A deuteron beam (deuteron rest energy ) strikes a target ( ) and produces via stripping

PHYSICS · Multi-part · Fermi University Physics · Nuclear Physics · The nuclear force and nuclear models · Nuclear reactions

The Nucleus Resolved, Ch Reaction mechanisms: resonances, the compound nucleus, direct reactions and the optical model

Problem

12.0\,\mathrm{MeV}

deuteron beam (deuteron rest energy

m_dc^2 = 1875.6\,\mathrm{MeV}

) strikes a

^{40}\mathrm{Ca}

target (

J^\pi=0^+

) and produces

^{41}\mathrm{Ca}

via

(d,p)

stripping. The measured proton angular distribution to a particular final state has its first maximum at

\theta_{\rm lab} \approx 60^\circ

, where the differential cross section is

4.4\,\mathrm{mb/sr}

. Distorted-wave calculations for pure

\ell

transfer, normalised to a spectroscopic factor

C^2S=1

, give

9.8

7.9

and

6.5\,\mathrm{mb/sr}

at that angle for

\ell=1

2

and

3

. ((a)) Using the target radius

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

with

A=40

, the non-relativistic deuteron wavenumber

k

q = 2k\sin(\theta/2)

, and the first maxima of

|j_\ell(qR)|^2

qR = 2.08

3.34

and

4.51

for

\ell = 1

2

and

3

, compute the angle of the first maximum for each

\ell

and identify which value of

\ell

matches the observed one. ((b)) State the parity of the transferred nucleon and, using the fact that

^{40}\mathrm{Ca}

is a doubly closed shell at

N=Z=20

, name the orbital and give

J^\pi

of the final state. ((c)) Compute the spectroscopic factor

C^2S

. ((d)) Compare

(2J_f+1)C^2S

with the sum-rule maximum

(2j+1)

for a neutron added to an initially empty orbital, and comment on the result.