From 99bc917232ab49df31e11cb7d13d8ed077854b66 Mon Sep 17 00:00:00 2001
From: "Joachim Wuttke (h)" <j.wuttke@fz-juelich.de>
Date: Wed, 7 Jun 2023 16:37:08 +0200
Subject: [PATCH] notation c_\pm instead of lambda_pm

---
 Doc/DevRef/DevRef.pdf     | Bin 2215697 -> 2212548 bytes
 Doc/DevRef/Macros.tex     |   1 +
 Doc/DevRef/Polarized.tex  |  84 +++++++++++++++-----------------------
 Doc/DevRef/Scattering.tex |   2 +-
 4 files changed, 34 insertions(+), 53 deletions(-)

diff --git a/Doc/DevRef/DevRef.pdf b/Doc/DevRef/DevRef.pdf
index d774482b227345b6cf56ce44bd1389bb9689e8f2..dd0b8cedb304376e6b0a84283f964fc9ba8a0f53 100644
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diff --git a/Doc/DevRef/Macros.tex b/Doc/DevRef/Macros.tex
index 80f1af9b01a..ace0e644eb5 100644
--- a/Doc/DevRef/Macros.tex
+++ b/Doc/DevRef/Macros.tex
@@ -63,6 +63,7 @@
 \def\etal{\E{et al.}}
 \def\e{\mathrm e}
 \def\eB{\v{e}_{\v{B}}}
+\def\evp{c} % eigenvalue of \hat p
 \def\hx{\v{\hat x}}
 \def\hy{\v{\hat y}}
 \def\idest{\E{i.e.\ }}
diff --git a/Doc/DevRef/Polarized.tex b/Doc/DevRef/Polarized.tex
index 60ae943ab00..501c4dac5e8 100644
--- a/Doc/DevRef/Polarized.tex
+++ b/Doc/DevRef/Polarized.tex
@@ -133,65 +133,45 @@ waves that propagate with two different vertical wavenumbers that correspond
 to the two eigenvalues of~\cref{EWaveZ2}.
 
 We now consider a homogeneous layer with constant potential.
-We write the formal solution of~\cref{EWaveZ2} as
+Similar to \cite{KeRT03,KeRT08},
+we write the formal solution of~\cref{EWaveZ2} as
 \begin{equation}
   \psi(z) = \e^{-i\hat p z} t + \e^{i\hat p z} r,
 \end{equation}
 where $t$ and~$r$ are the transmitted and reflected spinor amplitudes.
 By comparison with~\cref{EWaveZ2}, we see that the square of the operator~$\hat p$ is
-\begin{equation}
+\begin{equation}\label{Dhp2}
   \hat p^2 = \kappa_0^2-4\pi \hat \mv = \kappa_0^2-4\pi(\sldN + \sldM \eB \Pauli).
 \end{equation}
-With \cite[\S~55, Exercice~1, p.~198]{LL3}, we find
-the operator\footnote
+Without derivation,\footnote
+{To verify, use standard properties of Pauli matrices.
+Square~\cref{Ehp1} to reproduce~\cref{Dhp2}.
+Then confirm that $\lambda+\pm^2$ are eigenvalues of $\hat p^2$.
+See \cite[\S~55, Exercice~1, p.~198]{LL3}.}
+we state that the square root of $\hat p^2$ is the operator\footnote
 {Implemented in \T{MatrixFlux::computeP()}.}
-\begin{equation}\label{EPauliRep}
+\begin{equation}\label{Ehp1}
   \hat{p}
-  = \sqrt{\hat{p}^2}
-  = \frac{1}{2}\left[(\lambda_++\lambda_-) + (\lambda_+-\lambda_-) \eB \Pauli\right],
+  = \frac{1}{2}\left[(\evp_++\evp_-) + (\evp_+-\evp_-) \eB \Pauli\right],
 \end{equation}
 expressed through its eigenvalues
 \begin{equation}
-\lambda_\pm
-  \coloneqq \sqrt{ \kappa_0^2 - 4 \pi (\sldN \mp \sldM) }.
