diff --git a/Doc/UserManual/Assemblies.tex b/Doc/UserManual/Assemblies.tex index fa5030e252b78b4b766624414a7e19b3fed21ad0..c281f1916dc4c6abe59b7fd56d15bcbeebf04317 100644 --- a/Doc/UserManual/Assemblies.tex +++ b/Doc/UserManual/Assemblies.tex @@ -62,7 +62,7 @@ with the indicator function 1&\text{~if $\r$ in p,}\\[.2ex] 0&\text{~otherwise.} \end{array}\right. \end{equation} -\nomenclature[1χ032 2p000]{$\chi_\sp(\r)$}{indicates whether $\r$ is in particle of type~$\sp$}% +\nomenclature[1χ032 2p000]{$\chi_\sp(\r)$}{Indicates whether $\r$ is in particle of type~$\sp$}% \index{Indicator function}% The following formalism, however, shall not rely on \cref{EDvpHomo}, but also allow for soft particles for which $\Delta v_\sp(\r)$ is a continuous function of~$\r$. diff --git a/Doc/UserManual/FormFactors.tex b/Doc/UserManual/FormFactors.tex index 05da943ab0a29f1faac0358d89ad5f6ae0e6576d..fe0663ddeded67c6684e069df113367d5a31621b 100644 --- a/Doc/UserManual/FormFactors.tex +++ b/Doc/UserManual/FormFactors.tex @@ -730,7 +730,7 @@ for four different tilt angles~$\vartheta$ (rotation around the $y$ axis).} \end{figure} \paragraph{History and Derivation}\strut\\ -For real wave vectors, this form factor is well known; +For real wavevectors, this form factor is well known; it goes back to Lord Rayleigh. In \IsGISAXS, it has been implemented as form factor \E{Cylinder} \cite[Eq.~2.27]{Laz08} \cite[Eq.~223]{ReLL09}, @@ -960,7 +960,7 @@ computed with $R=3.9$~nm.} \end{figure} \paragraph{History and Derivation}\strut\\ -For real wave vectors, this form factor is well known; +For real wavevectors, this form factor is well known; it goes back at least to Lord Rayleigh. In \IsGISAXS, it has been implemented as form factor \E{Full sphere} \cite[Eq.~2.36]{Laz08} \cite[Eq.~226]{ReLL09}, diff --git a/Doc/UserManual/Instrument.tex b/Doc/UserManual/Instrument.tex index 7b78713cb1d6848821e2d4f2d65bc24780c01565..00172a2d96870e9e81b924a710725d1e553694d6 100644 --- a/Doc/UserManual/Instrument.tex +++ b/Doc/UserManual/Instrument.tex @@ -80,9 +80,9 @@ and the vertical glancing angle $\alpha_\sf$. The projection of $(\alpha_\sf,\phi_\sf)$ into the detector plane~$(y,z)$ is known as the \E{gnomonic projection}. \index{Gnomonic projection}% -\index{Projection!wave vector to pixel coordinate}% -\index{Mapping!wave vector to pixel coordinate}% -\index{Transformation!wave vector to pixel coordinate}% +\index{Projection!wavevector to pixel coordinate}% +\index{Mapping!wavevector to pixel coordinate}% +\index{Transformation!wavevector to pixel coordinate}% From elementary trigonometry one finds \begin{equation}\label{Eyzdet} \begin{array}{lcl} diff --git a/Doc/UserManual/Multilayers.tex b/Doc/UserManual/Multilayers.tex index 54a98b31bfe9bb9b29fb6c8f10b1a48b243a1163..034b8e533fbed80fde1cf763de69036e6b0b9148 100644 --- a/Doc/UserManual/Multilayers.tex +++ b/Doc/UserManual/Multilayers.tex @@ -139,7 +139,7 @@ The exciting wavefunction is \nomenclature[2k021\perp]{$k_\perp$}{Component of $\k$ along the sample normal}% \nomenclature[2k041\plll]{$\k_\plll$}{Projection of $\k$ onto the sample plane}% The subscripts $\plll$ and~$\perp$ refer to the sample $xy$ plane. -The wave vector components $\k_\plll$ and $k_{\perp}$ must fulfill +The wavevector components $\k_\plll$ and $k_{\perp}$ must fulfill \begin{equation} k(z)^2=\k_\plll^2+k_{\perp}^2. \end{equation} @@ -380,7 +380,7 @@ and the geometric interpretation of~$\psi_l(\r)$ less obvious. so that one has to fully rely on the algebraic formalism. With the indicator