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 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 .

* 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.