I've been reading lots of different textbooks on the parton model and deep inelastic scattering and the discovery of quarks, and most of the time whenever I see a plot, it's about the measurement of the structure functions, like $G_E, G_M$ or $F_1, F_2$. I can understand how they come up with the cross section (by dividing the number of particles shot vs the number detected perhaps?), but I can't see how derive the value of the structure functions, which live within the formula for the cross section. For example:

$$\frac{\mathrm{d} \sigma}{\mathrm{d} \Omega}=\left(\frac{G_{E}^{2}+\tau G_{M}^{2}}{(1+\tau)}+2 \tau G_{M}^{2} \tan ^{2} \frac{\theta}{2}\right) \cdot\left(\frac{\mathrm{d} \sigma}{\mathrm{d} \Omega}\right)_{0}$$ where $$\left(\frac{\mathrm{d} \sigma}{\mathrm{d} \Omega}\right)_{0}=\frac{\alpha^{2}}{4 E^{2} \sin ^{4}(\theta / 2)}\left(\frac{E^{\prime}}{E}\right) \cos ^{2} \frac{\theta}{2}$$ or hell, even $$\begin{aligned} \frac{d \sigma}{dE' d \Omega} &= \sum_{i} \int_{0}^{1} f_{i}(x) \frac{\alpha^{2} e_{i}^{2}}{4 E^{2} \sin ^{4} \frac{\theta}{2}}\left[\frac{2 M}{Q^{2}} x^{2} \cos ^{2} \frac{\theta}{2}+\frac{1}{M} \sin ^{2} \frac{\theta}{2}\right] \delta (\xi - x) dx \\ &= \frac{\alpha^{2}}{4 E^{2} \sin ^{4} \frac{\theta}{2}}\left[\frac{2 M}{Q^{2}} x^{2} \cos ^{2} \frac{\theta}{2}+\frac{1}{M} \sin ^{2} \frac{\theta}{2}\right]\sum_{i} e_{i}^{2} f_{i}(x) \end{aligned}$$

and they somehow manage to calculate the $f_i$ (I think?).

Can anyone maybe explain for me please? Thank you!


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