Jess at 18:30, 17 September 2022

2022-09-17T18:30:51Z

WikiSysop: /* Phase Shifts in Precession Experiments */

2022-08-29T17:27:01Z

Phase Shifts in Precession Experiments

WikiSysop: /* Delayed Muonium Formation */

2022-08-29T17:25:48Z

Delayed Muonium Formation

WikiSysop: Created page with "Relaxonomy --> here ---- A wide class of µSR problems correspond to the following simple picture: the muon spin evolves with time in an initial state $\vert..."$

2022-08-20T01:26:45Z

Created page with "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..."

New page

[[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
<center><math>
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></center>
or
<center><math>
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></center>

For further reference, this is the Rate Equation for this simple model.
For <math>q=1</math> (no depolarization on transition) it becomes
<center><math>
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></center>

=== Example: No Depolarization in Initial State ===

Here we have <math>\lambda_i = 0</math> and <math>\lambda_f = \lambda</math>, giving
<center><math>
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></center>
or, for <math>q=1</math>,
<center><math>
P(t) = \left( {\lambda \over \lambda - \Lambda} \right) e^{-\Lambda t}
+ \left( {\Lambda \over \Lambda - \lambda} \right) e^{-\lambda t}
</math></center>
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>).

<center>
[[Image:ResPol_q1_Ri0_Rf1.png|600px|inline image (click to see full size)]]
{| width="80%"
! align="left" |
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>.
|}

[[Image:ResPol_q.5_Ri0_Rf1.png|600px|inline image (click to see full size)]]
{| width="80%"
! align="left" |
Figure 1b:
Depolarization functions for <math>q=0.5</math> and other parameters the same as in Fig. 1a.
|}

</center>

=== Example: No Depolarization in Final State ===

Here we have <math>\lambda_f = 0</math> and <math>\lambda_i = \lambda</math>, giving

<center><math>
P(t) = q \left( {\Lambda \over \Lambda + \lambda} \right) +
\left[1 - q \left( {\Lambda \over \Lambda + \lambda} \right) \right]
e^{-(\Lambda + \lambda)t}
</math></center>
or, for <math>q=1</math>,
<center><math>
P(t) = { \Lambda \over \Lambda + \lambda } +
{\lambda \over \Lambda + \lambda} e^{-(\Lambda + \lambda)t}
</math></center>

<center>
[[Image:ResPol_q1_Ri1_Rf0.png|600px|inline image (click to see full size)]]
{| width="80%"
! align="left" |
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>.
|}</center>
<center>
[[Image:ResPol_q.5_Ri1_Rf0.png|600px|inline image (click to see full size)]]
{| width="80%"
! align="left" |
Figure 2b:
Depolarization functions for <math>q=0.5</math>
and other parameters the same as in Fig. 2a.
|}</center>

== 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
<center><math>
D_i = {a^2 \over 4\tau_i} ,
</math></center>
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>:
<center><math> \Lambda = kc = 4 \pi c R D ,</math></center>
giving
<center><math> k = { \pi R a^2 \over \tau_i } .
</math></center>
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
<center><math>
\lambda_i = { 2 \delta^2 \tau_i \over 1 + \omega^2 \tau_i^2 }
</math></center>
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
<center><math>
\lambda_f = { 2 \delta^2 \tau_f \over 1 + \omega^2 \tau_f^2 } ,
</math></center>
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"
<center>Mu + X <math>\to</math> MuX </center>
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
<center><math>
\tilde{P}_{\rm res} = {1 \over 2} {\Lambda \over \Lambda
- i(\gamma_{\rm Mu} + \gamma_\mu) B }
</math></center>
which adds to the original diamagnetic component to give a
diamagnetic signal
<center><math>
\tilde{P}_D(t) = \left[ f_D + \tilde{P}_{\rm res} \right]
e^{i \gamma_\mu B t} .
</math></center>

=== Delayed Muonium Formation ===

The opposite situation (in a sense) applies in some insulators and semicondustors
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:
<center><math> \mu^+ + e^- \to </math> Mu</center>
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
<center><math>
\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></center>
and the muonium signal (in low field)
<center><math>
\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></center>
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>.

