# Fusion energy gain factor

The explosion of the Ivy Mike hydrogen bomb. The hydrogen bomb is the only known man-made item to achieve fusion energy gain factor larger than 1.

The fusion energy gain factor, usually expressed with the symbol Q, is the ratio of fusion power produced in a nuclear fusion reactor to the power required to maintain the plasma in steady state. The condition of Q = 1 is referred to as breakeven.

In a fusion power reactor a plasma must be maintained at a high temperature in order that nuclear fusion can occur. Some of this power comes from the fraction fch of the fusion power contained in charged products Pfus which remain in the plasma. This power may be designated fchPfus. The rest, designated Pheat comes from external sources required for heating, some of which may also serve additional purposes like current drive and profile control. This power is lost through various processes to the walls of the plasma chamber. In most reactor designs, various constraints result in this heat leaving the reactor chamber at a relatively low temperature, so that little or none of it can be recovered as electrical power. In these reactors, electrical power is produced from the fraction of the fusion power contained in neutrons, (1 − fch)Pfus. The neutrons are not contained by the magnetic fields (in magnetic confinement fusion) nor the dense plasma (in inertial confinement fusion) but are absorbed in a surrounding walls (blanket). Due to various exothermic and endothermic reactions, the blanket may have a power gain factor a few percent higher or lower than 100%, but that will be neglected here. The neutron power would be used to heat a working medium such as helium gas or liquid lithium to a high temperature, and the working medium is then used to produce electricity at some efficiency ηelec, so that Pelec = ηelec(1 − fch)Pfus. A fraction frecirc of the electrical power is recirculated to run the reactor systems. Power is needed for lighting, pumping, producing magnetic fields, etc., but most is required for plasma heating so we can write Pheat = ηheatPelec, where ηheat is substantially the efficiency with which electrical power is converted to the form of power needed to heat the plasma.

The heating power can thus be related to the fusion power by the following equation:

$P_{heat} = \eta_{heat} \cdot f_{recirc}\cdot \eta_{elec}\cdot (1-f_{ch})\cdot P_{fus}$

The fusion energy gain factor is then defined as:

$Q \equiv \frac{P_{fus}}{P_{heat}} = \frac{1}{\eta_{heat} \cdot f_{recirc}\cdot \eta_{elec}\cdot (1-f_{ch})}$

For the D-T reaction, fch = 0.2. Efficiency values depend on design details but may be in the range of ηheat = 0.7 and ηelec = 0.4. The purpose of a fusion reactor is to produce power, not to recirculate it, so a practical reactor must have frecirc = 0.2 approximately. Lower would be better but will be hard to achieve. Using these values we find for a practical reactor Q = 22. Of course, Q = 15 might be enough and Q = 30 might be achievable, but this simple calculation shows the magnitude of fusion energy gain required.

The goal of ignition, a plasma which heats itself by fusion energy without any external input, corresponds to infinite Q. Note that ignition is not a necessary condition for a practical reactor. On the other hand, achieving Q = 20 requires quality of confinement almost as good as that required to achieve ignition, so the Lawson criterion is still a useful figure of merit. The condition of Q = 1 is referred to as breakeven. It is somewhat arbitrary, but it does mean that a significant fraction (20%) of the heating power comes from fusion, so that fusion heating can be studied. Above Q = 5 the fusion heating power is greater than the external heating power.

The one channel of energy loss that is independent of the confinement scheme and practically impossible to avoid is Bremsstrahlung radiation. Like the fusion power density, the Bremsstrahlung power density depends on the square of the plasma density, but it does not increase as rapidly with temperature. By equating the two power densities, one can determine the lowest temperature for which the fusion power can overcome the Bremsstrahlung power. This ignition temperature is about 4 keV for the D-T reaction and about 35 keV for the D-D reaction.