGj /dz50, then maximum ratio of sig-nal to thermal noise is obtained. Yet to optimize a detectioncoil system one should consider two other factors, that is,maximum induced voltage, and insensitivity of induced volt-age to perturbation due to sample displacement.

Now by Eq. ~11! we study effects of shape and size ofthe sample on the sensitivity functions. We designate theratio of peak value to volumeuGj (XM ,YM ,ZM)uM /(CVM),where @XM ,YM ,ZM#5@0,15~mm!, ZM ~mm!# are the coor-dinates of the peak. When the axial lengths of an ellipsoidalsample satisfiesa 5 b 5 c, anda changes from 0.1 mm to 5mm, thenuGj uM /(CVM) retains a constant 1.08831024, andthe coordinates of the peak also maintain constant values@0,

15, 5.88#, which implies that the surrounding field of spheri-cal samples is exactly alike. Whena 5 c 5 1.5 mm andbvaries from 0.1 to 5 mm,ZM changes from 5.91 to 5.23 mmwhile uGj uM /(CVM) changes from 1.05431024 to1.30531024. Whenb 5 c 5 1.5 mm and a varies from 0.1 to5 mm, ZM increases from 5.88 to 5.93 mm whileuGj uM /(CVM) decreases from 1.08131024 to1.00831024. Whena5b 51.5 mm andc varies from 0.1 to5 mm, ZM changes from 5.83 to 6.93 mm whileuGj uM /(CVM) changes from 0.96231024 to 1.08531024.These results indicate that the near-field due to an ellipsoidalsample deviates from that of a dipole, unlessa 5 b 5 c.Theshape and size errors may produce an influence on the mea-sured value of induced voltage, but the induced voltage doesnot approximate touGj uM /(CVM) since it depends on theaverage value of sensitivity function in the area encircled bypickup coils.

IV. CALCULATION OF INDUCED VOLTAGE IN ASINGLE MULTILAYER COIL

In the case of Fig. 2~a!, ni j , cosa5cosg50, cosb51,cosb dS5dx dz, Eq. ~1! can be simplified asV5( i51

N **SiGj (x,y,z)dx dz. A multilayer coil with N

turns can be thought of as superposition ofNl single layercoils, which have the same widthW, the same density ofturnsNc , but different radiusR(I ). It is obvious that the turnnumber of each layer is (Nc•W), and N5(NcW)Nl ,Nc5(2r c)

21, r c'0.05 mm, andr c!W, ( i51Nc•W can be re-

placed by an integrationNc*dy. Thus for each single layer,the induced voltage

VI5 (i51

Nc•W E E Gj~x,y,z!dx dz5NcE S E E Gjdx dzDdy5NcE E E

V

Gjdx dy dz,

whereV is the volume enclosed by the single layer coil. Fora densely wound multilayer coil, the distance between adja-cent layers isA3r , the total number of layers is the round-offnumber of (R22R1)/(A3r ), which is indicated as

Nl5R0S R22R1

A3r D . ~24!

The radius of thei th layer is

R~ I !5R11I •A3r , ~ I51,2,••••••,Nl !. ~25!

The total induced voltage is the sum of the contributions dueto individual layers,V5( I51

Nl VI , so the induced voltage in asingle multilayer coil shown in Fig. 2~a! is

V5Nc(I51

Nl E2R~ I !

R~ I ! Et

t1WEd1R22AR22x2

d1R21AR22x2Gj~x,y,z!dx dy dz.

~26!

The induced voltage of a multilayer coil can be calculated byEq. ~26!, whereNl andR(I ) are given by Eqs.~24! and~25!,respectively.

FIG. 7. The curved surfaceGj (x,y,z)/C and the positions ofuGj uM /C for(x50,y>0,z<0) plane.

