Electron scattering from a specimen in EM measurements

Electron Scattering from a Specimen in EM Measurements
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Electron-scattering from specimens in EM measurements can be grouped in different ways. However, the very common and important classification is sorted by elastic and inelastic scattering as listed in Table 1219.

Table 1219a. Electron scattering from a specimen in TEM measurements (Incoherent does not imply inelastic scattering; however, inelastic scattering is necessarily incoherent in EM measurements).

Scattering type	Energy loss	Wave (phase) property	Scattering direction	Scattering angle	Electron property	Full name
Elastic	No energy change of the wave after scattering	Usually coherent (when the specimen is thin and crystalline)	Forward scattering	1° ~ 10°	Wave	Coherent elastic scattering
		Incoherent	Forward scattering	> 10°	Particle	Incoherent elastic scattering
		Incoherent	Back scattering			Incoherent elastic scattering
Inelastic	There is energy change of the wave after scattering	Almost always incoherent	Forward scattering	< 1°		Incoherent inelastic scattering
Inelastic	There is energy change of the wave after scattering	Coherent	Does not exist in EM measurement, but it shows in neutron scattering			Coherent inelastic scattering

* Incoherence does not imply inelastic scattering, while inelastic electron scattering is incoherent in electron microscopy.
** Inelastic scattering is not necessarily incoherent in neutron scattering.

Table 1219b. Coherency of scattered electrons from a specimen in EM (electron microscope) measurements.

Coherency	Property
Coherent	In phase: preserves the relative phases of the wavelets scattered from different locations in a material; namely, well-defined phase relationship between incident and scattered radiation
	Fixed wavelength
	The resolution for coherent imaging is much worse than that for incoherent imaging
	Useful for diffraction experiments
Incoherent	No phase relationship: does not preserve a phase relationship between the incident wave and the scattered wavelets
	Not useful for diffraction experiments
	The spatial distribution of the scattered intensity is obtained by summing up the intensities from independent scattering events
	Incoherent elastic scattering follows the Rutherford scattering law
	The resolution for incoherent imaging is much greater than that for coherent imaging due to squaring an amplitude distribution
	Incoherent cases: e.g. difference in wavelength
More coherent beam	Sharper diffraction pattern can be obtained

* When the electrons are treated as waves, their coherency becomes important.

Table 1219c. Electron forward- and back-scattering from a specimen in EM (electron microscope) measurements.

Scattering direction	Scattering angle
Forward scattering	< 90°
Back scattering	> 90°

Table 1219d. Application/effects of electrons scattered from a specimen in forward direction in EM (electron microscope) measurements. The scattered wave is strongest in the forward direction.

Application/effects	Energy loss	Specimen	Coherency	Full name
Diffraction	Elastic scattering	Crystals	Coherent scattering	Coherent elastic scattering
		Amorphous solids	Incoherent scattering	Incoherent elastic scattering
		Disordered materials	Incoherent scattering
Diffuse background in diffraction pattern	Mixture of incoherent elastic and inelastic scattering	Thick specimen	Mainly created by incoherent elastic scattering
	Mixture of incoherent elastic and inelastic scattering	Thick specimen	Partially created by incoherent inelastic scattering	Incoherent inelastic scattering
	Thermal diffuse scattering		Incoherent
	Compton scattering		Incoherent and inelastic
HOLZ	Elastic scattering		Coherent	Coherent elastic scattering
HRTEM			Coherent	Coherent elastic scattering
Weak-beam dark-field images			Contrast after elimination of the incoherent inelastically scattered background
Kikuchi lines	Inelastic + elastic scattering	Thick specimen	Two electron scatterings: incoherent scattering (sometimes inelastic) is followed by coherent elastic scattering (Bragg diffraction).
Practical diffraction pattern			An incoherent summation of independent scattering for millions of electrons due to averaging atomic vibrations during acquisition
Thickness fringes in TEM images		Crystals	Fringes become less visible for thicker specimens due to effect of absorption or a loss of coherent
Fresnel fringes			Coherent
Electron holography	Elastic scattering		Coherent beam and scattering are needed	Coherent elastic scattering
Scattering from single atoms	Elastic scattering	Any	Incoherent
Scattering from small objectives of ≤1.5 nm in size		Crystals	Could be coherent
Spectroscopy, e.g. EELS	Inelastic scattering	Any	Incoherent scattering: occur with a transfer of energy from the wave to the material
Plasmon	Primay means of inelastic scattering	Any
Scattering angle < 3°	Elastic scattering	Crystals	Coherent
Bright field STEM with overlap of diffracted discs			Coherent
Larger scattering angle			As the scattering angle becomes larger, the coherency degree becomes less	Incoherent elastic scattering (Rutherford scattering)
> 50 mrad (~3°) for 300 kV		Any	Dominated by incoherent thermal diffuse scattering
Scattering angle > 5°			Incoherent and Bragg scattering is normally negligible
High angle Rutherford scattering			Incoherent
HAADF
Z-contrast STEM
Z-contrast STEM of silicon at θ > 40 mrad*
Mass-thickness contrast		Crystals
Secondary electrons		Any	Incoherent
Atomic vibration		Any	Create an incoherent or partially coherent intensity distribution
Fluctuations in lens currents			Causes incoherence
Energy spread of electrons from the gun
Fluctuations in the high voltage
Beam convergence
Smaller source size			Better coherency
Spatial coherence			Smaller electron sources give better coherency (at best only ~ 1 nm)
Spread of focus due to chromatic aberration			Causes incoherence
Field emission guns			High degree of coherence
LaB₆ electron guns			Low degree of coherence

