Difference between revisions of "NucPhys I HomeworkProblems"

From New IAC Wiki
Jump to navigation Jump to search
Line 105: Line 105:
1.) Solving the transcendental equation for the deuteron
1.) Solving the transcendental equation for the deuteron
a.) Rewrite the transcendental equation for the deuteron
a.) Rewrite the transcendental equation for the deuteron

Revision as of 18:02, 8 April 2008

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]V(x) = \infty \;\; x\lt 0[/math]
[math]V(x) =\left \{ {-V_0 \;\;\;\; 0\lt x \lt a \atop 0 \;\;\;\; x\gt a} \right .[/math]

[math]V_0 \gt 0[/math]. Assume particles are incident from [math]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]V(x) = 0 \;\; x\lt 0[/math]
[math]V(x) =\left \{ {V_o \;\;\;\; 0\lt x \lt a \atop 0 \;\;\;\; x\gt a} \right .[/math]

show that the transmission coefficient is

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

when [math]E \lt V_o[/math]

Assume particles are incident from [math]x = -\infty[/math] and [math]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]\lt r^2\gt = 3R^2/5[/math]

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

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

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

c.) [math]\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-line X-rays from Fe assuming a point nucleus and using a one-electron model..

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

3.) Find the binding energy using 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](\pi^-)[/math] in an [math]\ell =1[/math] orbital state.

What would be the orbital magnetic dipole moment of this system [math](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] R[/math] spinning about its axis with angular speed [math]\omega[/math].

a.) Integrate the charge distribution to show that :

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

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

b.) show that

[math]\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.) Rewrite the transcendental equation for the deuteron

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

in the form


and show that

[math]b \approx 0.46[/math]

when R = 2 fm.

Use the reduced mass for the deuteron system.

b.) Solve the transcendental equation for [math]x[/math] using an iterative technique.

I got x = 3.931

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

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

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

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

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

[math]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


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

Go Back