Theory ====== Polarization ------------- Steady-state solution ^^^^^^^^^^^^^^^^^^^^^^^^ .. autosummary:: mrfmsim_marohn.formula.polarization.rel_dpol_sat_steadystate .. _theory_irradation_section: Intermittent Irradiation ^^^^^^^^^^^^^^^^^^^^^^^^ .. autosummary:: mrfmsim_marohn.formula.polarization.rel_dpol_periodic_irrad 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 :math:`r0`. With the microwaves off, magnetization evolved towards equilibrium at a rate :math:`r1 = 1/T1`. The irradiation scheme and resulting magnetization dynamics are summarized in the following table .. _theory_arp_section: Adiabatic Rapid Passage ^^^^^^^^^^^^^^^^^^^^^^^^ .. autosummary:: mrfmsim_marohn.formula.polarization.rel_dpol_arp The Zeeman field :math:`B_z(\vec{r})` experienced by spins at location :math:`\vec{r}` is a sum of the :math:`z` component of the tip field and the static field, which we take to be oriented along the :math:`z` axis, .. math:: :label: Eq:Bz B_z(\vec{r}) = B_0 + B_z^{\mathrm{tip}}(\vec{r}) Here we assume that :math:`B_0 \gg B_z^{\mathrm{tip}}`, which requires us to consider only the :math:`z` component of the tip field and spin magnetization. The spin-dependent change in the spring constant is determined using .. math:: :label: Eq:DeltaK \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 :math:`\delta M_z` is the change in magnetization density and :math:`\Delta V_j` is the volume of each voxel (or volume element) in the sample. We have drawn equation :eq:`Eq:DeltaK` (with its leading minus sign) from reference [#Lee2012apra]_ (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 :math:`\delta M_z (\vec{r})` resulting from the adiabatic rapid passage. In a frame of reference rotating counter clockwise about the :math:`z` axis at frequency :math:`\omega`, the magnetization density evolves under the action of the effective field .. math:: :label: Eq:B_eff \vec{B}_{\mathrm{eff}} = (B_z(\vec{r}) - \frac{\omega}{\gamma}) \hat{z} + B_1 \hat{x}_{\mathrm{R}} with :math:`\hat{z}` a unit vector along :math:`z` axis, :math:`\hat{x}_{\mathrm{R}}` a transverse rotating basis vector, :math:`B_z(\vec{r})` given by equation :eq:`Eq:Bz`, :math:`\omega` the frequency of the applied oscillating field, :math:`\gamma` the gyromagnetic ratio, and :math:`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 :eq:`Eq:B_eff` to describe the evolution of magnetization under linearly polarized irradiation. It is useful to write down a unit vector parallel to the effective field: .. math:: :label: Eq:beff \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 :eq:`Eq:beff` by :math:`B_1`, this unit vector can be written as .. math:: :label: Eq:beff2 \hat{b}_{\mathrm{eff}}(\Omega) = \frac{\Omega}{\sqrt{\Omega^2+1}} \hat{z} + \frac{1}{\sqrt{\Omega^2+1}} \hat{x}_{\mathrm{R}} with .. math:: :label: Eq:Omega \Omega = \frac{\gamma B_z(\vec{r}) - \omega}{\gamma B_1} the (unitless) ratio of the resonance offset to the Rabi frequency; :math:`\Omega > 0` and :math:`\hat{b}_{\mathrm{eff},z} > 0` for spins at a field above the resonance field :math:`\omega/\gamma` while :math:`\Omega < 0` and :math:`\hat{b}_{\mathrm{eff},z} < 0` for spins at a field below the resonance field :math:`\omega/\gamma`. Now consider the evolution of sample magnetization during an adiabatic rapid passage through resonance. The magnetization is initially along the :math:`z` axis. Just before time :math:`t = 0`, .. math:: :label: Eq:Mz0_minus \vec{M}(0^{-}) = M_{z}(0) \: \hat{z} At time :math:`t = 0` the irradiation is turned on with an initial offset frequency of :math:`\Omega_{\mathrm{i}}`. Since this effective field is not quite parallel to the :math:`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 :math:`\hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{i}})` will quickly dephase, within in a time :math:`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 :math:`t = 0^{+}`, is given by the projection of :math:`\vec{M}_{z}(0^{-})` onto :math:`\hat{b}_{\mathrm{eff}}`, .. math:: :label: Eq:Mz0_plus \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 :math:`\vec{M}_{z}(0^{-})` and :math:`\hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{i}})` are parallel (:math:`\Omega >0`) or antiparallel (:math:`\Omega <0`). Substituting equations :eq:`Eq:beff2` and :eq:`Eq:Mz0_minus` into equation :eq:`Eq:Mz0_plus` .. math:: :label: Eq:Mz0+2 \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} We can see that equation :eq:`Eq:Mz0+2` captures :math:`\vec{M}_{z}(0^{+})` correctly for spins initially above and below resonance when the irradiation is turned on. For example, when :math:`\Omega = +10`, :math:`\vec{M}_{z}(0^{+}) = 0.99 \: \hat{z} + 0.01 \: \hat{x}_{\mathrm{R}}` while when :math:`\Omega = -10`, :math:`\vec{M}_{z}(0^{+}) = 0.99 \: \hat{z} - 0.01\: \hat{x}_{\mathrm{R}}`. In both cases, :math:`\vec{M}_{z}(0^{+})` points up as it should. The magnitude of :math:`\vec{M}_{z}(0^{+})` is .. math:: :label: Eq:AbsMz0+ \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} At a time just *after* :math:`t = 0^+`, the adiabatic rapid passage is initiated and :math:`\Omega` is swept from the initial offset :math:`\Omega_{\mathrm{i}}` to a final offset :math:`\Omega_{\mathrm{f}}`. At the end of the sweep, at time :math:`t_{\mathrm{f}}`, the magnetization density vector will have the same magnitude as it did at time :math:`t = 0^+`, but will be oriented parallel or antiparallel to the final effective field, :math:`\hat{b}_{\mathrm{eff}}(\Omega_{\mathrm{f}})`, .. math:: :label: Eq:vecMtf \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 :math:`\mathrm{sign}(\Omega_{\mathrm{i}})` accounts for the final magnetization being parallel (for positive initial offset, :math:`\mathrm{sign}(\Omega_{\mathrm{i}}) = +1`) or antiparallel (for negative initial offset, :math:`\mathrm{sign}(\Omega_{\mathrm{i}}) = -1`) to the final effective field. We are interested in the :math:`z`-component of the final magnetic field vector. Substituting equation :eq:`Eq:AbsMz0+` into equation :eq:`Eq:vecMtf` and using .. math:: \mathrm{sign}(\Omega_{\mathrm{i}}) \: \| \Omega_{\mathrm{i}} \| = \Omega_{\mathrm{i}} .. math:: :label: Eq:Mzf 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 :math:`z` component of magnetization following the adiabatic rapid passage is given by .. math:: :label: Eq:deltaMz \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 :math:`\Omega_{\mathrm{i}} \rightarrow +\infty` (way above resonance) to :math:`\Omega_{\mathrm{f}} \rightarrow -\infty` (way below resonance), then :math:`\delta M_{z} = -2 M_{z}(0)`. This is what we expect to see. For a swept-field or swept-tip experiment, .. math:: :label: Eq:omegas1 \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, .. math:: :label: Eq:omegas2 \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 :eq:`Eq:deltaMz` and either equation :eq:`Eq:omegas1` (for a swept-tip experiment) or equation :eq:`Eq:omegas2` (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 :math:`\vec{r}` in the sample the :math:`z` component of the magnetic field *only* at the beginning (:math:`\vec{r} = \vec{r}_{\mathrm{i}}`) and end (:math:`\vec{r} = \vec{r}_{\mathrm{f}}`) of the tip sweep. Adiabaticity ^^^^^^^^^^^^ We would also like to assess the adiabaticity of the sweep. With .. math:: M_{z}(t) = M_{z}(0) \: \cos{(\theta(t))}, the adiabaticity parameter is defined generally as .. math:: \alpha = \frac{\dot{\theta}}{\gamma B_1}. In a cryogenic ESR-MRFM observing :math:`\mathrm{E}^{\prime}` centers in quartz *via* cyclic adiabatic inversion, Wago and coworkers observed a peaking of signal when :math:`\alpha \sim 0.1` (note that they define the adiabaticity parameter as :math:`1/\alpha`) [#Wago1998jan]_. 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 :math:`\alpha \leq 0.1` [#Klein2000aug]_. In both of these experiments, a linear frequency sweep was used. We note that more efficient sweeps have been devised. [#Baum1985dec]_ [#Kupce1996feb]_ For a linear sweep, the adiabaticity parameter is *largest* near resonance, where .. math:: \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 :math:`\Omega` is given by equation :eq:`Eq:Omega` and :math:`\omega_1 = \gamma B_1` is the Rabi frequency. In a swept-tip experiment, the field at position :math:`\vec{r}` changes by an amount :math:`\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, :math:`\delta t = 1/(2 f_c)`. If we approximate the sweep as linear, then the adiabaticity parameter is given by .. math:: :label: Eq:alpha-swept-tip \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 :math:`\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 :math:`\vec{r}` in the sample. Equation :eq:`Eq:alpha-swept-tip` is only strictly valid at resonance and, moreover, does not account for the sinusoidal time dependence of :math:`\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 :eq:`Eq:alpha-swept-tip` 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 :math:`\omega_{\mathrm{i}}` to :math:`\omega_{\mathrm{f}}` in a time :math:` \Delta