A proton beam is fully stopped in a thick target ( )

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Nuclear Physics · Nuclear reactions

The Nucleus Resolved, Ch Reactions: four-vectors, amplitudes and partial waves

Problem

A proton beam is fully stopped in a thick

^{63}\mathrm{Cu}

target (

M = 63.0\,\mathrm{g\,mol^{-1}}

), entering at

E_1 = 8.00\,\mathrm{MeV}

and slowing to

E_2=2.00\,\mathrm{MeV}

before the reaction channel of interest switches off. Take the nuclear radius

R = 1.2\times63^{1/3}\,\mathrm{fm}

, the proton rest energy

m_pc^2 = 938.3\,\mathrm{MeV}

, and

\hbar c = 197.327\,\mathrm{MeV\,fm}

; treat the protons non-relativistically (

E \ll m_pc^2

) and use the de Broglie wavenumber

k=\sqrt{2m_pc^2E}/\hbar c

. ((a)) Calculate

kR

E_1

and at

E_2

, and state how many partial waves (

\ell=0

up to the nearest integer) contribute to the reaction at each energy. ((b)) At

E_1

the elasticity parameters for the contributing waves are

\eta_0=0.20

\eta_1=0.55

\eta_2=0.85

\eta_3=0.98

; at

E_2

only

\eta_0=0.20

and

\eta_1=0.55

contribute (higher waves do not reach the nucleus). Calculate the reaction cross section

\sigma_{\rm reac}=(\pi/k^2)\sum_\ell(2\ell+1)(1-|\eta_\ell|^2)

at each energy. ((c)) Using the mean of the two cross sections as a trapezoidal estimate of the excitation function, and a mean stopping power

(-\mathrm{d}E/\mathrm{d}x)/\rho = 60.0\,\mathrm{MeV\,cm^2\,g^{-1}}

over this energy range, estimate the thick-target yield

Y \approx \frac{N_A}{M}\,\bar\sigma_{\rm reac}\,\frac{E_1-E_2}{(-\mathrm{d}E/\mathrm{d}x)/\rho}

in reactions per incident proton, with

N_A = 6.022\times10^{23}\,\mathrm{mol^{-1}}