The Residual Polarization Model
Relaxonomy --> here
A wide class of µSR problems correspond to the following simple picture: the muon spin evolves with time in an initial state <math>\vert i \rangle</math> until time <math>t'</math>, when it makes an abrupt transition to state <math>\vert f \rangle</math>, in which it evolves subsequently. The transition times may in principle have any distribution <math>D(t')</math> such that <math>\int_0^\infty D(t') \, dt' \; = \; 1 </math>, but the mathematics will be simplest in cases where <math>D(t') = \Lambda \exp(-\Lambda t')</math> -- that is, where the transition occurs at a time that is distributed exponentially with a "reaction rate" <math>\Lambda</math>. This picture can be extended to two-transition and even repetitive-transition cases, but the salient features emerge most vividly from the simplest cases.
Because it was first used to describe the observable diamagnetic muon polarization <math>\vert f \rangle \equiv \mu_D</math> following chemical reaction of muonium <math>\vert i \rangle \equiv </math> Mu in weak transverse magnetic fields (wTF), I call this picture the "Residual Polarization Model." Since it can also be used to describe situations in which the time-dependence of the muon polarization in the initial state is observable, this nomenclature is logically imperfect; but it does have the desired connotations, so I will use it.
First let me be fairly formal and rigourous to show how general this formulation might be; then I will revert to simple cases and build up slowly to the more elaborate.
Formalism
The symbols <math>\vert i \rangle</math> and <math>\vert f \rangle</math> do not refer to energy eigenkets, of course, but to "superstates" -- generalizations of the notion of a "state" in which <math>\vert i \rangle</math> could refer to something as complicated as a "diffusing muonium atom" as long as we can get away with treating its spin polarization as a well-defined funtion of time. Similarly with <math>\vert f \rangle</math>, whose polarization would evolve in a well-defined manner if muons started in this state at <math>t=0</math>. The term "spin polarization" need not refer only to the muon polarization <math>\vec{P}_\mu(t)</math>, much less its scalar component <math>P^\mu_j(t)</math> along the <math>j^{\rm th}</math> axis; for the familiar case <math>\vert i \rangle \equiv</math> Mu, for instance, the initial polarization of the muon is subsequently shared between muon, electron and "off-diagonal" degrees of freedom of the spin density matrix. Whether all these need to be "tracked" through the "reaction" depends, of course, on whether the other spin degrees of freedom are passed coherently to analogous ones in <math>\vert f \rangle</math>, in which case their values at the time of reaction affect the subsequent evolution. This can get messy, and ought probably to be formulated in terms of the density matrix itself, but I will stick with a phenomenological formalism where <math>\Pi(t)</math> represents "the spin polarization" in the relevant superstate, however many components or degrees of freedom it must encompass to satisfy the requirements of the problem.
Hmmm... This has to be done with quantum states and operators for the general case. Later.
Simple Exponentially Relaxing States
We begin with what might be considered the trivial case: plain exponential muon relaxation in LF geometry so that only the longitudinal component of the muon polarization matters and the whole problem is a real, scalar one. Then <math>\vert i \rangle</math> has a time dependent polarization <math>P_i(t) = e^{-\lambda_i t}</math> and <math>\vert f \rangle</math> would have <math>P_f(t) = e^{-\lambda_f t}</math> if it started fully polarized at <math>t=0</math>.