<center>
[[Image:DMF_0-400ns.png|600px|inline image (click to see full size)]]
{| width="80%"
! align="left" |
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.
|}</center><center>
[[Image:DMF_0-4us.png|600px|inline image (click to see full size)]]
{| width="80%"
! align="left" |
Figure 3b:
First 4 µs of the same muon polarization functions as in Fig. 3a
}
|}</center>

<center>
[[Image:DMF_0.5G.png|600px|inline image (click to see full size)]]
{| width="80%"
! align="left" |
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.
|}</center>

=== 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>:
<center><math>
\tilde{P}_i^{\scriptscriptstyle\rm RRF}(t) = \tilde{P}_i(t) \cdot e^{i \omega_0 t} </math></center>
and <center><math>
\tilde{P}_f^{\scriptscriptstyle\rm RRF}(t) = \tilde{P}_f(t) \cdot e^{i \omega_0 t} .
</math></center>

(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>:
<center><math>
\tilde{P}_i^{\scriptscriptstyle\rm RRF}(t) =
\int_0^\infty a^{\scriptscriptstyle\rm RRF}_i(\omega) e^{i \omega t} d\omega
</math></center> and <center><math>
\tilde{P}_f^{\scriptscriptstyle\rm RRF}(t) =
\int_0^\infty a^{\scriptscriptstyle\rm RRF}_f(\omega) e^{i \omega t} d\omega
</math></center>

(3) Then some simple calculus produces the residual polarization in the RRF,
<center><math>
\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></center>

(4) Finally, transform the result back into the lab frame:
<center><math>
\tilde{P}(t) = \tilde{P}^{\scriptscriptstyle\rm RRF}(t) \cdot e^{-i \omega_0 t} .
</math></center>

@@ Line 155: / Line 155: @@
 <center>
-[[Image:ResPol_q1_Ri0_Rf1.png|600px|inline image (click to see full size)]]
+[[Image:ResPol_q1_Ri0_Rf1.png|400px|inline image (click to see full size)]]
 {| width="80%"
 ! align="left" |
@@ Line 166: / Line 166: @@
 |}
-[[Image:ResPol_q.5_Ri0_Rf1.png|600px|inline image (click to see full size)]]
+[[Image:ResPol_q.5_Ri0_Rf1.png|400px|inline image (click to see full size)]]
 {| width="80%"
 ! align="left" |
@@ Line 191: / Line 191: @@
 <center>
-[[Image:ResPol_q1_Ri1_Rf0.png|600px|inline image (click to see full size)]]
+[[Image:ResPol_q1_Ri1_Rf0.png|400px|inline image (click to see full size)]]
 {| width="80%"
 ! align="left" |
@@ Line 202: / Line 202: @@
 |}</center>
 <center>
-[[Image:ResPol_q.5_Ri1_Rf0.png|600px|inline image (click to see full size)]]
+[[Image:ResPol_q.5_Ri1_Rf0.png|400px|inline image (click to see full size)]]
 {| width="80%"
 ! align="left" |
@@ Line 385: / Line 385: @@
 <center>
-[[Image:DMF_0-400ns.png|600px|inline image (click to see full size)]]
+[[Image:DMF_0-400ns.png|400px|inline image (click to see full size)]]
 {| width="80%"
 ! align="left" |
@@ Line 398: / Line 398: @@
 Only the real part of the polarization is shown.
 |}</center><center>
-[[Image:DMF_0-4us.png|600px|inline image (click to see full size)]]
+[[Image:DMF_0-4us.png|400px|inline image (click to see full size)]]
 {| width="80%"
 ! align="left" |
@@ Line 407: / Line 407: @@
 <center>
-[[Image:DMF_0.5G.png|600px|inline image (click to see full size)]]
+[[Image:DMF_0.5G.png|400px|inline image (click to see full size)]]
 {| width="80%"
 ! align="left" |

@@ Line 287: / Line 287: @@
 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 <i>phase shifts}.
+terms are also complex and therefore include <i>phase shifts</i>.
 Although this extension is formally trivial, it introduces
 a wealth of additional phenomenology.

@@ Line 351: / Line 351: @@
 === Delayed Muonium Formation ===
-The opposite situation (in a sense) applies in some insulators and semicondustors
+The opposite situation (in a sense) applies in some insulators and semiconductors
 where the muon ensemble differentiates initially into
 a <i>stable</i> diamagnetic fraction <math>f_D</math>,

The Residual Polarization Model - Revision history

Jess at 18:30, 17 September 2022

WikiSysop: /* Phase Shifts in Precession Experiments */

WikiSysop: /* Delayed Muonium Formation */

WikiSysop: Created page with "Relaxonomy --> here ---- A wide class of µSR problems correspond to the following simple picture: the muon spin evolves with time in an initial state \vert..."

WikiSysop: Created page with "Relaxonomy --> here ---- A wide class of µSR problems correspond to the following simple picture: the muon spin evolves with time in an initial state $\vert..."$