3919Rev. Sci. Instrum., Vol. 67, No. 11, November 1996 Vibrating sample magnetometer This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

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V. CALCULATION OF INDUCED VOLTAGE IN ADETECTION SYSTEM CONSISTING OF MULTILAYERCOILS

For a detection system consisting of four multilayer coilsas shown in Fig. 2~a!, if the sample is at the center of thedetection system, the induced voltage of the system is simplyfour times of that in one coil. When the equilibrium positionof the sample is not at the origin of the coordinate system,but rather at point (x0 ,y0 ,z0), the induced voltages in thesecoils are different, so we need to calculate them separately.However, we can utilize symmetry to simplify the calcula-tion. A point (x0 ,y0 ,z0) has eight symmetry points includingitself in three-dimensional space, the same measured valueshould be received when the sample is centered at these sym-

metry points. Therefore ifF j@x,y,z,x8,y8(x8,z8),z8#, the in-tegrand of Gj , given by Eq. ~13! is substituted byF j (x,y,z,x0 ,y0 ,z0 ,x8,y8(x8,z8),z8),

F j51

8(l51

2

(m51

2

(n51

2

F j@x1~21! lx0 ,y1~21!my0 ,z

1~21!nz0 ,x8,y8~x8,z8!,z8#, ~27!

we can get the induced voltage in the detection coil systemby simply calculating that in one of the coils. Ifx05y05z050, F j5F j . Suppose the distances between ad-jacent coils are 2dand 2t as shown in Fig. 2~a!, the inducedvoltage of the detection system is received from Eq.~26! as

V~x0 ,y0 ,z0!5432NcC(I51

Nl E0

R~ I !Et

t1wEd1R22AR22x2

d1R21AR22x2E2 f X8

f X8 E2 f Z8

f Z8F jdx dy dz dx8 dz8, ~28!

where f X8 and f Z8 are given by Eqs.~15!–~17! for sampleswith different shape and size. The numerical factor 4 is in-troduced because of four detection coils in the pickupsystem, and factor 2 is used due to reduced integratingrange, since F j (2x,y,z)5F j (x,y,z), *2R(I )

R(I )F jdx

52*0R(I )

F j dx, exactly. Figures 8, 9, and 10 show the cal-culatedV(x0,0,0), V(0,y0,0), andV(0,0,z0) by Eq. ~28! forvarious x0 ,y0 ,z0 , respectively. Figure 8 indicates thatV(x0,0,0) decreases monotonically with the increase in2dor ux0u. Figure 9 indicates that in the range26 mm, y0,6mm, and when 2d'16 mm,V(0,y0,0) has a plateauon which V(0,y0,0) remains almost unchanged. Figure 10shows a plateau ofV(0,0,z0) located in 2d'12 mm and25 mm , z0,5 mm. The absolute value of the inducedvoltage in the detection system should be large, and thechange in the induced voltage should be small when the

equilibrium position of the sample deviates from the originof theO-XYZ system. It seems that 2d'12 mm may be thebest distance as far as this is concerned.

Sometimes, when calculating the induced voltage of adetection coil system consisting of several multilayer coils,and searching the optimum position of each coil, the shapeand volume effects of a sample can be neglected. In thiscase, the sensitivity functionGj (x,y,z,x8,y8,z8) can be ap-proximately substituted byGj

0(x,y,z) given by Eq.~2!. Inthe situation of Fig. 2~a!, we obtain

V~x0 ,y0 ,z0!

5432Nc(I51

Nl E0

R~ I !Et

t1wEd1R22AR22x2

d1R21AR22x2

3G j0 dx dy dz. ~29!

FIG. 8. The calculated induced voltageV(x0,0,0) in a system vs 2d andx0 . R2510 mm, R151.5 mm, r c50.05 mm,W58 mm, t516 mm,a5b5c5A51 mm.

FIG. 9. The calculated induced voltageV(0,y0,0) in a system vs 2dandy0 .

3920 Rev. Sci. Instrum., Vol. 67, No. 11, November 1996 Vibrating sample magnetometer This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

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The relationG j0(x,y,z,x0 ,y0 ,z0) with Gj

0(x,y,z) in Eq. ~2!has an expression similar to Eq.~27!. Figure 11 shows thecomparison of theoretical and experimental relative inducedvoltages; the sample used is a saturated magnetized Nisphere with a diameter of 1.5 mm. The figure shows that thecalculated results are in good agreement with experimentalones, and the accuracy of Eq.~28! is a little better than thatof Eq. ~29! as compared with the experimental results.

