dynamical diffraction

Electrons incident on a crystalline specimen are reflected (diffracted) by the lattice planes satisfying a Bragg condition. When the specimen is thick, many electrons are reflected. Then, the intensity of the incident wave decreases and that of the reflected wave exceeds that of the incident wave. At a certain thickness (half the extinction distance ξg), the incident wave intensity eventually becomes to zero, and the reflected wave reaches its maximum. Then, the reflected wave begins to be diffracted back toward the incident direction. When the thickness reaches the extinction distance ξg, the incident wave regains its maximum intensity, and the reflected wave intensity becomes back to zero. This interaction between the incident and diffracted waves is called dynamical diffraction.

When a specimen is very thin (<3 nm), diffraction phenomena can be treated under the assumption that only a minimal amount of reflection occurs without attenuation of the incident beam. This approximation is referred to as kinematical diffraction. It is noted that in X-ray and neutron diffraction the kinematical diffraction approximation can be applied due to their low scattering power, thereby complicated dynamical diffraction can be neglected.

The figure schematically illustrates kinematical diffraction, in which only minimal reflection occurs (Fig.(a)), and dynamical diffraction, in which many electrons are reflected and interactions arise between the incident and reflected waves (Fig. (b)).

Fig.(a) In kinematical diffraction, only minimal diffraction occurs from the incident wave, and the amplitude of the diffracted wave is very small compared to that of the transmitted wave, which is considered to experience no attenuation. Kinematical diffraction is applicable only when the specimen is very thin (<3 nm).
(b) In dynamical diffraction, the incident wave undergoes repeated diffraction. When a certain depth is reached, the transmitted wave is entirely transferred to the diffracted wave. Then, diffraction occurs in the reverse direction from the diffracted wave back to the transmitted wave. When the depth reaches a depth ξg (extinction distance), the intensity of the diffracted wave returns to zero, and that of the transmitted wave regains to maximum. This oscillation between the transmitted and diffracted waves is observed as the equal thickness fringes in a wedge-shaped crystal.

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