Plasma physics by Dr. imran aziz
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  • 1. Plasma Physics DR.MOHAMMAD IMRAN AZIZ Assistant Professor(Sr.) PHYSICS DEPARTMENTSHIBLI NATIONAL COLLEGE, AZAMGARH (India). aziz_muhd33@yahoo.co.in 1
  • 2. aziz_muhd33@yahoo.co.in 2
  • 3. Ionized Gases• An ionized gas is characterized, in general, by a mixture of neutrals, (positive) ions and electrons.• For a gas in thermal equilibrium the Saha equation gives the expected amount of ionization: ni T 3 / 2 −Ui / kBT 2.4 ⋅ 1021 e nn ni• The Saha equation describes an equilibrium situation between ionization and (ion-electron) recombination rates. aziz_muhd33@yahoo.co.in 3
  • 4. Example: Saha Equation• Solving Saha equation ni T 3/ 2 −U i / kBT 2.4 ⋅1021 e nn ni ni2 2.4 ⋅ 1021 nnT 3 / 2e −Ui / kBT aziz_muhd33@yahoo.co.in 4
  • 5. Example: Saha Equation (II) aziz_muhd33@yahoo.co.in 5
  • 6. Backup: The Boltzmann EquationThe ratio of the number density (in atoms per m^3) of atoms in energy state B to those in energy state A is given by NB / NA = ( gB / gA ) exp[ -(EB-EA)/kT ]where the gs are the statistical weights of each level (the number of states of that energy). Note for the energy levels of hydrogen gn = 2 n2which is just the number of different spin and angular momentum states that have energy En. aziz_muhd33@yahoo.co.in 6
  • 7. From Ionized Gas to Plasma• An ionized gas is not necessarily a plasma• An ionized gas can exhibit a “collective behavior” in the interaction among charged particles when when long-range forces prevail over short-range forces• An ionized gas could appear quasineutral if the charge density fluctuations are contained in a limited region of space• A plasma is an ionized gas that presents a collective behavior and is quasineutral aziz_muhd33@yahoo.co.in 7
  • 8. The “Fourth State” of the Matter• The matter in “ordinary” conditions presents itself in three fundamental states of aggregation: solid, liquid and gas.• These different states are characterized by different levels of bonding among the molecules.• In general, by increasing the temperature (=average molecular kinetic energy) a phase transition occurs, from solid, to liquid, to gas.• A further increase of temperature increases the collisional rate and then the degree of ionization of the gas. aziz_muhd33@yahoo.co.in 8
  • 9. The “Fourth State” of the Matter (II)• The ionized gas could then become a plasma if the proper conditions for density, temperature and characteristic length are met (quasineutrality, collective behavior).• The plasma state does not exhibit a different state of aggregation but it is characterized by a different behavior when subject to electromagnetic fields. aziz_muhd33@yahoo.co.in 9
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  • 17. The Particle Picture1 Unmagnetized Plasmas2 Magnetized Plasma aziz_muhd33@yahoo.co.in 17
  • 18. 1 Unmagnetized Plasmas1.1 Charge in an Electric Field1.2 Collisions between Charged Particles aziz_muhd33@yahoo.co.in 18
  • 19. 1.1 Charge in an Electric Field• Electric force: F=qE Dimensional analysis: N=C V/m• A positive isolated charge q will produce a positive electric field at a point distance r given by q r V = C 1  E=  m F / m m2  4πε 0 r 3  • The force on another positive charge will be repulsive aziz_muhd33@yahoo.co.in 19 since F=qE is directed as r
  • 20. 1.2 Collisions between Charged Particlesr0 v• Interaction time T=r0/v• Change in momentum: q1q2 1 r0 q1q2 1 ∆ (mv) mv = FT = = 4πε 0 r0 v 4πε 0 r0 v 2 aziz_muhd33@yahoo.co.in 20
  • 21. • Impact parameter: q1q2 1 r0 = 4πε 0 mv 2 • Collisional cross section: σ =π r02 = ( q1q2 )1 2 16πε 0 m v 2 2 4 aziz_muhd33@yahoo.co.in 21
  • 22. Charge in an Electric Field• Electric force: F=qE Dimensional analysis: N=C V/m• A positive isolated charge q will produce a positive electric field at a point distance r given by q r V = C 1  E=  m F / m m2  4πε 0 r 3  • The force on another positive charge will be repulsive aziz_muhd33@yahoo.co.in 22 since F=qE is directed as r
  • 23. 2 Magnetized Plasmas2.1 Charge in an Uniform Magnetic Field aziz_muhd33@yahoo.co.in 23
  • 24. 1.1 Charge in an an Uniform Magnetic Field• Magnetic force: F = mv = qv × B & Dimensional analysis: N=C T m/s• Equation of the motion for a positive isolated charge q in a magnetic field B: i j k   F = mv = qv × B = q  vx & vy vz   Bx  By Bz   aziz_muhd33@yahoo.co.in 24
  • 25. Charge in an an Uniform Magnetic Field (II)i j k  vx vy vz  = i (v y Bz − vz By ) − j(vx Bz − vz Bx ) + k (vx By − v y Bx ) Bx By Bz   • Case of a magnetic field B directed along z: mvx = qv y Bz & mv y = −qvx Bz & mvz = 0 & aziz_muhd33@yahoo.co.in 25
  • 26. Charge in an an Uniform Magnetic Field (III)• By taking the derivative of mvx = qv y Bz & mvx = qv y Bz && & • Then replacing : v y = −vx qBz / m & vx = −vx ( qBz / m ) 2 && • Analogously: v y = −v y ( qBz / m ) 2 && aziz_muhd33@yahoo.co.in 26
  • 27. Charge in an an Uniform Magnetic Field (III) • The equations for vx and vy are harmonic oscillator equations.• The oscillation frequency, called cyclotron frequency is defined as: ω c = q Bz / m aziz_muhd33@yahoo.co.in 27
  • 28. Charge in an an Uniform Magnetic Field (IV)• The solution of the harmonic oscillator equation is vx = A exp ( iω ct ) + B exp ( −iω ct ) aziz_muhd33@yahoo.co.in 28
