%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: magnetic susceptibility
%%% Primary Category Code: 00.
%%% Filename: MagneticSusceptibility.tex
%%% Version: 11
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}
\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}
\usepackage{html}
% 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{document}
\begin{definition}
In \htmladdnormallink{Electromagnetism}{http://planetphysics.us/encyclopedia/Electromagnetism.html}, 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 \htmladdnormallink{volume}{http://planetphysics.us/encyclopedia/Volume.html}, measured in amperes per meter), and H is the \emph{strength of the \htmladdnormallink{magnetic field}{http://planetphysics.us/encyclopedia/NeutrinoRestMass.html}} $\vec{H}$, 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 \htmladdnormallink{mass}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}.
\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 \htmladdnormallink{scalar}{http://planetphysics.us/encyclopedia/Vectors.html}, but it is instead representable by a \htmladdnormallink{tensor}{http://planetphysics.us/encyclopedia/Tensor.html} \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. Chem.} 65 (9): 1552-1555. doi:10.1021/j100905a020.
\bibitem{DF73}
R. Dupree and C. J. Ford (1973). Magnetic susceptibility of the noble metals around their melting points. {\em Phys. Rev. B} 8 (4): 1780–1782. doi:10.1103/PhysRevB.8.1780.
\subsubsection{Magnetic Moments and Nuclear Magnetic Resonance Spectrometry}
\bibitem{ZF57}
J. R. Zimmerman, and M. R. Foster (1957). Standardization of NMR high resolution spectra. {\em J. Phys. Chem.} 61: 282-289. $doi:10.1021/j150549a006$.
\bibitem{EHB73}
Robert Engel, Donald Halpern, and Susan Bienenfeld (1973). Determination of magnetic moments in solution by nuclear magnetic resonance spectrometry. Anal. Chem. 45: 367-369. doi:10.1021/ac60324a054.
\bibitem{KCBHDH2k3}
P. W. Kuchel, B. E. Chapman, W. A. Bubb, P. E. Hansen, C. J. Durrant, and M. P. Hertzberg (2003). Magnetic susceptibility: solutions, emulsions, and cells. {\em Concepts Magn. Reson.} A 18: 56-71. $doi:10.1002/cmr.a.10066$.
\bibitem{KB62}
K. Frei and H. J. Bernstein (1962). 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}
\end{document}
