Another unpolarized nucleus

Assuming a target consisting of 2 types of nuclei: n_D polarized deuteron nuclei with cross section and n_N unpolarized nuclei with cross section . Using eq. 5 for summation and normalization leads to:

$$N^{hP}\propto n_{D}\sigma _{D}(h,P,T)+n_{N}\sigma _{N}(h)$$

Beam-target asymmetry (eq. 2) with average polarizations according to equations 6-10:

$$A_{BT}=\frac{n_{D}\sigma ^{(D)}_{0}\left[ \left\langle hP\rig... ...angle A_{d}^{V}+\left\langle T\right\rangle A_{ed}^{T}\right] }$$

Assuming that the asymmetry can be expressed in another form:

$$A_{BT}=f\frac{\left\langle hP\right\rangle A_{ed}^{V}+\left\... ...right\rangle A_{d}^{V}+\left\langle T\right\rangle A_{ed}^{T}}$$ (11)

with the dilution factor f

$$f=\frac{n_{D}\sigma _{0}^{(D)}}{\frac{n_{N}\sigma _{0}^{(N)}... ...\left\langle T\right\rangle A_{d}^{T}}+n_{D}\sigma _{0}^{(D)}}$$ (12)

The concept of dilution factor according to equations 11 and 12 is only applicable if the terms of the ``diluting'' nucleus are limited to the denominator, otherwise the factorization does not work.