Theory

Polarization

Steady-state solution

mrfmsim_marohn.formula.polarization.rel_dpol_sat_steadystate(...)

Relative change in polarization for steady-state.

Intermittent Irradiation

mrfmsim_marohn.formula.polarization.rel_dpol_periodic_irrad(...)

Relative change in polarization for intermittent irradiation.

In our magnetic resonance experiment, we alternate microwave-on and microwave-off periods. With the microwaves on, magnetization evolves towards a saturated, steady-state value at a rate \(r0\). With the microwaves off, magnetization evolved towards equilibrium at a rate \(r1 = 1/T1\). The irradiation scheme and resulting magnetization dynamics are summarized in the following table

Adiabatic Rapid Passage

mrfmsim_marohn.formula.polarization.rel_dpol_arp(...)

Relative change in polarization for adiabatic rapid passage.

The Zeeman field \(B_z(\vec{r})\) experienced by spins at location \(\vec{r}\) is a sum of the \(z\) component of the tip field and the static field, which we take to be oriented along the \(z\) axis,

(1)\[B_z(\vec{r}) = B_0 + B_z^{\mathrm{tip}}(\vec{r})\]

Here we assume that \(B_0 \gg B_z^{\mathrm{tip}}\), which requires us to consider only the \(z\) component of the tip field and spin magnetization. The spin-dependent change in the spring constant is determined using

(2)\[\delta k_{\mathrm{spin}} = - \sum_j \delta M_z (\vec{r}_j) \frac{\partial^2 B_{z}^{\mathrm{tip}}({\vec{r}}_j)}{\partial z^2} \Delta V_j\]

where \(\delta M_z\) is the change in magnetization density and \(\Delta V_j\) is the volume of each voxel (or volume element) in the sample. We have drawn equation (2) (with its leading minus sign) from reference 2 (equation 8). We have taken care to implement the leading minus sign here.

Let us now derive the equations we will use to calculate the change in magnetization density \(\delta M_z (\vec{r})\) resulting from the adiabatic rapid passage. In a frame of reference rotating counter clockwise about the \(z\) axis at frequency \(\omega\), the magnetization density evolves under the action of the effective field

(3)\[\vec{B}_{\mathrm{eff}} = (B_z(\vec{r}) - \frac{\omega}{\gamma}) \hat{z} + B_1 \hat{x}_{\mathrm{R}}\]

with \(\hat{z}\) a unit vector along \(z\) axis, \(\hat{x}_{\mathrm{R}}\) a transverse rotating basis vector, \(B_z(\vec{r})\) given by equation (1), \(\omega\) the frequency of the applied oscillating field, \(\gamma\) the gyromagnetic ratio, and \(2 B_1\) the amplitude of the oscillating magnetic field, assumed to be linearly polarized. We are making the rotating wave approximation in using equation (3) to describe the evolution of magnetization under linearly polarized irradiation. It is useful to write down a unit vector parallel to the effective field:

(4)\[\vec{b}_{\mathrm{eff}} = \frac{B_z(\vec{r}) - \omega/\gamma} {\sqrt{(B_z(\vec{r}) - \omega/\gamma)^2 + B_1^2}} \hat{z} + \frac{B_1} {\sqrt{(B_z(\vec{r}) - \omega/\gamma)^2 + B_1^2}} \hat{x}_{\mathrm{R}}\]

Dividing both the numerators and denominators in the equation (4) by \(B_1\), this unit vector can be written as

(5)\[\hat{b}_{\mathrm{eff}}(\Omega) = \frac{\Omega}{\sqrt{\Omega^2+1}} \hat{z} + \frac{1}{\sqrt{\Omega^2+1}} \hat{x}_{\mathrm{R}}\]

with

(6)\[\Omega = \frac{\gamma B_z(\vec{r}) - \omega}{\gamma B_1}\]

the (unitless) ratio of the resonance offset to the Rabi frequency; \(\Omega > 0\) and \(\hat{b}_{\mathrm{eff},z} > 0\) for spins at a field above the resonance field \(\omega/\gamma\) while \(\Omega < 0\) and \(\hat{b}_{\mathrm{eff},z} < 0\) for spins at a field below the resonance field \(\omega/\gamma\).

