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{\large \bf Physics 499}\\
Homework Assignment 1\\
Finding Roots\\
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Due Wednesday April 17th
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Problem : \= $\Lambda$ hypernuclei \\
Reference: \> Am. J. Phys. 58, 1016 (Oct 1990).
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The energies of the bound states of a three dimensional
square well of radius $R$ are solutions of the equation:
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\noindent for $\ell$ = 0
\begin{equation}
tan \Big( \sqrt{ {{2m(V_0-|E|)} \over {\hbar^2}} } \; R \Big) = -\sqrt{{{V_0-|E|} \over {|E|}}}
\end{equation}
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\noindent Consider the following data for $\Lambda$ bound states in the
nucleus:
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\begin{tabular}{l|llllll}
Nucleus & $^{13}$C & $^{16}$O & $^{28}$Si & $^{40}$Ca & $^{51}$V &
$^{89}$Y \\
\hline
Mass Number (A): & 13 & 16 & 28 & 40 & 51 & 89 \\
$\ell$=0 binding energy (in MeV): & 10.5 & 12.1 & 17.1 & 18.5 & 18.0 &
23.0
\end{tabular}
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\noindent Useful information:
Nuclear radii are roughly given by: R = 1.1 A$^{1/3}$ in fm;
$m_\Lambda c^2 = 1115$ MeV ; $\hbar$\/c = 197.33 MeV-fm
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Using your favorite computer programming method for finding roots of
an equation, find values of the mean
field potential $V_0$ (in MeV) for each of these nuclei. Are they
approximately equal? Discuss the significance of this. What is
the average mean field potential that the lambda particle
experiences in the nuclear medium?
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You should turn in (e-mail) two files: your computer code that will run in
either gcc or ROOT, and a file discussing your results. The prefered
format for the discussion file is latex. Be sure your name is somewhere
in each file you e-mail to me.
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