Synchrotron Radiation

Synchrotron radiation is the name given to the radiation which occurs when charged particles are accelerated in a curved path or orbit. Classically, any charged particle which moves in a curved path or is accelerated in a straight-line path will emit electromagnetic radiation. Various names are given to this radiation in different contexts. For example, when it occurs upon electron impact with a solid metal target in an x-ray tube, it is called "brehmsstrahlung" radiation.

回旋辐射是指带电粒子在弯曲路径或轨道中加速时发出的辐射。经典物理中，任何在弯曲路径中运动或在直线上加速的带电粒子都会发出电磁辐射。在不同情境下，这种辐射有不同的名称。例如，当它在X射线管中发生于电子撞击固体金属靶时，称为‘制动辐射’。

Particularly in the application to circular particle accelerators like synchrotrons, where charged particles are accelerated to very high speeds, the radiation is referred to as synchrotron radiation. This radiated energy is proportional to the fourth power of the particle speed and is inversely proportional to the square of the radius of the path. It becomes the limiting factor on the final energy of particles accelerated in electron synchrotrons like the LEP at CERN. In other contexts like the detector arrays in accelerators, it can be detected and used as an aid to analyzing the products of the scattering event in the accelerator.

特别是在应用到如同步回旋加速器这样的圆周粒子加速器时，其中带电粒子被加速到非常高的速度，这种辐射被称为同步辐射。这种辐射的能量与粒子速度的四次方成正比，与粒子运动轨迹半径的平方成反比。在像位于欧洲核子研究中心（CERN）的LEP电子同步回旋加速器这样的装置中，它成为限制粒子最终能量的关键因素。在其他情境下，如加速器中的探测器阵列中，这种辐射可以被检测并用于分析加速器中散射事件的产物。

加速电子辐射功率的经典公式是

For a non-relativistic circular orbit, the acceleration is just the centripetal acceleration, v²/r. The orbits of interest in accelerators are highly relativistic, so the relativistic acceleration can be gotten from

对于非相对论性的圆周轨道，加速度就是向心加速度，即 v²/r。而加速器中感兴趣的轨道通常是高度相对论性的，因此可以利用相对论加速度来计算。

and m is the rest mass of the particle and the time rate of change of γ is neglected. That assumption really needs to be defended more rigorously*, but it appears that the γ⁴ term should dominate. The radiated power is then

且m是粒子的静止质量，时间变化率的γ项被忽略。这个假设确实需要更严谨地进行辩护*，但似乎γ的4项应占主导地位。辐射功率则为

Since the velocity becomes nearly constant for highly relativistic particles, the term γ⁴ becomes the dominant variable in determining loss rate. This means that the loss scales as the fourth power of the particle energy γmc². For an accelerator like a synchrotron, the radius is fixed after construction, but the inverse dependence of synchrotron radiation loss on radius argues for building the accelerator as large as possible.

由于高速相对论粒子的速度几乎保持不变，术语γ⁴成为确定损失率的主要变量。这意味着损失与粒子能量γmc²的四次方成正比。对于像回旋加速器这样的加速器，其半径在建造后是固定的，但回旋辐射损失与半径的反比关系表明应尽可能将加速器建造得更大。

The LEP electron synchrotron has a rated energy of 50 GeV and a radius of 4300 meters. This gives a relativistic gamma of about 98,000 compared to a gamma of 54 for a 50 GeV proton (see relativistic kinetic energy calculation). At that energy, the velocity is essentially c, so the synchrotron radiation power for a single electron can be calculated as follows:

LEP电子回旋加速器的额定能量为50 GeV，半径为4300米。这给出了相对于50 GeV质子的相对论gamma因子约为98,000（见相对论动能计算）。在该能量下，速度接近光速，因此单个电子的同步辐射功率可按以下方式计算：

Two-tenths of a microwatt may not sound like much loss, but per electron it is enormous! At this energy the proton velocity would also be essentially c, so the synchrotron radiation loss for the two particles scales like their gammas. So the loss rate for the electron is (97833/54)⁴ or over 10¹³ times the loss for a proton of the same energy in the same synchrotron.

两微瓦的损失似乎不多，但对每个电子来说却是巨大的！在这一能量下，质子的速度也几乎接近光速，因此这两个粒子的同步辐射损失率与它们的洛伦兹因子成正比。因此，电子的损失率是质子在相同同步回旋加速器中相同能量下损失率的（97833/54）4 或超过10¹³倍。

可以从每轨道能量损失计算得出

在这些电子所关联的高gamma值下，如果将该速度项视为与其他速度项相同，就不能忽略速度项。应用链式求导法则会使gamma的导数项变得比其他项更大。然而从物理角度来看，gamma中的速度是与动力学效应相关的速度，而在圆周运动中，给定的速度不会随时间变化。产生v²/r项的项与方向变化有关，必须包含在内。