Variable packing fraction

A change of target density at constant target volume corresponds to a change in n_(j) which can in general be different for each run i: n_(j)i.

In this case the averaging of the polarizations gets affected too and the formalism looks like the following:

Averaged n_(j)i:

$$\left\langle n_{(j)}\right\rangle =\frac{1}{2}\left( \frac{1... ...=1}^{k}n_{(j)i}+\frac{1}{n-k}\sum _{i=k+1}^{n}n_{(j)i}\right)$$ (13)

Averaged polarizations:

     

$$\left\langle hP\right\rangle _{(j)} \frac{1}{2\left\langle n_{(j)}\right\rangle }\left( \frac{1}{k}\s... ...{i=1}n_{(j)i}h_{i}P_{i}-\frac{1}{m-k}\sum _{i=k+1}^{m}n_{(j)i}h_{i}P_{i}\right)$$ = (14)

$$\left\langle hT\right\rangle _{(j)} \frac{1}{2\left\langle n_{(j)}\right\rangle }\left( \frac{1}{k}\s... ...{i=1}n_{(j)i}h_{i}T_{i}-\frac{1}{m-k}\sum _{i=k+1}^{m}n_{(j)i}h_{i}T_{i}\right)$$ = (15)

$$\left\langle P\right\rangle _{(j)} \frac{1}{2\left\langle n_{(j)}\right\rangle }\left( \frac{1}{k}\sum ^{k}_{i=1}n_{(j)i}P_{i}+\frac{1}{m-k}\sum _{i=k+1}^{m}n_{(j)i}P_{i}\right)$$ = (16)

$$\left\langle T\right\rangle _{(j)} \frac{1}{2\left\langle n_{(j)}\right\rangle }\left( \frac{1}{k}\sum ^{k}_{i=1}n_{(j)i}T_{i}+\frac{1}{m-k}\sum _{i=k+1}^{m}n_{(j)i}T_{i}\right)$$ = (17)

$$\left\langle h\right\rangle _{(j)} \frac{1}{2\left\langle n_{(j)}\right\rangle }\left( \frac{1}{k}\sum ^{k}_{i=1}n_{(j)i}h_{i}-\frac{1}{m-k}\sum _{i=k+1}^{m}n_{(j)i}h_{i}\right)$$ = (18)

Beam-target asymmetry (eq. 2) with averaged n_(j)iaccording to eq. 13 and averaged polarizations according to eq. 14 to 18:

$$A_{BT}=\frac{\sum ^{L}_{j=1}\left\langle n_{(j)}\right\rangle... ...{V(j)}+\left\langle T\right\rangle _{(j)}A_{ed}^{T(j)}\right] }$$