+\evp_\pm
+  \coloneqq \sqrt{ \kappa_0^2 - 4 \pi \sldN \pm 4 \pi \sldM }.
 \end{equation}
 For future reference, we note the inverse operator\footnote
 {Implemented in \T{MatrixFlux::computeInverseP()}.}
 \begin{equation}
   \hat{p}^{-1}
-  \coloneqq \frac{1}{2\lambda_+\lambda_-}
-            \left[(\lambda_++\lambda_-) - (\lambda_+-\lambda_-) \eB \Pauli\right].
+  \coloneqq \frac{1}{2\evp_+\evp_-}
+            \left[(\evp_++\evp_-) - (\evp_+-\evp_-) \eB \Pauli\right].
 \end{equation}
 
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{From old document ``Reflectivity''}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
-By choosing the convention that the plane of a multilayer lies within the $x$-$y$ plane, and using the appropriate boundary conditions, the three-dimensional Helmholtz equation reduced to the usual one-dimensional Helmholtz equation $\hat H \Psi = 0$.
-Following \cite{KeRT08} and \cite{KeRT03}, the Hamiltonian for a magnetized layer $m$ in a sample is given by
-\begin{align}
-\hat H_m &= \frac{\partial^2}{\partial z^2} + k_0^2 - \frac{2 \pi \hbar^2}{m_n} \left( b^{\mathrm{N}} + b^{\mathrm{M}} \vec \sigma \cdot \vec b\right) = \frac{\partial^2}{\partial z^2} + k_0^2 - 4 \pi \underbrace{\left( \sldN + \sldM \vec \sigma \cdot \vec b \right)}_\rho = \frac{\partial^2}{\partial z^2} + p_m^2
-\end{align}
-where $\vec b$ is the unit vector of the magnetic field.
-Within this convention, the interaction is parametrized by the usual nuclear \textsc{Sld} $\sldN$ and the magnetic \textsc{Sld} $\sldM$, which are both scalar quantities.
-For convenience, we then introduce the operator $\hat p_m$ in layer $m$
-\begin{align}
-\hat p_m &\coloneqq \frac{1}{2} \left( \left( \lambda_+ + \lambda_- \right) + \left( \lambda_+ - \lambda_- \right) \vec \sigma \cdot \vec b \right) \,,
-\end{align}
-with the eigenvalues given by
-\begin{align}
-\lambda_\pm &= \sqrt{ k_0^2 - 4 \pi \left( \rho^\mathrm{N} \mp \rho^\mathrm{M} \right) }  \,.
-\end{align}
-The ansatz for the wave function is then
-\begin{align}
-\Ket{\Psi_m\left( z \right)} &= \hat S_m \left( z \right) \Ket{\Psi_m(0)} \,.
-\end{align}
-Here, the propagator $\hat S$ is a $2\times 2$ matrix given by the expression
-\begin{align}
-\hat S\left( z \right) &= \exp{i \hat p_m \left(z - z_{m-1} \right) } \hat t_m + \exp{-i \hat p_m \left(z - z_{m-1} \right) } \hat r_m \,,
-\end{align}
-and it propagates the wave function $\Ket{\Psi_m(0)}$ at the bottom of layer $m$ to position $z$.
-This notation is identical to the one introduced by \cite{KeRT08} and \cite{KeRT03} and $\hat t_m$ and $\hat r_m$ are operators that can be written as $2 \times 2$ matrices.
 
 %===============================================================================
 \subsection{Transfer Matrix Method}
@@ -386,7 +366,7 @@ The matrix $\eB\Pauli$ has the eigenvalues $\pm 1$ and the normalized eigenvecto
  -\eBcomp_x+i\eBcomp_y\\ 1 +\eBcomp_z \end{pmatrix}.
