An important consequence of scattering in fusion plasmas is bremsstrahlung radiation, which refers to the process of radiation emission when a charged particle accelerates or decelerates. It involves the transformation of some particle kinetic energy into radiation energy which-due to its relatively high frequency (X-ray wavelength range of ~ 10’9 m)-may readily escape from a plasma; thus, the kinetic energy of plasma particles is reduced, plasma cooling occurs, and a compensating energy supply may be required in order to maintain the desired plasma temperature.

A rigorous derivation of bremsstrahlung power emission in a hot plasma involves considerations of quantum mechanics and relativistic effects and is both tedious and time consuming. Indeed, even advanced methods of analysis require the imposition of simplifying assumptions if the formulation is to be at all tractable. It is, however, less difficult to develop an approximation of the dominant effect of the bremsstrahlung processes by the following considerations. A particle of charge qe and moving with a time varying velocity v(t) will — according to classical electromagnetic theory in the nonrelativistic limit-emit radiation at a power

In Fig. 3.6, we suggest this process for an electron moving in the electrostatic field of a heavy ion. To estimate the energy radiated per encounter, we replace Idv/dtl in Eq.(3.35) by an average acceleration a to be calculated from Newton’s Law, a = Fc / m, with Fc representing the magnitude of the average Coulomb

force approximated here by the electrostatic force between the two interacting particles when separated by the impact parameter r0. That is, we take

7 Fc(r„) Zq a ~——- = ;

for the average acceleration experienced by a particle of mass m and charge qe in the field of a charge qi = — Zqe. Since an electron possesses a mass 1/1836 of that of a proton-with an even smaller ratio existing when compared to a deuteron or triton-the electrons in a plasma will therefore be the main contributors to bremsstrahlung radiation. We suggest this in Fig. 3.7 for a representative electron trajectory undergoing significant directional changes in a background of sluggish ions.

Imagining an electron (me, qe) to be in the vicinity of an ion (m„ |q;| = IZqel) for a time interval

(3.37)

with vr denoting the average relative speed between the electron and ion, we combine Eqs.(3.35) to (3.37) to assess the bremsstrahlung radiation energy per collision event as

.. ~^r. (3-38)

collision nie Го Vr

Note this expression refers to the collision of a single electron with an ion at the specific impact parameter r0. Extending these considerations to a bulk of electrons of density Ne all approaching the ion with r, then the number of electrons colliding per unit time with the same ion at the same impact parameter r0 is given by

— ~Nevr2Krodr0. (3.39)

ion

Multiplying this relation by the ion density Nj we obtain the differential electron — ion collision rate density

dR0~ NeNiVr2Krodr0 (3.40)

thereby accounting for all electron-ion interactions per unit volume and per unit time occurring about a specific impact parameter r0.

Fig. 3.6: Depiction of photon emission from an accelerated electron passing near an ion.

Next, in order to obtain the total bremsstrahlung radiation power density associated with the entire electron-ion force impact area, Eq. (3.38) needs to be multiplied by Eq.(3.40) and integrated over the range of the impact parameter Го. тіп to romax; that is we now obtain for the bremsstrahlung power the proportionality

While r0>max = 00 can be imposed, a finite minimum impact parameter needs to be specified due to the integral singularity for r0 —> 0. We choose to identify ro min with the DeBroglie wavelength of an electron, that is,

г_ = — Л — (3.42)

me ve

where h is Planck’s constant and ye represents the average thermal electron speed defined by Eq.(2.19b):

With the integration limits thus specified, the integral of Eq.(3.41) is readily evaluated so that the electron bremsstrahlung radiation power density is found to exhibit the following dominant dependencies:

Ры = Аы Ni Ne Z2 JkT (3.44)

where Abr is a constant of proportionality. For kT in units of eV and particle densities in units of m’3,

which yields Pbr in units of Wm. An important point to note here is that Pbr is proportional to (kT)1/2.

Problems

3.1 Examine the relationship between 0C and 0L for the limiting cases of

mb/ma = {0, 1, =»}.

3.2 For a typical case of d-t interaction at 5 keV, plot as(0c), 0 < 0C < n.

3.3 For the conditions in problem 3.2, calculate as assuming a background particle density of 1020 m’3.

3.4 Calculate the bremsstrahlung power increase if a d-t plasma were to contain totally stripped oxygen ions at a concentration of 1% of the electron density.

3.5 Calculate the ratio of bremsstrahlung power to fusion power for a d-t plasma with N; = Ne = 1020 rn3 at 2 keV and 20 keV.

3.6 Determine from a plot of power density versus kinetic temperature (use a double logarithmic scale) for a 50:50% D-T fusion plasma

(a) the so called ideal d-t ignition temperature T, i. e. the temperature at which the plasma fusion power density equals the loss power density due to bremsstrahlung.

(b) the temperature T jgn more relevant to fusion ignition since it refers to the plasma operational state where the bremsstrahlung loss power is balanced by the energy transfer from the fusion alphas to the background plasma per unit time by Coulomb collisions. Assuming that the charged particle fusion power is entirely transferred, T ign is found from

with fcdt representing the fraction of d-t fusion power allocated to charged particles (in this case a’s). What does this condition mean for the overall power balance of the fusion plasma if other heat losses were neglected?

3.7 A fusion reactor using two opposing accelerators is proposed, where a 30 keV tritium beam from one is aimed head on at a 30 keV deuterium beam from the other. Would this work, and can you suggest improvements? What would you estimate is the maximum energy gain possible with this system (see Ch. 8)?

Part II Confinement, transport, Burn