View source for Quantum mechanical derivation of neutron-phonon scattering

We will now make a quantum mechanical derivation of the inelastic scattering cross section (\ref{eq:sum_lambda_f})
for the case of phonons in a crystal.
Along the way, we touch upon a proper
derivation of the Debye-Waller factor from the description of neutron diffraction (\ref{eq:diffraction_DW}).

==* Inelastic cross section of atoms in a lattice==
We make the usual assumption that the nuclei vibrate around their equilibrium positions,
eq.~(\ref{eq:vibration}), and we use the translational symmetry of the crystal, so that (as for crystal diffraction) 
the sum over \(j\) and \(j'\) equals \(N\) times the sum over \(j\):

\begin{align}
  &\sum_{j,j',i,i'} b_{j,i} b_{j',i'} 
   \big\langle \exp(-i {\bf q} \cdot {\bf R}_{j,i}(0))
     \exp(i {\bf q} \cdot {\bf R}_{j',i'}(t)) \big\rangle  \\ \nonumber
 &\quad= N \sum_{i,i'} b_i b_{i'} \exp(i {\bf q} \cdot ({\bf \Delta}_{i'}-{\bf \Delta}_i)) ,
\sum_{j'} \exp(i {\bf q} \cdot {\bf r}_{j'})  \\
 &\quad\quad \times \big\langle \exp(-i {\bf q} \cdot {\bf u}_{0,i}(0))
     \exp(i {\bf q} \cdot {\bf u}_{j',i'}(t)) \big\rangle \nonumber \\
&\quad=  N |F_{\rm N}({\bf q})|^2
\sum_{j',i,i'} \exp(i {\bf q} \cdot {\bf r}_{j'}) 
  \big\langle \exp(-i {\bf q} \cdot {\bf u}_{0,i}(0))
     \exp(i {\bf q} \cdot {\bf u}_{j',i'}(t)) \big\rangle \nonumber
\end{align}

where we recognize the nuclear structure factor, \(F_{\rm N}({\bf q})\), from (\ref{eq:structurefactor}).
The scattering cross section now becomes
\begin{align}
\frac{d^2\sigma}{d\Omega dE_{\rm f}}  
 &=  \frac{k_{\rm f}}{k_{\rm i}} \frac{N |F_{\rm N}({\bf q})|^2}{2 \pi \hbar}
\sum_{j,i,i'} \exp(i {\bf q} \cdot {\bf r}_j) \\
&\quad\times \int_{-\infty}^{\infty} 
  \big\langle \exp(-i {\bf q} \cdot {\bf u}_{0,i}(0))
     \exp(i {\bf q} \cdot {\bf u}_{j,i'}(t)) \big\rangle \,
\exp(-i \omega t) dt \nonumber .
\end{align}

For a Bravais lattice (one atom per unit cell), the sum over the \(i\)'s disappear, and \(F_{\rm N}({\bf q}) = b\).
Here, we can specialize the cross section to yield

\begin{align} \label{eq:cross_phonon_1}
\left. \frac{d^2\sigma}{d\Omega dE_{\rm f}}\right|_{\rm Bravais}  
  &=  \frac{k_{\rm f}}{k_{\rm i}} \frac{Nb^2}{2 \pi \hbar} 
      \sum_{j} \exp(i {\bf q} \cdot {\bf r}_j) \\
  &\quad  \times  \int_{-\infty}^{\infty}
    \big\langle \exp(-i {\bf q} \cdot {\bf u}_0(0))
     \exp(i {\bf q} \cdot {\bf u}_{j}(t)) \big\rangle
     \exp(-i\omega t) dt . \nonumber
\end{align}

==Details of phonon operators==

The difficult portion of the phonon cross section (\ref{eq:cross_phonon_1}) 
is now the operator exponential functions,
which are of the type \(\langle \exp(U) \exp(V) \rangle\). 