* θ is the inner angle of HAADF.
** For a LaB₆ electron gun with an emission spot of 10 μm in diameter to achieve the same coherence length as a field emission source additionally requires a source demagnification of several hundred times.

Table 1219e. Thickness dependence of electron scattering from a specimen in EM (electron microscope) measurements.

Scattering direction	EM method	Scattering event/angle	Scattering direction	Electrons	Wave (phase) property	Electron property	Contrast interpretation
Thin specimen	TEM	Less scattering events and then smaller scattering angle	Forward scattering	More electrons	Strong coherent signal	Strong wave property	Easy
Thin specimen	TEM		Back scattering	Fewer electrons	Weak incoherent signal	Weak particle property
Thick specimen	TEM/SEM		Forward scattering	Fewer electrons	Weak coherent signal	Weak wave property	Middle
Thick specimen	TEM/SEM	Electron is scattered more than once (plural scattering), and then scattering angle is greater	Back scattering	More electrons	Strong incoherent signal	Strong particle property
Very thick or bulk specimen	SEM	More scattering events and then greater scattering angle	Forward scattering	Fewest electrons	Weakest coherent signal	Weakest wave property	Difficult
Very thick or bulk specimen	SEM	Electron is scattered >20 times (multiple scattering), and then scattering angle is greatest	Back scattering	Most electrons	Strongest incoherent signal	Strongest particle property

Table 1219f. Factors which affects coherency in EM (electron microscope) measurements.

Factors	Coherency
Larger angular spread of illumination	Less coherent
Smaller angular spread of illumination	More coherent
Larger chromatic aberration	Less coherent
More beam convergence	Less coherent
Smaller collector aperture in STEM	More coherent
The spacing of Bragg planes is less than the atomic vibration amplitude	Coherent Bragg scattering cannot occur

* In STEM mode, the electron coherence is defined by the coherence length seen at the condenser aperture.

Table 1219g. Coherency effects in EM (electron microscope) measurements.

Coherency	Phase contrast	Image sensitivity
More coherent	Enhanced	Improved
Partially coherent	Limit interference effect

Table 1219h. Types of coherencies in EM (electron microscope) measurements.

Types of coherencies	Mechanisms
Temporal Coherency	All the electrons have the same wavelength, just like monochromatic light.
Spatial Coherency	Spatial coherency is related to the size of the electron source. Perfect spatial coherence would imply that the electrons were all emanating from the same point at the source. Therefore, source size governs spatial coherence and smaller source size gives better coherency.

Figure 1219. Coherence of various scattered electron rays: (a) Thin film specimen in TEM and/or STEM modes, and (b) Bulk specimen in SEM mode. The electron rays in red are incoherent, while the rays in green are coherent.

Table 1219i. Single and multiple electron scattering.

Scattering	Models
Single scattering	Lenz model, Mott cross-sections
Multiple scattering	Poisson distribution

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