function -\nomenclature[1χ032 2l010]{$\chi_l(z)$}{indicates whether $z$ is in layer~$l$}% +\nomenclature[1χ032 2l010]{$\chi_l(z)$}{Indicates whether $z$ is in layer~$l$}% \index{Indicator function}% \begin{equation}\label{Echildef} \chi_l(\r)\coloneqq\left\{\begin{array}{ll} diff --git a/Doc/UserManual/Scattering.tex b/Doc/UserManual/Scattering.tex index 6dd6d0d6ea15549d94444bad7aa3f18f80934727..22f705ce63ae502342b884d45e86388aa31a34ac 100644 --- a/Doc/UserManual/Scattering.tex +++ b/Doc/UserManual/Scattering.tex @@ -250,12 +250,14 @@ Mostly we will assume pure states to be \E{plane waves} \index{Plane wave}% \index{Wave!plane}% \begin{equation}\label{EplaneWave} - \psi_\k(\r)\coloneqq\e^{i\k\r}. + \psi_\k(\r)\coloneqq\e^{i\k\r}, \end{equation} -In this case, the sum in~\cref{EdefRho} must be replaced by an integral, -and the flux is simply +where the wavevector~$\k$ is allowed to have an imaginary part that describes damping. +\index{Damping}% +We replace the sum in~\cref{EdefRho} by an integral, +and find that the flux is simply \begin{equation} - \v{J}(\r) = \int\!\d^3k\; p_\k\, \k. + \v{J}(\r) = \int\!\d^3k\; p_\k\, |\psi_\k(\r)|^2 \Re\k. \end{equation} %=============================================================================== @@ -398,7 +400,7 @@ of the standard form of the Schrödinger equation.% \index{Sign convention!wave propagation|)}% } \index{Phase factor}% -In the following, we will concentrate on the electric field +We will formulate the following in terms of the electric field \index{Electric field}% \begin{equation}\label{EstationaryX} \v{E}(\r,t) = \v{E}(\r)\e^{-i\omega t}. @@ -456,22 +458,38 @@ Using a standard identity from vector analysis, it can be brought into the more \end{equation} \vspace*{-5pt}} -Electromagnetic flux is given by Poynting's vector +It is well known that the electromagnetic energy flux is given by the Poynting vector. +\index{Poynting vector}% +\index{X-ray!flux}% +\index{Flux!X-rays}% +However, its standard definition, $\v{S}\coloneqq\v{E}\times\v{H}$, +is not applicable here because it only holds for \E{real} fields. +With our complex notation, it must be replaced by +\begin{equation} + \v{S}\coloneqq \Re\v E(\r,t)\times\Re\v H(\r,t). +\end{equation} +\nomenclature[2s150]{$\v S$}{Poyinting vector}% +For stationary oscillations~\cref{EstationaryX}, +the time average is \begin{equation} - \v{S}\coloneqq \v{E}\times\v{H}. + \braket{\v S} = \frac{1}{4}\braket{\v E(\r) \times \v H(\r)^* + \text{c.~c.}}. \end{equation} -Under our assumptions $\omega=\text{const}$ and $\TENS\mu(\r)=1$, -the $H$~field is +\nomenclature[2c000 2c000]{$\text{c.~c.}$}{Complex conjugate}% +We specialize to vacuum with $\TENS\mu(\r)=1$ and $\TENS\eps(\r)=1$, +and obtain \begin{equation} - \v{H} = \frac{i}{\omega\mu_0}\Nabla\times\v{E}. + \braket{\v S} + = \frac{1}{4i\omega\mu_0} + \left( \v E^*(\r)\times\left(\Nabla\times\v{E}(\r)\right) + \text{c.~c.} \right). \end{equation} -For the vacuum field (which -It is very unusual to describe a classical electromagnetic field by a density matrix, -but nonetheless useful. +For a plane wave $\v E(\r)=\v E_\k \e^{i\k\r}$, we find +\begin{equation} + \braket{\v S} + = \frac{1}{2\omega\mu_0} |\v E|^2 \Re \k, +\end{equation} +which confirms the common knowledge that the radiation intensity +counted in a detector is proportional to the squared electric field amplitude. -\index{Poynting's vector}% -\index{X-ray!flux}% -\index{Flux!X-rays}% \index{Wave propagation!X-ray|)}% \index{X-ray!wave propagation|)}%