A quantum mechanical theorem <ref name="squires">G.L. Squires. Thermal Neutron Scattering. Cambridge University Press, 1978.</ref> states that if \([U,V]\) 
is a c-number (in contract to being another operator), then 
\begin{equation}
\langle \exp(U) \exp(V) \rangle = \langle \exp(U+V) \rangle 
    \exp\left(\frac{1}{2}[U,V]\right) .
\end{equation}

Let us take a closer look at \(U\) and \(V\). Both can be put in the form:
\begin{align}
U &= -i {\bf q} \cdot {\bf u}_0(0) 
    = -i \sum_{q,p} (g_{q,p} a_{q,p} + g^*_{q,p} a_{q,p}^\dagger) , \\
V &= i {\bf q} \cdot {\bf u}_j(t) 
    = i \sum_{q,p} (h_{q,p} a_{q,p} + h_{q,p}^* a_{q,p}^\dagger) ,
\end{align}
where the coefficients are given by
\begin{align}
g_{q,p} &= \sqrt{\frac{\hbar}{2MN}} 
           \frac{{\bf q} \cdot {\bf e}_{q,p}}{\sqrt{\omega_{q,p}}} . \\
h_{q,p} &= \sqrt{\frac{\hbar}{2MN}} 
           \frac{{\bf q} \cdot {\bf e}_{q,p}}{\sqrt{\omega_{q,p}}} 
           \exp(i {\bf q} \cdot {\bf r}_j) \exp(-i \omega_{q,p}t) .
\end{align}
Now, we calculate \([U,V]\), which becomes a quadruple sum.
To ease our task, we note that \([a_{q,p},a^\dagger_{q',p'}]\) is
nonzero only if \(q=q'\) and \(p=p'\). We can now show that
\([U,V]\) is a c-number and calculate its value:
\begin{align}
[U,V] &= \sum_{q,p,q',p'} 
  [g_{q,p} a_{q,p} + g_{q,p} a_{q,p}^\dagger, 
     h_{q',p'} a_{q',p'} + h_{q',p'}^* a_{q',p'}^\dagger] \nonumber \\
  &= \sum_{q,p} g_{q,p}h_{q,p}^* [a_{q,p},a^\dagger_{q,p}]
              + g_{q,p}h_{q,p}   [a^\dagger_{q,p},a_{q,p}] \nonumber \\
  &= \sum_{q,p} (g_{q,p}h_{q,p}^* - g_{q,p}h_{q,p}).
\end{align}

Another theorem proves that for any harmonic oscillator operator 
(e.g. \(U\) or \(V\)):
\begin{equation}
\left\langle \exp(U) \right\rangle 
  = \exp\left(\frac{1}{2}\left\langle U^2 \right\rangle\right) .
\end{equation}

We now use this theorem to reach
\begin{align}
\left\langle \exp(U+V) \right\rangle \exp(\frac{1}{2}[U,V])
  &= \exp\left(\frac{1}{2}\left\langle U^2+V^2+UV+VU+UV-VU \right\rangle\right) \nonumber \\
 &=  \exp(\langle U^2 \rangle) 
           \exp(\langle UV \rangle) .
\end{align}
In the last step, we have used that \(\langle U^2 \rangle = \langle V^2 \rangle\).
This can be argued by noting that the only differences 
between these harmonic operators are their time and position. 
Since the system is translation- 
and time invariant, the two expectations values 
must be identical.

==* The phonon expansion==

We first summarize the expression for the (Bravais lattice) phonon cross section so far:
\begin{align} \label{eq:cross_phonon_2}
\left.\frac{d^2\sigma}{d\Omega dE_{\rm f}}\right|_{\rm Bravais} &=
   \frac{k_{\rm f}}{k_{\rm i}} \frac{Nb^2}{2 \pi \hbar}  
    \exp(\left\langle U^2 \right\rangle)  \\
  &\quad\times    
      \sum_{j} \exp(i {\bf q} \cdot {\bf r}_j)
      \int_{-\infty}^{\infty}
    \exp(\left\langle UV \right\rangle) \exp(-i\omega t) dt . \nonumber
\end{align}
The most significant part of this expression is the operator 
\(\exp(\langle UV \rangle)\), which can create or annihilate a number of phonons.
We proceed by a series expansion of the exponential:

\begin{equation} \label{eq:UVexpansion}
\exp(\langle UV \rangle) 
  \approx 1 + \langle UV \rangle + \frac{1}{2} \langle (UV)^2 \rangle + \,\cdots
\end{equation}
The zero'th order term we have dealt with before.
Here, the time integral in (\ref{eq:cross_phonon_2}) results in \(2\pi\hbar \delta(\hbar\omega)\), 
i.e. elastic scattering. The cross section resulting from
this term essentially equals (\ref{eq:diffract2}), which for crystalline materials leads to the
Bragg law, modified by \(\exp(\left\langle U^2 \right\rangle) \), 
which is the Debye-Waller factor, to be elaborated in section~\ref{subsect:DW}.

