Photons, deBroglie wavelength, electron diffraction

Photons

Problem:

Calculate the long-wavelength limit of the sensitivity of a photocathode layer with work function of 1.5  eV.

Solution:

Problem:

Photons of energy 0.2 MeV are being scattered by a free electrons (initially at rest).
(a)  At what angle must the scattering occur so that photons lose 10% of its energy?
(b)  What is the momentum of corresponding electrons after scattering?

Given:  Δλ = (λf - λi) = [h/(mec)](1 - cosθ) for Compton scattering.

Solution:


deBroglie wavelength

Problem:

N indistinguishable particles of mass m are confined to a box of volume V.  The temperature of this gas is T.
(a)  Above what temperature do you expect relativistic effects to become important?
(b)  Below which temperature do you expect quantum mechanical effects to become important?

Solution:

Problem:

Compute the speed and the deBroglie wavelength of
(a)  an electron whose kinetic energy is 5 eV.
(b)  an electron whose kinetic energy is 5 MeV.
(c)  a proton whose kinetic energy is 5 MeV.
(d)  a baseball ( m = 150 g) whose kinetic energy is 5 MeV.

Solution:

Problem:

Consider thermal neutrons in equilibrium at temperature T = 300 K.
(a)  Calculate its deBroglie wavelength.  State whether a beam of these neutrons could be diffracted by a crystal, and why?
(b)  Use Heisenberg's uncertainty principle to estimate the kinetic energy (in MeV) of a nucleon bound within a nucleus of radius 10−15 m.

Solution:

Problem:

(a)  Show that the deBroglie wavelength of a particle approaches zero faster than 1/v as its speed v approaches the speed of light.
(b)  Find the deBroglie wavelength of a 15 eV proton.
(c)  Find the deBroglie wavelength of a 15 keV electron.
(d)  What is the deBroglie wavelength of an electron whose speed is 9*107 m/s?
(e)  Find the wavelength of a 1 kg object whose speed is 1 m/s.
(f)  Neutrons in equilibrium with matter at room temperature (300 K) have average energy of about ½5 eV.  Such neutrons are often called "thermal" neutrons.  Find their deBroglie wavelength.
(g)  Derive a formula expressing the deBroglie wavelength (in Å) of an electron in terms of the potential difference U (in Volts) through which it is accelerated.
(h)  Derive a formula for the deBroglie wavelength of a particle in terms of its kinetic energy T and its rest energy mc2.

Solution:


Electron diffraction

Problem:

If we send electrons with energy of 25 eV through the same two slits (d = 0.5 mm) we use to produce a visible light interference pattern, what is the spacing between maxima on a screen 3 m away?

Solution:

Problem:

Consider a crystal with the atoms arranged in cubic array, each atom at distance 0.91 Å from its nearest neighbor.
Examine the conditions for Bragg reflection from atomic planes connecting diagonally placed atoms.
image
(a)  Find the lowest energy of electrons that can produce a first order maximum.
(b) If 300 eV electrons are used, at what angle from the crystal normal must they be incident to produce a first order maximum?

Solution: