PHYSICAL REVIEW B VOLUME 50, NUMBER 13 1 OCTOBER 1994-I KfFect of a magnetic field on a vibrating reed with anisotropic susceptibility R. L. Jacobsen, ' A. C. Ehrlich, T. M. Tritt, and D. J. Gillespie Materials Physics Branch, Naval Research Laboratory, Washington, D. C. 20375 (Received 27 December 1993; revised manuscript received 29 June 1994) A magnetic field may alter the flexural resonant frequencies of a vibrating reed that possesses an aniso- tropic magnetic susceptibility. The phenomenon s magnitude is a function of the field strength, the length of the reed, the radius of gyration of the reed's cross section, the Young's modulus of the material, the vibrational mode observed, the magnetic susceptibility tensor, and the field s orientation relative to the principal susceptibility directions and the reed s plane of vibration. This efFect can allow a direct and extremely sensitive measure of magnetic anisostropy of nonferromagnetic materials. I. INTRODUCTION The vibrating-reed technique probes the elastic proper- ties of a material by exploring the behavior of resonant flexural modes of a specimen. Increasingly the method is being used in conjunction with applied magnetic fields to investigate magnetic and magnetoelastic behavior. In some experiments a test specimen is mounted at the end of a reed, as with the vibrating-reed magnetometer, ' or the flux-lattice melting studies of Gammel et al. In oth- er experiments the reed itself is of interest, as with super- conducting vibrating-reed experiments, " and also the effort of Hoen et al. to detect field-induced condensation of a charge-density wave in NbSe3. ' However, no con- sideration has yet been given to the effect of magnetiza- tion per se on the reed's motion. Here we investigate the effect of magnetic field, through induced magnetization, on the resonance fre- quencies of reeds, with the finding that in anisotropic sys- tems (as reeds are likely to be) these frequencies can be substantially altered by moderate to high magnetic fields even if the reed is only very weakly para- or diamagnetic. We begin with a general derivation of the equation of motion for a reed in an applied magnetic field. We then solve the equation and use perturbation theory to deduce the dependence of the resonant frequencies on field strength and a variety of other experimental parameters. Next we give results of experimental measurements in an effort to verify the theoretical predictions. Finally we dis- cuss peripheral issues, including the elimination of possi- ble alternative interpretations of the data, and analysis of the sensitivity and utility of the vibrating reed as the basis for a susceptometer to probe magnetic anisotropy direct- ly. Because the alteration of a reed's resonant frequencies by a magnetic field can occur only if the reed possesses anisotropic magnetic susceptibility (AMS), we will refer to this phenomenon as the AMS effect. II. THEORY The equation of motion for a reed with an anisotropic magnetic susceptibility vibrating in flexure in an applied magnetic field H can be derived in the same manner as for the usual equation for a vibrating reed or bar, but with the addition of a field-induced torque acting on each element of the reed as it turns in the magnetic field. Solu- tion of this equation under appropriate boundary condi- tions will reveal the effect of the field on the resonant vi- bration frequencies of the reed. Figure 1 shows a schematic diagram of the physical sit- uation. The reed is of length I and uniform cross- sectional area S. It is mounted as a cantilever, oriented along the x direction. Vibration is taken to lie in the x-y plane, y (x) denoting the deflection of the element at x from its equilibrium position. When the reed deflects from its equilibrium position, a volume element of the reed, S dx, may be tilted from the x orientation. This tilt is measured by the angle 8. H is the applied magnetic field, and po the permeability of free space. g measures the azimuthal angle of H with respect to x, and (() the po- lar angle, with respect to y, of the projection of H into the y-z plane. y is the volume magnetic susceptibility, and is a second-rank tensor. An elementary treatment of the vibrating reed in the absence of a magnetic field is given by Morse. ' The mechanically resonant frequency of vibration of a given mode of the reed at zero field is given by 1/2 ~'r (l) l where ~ is the radius of gyration of the reed's cross sec- tion, p its density, F its longitudinal Young s modulus, and P„a constant determined by the boundary conditions of the vibration and the mode n (n = I indicates a funda- mental, n =2 a first overtone, etc; see Ref. 15 for a tabu- lation of P„). If, however, a magnetic field is present, and if g is anisotropic, the resonant frequency will be altered in the manner described below. A uniform magnetic field applied to the reed will in- duce a magnetization M in each element of the reed, M=y. H . (2) During vibration the reed is not usually straight, and thus the relative orientation of H with respect to the reed is a function of position. Any given element inclined at an angle 8 about the z axis, in the laboratory reference frame, appears to experience a field rotated by — 8 in the reference frame that moves with the element. A.cting first on H with the appropriate rotation matrix, M can be ex- pressed in terms of the susceptibility tensor components 0163-1829/94/50(13)/9208(7)/$06. 00 50 9208 1994 The American Physical Society