Likewise, in the situation shown in Fig. 2~b!, we can getthe formula for calculating induced voltage in the detectionsystem

Vi~x0 ,y0 ,z0!

5234NcC(I51

Nl Et

t1wE0

R~ I !E0

AR22x2E2 f X8

f X8 E2 f Y8

f Y8

3F ki dz dx dy dx8 dy8, ~30!

whereNl , R(I ) are given by Eqs.~24! and ~25!, f X8 andf Y8 are given by Eqs.~21!–~23! for samples with a differentshape and size, two numerical factors, 2 and 4, come fromtwo detection coils in the pickup system and half-integratingrange for * dx and * dy, respectively.F ki@x,y,z,x0 ,y0 ,z0 ,x8,y8(x8,z8),z8] has an expression similar to Eq.~27!with the exception thatF j is substituted byFki given by Eq.~20!. In calculation, the condition that the distancet . z01A1c should be satisfied, wherec is the half-length ofsample ink direction, otherwiseF ki may be infinitely largeat some place in the integrating region. Calculating results ofEq. ~30! indicates that the optimum distance between twodetection coils is about 10–11 mm for detection system pa-rameterR2514 mm, R157.5 mm, r c50.05 mm,W512mm, anda5b5c5A51 mm because at this distance, forsampleposition(x05 0,y05 0,2 5.5 mm, z0,15.5mm!or(z050,Ax021y0

2 , 4 mm!, the induced voltage almost re-mains constant. When the shape effect of a sample can beignored, Eq.~30! also can be reduced to a simple form simi-lar to Eq.~29!.

VI. SHAPE AND SIZE EFFECTS

Now we study the effects which the shape and size of asample exert on the induced voltage. Figure 12 shows theinduced voltageV(0,0,0) under the situation shown in Fig.2~a!, which is calculated using Eq.~28!. This figure indicatesthat the ratio of induced voltage to sample volumeV/(CVM) is not a constant for ellipsoidal samples other thanspheres, and that the errors caused by shape and size alsodepend on distance 2d between the coils. Figure 12 showsthat when 2d 5 0, the induced voltage has both large valueand large shape and size errors, especially an ellipsoidalsample with a longer axisbi j has the largest shape and sizeeffects as shown by the curve 2 of the figure. When 2d5 16 mm, curve 2 coincides with curve 1, the shape and sizeerrors atj direction are eliminated to a great extent; when2d 5 12 mm, curve 4 and curve 1 nearly coincide with eachother, the shape and size error atk direction are greatly de-creased. These results are consistent with the plateaus inFigs. 9 and 10: aV(0,y0,0) plateau round 2d'16 mm and aV(0,0,z0) plateau round 2d'12 mm but shape and size er-rors in i direction cannot be eliminated completely by adjust-ing distance 2d, because there is no plateau ofV(x0,0,0) asshown in Fig. 8. The shape and size effects become moreimportant with decreasing distance between sample andpickup coils. If a difference exists in the shape and size of

FIG. 10. The calculated induced voltageV(0,0,z0) in a system vs 2d andz0 .

FIG. 11. The relative induced voltagesV(0,0,z0)/V(0,0,0) calculated fromEq. ~28! ~solid line! and Eq. ~29! ~circle!, and measured experimentally~square! vs z0 , the spherical sample diameterr 5 0.75 mm, 2d55 mm,r c50.03 mm,R259 mm,R152.5 mm,W 5 4.5 mm.

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both calibration and measured samples, the measurement er-ror is introduced. Using calibration and measured sampleswith identical geometry is the best method to avoid shapeand size errors. If this error cannot be ignored, the measuredmagnetizationM 8 should multiply a correcting coefficientQ, which can be calculated from the shape and size of both

calibration and measured samples, together with the param-eters of the pickup system. LetVC be the volume of thecalibration sample andVM be that of the measured one. If asample is centered at (x0 ,y0 ,z0)5(0,0,0), then by Eq.~27!F j@x ,y ,z,x0 ,y0 ,z0 ,x8, y8(x8, z8), z8# 5 F j@x, y, z, x8,y8(x8,z8),z8], and using Eq.~28! Q can be expressed as