Now consider the evolution of sample magnetization during an adiabatic rapid passage through resonance. The magnetization is initially along the \(z\) axis. Just before time \(t = 0\),

(7)\[\vec{M}(0^{-}) = M_{z}(0) \: \hat{z}\]

At time \(t = 0\) the irradiation is turned on with an initial offset frequency of \(\Omega_{\mathrm{i}}\). Since this effective field is not quite parallel to the \(z\) axis in the rotating frame, the initial magnetization vector will precess around it. The component of the initial magnetization perpendicular to the initial effective field \(\hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{i}})\) will quickly dephase, within in a time \(T_2 \sim 5 \: \mu\mathrm{s}\). The component of the initial magnetization parallel to the initial effective field will survive this dephasing. The magnetization after this dephasing, at time \(t = 0^{+}\), is given by the projection of \(\vec{M}_{z}(0^{-})\) onto \(\hat{b}_{\mathrm{eff}}\),

(8)\[\vec{M}(0^{+}) = M_{z}(0) \left( \hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{i}}) \cdot \vec{M}_{z}(0^{-}) \right) \: \hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{i}})\]

The prefactor in parenthesis may be positive or negative, depending on whether \(\vec{M}_{z}(0^{-})\) and \(\hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{i}})\) are parallel (\(\Omega >0\)) or antiparallel (\(\Omega <0\)). Substituting equations (5) and (7) into equation (8)

(9)\[\begin{split}\begin{align} \vec{M}(0^{+}) & = M_{z}(0) \frac{\Omega_{\mathrm{i}}}{\sqrt{\Omega_{\mathrm{i}}^2+1}} \left( \frac{\Omega_{\mathrm{i}}}{\sqrt{\Omega_{\mathrm{i}}^2+1}} \hat{z} + \frac{1}{\sqrt{\Omega_{\mathrm{i}}^2+1}} \hat{x}_{\mathrm{R}} \right) \\ & = M_{z}(0) \left( \frac{\Omega_{\mathrm{i}}^2}{\Omega_{\mathrm{i}}^2+1} \hat{z} + \frac{\Omega_{\mathrm{i}}}{\Omega^2_{\mathrm{i}}+1} \hat{x}_{\mathrm{R}} \right) \end{align}\end{split}\]

We can see that equation (9) captures \(\vec{M}_{z}(0^{+})\) correctly for spins initially above and below resonance when the irradiation is turned on. For example, when \(\Omega = +10\), \(\vec{M}_{z}(0^{+}) = 0.99 \: \hat{z} + 0.01 \: \hat{x}_{\mathrm{R}}\) while when \(\Omega = -10\), \(\vec{M}_{z}(0^{+}) = 0.99 \: \hat{z} - 0.01\: \hat{x}_{\mathrm{R}}\). In both cases, \(\vec{M}_{z}(0^{+})\) points up as it should. The magnitude of \(\vec{M}_{z}(0^{+})\) is

(10)\[\begin{split}\begin{align} \| \vec{M}(0^{+}) \| & = M_{z}(0) \left( \frac{\Omega_{\mathrm{i}}^4}{(\Omega_{\mathrm{i}}^2+1)^2} + \frac{\Omega_{\mathrm{i}}^2}{(\Omega_{\mathrm{i}}^2+1)^2} \right)^{1/2} \\ & = M_{z}(0) \left( \frac{\Omega_{\mathrm{i}}^2 (\Omega_{\mathrm{i}}^2 + 1) } {(\Omega_{\mathrm{i}}^2+1)^2} \right)^{1/2} \\ & = M_{z}(0) \frac{\| \Omega_{\mathrm{i}} \|}{\sqrt{\Omega_{\mathrm{i}}^2 + 1}} \end{align}\end{split}\]