  • 29. The Kinetic Theory1 The Distribution Function2 The Kinetic Equations3 Relation to Macroscopic Quantities aziz_muhd33@yahoo.co.in 29
  • 30. The Distribution Function1 The Concept of Distribution Function2 The Maxwellian Distribution aziz_muhd33@yahoo.co.in 30
  • 31. 1.1 The Concept of Distribution Function• General distribution function: f=f(r,v,t)• Meaning: the number of particles per m3 at the position r, time t and velocity between v and v+dv is f(r,v,t) dv, where dv= dvx dvy dvz• The density is then found as ∞ ∞ ∞ ∞n(r, t ) = ∫ dvx ∫ dv y ∫ dvz f (r, v, t ) = ∫ 3 f (r, v, t )d v −∞ −∞ −∞ −∞ • If the distribution is normalized as ∞ ∫ f (r, v, t ) dv = 1 ˆ f (r, v, t ) = n(r, t ) f (r, v, t ) ˆ −∞ then f^ represents a probability distribution aziz_muhd33@yahoo.co.in 31
  • 32. The Maxwellian Distribution• The maxwellian distribution is defined as: 3/ 2  m   −v 2  fm =  ˆ  exp  2   2π k BT   vth  where v= 2 vx + vy 2 + vz 2 vth = 2k BT / m • The known result ∞ ∫ exp(− x )dx = π 2 −∞ yields ∞ ˆ ( v ) dv = 1 ∫ f maziz_muhd33@yahoo.co.in 32 −∞
  • 33. The Maxwellian Distribution (II)• The root mean square velocity for a maxwellian is: v 2 = 3k BT / m recall W = 1 mv 2 = 3k BT 2 • The average of the velocity magnitude v=|v| is: ∞ v = ˆm ( v )dv3 = 2vth = 2 2k BT / π m ∫ vf π −∞ • In one direction: ∞vx = 0 vx = ∫ vf m ( v )dv = vth / π = 2k BT / π m ˆ −∞ aziz_muhd33@yahoo.co.in 33
  • 34. The Maxwellian Distribution (III)• The distribution w.r.t. the magnitude of v ∞ ∞ ∫ g ( v)dv = ∫ f ( v ) dv 0 −∞ • For a Maxwellian 3/ 2  m   −v 2  g m = 4π n   v 2 exp  2   2π k BT   vth  aziz_muhd33@yahoo.co.in 34
  • 35. The Kinetic Equations1 The Boltzmann Equation2 The Vlasov Equation3 The Collisional Effects aziz_muhd33@yahoo.co.in 35
  • 36. 1. The Boltzmann Equation • A distribution function: f=f(r,v,t) satisfies the Boltzmann equation ∂f F ∂f  ∂f  + v ⋅ ∇f + ⋅ =   ∂t m ∂v  ∂t c • The r.h.s. of the Boltzmann equation is simply the expansion of d f(r,v,t)/dt • The Boltzmann equation states that in absence of collisions df/dt=0 vx Motion of a group of t+∆tparticles with constant density t in the phase space: aziz_muhd33@yahoo.co.in 36 x
  • 37. 2. The Vlasov Equation• In general, for sufficiently hot plasmas, the effect of collisions are less and less important• For electromagnetic forces acting on the particles and no collisions the Boltzmann equation becomes ∂f q ∂f + v ⋅ ∇f + ( E + v ⋅ B ) ⋅ = 0 ∂t m ∂v that is called the Vlasov equation aziz_muhd33@yahoo.co.in 37
  • 38. 3. The Collisional Effects• The Vlasov equation does not account for collisions  ∂f    =0  ∂t c • Short-range collisions like charged particles with neutrals can be described by a Boltzmann collision operator in the Boltzmann equation• For long-range collisions, like Coulomb collisions, a statistical approach yields the Fokker-Planck collision term • The Boltzmann equation with the Fokker-Planck collision term is simply named the Fokker-Planck aziz_muhd33@yahoo.co.in 38 equation.
  • 39. 4. Relation to Macroscopic Quantities1 The Moments of the Distribution Function2 Derivation of the Fluid Equations aziz_muhd33@yahoo.co.in 39
  • 40. 1. The Moments of the Distribution Function • Notation: define ∞ ∞ ∞ ∞ ∫ dvx ∫ dv y ∫ dvz = ∫ d 3v −∞ −∞ −∞ −∞• If A=A(v) the average of the function A for a distribution function f=f(r,v,t) is defined as ∞ ∫ A(r, v, t ) f (r, v, t )d 3v A(r, t ) v = −∞ ∞ = 3 ∫ f (r, v, t )d v −∞ 1 ∞ = ∫ A(r, v, t ) f (r, v, t )d 3v n(r, t ) −∞aziz_muhd33@yahoo.co.in 40
  • 41. The Moments of the Distribution Function (II)• General distribution function: f=f(r,v,t)• The density is defined as the 0th order moment and was found to be ∞ ∞ ∞ ∞n(r, t ) = ∫ dvx ∫ dv y ∫ dvz f (r, v, t ) = ∫ f (r, v, t )d 3v −∞ −∞ −∞ −∞ • The mass density can be then defined as ∞ ρ (r, t ) = mn(r, t ) = m ∫ f (r, v, t )d 3v −∞ aziz_muhd33@yahoo.co.in 41
  • 42. The Moments of the Distribution Function (III)• The 1st order moment is the average velocity or fluid velocity is defined as 1 ∞ u(r, t ) = 3 ∫ vf (r, v, t )d v n(r, t ) −∞• The momentum density can be then defined as ∞ r , t ) = m ∫ vf n(r, t )mu(aziz_muhd33@yahoo.co.in(r, v, t )d 3v 42 −∞
  • 43. The Moments of the Distribution Function (IV)• Higher moments are found by diadic products with v• The 2nd order moment gives the stress tensor (tensor of second order) ∞ Π (r, t ) = m ∫ vvf (r, v, t )d 3v −∞• In the frame of the moving fluid the velocity is w=v-u. In this case the stress tensor becomes the pressure tensor ∞ P (r, t ) = m ∫ wwf (r, v, t )d 3v aziz_muhd33@yahoo.co.in 43 −∞
  • 44. 2 Derivation of the Fluid Equations • Boltzmann equation written for the Lorentz force ∂f q ∂f  ∂f  + v ⋅ ∇f + ( E + v × B ) ⋅ =   ∂t m ∂v  ∂t c • Integrate in velocity space: ∂f 3 q ∂f 3  ∂f  d 3v∫ ∂t d v + ∫ v ⋅ ∇f d v + m ∫ ( E + v × B ) ⋅ ∂vd v = ∫  ∂t  3   c • From the definition of density ∂f 3 ∂ ∂n ∫ ∂t d v = ∂t ∫ fd v = ∂t 3 aziz_muhd33@yahoo.co.in 44
