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HRTEM Image Formation

The HRTEM image is formed from the exit-surface wavefunction — the transmission coefficients \(T_{\mathbf g}\) obtained from the dynamical core — by passing it through the objective lens. ReciPro offers two models: the fast quasi-coherent approximation and the more rigorous transmission cross coefficient (TCC) model. See also the HRTEM simulator GUI page.

Symbols

Symbol Meaning
\(\mathbf R\) X–Y component in real space (image plane)
\(\mathbf K\) X–Y component of the incident wavevector
\(\mathbf G, \mathbf H\) X–Y components of reciprocal-lattice vectors
\(\mathbf u\) spatial frequency (e.g. \(\mathbf K+\mathbf G\))
\(\chi(\mathbf u)\) lens aberration function
\(A(\mathbf u)\) objective-aperture function
\(\Delta f\) defocus value
\(C_s\) spherical aberration coefficient
\(C_c\) chromatic aberration coefficient
\(\beta\) illumination semi-angle (finite source size)
\(\Delta E\) \(1/e\) width of the electron energy fluctuations
\(\Delta_0\) \(1/e\) width of the defocus spread (Gaussian), \(\Delta_0 = C_c\,\Delta E / E\)

Lens aberration function and aperture

\[\chi(\mathbf u) = \pi\lambda\Delta f\, u^2 + \tfrac{1}{2}\pi\lambda^3 C_s\, u^4 = \pi\lambda u^2\!\left(\Delta f + \tfrac{1}{2}\lambda^2 C_s u^2\right)\]
\[A(\mathbf u) = \begin{cases} 1 & (\mathbf u\ \text{inside the objective aperture})\\[2pt] 0 & (\mathbf u\ \text{outside the objective aperture})\end{cases}\]

Quasi-coherent model

A fast approximation: each diffracted beam is modulated by the lens transfer and damped by coherence envelopes, then summed coherently.

\[I(\mathbf R) = |\psi(\mathbf R)|^2\]
\[\psi(\mathbf R) = \sum_{\mathbf g} T_{\mathbf g}\,\exp\!\left[2\pi i(\mathbf K+\mathbf G)\cdot\mathbf R\right]\exp\!\left[-i\chi(\mathbf K+\mathbf G)\right]A(\mathbf K+\mathbf G)\,E_c(\mathbf K+\mathbf G)\,E_s(\mathbf K+\mathbf G)\]

with the temporal and spatial coherence envelopes

\[E_c(\mathbf u) = \exp\!\left[-\tfrac{1}{2}\left(\pi\lambda\Delta_0\, u^2\right)^2\right], \qquad E_s(\mathbf u) = \exp\!\left[-\pi^2\beta^2 u^2\!\left(\Delta f + \lambda^2 C_s u^2\right)^2\right]\]

Transmission cross coefficient (TCC) model

The rigorous treatment of partial coherence: every pair of beams \((\mathbf g, \mathbf h)\) interferes through the transmission cross coefficient.

\[I(\mathbf R) = \sum_{\mathbf g}\sum_{\mathbf h} T_{\mathbf g}\,T_{\mathbf h}^{*}\,\exp\!\left[2\pi i(\mathbf G-\mathbf H)\cdot\mathbf R\right]\mathrm{TCC}(\mathbf K+\mathbf G,\ \mathbf K+\mathbf H)\]
\[\mathrm{TCC}(\mathbf u, \mathbf u') = A(\mathbf u)\,A(\mathbf u')\,\exp\!\left[-i\{\chi(\mathbf u)-\chi(\mathbf u')\}\right]E_c(\mathbf u, \mathbf u')\,E_s(\mathbf u, \mathbf u')\]

with the mixed coherence envelopes

\[E_c(\mathbf u, \mathbf u') = \exp\!\left[-\tfrac{1}{2}\left(\pi\lambda\Delta_0\right)^2\!\left(u^2 - u'^2\right)^2\right]\]
\[E_s(\mathbf u, \mathbf u') = \exp\!\left[-\pi^2\beta^2\left\{\Delta f(\mathbf u-\mathbf u') + \lambda^2 C_s\!\left(u^2\mathbf u - u'^2\mathbf u'\right)\right\}^2\right]\]

In the limit \(\mathbf u' \to \mathbf u\) the TCC reduces to the quasi-coherent envelopes above.

See also