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11
is the expansion of the Fourier transform of the transition three-current in terms of
the standard unit spherical vectors ⃗e(⃗q; 1, m) defined by
⃗e(⃗q; 1, 0) = ⃗e z (2.22)
⃗e(⃗q; 1, ±1) = ∓√ 1 (⃗e x ± i⃗e y ). 2
(2.23)
Furthermore, current conservation imposes that only three components of the fourcurrent
J µ are independent :
q µ J µ (⃗q) fi = ωρ(⃗q) fi − qJ(⃗q; 0) fi = 0 , (2.24)
so that J(⃗q; 0) fi = (ω/q)ρ(⃗q) fi . The unpolarized structure functions are then defined
in a standard fashion as
R L fi = |ρ(⃗q) fi | 2 , (2.25)
R T fi = |J(⃗q; +1) fi | 2 + |J(⃗q; −1) fi | 2 , (2.26)
R T fi T = 2 Re {J ⋆ (⃗q; +1) fi J(⃗q; −1) fi } , (2.27)
R T fi L = −2 Re {ρ ⋆ (⃗q) fi (J(⃗q; +1) fi − J(⃗q; −1)) fi } . (2.28)
Similarly, for the polarized structure functions we have
R T ′
fi = |J(⃗q; +1) fi | 2 − |J(⃗q; −1) fi | 2 , (2.29)
R T L′
fi = −2 Re {ρ ⋆ (⃗q) fi (J(⃗q; +1) fi + J(⃗q; −1)) fi } . (2.30)
From here on one can proceed and perform a multipole expansion for the electromagnetic
transition operators. At higher momentum transfer this method becomes
cumbersome as a large amount of multipoles has to be considered. Therefore, we will
not consider these expansions here, and proceed with the full transition operators.