  • 45. Derivation of the Fluid Equations (II)• Since the gradient operator is independent from v: ∫ v ⋅ ∇f d 3v = ∇ ⋅ ∫ vf d 3v = ∇ ⋅ ( nu )• Through integration by parts it can be shown that q ∂f 3 m ∫ ( E + v × B ) ⋅ ∂vd v = 0 • If there are no ionizations or recombination the collisional term will not cause any change in the number of particles (no particle sources or sinks) therefore  ∂f  d 3v = 0 ∫  ∂t    aziz_muhd33@yahoo.co.in c 45
  • 46. Derivation of the Fluid Equations (III) • The integrated Boltzmann equation then becomes ∂n + ∇ ⋅ ( nu ) = 0 ∂t that is known as equation of continuity• In general moments of the Boltzmann equation are taken by multiplying the equation by a vector function g=g(v) and then integrating in the velocity space • In the case of the continuity equation g=1 • For g=mv the fluid equation of motion, or momentum equation can be obtained aziz_muhd33@yahoo.co.in 46
  • 47. Derivation of the Fluid Equations (IV) • Integrate the Boltzmann equation in velocity space with g=mv ∂f 3 ∂f 3 m ∫ v d v + m ∫ vv ⋅ ∇f d v + q ∫ v ( E + v × B ) ⋅ d v = 3 ∂t ∂v  ∂f  d 3v = ∫ mv    ∂t c • The first term is ∂f 3 ∂ ∂  3 ∫ vfd 3v  ∂m ∫ v d v = m ∫ vfd v = m  ∫ fd v 3 3  = m ( nu ) ∂t ∂t ∂t   ∫ fd v  ∂t aziz_muhd33@yahoo.co.in 47
  • 48. Derivation of the Fluid Equations (V) • Further simplifications yield the final fluid equation of motion  ∂u + u ⋅ ∇ u  = qn E + u × B − ∇ ⋅ P + P mn  ( )  ( )  ∂t coll where u is the fluid average velocity, P is the stress tensor and Pcoll is the rate of momentum change due to collisions • Integrating the Boltzmann equation in velocity space with g=½mvv the energy equation is obtained aziz_muhd33@yahoo.co.in 48
  • 49. The Kinetic Theory1 The Distribution Function2 The Kinetic Equations3 Relation to Macroscopic Quantities4 Landau Damping aziz_muhd33@yahoo.co.in 49
  • 50. 4 Landau Damping1 Electromagnetic Wave Refresher2 The Physical Meaning of Landau Damping3 Analysis of Landau Damping aziz_muhd33@yahoo.co.in 50
  • 51. 1 Electromagnetic Wave Refresher aziz_muhd33@yahoo.co.in 51
  • 52. Electromagnetic Wave Refresher (II) • The field directions are constant with time, indicating that the wave is linearly polarized (plane waves).• Since the propagation direction is also constant, this disturbance may be written as a scalar wave: E = Emsin(kz-ωt) B = Bmsin(kz-ωt) k is the wave number, z is the propagation direction, ω is the angular frequency, Em and Bm are the amplitudes of the E and B fields respectively.• The phase constants of the two waves are equal (since they are in phase with one another) and have been arbitrarily set to 0. aziz_muhd33@yahoo.co.in 52
  • 53. The Physical Meaning of Landau Damping• An e.m. wave is traveling through a plasma with phase velocity vφ• Given a certain plasma distribution function (e.g. a maxwellian), in general there will be some particles with velocity close to that of the wave.• The particles with velocity equal to vφ are called resonant particles aziz_muhd33@yahoo.co.in 53
  • 54. The Physical Meaning of Landau Damping (II)• For a plasma with maxwellian distribution, for any given wave phase velocity, there will be more “near resonant” slower particles than “near resonant” fast particles• On average then the wave will loose energy (damping) and the particles will gain energy• The wave damping will create in general a local distortion of the plasma distribution function• Conversely, if a plasma has a distribution function with positive slope, a wave with phase velocity within that positive slope will gain energy aziz_muhd33@yahoo.co.in 54
  • 55. The Physical Meaning of Landau Damping (III)• Whether the speed of a resonant particle increases or decreases depends on the phase of the wave at its initial position• Not all particles moving slightly faster than the wave lose energy, nor all particles moving slightly slower than the wave gain energy.• However, those particles which start off with velocities slightly above the phase velocity of the wave, if they gain energy they move away from the resonant velocity, if they lose energy they approach the resonant velocity. aziz_muhd33@yahoo.co.in 55
  • 56. The Physical Meaning of Landau Damping (IV)• Then the particles which lose energy interact more effectively with the wave• On average, there is a transfer of energy from the particles to the electric field.• Exactly the opposite is true for particles with initial velocities lying just below the phase velocity of the wave. aziz_muhd33@yahoo.co.in 56
  • 57. The Physical Meaning of Landau Damping (V)• The damping of a wave due to its transfer of energy to “near resonant particles” is called Landau damping• Landau damping is independent of collisional or dissipative phenomena: it is a mere transfer of energy from an electromagnetic field to a particle kinetic energy (collisionless damping) aziz_muhd33@yahoo.co.in 57
  • 58. Analysis of Landau Damping• A plane wave travelling through a plasma will cause a perturbation in the particle velocity distribution: f(r,v,t) =f0(r,v,t) + f1(r,v,t)• If the wave is traveling in the x direction the perturbation will be of the form f1 ∝ exp [i ( kx − ω t )] • For a non-collisional plasma analysis the Vlasov equation applies. For the electron species it will be ∂f e ∂f + v ⋅ ∇f − ( E + v × B ) ⋅ = 0 ∂t m aziz_muhd33@yahoo.co.in ∂v 58
