Wikiadmin: Wikiadmin moved page Problem: Simple Bragg scattering, the monochromator to Problem:Simple Bragg scattering, the monochromator

2020-09-20T15:47:15Z

Wikiadmin moved page Problem: Simple Bragg scattering, the monochromator to Problem:Simple Bragg scattering, the monochromator

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ucph>Tommy: Created page with "Consider the Bragg law for scattering of waves by a crystal. A reciprocal lattice vector, \({\boldsymbol\t..."

2019-07-14T21:32:38Z

Created page with "Consider the Bragg law  for scattering of waves by a crystal. A reciprocal lattice vector, \({\boldsymbol\t..."

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Consider the [[Diffraction from crystals#label-eq:bragg|Bragg law  for scattering of waves by a crystal]]. A reciprocal lattice vector, \({\boldsymbol\tau}\), is always perpendicular to a set of lattice planes, and has the length

\begin{equation}\label{eq:taubragg}
|{\boldsymbol\tau}| = n \dfrac{2 \pi}{d} ,
\end{equation}

where \(n\) is an integer.

=====Question 1=====
Argue why \(n\) is needed in equation \eqref{eq:taubragg}.

{{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}
The \(n\) is needed in equation \eqref{eq:taubragg} since Bragg-scattering from the same set of crystal planes can occur also for neutrons of double (triple etc) wavevector. I.e. the crystal can scatter wavelengths <br> \(\lambda, \lambda/2, \lambda/3, \ldots, \lambda/n\) if they are present in the incoming neutron spectrum.
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=====Question 2=====
Show that the Bragg law can be written as \(\tau = 2 k \sin \theta\), following the [[Diffraction from crystals#label-fig:bragg|diffraction in Bragg geometry figure]] on the [[Diffraction from crystals]] page.

{{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}

<figure id="fig:Bragg"> [[File:Bragg.png| thumb | 200px | <caption> A sketch of the SANS instrument in terms of \(q\)-range and resolution.</caption>]] </figure>

On <xr id="fig:Bragg">Figure %i</xr> it is seen that \(\sin{\theta}=\dfrac{l}{d}\). The condition for constructive interference between rays of wavelength \(\lambda=2\pi/k\) scattered from the upper and lower crystalplane is \(2l=n\lambda\). Combining the two equations and rewriting \(d= n \frac{2\pi}{\tau}\) we reach at

\(
n\lambda = 2l = 2d\sin{\theta} \Rightarrow \tau = 2k\sin{\theta}
\)

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=====Question 3=====
Show that the Bragg law can be derived from the (crystal) momentum conservation law \(\hbar {\bf k}_{\rm i} = \hbar {\bf k}_{\rm f} + \hbar {\boldsymbol\tau}\).

{{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}
For elastic scattering \(|\mathbf{k}_i| =|\mathbf{k}_f| = k \), hence by trigonometry \(q/2=k\sin{\theta}\)

\(
{\boldsymbol\tau} = \mathbf{k}_i - \mathbf{k}_f \Rightarrow |{\boldsymbol\tau}| = |\mathbf{k}_i - \mathbf{k}_f| = q= 2k \sin{\theta}
\)
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=====Question 4=====
Determine the scattering angle, \(2 \theta\), for 5 meV neutrons scattering off a pyrolytic graphite monochromator crystal with \(\tau_{(002)} = 1.8734\) Å\(^{-1}\).

{{hidden begin|toggle=right|title=Solution|titlestyle=background:#ccccff}}

It is useful to have the relation \(E\)[meV]\(=\frac{\hbar^2 k^2}{2m_n}= 2.072 k^2\) where \(k\) is in reciprocal Ångstrøm. Hence \(k=\sqrt{\frac{5.00}{2.072}}= 1.553\) Å\(^{-1}\). For \(\tau_{(002)}=1.8734\) Å\(^{-1}\) we get

\(
2\theta = 2 \sin^{-1}\left(\dfrac{\tau}{2k}\right) = 74.19^\circ
\)
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Problem:Simple Bragg scattering, the monochromator - Revision history

Wikiadmin: Wikiadmin moved page Problem: Simple Bragg scattering, the monochromator to Problem:Simple Bragg scattering, the monochromator

Wikiadmin: 1 revision imported

ucph>Tommy: Created page with "Consider the Bragg law for scattering of waves by a crystal. A reciprocal lattice vector, \({\boldsymbol\t..."