At a time just after \(t = 0^+\), the adiabatic rapid passage is initiated and \(\Omega\) is swept from the initial offset \(\Omega_{\mathrm{i}}\) to a final offset \(\Omega_{\mathrm{f}}\). At the end of the sweep, at time \(t_{\mathrm{f}}\), the magnetization density vector will have the same magnitude as it did at time \(t = 0^+\), but will be oriented parallel or antiparallel to the final effective field, \(\hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{f}})\),

(11)\[\vec{M}(t_{\mathrm{f}}) = \| \vec{M}_{z}(0^{+}) \| \: \mathrm{sign}(\Omega_{\mathrm{i}}) \left( \frac{\Omega_{\mathrm{f}}}{\sqrt{\Omega_{\mathrm{f}}^2+1}} \hat{z} + \frac{1}{\sqrt{\Omega_{\mathrm{f}}^2+1}} \hat{x}_{\mathrm{R}} \right)\]

Here \(\mathrm{sign}(\Omega_{\mathrm{i}})\) accounts for the final magnetization being parallel (for positive initial offset, \(\mathrm{sign}(\Omega_{\mathrm{i}}) = +1\)) or antiparallel (for negative initial offset, \(\mathrm{sign}(\Omega_{\mathrm{i}}) = -1\)) to the final effective field. We are interested in the \(z\)-component of the final magnetic field vector. Substituting equation (10) into equation (11) and using

\[\mathrm{sign}(\Omega_{\mathrm{i}}) \: \| \Omega_{\mathrm{i}} \| = \Omega_{\mathrm{i}}\]

(12)\[M_{z}(t_{\mathrm{f}}) = M_{z}(0) \dfrac{\Omega_{\mathrm{i}}}{\sqrt{\Omega_{\mathrm{i}}^2+1}} \dfrac{\Omega_{\mathrm{f}}}{\sqrt{\Omega_{\mathrm{f}}^2+1}}\]

At each point in the sample, the change, final minus initial, in \(z\) component of magnetization following the adiabatic rapid passage is given by

(13)\[\delta M_{z} = M_{z}(t_{\mathrm{f}}) - M_{z}(0) = M_{z}(0) \left( \dfrac{\Omega_{\mathrm{i}}}{\sqrt{\Omega_{\mathrm{i}}^2+1}} \dfrac{\Omega_{\mathrm{f}}}{\sqrt{\Omega_{\mathrm{f}}^2+1}} -1 \right)\]

If we sweep from \(\Omega_{\mathrm{i}} \rightarrow +\infty\) (way above resonance) to \(\Omega_{\mathrm{f}} \rightarrow -\infty\) (way below resonance), then \(\delta M_{z} = -2 M_{z}(0)\). This is what we expect to see. For a swept-field or swept-tip experiment,

(14)\[\Omega_{\mathrm{i}} = \frac{B_z(\vec{r}_{\mathrm{i}}) - \omega/\gamma}{B_1} \: \: \: \mathrm{and} \: \: \: \Omega_{\mathrm{f}} = \frac{B_z(\vec{r}_{\mathrm{f}}) - \omega/\gamma}{B_1}\]

while for a swept-frequency experiment,

(15)\[\Omega_{\mathrm{i}} = \frac{B_z(\vec{r}) - \omega_{\mathrm{i}}/\gamma}{B_1} \: \: \: \mathrm{and} \: \: \: \Omega_{\mathrm{f}} = \frac{B_z(\vec{r}) - \omega_{\mathrm{f}}/\gamma}{B_1}\]

To compute the change in magnetization contributing to signal at each position, we will use the equation (13) and either equation (14) (for a swept-tip experiment) or equation (15) (for a swept-frequency experiment). In the swept-frequency calculation we need to compute the field at each point. In the swept-tip calculation, we need to compute at each position \(\vec{r}\) in the sample the \(z\) component of the magnetic field only at the beginning (\(\vec{r} = \vec{r}_{\mathrm{i}}\)) and end (\(\vec{r} = \vec{r}_{\mathrm{f}}\)) of the tip sweep.

