A beam of protons with kinetic energy strikes a thick target of natural nickel (

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 beam of protons with kinetic energy

T = 5.00\ \mathrm{MeV}

strikes a thick target of natural nickel (

A = 58

, molar mass

M = 58.7\ \mathrm{g\,mol^{-1}}

), inducing the

(p,n)

reaction. The target is thick enough that the protons come to rest inside it, and the reaction has an effective threshold near

T = 2.00\ \mathrm{MeV}

, below which its cross section is negligible. ((a)) Estimate the number of partial waves that contribute to the reaction by finding

\ell_{\max} \approx kR

, using the equivalent sharp radius

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

and the relativistic relation between kinetic energy and momentum for the incident proton (rest energy

m_pc^2 = 938.3\ \mathrm{MeV}

; the reduced mass may be taken equal to the proton mass, since the target is far heavier). ((b)) At

T = 5.00\ \mathrm{MeV}

the elasticity parameters are measured to be

\eta_0 = 0.20

\eta_1 = 0.50

\eta_2 = 0.80

, with

\eta_\ell = 1

(no absorption, negligible scattering into this channel) for

\ell \ge 3

. Calculate the reaction cross section

\sigma_{\rm reac}

from the partial-wave sum. ((c)) Treating

\sigma_{\rm reac}

found in part (b) as roughly constant over the whole range

2.00\ \mathrm{MeV} \le T \le 5.00\ \mathrm{MeV}

that the beam traverses inside the target, and taking the mass stopping power of nickel for protons in this range as the constant value

(-\mathrm{d}E/\mathrm{d}x)/\rho = 58.0\ \mathrm{MeV\,cm^2\,g^{-1}}

, calculate the thick-target yield in reactions per incident proton.