DEPOLARIZATION EFFECTS IN THERMONUCLEAR PLASMAS

Transcription

1 DEPOLARIZATION EFFECTS IN THERMONUCLEAR PLASMAS R. Gatto Dept. Energy Engineering Sapienza University of Rome INFN Ferrara Meeting - 23 July 2015

2 Outline In this talk I will review the mechanisms that can lead to depolarization of a polarized D-T plasma Properties of polarized D-T fuel Kulsrud scenario Depolarization problem Depolarization mechanisms: Static inhomogeneous magnetic field Binary collisions Spin-orbit interaction of T and e Spin-orbit interaction of D with e elastic nuclear scattering Magnetic fluctuations Collective mode driven by alpha particles

3 Outline Moreover, I will present few considerations on p 11 B plasma

4 References Kulsrud, Furth, Valeo, Goldhaber, Fusion Reactor with Polarized Nuclei, PRL Kulsrud, Valeo, Cowley, Physics of Spin-Polarized Plasmas, NF Coppi, Pegoraro, Ramos, Instability of Fusing Plasmas and Spin-Depolarization Processes, PRL 51, 1983 Coppi, Cowley, Detragiache, Kulsrud, Pegoraro, Collective Effects in Spin Polarized Plasmas, CPPCF 9, 1985 Coppi, Cowley, Kulsrud, Detragiache, Pegoraro, High-energy components and collective modes in thermonuclear plasmas, PoF 29, 1986

5 Properties of polarized D-T fuel Nuclear reaction rates in a magnetically confined plasma can be modified by taking advantage of their dependence on the nuclear spin of the reacting particles, i.e. on the orientation of the spin relative to the local equilibrium magnetic field

6 Properties of polarized D-T fuel D +T ( 5 He) 4 He +n D has s = 1 and m s = 1,0,+1 (antiparallel, transverse, parallel) T has s = 1/2 and m s = 1/2,+1/2 (antiparallel, parallel) D +T = ( 5 He) has J = l,s (l+s) s s = 1/2,3/2 (we ll consider only l = 0 as for low-energy fusion reactions, i.e. with impact parameters small enough)

7 Properties of polarized D-T fuel D +T ( 5 He) 4 He +n When D and T collide and penetrate the Coulomb barrier, their energy is very close to that of an excited state of the compound nucleus 5 He (unstable but with long lifetime) with energy 107 kev above the energy of unbound D and T at zero kinetic energy. This excited state has J = 3/2 + If the colliding D-T system has J = 3/2 + then this ( 5 He) is formed with high probability and decays to 4 He +n+17 MeV Experimentally, the fraction of reactions occurring through the J = 3/2 + excited state is > 0.99

8 Properties of polarized D-T fuel There are 6 way to combine the spins of D and T, and the statistical weight of J = 3/2 is 4/6=2/3 while that of J = 1/2 is 2/6=1/3: thus in an unpolarized plasma only 2/3 of interactions contribute to the reaction rate If D and T are polarized parallel to B and to each other, all interactions contribute to the reaction rate, with an increment equal to (1 2/3)/2/3 = 1/2, i.e. 50%

9 Kulsrud scenario Three different polarization modes: (a) Enhanced mode: both s(d) and s(t) are to B 0 and parallel to each other σ DT up by 50% and α,n s are emitted with dσ/dω sin 2 θ (roughly perpendicular) (b) Unenhanced mode: s(d) to B 0, s(t) unpolarized: σ DT unchanged and α, n s are emitted with dσ/dω (1+3cos 2 θ) (roughly parallel) (c) Suppressed mode: both s(d) and s(t) are to B 0 and antiparallel to each other σ DT down by a factor of two and α,n s are emitted with dσ/dω sin 2 θ (roughly perpendicular)

10 Kulsrud scenario θ B

11 Kulsrud scenario (Step 1) Polarization should be accomplished by first orienting or polarizing the nuclei outside the fusion device where they are in a atomic state (optical pumping, cryogenic methods using molecular formation or employing Boltzmann equilibrium, polarization of energetic neutral beams) (Step 2) Polarized nuclei are then introduced into the reactor as polarized fuel (gas puffing, injection of energetic neutral beams or pellet injection) Key questions: Can atoms be polarized at a rate comparable to that at which plasma ions are recirculated in a fusion device? Can the polarization state of the nuclei be increased to almost 100% Do polarized nuclei in a fusion plasma remain so for a time long enough for them to react? Is polarization economically convenient?

