MagneticSusceptibility 2009-02-17 18:49:31 2009-02-17 19:35:47 Topic diamagnetic susceptibility magnetic field strength volume magnetic susceptibility mass magnetic susceptibility \chi_g \chi_m molar magnetic susceptibility $\chi_(mol) magnetic susceptibility tensor \chi magnetization magnetic induction vector magnetic moments in solution by nuclear magnetic resonance spectrometry NMR % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@ }}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \begin{definition} In electromagnetism, the \emph{volume magnetic susceptibility}, represented by the symbol $ \chi_{v} $ is defined by the following equation $$ \vec{M} = \chi_{v} \vec{H},$$ where in SI units $\vec{M}$ is the \emph{magnetization} of the material (defined as the magnetic dipole moment per unit volume, measured in amperes per meter), and H is the \emph{strength of the magnetic field} $\vec{H}$, also measured in amperes per meter. \end{definition} On the other hand, the magnetic induction $\vec{B}$ is related to $\vec{H}$ by the equation $$\vec{B} \ = \ \mu_0(\vec{H} + \vec{M}) \ = \ \mu_0(1+\chi_{v}) \vec{H} \ = \ \mu \vec{H},$$ where $\mu_0$ is the magnetic constant, and $ \ (1+\chi_{v}) $ is the relative permeability of the material. Note that the magnetic susceptibility $\chi_v$ and the magnetic permeability $\mu$ of a material are related as follows: $$ \mu = \mu_0(1+\chi_v) \, .$$ \begin{remark} There are two other measures of susceptibility, the \emph{mass magnetic susceptibility}, $\chi_g$ or $\chi_m$, and the \emph{molar magnetic susceptibility}, $\chi_{mol}:$ $$ \chi_{\text{mass}}= \chi_v/\rho ,$$ $$ \chi_{mol} \, = \, M\chi_m = M \chi_v / \rho, $$ where $\rho$ is the density and M is the molar mass (in kg·mol−1 (in SI) or g·mol−1 (in cgs)) \end{remark} \subsubsection{Susceptibility Sign convention} If $\chi$ is positive, then $(1+\chi_v)> 1$ (or, in cgs units, $(1+4 \pi \chi_v) > 1)$ and the material can be paramagnetic, ferromagnetic, ferrimagnetic, or anti-ferromagnetic; then, the magnetic field inside the material is strengthened by the presence of the material, that is, the magnetization value is greater than the external H-value. On the other hand there are certain materials--called \emph{diamagnetic}-- for which $\chi$ negative, and thus $(1+χv) < 1$ (in SI units). \subsection{Magnetic Susceptibility Tensor, $\chi$} The magnetic susceptibility of most crystals (that are anisotropic) cannot be represented only by a scalar, but it is instead representable by a tensor \textbf{$\chi$}. Then, the crystal magnetization $\vec{M}$ is dependent upon the orientation of the sample and can have non-zero values along directions other than that of the applied magnetic field $\vec{H}$. Note that even non-crystalline materials may have a residual anisotropy, and thus require a similar treatment. In all such magnetically anisotropic materials, the volume magnetic susceptibility tensor is then defined as follows: $$ M_i=\chi_{ij}H_j , $$ where $i$ and $j$ refer to the directions (such as, for example, x, y, z in Cartesian coordinates) of, respectively, the applied magnetic field and the magnetization of the material. This rank 2 tensor (of dimension (3,3)) relates the component of the magnetization in the $i$-th direction, $M_i$ to the component $ H_j$ of the external magnetic field applied along the $j$-th direction. \begin{thebibliography}{99} \subsubsection{Magnetic Properties of Materials} \bibitem{AMM68} G. P. Arrighini, M. Maestro, and R. Moccia (1968). Magnetic Properties of Polyatomic Molecules: Magnetic Susceptibility of $H_2O, NH_3, CH_4, H_2O_2$. {\em J. Chem. Phys.} 49: 882-889. doi:10.1063/1.1670155. \bibitem{OMM80} S. Otake, M. Momiuchi and N. Matsuno (1980). Temperature Dependence of the Magnetic Susceptibility of Bismuth. J. Phys. Soc. Jap. 49 (5): 1824-1828. doi:10.1143/JPSJ.49.1824. \bibitem{HOM94} J. Heremans, C. H. Olk and D. T. Morelli (1994). Magnetic Susceptibility of Carbon Structures. {\em Phys. Rev. B} 49 (21): 15122-15125. doi:10.1103/PhysRevB.49.15122. \bibitem{OMM80} R. E. Glick (1961). On the Diamagnetic Susceptibility of Gases. {\em J. Phys. 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Method for determining magnetic susceptibilities by NMR. J. Chem. Phys. 37: 1891-1892. $doi:10.1063/1.1733393$. \bibitem{H2k3} R. E. Hoffman (2003). Variations on the chemical shift of TMS. {\em J. Magn. Reson.} 163: 325-331. $doi:10.1016/S1090-7807(03)00142-3$. \end{thebibliography}