Chapter 5

Notes on fusion reactionsand power balance of a thermonuclear plasma!

Stefano Atzeni

See S. Atzeni and J. Meyer-ter-Vehn, “The Physics of Inertial Fusion, Oxford University Press (2004, 2009), Chapter 1 and Chapter 2.

SA, 3/2017

Exoenergetic reactions: fusion and fission reactions

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Exoenergetic reactions = energy releasing reactions

1 + 2 = > 3 + 4 + ... + Q, with energy Q > 0

For Einstein mass-energy relation, Q > 0 if the total mass of the reaction products is smaller than the mass of the reactants

Equivalently, if we consider the nuclear binding energies B:

Exoenergetic reactions < = => average binding energy per nucleon increases

D + 3He --> α (3.67 MeV) + p (14.7 MeV)

a few fusion reactions

the easiest

deuterium cycle

a dream

in the Sun(first of a chain)

p + p --> D + e+ + ν

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Fusion reaction cross sections

pp cross-section extremely small: 3 x 10-26 barn at energy of 10 keV

For energy production: thermonuclear fusion

•  beam-target reactions (between an accelerated nucleus and target at rest) or!beam-beam reactions cannot be used for net energy production,because scattering and slowing down dominate over fusion(accelerated nuclei are slowed down before they fuse; the energy they lose cannot be recovered)

•  energy production requires a hot plasma => thermonuclear fusion !

In a hot plasma fast nuclei (ions) collide with other nuclei and electrons; the energy lost by one particle is gained by another particle within the same thermal distribution

Thermonuclear reaction rate

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where•  n1 = number density of nuclei “1”•  n2 = number density of nuclei “2”•  δ12 = Kronecker δ •  = Mawellian averaged reactivity

Maxwellian averaged reactivity:product of the cross section σ(v) times the relative velocity of the reacting nuclei, averaged over the Maxwellian velotityy distribution

depends on temperature only (for a given reaction)

Volumetric reaction rate for the reaction between nuclei of species “1” and “2” = number of reactions per unit volume and unit time

Maxwellian averaged reactivities

D + T => α + n + 17.6 MeV has by far the largest reactivity

DT reactivity is maximum at T = 64 keV

At T < 60 keV is at least 10 times larger than the reactivity of any other reaction

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Even for DT, temperature > 5 keV required for thermonuclear plasma self-sustainement

A (simplified) necessary condition for DT plasma self-sustainement:

D + T => α + n + 17.6 MeV

14.1 MeV neutrons: leave the plasma;3.5 MeV α-particles: can be slowed in the

plasma.

Plasma emits bremsstrahlung

Plasma temperature self-sustainement:fusion α-particle power ≥ Bremsstrahlung

at 4.2 keV fusion alpha particle power exceeds bremsstrahlung power

11ideal ignition temperature

€

nDnT < sv >QDT5

> neniCb T

< sv > QDT20

> Cb T

(assuming equimolar pure DT plasma)

Steady-state plasma power balance

power balance (per unit volume):

•  losses: •  bremsstrahlung ∝ n2 T1/2 , where n is the electron density•  other, characterized by an energy confinement time τ:

3 n T/τ (assuming equal densities and temperatures for electron and ions)

•  input:•  fusion-α power = (1/5) fusion power ∝ n2 •  auxiliary heating: (fusion power)/Q

Q: energy multiplication [1/Q: fraction of re-cycled fusion energy]Q = ∞ : ignition (self-sustainment of plasma temperature)

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Lawson criterion

T = 10 - 20 keVnτ ≥ 2 x 1020 m-3 s

Triple product n τ T

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Magnetic and inertial plasma confinement

•  magnetic confinement fusion (MCF):

quasi-steady state fusion power release from low density plasma (1020-1021 m-3), in quasi steady-state, contained by appropriately shaped magnetic fields

•  inertial confinement fusion (ICF):

explosive fusion energy release from a strongly compressed plasma; theplasma pressure cannot be sustained by any means. Therefore the compressed plasma is “confined” by its inertia only, and remains compressed for a time τ ≈ R/cs, where R is a characteristic dimension (e.g. radius of a plasma sphere, and cs is the sound speed).

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Magnetically confined thermonuclear fusion plasma

•  T > 10 keV

•  plasma pressure (p = 2nkBT) contained by magnetic field pressure B2/2µ0;

in practice p = 2 n kB T = β B2/2µ0 with β ≤ 0.1

⇒  for T = 5 tesla, n ≤ few units x 1020 m-3

⇒  energy confinement times τΕ ≈ 1 - 5 s

T = 10 - 20 keVn ≈ 1014 cm-3τ ≈ 1 - 5 s

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