12 Depolarization problem Consider a DT plasma prepared in enhanced mode (a): m D = +1,m T = +1/2 (spins parallel to B 0 = B 0 ẑ and to each other): Relevant frequencies. The gyration (Ω) and the precession (Ω p ) frequencies in a static magnetic field B 0 are: Deuteron: Ω D = eb0 2m and pc Ωp D = E = g DΩ D = 0.86Ω D Ω p D Ω D Energies. Zeeman energy for a change of spin orientation m s = 1: E p ev Average energy of a Maxwellian plasmas: kt ev E p kt it seems that an unpolarized equilibrium would be rapidly established. However, Kulsrud: the mechanisms for depolarization are surprisingly weak

13 Static inhomogeneous magnetic field Consider a static magnetic field with inhomogeneity scale length S B = B 0 /B 0 1, and a nucleus moving with velocity v = ρω (gyration velocity) The nucleus sees the inhomogeneity at a frequency v /S B which needs to be compared with the precession frequency: v S B = ρ D,TΩ D,T S B Ω p D,T or ρ D,T Ω p S B D,T Ω D,T Resonance possible only if ρ D,T /S B 1 which is not the case in a tokamak: ρ D,T /S B 1

14 Binary collisions: Coulomb scattering Simple electrostatic Coulomb scattering does not affect the nuclear spin But the following binary collision processes could: T: Spin-orbit and spin-spin interaction between T-e, T-D, T-T: case T-e dominant because relative velocity is higher D: Spin-orbit and spin-spin interaction between D-e, D-D, D-T D: Quadrupole moment interaction of D-e

15 Binary collisions: Spin-orbit interaction T-e α,β = initial, final polarization status of T: dβ 2 dt = n T σ s o T e v rel Using σ s o T e = (4π/3)g2 T r2 p ln(c/ω e λ) = cm2 [where r p = e 2 /m p c 2, ω e = 4πne 2 /m e, λ = /m e v] n T = /cm3 v rel = v e v T where v = 2E/m we obtain n T σ s o D e v rel [1/sec] much less that fusion-energy-multiplication (1 [1/sec]) and complete-fuel-burnup (0.01 [1/sec]) time scales

16 Binary collisions: Spin-spin interaction of T For spin-spin depolarization σ s s T e = 11 9 πg2 T r2 p cm2 even smaller that spin-orbit interaction

17 Binary collisions: Spin-orbit interaction of D Binary collision cross-section for D are smaller: For m s = 0: σ smaller than for T by (g D /g T ) 2 = For m s = ±1: σ smaller than for T by (g D /g T ) 2 /2 = negligible

18 Binary collisions: Elastic nuclear scattering Contribution from elastic nuclear scattering β per fusion event negligible

19 Magnetic fluctuations An effective mechanism for depolarization would be the presence of magnetic fluctuations above thermal level, that can resonate with the spin precession frequency and thus induce spin-flip transitions H = µ B Such e.m. mode could be driven unstable by the velocity-space anisotropy of energetic (hot) ions (neutral beam injection or fusion products) modes with: ω p Ω h and v h = ω/k Ẽ 0 to avoid e-landau damping and transit-time damping propagation almost B 0 left circular polarized component of B

20 Magnetic fluctuations Circularly polarized em wave Polarization convention for a wave traveling toward us: ~ ~ E y E y Bo=Bo z x Bo x Ion gyration while streaming along B y precession + B 0 gyration x RIGHT POLARIZATION (counter clock wise) LEFT POLARIZATION (clock wise)

21 Magnetic fluctuations z + B J 0 +

22 Magnetic fluctuations A polarized nucleus (e.g. T) streaming in the z direction with velocity v z (as the wave) will tend to be depolarized by those harmonics of the fluctuating magnetic field with frequency ω which are left-circularly polarized with respect to B 0 and with frequency satisfying ω k z v z +nω T = Ω p T n = 0 (direct interaction),±1,±2, where ω k z v z +nω T is the longitudinal (translation-motion) and transverse (gyro-motion) Doppler-shifted fluctuation frequency (fluctuation frequency seen by the particle), Ω p T is the precession frequency, and k z is the z-component of the wave number of the fluctuation

