Scattering Cross Section

If you are firing projectiles which you cannot see at targets which are much smaller than atoms, how can you make sense out of what you detect on the other side of the target? This kind of scenario highlights the value of the concept of cross-section.

如果你在射击那些你看不到的弹丸，而这些弹丸比原子还要小，你如何解释穿过目标后检测到的结果？这种情形突显了‘截面’概念的重要性。

If you had a uniform beam of projectiles and encountered a target which intercepted 1% of the area of the beam, you would expect 1% of the projectiles to be scattered. That basic concept can be extended to the calculation of the fraction of alpha particles scattered from a gold foil.

如果你有一束均匀的弹丸，并遇到一个拦截了该束弹丸1%面积的目标，你预计有1%的弹丸会被散射。这个基本概念可以扩展到计算从金箔上散射的α粒子的分数。

Suppose you had a beam of incident alpha particles with rate R_i = 10,000 /s on a gold foil of thickness 10^-6 meters and the effective area (cross-section) for each gold nucleus was 100 barns. The fraction of those 10,000 particles scattered would be the fraction of the beam area which is covered by one of the scattering gold nuclei. The fraction of the beam covered can be calculated as follows

假设有束入射的α粒子，其通量为R_i = 10,000 /s，照射到厚度为10⁻⁶米的金箔上，每个金核的有效面积（截面）为100 barns。被散射的10,000个粒子中的比例，即为该束流中被一个散射的金核覆盖的区域比例。该束流被覆盖的比例可通过以下方式计算：

Multiplying the number of targets per square meter times the cross-section for each target gives you the fraction of the area which is covered by target and therefore the fraction of the beam which would be scattered:

将每平方米的目标数乘以每个目标的截面，你就能得到被目标覆盖的面积分数，从而得到被散射的束流分数：

For an incident rate of 10,000/s, an average of 5.88 particles/s would be expected to be scattered.

对于10000/s的入射率，预计有5.88个粒子/s会被散射。

To put this example more directly in the context of Rutherford scattering, you can use the cross-section calculation to verify that for 6 MeV alpha particles incident upon this gold film, the cross-section would be 100 barns for scattering above about 37 degrees. Another context would be the calculation of scattering within an angular window. For example, suppose you start at 20 degrees, where the calculated cross-section is 363 barns for scattering above that angle. The scattering cross-section for scattering at 23.4 degrees is about 263 barns, so the cross-section for scattering between 20 and 23.4 degrees is about 100 barns. Using finite detectors and windows like this, the Rutherford team was able to compare the number of scattered alpha particles at different scattering angles and confirm that they followed the calculated pattern for coulomb scattering. Then at higher energies, when they detected a departure from the pattern, they could judge that they had interacted with a different kind of force (the strong force) and could then imply a radius for the nucleus.

为了将这个例子更直接地放在卢瑟福散射的背景下，你可以用截面计算来验证：对于6 MeV的α粒子照射在金膜上，当散射角度超过约37度时，截面应为100个拜耳。另一个情境是计算某个角度窗口内的散射。例如，假设你从20度开始，计算出该角度以上散射的截面为363个拜耳。在23.4度处的散射截面约为263个拜耳，因此在20度到23.4度之间的散射截面约为100个拜耳。利用这样的有限探测器和窗口，卢瑟福团队能够比较不同散射角度下散射的α粒子数量，并确认它们遵循库仑散射的计算模式。当在更高能量下检测到模式偏离时，他们可以判断发生了与不同种类的力（强作用力）的相互作用，并据此推断出核的半径。

The Value of the Cross-Section Concept