NucPhys I HomeworkProblems

Chapter 2

Set 1

1.) In your own words, describe what the "Standard Model" of physics is using 2 to 5 paragraphs. Your paragraphs should include the concepts of the 4 fundamental forces of physics, QED, and QCD at a minimum. Any sentences in which a string of 5 or more words match a sentence that is found on the internet will be evidence of cheating.

2.) Solve the Schrodinger equation for the following potential:

[math]\displaystyle{ V(x) = \infty \;\; x<0 }[/math]

[math]\displaystyle{ V(x) =\left \{ {-V_0 \;\;\;\; 0<x <a \atop 0 \;\;\;\; x>a} \right . }[/math]

[math]\displaystyle{ V_0 > 0 }[/math]. Assume particles are incident from [math]\displaystyle{ x = -\infty }[/math], don't try to normalize but do express the wave function in terms of one coefficient.

Set 2

1.) Given the following barrier potential

[math]\displaystyle{ V(x) = 0 \;\; x<0 }[/math]

[math]\displaystyle{ V(x) =\left \{ {V_o \;\;\;\; 0<x <a \atop 0 \;\;\;\; x>a} \right . }[/math]

show that the transmission coefficient is

[math]\displaystyle{ T = \frac{1}{1+ \frac{V_o^2 \sinh^2(k_2a)}{4E(V_o-E)}} }[/math]

when [math]\displaystyle{ E < V_o }[/math]

Assume particles are incident from [math]\displaystyle{ x = -\infty }[/math] and [math]\displaystyle{ k_2^2 = 2m(V_o-E)/\hbar^2 }[/math].

Set 3

1.) Starting with Shrodinger's time-independent equation, derive the wave functions for a 2-D simple harmonic oscillator. Your derivation should take advantage of separation of variables and you are not required to normalize the wave function.

Set 4

1.) Show that the mean-square charge radius of a uniformly charged sphere is [math]\displaystyle{ <r^2> = 3R^2/5 }[/math]

2.) Using the definition of the form factor [math]\displaystyle{ F(q) }[/math] and probably an integral table, calculate [math]\displaystyle{ F(q) }[/math] when

a.):[math]\displaystyle{ \rho(r) =\left \{ {\rho_0 \;\;\;\; r<R \atop 0 \;\;\;\; r>R} \right . }[/math]

b.) [math]\displaystyle{ \rho(r) = \rho_0 e^{- \alpha r} }[/math]

c.) [math]\displaystyle{ \rho(r) = \rho_0 e^{- \alpha^2 r^2} }[/math]

Set 5

1.) a.) find the binding energy difference between O-15 and N-15

b.) compute the nuclear radius of O-15 and N-15 assuming the above binding energy is due to the coulomb energy.

2.) Muonic X-rays

a.) Calculate the energies of muonic K[math]\displaystyle{ _{\alpha} }[/math]-line X-rays from Fe assuming a point nucleus and using a one-electron model..

b.) Calculate the energy correction [math]\displaystyle{ (\Delta E) }[/math] due to the finite nuclear size.

3.) Find the binding energy using the fit equation B(Z,A) from the semi-empirical mass formula for

a.) Ne-21

b.) Fe-57

c.) Bi-209

d.) Fm-256

4.) Find the neutron separation energies for

a.) Li-7

b.) Zr-91

c.) U-236

5.) Find the proton separation energies for

a.) Ne-20

b.) Mn-55

c.) Au-197

Set 6

1.) Assume a neutron may be described as a proton with a negative pion [math]\displaystyle{ (\pi^-) }[/math] in an [math]\displaystyle{ \ell =1 }[/math] orbital state.

What would be the orbital magnetic dipole moment of this system [math]\displaystyle{ (s_{\pi} = 0) }[/math]?

2.) Assume that the proton magnetic moment is due to the rotational motion of a positive spherical uniform charge distribution of radius[math]\displaystyle{ R }[/math] spinning about its axis with angular speed [math]\displaystyle{ \omega }[/math].

a.) Integrate the charge distribution to show that :

[math]\displaystyle{ \mu = \frac{1}{5} e \omega R^2 }[/math]

(hint: [math]\displaystyle{ \mu = i A }[/math])

b.) show that

[math]\displaystyle{ \mu = \frac{e s}{2 m} }[/math]

using the classical relationship between angular momentum and rotational speed for the spin.

Set 7

1.) Solving the transcendental equation for the deuteron

a.) Assume the 3-D square well approximates the deuteron system such that the well width is 2.2 fm. Using boundary conditions show that

[math]\displaystyle{ \frac{k_1}{k_2} = -\tan(k_1R) }[/math]

where

[math]\displaystyle{ k_1^2 = \frac{2 m}{\hbar^2} (V+E) }[/math]

and

[math]\displaystyle{ k_2^2 = -\frac{2 m}{\hbar^2}E }[/math]

[math]\displaystyle{ E = -2.224 MeV < 0 }[/math] : bound state

b.) Rewrite the transcendental equation for the deuteron in the form

[math]\displaystyle{ x=-tan(bx) }[/math]

and show that

[math]\displaystyle{ b \approx 0.46 }[/math]

when R = 2 fm.

Use the reduced mass for the deuteron system.

c.) Solve the transcendental equation for [math]\displaystyle{ x }[/math] using an iterative technique.

I got x = 3.93xxxxxx

Set 500

5.) Several nuclei decay by the emmission of an alpha particle. An alpha particle (He-4) is a tighlty bound nuclear containing 2 protons and 2 neutrons in which the energy needed to remove one neutron is 20.5 MeV. One model for this decay process views the alpha particle as being bound to the nucleus via a spherical potential well.

[math]\displaystyle{ V_{bound} =\left \{ {-V_o \;\;\;\; r <r_o \atop \frac{A}{r} \;\;\;\; r>r_o} \right . }[/math]

Once outside the nucleus, the alpha particle is repelled via Coulombs law

[math]\displaystyle{ V_{outside} \approx \frac{2(Z-2)e^2}{r} \equiv \frac{A}{r} }[/math]

The original nucleus had a charge [math]\displaystyle{ Ze }[/math] and the alpha particle has a charge [math]\displaystyle{ 2e }[/math].

Use the WKB approximation to show that the transmissivity (T : transmission coefficient) is:

[math]\displaystyle{ T = e^{-\frac{2}{\hbar} \int_{r_o}^{r_1} \sqrt{2m \left ( \frac{A}{r} - E\right )}dr} = e^{-\frac{2A}{\hbar v} (2W-\sin(2W) )} }[/math] Gamow's formula Media:GamowFormula.pdf

where

[math]\displaystyle{ (v=\sqrt{2E/m}) }[/math] and [math]\displaystyle{ \sqrt{r_o/r}\equiv cos W }[/math] and [math]\displaystyle{ r_1 = \frac{A}{E} }[/math].

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