It is often desirable to use easily understood and readily recognizable parameters as indicators of the merit of a particular system or process. A widely used formulation that provides a test of the energy balance for fusion devices is known as the Lawson Criterion. It is an algebraic relationship based on the assertion that the recoverable energy from a fusion reactor must exceed the energy which is supplied to sustain the fusion reaction; that is, Eout* > Ein* must hold during a representative time interval X which may be the cycling time for a pulsed fusion device or a typical period of time if steady-state operation is envisaged.

We may associate the recoverable energy from a fusion device with three forms of energy: the fusion reaction energy release, Efu, all radiation losses, Erad and the thermal motion of the particles, E, h, Fig. 8.2. Since no energy conversion can take place with 100% efficiency, the fraction of the energy that is recovered can be written as

Eout = Tlfi, E*fu + Лrad Erad + Лт Ел (8-19)

where the TJ’s are the respective conversion efficiencies for each energy component.

Fig. 8.2: Energy flow for an arbitrary fusion reactor.

The energy supplied to the device in order to sustain the fusion reactions, E, n* in Fig.8.2, appears as the thermal motion of the particles and as radiation:

Лы E*n = Erad + E*th • (8.20)

Here Т|іп is the fraction of Ein* which is coupled to the plasma and amounts to Eth* and Erad*; that is, only the fraction Г|ІПЕІП* is actually deposited to heat and sustain the fuel at a constant temperature.

The energy viability statement, Eq.(8.1), can now be written as

* *

t?/H E}u + Лrad Erad + Л. Н Ел > ~ad + ~ (8.21)

Ліп

or

ЛіпЛош{е% + E*ad + E*h) > Emd + Eth (8.22)

where, for reasons of algebraic convenience, we have taken Tm as an average conversion efficiency

Лот = ——- у 1 = fU> rad> th • (8-23)

2jEe

l

Consider now the idealization that the system operates in a mode characterized by a constant power production during a global energy confinement

time, Те* such that the global energy terms, E, can be readily evaluated from the respective power densities by

E*t = TE<Pe(r)d. (8.24)

v

Also, for present purposes it is assumed that radiation losses are due to bremsstrahlung only so that Prad = Ръг. Equation (8.22) can therefore be expressed in terms of these steady power flows for time Те* to give

Ліп Лот J d3 r( t£* Pfu + т£> Ры + 3NT) > j d3 r( т£* Ры + 3NT) (8.25)

V V

where we have substituted for the thermal energy density according to Eq. (8.13), Eth(r) = j(NiTi+ NeTe) = 3NT (8.26)

assuming Ni = Ne = N and T, = Te = T. Further, we have suppressed Boltzmann’s constant к in our notation, since from now on we will always refer to T as kinetic temperature in units of eV or keV and hence already incorporate the constant k.

Next we introduce in Eq. (8.25) the density and temperature dependent expressions for Ре, and Pbr discussed in preceding chapters and-for reasons of a simplified demonstration of the parameter interrelation-we assume homogeneity throughout the plasma volume V to obtain

7^r<ov>ab QabTE.+AbrN2^TE. +3NT>AbrN24rxE.+3NT

f + Oab У

(8.27)

with the Kronecker-8 introduced in the fusion rate density in order to account for the case of indistinguishable reactants, i. e. when a = b. Here T(), Qab and Abr are constants, whereas <Gv>ab is a function of temperature (see Fig.7.5). Taking a 50:50% fuel mixture, i. e., Na= Nb = N/2, and solving the above inequality for the product NTe* yields the well-known Lawson Criterion for energy viability

3(1 ~ Ліп Лот)Т

<crv >ab(T)Qab .. A ГЕ

Ліп Лош — An^ g -(J-Ліп Лот )Abr V Т 1 ~r Oab )

representing a rough, but useful reactor criterion.

Thus, energy viability requires that the density N times the global energy confinement time, Те*, must exceed a particular function of ion temperature. With all constants known and for illustrative purposes taking T|in • T|out ~ 1/3, as was suggested by Lawson, it becomes possible to determine this lower bound of Nte* as a function of the plasma temperature as shown in Fig.8.3.

Fig. 8.3: Lawson criterion bounds for d-t and d-d fusion.

As indicated in Fig.8.3, the lower bound of Nte. attains a minimum value at a specific temperature. This is a useful result and defines the combination of particle densities, confinement times, and temperature needed for reaction energy break-even conditions. The extremum point in the case of d-t fusion, Nte. ~ 1020 m’3s at approximately 15 keV was commonly cited as an initial experimental objective. Note also that no particular fusion design was necessary in the derivation of this criterion.

Equation (8.28) does not contain all relevant processes. For example, cyclotron radiation emission is not included and it may be difficult to sustain the various operational parameters as constants during the time interval TE*. Nevertheless, Eq.(8.28) is a useful and widely employed criterion. For commercial power applications, it would be necessary to exceed the minimum Lawson limit by perhaps a factor of ten or better.