2002 Autumn Modern Physics
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(You only have to answer two of the three questions below.)

1. [50 points total] Relativity

A. A particle of mass $m_1$ decays at rest into two particles with rest masses $m_2$ and $m_3$ respectively. The i-th particle has 4-momentum $p_i=m_i\gamma_i\{1,0,0,\beta_i\}$ where $\gamma_i=(1-\beta_i^2)^{-1/2},i=1,2,3,\beta_1=0$.

i. [5 points] Give an expression for the relation between the three 4-vectors. What conservation law(s) does it represent?

Greg’s Solution:

(1)
\begin{equation} p_1 = p_2+p_3 \end{equation}

This represents conservation of 4-momentum.

ii. [5 points] Find the invariant mass of the two-particle system following the decay.

Greg’s Solution:

(2)
\begin{align} \sqrt{(p_2+p_3)^2} = \sqrt{m_2^2 + m_3^2 + m_2m_3\gamma_2\gamma_3(1-\beta_2\beta_3)} = m_1^2 \end{align}

iii. [5 points] Find the kinetic energy of particle 3 in terms of $m_1,m_2,m_3,\beta$.

Greg’s Solution:

(3)
\begin{aligned} K &= \sqrt{p_3^3-m_3^2} \\ K &= \sqrt{(p_1-p_2)^2-m_3^2} \\ K &= \sqrt{m_1^2+m_2^2-2\frac{m_1m_2}{\sqrt{1-\beta_2^2}}-m_3^2} \\ \end{aligned}

B. [15 points] In system S, two events at coordinates $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ occur simultaneously at time $t_0$. System S’ moves at a velocity $\beta_c$ along the x-axis with respect to system S. Show that the two events are not simultaneous in S’, but instead have a time separation of $\Delta t=-\beta \gamma \Delta x/c$, where $\gamma=1/(1-\beta^2)^{1/2}$ and $\Delta x=x_1-x_2$.

Greg’s Solution:

The Lorentz transform is,

(4)
\begin{align} L = \begin{bmatrix} \gamma & -\gamma\beta & 0 & 0 \\ -\gamma\beta & \gamma & 0 & 0 \\ 0 & 0 & \gamma & 0 \\ 0 & 0 & 0 & \gamma \end{bmatrix} \end{align}

Thus, the two points are transformed to,

(5)
\begin{aligned} (ct,x_1,y_1,z_1) &\to \gamma (ct- \beta x_1,x_1-\beta c t,y_1,z_1) \\ (ct,x_2,y_2,z_2) &\to \gamma (ct- \beta x_2,x_2-\beta c t,y_2,z_2) \\ \end{aligned}

And so we see that the difference between the two time components is indeed $\gamma\beta (x_2-x_1)/c$.

C. [20 points] The CERN SPS accelerator produces a beam of $\,^{208}\text{Pb}$ nuclei with a laboratory total energy of 158 GeV/nucleon. This beam strikes a fixed target of $\,^{208}\text{Pb}$ atoms. (Note: the rest mass of a nucleon is 0.938 GeV. Ignore effects of binding energy.)

i. [6 points] Find the Lorentz $\gamma$ factor of the beam.

Greg’s Solution:

The total energy is $\gamma m$, so we have that

(6)
\begin{align} \gamma = \frac{158}{.938} \sim 175 \end{align}

ii. [7 points] For Pb-Pb collisions, find the invariant mass per nucleon of the system and the center of mass collision energy per nucleon.

Greg’s Solution:

(7)
\begin{aligned} m_{\text{tot}}^2 &= \left[ \underbrace{\gamma m (1,0,0,\beta)}_{\text{incoming particle}} +\underbrace{ m(1,0,0,0)}_{fixed particle}\right]^2 \\ &= \left[ \gamma m (1,0,0,\beta) + m(1,0,0,0)\right]^2 \\ &= m^2 (\gamma+1,0,0,\gamma\beta)^2 \\ &= m^2 [\gamma^2 + 2\gamma + 1 - \beta^2] \\ &= m^2 [\gamma^2 + 2\gamma + 1 - \gamma^2\beta^2] \\ &= m^2 [\gamma^2(1-\beta^2) + 2\gamma + 1 ] \\ &= m^2 [1+ 2\gamma + 1 ] \\ &= 2m^2 [\gamma + 1 ] , \end{aligned}

where $m$ is the rest mass of a nucleon of lead; note that this has the correct limit that as $\gamma\to 1$, $m_{\text{tot}}^2\to (2m_{\text{nucleon}})^2$.

The first component of the momentum tensor is is the total energy; to obtain the kinetic energy we just subtract the mass of the two nucleons from it,

(8)
\begin{align} K = m\left[\gamma-1\right] \end{align}

iii. [7 points] If a collider produced colliding beams of lead ions at the same energy, what would be the center of mass collision energy per nucleon?

Greg’s Solution:

In this case, the total 4-momentum of the system would be

(9)
\begin{align} p_{\text{tot}} = m\gamma(1,0,0,+\beta) + m\gamma(1,0,0,-\beta) = 2m\gamma(1,0,0,0) \end{align}

Subtract the mass of the two nucleons to obtain the collision energy,

(10)
\begin{align} K = 2m(\gamma-1) \end{align}

2. [50 points total] Astrophysics

A. [25 points] Dark Matter

3. [50 points total] Nuclear Physics

A. [20 points] The atom $\,^{27}\text{Si}$ (Z=14,N=13) has a mass of 26.986704i and its mirror nucleus’‘ twin [[$\,^{27}\text{Al$]] (Z=13,N=14) has a mass of 26.981539u. Assume that nuclear radii obey the volume rule [[$R=r_0 A^{1/3}$]], where A is the mass number and [[$r_0$]] is a constant. Estimate the value of [[$r_0$]] from the data above. [Note: the classical electron radius is [[$r_c=e^2/m_ec^2=e^2/0.511\text{MeV}=2.8\times 10^{-13}\text{cm}$]], and [[$1\quad u=931.503 \,\, \text{MeV}/c^2$]].] ++ **B.** [30 points] The nuclear shell model describes the neutrons and protons of a nucleus as moving in independent nuclear potential wells, one for neutrons and the other for protons, with the spin [[$\textbf{S}$]] and angular momentum [[$\textbf{L}$]] coupled by an interaction [[$H_{\text{SO}}=-2\alpha \textbf{S}\cdot\textbf{L}$]], where [[$\alpha$]] is a positive constant. +++ **i.** [7 points] What is the difference in the shapes of the shell-model potential wells used for the neutrons and for the protons? Why must they be treated as separate, rather than putting all nucleons into one well? +++ **ii.** [7 points] Discuss the ordering of the first three shells for neutrons, and the number of neutrons that will close each shell. +++ **iii.** [7 points] The nucleus [[$\,^{16}O$]] is a closed-shell nucleus in the sense that its neutron and proton shells are both closed. Explain why and give its spin and parity. +++ **iv.** [9 points] Consider the three [[$Z=N-1$]] nuclei that have a closed proton shell and a single neutron hole’‘ in the similar neutron shell. The shell closures are for the first, second, and third closed proton shells. What are these nuclei and what are their spins and parities?

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