Adiabaticity

We would also like to assess the adiabaticity of the sweep. With

\[M_{z}(t) = M_{z}(0) \: \cos{(\theta(t))},\]

the adiabaticity parameter is defined generally as

\[\alpha = \frac{\dot{\theta}}{\gamma B_1}.\]

In a cryogenic ESR-MRFM observing \(\mathrm{E}^{\prime}\) centers in quartz via cyclic adiabatic inversion, Wago and coworkers observed a peaking of signal when \(\alpha \sim 0.1\) (note that they define the adiabaticity parameter as \(1/\alpha\)) 3. In a room temperature NMR-MRFM experiment observing proton magnetization in an ammonium nitrate crystal via cyclic adiabatic inversion and force detection, Klein and coworkers observed lossless inversion of magnetization when \(\alpha \leq 0.1\) 1. In both of these experiments, a linear frequency sweep was used. We note that more efficient sweeps have been devised. 4 5 For a linear sweep, the adiabaticity parameter is largest near resonance, where

\[\alpha_{\mathrm{res}} = \frac{1}{\gamma B_1^2} \frac{d B_{\mathrm{eff}}}{d t} = \frac{1}{\omega_1} \frac{d \Omega}{d t}\]

Here \(\Omega\) is given by equation (6) and \(\omega_1 = \gamma B_1\) is the Rabi frequency. In a swept-tip experiment, the field at position \(\vec{r}\) changes by an amount \(\delta B = B_{z}^{\mathrm{tip}}(\vec{r}_{\mathrm{f}}) - B_ {z}^{\mathrm{tip}}(\vec{r}_{\mathrm{i}})\) in a time equal to half of a cantilever period, \(\delta t = 1/(2 f_c)\). If we approximate the sweep as linear, then the adiabaticity parameter is given by

(16)\[\alpha_{\mathrm{res}}(\vec{r}) \approx \frac{1}{\gamma B_1^2} \frac{\delta B}{\delta t} = \frac{2 f_c}{\gamma B_1^2} \| B_{z}^{\mathrm{tip}}(\vec{r}_{\mathrm{f}}) - B_{z}^{\mathrm{tip}}(\vec{r}_{\mathrm{i}}) \|\]

We have introduced an absolute value sign to guarantee that \(\alpha\) is positive and independent of the sweep direction. We write :math:` alpha_{mathrm{res}}(vec{r})` to emphasize that the adiabaticity parameter should be evaluated at each position \(\vec{r}\) in the sample. Equation (16) is only strictly valid at resonance and, moreover, does not account for the sinusoidal time dependence of \(\vec{r}(t)\) during the cantilever motion. Nevertheless, we will use it to access the adiabaticity of the spin inversion in the swept-tip experiment. Since :math:` alpha` is smaller for sample spins that do not pass through resonance, equation (16) provides an upper-bound estimate for the adiabaticity parameter at any location. The spins which contribute most to the signal are those which pass through resonance; for these spins, equation :eq:` Eq:alpha-swept-tip` should be reasonably accurate.

In the swept-frequency experiment, the irradiation frequency is ramped from \(\omega_{\mathrm{i}}\) to \(\omega_{\mathrm{f}}\) in a time :math:` Delta T_{mathrm{sweep}}`; the period of the sweep \(\Delta T_ {\mathrm{sweep}}\) is not restricted to be half a cantilever period. The adiabaticity parameter is independent of position \(\vec{r}\) and, assuming a linear frequency sweep, equal to

(17)\[\alpha_{\mathrm{res}} = \frac{1}{\gamma^2 B_1^2} \frac{\| \omega_{\mathrm{f}} - \omega_{\mathrm{i}} \|} {\Delta T_{\mathrm{sweep}}}\]

As with equation (16), equation (17) is only strictly valid for spins that pass through resonance. Spins far away from the resonant slice will experience an \(\alpha\) even smaller and \(\alpha_{\mathrm{res}}\). We can therefore regard equation :eq:` Eq:alpha-swept-freq` as an upper bound for the adiabaticity parameter experienced by any spin in the sample.

Nutation

mrfmsim_marohn.formula.polarization.rel_dpol_nut(...)

Relative change in polarization under the evolution of irradiation.

