Thermal Bremsstrahlung

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Bremsstrahlung (braking radiation)

Bremsstrahlung is the continuum of emission from a plasma caused by the deflection of charged particles off of one another. It is a common source of X-ray emission. The deflection we will discuss, which is the most important, is of an by a positive nucleus, where we assume the nucleus to be stationary.

Thermal Bremsstrahlung

Recall that the power radiated by an accelerated is (see Larmor Formula):

In this case, the acceleration is due to the electric force of a nucleus, so , where is the distance of closest approach between the and the nucleus, known as the impact parameter. Thus:

The energy released by this encounter is given by , where is the time of interaction. We can approximate the interaction to occur roughly over a distance to either side of the nucleus, and so, if the is traveling at a speed , then . Therefore:

From the ’s point of view, the force of the nucleus’s is initially pulling the almost directly forward, and ends up pulling the nearly completely backwards. At the point of closest approach, the is not being pulled forwards or backwards, the electric force is solely perpendicular to the original direction of motion. We can therefore see that if we plot the parallel component of the force, we end up with something that looks a lot like a sine wave of period .

Region in which the electron and nucleus are ”interacting.”
Parallel component of the Electric Force from the electron’s point of view.

From earlier, we know though that . Multiplying this by 2 in order to get a period, and inverting it to get a frequency, we have . This enables us to relate to , which will be useful to us in a minute:

Now let’s consider a nucleus in a sea of electrons.

Sea of Electrons

The electrons of interest to us are those a distance from the nucleus. This fraction of electrons is given by the number density of electrons multiplied by the area of the ring: . Thus, the power radiated by that ring is:

where we need to include the velocity to account for the increased interaction rate for fast moving electrons. Using our relation between and , and our expression for we arrive at:

So the power per frequency interval is independent of distance!

The remaining term left to define is the velocity. We have seen that in a plasma, the collision rates between electrons are high enough to relax into a Maxwellian velocity distribution fairly quickly:

However, we are considering free-free emission, and so we must define a minimum velocity below which the electron would otherwise get captured by the nucleus. An with this minimum velocity will have a kinetic energy of order the energy radiated by the acceleration, which is the energy of a photon: . Then the average total power released over all velocities due to the interaction with one nucleus is:

where the in the exponential is from the lower velocity bound which we just defined. We now define to be the volume emissivity for free-free interactions. That is, measures the power radiated by plasma, per volume, into a solid angle . We can calculate the “per ” because the radiation is isotropic:

where we have to include the # density of ions, , as well to account for the contribution from multiple nuclei. This is the expression for Thermal Bremsstrahlung. Note that the definition of in Rybicki & Lightman has an additional factor of , which is a quantum mechanical correction factor of order unity. It’s called the “Gaunt factor”.

Inverse Bremsstrahlung

We now consider the inverse effect, an can also absorb a photon and become more energetic and “free”.

Inverse Bremsstrahlung

We define the coefficient for thermal free-free absorption as:

where we use the Planck function because we know that in the optically thick case the specific intensity, , becomes the source function, , and in the case of optically thick thermal radiation, the source function becomes the Planck function. And so, just as the expression for was the definition of Thermal Bremsstrahlung, the expression for is the definition for Inverse Bremsstrahlung.

where we can once again include a factor of to obtain the same result as Rybicki & Lightman. In the regime where the exponential term becomes negligible and the absorptivity will go as . The regime is the Rayleigh-Jeans regime and the expression for absorptivity goes as , where we have lost a factor of due to the taylor expansion of the exponential.