Chapter 39: Problem 15
A CD-ROM is used instead of a crystal in an electrondiffraction experiment. The surface of the CD-ROM has tracks of tiny pits with a uniform spacing of 1.60 \(\mu{m}\). (a) If the speed of the electrons is 1.26 \(\times\) 10\(^4\) m/s, at which values of \(\theta\) will the \(m\) = 1 and \(m\) = 2 intensity maxima appear? (b) The scattered electrons in these maxima strike at normal incidence a piece of photographic film that is 50.0 cm from the CD-ROM. What is the spacing on the film between these maxima?
Short Answer
Step by step solution
Understanding Electron Wavelength
Calculate De Broglie Wavelength
Determine Angles for Maxima
Solve for \\(\theta\\)
Calculate Spacing on Film
Calculate Film Separation
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
de Broglie Wavelength
- This wavelength is calculated using the formula: \( \lambda = \frac{h}{mv} \).
- \(h\) represents Planck's constant, a fundamental quantity in quantum mechanics known to be approximately \(6.626 \times 10^{-34} \, \text{Js}\).
- \(m\) refers to the mass of the particle, in this case, the electron, which has a mass of about \(9.11 \times 10^{-31} \, \text{kg}\).
- \(v\) is the velocity of the particle.
Constructive Interference
- \(d \sin \theta = m \lambda\),
- \(d\) is the distance between diffraction lines, in this exercise, it's the pit spacing on the CD-ROM, \(1.60 \times 10^{-6} \, \text{m}\).
- \(m\) is the order of the maxima, which can be any integer (1 for the first maxima, 2 for the second, etc.).
- \(\lambda\) is the de Broglie wavelength of the electrons.
- \(\theta\) is the angle at which these maxima occur.
Angle of Diffraction
- For \(m = 1\), the angle is calculated as \(\theta_{m=1} \approx 2.08^\circ\).
- For \(m = 2\), \(\theta_{m=2} \approx 4.15^\circ\).
Electron Velocity
- The energy of the electrons, which is proportional to the square of the velocity.
- The resolution of the diffraction pattern, as changes in speed can shift the positions of interference maxima.