Electron diffraction can also take place when there is...

Electron diffraction can also take place when there is interference between electron waves that scatter from atoms on the surface of a crystal and waves that scatter from atoms in the next plane below the surface, a distance $d$ from the surface (see Fig.36.23 $\mathrm{c} ) .$ (a) Find an equation for the angles $\theta$ at which there is an intensity maximum for electron waves of wavelength $\lambda$ . (b) The spacing between crystal planes in a certain metal is 0.091 $\mathrm{nm} .$ If 71.0 ev electrons are used, find the angle at which there is an intensity maximum due to interference between scattered waves from adjacent crystal planes. The angle is measured as shown in Fig. 36.23 $\mathrm{c} .$ (c) The actual angle of the intensity maximum is slightly different from your result in part (b). The reason is the work function $\phi$ of the metal (see Section $38.1 ),$ which changes the electron potential energy by $-e \phi$ when it moves from vacuum into the metal. If the effect of the work function is taken into account, is the angle of the intensity maximum larger or smaller than the value found in part (b)? Explain.

Electron diffraction can also take place when there is interference between electron waves that scatter from atoms on the surface of a crystal and waves that scatter from atoms in the next
plane below the surface, a distance $d$ from the surface (see Fig.36.23 $\mathrm{c} ) .$ (a) Find an equation for the angles $\theta$ at which there is an intensity maximum for electron waves of wavelength $\lambda$ . (b) The spacing between crystal planes in a certain metal is 0.091 $\mathrm{nm} .$ If 71.0 ev electrons are used, find the angle at which there is an intensity maximum due to interference between scattered waves from adjacent crystal planes. The angle is measured as shown in Fig. 36.23 $\mathrm{c} .$ (c) The actual angle of the intensity maximum is slightly different from your result in part (b). The reason is the work function $\phi$ of the metal (see Section $38.1 ),$ which changes the electron potential energy by $-e \phi$ when it moves from vacuum into the metal. If the effect of the work function is taken into account, is the angle of the intensity maximum larger or smaller than the value found in part (b)? Explain.

Key Concepts

Crystal Plane Spacing

Crystal plane spacing refers to the distance between parallel planes of atoms within a crystal lattice. It is an important parameter in diffraction experiments because the geometric arrangement and spacing of these planes determine the interference conditions and the resulting diffraction pattern. The interplanar spacing is crucial for applying Bragg's diffraction principles to determine structural parameters.

Work Function

The work function is the minimum energy required for an electron to escape from the surface of a metal to the vacuum. In electron diffraction, the work function plays a significant role because it effectively alters the kinetic energy and hence the wavelength of electrons as they enter the material. This change in wavelength can shift the position of the observed diffraction maxima.

De Broglie Wavelength

The de Broglie wavelength is a key concept in quantum mechanics which assigns a wavelength to a moving particle, such as an electron, based on its momentum. This relationship is given by ? = h/p, where h is Planck’s constant. The concept underscores the wave-like behavior of particles and is fundamental to understanding electron diffraction phenomena.

Wave Interference

Wave interference refers to the phenomenon that occurs when two or more waves overlap and combine, leading to regions of constructive (high intensity) or destructive (low intensity) interference. In diffraction experiments, interference between waves scattered at different points or from different lattice planes determines the positions of intensity maxima and minima.

Electron Diffraction

Electron diffraction is the phenomenon where electrons behave as waves and exhibit interference patterns when they encounter obstacles or periodic structures. This behavior is based on the wave-particle duality of matter and is analogous to light diffraction, helping us to probe the atomic-scale structure of materials.

Constructive Interference Condition

The constructive interference condition is a rule that determines when waves will sum to produce a maximum in intensity. In the context of electron diffraction, this condition is typically expressed as a relationship between the path difference from adjacent scattering centers (such as atomic planes) and the electron wavelength, ensuring that the phase difference is an integer multiple of 2?, which results in observable diffraction peaks.

Transcript

00:01 In this question, we're looking at x -ray diffraction of a surface of a crystal.

00:08 And the way that the interference actually occurs is because we have multiple rays coming in.

00:16 We have the first ray reflecting of the surface, of the first surface layer, whereas some of them might go down to the second layer, right before it reflects off.

00:34 So as a result of this too, there will be some difference in path that is traveled, which is this path over here, which i'm going to highlight as blue.

00:47 This is the extra distance that is traveled by the second ray.

00:57 This is this ray.

00:59 Travels an extra distance given over here.

01:04 I have not really drawn it very properly but it's supposed to be symmetric around the center.

01:13 Here, right, it's around like that.

01:20 And this extra path difference, if it is an integer multiple of the wavelength, then when these two coincide at a very, very far distance, when they meet together, they will have, no face difference, right? they will be in face with each other, they will have constructive interference, and this will give us a maximum.

01:47 So what is this path difference? well, using the angle theta, right, with respect to the central line over here, this is detour.

01:59 It means that this distance, if we take this as the distance between, two layers of crystal as a d then this distance over here plus this over here will be two times of d sine theta so this is our formula for this electron surface crystal surface diffraction we're going to use this to find what is the angle theta where we get our first maximum right this what we want to find but your first thing to find what is the wavelength of our electron so given that it is 71 e v is the kinetic energy so it goes to p square over 2m can get the wavelength by h over p h over square roots of 2m times kinetic energy is the of the electron...

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