  • 59. Analysis of Landau Damping (II)• A linearization of the Vlasov equation considering f = f 0 + f1 E = E0 + E1 ; B = B0 + B1 ; E0 = 0; B 0 = 0 v × B = 0 (since only contributions along v are studied) yields ∂f1 e ∂f 0 + v ⋅ ∇f1 − E1 ⋅ =0 ∂t m ∂v or, considering the wave along the dimension x, e ∂f 0 iω f1 + ikvx f1 = − E1x m aziz_muhd33@yahoo.co.in ∂vx 59
  • 60. Analysis of Landau Damping (III)• The electric field E1 along x is not due to the wave but to charge density fluctuations• E1 be expressed in function of the density through the Gauss theorem (first Maxwell equation) ∇ ⋅ E1 = −en ε 0 or, in this case, considering a perturbed density n1 equivalent to the perturbed distribution f1 ikE x = −en ε 0 • Finally the density can be expressed in terms of the distribution function as ∞ , t ) = ∫ f1 (r, v n1 (raziz_muhd33@yahoo.co.in , t )d 3v 60 −∞
  • 61. Analysis of Landau Damping (IV)• The linearized Vlasov equation for the wave perturbation e ∂f 0 iω f1 + ikvx f1 = − E1x m ∂vx can be rewritten, after few manipulations as a relation between ω, k and know quantities: ω2 p ∞ ∂f 0 (vx ) ∂vx ˆ 1= 2 ∫ dvx k −∞ vx − (ω k ) where f 0 = f 0 / n0 ˆ aziz_muhd33@yahoo.co.in 61
  • 62. Analysis of Landau Damping (V)• For a wave propagation problem a relation between ω and k is called dispersion relation• The integral in the dispersion relation ω 2 ∞ ∂fˆ0 (vx ) ∂vx p 1= 2 ∫ dvx k −∞ vx − (ω k ) can be computed in an approximate fashion for a maxwellian distribution yielding  π ω p ∂fˆ0 (vx )  2 ω = ω p 1 + i   2k 2  ∂vx v =ω / k   aziz_muhd33@yahoo.co.in 62
  • 63. Analysis of Landau Damping (VI)• For a one-dimensional maxwellian along the x direction ∂f 0 (vx ) ˆ 2v x  vx  2 = − 1 2 3 exp  − 2  ∂vx π vth  vth  • This will cause the imaginary part of the expression  ω 2 ∂fˆ0 (vx ) π p  ω = ω p 1 + i   2k 2  ∂vx v =ω / k   to be negative (for a positive wave propagation direction) aziz_muhd33@yahoo.co.in 63
  • 64. Analysis of Landau Damping (VII)• For a wave is traveling in the x direction the of the formf1 ∝ exp [i ( kx − ω t )] = exp ( ikx ) exp [ −i (ω R + iω I ) t ] == exp ( ikx ) exp [( −iω R + ω I ) t ] == exp ( ikx ) exp ( −iω R t ) exp (ω I t ) a negative imaginary part of ω will produce an attenuation, or damping, of the wave. aziz_muhd33@yahoo.co.in 64
  • 65. The Fluid Description of PlasmasThe Fluid Equations for a Plasma aziz_muhd33@yahoo.co.in 65
  • 66. Plasmas as Fluids: Introduction• The particle description of a plasma was based on trajectories for given electric and magnetic fields• Computational particle models allow in principle to obtain a microscopic description of the plasma with its self-consistent electric and magnetic fields• The kinetic theory yields also a microscopic, self- consistent description of the plasma based on the evolution of a “continuum” distribution function• Most practical applications of the kinetic theory rely also on numerical implementation of the kinetic equations aziz_muhd33@yahoo.co.in 66
  • 67. Plasmas as Fluids: Introduction (II) • The analysis of several important plasma phenomena does not require the resolution of a microscopic approach• The plasma behavior can be often well represented by a macroscopic description as in a fluid model • Unlike neutral fluids, plasmas respond to electric and magnetic fields• The fluidodynamics of plasmas is then expected to show additional phenomena than ordinary hydro, or gasdynamics aziz_muhd33@yahoo.co.in 67
  • 68. Plasmas as Fluids: Introduction (III) • The “continuum” or “fluid-like” character of ordinary fluids is essentially due to the frequent (short-range) collisions among the neutral particles that neutralize most of the microscopic patterns• Plasmas are, in general, less subject to short-range collisions and properties like collective effects and quasi-neutrality are responsible for the fluid-like behavior aziz_muhd33@yahoo.co.in 68
  • 69. Plasmas as Fluids: Introduction (IV) • Plasmas can be considered as composed of interpenetrating fluids (one for each particle species)• A typical case is a two-fluid model: an electron and an ion fluids interacting with each other and subject to e.m. forces• A neutral fluid component can also be added, as well as other ion fluids (for different ion species or ionization levels) aziz_muhd33@yahoo.co.in 69
  • 70. The Fluid Description of Plasmas1 The Fluid Equations for a Plasma2 Plasma Diffusion3 Fluid Model of Fully Ionized Plasmas aziz_muhd33@yahoo.co.in 70
  • 71. Fluid Model of Fully Ionized Plasmas. The Magnetohydrodynamic Equations.Diffusion in Fully Ionized Plasmas. Hydromagnetic Equilibrium. Diffusion of Magnetic Field in a Plasma aziz_muhd33@yahoo.co.in 71
  • 72. Magnetohydrodynamic Equations • Goal: to derive a single fluid description for a fully ionized plasma• Single-fluid quantities: define mass density, fluid velocity and current density from the same quantities referred to electrons and ions: ρ m = mi ni + me ne ≈ n( mi + me ) 1 ( mi ui + meue ) u= ( mi ni ui + me neue ) ≈ ρm (mi + me ) j = e ( ni ui − ne u e ) ≈ ne ( ui − u e ) aziz_muhd33@yahoo.co.in 72
