Equilibrium magnetization and variance#

Equilibrium magnetization#

formula.magnetization.mz_eq(): equilibrium magnetization per spin [1]

From the sample properties, we compute the magnetic moment \(\mu\) of the state with the largest \(m_J\) quantum number,

\[\mu = \hbar\gamma J [\mathrm{aN}\:\mathrm{nm}\:\mathrm{mT}^{-1}]\]

We calculate the ratio of the energy level splitting of spin states to the thermal energy,

\[x = \dfrac{\mu B_0}{k_b T} \: [\mathrm{unitless}],\]

and define the following two unitless numbers:

\[\begin{split}a &= \dfrac{2 \: J + 1}{2 \: J} \\ b &= \dfrac{1}{2 \: J}\end{split}\]

In terms of these intermediate quantities, the thermal-equilibrium polarization is given by

\[p_{\text{eq}} = a \coth{(a x)} - b \coth{(b x)} \: [\mathrm{unitless}].\]

The equilibrium magnetization is given by

\[\mu_z^{\text{eq}} = p_{\text{eq}} \: \mu \: [\mathrm{aN} \: \mathrm{nm} \: \mathrm{mT}^{-1}].\]

In the limit of low field or high temperature, the equilibrium magnetization tends towards the Curie-Weiss law,

\[mu_z^{\text{eq}} \approx \dfrac{\hbar^2 \gamma^2 \: J (J + 1)}{3 \: k_b T} B_0\]

Equilibrium variance#

formula.magnetization.mz2_eq(): magnetization variance per spin, magnetization variance density [2]

Compute the magnetization variance per spin and the magnetization variance density for spins fluctuating at thermal equilibrium.

Mz2_eq: magnetization variance per spin [aN^2 nm^2/mT^2] times gradient

The variance in a single spin’s magnetization in the low-polarization limit is given by [2]

\[\sigma_{{\cal M}_{z}}^{2} = \hbar^2 \gamma^2 \dfrac{J \: (J + 1)}{3}\]

The magnetization variance density is obtained from \(\sigma_{{\cal M}_{z}}^{2}\) by multiplying by the sample’s spin density \(\rho\).

Note

We assume for simplicity that the root mean square polarization fluctuations are much larger than the equilibrium polarization. In this limit, the polarization fluctuations are independent of applied field \(B_0\) and temperature \(T\). This approximation will not be valid for \(p \sim 1\) electrons.

`formula.magnetization` module#

mrfmsim.formula.magnetization.mz2_eq(Gamma, J)[source]#

Compute the magnetization variance per spin.

Parameters:

gamma (float) – the gyromagnetic ratio. [rad/s.mT]
J (float) – total spin angular momentum

Returns:

Mz2_eq, rhoMz_eq

mrfmsim.formula.magnetization.mz_eq(B_tot, Gamma, J, temperature)[source]#

Magnetization per spin at the thermal equilibrium using the Brillouin function.

Parameters:

gamma (float) – the gyromagnetic ratio. [rad/s.mT]
j (float) – total spin angular momentum
temperature (float) – the spin temperature [K]
spin_density (float) – the sample spin density \(\rho\) [1/nm^3]
b0 (float) – the external magnetic field [mT]
bz (float) – tip magnetic field in z [mT]

Returns:

equilibrium per-spin magnetization [aN.nm/mT]

The outputs are calculated from the sample properties

\[\begin{split}J &= \text{spin angular momentum quantum number}\: [\mathrm{unitless}]\\ \gamma & = \text{gyromagnetic ratio} \: [\mathrm{s}^{-1} \mathrm{mT}^{-1}] \\ B_0 &= \text{applied magnetic field} \: [\mathrm{mT}] \\ T &= \text{temperature} \: [\mathrm{K}] \\ \rho &= \text{spin density} \: [\mathrm{nm}^{-1}]\end{split}\]

as follows. From the sample properties, we compute the magnetic moment \(\mu\) of the state with the largest \(m_J\) quantum number,

\[\mu = \hbar\gamma J \: [\mathrm{aN} \: \mathrm{nm} \: \mathrm{mT}^{-1}]\]