Physics 309

The Rectangular Barrier

1. The propagation and discontinuity matrices for the rectangular barrier are
   
   $${\bf d_{12}} = {1 \over 2} \pmatrix{ 1 + {k_2 \over k_1} & ... ...ver k_2} \cr 1 - {k_1 \over k_2} & 1 + {k_1 \over k_2} \cr }$$
   
   
   
   
   $${\bf p_1} = \pmatrix{ e^{ i k_1 2 a} & 0 \cr 0 & e^{- i k... ... \pmatrix{ e^{-i k_2 2 a} & 0 \cr 0 & e^{+ i k_2 2 a} \cr } .$$
   
   
   What is the transfer matrix for the rectangular barrier in terms of the appropriate wave numbers? What is the transmission coefficient? Is your expression consistent with the one in your text? Why or why not?

   
   

2. Fusion reactions in the Sun are responsible for the production of solar energy. One of these reactions is the fusion of a proton with a carbon nucleus ( $\rm p + ^{12}C \rightarrow ^{13}N$). The carbon nucleus has a radius of about $2 fm$ and the proton has a radius of about $1 fm$ where $1 fm = 10^{-15}m$.
   1. Calculate the inter-nuclear distance when the proton and the $\rm ^{12}C$ are just touching and call this distance $r^\prime$. Estimate the Coulomb barrier, $V_C$, between the proton and the carbon nucleus.

   2. The proton is incident on the carbon nucleus because of its thermal motion in the Sun's core. Estimate the incident energy of the proton by assuming that its energy is ${3\over 2} kT$ where $k$ is Boltzman's constant ( $8.62\times 10^{-11} MeV/K$) and $T$ is the temperature of the solar core ( $\approx 10^7 K$).

   3. Fusion will occur when the proton can tunnel through the barrier. Treat the Coulomb barrier as a rectangular barrier and calculate the transmission coefficient. Justify your choice for the width and the height of the barrier. Do you expect solar fusion to be a speedy process? Cite any other evidence that would support your result.