If the rf is turned on suddenly at \(t =0\), the magnetization will nutate around the effective field. Neglecting relaxation, the magnetization vector a time \(t_{\mathrm{p}}\) after the start of the rf pulse is

\[\begin{split}\begin{align} \frac{\vec{M}(t_p)}{M_{z}(0)} = & \left( \frac{\Omega}{\Omega^2+1} - \frac{\Omega}{\Omega^2+1} \cos{(\Omega_1 t_p \sqrt{\Omega^2+1} \: )} \right) \hat{x}_{\mathrm{R}} \\ & - \left( \frac{1}{\Omega^2+1} \sin{(\Omega_1 t_p \sqrt{\Omega^2+1} \: )} \right) \hat{y}_{\mathrm{R}} \\ & + \left( \frac{\Omega^2}{\Omega^2+1} + \frac{1}{\Omega^2+1} \cos{(\Omega_1 t_p \sqrt{\Omega^2+1} \: )} \right) \hat{z} \end{align}\end{split}\]

with \(\Omega\), the unitless resonance offset, given by Eq. (6) . Since we observe the \(z\) component of magnetization, we are interested in:

\[\rho_{\mathrm{rel}} = \frac{M_z}{M_z(0)} = \frac{\Omega^2}{\Omega^2+1} + \frac{1}{\Omega^2+1} \cos{(\Omega_1 t_p \sqrt{\Omega^2+1} \: )}\]

The change in the \(z\) component of the magnetization due to the pulse is, after some simplification,

(18)\[\delta M_{z}(\theta, \Omega) = M_{z}(t_{\mathrm{p}}) - M_{z}(0) = - 2 \, M_{z}(0) \frac{1}{\Omega^2+1} \sin^2{\left( \frac{\theta_p}{2} \sqrt{\Omega^2 + 1} \: \right)}\]

where \(\theta_p \equiv \omega_1 t_p\) is the pulse angle. Plotting this function, we see that the biggest change in magnetization, \(\delta M_{z}(\theta_p,\Omega)/M_{z}(0) = -2\), occurs on resonance (\(\Omega = 0\) ) when the pulse angle is set to \(\theta_p = \pi\). When \(\theta_p = \pi\), the full width at half max (FWHM) of the \(\delta M_ {z}(\pi,\Omega)/M_{z}(0)\) function is approximately \(1.597 \Omega\). In other words, a \(\pi\) pulse will invert magnetization over a resonant slice of whose width, in field units, is approximately \(1.597 \: B_1\).

Equilibrium magnetization and Variance

equilibrium magnetization

formula.magnetization.mz_eq(): equilibrium magnetization per spin 6

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 7

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 7

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

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.

Reference

1

Klein, O.; Naletov, V. & Alloul, H. “Mechanical Detection of Nuclear Spin Relaxation in a Micron-size Crystal”, Eur. Phys. J. B, 2000, 17, 57 - 68 [10.1007/s100510070160].

2

Lee, S.-G.; Moore, E. W. & Marohn, J. A. “A Unified Picture of Cantilever Frequency-Shift Measurements of Magnetic Resonance”, Phys. Rev. B, 2012, 85, 165447 [10.1103/PhysRevB.85.165447].

3

Wago, K.; Botkin, D.; Yannoni, C. & Rugar, D. “Force-detected Electron-spin Resonance: Adiabatic Inversion, Nutation, and Spin Echo”, Phys. Rev. B, 1998, 57, 1108 - 1114 [10.1103/PhysRevB.57.1108].

4

Baum, J.; Tycko, R. & Pines, A. “Broadband and Adiabatic Inversion of a Two-level System by Phase-modulated Pulses”, Phys. Rev. A , 1985, 32, 3435 - 3447 [10.1103/PhysRevA.32.3435].

5

Kupce, E. & Freeman, R. “Optimized Adiabatic Pulses for Wideband Spin Inversion”, Journal of Magnetic Resonance, Series A, 1996, 118, 299 - 303 [10.1006/jmra.1996.0042].

