1. Physics of fusion power
Lecture 2: Lawson criterion / Approaches to fusion.

2. Key problem of fusion
…. Is the Coulomb barrier

3. Cross section
The cross section is the effective area connected with the likelihood of occurrence of a reaction
For snooker balls the cross section is pr2 (with r the radius of the ball)
1 barn = m2
The cross section of various fusion reactions as a function of the energy. (Note logarithmic scale)

4. Averaged reaction rate
One particle (B) colliding with many particles (A)
Number of reactions in Dt is
Both s as well as v depend on the energy which is not the same for all particles. One builds the average
The cross section s
Schematic picture of the number of reactions in a time interval Dt

5. Averaged reaction rate …..
The cross section must be averaged over the energies of the particles. Assume a Maxwellian
Averaged reaction rates for various fusion reactions as a function of the temperature (in keV)

6. Sizeable number of fusion reactions even at relatively low temperatures
The Maxwellian (multiplied by the velocity)
Even for temperatures below the energy at which the cross section reaches its maximum, there is a sufficient amount of fusion reactions due to the number of particles in the tail of the Maxwell distribution
The cross section
The product of distribution and cross section
Schematic picture of the calculation of the averaged reaction rate (Integrand as a function of energy)

7. Compare the two
The averaged reaction rate does not fall off as strongly when going to lower energies
Cross section as a function of energy
Averaged reaction rate as a function of Temperature

8. Current fusion reactor concepts
are designed to operate at around 10 keV (note this is still 100 million Kelvin, matter is fully ionized or in the plasma state)
Are based on a mixture of Deuterium and Tritium
Both decisions are related to the cross section
Averaged reaction rates for various fusion reactions as a function of the temperature (in keV)

9. Limitations due to the high temperature
10 keV is still 100 million Kelvin (matter is fully ionized, i.e. in the plasma state)
Some time scales can be estimated using the thermal velocity
This is 106 m/s for Deuterium and m/s for the electrons
In a reactor of 10m size the particles would be lost in 10 ms.

10. Two approaches to fusion
One is based on the rapid compression, and heating of a solid fuel pellet through the use of laser or particle beams. In this approach one tries to obtain a sufficient number of fusion reactions before the material flies apart, hence the name, inertial confinement fusion (ICF).

11. Week five ……
Guest lecturer and international celebrity Dr. D. Gericke will give an overview of inertial confinement fusion …..

12. Magnetic confinement ..
The Lorentz force connected with a magnetic field ensures that the charged particles cannot move over large distances across the magnetic field
They gyrate around the field lines with a typical radius
At 10 keV and 5 Tesla this radius of 4 mm for Deuterium and 0.07 mm for the electrons

13. Lawson criterion
Derives the condition under which efficient production of fusion energy is possible
Essentially it compares the generated fusion power with any additional power required
The reaction rate of one particle B due to many particles A is
In the case of more than one particle B one obtains

14. Fusion power The total fusion power then is Using quasi-neutrality
For a 50-50% mixture of Deuterium and Tritium

15. Fusion power
To proceed one needs to specify the average of the cross section. In the relevant temperature range 6-20 keV
The fusion power can then be expressed as

16. The power loss
The fusion power must be compared with the power loss from the plasma
For this we introduce the energy confinement time tE
Where W is the stored energy

17. Ratio of fusion power to heating power
If the plasma is stationary
Compare this with the fusion power
One can derive the so called n-T-tau product

18. Break-even
The break-even condition is defined as the state in which the total fusion power is equal to the heating power
Note that this does not imply that all the heating power is generated by the fusion reactions

19. Ignition condition
Ignition is defined as the state in which the energy produced by the fusion reactions is sufficient to heat the plasma.
Only the He ions are confined (neutrons escape magnetic field and plasma) and therefore only 20% of the total fusion power is available for plasma heating

20. n-T-tau
Difference between inertial confinement and magnetic confinement: Inertial short tE but large density. Magnetic confinement the other way around
Magnetic confinement: Confinement time is around 3 seconds
Note that the electrons move over a distance of km in this time

21. Energy Cycle in a Power Plant
Electrical power out
External Power In
1GW
0.2GW
Steam Turbine D
1.8GW
He (20% of power)
3GW
Waste heat T
Energy losses due to imperfect confinement
Neutrons (80% of fusion power)

22. n-T-tau is a measure of progress
Over the years the n-T-tau product shows an exponential increase
Current experiments are close to break-even
The next step ITER is expected to operate well above break-even but still somewhat below ignition

23. Some landmarks in fusion energy Research
Initial experiments using charged grids to focus ion beams at point focus (30s).
Early MCF devices: mirrors and Z-pinches.
Tokamak invented in Russia in late 50s: T3 and T4
JET tokamak runs near break-even 1990s
Other MCF concepts like stellarators also in development.
Recently, massive improvements in laser technology have allowed ICF to come close to ignition: planned for this year.
Some landmarks in fusion energy Research

24. Alternative fusion concepts

25. End of lecture 2

26. Force on the plasma
The force on an individual particle due to the electro-magnetic field (s is species index)
Assume a small volume such that
Then the force per unit of volume is

27. Force on the plasma For the electric field Define an average velocity
Then for the magnetic field

28. Force on the plasma Averaged over all particles
Now sum over all species
The total force density therefore is

29. Quasi-neutrality
For length scales larger than the Debye length the charge separation is close to zero. One can use the approximation of quasi-neutrality
Note that this does not mean that there is no electric field in the plasma.
Under the quasi-neutrality approximation the Poisson equation can no longer be used directly to calculate the electric field
The charge densities are the dominant terms in the equation, and implicitly depend on E.

30. Divergence free current
Using the continuity of charge
Where J is the current density
One directly obtains that the current density must be divergence free

31. Also the displacement current must be neglected
From the Maxwell equation
Taking the divergence and using that the current is divergence free one obtains the unphysical result:
To avoid this, the displacement current must be neglected, so the Maxwell equation becomes

32. Quasi-neutrality
The charge density is assumed zero (but a finite electric field does exist)
One can not use the Poisson equation to calculate this electric field (since it would give a zero field)
Valid when length scales of the phenomena are larger than the Debye length
The current is divergence free
The displacement current is neglected (this assumption restricts us to low frequency waves: no light waves).

33. Force on the plasma
This force contains only the electro-magnetic part. For a fluid with a finite temperature one has to add the pressure force

34. Reformulating the Lorentz force
Using
The force can be written as
Then using the vector identity

35. Force on the plasma One obtains
Important parameter (also efficiency parameter) the plasma-beta
Magnetic field pressure Magnetic field tension