91Ó°ÊÓ

The concept of de Broglie wavelength links the wave-particle duality of matter, particularly for particles like electrons. According to the de Broglie hypothesis, every moving particle or object has an associated wave, just like light exhibits both wave-like and particle-like properties. The de Broglie wavelength \( \lambda \) of a particle is expressed by the equation:

• \( \lambda = \frac{h}{mv} \)

where:

• \( h \) is Planck's constant \( 6.626 \times 10^{-34} \, \text{m}^2 \, \text{kg/s} \)
• \( m \) is the particle's mass; for electrons, \( 9.11 \times 10^{-31} \, \text{kg} \)
• \( v \) is the particle's velocity

The smaller the mass and the faster the speed of a particle, the shorter is its wavelength. When electrons move at significant speeds, as in this problem, the wavelength becomes small enough to interact with structures roughly the size of that wavelength, like the tracks on a CD-ROM.
The ability to calculate and predict such wavelengths is crucial in understanding the behavior of electrons in diffraction experiments.

Constructive interference occurs when wave patterns overlap in such a way that their amplitudes reinforce each other. In electron diffraction, constructive interference leads to intensity maxima, where waves are in phase and combine to form brighter regions on a detecting screen like photographic film.
The condition for constructive interference in diffraction patterns is given by:

• \( d \sin \theta = m \lambda \)

where:

• \( d \) is the spacing between scattering centers, in this case, the pits on a CD-ROM
• \( \theta \) is the diffraction angle, the angle at which the constructive peaks occur
• \( m \) is the order of the maximum (an integer representing the series of spots)

Understanding this concept helps us predict where the intensity maxima, that is, the bright spots in a diffraction pattern, will appear based on the structure of the diffracting surface and the properties of the incoming electron waves.

The diffraction angle \( \theta \) represents the angle at which waves scatter from a surface and still manage to interfere constructively, creating bright spots (maxima) in the pattern that appears. In the experiment described, finding \( \theta \) allows us to determine where on the film these maxima will appear relative to the centerline of the electron beam.
The diffraction condition for finding \( \theta \) is again:

• \( d \sin \theta = m \lambda \)

where the values of \( m \) (order of maxima) are crucial. For the first maximum, \( m = 1 \), and for the second, \( m = 2 \). By solving for \( \theta \) using given values of \( d \) and the de Broglie wavelength \( \lambda \), we can precisely locate the maxima on the detection material, such as photographic film.

Maxima spacing is the distance between successive bright spots or maxima on a detecting surface, such as photographic film. This geometric setup allows us to analyze the results of the diffraction experiment and is crucial for interpreting the pattern produced by scattering electrons.
In the described experiment, the spacing \( x \) between the intensity maxima on the film can be calculated using trigonometric relations due to the known distance to the film \( L \):

• \( x = L ( \tan\theta_{m2} - \tan\theta_{m1} ) \)

where:

• \( \theta_{m1} \) and \( \theta_{m2} \) are diffraction angles corresponding to the first and second maxima, respectively.
• \( L \) is the distance from the CD-ROM to the film (50.0 cm)

Using this equation and inserting the calculated angles ensures precise measurement of the maxima spacing. This provides insights into the electron diffraction pattern and verifies the wave-like behavior of electrons.