The depolarization rate in state <math>\vert i \rangle</math> is <math>\lambda_i</math>. The depolarization rate in state <math>\vert f \rangle</math> is <math>\lambda_f</math>. For generality we may let a fraction <math>(1-q)</math> of the polarization be "abruptly" lost at the time of transition between <math>\vert i \rangle</math> and <math>\vert f \rangle</math> -- for muonium formation in low field, this fraction would be effectively 50%. (If <math>\Lambda > \sim 4.463</math> GHz, this approximation is invalid.) If <math>\vert i \rangle \to \vert f \rangle</math> at time <math>t'</math> with <math>D(t') \; = \; \Lambda \, e^{-\Lambda \, t'}</math> so that <math>\int_0^\infty D(t') \, dt' = 1 </math>, we have
P(t) =
e^{-\Lambda t} \cdot e^{-\lambda_i t} + q \Lambda \, \int_0^t
e^{-\lambda_i t'} \cdot e^{-\lambda_f(t-t')} \cdot e^{-\Lambda t'} dt'
</math> <math>
= e^{-(\Lambda + \lambda_i)t} + q \Lambda \, e^{-\lambda_f t}
\int_0^t e^{-(\Lambda + \lambda_i - \lambda_f)t'} dt'
</math> <math>
= e^{-(\Lambda + \lambda_i)t} -
q {\Lambda \over \Lambda + \lambda_i - \lambda_f} \, e^{-\lambda_f t}
\int_0^{-(\Lambda + \lambda_i - \lambda_f)t} e^u \, du
</math> <math>
= e^{-(\Lambda + \lambda_i)t} -
q {\Lambda \over \Lambda + \lambda_i - \lambda_f} \, e^{-\lambda_f t}
\left[e^{-(\Lambda + \lambda_i - \lambda_f)t} - 1 \right]
</math> <math>
= \; e^{-(\Lambda + \lambda_i)t} -
q {\Lambda \over \Lambda + \lambda_i - \lambda_f}
\left[e^{-(\Lambda + \lambda_i)t} - e^{-\lambda_f t} \right]
</math>or
P(t) = \left[ 1 - q \left( {\Lambda \over \Lambda + \lambda_i - \lambda_f} \right) \right] e^{-(\Lambda + \lambda_i)t}
+ q \left( {\Lambda \over \Lambda + \lambda_i - \lambda_f} \right) e^{-\lambda_f t} .
</math>For further reference, this is the Rate Equation for this simple model. For <math>q=1</math> (no depolarization on transition) it becomes
P(t) =
\left( {\lambda_i - \lambda_f \over \Lambda + \lambda_i - \lambda_f} \right) e^{-(\Lambda + \lambda_i)t}
+ \left( {\Lambda \over \Lambda + \lambda_i - \lambda_f} \right) e^{-\lambda_f t}
</math>Example: No Depolarization in Initial State
Here we have <math>\lambda_i = 0</math> and <math>\lambda_f = \lambda</math>, giving
P(t) =
\left[ 1 - q \left( {\lambda \over \Lambda - \lambda} \right) \right] e^{-\Lambda t}
+ q \left( {\Lambda \over \Lambda - \lambda} \right) e^{-\lambda t}
</math>or, for <math>q=1</math>,
P(t) = \left( {\lambda \over \lambda - \Lambda} \right) e^{-\Lambda t}
+ \left( {\Lambda \over \Lambda - \lambda} \right) e^{-\lambda t}
</math>Now, this may look unconvincing, since it appears to diverge as <math>\Lambda \to \lambda</math>. However, appearances are deceiving! The reader may wish to show as an exercise that there is no anomaly at <math>\Lambda = \lambda</math>; I will let the computer speak for itself.
Some sample relaxation functions from this formula are shown in Fig. 1.
Note the similarity of the <math>q=1</math> relaxation functions to the widely overused "stretched exponential" function <math>G_{\rm SE}(t) = \exp(-[\lambda t]^\beta/\beta)</math> (shown in red for <math>\lambda=1</math> µs<math>^{-1}</math> and <math>\beta=1.5</math>).
