We have established that the helical motion of a charged particle, guided by magnetic field lines as suggested in Fig. 5.5, involves a centripetal acceleration and therefore leads to the emission of radiation called cyclotron radiation and evidently involves an energy loss for the particle. We assess the associated power loss starting with the classical expression for the radiation emission rate of ions and electrons moving in an accelerating field which is known to exhibit the proportionality

Pcycx Ni q] af + Ne q] al = Neq] ae (5.82)

where a() denotes the respective accelerations and q( , the respective electrical charges with qj = | Zqe |. Here, the single particle radiation as given by Eq.(3.40) has been multiplied by the particle number density in order to provide an expression for power density. Note also that the smaller mass of the electrons will ensure that with their attendant higher acceleration-recall = F/me-the electrons will be the predominant contributors to this cyclotron power loss. Ion cyclotron radiation is hence neglected. For cyclical motion in a magnetic field, the acceleration of electrons is a constant given by

Further, knowing the electron gyrofrequency as (0g, e = lqeIB/nie with В as the controlling magnetic field, the relation of Eq. (5.41) allows us to write

vL _ я] В2

2

me

Finally, assuming a Maxwellian distribution of electrons, we may take from kinetic energy considerations or Eq.(2.19b)

where a constant of proportionality is explicitly introduced. With Ne in units of m"3, В in units of Tesla, and kT in units of eV, the constant is given by Acyc = 6.3xlO’20 J eV"1- Tesla"2 s_1 for Pcyc in units of W m"3. Thus, cyclotron radiation is most important at very high magnetic fields and high electron temperatures. Indeed, with electrons at a possibly very high energy, they may need to be subjected to a relativistic treatment.

Cyclotron radiation exists in the far infrared radiation spectrum with a

wavelength of 10’3 — 1СГ4 m and is therefore partially re-absorbed in a plasma. Further, the emitted radiation may be reflected from the surrounding wall in a magnetic confinement fusion device and thereby re-enter the plasma. Hence, we choose to write the net cyclotron power finally lost from a plasma as

Pcyc = Acyc Ne в2 к Те V (5.88)

where If/ accounts for the complex processes of reflection and reabsorption of cyclotron radiation and is a dimensionless function involving several plasma parameters, including the magnetic field strength and the reflectivity of the surrounding wall. For a reasonable reflectivity of 90% and conditions expected in a fusion reactor, the range of l// could typically be from 10"4 (for low Te) to 10‘2 (for very high Te).

Problems

5.1 The maximum attainable magnetic field В is expected to be about 20 Tesla. For V|| = 0, estimate the associated gyrofrequency, cog, and gyroradius, rg, for electrons and protons given kTe = kTj = 5 keV.

5.2 Graphically depict the motion of an ion in a combined E-field and B — field. Display the results in an isometric representation.

5.3 (a) Determine the motion and position of a positive test charge in an electric field given by E = (E cos(cot), 0, 0) with initial velocity v(0) = (vx>0, 0, vz>0) and initial position at the origin.

(b) For a similar positive test charge in only a magnetic field given by В = (B cos(cot), 0, 0) with initial velocity v(0) = (vx0, vy>0, vz>0) and initial position at the origin, describe qualitatively and sketch the motion for 0 < t < 27t/co and co « cog.

5.4 Perform the integration suggested in Eq.(5.49).

5.5 Confirm the statement in Sec. 5.6 that integrating the solution of Eq. (5.56c), given appropriate initial conditions, over one gyration period yields a drift in the negative-V|| direction dependent upon vx.

5.6 As discussed at the end of Sec. 5.6, curvature drift and VB-drift will always occur together. Using the co-ordinate system of Fig. 5.10, approximate-in analogy to Sec. 5.5 and 5.6-the magnetic field strength B(y, z) = (0, By(y, z), Bz(y, z)) with a first order Taylor-expansion at the у-axis in order to account for both weak curvature and weak inhomogeneity (note that ЭВу/Эу = 9Bz/9z = 0 at the reference point). What simple condition relating the degree of curvature an of inhomogeneity is then required to satisfy Maxwell’s Eqs. (5.62) and (5.63)?

5.7 Demonstrate that the expression for the magnetic moment of an ion, Eq.(5.76), also follows from the basic definition of a current loop I encircling an area A (i. e., ц = IA), and also identify the units for the magnetic moment Ц.

5.8 Compare PnCyC with Pbr for a deuterium-tritium plasma of density Ne = N, = 1020 m’3 confined by a magnetic field of magnitude В = 6 T.

Consider the specific cases:

(i) Te= lOkeV; ул= 10’3

(ii) Te = 50 keV; yr= 10’2

and also compare Pbr + P%’c with P^ at the given conditions.

5.9 Consider a 50:50% MCF device having Tj = Te and a plasma beta value of 0.2, and for which ул ~ 10’3. Compute and compare the modified ignition temperature, i. e. the temperature at which Pbr + PTyC = С, л Рл, to T*ign of problem 3.6.

5.10 Calculate an expression for the power density of cyclotron radiation emitted from a d-h fusion plasma of density N, + Ne and of kinetic temperature Т;~Те=Т by knowing that a particle of charge q and velocity v will-according to classical electromagnetic theory-emit radiation at a power Prad q2 | dv/dt |2.

(a) How much smaller is the cyclotron radiation power of ions than that of electrons?

(b) Compare the primary cyclotron radiation power density (that with no reabsorption) to the fusion energy release per unit time for the case of Nd = Nh = 2 x 1020 m’3, T = 80 keV and В = 6 Tesla, and discuss the importance of radiation reflecting walls in an MCF reactor.

(c) Plot PnCyC (accounting for reflection and reabsorption by the approximation

t//(T) ~ 10’4 2 [T(keV) / 1 keV]14) and Pbr as functions of T over the temperature range 10 to 120 keV.