= Electron microscope resolution
<Electron microscope resolution> concerns distinguishable specimen detail under a specified contrast criterion. It depends on <diffraction>, lens defects, <Electron> energy spread, specimen motion and detection. A smaller <de Broglie wavelength> permits finer detail in principle, but neither <wavelength> nor <magnification> is the instrument's measured resolution.
For an <Electron> accelerated from negligible initial <kinetic energy> through a positive <electric potential difference> $U$, combine $K=eU$, the <relativistic energy-momentum relation> and the <de Broglie wavelength>:
$$
p^2c^2=K(K+2m_ec^2),\qquad
\lambda(U)=\frac{h}{\sqrt{2m_e eU\left(1+\frac{eU}{2m_ec^2}\right)}}.
$$
Here $h$ is the <Planck constant>, $m_e$ the <Electron> <mass>, $e$ the <elementary charge>, and $c$ the <speed of light>. This includes the relativistic correction omitted by the low-<voltage> <electron accelerating-voltage wavelength> approximation. With $m_e\simeq9.1093837\times10^{-31}\,\mathrm{kg}$ and SI defining values of $h,e,c$, calculated <wavelengths> are approximately $6.979$, $3.701$ and $1.969\,\mathrm{pm}$ at $30$, $100$ and $300\,\mathrm{kV}$ respectively. The rounded <mass> agrees with the recommended value.<CODATA Value: electron mass by National Institute of Standards and Technology>[CODATA electron mass, 2022 adjustment]{ref}
For a circular aperture admitting a small semi-angle $\alpha$, the <Rayleigh criterion> suggests a <diffraction-limited resolution> scale $d_D=0.61\lambda/\alpha$. At $100\,\mathrm{kV}$ and $\alpha=10\,\mathrm{mrad}$, this gives $0.226\,\mathrm{nm}$, already much larger than the $0.00370\,\mathrm{nm}$ <wavelength>. This is an ideal aperture estimate. <Transmission electron microscopy> with coherent illumination and a particular specimen requires its own image-contrast analysis; this scale alone is not a universal two-feature resolution test.
An original illustrative blur model adds competing effects:
$$
b(\alpha)^2=\left(\frac{0.61\lambda}{\alpha}\right)^2
+(C_s\alpha^3)^2
+\left(C_c\alpha\frac{\Delta K}{K}\right)^2+b_0^2.
$$
Define $C_s\alpha^3$ as a characteristic transverse blur from the electron-lens analogue of <spherical aberration>, and $C_c\alpha\Delta K/K$ as one from energy-dependent focusing, the analogue of <chromatic aberration>. The coefficients $C_s,C_c$ have units of length, $\Delta K$ is a chosen characteristic <kinetic energy> spread, and $b_0$ is a fixed residual blur. Adding their squares is a declared heuristic, not a measured <point spread function> or a rigorous resolution theorem. The model omits contrast transfer, specimen scattering and signal statistics, so its minimum is only an estimate within its assumptions.
Choose $C_s=C_c=1\,\mathrm{mm}$, $\Delta K=0.7\,\mathrm{eV}$ and $b_0=0.10\,\mathrm{nm}$ at $K=100\,\mathrm{keV}$. At $5\,\mathrm{mrad}$, the four contributions are about $0.452$, $0.125$, $0.035$ and $0.100\,\mathrm{nm}$, giving $b\simeq0.480\,\mathrm{nm}$. At $10\,\mathrm{mrad}$, the <diffraction> contribution falls to $0.226\,\mathrm{nm}$ but the assumed <spherical aberration> contribution grows to $1.00\,\mathrm{nm}$. Thus opening the aperture indefinitely worsens this model's image. All these values are calculations with chosen parameters, not observations of any microscope.
\Image[/electron-wavelength-resolution-budget.png]
{title=Short <Electron> <wavelength> and competing image-blur scales}
{description=Original calculations, not measurements. Left: relativistic <de Broglie wavelength> versus accelerating <voltage>. Right: ideal <diffraction> and assumed electron-lens blur contributions at $100\,\mathrm{kV}$, with their quadrature total. The marked minimum belongs only to the stated heuristic model and does not specify a historical or modern instrument's resolution.}
{source=electron-wavelength-resolution-budget.py}
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