  • 73. Magnetohydrodynamic Equations (II) • Equation of motion for electron and ions with Coulomb collisions, ne=ni and a gravitational term (that can be used to represent any additional non e.m. force):  ∂ui nmi  + ( ui ⋅ ∇ ) ui  = qi n ( E + ui × B ) − ∇pi + Pie + mi ng  ∂t   ∂u e nme  + ( u e ⋅ ∇ ) u e  = qe n ( E + u e × B ) − ∇pe + Pei + me ng  ∂t  • Approximation 1: the viscosity tensor has been neglected, acceptable for Larmor radius small w.r.t. the scale length of variations of the fluid quantities. aziz_muhd33@yahoo.co.in 73
  • 74. Magnetohydrodynamic Equations (III) • Approximation 2: neglect the convective term, acceptable when the changes produced by the fluid convective motion are relatively small  ∂ui nmi  + ( ui ⋅ ∇ ) ui  = qi n ( E + ui × B ) − ∇pi + Pie + mi ng  ∂t   ∂u e nme  + ( u e ⋅ ∇ ) u e  = qe n ( E + u e × B ) − ∇pe + Pei + me ng  ∂t  • These equation can be added and by setting p=pe+pi, -qi=qe=e and Pei=-Pie obtaining: ∂ n ( mi ui + me u e ) = en ( ui − u e ) × B − ∇p + n ( mi + me ) g ∂t aziz_muhd33@yahoo.co.in 74
  • 75. Magnetohydrodynamic Equations (IV) • By substituting the definition of the single fluid variables r, u and j the equation ∂n ( mi ui + me u e ) = en ( ui − u e ) × B − ∇p + n ( mi + me ) g ∂t can be written as ∂u ρm = j × B − ∇p + ρ m g ∂t that is the single fluid equation of motion for the mass flow. There is no electric force because the fluid is globally neutral (ne=ni). aziz_muhd33@yahoo.co.in 75
  • 76. Magnetohydrodynamic Equations (V) • To characterize the electrical properties of the single-fluid it is necessary to derive an equation that retains the electric field• By multiplying the ion eq. of motion by me, the electron one by mi, by subtracting them and taking the limit me/ mi=>0, d/dt=>0 it is obtained 1 E + u × B = η j + ( j × B ) − ∇pe en that is the generalized Ohm’s law that includes the Hall term (jxB) and the pressure effects aziz_muhd33@yahoo.co.in 76
  • 77. Magnetohydrodynamic Equations (VI)• Analogous procedures applied to the ion and electron continuity equations (multiplying by the masses, adding or subtracting the equations) leadto the continuity for the mass density rm or for the charge density r: ∂ρ m + ∇ ⋅ ( ρmu ) = 0 ∂t ∂ρ +∇⋅j= 0 ∂t • The single-fluid equations of continuity and motion and the Ohm’s law constitute the set of magnetohydrodynamic (MHD) equations. aziz_muhd33@yahoo.co.in 77
  • 78. Diffusion in Fully Ionized Plasmas• The MHD equations, in absence of gravity and for steady-state conditions, with a simplified version of the Ohm’s law, are 0 = j × B − ∇p E + u × B =ηj• The parallel (to B) component of the last equation reduce simply to the ordinary Ohm’s law: E =η j aziz_muhd33@yahoo.co.in 78
  • 79. Diffusion in Fully Ionized Plasmas (II)• The component perpendicular to B is found by taking the the cross product with B E × B + ( u ⊥ × B ) × B = η⊥ j × B that is E × B − u ⊥ B 2 = η ⊥ j × B = η ⊥ ∇p and finally E × B η⊥ u ⊥ = 2 − 2 ∇p B B• The first term is the usual ExB drift (for bothspecies together), the second is a diffusion driven by the gradient of the pressure aziz_muhd33@yahoo.co.in 79
  • 80. Diffusion in Fully Ionized Plasmas (III)• The diffusion in the direction of -grad p produces a fluxη Γ ⊥ = nu ⊥ = −n ⊥ ∇p 2 B • For isothermal, ideal gas-type plasma the perpendicular flux can be written as η⊥ n(k BTi + k BTe ) Γ⊥ = − 2 ∇n B that is a Fick’s law with diffusion coefficient η⊥ n(k BTi + k BTe ) D⊥ = B2 named classical diffusion coefficient aziz_muhd33@yahoo.co.in 80
  • 81. Diffusion in Fully Ionized Plasmas (IV)• The classical diffusion coefficient is proportional to 1/B2 as in the case of weakly ionized plasmas: it is typical of a random-walk type of process with characteristic step length equal to the Larmor radius• The classical diffusion coefficient is proportional to n, not constant, because does not describe the scattering with a fixed neutral background• Because the resistivity decreases with T3/2 so does the classical diffusion coefficient (the opposite of a partially ionized plasma) aziz_muhd33@yahoo.co.in 81
  • 82. Diffusion in Fully Ionized Plasmas (IV)• The classical diffusion is automatically ambipolar, as it was derived for a single fluid (both species are diffusing at the same rate) • Since the equation for the perpendicular velocity does not contain any term along E that depend on E itself, it can be concluded that there is no perpendicular mobility: an electric field perpendicular to B produces just a ExB drift. aziz_muhd33@yahoo.co.in 82
  • 83. Diffusion in Fully Ionized Plasmas (V)• Experiments with magnetically confined plasmas showed a diffusion rate much higher than the one predicted by the classical diffusion• A semiempirical formula was devised: this is the Bohm diffusion coefficient that goes like 1/B and increases with the temperature: 1 k BTe D⊥ Bohm = 16 eB• Bohm diffusion ultimately makes more difficult to reach fusion conditions in magnetically confined plasma aziz_muhd33@yahoo.co.in 83
  • 84. Hydromagnetic Equilibrium• The MHD momentum equation, in absence of gravity and for steady-state conditions is considered to describe an equilibrium condition for a plasma in a magnetic field. ∇p = j × B• The momentum equation expresses the force balance between the pressure gradient and the Lorentz force • In force balance both j and B must beperpendicular to grad p: j and B must then lie on constant p surfaces aziz_muhd33@yahoo.co.in 84
  • 85. Hydromagnetic Equilibrium (II) j B grad p • For an axial magnetic field in a cylindrical configuration with radial pressure gradient, the current must be azimuthal• The momentum equation in the perpendicular plane (w.r.t. B) will then give an expression for j aziz_muhd33@yahoo.co.in 85
  • 86. Hydromagnetic Equilibrium (II)• The cross product of the momentum with B yields B × ∇p = B × j × B = jB 2 and, in the usual approximations, solving for j yield again the expression for the diamagnetic current B × ∇p B × ∇n j= 2 = ( k BTi + k BTe ) B B2 • From the MHD point of view the diamagnetic current is generated by the grad p force that interacts (via a cross product) with B aziz_muhd33@yahoo.co.in 86
  • 87. Hydromagnetic Equilibrium (IV)• The connection between the fluid and the particle point of view was previously discussed: the diamagnetic current arises from an unbalance of the Larmor gyration velocities in a fluid element • From a strict particle point of view the confinement of the plasma with a gradient of pressure occurs because each particle guiding center is tight to a line of force and diffusion is not permitted (in absence of collisions) aziz_muhd33@yahoo.co.in 87
  • 88. Hydromagnetic Equilibrium (V) • For the equilibrium case under consideration, the momentum equation in the direction parallel to B will be simply ∂p ∇p = 0 = ∂s where s is a generalized coordinate along the lines of force. ∂n • For isothermal plasma it will be = 0 ∂s then the density is constant along the lines of force• This condition is valid only for a static case (u=0). • For example in a magnetic mirror there are more particles trapped at the midplane (lower line of force density) than at the mirror end sections 88 aziz_muhd33@yahoo.co.in
  • 89. Waves in Plasmas1 Electrostatic Waves in Non-Magnetized Plasmas2 Electrostatic Waves in Magnetized Plasmas aziz_muhd33@yahoo.co.in 89
  • 90. E.S. Waves in Non-Magnetized Plasmas1. Wave fundamentals2. Electron Plasma Waves3. Sound waves4. Ion Acoustic Waves aziz_muhd33@yahoo.co.in 90
  • 91. Wave Fundamentals • Any periodic motion of a fluid can be decomposed, through Fourier analysis, in a superposition of sinusoidal components, at different frequencies• Complex exponential notation is a convenient way to represent mathematically oscillating quantities: the physical quantity will be obtained by taking the real part • A sinusoidal plane wave can be represented as f (r, t ) = f 0 exp i ( k ⋅ r − ω t )    where f0 is the maximum amplitude, k is the propagation constant, or wave vector (k is the wavenumber) and w the angular frequency aziz_muhd33@yahoo.co.in 91
  • 92. Wave Fundamentals (II) • If f0 is real then the wave amplitude is maximum (equal to f0) in r=0, t=0, therefore the phase angle of the wave is zero • A complex f0 can be used to represent a non zero phase angle:f 0 exp i ( k ⋅ r − ω t + δ )  = f 0 exp ( iδ ) exp i ( k ⋅ r − ω t )      • A point of constant phase on the wave will travel along with the wave front • A constant phase on the wave implies d (k ⋅ r − ωt ) = 0 dt aziz_muhd33@yahoo.co.in 92
  • 93. Wave Fundamentals (III) • In one dimension it will be d dx ω ( kx − ω t ) = 0 ⇒ = vϕ dt dt k where vf is defined as the wave phase velocity • The wave can be then also expressed by f ( x, t ) = f 0 exp ik ( x − vϕ t )    • The phase velocity in a plasma can exceed the velocity of the light c, however an infinitely longwave train that maintains a constant velocity does notcarry any information, so the relativity is not violated. aziz_muhd33@yahoo.co.in 93
  • 94. Wave Fundamentals (IV) • A wave carries information only with some kind of modulation• An amplitude modulation is obtained for example by adding to waves of different frequencies (wave “beating”) • If a wave with phase velocity vf is formed by two waves with frequency separation 2Dw , both the two components must also travel at vf• The two components of the wave must then also have a difference in their propagation constant k equal to 2Dk aziz_muhd33@yahoo.co.in 94
  • 95. Wave Fundamentals (V) • For the case of two wave beating it can be written f A ( x, t ) = f 0 cos ( k + ∆k ) x − (ω + ∆ω ) t    f B ( x, t ) = f 0 cos ( k − ∆k ) x − (ω − ∆ω ) t    • By summing the two waves and expanding with trigonometric identities it is foundf A ( x, t ) + f B ( x, t ) = 2 f 0 cos ( ∆k ) x − ( ∆ω ) t  ⋅ cos [ kx − ω t ]   • The first term of the r.h.s. is the modulating component (that does carry information) • The second term of the r.h.s. is just the “carrier” component of aziz_muhd33@yahoo.co.in does not carry 95 the wave (that information)
  • 96. Wave Fundamentals (VI) • The modulating component travels at the group velocity defined as ∆ω dω vg = ⇒ vg = ∆k ∆ω →0 dk• The group velocity can never exceed c aziz_muhd33@yahoo.co.in 96
  • 97. Electron Plasma Waves• Thermal motions cause electron plasma oscillations to propagate: then they can be properly called (electrostatic ) electron plasma waves• By linearizing the fluid electron equation of motion with respect equilibrium quantities according to ne = ne 0 + ne1 ue = ue 0 + ue1 E = E0 + E1 the frequency of the oscillations can be found as 3 2 2 ω 2 = ω2 pe + k vth 2 where vth = 2k BTe me 2 aziz_muhd33@yahoo.co.in 97
  • 98. Electron Plasma Waves (II)• Electron plasma waves have a group velocity equal to dω 3 k 2 3 k 2 = vth = vth dk 2 ω 2 vϕ • In general a relation linking w and k for a wave is called dispersion relation• The slope of the dispersion relation on a w-k diagram gives the phase velocity of the wave aziz_muhd33@yahoo.co.in 98
  • 99. Sound Waves• For a neutral fluid like air, in absence of viscosity, the Navier-Stokes equation is  ∂u + u ⋅ ∇ u  = −∇p ρm  ( )   ∂t  γp • From the equation of state ∇p = ρm then  ∂u + u ⋅ ∇ u  = − γ p ρm  ( )   ∂t  ρm • Continuity equation yields ∂ρ m + ∇ ⋅ ( ρmu ) = 0 ∂aziz_muhd33@yahoo.co.in t 99
  • 100. Sound Waves (II) • Linearization of the momentum and continuity equations for stationary equilibrium yield 12 12 ω  γ p0   γ k BT  =  =  m  = cs k  ρm0   N  where mN is the neutral atom mass and cs is the sound speed.• For a neutral gas the sound waves are pressure waves propagating from one layer of particles to another one • The propagation of sound waves requires collisions among the neutrals aziz_muhd33@yahoo.co.in 100
  • 101. Electromagnetic Waves in Plasmas1E.M. Waves in a Non-Magnetized Plasma2 E.M. Waves in a Magnetized Plasma3Hydromagnetic (Alfven) Waves4Magnetosonic Waves aziz_muhd33@yahoo.co.in 101
  • 102. Electromagnetic Waves in a Plasma• In a plasma there will be current carriers, therefore the curl of Ampere’s law is ∂D ∇×H = j+ ∂t • By taking the curl of Faraday’s law ∂ ∇ × ∇ × E = ∇ ( ∇ ⋅ E ) − ∇ E = −µ0 ( ∇ × H ) 2 ∂t and eliminating the curl of H ∂ ∂2D  ∇ ( ∇ ⋅ E ) − ∇2 E = −µ0  j + 2   ∂t ∂t  aziz_muhd33@yahoo.co.in 102
  • 103. Electromagnetic Waves in a Plasma (II)• If a wave solution of the form exp(k·r-wt) is assumed it can be written (D=e0E) ik ( ik ⋅ E ) + k 2 E = iωµ 0 j + ω 2 µ 0ε 0 E• By recalling that an e.m. must be transverse (k·E =0) and that c2=1/(m0e0) it follows ( ω 2 − c 2 k 2 ) E = −iω j / ε 0 • In order to estimate the current the ions are considered fixed (good approximation for high frequencies) and the current is carried by electrons with density n0 and velocity u: j = − n0 eu e aziz_muhd33@yahoo.co.in 103
  • 104. Electromagnetic Waves in a Plasma (III) • The electron equation of motion is ∂u me = −eE − eu × B ∂t• The motion of the electrons here is the self-consistent solution of u, E, B (E and B are not external imposed field like in the particle trajectory calculations) • A first-order form of the equation of motion is then ∂u me = −eE ∂t then 2 −eE n0 e E u= ⇒ j= aziz_muhd33@yahoo.co.in 104 −iω me iω me
  • 105. Electromagnetic Waves in a Plasma (IV) • Finally, substituting the expression of j in ( ω 2 − c 2 k 2 ) E = −iω j / ε 0 it is found n0 e 2 ( ω 2 − c2 k 2 ) E = ε0m E ⇒ ω 2 = ω p + c2 k 2 2 that is the dispersion relation for e.m. waves in a plasma (without external magnetic field)• The phase velocity is always greater than c while the group velocity is always less than c: ω 2 ωp 2 dω c 2 vϕ = 2 = 2 + c aziz_muhd33@yahoo.co.in vg = 2 2 = k k dk vϕ 105
  • 106. Electromagnetic Waves in a Plasma (V) • For a given frequency w the dispersion relation ω 2 = ω p + c2 k 2 2 gives a particular k or wavelength (k=2p/l) for the wave propagation• If the frequency is raised up to w=wp then it must be k=0. This is the cutoff frequency (conversely, cutoff densitywill be the value that makes wp equal to w)• For even larger densities, or simply w<wp there is no real k that satisfies the dispersion relation and the wave cannot propagate through the plasma aziz_muhd33@yahoo.co.in 106
  • 107. Electromagnetic Waves in a Plasma (VI)• When k becomes imaginary the wave is attenuated • The spatial part of the wave can be written as exp ( ikx ) = exp ( − k x ) exp ( − x / δ ) where d is the skin depth defined as −1 c δ=k = (ω p − ω ) 2 2 1/ 2 aziz_muhd33@yahoo.co.in 107
  • 108. E.M. Waves in a Magnetized Plasma • The case of an e.m. wave perpendicular to an external magnetic field B0 is considered • If the wave electric field is parallel to B0 the same derivation as for non magnetized plasma can be applied (essentially because the first-order electron equation of motion is not affected by B0) • The the wave is called ordinary wave and the dispersion relation in this case is still z ω =ω +c k 2 2 p 2 2 E B0k y aziz_muhd33@yahoo.co.in 108 x
  • 109. E.M. Waves in a Magnetized Plasma (II) • The case of the wave electric field perpendicular to B0 requires both x and y components of E since the wave becomes elliptically polarized z E B0k y x • A linearized (first-order) form of the equation electron equation of motion is then ∂u me = −eE − eu × B 0 ⇒ −iω me u = −eE − eu × B 0 ∂t aziz_muhd33@yahoo.co.in 109
  • 110. E.M. Waves in a Magnetized Plasma (III)• The wave equation now must keep the longitudinal electric field k·E=kEx ik ( ik ⋅ E ) + k 2 E = iωµ 0 j + ω 2 µ 0ε 0 E or( ω 2 − c 2 k 2 ) E + c 2 kEx k = −iω j / ε 0 = −in0ω eu / ε 0 • By solving for the separate x and y components a dispersion relation for the extraordinary wave is found as 2 2 c k ωp 2 ω −ωp 2 2 =1− 2 2 ω 2 ω ω − (ω p + ω c2 ) 2 aziz_muhd33@yahoo.co.in 110
  • 111. E.M. Waves in a Magnetized Plasma (IV) • The case of the wave vector parallel to B0 also requires both x and y components of E k zE B0 y x • The same derivation as for the extraordinary wave can be used by simply by changing the direction of k aziz_muhd33@yahoo.co.in 111
  • 112. E.M. Waves in a Magnetized Plasma (V) • The resulting dispersion relation is ck2 2 ωp ω2 2 =1− ω 2 1 m (ω c ω ) or the choice of sign distinguish between a right-hand circular polarization (R-wave) and a left hand circular polarization (L-wave) • The R-wave has a resonance corresponding to the electron Larmor frequency: in this case the wave looses energy by accelerating the electrons along the Larmor orbit• It can be shown that the L-wave has a resonance in correspondence to the ion Larmor frequency 112 aziz_muhd33@yahoo.co.in
  • 113. Hydromagnetic (Alfven) Waves• This case considers still the wave vector parallel to B0 but includes both electrons and ion motions and current j and electric field E perpendicular to B0 k z B0 E,j y x • The solution neglects the electron Larmor orbits, leaving only the ExB drift and considers propagation frequencies much smaller than the ion cyclotron frequency aziz_muhd33@yahoo.co.in 113
  • 114. Hydromagnetic (Alfven) Waves (II) • The dispersion relation for the hydromagnetic (Alfven) waves can be derived as ω 2 c 2 c 2 = = k 2 ( 1 + ρ ( ε 0 B0 ) 2 )1 + c 2 ( ρµ 0 B02 ) where r is the mass density• It can be shown that the denominator is the relative dielectric constant for low-frequency perpendicular motion in the plasma• The dispersion relation for Alfven waves gives the phase velocity of e.m. waves in the plasma considered as a dielectric medium aziz_muhd33@yahoo.co.in 114
  • 115. Hydromagnetic (Alfven) Waves (III)• In most laboratory plasmas the dielectric constant is much larger than unity, therefore, for hydromagnetic waves, ω B02 c 2 ≈ ≡ vA k ( ρµ 0 ) 1/ 2 where vA is the Alfven velocity• The Alfven velocity can be considered the velocity of the perturbations of the magnetic lines of force due to the wave magnetic field in the plasma • Under the approximations made the fluid and the field lines oscillate as they were “glued” together aziz_muhd33@yahoo.co.in 115
  • 116. Magnetosonic Waves• This case considers the wave vector perpendicular to B0 and includes both electrons and ion motions (low- frequency waves) with E perpendicular to B0 k z B0 E y x • The solution includes the pressure gradient in the (fluid) equation of motion since the oscillating ExB0 drifts will cause compressions in the direction of the wave aziz_muhd33@yahoo.co.in 116
  • 117. Magnetosonic Waves (II)• For frequencies much smaller than the ion cyclotron frequency the dispersion relation for magnetosonic waves can be derived as ω 2 2 vs + v A 2 2 =c 2 k 2 c + vA 2 where vs is the sound speed in the plasma• The magnetosonic wave is an ion-acoustic wave that travels perpendicular to the magnetic field • Compressions and rarefactions are due to the ExB0 drifts aziz_muhd33@yahoo.co.in 117
  • 118. Magnetosonic Waves (III) • In the limit of zero magnetic field the ion-acoustic dispersion relation is recovered• In the limit of zero temperature the sound speed goes to zero and the wave becomes similar to an Alfven wave aziz_muhd33@yahoo.co.in 118
  • 119. APPLICATION OF PLASMA PHYSICS 1. Magnetohydrodynamic Generator 2. Thermonuclear fusion reactor aziz_muhd33@yahoo.co.in 119
  • 120. 1.Magneto hydrodynamic Generator• MHD power generation uses the interaction of an electrically conducting fluid with a magnetic field to convert part of the energy of the fluid directly into electricity• Converts thermal or kinetic energy into electricity aziz_muhd33@yahoo.co.in 120
  • 121. WhereLorentz Force Law: is the force of the acting particle (vector) • F F = QvB V is the velocity of the particle (vector) • • Q is the charge of the particle (scalar) • B is the magnetic field (vector) aziz_muhd33@yahoo.co.in 121
  • 122. Conversion Efficiency• MHD generator alone: 10-20%• Steam plant alone: ≈ 40%• MHD generator coupled with a steam plant: up to 60% aziz_muhd33@yahoo.co.in 122
  • 123. Losses• Heat transfer to walls• Friction• Maintenance of magnetic field aziz_muhd33@yahoo.co.in 123
  • 124. 2. Thermonuclear fusion reactor aziz_muhd33@yahoo.co.in 124
  • 125. aziz_muhd33@yahoo.co.in 125
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  • 130. Advantages of Fusion • Inexhaustible Supply of Fuel • Relatively Safe and Clean • Possibility of Direct Conversion aziz_muhd33@yahoo.co.in 130
  • 131. aziz_muhd33@yahoo.co.in 131
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  • 133. Requirements for Fusion • High Temperatures • Adequate Densities • Adequate Confinement • Lawson Criterion: nτ > 10 20 s/m3 aziz_muhd33@yahoo.co.in 133
  • 134. aziz_muhd33@yahoo.co.in 134
  • 135. Two Approaches • Inertial Confinement: – n ≈ 1030 / m3 τ ≈ 10-10 s • Magnetic Confinement: – n ≈ 1020 / m3 τ≈1s aziz_muhd33@yahoo.co.in 135
  • 136. Magnetic Confinement• Magnetic Field Limit: B < 5 T• Pressure Balance: nkT ≈ 0.1B2/2µ0• ==> n ≈ 1020 / m3 @ T = 108 K• Atmospheric density is 2 x 1025 / m3• Good vacuum is required• Pressure: nkT ≈ 1 atmosphere• Confinement: τ ≈ 1 s• A 10 keV electron travels 30,000 miles in 1 s aziz_muhd33@yahoo.co.in 136
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