Wikiadmin: 1 revision imported

2020-02-18T22:15:08Z

1 revision imported

← Older revision	Revision as of 22:15, 18 February 2020
(No difference)

ucph>Tommy: Created page with "If the crystal grains are very small, often meaning below 100-500 nm, the "infinite crystal" approximation leading to the Bragg law in equation \eqref{eq:bragg} breaks down. I..."

2019-07-14T21:06:18Z

Created page with "If the crystal grains are very small, often meaning below 100-500 nm, the "infinite crystal" approximation leading to the Bragg law in equation \eqref{eq:bragg} breaks down. I..."

New page

If the crystal grains are very small, often meaning below 100-500 nm, the "infinite crystal" approximation leading to the Bragg law in equation \eqref{eq:bragg} breaks down. It is then important to reconcile equation \eqref{eq:diffract2} for a nano-sized particle. The resulting mathematics is tedious, but the result is rather simple. The Bragg peaks broaden to a width (FWHM) of approximately

\begin{equation}\label{eq:nano_broad}
\Delta q \approx \dfrac{2\pi}{L} ,
\end{equation}

where \(L\) here is a typical dimension of the particle. We now dig a little deeper into the effect nano-sizes have on the diffraction signal.

==A cubic nanoparticle==
We first illustrate equation \eqref{eq:nano_broad} with an example. Consider a simple cubic Bravais crystal with lattice constant \(a\). Let also the outer shape of the particle be cubic with side length \(d\). The atoms along a side are numbered from \(0\) to \(m-1\), where \(m=d/a\). The scattering cross section for \({\mathbf q}\) parallel to one side of the cube (here taken as the \(x\)-direction) reads:

\begin{align}\label{eq:diffract_nano}
\dfrac{d\sigma}{d\Omega}
&= \exp(-2W) \biggr| b m^2 \displaystyle\sum_{n=0}^{m-1} \exp(i q_x n a) \biggr|^2 \\
&= \exp(-2W) m^4 b^2 \left| \dfrac {1-\exp(i q_x m a)}{1-\exp(i q_x a)} \right|^2 \nonumber\\
&= \exp(-2W) m^4 b^2 \left| \dfrac{\sin(m q_x a/2)}{\sin(q_x a/2)} \right|^2 .\nonumber
\end{align}

This expression peaks around \(q_x = 2\pi/a\), which is just the Bragg condition. However, the peak has a width of \(\Delta q_x = 2\pi/(ma) = 2\pi/L\) as anticipated above. (The width of the reflection in the \(y\) and \(z\) directions are identical.)

It should be noticed that the squared term in equation \eqref{eq:diffract_nano} has a peak amplitude of \(m^2\), meaning that the total peak amplitude is proportional to \(m^6=N^2\). We should, however, also take into account that the total broadening of the reflection in reciprocal space scales as \(m^{-3}=N^{-1}\). Hence, the integrated intensity of the diffraction peak is proportional to \(N\) - and hence to the particle volume, \(V\) - as was also found in the infinite system, ''c.f.'' \eqref{eq:diffract}.

==The Scherrer equation==
The literature goes one step further into this problem. We will not go into any details with the derivation, but the "apparent" size of the particle is found to be given by the ''Scherrer'' equation originally derived for X-ray diffraction<ref name="langford78">J.I.L Langford and A.J.C. Wilson, ''J. Appl. Cryst.'', vol. 11, p. 102 (1978)</ref>:

\begin{equation}\label{dummy2079019572}
\epsilon = \dfrac{\lambda}{b \cos(\theta)} ,
\end{equation}

where \(b\) is the angular broadening of the peak in radians. Identifying \(b = \delta(2\theta) = 2 \delta\theta\), we reach \(\epsilon = \pi / (k \cos(\theta) \delta\theta)\). From the identity \(q=2 k \sin(\theta)\) we reach \(\delta q = 2 k \cos(\theta) \delta \theta\), leading to

\begin{equation}\label{dummy1764686862}
\epsilon = \dfrac{2 \pi}{\delta q} ,
\end{equation}

as we anticipated in equation \eqref{eq:nano_broad}.

The relation between the "true" particle size, \(p = (V)^{1/3}\) and the "apparent" size, \(\epsilon\) is given by

\begin{equation}\label{dummy1773762082}
p = K \epsilon ,
\end{equation}

where \(K\) is a constant of the order unity. This is discussed in great detail for different particle shapes in Ref. <ref name="langford78" />.

<references />

Diffraction from nano-sized systems - Revision history

Wikiadmin: 1 revision imported

ucph>Tommy: Created page with "If the crystal grains are very small, often meaning below 100-500 nm, the "infinite crystal" approximation leading to the Bragg law in equation \eqref{eq:bragg} breaks down. I..."