 \end{equation}
 For readability, we have omitted the subscript $\v{B}$ from the components of~$\eB$.
-The matrix $\UU{p}$ has the eigenvalues $\lambda_\pm$,
+The matrix $\UU{p}$ has the eigenvalues $\evp_\pm$,
 and the same eigenvectors~\cref{Ev1v2} as $\eB\Pauli$.
 We introduce the eigenvector matrix
 \begin{equation}
@@ -395,16 +375,16 @@ We introduce the eigenvector matrix
 Then $\UU{p}$ has the eigenvalue decomposition
 \begin{equation}
   \UU{p}
-  = \UU{Q} \, \begin{pmatrix}\lambda_+ & 0 \\ 0 & \lambda_-\end{pmatrix} \UU{Q}^\dagger,
+  = \UU{Q} \, \begin{pmatrix}\evp_+ & 0 \\ 0 & \evp_-\end{pmatrix} \UU{Q}^\dagger,
 \end{equation}
 and the phase-shift matrix \cref{EdeltaP} can be written
 \begin{equation}\label{EdP2}
   \UU\delta
   =
-    \UU{Q} \, \begin{pmatrix}\e^{id\lambda_+} & 0 \\ 0 & \e^{id\lambda_-}\end{pmatrix} \UU{Q}^\dagger.
+    \UU{Q} \, \begin{pmatrix}\e^{id\evp_+} & 0 \\ 0 & \e^{id\evp_-}\end{pmatrix} \UU{Q}^\dagger.
 \end{equation}
 
-In the case $\v{B} = 0$, we have $\lambda_+=\lambda_-$,
+In the case $\v{B} = 0$, we have $\evp_+=\evp_-$,
 which has for consequence that \cref{EdP2} holds for whatever direction vector~$\eB$;
 the simplest choice is $\eB(B\!=\!0)\coloneqq\v{\hat z}$ so
 that $\UU{Q}=\UU{1}$.
@@ -797,9 +777,9 @@ Using the well-known representation of the Pauli matrices, the matrix $\UU{p_a}$
 \end{align}
 with
 \begin{align}
-\alpha &\coloneqq \lambda_+ + \lambda_- \,, & \beta &\coloneqq \lambda_+ - \lambda_- \,.
+\alpha &\coloneqq \evp_+ + \evp_- \,, & \beta &\coloneqq \evp_+ - \evp_- \,.
 \end{align}
-$\lambda_+$ and $\lambda_-$ are the eigenvalues of the operator $\UU{p}$.
+$\evp_+$ and $\evp_-$ are the eigenvalues of the operator $\UU{p}$.
 The inverse is given by
 \begin{align}
 \UU{p}^{-1} &= \frac{2}{\alpha^2 - \beta^2}
@@ -862,9 +842,9 @@ Furthermore, this form is not very convenient for numerical evaluations since th
 For this reason, the first diagonal exponential matrix can be multiplied inside to obtain
 \begin{align}
 \exp{i \UU{p} d} &= \UU{Q} \begin{pmatrix}
-\exp{ i d \lambda_+ } & 0
+\exp{ i d \evp_+ } & 0
  \\
-0 & \exp{ i d \lambda_- }
+0 & \exp{ i d \evp_- }
 \end{pmatrix} \UU{Q}^\dagger
 \end{align}
 The case $\eB = 0$ needs to be considered separately, then $\beta = 0$ and obviously
@@ -1048,21 +1028,21 @@ This requires the decomposition of
 \U{\Phi_j}\left( z \right) &= \exp{i \UU{p_a} \left(z - z_{j-1} \right) } \U{t_j} + \exp{-i \UU{p_a} \left(z - z_{j-1} \right) } \U{r_j} \,,
 \end{align}
 into its eigenmodes.
