For collisions with free electrons, compare the Compton shift of a photon scattered as an angle of \(30^{\circ}\) to that of a photon scattered at \(45^{\circ}\).

We use the formula for the Compton shift: \[\Delta\lambda_{30} = \frac{h}{m_ec}(1 - \cos30^{\circ})\] The cosine of \(30^{\circ}\) is \(\frac{\sqrt{3}}{2}\), so we have: \[\Delta\lambda_{30} = \frac{h}{m_ec}(1 - \frac{\sqrt{3}}{2})\]

We use the formula for the Compton shift: \[\Delta\lambda_{45} = \frac{h}{m_ec}(1 - \cos45^{\circ})\] The cosine of \(45^{\circ}\) is \(\frac{1}{\sqrt{2}}\), so we have: \[\Delta\lambda_{45} = \frac{h}{m_ec}(1 - \frac{1}{\sqrt{2}})\]

We now compare the Compton shift values for the two angles. Let's divide \(\Delta\lambda_{30}\) by \(\Delta\lambda_{45}\), for comparison: \[\frac{\Delta\lambda_{30}}{\Delta\lambda_{45}} = \frac{\frac{h}{m_ec}(1 - \frac{\sqrt{3}}{2})}{\frac{h}{m_ec}(1 - \frac{1}{\sqrt{2}})}\] Notice that the \(\frac{h}{m_ec}\) terms will cancel out: \[\frac{\Delta\lambda_{30}}{\Delta\lambda_{45}} = \frac{1 - \frac{\sqrt{3}}{2}}{1 - \frac{1}{\sqrt{2}}}\]

Now, we need to find the value of the fraction: \[\frac{\Delta\lambda_{30}}{\Delta\lambda_{45}} = \frac{1 - \frac{\sqrt{3}}{2}}{1 - \frac{1}{\sqrt{2}}}\] Simplifying: \[\frac{\Delta\lambda_{30}}{\Delta\lambda_{45}} = \frac{2 - \sqrt{3}}{(\sqrt{2}-1) \cdot 2}\] Now, we multiply the numerator and denominator of the fraction by its conjugate \((\sqrt{2}+1)\): \[\frac{\Delta\lambda_{30}}{\Delta\lambda_{45}} = \frac{(2-\sqrt{3})(\sqrt{2}+1)}{(\sqrt{2}-1)(\sqrt{2}+1) \cdot 2}\] \[\frac{\Delta\lambda_{30}}{\Delta\lambda_{45}} = \frac{2\sqrt{2} - 2 + \sqrt{6}-\sqrt{3}}{(2-1) \cdot 2}\] \[\frac{\Delta\lambda_{30}}{\Delta\lambda_{45}} = \frac{2\sqrt{2} - 2 + \sqrt{6}-\sqrt{3}}{2}\] Finally, \[\frac{\Delta\lambda_{30}}{\Delta\lambda_{45}} = \sqrt{2} - 1 + \frac{\sqrt{6}}{2}-\frac{\sqrt{3}}{2}\] This expression shows the comparison of the Compton shift for a photon scattered at an angle of \(30^{\circ}\) to that of a photon scattered at an angle of \(45^{\circ}\).