6

“Brillouin and Langevin functions”

7(1,2)

Equations 1 and 2 in Xue, F.; Weber, D.; Peddibhotla, P. & Poggio, M. “Measurement of statistical nuclear spin polarization in a nanoscale GaAs samples”, Phys. Rev. B, 2011, 84, 205328 [10.1103/PhysRevB.84.205328].

formula.polarization module

Collection of calculations of relative changes in polarization.

mrfmsim_marohn.formula.polarization.rel_dpol_arp(B_offset, B1, df_fm, Gamma)

Relative change in polarization for adiabatic rapid passage.

Compute the resonance offset at each point in the sample and compute the change in the polarization at each point in the sample following the adiabatic rapid passage. The experiment here is a swept-field experiment or a swept-tip experiment, where

\[\Delta B_\text{offset} = B_0 + B_\text{tip} - 2 \pi f_\text{rf}/\gamma\]

and

\[\Omega_\text{initial} = B_0 + B_\text{tip} - (2 \pi f_\text{rf}/\gamma - 2 \pi \Delta f_\text{FM}/\gamma) = \Delta B_\text{offset} + 2 \pi \Delta f_\text{FM} / \gamma\]

Parameters

• Gamma (float) – the gyromagnetic ratio [rad/s.mT]

• B_offset (float) – resonance offset field \(\Delta B_{\text{offset}}\) [mT]

• B_1 (float) – amplitude \(B_1\) of the applied transverse field [mT]

• df_fm (float) – the peak-to-peak frequency modulation \(\Delta f_{\text{FM}}\) of the applied transverse radio frequency magnetic field [Hz]

Returns

relative change in polarization

Return type

np.array, the same shape as B_offset

mrfmsim_marohn.formula.polarization.rel_dpol_arp_ibm(B_offset, df_fm, Gamma)

Relative change in polarization for IBM adiabatic rapid passage.

\[\begin{split}\eta \, (\Delta B_{\text{offset}}) = \begin{cases} \cos^2 \left(\dfrac{\gamma \Delta B_{\text{offset}}} {2 \: \Delta f_{\text{FM}}} \right) & \text{for } \Delta B_{\text{offset}} \leq \pi \Delta f_{\text{FM}} / \gamma, \\ 0 & \text{otherwise.} \end{cases}\end{split}\]

with

\[\Delta B_{\text{offset}} = B_0 - 2 \pi f_{\text{rf}} / \, \gamma [\mathrm{mT}]\]

The result added in pol_arp the negative sign for it is the signal of final - initial.

Parameters

• Gamma (float) – the gyromagnetic ratio [rad/s.mT]

• B_offset (float) – resonant offset \(\Delta B_{\text{offset}}\) [mT]

• df_fm (float) – the peak-to-peak frequency modulation \(\Delta df_{\text{FM}}\) of the applied transverse radio frequency magnetic field [Hz]

Returns

relative change in polarization

Return type

np.array, the same shape as B_offset

mrfmsim_marohn.formula.polarization.rel_dpol_multipulse(rel_dpol, T1, dt_pulse)

Calculate the average relative change in polarization after multiple pulses.

The formula ignores relaxation during pulses. :param float dt_pulse: time between pulses

mrfmsim_marohn.formula.polarization.rel_dpol_nut(B_offset, B1, Gamma, t_p)

Relative change in polarization under the evolution of irradiation.

Equations:

\[\rho_{\mathrm{rel}} = \frac{\Delta M_{z}}{M_{z}(0)} = \frac{1}{\Omega^2+1} (1 + \cos{(\Omega_1 t_p \sqrt{\Omega^2+1})})\]

with

\[\Delta B_{\mathrm{offset}} = B_z(\vec{r}) - \omega/\gamma\]

the resonance offset field and

\[\Omega_1 = \gamma B_1\]

\[\Omega = \frac{\Delta B_{\mathrm{offset}}}{B_1}\]

the unitless resonance offset.