-In order to achieve this, we again apply the eigenvalue decomposition of $\UU{p_a}$, as it is also used in \cref{eq:roughness_tanh_eigenvalue_decomposition}, where the transformation matrix is given by \cref{eq:transformation_matrix_q} and the eigenvalues are of course $\lambda_+$ and $\lambda_-$ and we have
+In order to achieve this, we again apply the eigenvalue decomposition of $\UU{p_a}$, as it is also used in \cref{eq:roughness_tanh_eigenvalue_decomposition}, where the transformation matrix is given by \cref{eq:transformation_matrix_q} and the eigenvalues are of course $\evp_+$ and $\evp_-$ and we have
 \begin{align}
 \exp{i \UU{p_j} z} &= \UU{Q} \cdot \exp {i \Lambda z } \cdot \UU{Q}^\dagger
 \\
 &= \UU{Q} \cdot \begin{pmatrix}
-\exp{i \lambda_+ z} & 0 \\ 0 & 0
+\exp{i \evp_+ z} & 0 \\ 0 & 0
 \end{pmatrix} \cdot \UU{Q}^\dagger +
 \UU{Q} \cdot \begin{pmatrix}
-0 & 0 \\ 0 & \exp {i \lambda_- z}
+0 & 0 \\ 0 & \exp {i \evp_- z}
 \end{pmatrix} \cdot \UU{Q}^\dagger
 \\
-&= \exp{i \lambda_+ z} \UU{Q} \cdot \begin{pmatrix}
+&= \exp{i \evp_+ z} \UU{Q} \cdot \begin{pmatrix}
 1 & 0 \\ 0 & 0
 \end{pmatrix} \cdot \UU{Q}^\dagger +
-\exp {i \lambda_- z}
+\exp {i \evp_- z}
 \UU{Q} \cdot \begin{pmatrix}
 0 & 0 \\ 0 & 1
 \end{pmatrix} \cdot \UU{Q}^\dagger \,.
@@ -1070,17 +1050,17 @@ In order to achieve this, we again apply the eigenvalue decomposition of $\UU{p_
 This decomposition is valid unless the magnetic field vanishes.
 In the latter case, we have
 \begin{align}
-\exp{i \UU{p_a} z} &= \exp{i \lambda_+ z} \cdot \begin{pmatrix}
+\exp{i \UU{p_a} z} &= \exp{i \evp_+ z} \cdot \begin{pmatrix}
 1 & 0 \\ 0 & 0
 \end{pmatrix} +
-\exp {i \lambda_- z}
+\exp {i \evp_- z}
 \cdot \begin{pmatrix}
 0 & 0 \\ 0 & 1
 \end{pmatrix} \,.
 \end{align}
 The resulting matrix can be written as a sum of two matrices
 \begin{align}
-\exp{i \UU{p_a} z} &= \exp{i \lambda_+ z} \UU{T_2} + \exp {i \lambda_- z} \UU{T_1} \,,
+\exp{i \UU{p_a} z} &= \exp{i \evp_+ z} \UU{T_2} + \exp {i \evp_- z} \UU{T_1} \,,
 \end{align}
 and the needed amplitudes are then given by
 \begin{align}
diff --git a/Doc/DevRef/Scattering.tex b/Doc/DevRef/Scattering.tex
index b89d0629579..c82f94c7c7a 100644
--- a/Doc/DevRef/Scattering.tex
+++ b/Doc/DevRef/Scattering.tex
@@ -100,7 +100,7 @@ given by the dispersion relation%
   \hbar\omega = \frac{(\hbar K)^2}{2m}.
 \end{equation}
 The Schrödinger equation~\cref{ESchrodi1} then takes the simple form
-\begin{equation}\label{ESchrodi2}
+\begin{equation}\label{EWvEq}
   \left\{\Nabla^2+K^2-4\pi v_\snuc(\r)\right\} \psi(\r) = 0
 \end{equation}
 with the rescaled form of Fermi's pseudopotential
-- 
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