Hall

Shorting and Field Corrections in Hall Measurements W. F. Flanagan, P. A. Flinn, and B. L. Averbach Citation: Review of Scientific Instruments 25, 593 (1954); doi: 10.1063/1.1771138 View online: http://dx.doi.org/10.1063/1.1771138 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/25/6?ver=pdfcov Published by the AIP Publishing 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: 155.97.178.73 On: Sat, 22 Mar 2014 16:32:21 REG ENE RAT I V E D E F LEe T ION A S RES 0 NAN C E P HEN 0 MEN 0 N 593 first harmonic component of field gradient inhomo­ geneity is required with optimum coupling obtained for (n)=l It has been suggested3 that such a mechanism, existing for a very short time during the period of injec­ tion, is responsible for the damping of radial oscillations necessary if the electrons are to miss the structure of the electron gun from which they were injected. Its applica­ tion to synchrotrons would, under optimum coupling conditions, make it possible to obtain wide output pulses of gamma rays or electrons with relatively small current pulses through a suitable array of conductors which are necessary to produce the coupling inhomogeneity. It is hoped to use such means to extract electrons from their phase stable orbits and enable them to strike an internal target in order to produce a beam of photons. The scheme proposed for extraction of synchrotron beams by Clark, Getting, and Thomas,5 did not include the basic ideas of parametric excitation of radial oscilla­ tions. However, near (n)=f a first harmonic field inhomogeneity as considered by them would certainly produce parametric excitation.3 5 Clark, Getting, and Thomas, Phys. Rev. 70, 562 (1946). THE REVIEW OF SCIENTIFIC INSTRUMENTS 8. CONCLUSION It has been shown that regenerative deflection forms one of a class of parametrically excited resonance phe­ nomena. Numerical results are given for optimum coupling conditions, and a method of increasing the width of the extracted pulse, while at the same time increasing the efficiency of extraction by reducing the axial spread of the beam, is described. The coupling inhomogeneities are produced electrodynamically. It is shown that phase stability of the particle motion is not seriously affected by passage of the particles in the deflector fields. ACKNOWLEDGMENTS The author wishes to thank Miss B. M. Dent for performing the Fourier analysis used in this work; also Dr. C. Dannatt, M.C., O.B.E., M.I.E.E., Director of Research and Education, and Mr. B. G. Churcher, M.Sc., M.I.E.E., Manager of the Research Department, Metropolitan-Vickers Electrical Company, Ltd. for per­ mission to publish this paper. VOLUME 25. NUMBER 6 JUNE. 1954 Shorting and Field Corrections in Hall Measurements* W. F. FLANAGAN, P. A. FLINN,t AND B. L. AVERBACH Department of Metallurgy, Massachusetts Institute of Technology, Cambridge, Massachusetts (Received July 10, 1953) A method is given for correcting the measured Hall voltage for shorting by the current contacts and for inhomogeneities in the applied magnetic field. This method is applied to measurements of the Hall coefficient in gold-silver alloys. INTRODUCTION HALL effect measurements in metallic solid so­ lutions are in progress in order to investigate the electronic changes on alloy formation. These measurements are being made on thin alloy foils which were fabricated by rolling and sintering together foils of the pure metals and then SUbjecting them to a diffusion heat treatment with intermediate roIlings to assure a homogeneous composition. These foils are 2X3Xapproximately 0.0005 inch thick and of the geometry shown in Fig. 1. 