Parameters

• Gamma (float) – the gyromagnetic ratio. [rad/s.mT]

• B_offset (float) – resonance offset field \(\Delta B_{\text{offset}}\) [mT]

• B_1 (float) – amplitude of the applied transverse field \(B_{\text{1}}\) [mT]

• t_p (float) – pulse time \(t_{\mathrm{p}}\) [s]

Returns

relative polarization

Return type

np.array, the same shape as B_offset

mrfmsim_marohn.formula.polarization.rel_dpol_nut_multi_freq_pulse(B_tot, B1, f_rf_array, Gamma, t_p)

Nutation experiments where different frequencies are applied in steps.

The polarization is aggregated as a product.

mrfmsim_marohn.formula.polarization.rel_dpol_periodic_irrad(B_offset, B1, dB_sat, dB_hom, T1, t_on, t_off)

Relative change in polarization for intermittent irradiation.

\[\langle \Delta M_z \rangle = - \frac{S^2 \, M_0}{1 + S^2 + \Omega^2} \left(\frac{1}{r_1} \frac{1}{\tau_\text{on}+\tau_\text{off}} \frac{(1 - E_\text{on})(1 - E_\text{off})} {1 - E_\text{on} \, E_\text{off}} \frac{S^2}{1 + S^2 + \Omega^2} + \frac{\tau_\text{on}} {\tau_\text{on}+\tau_\text{off}}\right)\]

Parameters

• dB_hom (float) – the homogeneous linewidth [mT]

• dB_sat (float) – the saturation linewidth [mT]

• B_offset (float) – resonant offset \(\Delta B_{\text{offset}}\) [mT]

• B_1 (float) – the amplitude \(B_1\) of the applied transverse field [mT]

• t_1 (float) – spin-lattice relaxation [s]

• t_on (float) – time with the microwaves on [s]

• t_off (float) – time with the microwaves off [s]

Returns

relative change in polarization

Return type

np.array, the same shape as B_offset

mrfmsim_marohn.formula.polarization.rel_dpol_sat_steadystate(B_offset, B1, dB_sat, dB_hom)

Relative change in polarization for steady-state.

Compute and return the sample’s relative steady-state spin polarization. As given by the Bloch equations,

\[\Delta\rho_{\text{rel}} = \dfrac{\Delta {\cal M}_z} {{\cal M}_{z}^{\text{eq}}} = - \dfrac{(B_1 / \Delta B_{\text{sat}})^2} {1 + (\Delta B_{\text{offset}} / \Delta B_{\text{homog}})^2 + (B_1 / \Delta B_{\text{sat}})^2}\]

At resonance,

\[\rho_{\text{rel}} = \dfrac{S}{1+S}\]

and the change in polarization is governed by the saturation factor

\[S = (B_1 / \Delta B_{\text{sat}})^2\]

\(\rho_{\text{rel}}\) under irradiation.

Parameters

• dB_hom (float) – the homogeneous linewidth \(\Delta B_{\text{homog}}\) [mT]

• dB_sat (float) – the saturation linewidth \(\Delta B_{\text{sat}}\) [mT]

• B_offset (float) – resonant offset \(\Delta B_{\text{offset}}\) [mT]

• B_1 (float) – the amplitude of the applied transverse field \(B_1\) [mT]

Returns

relative polarization

Return type

np.array, the same shape as B_offset

mrfmsim_marohn.formula.polarization.rel_dpol_sat_td(Bzx, B1, ext_B_offset, ext_pts, Gamma, T2, tip_v)

Relative change in polarization for time-dependent saturation.

The result is not a steady-state solution because it ignores T1 relaxation.

mrfmsim_marohn.formula.polarization.rel_dpol_sat_td_smallsteps(B1, ext_Bzx, ext_B_offset, ext_pts, Gamma, T2, tip_v)

Small step approximation of the time-dependent relative change in polarization.

mrfmsim_marohn.formula.magnetization.mz2_eq(Gamma, J)

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_marohn.formula.magnetization.mz_eq(B_tot, Gamma, J, temperature)

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,

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

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

\[{\cal M}_{z}^{\text{eq}} = p_{\text{eq}} \: \mu \: [\mathrm{aN} \: \mathrm{nm} \: \mathrm{mT}^{-1}].\]

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

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