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Figure 1a: Depolarization functions for <math>q=1</math> and a relaxation rate of <math>\lambda = 1</math> µs<math>^{-1}</math> in the final state with no relaxation in the initial state, for various choices of the "reaction" rate <math>\Lambda = \{0.5, 1.01, 3, 10, 100\} </math> µs<math>^{-1}</math>. |
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Figure 1b: Depolarization functions for <math>q=0.5</math> and other parameters the same as in Fig. 1a. |
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Example: No Depolarization in Final State
Here we have <math>\lambda_f = 0</math> and <math>\lambda_i = \lambda</math>, giving
P(t) = q \left( {\Lambda \over \Lambda + \lambda} \right) +
\left[1 - q \left( {\Lambda \over \Lambda + \lambda} \right) \right]
e^{-(\Lambda + \lambda)t}
</math>or, for <math>q=1</math>,
P(t) = { \Lambda \over \Lambda + \lambda } +
{\lambda \over \Lambda + \lambda} e^{-(\Lambda + \lambda)t}
</math>|
Figure 2a: Depolarization functions for <math>q=1</math> with a relaxation rate of <math>\lambda = 1</math> µs<math>^{-1}</math> in the initial state and no relaxation in the final state, for various choices of the "reaction" rate <math>\Lambda = \{0.1, 0.5, 1.01, 3, 10, 200\} </math> µs<math>^{-1}</math>. |
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Figure 2b: Depolarization functions for <math>q=0.5</math> and other parameters the same as in Fig. 2a. |
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Application to Muonium Diffusion and Trapping
Suppose Mu initially hops about in the host lattice at a rate <math>\tau_i^{-1}</math>, giving an "intrinsic" diffusion constant
D_i = {a^2 \over 4\tau_i} ,
</math>where <math>a</math> is the lattice constant. If the concentration of point defects (e.g., impurities) is <math>c</math> and the effective radius of their strain fields (defined vaguely as the radius at which the energy difference between sites due to the strain fields is sufficient to slow down the Mu diffusion enough that I may regard Mu as "nearly static") is <math>R</math>, the Mu trapping rate (defined in this vague way) is some "rate constant" <math>k</math> times the defect concentration <math>c</math>:
giving
Meanwhile if a weak longitudinal field (wLF) of <math>B</math> is applied, the relaxation rate in state <math>\vert i \rangle</math> is given by
\lambda_i = { 2 \delta^2 \tau_i \over 1 + \omega^2 \tau_i^2 }
</math>where <math>\delta</math> is the static dipolar width (which can sometimes be determined directly from wTF\ measurements in the static limit) and <math>\omega = \gamma_{\rm Mu} B</math> is the muonium Larmor frequency (I assume the low-field limit where hyperfine splittings can be neglected).
Once trapped, the Mu still hops slowly. Of course it hops at different rates when trapped at different "depths" in the strain fields, but I shall just pretend for now that the hopping around in the strain field can be characterized by a slow hop rate <math>\tau_f^{-1}</math> in the state <math>\vert f \rangle</math> [trapped state]. Then the final state relaxation rate is
\lambda_f = { 2 \delta^2 \tau_f \over 1 + \omega^2 \tau_f^2 } ,
</math>where I have tacitly assumed that <math>\delta</math> is the same in both states [<math>\vert i \rangle</math> and <math>\vert f \rangle</math>] because <math>R \gg a</math> and the Mu atom always sees the same types of neighbours.
Then, in principle, the relaxation function depends only upon <math>\tau_i</math>, <math>\tau_f</math>, <math>R</math> and the known or (sometimes) independently measurable parameters <math>c</math>, <math>a</math>, <math>B</math> and <math>\delta</math>. It may or may not be worth fitting the data directly to this formula with <math>\tau_i</math>, <math>\tau_f</math> and <math>R</math> as free parameters.
Phase Shifts in Precession Experiments
The conventional way to represent precession in transverse field is to let <math>\tilde{P}^\mu(t)</math> be a complex muon polarization whose real part is the projection of <math>\vec{P}^\mu(t)</math> along its initial direction (assumed perpendicular to the applied field) and whose imaginary part is the projection along the orthogonal axis -- e.g., if <math>\vec{B} \parallel \hat{z}</math> and <math>\vec{P}^\mu(0) \parallel \hat{x}</math>, <math>\tilde{P}(t) \equiv P^\mu_x(t) + i P^\mu_y(t)</math>.