The first order term \(UV\) in (\ref{eq:UVexpansion}) corresponds to creation 
or annihilation of one single phonon. This will be discussed in
section \ref{sect:cross_single_phonon}.
Likewise, the term \((UV)^2\)
describes processes involving two phonons, and so on. We will here
discuss only one-phonon processes. The multi-phonon processes 
will lead to a continuum of scattering, difficult to detect and analyze. 
This is to some extent discussed in Squires <ref name="squires" />.

==* The Debye-Waller factor==

We now look closer at the term \(\langle U^2 \rangle\). Using the definition
of \(U\), we can write
\begin{align}
\langle U^2 \rangle 
  &= - \sum_i p_i \left\langle \lambda_i \left| 
    U^2 \right| \lambda_i \right\rangle \nonumber \\
  &= - \sum_i p_i \biggr\langle \lambda_i \biggr| 
  \sum_{q,p,q',p'} g_{q,p} g_{q',p'} 
        (a_{q,p}+a^\dagger_{q,p})(a_{q',p'}+a^\dagger_{q',p'}) 
 \biggr| \lambda_i \biggr\rangle \nonumber \\
  &= - \sum_{q,p} g_{q,p}^2 \sum_i p_i \left\langle \lambda_i \left| 
        a_{q,p}a^\dagger_{q,p}+a^\dagger_{q,p}a_{q,p} 
 \right| \lambda_i \right\rangle \nonumber \\
 &= - \sum_{q,p} g_{q,p}^2 \left( 2 n_{\rm B}\left( \frac{\hbar \omega_{q,p}}{k_{\rm B}T} \right) + 1 \right) \nonumber \\
 &= - \frac{\hbar}{2 M N} \sum_{q,p} 
        \frac{\left( {\bf q} \cdot {\bf e}_{q,p}\right)^2}{\omega_{q,p}}
         \left( 2 n_{\rm B}\left( \frac{\hbar\omega_{q,p}}{k_{\rm B}T} \right) + 1 \right) , 
\end{align}
where the expectation value \(\left\langle a_{q,p}^\dagger a_{q,p} \right\rangle\) 
has been replaced by its value \(n_{\rm B}\), the Bose occupation number (\ref{eq:nB}).
In the derivation, we have also used that the expectation value is non-zero 
only when the number of \(a\) and \(a^\dagger\) operators is the same for each \((q,p)\).
For example, the expression
\(a_{q,p}^\dagger a_{q',p'}^\dagger |\lambda\rangle\) gives you the state 
\(|\lambda\rangle\) with two additional phonons, whence 
\(\langle\lambda|a_{q,p}^\dagger a_{q',p'}^\dagger | \lambda\rangle = 0\).

It is customary to define \(2W = -\langle U^2 \rangle\) and then 
define the Debye-Waller factor
\begin{equation}

\exp(-2W) \equiv \exp(\langle U^2 \rangle)  \, .
\end{equation}
This is used to describe the reduction in diffraction intensity due to lattice vibrations, 
as was anticipated in chapter~\ref{ch:powder}.
For a cubic crystal it can be shown that

\begin{equation}
2W = \frac{1}{3} q^2 \langle u^2 \rangle ,
\end{equation}
where \(u\) is the mean atomic displacement from equilibrium.
Using this as representative for all crystals, we can see that \(2W\) 
is non-zero at zero temperature (due to zero-point motion)
and increases with temperature. 
As a consequence, the resulting Debye-Waller factor is slightly below unity
at low temperatures, decreasing at higher temperatures.

Since \(2W\) is proportional to \(q^2\), 
the Debye-Waller factor can be approximated by unity
in small-angle scattering; as we did implicitly in [[Small angle neutron scattering, SANS]].

