Physics 251 Spring 1999
Modern Physics
Homework Assignment # 6, Due (in class) Monday March 8

Reading Assignment: Finish reading Krane, Chapter 6; start Chapter 7 before lecture Friday March 5. Problem Assignment:

1.

Krane, Chapter 6, p.202, #6. Clarification: the minimum approach distance can be worked out for alphas which scatter at 90$^{\circ}$, but this is both harder and less interesting than the absolute closest approach distance which is achieved with b=0. This is the number you should calculate.

2.

Using the Van de Graaf accelerator and the superconducting LINAC post-accelerator here at Stony Brook, Profs. Orozco and Sprouse make the unstable atom francium (²¹²Fr, Z=87, half-life 20min) by colliding oxygen nuclei (¹⁸O) with a gold foil. Oxygen has Z=8 and gold has Z=79. Some neutrons are released in the collision. What is the minimum energy that the oxygen (O⁸⁺ ions) can have so that they approach the gold nuclei to a distance of the gold nuclear radius, $7.0\times 10^{-15}$m? (This is called ``penetrating the Coulomb barrier.'')

3.

The treatment of Rutherford scattering in Krane's book ignores the recoil of the nucleus when the alpha particle scatters from it. Suppose the nucleus is in free space rather than being embedded in a solid. The maximum recoil velocity occurs for a head-on collision (b=0.) Show that for alphas with kinetic energy small compared with 4000 MeV, scattering from gold nuclei, the maximum kinetic energy absorbed by the recoiling gold nuclei is about 15% of the incident alpha particle kinetic energy.

4.

In a head-on collision between a non-relativistic alpha particle and an electron at rest, the result is that the electron will recoil with twice the velocity of the incident alpha. The same physics applies to a baseball at rest and a massive bat, and is most easily derived by looking at the collision from the reference frame of the moving bat. Explain this argument. Use this to calculate what fraction of the energy of an incident alpha particle is lost in a head-on collision with an electron, and verify that the recoiling electron is non-relativistic if the alpha particle has kinetic energy 100MeV or less.

5.

For the lowest 3 levels (n=1,2,3) of the H atom, find the radius of the orbit (in nm), the kinetic energy of the electron (in eV) and the potential energy of the electron (in eV.)

6.

For the lowest 3 levels (n=1,2,3) of the H atom, derive the formula for the electron's velocity divided by the speed of light. For the lowest (n=1) level, the ratio v/c is equal to a number, denoted $\alpha$ and called the ``fine structure constant.''

7.

Krane, Chapter 6, p.203, #19.

8.

Ultraviolet light of wavelength 21.0nm impinges on (a) a beam of H atoms, or (b) a beam of He⁺ ions. Find the kinetic energy of the photo-emitted electrons.

9.

Krane, Chapter 6, p.204, #29. Clarification: The ground state is stable; only the excited states have non-infinite lifetimes.