T_{\mathrm{sweep}}`; the period of the sweep :math:`\Delta T_ {\mathrm{sweep}}` is not restricted to be half a cantilever period. The adiabaticity parameter is independent of position :math:`\vec{r}` and, assuming a linear frequency sweep, equal to .. math:: :label: Eq:alpha-swept-freq \alpha_{\mathrm{res}} = \frac{1}{\gamma^2 B_1^2} \frac{\| \omega_{\mathrm{f}} - \omega_{\mathrm{i}} \|} {\Delta T_{\mathrm{sweep}}} As with equation :eq:`Eq:alpha-swept-tip`, equation :eq:`Eq:alpha-swept-freq` is only strictly valid for spins that pass through resonance. Spins far away from the resonant slice will experience an :math:`\alpha` even smaller and :math:`\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. .. _theory_nutation_section: Nutation ^^^^^^^^^^^^^^^^^^^^^^^^ .. autosummary:: mrfmsim_marohn.formula.polarization.rel_dpol_nut If the rf is turned on suddenly at :math:`t =0`, the magnetization will nutate around the effective field. Neglecting relaxation, the magnetization vector a time :math:`t_{\mathrm{p}}` after the start of the rf pulse is .. math:: \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} with :math:`\Omega`, the unitless resonance offset, given by Eq. :eq:`Eq:Omega` . Since we observe the :math:`z` component of magnetization, we are interested in: .. math:: \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 :math:`z` component of the magnetization due to the pulse is, after some simplification, .. math:: :label: Eq:deltaMz-pulse \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 :math:`\theta_p \equiv \omega_1 t_p` is the pulse angle. Plotting this function, we see that the biggest change in magnetization, :math:`\delta M_{z}(\theta_p,\Omega)/M_{z}(0) = -2`, occurs on resonance (:math:`\Omega = 0` ) when the pulse angle is set to :math:`\theta_p = \pi`. When :math:`\theta_p = \pi`, the full width at half max (FWHM) of the :math:`\delta M_ {z}(\pi,\Omega)/M_{z}(0)` function is approximately :math:`1.597 \Omega`. In other words, a :math:`\pi` pulse will invert magnetization over a resonant slice of whose width, in field units, is approximately :math:`1.597 \: B_1`. Equilibrium magnetization and Variance ----------------------------------------- equilibrium magnetization ^^^^^^^^^^^^^^^^^^^^^^^^^ :py:func:`formula.magnetization.mz_eq`: equilibrium magnetization per spin [#Brill]_ From the sample properties, we compute the magnetic moment :math:`\mu` of the state with the largest :math:`m_J` quantum number, .. math:: \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, .. math:: x = \dfrac{\mu B_0}{k_b T} \: [\mathrm{unitless}], and define the following two unitless numbers: .. math:: a &= \dfrac{2 \: J + 1}{2 \: J} \\ b &= \dfrac{1}{2 \: J} In terms of these intermediate quantities, the thermal-equilibrium polarization is given by .. math:: p_{\text{eq}} = a \coth{(a x)} - b \coth{(b x)} \: [\mathrm{unitless}]. The equilibrium magnetization is given by .. math:: \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, .. math:: mu_z^{\text{eq}} \approx \dfrac{\hbar^2 \gamma^2 \: J (J + 1)}{3 \: k_b T} B_0 equilibrium variance ^^^^^^^^^^^^^^^^^^^^ :py:func:`formula.magnetization.mz2_eq`: magnetization variance per spin, magnetization variance density [#Xue2011nov]_ 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 [#Xue2011nov]_ .. math:: \sigma_{{\cal M}_{z}}^{2} = \hbar^2 \gamma^2 \dfrac{J \: (J + 1)}{3} The magnetization variance density is obtained from :math:`\sigma_{{\cal M}_{z}}^{2}` by multiplying by the sample's spin density :math:`\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 :math:`B_0` and temperature :math:`T`. This approximation will *not* be valid for :math:`p \sim 1` electrons. Reference ---------- .. [#Klein2000aug] 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 `__]. .. [#Lee2012apra] 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 `__]. .. [#Wago1998jan] 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 `__]. .. [#Baum1985dec] 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 `__]. .. [#Kupce1996feb] 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 `__]. .. [#Brill] `"Brillouin and Langevin functions" `__ .. [#Xue2011nov] 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 `__]. :py:mod:`formula.polarization` module ---------------------------------------------------- .. automodule:: mrfmsim_marohn.formula.polarization :members: :undoc-members: :show-inheritance: .. automodule:: mrfmsim_marohn.formula.magnetization :members: :undoc-members: :show-inheritance: