Magnetization of a Diamagnetic Particle Measured From its Parabolic Movement Caused by Field-Gradient Force in Terrestrial Gravity

Abstract

Diamagnetic susceptibility <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi _{\mathrm {DIA}}$ </tex-math></inline-formula> of a small particle is determined from its parabolic translation induced by a field gradient generated by a pair of neodymium magnet in normal gravity condition. Specifically, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi _{\mathrm {DIA}}$ </tex-math></inline-formula> of a single particle is determined from the relationships between velocity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v(x_{i})$ </tex-math></inline-formula> of particle and field intensity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$B(x_{i})$ </tex-math></inline-formula> that are observed at different positions <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x_{i}$ </tex-math></inline-formula> of a horizontal coordinate, without measuring mass <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula> of particle. The method was based on a recent finding that the variance of acceleration observed between the translating particles induced in a common area of field gradient is uniquely determined by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi _{\mathrm {DIA}}$ </tex-math></inline-formula> of material. Weak magnetic (i.e., diamagnetic and paramagnetic) materials were previously not considered to show translation by a low magnetic field produced by a permanent magnet. Using a conventional device, measurement of diamagnetic susceptibility <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi _{\mathrm {DIA}}$ </tex-math></inline-formula> generally requires a bulk sample that weight above <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sim 1$ </tex-math></inline-formula> mg. In the new apparatus, the accuracy of obtained <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi _{\mathrm {DIA}}$ </tex-math></inline-formula> value was significantly improved by optimizing the field distribution of the magnetic circuit, thus decreasing the fluctuation of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$v(x_{i})$ </tex-math></inline-formula> values, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi _{\mathrm {DIA}}$ </tex-math></inline-formula> of a submillimetre-size sample is measurable with an uncertainty below <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$5\times 10^{-7}$ </tex-math></inline-formula> emu/g. Approximate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi _{\mathrm {DIA}}$ </tex-math></inline-formula> value of the translated particle is swiftly estimated from its position recovered on a collecting plate that is set at bottom of the translating area. Material identification of the translated particle is possible by collating the obtained <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi _{\mathrm {DIA}}$ </tex-math></inline-formula> value with a list of published data; unlike most conventional methods of chemical analysis, the identification is conducted without consuming the sample. Hence, the setup will extend as an effective technique of pre-treatment in analyzing a mixture of heterogeneous particles; such a technique is useful in various types of material exploration.