---
title: Cowley–Moodie Multislice Solution
url: https://www.emergentmind.com/topics/cowley-moodie-multislice-solution
type: topic
---

# Cowley–Moodie Multislice Solution

The Cowley–Moodie multislice solution is a foundational theoretical and computational framework for modeling dynamical electron scattering in crystalline materials, particularly within transmission electron microscopy (TEM). It systematically decomposes the three-dimensional crystal potential into consecutive thin slices, capturing both phase modulation by the local projected potential and Fresnel propagation between slices. Recent reformulations cast the multislice algorithm into a “transmission-matrix” formalism, enabling rigorous eigenstructure analysis and direct comparison with the Bloch-wave scattering matrix. This establishes the conditions under which both formulations are formally equivalent (modulo $2\pi$ phase ambiguities), and supports physically meaningful parameter extraction, such as the mean inner potential, directly from transmission matrix determinants [2412.21119].

## 1. Multislice Decomposition and Transmission Operators

The Cowley–Moodie approach partitions the specimen’s three-dimensional electrostatic potential $V(x,y,z)$ into $M$ slices of uniform thickness $\Delta z$. The projected potential for slice $n$ is
\[
V_n(x,y) = \int_{z_n}^{z_n+\Delta z} V(x,y,z)\,dz.
\]
Each slice imparts a phase shift to the traversing electron wave, described by the transmission operator
\[
T_n(x,y) = \exp[i\,\sigma\,V_n(x,y)], \qquad n=1,\ldots,M,
\]
where the interaction constant is
\[
\sigma = \frac{2\pi m e}{h^2 k_0},
\]
with $m$ as the relativistic electron mass, $e$ the elementary charge, $h$ Planck’s constant, and $k_0=2\pi/\lambda$ the relativistically-corrected electron wavenumber.

Upon traversing slice $n$, the wave immediately after is
\[
\Psi_n^+(x,y) = T_n(x,y)\,\Psi_{n-1}(x,y).
\]

## 2. Propagation Between Slices: Fresnel Operator

Electron propagation between adjacent slices over distance $\Delta z$ is described as Fresnel (paraxial) propagation. In Fourier space, this is a multiplication:
\[
\Psi^-(q_x,q_y) = \exp\Big[-i\pi\lambda\Delta z(q_x^2 + q_y^2)\Big]\,\Psi^+(q_x,q_y).
\]
The equivalent real-space operation is implemented efficiently via fast Fourier transforms (FFT) as
\[
\Psi^-(x,y) = \mathcal{F}^{-1}_{2D} \left[ P(q_x,q_y)\,\mathcal{F}_{2D}[\Psi^+(x,y)] \right],
\]
with $P(q_x,q_y)$ as the Fresnel propagator.

## 3. Multislice Transmission Matrix Construction

The complete multislice algorithm applies alternating transmission and propagation operators for $M$ slices, such that the exit wave is
\[
\Psi_M^-(x,y) = \Big[P\,T_M\,P\,T_{M-1}\cdots P\,T_1\Big]\Psi_{\text{in}}(x,y).
\]
Upon discretizing the lateral plane to an $N \times N$ grid, each transmission operator $T_n$ becomes a diagonal $N^2 \times N^2$ matrix $O_n$, while propagation between slices is mediated by the matrix $G = F_{2D}^{-1} D F_{2D}$, where $D$ is a diagonal matrix of Fresnel phase factors.

The full real-space multislice operator is
\[
A = \prod_{n=1}^{M} G\,O_n \in \mathbb{C}^{N^2 \times N^2}.
\]
For reciprocal-space (diffraction) analysis, the transmission matrix is conjugated by the $2$D Fourier transform:
\[
\widehat{S} = F_{2D} A F_{2D}^{-1}.
\]

## 4. Bloch-Wave Scattering Matrix and Eigenstructure

The Bloch-wave formulation constructs the scattering matrix by exponentiating the crystal “structure matrix” $B$, assembled from Fourier components of $V$:
\[
S = \exp\left(i2\pi T B\right) = C\,\Lambda\,C^{-1},
\]
with $T = M\Delta z$ the total thickness, $\{\gamma_j\}$ the eigenvalues of $B$, and $C$ the eigenvector matrix of Bloch-wave Fourier coefficients. The diagonal matrix $\Lambda$ contains eigenphases $\exp(2\pi i \gamma_j T)$. Both $S$ and $\widehat{S}$ are unitary.

Diagonalization of the multislice transmission matrix yields
\[
\widehat{S} = W V W^{-1}, \quad V_{jj} = e^{i2\pi\theta_j},
\]
with $|V_{jj}| = 1$.

## 5. Formal Equivalence: Spectral and Structural Analysis

Quantitative comparison of $S$ and $\widehat{S}$ uses the Frobenius norm,
\[
\|S-\widehat S\|_F^2 = 2N^2 - 2\sum_{j=1}^{N^2} \cos\left[2\pi(\gamma_j T - \theta_j)\right].
\]
The matrices are equivalent if the following hold:
- The eigenvector bases coincide up to a $2$D Fourier transform: $C = F_{2D} W$.
- The eigenphase differences are integer multiples of $2\pi$: $\gamma_j T - \theta_j = 2\pi n_j$, $n_j\in\mathbb{Z}$.

Thus, the spectral representations of multislice and Bloch-wave formulations are congruent modulo $2\pi$ phase factors under these conditions [2412.21119].

## 6. Physical Parameter Extraction: Mean Inner Potential

The determinant of the transmission matrix encodes the total projected potential:
\[
\det(\widehat{S}) = \prod_{n=1}^{M} \det(G\,O_n) = \exp\Big[-i\big(\pi\lambda Q T - \sigma V_{\text{total}}\big)\Big],
\]
where $Q = \sum_{q_x,q_y} (q_x^2 + q_y^2)$ and $V_{\text{total}} = \sum_n \sum_{x,y} V_n(x,y)$. The mean inner potential (MIP), a quantity of central importance in electron scattering, is then extracted as
\[
V_{\text{total}} = \frac{\pi\lambda Q T - \arg\det(\widehat S)}{\sigma} + 2\pi k, \qquad k \in \mathbb{Z},
\]
and the normalized MIP for the unit-cell volume $\Omega$ as $V_0 = V_{\text{total}}/\Omega$.

## 7. Computational Considerations and Underlying Assumptions

The Cowley–Moodie multislice approach relies on several critical approximations:
- The thin-slice (phase-grating) approximation: $T_n\approx\exp[i\sigma V_n]$ assumes linearizable intra-slice multiple scattering.
- Paraxial propagation validity: $\lambda\Delta z(q_x^2 + q_y^2) \ll 1$ and small electron tilt angles are required.
- Efficient implementation: Each slice update via $G\,O_n$ exploits FFTs for computational scaling of $O(N^2\log N)$, which is tractable for large grids. By contrast, direct Bloch-wave eigendecomposition is $O(N^6)$, making the matrix multislice formulation preferable for practical simulation at scale.

The formalism thus not only establishes a rigorous equivalence between widely used dynamical electron scattering models but also provides an efficient, physically insightful, and numerically robust method for TEM simulation and analysis [2412.21119].

Source: https://www.emergentmind.com/topics/cowley-moodie-multislice-solution