==* The final scattering cross section for phonons==

Using (\ref{eq:cross_phonon_2}) as a starting point, we continue to 
develop the operator values.

\begin{align}
\langle UV \rangle &= \sum_i p_i 
  \left\langle \lambda_i \left| UV \right| \lambda_i \right\rangle \nonumber \\
 &= \sum_i p_i \biggr\langle \lambda_i \biggr| 
  \sum_{q,p} \left(g_{q,p}a_{q,p}+g_{q,p}a^\dagger_{q,p}\right)
             \left(h_{q,p}a_{q,p}+h^*_{q,p}a^\dagger_{q,p}\right)
     \biggr| \lambda_i \biggr\rangle \nonumber \\
 &= \sum_{q,p} \sum_i p_i \left\langle \lambda_i \left| 
  \left(g_{q,p}a_{q,p}h^*_{q,p}a^\dagger_{q,p}
       +g_{q,p}a^\dagger_{q,p}h_{q,p}a_{q,p}\right)
     \right| \lambda_i \right\rangle \nonumber \\
 &= \sum_{q,p} g_{q,p}h^*_{q,p} (n_{q,p}+1) + g_{q,p}h_{q,p}n_{q,p} ,
\end{align}
where we have used that the expectation values for a pair of 
creation-annihilation operators is non-zero only if they have the same quantum
numbers. Using the expression for the \(g\)'s and \(h\)'s, we reach
\begin{align}
\langle UV \rangle &= \frac{\hbar}{2MN} \sum_{q,p} 
  \frac{({\bf q}\cdot {\bf e}_{q,p})^2}{\omega_{q,p}}  \\
&\quad \times 
   \big( \exp(-i {\bf q} \cdot {\bf r}_j+i \omega_{q,p}t) (n_{q,p}+1)
      + \exp(i {\bf q} \cdot {\bf r}_j-i \omega_{q,p}t) n_{q,p} \big) . \nonumber
\end{align}
We now insert this into (\ref{eq:cross_phonon_2}). To avoid confusion of
labels, we use \({\bf q}'\) for the phonon wave vector.
\begin{align} \label{eq:cross_phonon_3}
\frac{d^2\sigma}{d\Omega dE_{\rm f}} &=
   \frac{k_{\rm f}}{k_{\rm i}} 
    \frac{b^2}{4 \pi M}  
    \exp(-2W) \sum_{q',p} \frac{({\bf q}'\cdot {\bf e}_{q',p})^2}{\omega_{q',p}}
  \sum_{j} \exp(i {\bf q} \cdot {\bf r}_j)  \\
  &\quad\times 
      \int_{-\infty}^{\infty}
  \big[ \exp(-i {\bf q}' \cdot {\bf r}_j+i \omega_{q',p}t) (n_{q',p}+1) \nonumber \\
 &\qquad\qquad+ \exp(i {\bf q}' \cdot {\bf r}_j-i \omega_{q',p}t) n_{q',p} \big]
     \exp(-i\omega t) dt . \nonumber
\end{align}
The Fourier transformations in time and space are immediately calculated,
giving the final one-phonon cross section:

\begin{align} \label{eq:cross_one_phonon_QM}
\frac{d^2\sigma}{d\Omega dE_{\rm f}} &=
   \frac{k_{\rm f}}{k_{\rm i}} 
    \frac{b^2(2\pi)^3}{2 M V_0}  
    \exp(-2W) \sum_{q,p,\tau} \frac{({\bf q}\cdot {\bf e}_{q,p})^2}{\omega_{q,p}}  \\
  &\quad\times 
  \left[ (n_{q,p}+1) \delta(\omega-\omega_{q,p}) \delta({\bf q}-{\bf q}'+{\bf \tau}) \right. \nonumber \\
  &\qquad  \left.  + n_{q,p} \delta(\omega+\omega_{q,p}) \delta({\bf q}+{\bf q}'+{\bf \tau})  \right]
     . \nonumber
\end{align}
This equation is essentially equal to the classical equation (\ref{eq:cross_one_phonon}), although derived
in a much more rigorous way.