LEED spectroscopy records the intensities and locations of...

LEED spectroscopy records the intensities and locations of electrons that are diffracted from a surface. For an electron to diffract, its de Broglie wavelength must be less than twice the distance between the atomic planes in the solid (see Section 29-9). Show that the de Broglie wavelength of an electron accelerated through a potential difference of $\phi$ volts is given by $$ \lambda / \mathrm{pm}=\left(\frac{1.504 \times 10^{6} \mathrm{~V}}{\phi}\right)^{1 / 2} $$

LEED spectroscopy records the intensities and locations of electrons that are diffracted from a surface. For an electron to diffract, its de Broglie wavelength must be less than twice the distance between the atomic planes in the solid (see Section 29-9). Show that the de Broglie wavelength of an electron accelerated through a potential difference of $\phi$ volts is given by
$$
\lambda / \mathrm{pm}=\left(\frac{1.504 \times 10^{6} \mathrm{~V}}{\phi}\right)^{1 / 2}
$$

Key Concepts

Electron Diffraction

Electron diffraction is a phenomenon that emerges from the wave nature of electrons, where electrons display interference patterns when they encounter obstacles or apertures. In surface analysis techniques, such as LEED spectroscopy, these diffraction patterns provide information about the atomic structure of surfaces, as the pattern is directly influenced by the spacing between atomic planes.

de Broglie Wavelength

The de Broglie wavelength is a fundamental concept in quantum mechanics relating the wave-like behavior of matter. According to de Broglie, every particle has an associated wavelength, given by ? = h/p, where h is Planck's constant and p is the momentum of the particle. This principle is essential for understanding electron diffraction phenomena, as it determines whether electrons have wavelengths compatible with the spacing of atoms in a material.

Accelerating Voltage and Kinetic Energy

When an electron is accelerated through a potential difference, it gains kinetic energy equal to the charge of the electron multiplied by the potential difference (e?). This conversion of electrical energy into kinetic energy enables the calculation of the electron's momentum using classical mechanics (non-relativistic approximation) since p = ?(2m e?), where m is the electron mass.

Relationship Between Kinetic Energy and de Broglie Wavelength

The relationship between an electron’s kinetic energy and its de Broglie wavelength is derived by combining the definitions of kinetic energy and momentum. By substituting the momentum found from the kinetic energy relation into the de Broglie equation, one obtains the formula for the wavelength as a function of the accelerating voltage. This relationship is critical in determining the conditions under which electron diffraction occurs on periodic atomic structures.

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