Q5VM

VC

S I51Nl *0

R~ I !* tt1w*

d1R21AR22x2d1R22AR22x2

*2 f

X8

f X8f c

*2 f

Z8c

f Z8c

F j dx dy dz dx8 dz8

S I51Nl *0

R~ I !* tt1w*

d1R22AR22x2d1R21AR22x2

*2 f

X8m

fX8m

*2 f

Z8m

fZ8m

F jdx dy dz dx8 dz8, ~31!

whereNl , R(I ) are given by Eqs.~24! and ~25!, f X8c and

f Y8m are given by Eqs.~15!–~17!, respectively, depending onthe shape and size of the calibration and measured samples.The revised measured result is

M5M 8•Q. ~32!

Similar results also can be received in the situation ofv'Hi j , while ni i or nik, or in other situations as Fig. 2~b!.

VII. AMPLITUDE EFFECT

Under the situation shown in Fig. 2~a!, using Eq.~26! wecan study effects of the amplitude on the induced voltage.The relation between the amplitudeA of the sample with the

sensitivity functionsGj is given by Eq.~13!. For a sphericalsample of diameter 1 mm, letuGj (A)uM /(CA) be the ratio ofthe peak value ofGj (A) to vibrating amplitude ony 5 15mm plane (XM ,YM ,ZM) be the coordinates of the peak po-sition of uGj (A)u. When amplitudeA changes from 0.01 mmto 2.5 mm, the coordinates (XM ,YM ,ZM) change from~0,15, 5.84 mm! to ~0, 15, 6.10 mm!, anduGj (A)uM /(CA) de-crease by 5.3%.uGj uM /(CA) almost remains unalteredwhenA<0.3 mm. The induced voltageV is a function ofamplitudeA, and distance 2d. We introduce a coefficient

j~A,2d!5V~A,2d!

A F limA→0

V~A,2d!

A G21

, ~33!

as an indication of linearity with whichV(A,2d) responds toA. If V(A,2d) } A exactly,V(A,2d)/A should be a constant,thusj(A,2d)51. Figure 13 showsj(A,2d) calculated by Eq.~28!, j varies from 0.97 to 1.03 whenA<3 mm. Errors dueto nonlinearity are much larger than those introduced by ne-glecting aj ,3 ,aj ,5 . For the detection system shown in Fig.2~a!, 2d'11 mm is the optimum distance for the induced

FIG. 12. The theoretical induced voltageV(0,0,0)/(CVM) vs axial lengthsof ellipsoidal samples with coil distance 2d as a parameter.~1!a5b5c51–5 mm ~circle!. ~2! a5c51.5 mm,b50.1–5 mm~triangle!.~3! b5c51.5 mm,a50.1–5 mm~square!. ~4! a5b51.5 mm,c50.1–5mm ~diamond!. Other coil parameters are the same as in the caption ofFig. 8.

FIG. 13. The linearity coefficientj(A,2d) vs sample amplitudeA with coildistance 2d as a parameter. Other coil parameters are the same as in thecaption of Fig. 8.

3922 Rev. Sci. Instrum., Vol. 67, No. 11, November 1996 Vibrating sample magnetometer This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitationnew.aip.org/termsconditions. Downloaded to IP:

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voltage to obtain the best linear response to amplitude. In-creasing vibrating amplitude may increase the induced volt-age, yet for precise measurement, nonlinearity effect of am-plitude should be considered.

VIII. DISCUSSION

By the method of calculating the sensitivity functionspresented in this article, we can theoretically predict the op-timum coil configuration and positions according to the spe-cific space available for the detection coil system of VSM.Nevertheless, if the body magnetic charge, which resultsfrom nonuniform magnetization of the sample exists thesample, additional induced voltage contributed by the bodymagnetic charge should be considered. In order to get opti-mum signal-to-noise ratio and the largest possible inducedvoltage which is not sensitive to the sample position, thestability of electric current providing the applied magneticfield, and the uniformity and symmetry between coils con-sisting of the detection system are also very important.

ACKNOWLEDGMENTS

Supported by the National Foundation for Doctoral Edu-cation and the National Center for Research and Develop-ment on Superconductivity, People’s Republic of China.

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