Editing Paramagnetism (section)

===Curie's law ===
{{main|Curie's law}}
For low levels of magnetization, the magnetization of paramagnets follows what is known as [[Curie's law]], at least approximately.  This law indicates that the susceptibility, <math> \chi</math>, of paramagnetic materials is inversely proportional to their temperature, i.e. that materials become more magnetic at lower temperatures.  The mathematical expression is:
<math display="block"> \mathbf{M} = \chi\mathbf{H} = \frac{C}{T}\mathbf{H}</math>
where:
*<math>\mathbf M</math> is the resulting magnetization, measured in [[ampere]]s/meter (A/m),
*<math>\chi</math> is the [[magnetic susceptibility|volume magnetic susceptibility]] ([[Dimensionless quantity|dimensionless]]),
*<math>H</math> is the auxiliary [[magnetic field]] (A/m),
*<math>T</math> is absolute temperature, measured in [[kelvin]]s (K),
*<math>C</math> is a material-specific [[Curie constant]] (K).

Curie's law is valid under the commonly encountered conditions of low magnetization (''μ''<sub>B</sub>''H'' ≲ ''k''<sub>B</sub>''T''), but does not apply in the high-field/low-temperature regime where saturation of magnetization occurs (''μ''<sub>B</sub>''H'' ≳ ''k''<sub>B</sub>''T'') and magnetic dipoles are all aligned with the applied field. When the dipoles are aligned, increasing the external field will not increase the total magnetization since there can be no further alignment.

For a paramagnetic ion with noninteracting magnetic moments with angular momentum ''J'', the Curie constant is related to the individual ions' magnetic moments,
<math display="block">C=\frac{n}{3k_\mathrm{B}}\mu_{\mathrm{eff}}^2 \text{ where } \mu_{\mathrm{eff}} = g_J \mu_\mathrm{B} \sqrt{J(J+1)}.</math>

where ''n'' is the number of atoms per unit volume. The parameter ''μ''<sub>eff</sub> is interpreted as the effective magnetic moment per paramagnetic ion.  If one uses a classical treatment with molecular magnetic moments represented as discrete magnetic dipoles, ''μ'', a Curie Law expression of the same form will emerge with ''μ'' appearing in place of ''μ''<sub>eff</sub>.

{{math proof | title = Derivation | proof =
Curie's Law can be derived  by considering a substance with noninteracting magnetic moments with angular momentum ''J''.  If orbital contributions to the magnetic moment are negligible (a common case), then in what follows ''J'' = ''S''.  If we apply a magnetic field along what we choose to call the ''z''-axis, the energy levels of each paramagnetic center will experience [[Zeeman splitting]] of its energy levels, each with a ''z''-component labeled by ''M<sub>J</sub>'' (or just  ''M<sub>S</sub>'' for the spin-only magnetic case).  Applying semiclassical [[Boltzmann statistics]], the magnetization of such a substance is
<math display="block">n\bar{m} = \frac{n\sum\limits_{M_{J} = -J}^{J}{\mu_{M_{J}}e^{{-E_{M_{J}}}/{k_{\mathrm{B}}T}\;}}}{\sum\limits_{M_{J} = -J}^{J}{e^{{-E_{M_{J}}}/{k_{\mathrm{B}}T}\;}}} = \frac{n\sum\limits_{M_{J} = -J}^{J}{M_{J}g_{J}\mu_{\mathrm{B}}e^{{M_{J}g_{J}\mu_{\mathrm{B}}H}/{k_{\mathrm{B}}T}\;}}}{\sum\limits_{M_{J} = -J}^{J}{e^{{M_{J}g_{J}\mu_{\mathrm{B}}H}/{k_{\mathrm{B}}T}\;}}}.</math>

Where <math>\mu_{M_J}</math> is the ''z''-component of the magnetic moment for each Zeeman level, so <math>\mu _{M_J} = M_J g_J\mu_\mathrm{B} - \mu_\mathrm{B}</math> is called the [[Bohr magneton]] and ''g''<sub>''J''</sub> is the [[Landé g-factor]], which reduces to the free-electron g-factor, ''g''<sub>''S''</sub> when&nbsp;''J''&nbsp;=&nbsp;''S''.  (in this treatment, we assume that the ''x''- and ''y''-components of the magnetization, averaged over all molecules, cancel out because the field applied along the ''z''-axis leave them randomly oriented.)  The energy of each Zeeman level is <math>E_{M_J} = -M_J g_J \mu_\mathrm{B} H</math>.  For temperatures over a few ''K'', <math>M_J g_J \mu_\mathrm{B}H/k_\mathrm{B} T \ll 1</math>, and  we can apply the approximation <math>e^{M_J g_J \mu_\mathrm{B} H /k_\mathrm{B} T\;} \simeq 1 + M_J g_J \mu_\mathrm{B} H/k_\mathrm{B} T\;</math>:
<math display="block">\bar{m}=\frac{\sum\limits_{M_J=-J}^J {M_J g_J \mu_\mathrm{B} e^{M_J g_J \mu_\mathrm{B} H/k_\mathrm{B} T\;}}}{\sum\limits_{M_J=-J}^J e^{M_Jg_J\mu_\mathrm{B} H/k_\mathrm{B} T\;}}\simeq g_J\mu_\mathrm{B} \frac{\sum\limits_{M_J=-J}^J M_J \left( 1+M_J g_J\mu_\mathrm{B} H/k_\mathrm{B} T\; \right)}{\sum\limits_{M_J=-J}^J \left( 1+M_J g_J \mu_\mathrm{B} H/k_\mathrm{B} T \; \right)}=\frac{g_J^2 \mu_\mathrm{B}^2 H}{k_\mathrm{B} T} \frac{\sum\limits_{-J}^J M_J^2}{\sum\limits_{M_J=-J}^J{(1)}},</math>
which yields:
<math display="block">\bar{m}=\frac{g_J^2 \mu_\mathrm{B}^2 H}{3k_\mathrm{B} T} J(J+1).</math>
The bulk magnetization is then <math>M = n\bar{m} = \frac{n}{3k_\mathrm{B}T} \left[ g_J^2 J(J+1) \mu_\mathrm{B}^2 \right]H,</math> and the susceptibility is given by
<math display="block">\chi=\frac{\partial M_{\rm m}}{\partial H} = \frac{n}{3k_{\rm B} T} \mu_{\mathrm{eff}}^2 \text{  ;    and     } \mu_{\mathrm{eff}} = g_J \sqrt{J(J+1)} \mu_{\mathrm B}.</math>
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When orbital angular momentum contributions to the magnetic moment are small, as occurs for most [[Radical (chemistry)|organic radicals]] or for octahedral transition metal complexes with d<sup>3</sup> or high-spin d<sup>5</sup> configurations, the effective magnetic moment takes the form ( with [[G-factor (physics)|g-factor]] ''g''<sub>e</sub> = 2.0023... ≈ 2),
<math display="block">\mu_{\mathrm{eff}}\simeq 2\sqrt{S(S+1)} \mu_\mathrm{B} =\sqrt{N_{\rm u}(N_{\rm u}+2)} \mu_\mathrm{B},</math>
where ''N''<sub>u</sub> is the number of [[unpaired electron]]s.  In other transition metal complexes this yields a useful, if somewhat cruder, estimate.

When Curie constant is null, second order effects that couple the ground state with the excited states can also lead to a paramagnetic susceptibility independent of the temperature, known as [[Van Vleck paramagnetism|Van Vleck susceptibility]].