Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 141
For the spin-density structure function one can go one step further
by expressing the most general nonrelativistic interaction potential as
(Sigl 1995b)
V (r ij , σ i , σ j ) = U 0 (r ij ) − U S (r ij ) σ i · σ j
− U T (r ij ) [ 3(σ i · ˆr ij )(σ j · ˆr ij ) − σ i · σ j
]
, (4.55)
where U 0 is a spin-independent potential of the interparticle distance
r ij = |r ij |, U S is the scalar, and U T the tensor part of the spin-dependent
potential. Of these terms, only the tensor part does not conserve the
total spin in nucleon-nucleon collisions and thus is the only part contributing
to spin fluctuations. With Vij
S
U T (r ij ) [3(σ i · ˆr ij )(σ j · ˆr ij ) − σ i · σ j ] one finds
∫ +∞
−∞
dω
2π ω S σ(ω, k) =
+ 4 ⟨
∑ NB
3n B
i,j=1
i≠j
k2
2m N
≡ U S (r ij ) σ i · σ j and V T
ij
[
1 − cos(k · rij ) ] Vij S + [ 1 + 1 cos(k · r 2 ij) ] Vij
T
⟩
≡
. (4.56)
The f-sum of the spin-density structure function is thus closely related
to the average spin-spin interaction energy in the medium.
In order for the eigenvalues of the Hamiltonian to be bounded from
below one must require that the potentials U 0,S,T (r) are not more singular
than 1/r 2 . In this case the r.h.s. of the spin-density f-sum rule
exists as a nondivergent expression. In our representation S σ (|ω|) =
(Γ σ /ω 2 ) s(ω/T ), the necessary existence of the l.h.s. of Eq. (4.56) implies
that s(x) must be a decreasing function of x for large x. This is
not the case for the s(x) derived from the OPE potential, indicating
that this interaction model is pathological in the sense that it is too
singular. Indeed, it corresponds to a dipole potential and thus varies
as 1/r 3 . Real nucleon-nucleon interaction potentials have a repulsive
core and thus do not exhibit this pathology.
4.6.4 Long-Wavelength Properties
In calculations involving the emission or scattering of neutrinos or axions
as in Sect. 4.3.1 one usually neglects the momentum transfer k
because the medium constituents are so heavy that recoil effects are