If we replace <math>\lambda_i</math> and/or <math>\lambda_f</math> with <math>\kappa_i = \lambda_i - i \omega_i</math> or <math>\kappa_f = \lambda_f - i \omega_f</math>, so that the argument of the exponential function has an imaginary part, then the description given earlier covers TF-µSR or wTF-MSR precession experiments -- e.g., for Mu <math>\to</math> diamagnetic compound in wTF\ muonium chemistry, for <math>F^+ \to F^-</math> hyperfine states of muonic atoms or for <math>\mu^+ \to</math> Sn traps in Sb. Equation (\ref{eq:GeneralRateEq}) is unchanged, but since <math>\kappa_i</math> and/or <math>\kappa_f</math> are complex, the prefactors in front of the time-dependent terms are also complex and therefore include phase shifts. Although this extension is formally trivial, it introduces a wealth of additional phenomenology.
In the first two examples outlined briefly below, the exponential relaxation <math>(e^{- \lambda t})</math> picture used to derive the General Rate Equation is replaced by a pure precession picture <math>(e^{i \omega t})</math> with no relaxation -- that is, <math>(-\lambda \to i \omega)</math>. Relaxation can be re-introduced by allowing <math>\omega</math> to have a positive imaginary component. For simplicity we will restrict ourselves to low magnetic field where muonium precession can be approximated by a single frequency <math>\omega_{\rm Mu} = - \gamma_{\rm Mu} B</math> (where <math>H</math> is the applied transverse field) and is in the opposite sense to the diamagnetic precession signal at the much lower frequency <math>\omega_\mu = \gamma_\mu B</math>. (Both magnetogyric ratios <math>\gamma</math> are taken to be positive constants.)
Fast-Reacting Muonium
The oldest application of this model is to the residual polarization in a system where muons differentiate at <math>t=0</math> into a "prompt" diamagnetic fraction <math>f_D</math> and an initial muonium fraction <math>f_{\rm Mu}</math>. (We will neglect any "missing polarization" as this enters simply as an overall reduction of all amplitudes.) The Mu fraction reacts chemically with reagent "X"
at exponentially distributed times (rate <math>\Lambda</math>) to form a "delayed" diamagnetic fraction which is reduced in magnitude and rotated in phase due to the fast precession of Mu prior to chemical reaction. In strong magnetic fields where the Mu hyperfine splitting becomes important, this can be somewhat more complicated; and if the reaction rate <math>\Lambda</math> is fast enough to compete with the Mu hyperfine frequency <math>\omega_0 \approx 4.463</math> GHz itself, then complications again arise; but for low fields and modest reaction rates we may treat the evolution of the muon spin in Mu as a simple precession with frequency <math>\omega_{\rm Mu} = - \gamma_{\rm Mu} B</math> (in the opposite sense to the diamagnetic precession at <math>\omega_\mu = + \gamma_\mu B</math>) and the effect of "hyperfine oscillations" can be treated as a simple loss of half the muon polarization in Mu. In that case we can simply replace <math>-\lambda_i</math> by <math>-\gamma_{\rm Mu} B</math> and <math>-\lambda_f</math> by <math>+\gamma_\mu B</math> in the General Rate Equation, giving
\tilde{P}_{\rm res} = {1 \over 2} {\Lambda \over \Lambda
- i(\gamma_{\rm Mu} + \gamma_\mu) B }
</math>which adds to the original diamagnetic component to give a diamagnetic signal
\tilde{P}_D(t) = \left[ f_D + \tilde{P}_{\rm res} \right]
e^{i \gamma_\mu B t} .
</math>Delayed Muonium Formation
The opposite situation (in a sense) applies in some insulators and semiconductors where the muon ensemble differentiates initially into a stable diamagnetic fraction <math>f_D</math>, a stable muonium fraction <math>f_{\rm Mu}</math> and an unstable diamagnetic fraction <math>f_X</math> which captures an electron at exponentially distributed times (rate <math>\Lambda</math>) to form delayed muonium:
where the free electron presumably comes from the incoming muon's ionization track. (This can only work in systems with substantial <math>e^-</math> mobilities.) In this case there are two detectable precessing components, the diamagnetic signal
\tilde{P}_D(t) = \left[ f_D + f_X \left( 1 - {1 \over 2}
{ \Lambda \over \Lambda - i(\gamma_\mu + \gamma_{\rm Mu}) B }
\right) e^{-\Lambda t} \right] e^{i \gamma_\mu B t}
</math>and the muonium signal (in low field)
\tilde{P}_{\rm Mu}(t) = {1 \over 2} \left[ f_{\rm Mu} + f_X
{ \Lambda \over \Lambda - i(\gamma_\mu + \gamma_{\rm Mu}) B }
\right] e^{-i \gamma_{\rm Mu} B t} .