'Although the Hall voltage is a result of the surface charge established on the conductor by the interaction of the magnetic field with the charge carriers, the voltage one measures is not directly the Hall voltage, but one which has been modified by the geometry of the sample and the method of measurement. In our .. This work was performed under the auspices of the U. S. Atomic Energy Commission. t Now at Department of Physics, Wayne University, Detroit, Michigan. measurements the Hall current was shorted by the heavy current leads extending the entire width of the specimen, and the applied magnetic field was inhomogeneous. A correction has been described for the shorting effect of the current contracts, but the solution assumed a homogeneous magnetic field.! This paper describes a method of making the shorting correction for a FIG. 1. Hall specimen geometry. tY ~ 'r-___ --';I..::O=--___ ,_~ I I I -----+------ X I " I .~ ,II-__ ii_" ~ _b _-----ll...l 1 Isenberg, Russell, and Greene, Rev. Sci. Instr. 19,685 (1948). 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: 155.97.178.73 On: Sat, 22 Mar 2014 16:32:21 594 FLANAGAN, FLINN, AND AVERBACH specimen in a non-uniform field and gives the results of Hall measurements in a series of gold-silver alloys. SHORTING CORRECTION Inside the specimen Laplace's equation is satisfied. Assuming the main current through the sample to be constant (Le., the Hall current is negligible) one can disregard its effect on the final field distribution. Considering just the effect of the Hall field, the related Laplacian equation may be written as: V'2V(X,y) =0, (1) where V represents the voltage contribution from the Hall effect only, and x and yare directions shown in Fig. 1. Assuming the variation in the applied magnetic field to be in the x direction only (this has been found to be true for our case), with the magnet geometry shown in the inset of Fig. 2, the Hall field at any distance x along the direction of the current density, J, can be represented by EH(X) = RoJB(x), (2) where Ro is the Hall coefficient, and B(x) is the magnetic induction at any point x. By introducing the relation, B(x)= B(L/2)cp(x), (3) where cp(x) is normalized to unity at X= L/2, the center of the specimen, and representing V H by the expression V H/w=RoJB(L/2), (4) where V H is the true Hall voltage and W the width of the foil, the field established by the surface charge along the edges of the sample can be represented by VH RoJB(x)=--cp(x). W (5) Since this field is in the y direction, the boundary con­ dition along the edges of the specimen becomes V(x,±w/2) VHali ± =f------cfJ(x). (6a) y W " ----~--------~--------~--x By symmetry one can assume that V(x,O)=O, (6b) and consequently the shorting effect of the current contacts at the ends of the specimen imposes the conditions V (O,y)= V(L,y)=O. (6c) Solving Eq. (1) with the boundary conditions (6a,b,c), one obtains: 00 2 {fL V H (mrx)} V(x,y)= L ----cf>(x) sin -dx n=l mr 0 W L sinh (mry/L) . (n7rx) X SID -• cosh (n7rW/2L) 2L (7) Since the voltage is measured between points a and b (see Fig. 1), the measured Hall voltage, V m is given by: 4 00 { 1 (n7rW) Vm=2V(L/2,w/2)=- L (-1)(n-lllLtanh- 7r n=odd n 2L fL V H (mrx)} X 0 --;;;,>(x) sin L dx . (8) This solution assumes complete shorting at the ends, and a surface charge proportional to the magnetic field B(x) along the sides. At the corners these assump­ tions are in conflict, but in practice the error introduced is small if the length-to-width ratio is at least !. This is especially true in the present case, where cp(x) has a very low value at both ends. The integral can be evaluated numerically after cf>(x) has been determined, and the summation is then straightforward. A rotating coil fluxmeter was used to obtain cp(x), which is shown in Fig. 2, and which was found to be independent of the magnitude of the magnetic field. This resulted in a constant correction term for all fields. The integral in Eq. (8) was evaluated numerically as a summation making use of Lipson-Beevers strips,2 and cp(x) was assumed to vary slowly enough so that this integral was well represented by a summation whose increments were 1/30th of the total length of the sample. Thus, 1L (m7rx) o cp(x) sin L dx 60 (n7rh) (L ) = L cp" sin --, "-ven 60 30 (9) FIG. 2. Experimentally determined magnetic field shape function, Z H. Lipson and C. A. Beevers, Proc. Phys. Soc. (London) 48, ",(x), and magnet geometry. 772 '(1936). 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: 155.97.178.73 On: Sat, 22 Mar 2014 16:32:21 S H 0 R TIN G AND FIE L D COR R E C T ION SIN HAL L MEA SUR E MEN T S 595 with q,,,= q, (x;) , where Xi= hL/60. Substituting Eq. (9) into (8): V m 2L 00 { 1 (mrw) j=-=-- L (-1)(n-l)f2-tanh-V H 15?rW n=odd n 2L x I: q,,, sin (mrh) }. h=O 60 (10) even For a length to width ratio of! (see Fig. 1) and for the field shape shown in Fig. 2, the correction becomes V m/V H=O.743. In practice, Eq. (10) was found to converge fast enough so that 15 terms gave sufficient accuracy. V m refers to the measured voltage between points a and b, and V H refers to the Hall voltage which would be present in the absence of shorting and inhomo­ geneity. It should be noted that in the case of ferromagnetics some portions of the sample may not be above satura­ tion because of the variation in magnetic field with ~I~ 10 I ::::l • G;; ORNSTEIN AND VAN GEEt!~~--- N e__ .___e- -~ t= --C~R:~~;--·- . x-X!-Xx~ 0:0 0.5l1=- - --Xx-JX-X_X- I UNCORRECTED FIG. 3. Hall constants for gold-silver alloys. distance along the sample. This introduces an error because of an additional contribution to the slope of the V m vs B curves due to the variation of the mag­ netization with B.3 For this reason, this correction is only applicable to nonferromagnetic alloys, although the error introduced for ferromagnetics is probably small under conditions where the samples are easily saturated. HALL MEASUREMENTS IN GOLD-SILVER ALLOYS For the measurements reported here a main current of 2.5 amperes at 250 cycles per second was supplied by a class B push-pull power amplifier which had sufficient feedback to make the output amplitude insensitive to line voltage changes. In measuring the Hall voltage, a microphone transformer with a high input impedance was used to isolate the specimen from the measuring system. The transverse voltage was fed a A. I. Schindler and E. M. Pugh, Phys. Rev. 89, 295 (1953). ,; ~ .... I/l Z o 0: I­o III oJ 1&.1 • ---- c: 0.5 O~~--~~--~~--~~--~~--J Au 40 60 80 Ag At. % Au FIG. 4. Effective electron concentration in gold-silver alloys. into a two-stage amplifier which had a bridge-T network peaked at the main current frequency, and the resultant voltage was measured with a vacuum tube voltmeter. The amplification was determined by means of a stand­ ard resistance placed in series with the specimen, and voltages were measured as a function of B(L/2). For these experiments the Hall constant was given by the relation: Ro=mR.d/V.j, (11) where m is the slope of the V m vs B(L/2) curve, R. is the standard calibrating resistance, d is the thickness of the sample; V. is the voltage obtained with the standard resistance, and j is the correction term, V m/V H. The sign of the effect was determined by observing the phase of the measured signal with respect to the input signal by means of Lissajous patterns, and comparing the phase change with that observed on pure gold, which is known to have a negative sign. Measurements were made at room temperature with magnetic fields varying from ± 11 000 gauss. The slopes obtained for the two directions of the field were averaged to minimize the Nernst and Richi-Leduc effects, and the Ettinghausen effect was absent because of the ac method used. The uncorrected Hall constants for Ag-Au alloys, together with their corrected values, are shown in Fig. 3 along with values published by Ornstein and van Geel.' Ornstein and van Geel did not apply a correction term, but this term would have been small for their geometry. The effective number of conduction electrons per atom calculated on the basis of free electronst is given in Fig. 4. Similar measurements are being made on Au-Cu, Au-Ni, Cu-Ni, Cu-Pt, and Ag-Pd, and the effect of ordering is also being investigated. 4 L. S. Ornstein and W. Ch. van Geel, Z. Physik 72, 488 (1931). tn=1J/Roq oNo, where n=number of free conduction electrons per atoms; V= molar volume; Ro= Hall coefficient; q= charge per electron; No=Avogadro's number. 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: 155.97.178.73 On: Sat, 22 Mar 2014 16:32:21