# Difference between revisions of "NucPhys I HomeworkProblems"

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=Chapter 2= | =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. | 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. | ||

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

− | V_0 > 0 | + | <math>V_0 > 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) = | + | :<math>V(x) = 0 \;\; x<0</math> |

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

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Assume particles are incident from <math>x = -\infty</math> and <math>k_2^2 = 2m(V_o-E)/\hbar^2</math>. | Assume particles are incident from <math>x = -\infty</math> and <math>k_2^2 = 2m(V_o-E)/\hbar^2</math>. | ||

− | 4.) 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. | + | ==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><r^2> = 3R^2/5</math> | ||

+ | |||

+ | 2.) Using the definition of the form factor <math>F(q)</math> and probably an integral table, calculate <math>F(q)</math> when | ||

+ | |||

+ | a.):<math>\rho(r) =\left \{ {\rho_0 \;\;\;\; r<R \atop 0 \;\;\;\; r>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<math>_{\alpha}</math>-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 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>(\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.) 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>\frac{k_1}{k_2} = -\tan(k_1R)</math> | ||

+ | |||

+ | where | ||

+ | :<math>k_1^2 = \frac{2 m}{\hbar^2} (V+E)</math> | ||

+ | and | ||

+ | :<math>k_2^2 = -\frac{2 m}{\hbar^2}E</math> | ||

+ | :<math>E = -2.224 MeV < 0</math> : bound state | ||

+ | |||

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

+ | |||

+ | :<math>x=-tan(bx)</math> | ||

+ | |||

+ | and show that | ||

+ | :<math>b \approx 0.46</math> | ||

+ | |||

+ | when R = 2 fm. | ||

+ | |||

+ | |||

+ | ;Use the reduced mass for the deuteron system. | ||

+ | |||

+ | c.) Solve the transcendental equation for <math>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>V_{bound} =\left \{ {-V_o \;\;\;\; r <r_o \atop \frac{A}{r} \;\;\;\; r>r_o} \right .</math> | :<math>V_{bound} =\left \{ {-V_o \;\;\;\; r <r_o \atop \frac{A}{r} \;\;\;\; r>r_o} \right .</math> | ||

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The original nucleus had a charge <math>Ze</math> and the alpha particle has a charge <math>2e</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 | + | 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]] | ||

+ | |||

+ | where | ||

+ | :<math>(v=\sqrt{2E/m})</math> and <math>\sqrt{r_o/r}\equiv cos W</math> and <math>r_1 = \frac{A}{E}</math>. | ||

− | [http:// | + | [http://wiki.iac.isu.edu/index.php/Forest_NucPhys_I Go Back][[ Forest_NucPhys_I]] |

## Latest revision as of 17:48, 18 March 2009

# 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:

. Assume particles are incident from , don't try to normalize but do express the wave function in terms of one coefficient.

## Set 2

1.) Given the following barrier potential

show that the transmission coefficient is

when

Assume particles are incident from

and .## 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

2.) Using the definition of the form factor

and probably an integral table, calculate whena.):

b.)

c.)

## 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

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

in an orbital state.What would be the orbital magnetic dipole moment of this system

?2.) Assume that the proton magnetic moment is due to the rotational motion of a positive spherical uniform charge distribution of radius

spinning about its axis with angular speed .a.) Integrate the charge distribution to show that :

(hint:

)b.) show that

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

where

and

- : bound state

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

and show that

when R = 2 fm.

- Use the reduced mass for the deuteron system.

c.) Solve the transcendental equation for

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.

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

The original nucleus had a charge

and the alpha particle has a charge .Use the WKB approximation to show that the transmissivity (T : transmission coefficient) is:

- Media:GamowFormula.pdf Gamow's formula

where

- and and .