Subatomic Physics

14.5. Low-Energy Nucleon–Nucleon Force 441
Figure 14.14: Scattering of polarized protons from a spinless nucleus. (a) The trajectories in the
scattering plane. (b) The spins and the orbital angular momenta of nucleons 1 and 2.
Experimentally, such left–right asymmetries are observed (22) and provide evidence
for the existence of a spin–orbit force.
The information obtained in the present section can be summarized by writing
the potential energy between two nucleons 1 and 2 as
VNN = Vc + Vscσ1 · σ 2· + VT S12 + VLSL · 1
2 (σ1 + σ2), (14.36)
where σ1 and σ2 are the spin operators of the two nucleons and L is their relative
orbital angular momentum,
L = 1
2 (r1 − r2) × (p 1 − p 2). (14.37)
Vc in Eq. (14.36) describes the ordinary central potential energy, Vsc is the spindependent
central term discussed above. VT gives the tensor force; the tensor
operator S12 is defined in Eq. (14.33). VLS characterizes the spin–orbit force introduced
in Eq. (14.35). VNN in Eq. (14.36) is nearly the most general form allowed
by invariance laws. (23)
Charge independence of the strong force implies invariance under rotation in
isospin space. The two isospin operators � I1 and � I2 of the two nucleons can only
occur in the combinations
1 and � I1 · � I2.
Thus each coefficient Vi in VNN can still be of the form
Vi = V ′
i + V ′′
i � I1 · � I2, (14.38)
where V ′ and V ′′ can be functions of r ≡|r1 − r2|,p= 1
2 |p 1 − p 2|, and|L|.
The coefficients Vi are determined by a mixture of theory and phenomenology.
The features that are reasonably well understood are incorporated in the
potential to begin with. An example is the one-pion exchange potential. Other
features are added to reach agreement with experiment. (24) A large (25) number
23S. Okubo and R. E. Marshak, Ann. Physik 4, 166 (1958). Actually one term allowed by
invariance arguments, the quadratic spin-orbit term, is missing in Eq. (12.37).
24K. Holinde, Phys. Rep. 68, 121 (1981); S.-O. Backman, G. E. Brown, and J. A. Niskanen,
Phys. Rep. 124, 1 (1985).