9.10 Assignments 297ne/ 2 .(Hint: the mean energy per degree of freedom isκT/ 2 so the mean energy
of a particle moving in three dimensional space is 3 κT/ 2 ).(b) What is the fusion reaction rateRfor a single incident ion of velocityvpass-
ing through stationary target ions whereσ(v)is the fusion reaction rate cross-
section andntis the density of the target ions (assumed stationary). If<
σ(v)v >denotes the velocity average ofσ(v)vwhat is the velocity-averaged
reaction rate?
(c) Assume that the reaction of hot tritium with hot deuterium can be approximated
as the sum of hot deuterium interacting with stationary tritium and hot tritium
interacting with stationary deuterium. Show that the fusion output powerin the
volume is
P=nDnT<σ(v)v>VEreaction
whereEreaction=17. 6 × 103 keV.
(d) Show that the ratioQof the fusion output energy to the input energy satisfies the
relation
neτE∼
6. 8 × 10 −^4 TkeV
<σ(v)v>Q
At break-evenQ=1.(e) For a thermal ion distribution and the experimentally measured reaction cross-
sectionσ(v)the velocity averaged rate integral has the following values
Temperature in keV <σ(v)v>inm^3 /s neτE
1 5. 5 × 10 −^27
2 2. 6 × 10 −^25
5 1. 3 × 10 −^23
10 1. 1 × 10 −^22
20 4. 2 × 10 −^22
50 8. 7 × 10 −^22
100 8. 5 × 10 −^22
Calculate theneτErequired for break-even at the listed temperatures (i.e. fill in
the last column).
(f) The Joint European Tokamak (JET) located in Culham, UK (in operationduring
the latter 20th and early 21st century) has a plasma volume of approximately 80
cubic meters. Suppose JET has an energy confinement time ofτE∼ 0. 3 s at
ne∼ 1020 m−^3 andT∼ 10 keV. If the plasma is a 50%-50% mixture of D-T
how much fusion power would be produced in JET and what isQ?What is the
weight of the plasma? How much does the confinement time have to be scaled
up to achieve break-even? How much does the volume have to be scaled up to
achieve 2000 MWt power output at breakeven? In order to achieve this volume
scale-up, how much does the linear dimension have to be scaled up?