</math>In this case both components (D and Mu) show a phase shift in general and the reaction rate can in many cases be measured directly from the decaying component of the diamagnetic signal. Here I have neglected any relaxation in either the initial or the final states; this can be put in easily enough, just by letting the frequencies be complex: <math>\exp(i \omega t) \to \exp(i [\omega + i \lambda]t)</math>.
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Figure 3a: First 400 ns of muon polarization functions for delayed muonium formation (<math>q=0.5</math>) in an applied magnetic field of 20 G: a non-relaxing initial diamagnetic fraction captures radiolysis electrons to form muonium at rates <math>\Lambda = \{0,1,3,7,20\}</math> µs<math>^{-1}</math> after which Mu relaxes at 0.1 µs<math>{-1}</math>. Only the real part of the polarization is shown. |
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Figure 3b: First 4 µs of the same muon polarization functions as in Fig. 3a } |
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Figure 4: Muon polarization functions for delayed muonium formation (<math>q=0.5</math>) in an applied magnetic field of 0.5 G: a non-relaxing initial diamagnetic fraction captures radiolysis electrons to form non-relaxing muonium at rates <math>\Lambda = \{0.5,1,3,7,20,100\}</math> µs<math>^{-1}</math>. Only the real part of the polarization is shown. Note the obvious phase shifts of the Mu signal due to delayed formation. |
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General Use with Fourier Transforms
The complex formulation allows treatment of yet more general cases in which the time dependence of the muon polarization in <math>\vert i \rangle</math> or <math>\vert f \rangle</math> is non-exponential, using the following procedure:
(1) If the Fourier transform of <math>\tilde{P}_i(t)</math> or <math>\tilde{P}_f(t)</math> contains negative frequency components, most FFT algorithms will get confused; to avoid this, first transform into a rotating reference frame (RRF) at frequency <math>\omega_0</math>:
and
\tilde{P}_f^{\scriptscriptstyle\rm RRF}(t) = \tilde{P}_f(t) \cdot e^{i \omega_0 t} .
</math>(2) Next, Fourier transform <math>\tilde{P}_i^{\scriptscriptstyle\rm RRF}(t)</math> and <math>\tilde{P}_f^{\scriptscriptstyle\rm RRF}(t)</math> to obtain their frequency components <math>a(\omega)</math>:
\tilde{P}_i^{\scriptscriptstyle\rm RRF}(t) = \int_0^\infty a^{\scriptscriptstyle\rm RRF}_i(\omega) e^{i \omega t} d\omega
</math>and
\tilde{P}_f^{\scriptscriptstyle\rm RRF}(t) = \int_0^\infty a^{\scriptscriptstyle\rm RRF}_f(\omega) e^{i \omega t} d\omega
</math>(3) Then some simple calculus produces the residual polarization in the RRF,
\tilde{P}^{\scriptscriptstyle\rm RRF}(t) = e^{-\Lambda t} \tilde{P}_i^{\scriptscriptstyle\rm RRF}(t)
+ \Lambda \int_0^\infty d\omega a^{\scriptscriptstyle\rm RRF}_i(\omega)
\int_0^\infty d\omega' a^{\scriptscriptstyle\rm RRF}_f(\omega') \left[ {
e^{-\Lambda t} \cdot e^{i \omega t} - e^{i \omega' t}
\over i(\omega - \omega') - \Lambda} \right] .
</math>(4) Finally, transform the result back into the lab frame:
\tilde{P}(t) = \tilde{P}^{\scriptscriptstyle\rm RRF}(t) \cdot e^{-i \omega_0 t} .
</math>
