Stars as Laboratories for Fundamental Physics - MPP Theory Group

Processes in a Nuclear Medium 139
For the static spin-density structure function the same steps can be
performed with the inclusion of the spin operators 1 2 σ i for the individual
nucleons. This takes us to the equivalent of Eq. (4.47)
⟨Ψ|s(k) · s(−k)|Ψ⟩ = 1 V
= 1 V
N B ∑
i=1
1
4 ⟨Ψ|σ2 i |Ψ⟩ + 1 V
N B ∑
i,j=1
N B ∑
i,j=1
i≠j
e k·r ij 1
4 ⟨Ψ|σ i · σ j |Ψ⟩ =
e ik·r ij 1
4 ⟨Ψ|σ i · σ j |Ψ⟩. (4.48)
Noting that ⟨( 1 2 σ i) 2 ⟩ = 1 2 (1 + 1 2 ) = 3 4 the first term is 3 4 N B/V = 3 4 n B.
Therefore, in the absence of correlations one finds S σ (k) = 1.
Even in a noninteracting medium there exist anticorrelations between
degenerate nucleons. Standard manipulations yield in this case
(e.g. Sawyer 1989)
S ρ,σ (k) = 1
n B
∫
2d 3 p
(2π) 3 f p (1 − f p+k ), (4.49)
with the Fermi-Dirac occupation number f p for the nucleon mode p.
For small temperatures this result can be expanded to yield S ρ,σ (k) =
3k/2p F + 3T m N /p 2 F.
4.6.3 The f-Sum Rule
The structure functions have a number of general properties, independently
of details of the interactions of the medium constituents. We
have already seen that they must obey the normalization condition
∫ +∞
−∞
∫ +∞
−∞
dω
2π S ρ(ω, k) = 1 + 1 ⟨
∑ NB
n B
i,j=1
i≠j
dω
2π S σ(ω, k) = 1 + 4 ⟨
∑ NB
3n B
i,j=1
i≠j
cos(k · r ij )
⟩
σ i · σ j cos(k · r ij )
,
⟩
. (4.50)
This can be referred to as a “sum rule” because the strength of S ρ,σ is
“summed” (integrated) over all frequencies ω. Usually we will assume
that the correlation expressions on the r.h.s. are negligible.
A more nontrivial sum rule obtains when a factor ω is included
under the integral. The definition of the density structure function