1. The average energy of an electron on the surface of a slab of metal held at temperature T is proportional to kT, where k is the Boltzman constant (1.38 x 10-23 J/K) and T is measured in degrees Kelvin.
(a) What is the average energy of the electron when the metal is at room temperature (i.e. at about 300 K)?
(b) How hot (in Fahrenheit scale) do we need to get the metal to free the electron to get from the metal? (Assume a work function of roughly 2 eV.)
Hint: for conversions use the site: MegaConverters
2. The green light of an argon-ion laser (514.5 nm) falls on a cesium metal. Will any photoelectrons get produced? If so, what is their maximum kinetic energy?
3. A beam of electrons is accelerated by passing it across a 10 kV potential. It is then struck across a molybdenum target. What is the minimum wavelength of the X-ray that are produced? Could these X-rays be used to examine the structure of a nanocrystal such as the ones created at NIST (see their Web page!)?
4. Consider the diffraction of waves (X-ray or particle's deBroglie) from an array of atoms whose planes are separated by a distance d, as shown below:
(a)Prove that the path-length-difference between the two "reflected" rays (scattered rays) is equal to 2dsin(q). Notice that the photon is "reflected" off the plane (called Bragg scattering) and not off individual atoms. The red and blue rays are the reflections from two (arbitrary) Bragg planes. There are of course many such planes that make different angles of reflection with the ray. Still, the possibilities are finite.
(b) Show that the condition for constructive interference for Bragg scattering is:
2 d sin (q) = m l : m= 0, 1, 2,...
m is the order of reflection and represents interference between consecutive Bragg planes.
5. In the following table V is the electric potential used in accelerating electrons, in volts. Determine the de Broglie wavelength for each accelerating potential, to fill in the second column, and in the third column write what is that such an electron beam could probe, were it to be used in a TEM.
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V (volts)
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l (m)
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Useful to probe
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1,000
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5,000
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8,000
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Hint: to obtain values of constants use the NIST site on Fundamental Physical Constants
6. (a) Calculate the de Broglie wavelength of a Na atom in the vapor of a sodium street lamp, say at 300 degrees Celsius.
(b) Repeat the calculation for part (a), but now for a vapor at 0.001 degree Kelvin; i.e. in a typical atom trap.
7. Consider a photon that is trapped between two highly reflective "mirrors" in a microcavity of the dimension of 10 nm. Apply the bounday condition to determine the photon's energy levels.
8. What are the first three energy levels for an electron confined to a lenth of 0.1 nm?
9. A particle confined to a one-dimensional box of length L is found to be in the state n=3. Where is the most likely position(s) that we could locate the particle?
10. Fill in the following table for the four color lines in the spectrum of the hydrogen
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Line #
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Color
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Wavelength (nm)
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Energy (eV)
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red
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teal
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violet
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UV
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Last Modified January 7, 2004 malekis@union.edu