23 Magnetic fluctuations Defining: I ω (Ω p )=magnetic fluctuation intensity frequency spectrum around the precession frequency Ω p fluctuation strength: (δ B) 2 = dω I ω (Ω p ) the change in polarization is dβ 2 dt = ( ) ge 2 ( ) ge 2 I ω (Ω p (δ B) 2 ) 2m p c 2m p c ω where ω is the bandwidth around Ω p over which δ B extends in the frame of the nucleus

24 Magnetic fluctuations In thermal equilibrium plasma fluctuations are very small For a T = 10 4 ev Plank spectrum of em waves we find: dβ 2 dt s 1 a depolarization rate sufficiently large so as to prevent reactor operation [i.e., dβ 2 /dt 1 1/s] would imply in case of either D or T δ B 3( ω/ω p ) 1/2 G

25 Magnetic fluctuations For a highly non-maxwellian plasma velocity distribution, or in the presence of spatial gradients of T and/or n microinstabilities (high frequency, short wavelength) driven by velocity space gradients could depolarize through direct (n = 0) interaction: ω k z v z = Ω p D = 0.86Ω D macroinstabilities (low frequency, long wavelength) driven by spatial gradients of plasma temperature and density could depolarize through higher cyclotron harmonic resonance: ω k z v z +nω D/T = Ω p D/T Examples: transverse Alfvén wave with resonance ω +nω D 0.86Ω D 0; whistler mode with resonance ω +nω T 5.96Ω T 0

26 Magnetic fluctuations Kulsrud et al.: Because of the complexity of the plasma wave spectrum, it is difficult to place a detailed upper limit on anomalous depolarization in a magnetic fusion reactor, but for a moderately close approach to thermal equilibrium (i.e., avoidance of steep gradients, especially in velocity space), the desired degree of quiescence seems likely to be attainable

27 Collective mode driven by α-particles - I PART I: MODE EXCITATION Noting that Ω α = Ω D = Ω p D /0.857 look for a mode with frequency that can resonate with both the D spin precession frequency and the distribution of the α particles α-particle free energy associated to anisotropic emission resonant mode ω k v α = Ω α where v α = 2 ε α /m α and ε α is representative of the slowing down distribution ( MeV) ~ ~ E, B Depolarization resonance: ω k v D nω D = Ω p D ω Ω p D

28 Collective mode driven by α-particles - I (Ê, ˆB) = (Ẽ, B)exp( iωt +ik x +ik z) In a fusion DT plasma, such a mode driven by the anisotropy of the velocity distribution of α-particles exists: is a magnetosonic (compressional Alfvén ) wave modified by the two-ions hybrid resonance, propagating almost in the perpendicular direction s B 0 Ω p D ~ B ~ E, z ω,k x y

29 Collective mode driven by α-particles - I The mode frequency ω and wavevector k = (k = k x,k = k z ) with k /k >> 1 has to satisfy some conditions: v 2 th,e ω k v α = Ω α Ẽ 2 << ω φ 2 << Ω 2 e mode excitation by α gyromotion to avoid parallel and perpendicular Landau damping by electrons

30 Collective mode driven by α-particles - I Procedure: linearization of plasma fluid equations (for D,T and e) in the cold-plasma approximation ñ e = ñ D +ñ T J = e(n D ũ D +n T ũ T n e ũ e ) = c 4π k2 Ã ωñ D,T,e = k ũ D,T,e iωm D,T ũ D,T = e ( iωm e ũ e = eẽ Ẽ+ũ e B c = 0 Ẽ+ũ D,T B c )

31 Collective mode driven by α-particles - I Expansion parameters (d e : electron inertial skin depth): d e c k 2 d e 1 or λ 2 d 2 long wavelength mode e ω pe k nearly perpendicular propagation (B 0 k 1 or λ λ strong; magnetic shear) Ω D ω Ω D 1 or ω Ω D needed resonance

32 Collective mode driven by α-particles - I Expanding in these small parameters we obtain the dispersion relation: ω 2 (ω 2 Ω 2 ) ω 2 Ω 2 H = k 2 v2 A magnetosonic (compressional Alfvén ) wave modified by the two-ion hybrid resonance where (isotopic ratios: a D = n D /n e, a T = n T /n e ; electron inertial skin depth; d e = c/ω pe ): Ω a T Ω T +a D Ω D : weighted cyclotron frequency Ω a T Ω D +a D Ω T : mixed-weighted cutoff cyclotron frequency Ω H (Ω D Ω T (Ω/Ω)) 1/2 : resonance ion hybrid frequency (v A,v A ) (d e Ωe Ω,v A Ω H /Ω): relevant Alfvén velocities

33 Collective mode driven by α-particles - I SPECIAL CASE: a D = a T = 1/2 Ω = Ω = (5/6)Ω D, Ω H = 2/3Ω D Note close proximity of Ω H, Ω and Ω p D

34 Collective mode driven by α-particles - I k v A : MHD wave with ω 2 Ω 2 D,Ω2 T k v A : MHD wave with ω 2 Ω 2 D,Ω2 T A frequency exactly equal to Ω p D can be obtained from the upper branch for finite values of k d i k v A /Ω

35 Collective mode driven by α-particles - I polarization parameter: λ iẽ y /Ẽ x High-frequency range (ω Ω D ): MHD wave ω 2 k 2 v2 A ; elliptical polarization with λ ω/ω

36 Collective mode driven by α-particles - I Low-frequency range (ω Ω D ): polarization varies rapidly from right circular λ = +1 at ω = Ω D, to left circular λ = 1 for ω = Ω, to linear λ = 0 for ω 2 = a D Ω 2 T +a TΩ 2 D, to λ = for ω = Ω H In a realistic tokamak plasma, if a mode with ω Ω is excited, the magnetic field inhomogeneity will make ω equal to Ω p D and Ω H in relatively close regions of space

37 Collective mode driven by α-particles - II PART II: Field polarization Polarization of the fluctuating fields in the plane B 0 : the mode propagates nearly perpendicular (k 2/k2 1) along the x (radial) direction B x B = k B k B, By B = iλ(ω)k B k B where B /B = ñ e /n e and λ(ω) = B y i B x = iẽx Ẽ y = ω[(a D(Ω 2 T ω2 )+a T (Ω 2 D ω2 )] a D Ω D (Ω 2 T ω2 )+a T Ω T (Ω 2 D ω2 )

38 Collective mode driven by α-particles - II From λ(ω) = B y i B x = iẽx Ẽ y = ω[a D(Ω 2 T ω2 )+a T (Ω 2 D ω2 )] a D Ω D (Ω 2 T ω2 )+a T Ω T (Ω 2 D ω2 ) we see that (a D = a T = 1/2) λ(ω D ) = λ(ω T ) = 1 right-circular polarization λ(ω H ) = linear polarization λ(ω) = 1 left-circular polarization at the cutoff frequency λ(ω p D ) 0.2, and λ(ω) varies rapidly due to the pole at Ω H

39 Collective mode driven by α-particles - IV PART III: Transition probability per precession period The resonant interaction of the mode with the deuteron precession ω k v nω D = Ω p D where the frequency seen by the deuteron is obtained subtraction the longitudinal and transverse Doppler shift. Using λ ρ D ( B seen as uniform) and ω/k v th,d we relevant resonance reduces to ω Ω p D

40 Collective mode driven by α-particles - IV Transition probability per precession period between spin states m > and m±1 >: w = ( ) 2π < m±1 µ B m > 2 δ(ω p D ω) Ω p D where the interaction Hamiltonian Ĥ = µ ˆB = µ Be iωt and the deuteron magneton µ = g D (e/(2m p c)s = (Ω p D /B)s with s the spin operator

41 Collective mode driven by α-particles - IV Using the spin operator decomposition in terms of the raising and lowering spin operators s ± = s x ±is y : and their matrix elements s = s z b+ 1 2 s (ˆx+iŷ)+ 1 2 s +(ˆx iŷ) < m s + m 1 >=< m 1 s m >= [(s +m)(s m+1)] 1/2 and B y = i B x λ(ω) = i B λ(ω) and considering a deuteron s = 1 and only the transition m = +1 m = 0 we obtain w = 2π 2 Ω p D B B 2 [1 λ(ω p D )]2 1+λ(Ω p D )2 δ(ωp D ω)

42 Collective mode driven by α-particles - IV To get a numerical estimate, consider two spectrum of the excited fluctuations: (i) narrow spectrum of frequency width ω around the spin resonance, so that the density of states is ρ(ω p D ) 1 ω (ii) counting the number of standing waves in a 2-D cavity B with cross-sectional area πd 2 i: ρ(ω p D ) 1 2 d2 iω p dk 2(Ωp D ) D dω 2

43 Collective mode driven by α-particles - IV For a ratio k /k 0.01 and normalizing the fluctuation level to B/B ñ e /n e 10 2, the two spectrum approximations leads to the transition probability per unit time ν dep = wω p D /2π equal to ν dep = B 2( ) B a B s 1 G where { Ω p a D /2 ω case (i) 1 case (ii) For B 1 G and B G: ν dep 0.4 [1/s] [case (ii)] the rate of spin depolarization can be significantly faster than that of fusion reaction

44 Collective mode driven by α-particles - IV PART IV: Mode growth rate The resonant interaction between the waves and the α population, and the damping attributable to the bulk ions and electrons, give an imaginary to the frequency: ω = ω 0 +iγ with γ ω 0. The growth rate is: γ = R 4πω2 (1+λ 2 ) k 2 c 2 ǫ δσ ǫ ω[ K(ω)/ ω] where ǫ = (1,iλ)/(1+λ 2 ) 1/2 is the dielectric function, δσ is the resonant correction to the conductivity tensor, and ω K ω = 2ω2 1+Ω 2 H (Ω2 Ω 2 H k 2v2 A (ω 2 Ω 2 > 0 H )2

45 Collective mode driven by α-particles - IV Electron contribution δσ e : important when parallel phase velocity ω/k is not sufficiently larger than the electron thermal velocity v th,e γ π1/2 e ω = β e ζ e exp( ζe) 2 ω K(ω)/ ω where ζ e = ω/ k v th,e This is the transit time damping due to F = µ B

46 Collective mode driven by α-particles - III Bulk ion contribution δσ D +δσ T : harmonics of the cyclotron resonance of the mode with the ions. Assuming a Maxwellian distribution γ p i ω = n i n e π 1/2 ( bi 2 Ω i Ω ω K(ω)/ ω ) bp 1 1 (p 1)! k 2 v2 A ( ) ω exp b i v2 R k v th,i vth,i 2 ( ) ] λ 2 1 +p +1 [ pλ where v R = (ω pω i )/k and b i = (k v th,i /Ω i ) 2 /2 1

47 Collective mode driven by α-particles - IV Alpha particle contribution: since n α /n e is small, only the resonant part of δσ α need be considered. Angular distribution is sin 2 θ = v 2 /v2 f α = (v 2 /v2 ) fα(v) 0 (v 2 1 /v2 ) v 3 +vc 3 where f 0 α is isotropic, in the range m α v 2 c/2 0.6 MeV ε = m α v 2 α/2 3.5 MeV

48 Collective mode driven by α-particles - IV Alpha particle contribution: due to smallness of n α /n e, only resonant contribution is important γ α = n α ω 2 Ω α Ω n e ω K(ω)/ ω k 2 π v2 A + ( ) 2 pωα d 2 vδ(ωk v pω α ω cd,α ) p= 2ǫ α n α k v α ( λj p (ξ α ) ξ α p J p(ξ α ) ) 2 Πf 0 where Π = (2/m α )[ / v 2 (ω α/ω) / v 2 ], and ξ α = k v /Ω α ; ω cd,α the curvature drift frequency; ω α the drift frequency due to the density gradient α

49 Collective mode driven by α-particles - IV Alpha particle contribution: γ α = n α n e (positive definite terms) 2 m α [ v 2 ( ω α ω ) γ α proportional to n α /n e (small) and velocity-space and density gradients v 2 ] f 0 α(v,v ) Numerical estimate: taking B G, n e /cm, ω Ω p D, and a D = a T = 1/2 and k /k 7, an unstable mode with γ 10 2n α Ω α n e

50 Collective mode driven by α-particles - IV Alpha particle contribution: Comments: This growth rate is a relatively small quantity, and moreover must be derived for more realistic toroidal geometries must verify if there are damping mechanisms: energy convection damping processes on the bulk plasma species (ion-cyclotron damping) and electrons (transit time damping)

51 Collective mode driven by α-particles - V PART V: Conclusions of Coppi s work Modes resonating with D (low frequency, ω Ω p D = 0.86Ω D) are unlikely to be excited, due to stabilizing effect of electron transit time damping the D polarization is not affected Modes resonating with T (high frequency, ω Ω p T 6Ω D) could be excited only when the α particle density gradient is included. The depolarization rate is ν Tdep π 2 Ωp T ( k k ) 2 ( B B 0 ) 2(Ω p ) T 1 ω s

52 Collective mode driven by α-particles - V Introducing the efficiency of the energy transfer from the α particles to the mode η n α ǫ α /( B /8π) and for realistic parameters: where τ diff 5τ E ν Tdep τ diff 1 for η 10 4 only a small efficiency η is sufficient for yielding a depolarization time as short as the diffusion time. Since in practice we must consider many recycling times, the T depolarization is significant

53 p B 11 plasma The p B 11 is a dream reaction because it does not produce neutrons: p+b 11 ( 12 C) 4 He+( 8 Be) 4 He+2 4 He MeV From Caughlan and Fowler (1988) we read the reaction rate for the p-b11 fusion reaction: N A σv pb = exp[ /T] exp[ /T] T 3/2 + T 3/ exp[ /T] exp[ /T 1/ T 2 ] T T 2/3 ( T 1/ T 2/ T T 4/ T 5/3) where N A is the Avogadro s number, the temperature is expressed in kev, and σv pb is in cm 3 /s.

54 p B 11 plasma Reactivity vs T: sigma velocity pb11 cm^3 s REACTIVITY for pb Ti kev The maximum value of cm3/s is reached for T 547 kev, but at 200 kev its value is already very close, σv pb 11(T = 200 kev) =

55 p B 11 plasma Ideal ignition: P br and fusion power P Th versus T [Ref: Marco Cavallone, thesis, Sapienza University of Rome, 2014] Pbremss and Palpha for pb11 Pbr and Palpha for pb11 W m Ti kev At about T 200 kev, the separation between the two powers is the smallest

56 p B 11 plasma Cross-section enhancement I heard that the formula for the reaction rate has been recently revisited, and it looks more favorable Appropriate polarization of the spins of p and B 11 could enhance the reaction rate up to 60% Possible additional enhancements due to the prolate, non-spherical shape of the B 11 nucleus

57 Appendix I: DT total cross section Denoting with d +,d 0,d the fraction of D-nuclei polarized parallel, transverse and antiparallel to B 0, and with t +,t the fraction of T-nuclei polarized parallel or antiparallel to B 0, the total DT fusion cross-section is σ DT = (a+ 23 b + 13 c ) fσ 0 + ( 2 3 b + 4 ) 3 c (1 f)σ 0 where a = d + t + +d t, b = D 0, c = d + t +d t + and where fσ 0 the cross section for the 3/2 + ( 5 He) state (f ) unpolarized plasma: a = b = c = 1/3 so that σ DT = (2/3)σ 0 all nuclei with parallel polarization: a = 1, b = c = 0, and so σ DT = fσ 0 enhancement of 3/2f

58 Appendix II: DT direction of emission distribution Conservation of parity and of angular momentum require n and α produced by the decay of 5 He to be in a d state of orbital angular momentum anisotropic distribution in velocity space If θ is the pitch angle relative to B 0 and dω is the incremental solid angle, the angular distributions of the n and α are: dσ dω = fσ [ ( π 4 asin2 θ + 3 b + 1 )( (4/f) 3+3cos 2 )] 3 c θ 4 if all nuclei are polarized parallel to B 0, dσ/dω sin 2 θ perpendicular emission if the D nuclei are polarized transverse to B 0, dσ/dω (4/f) 3+3cos 2 θ parallel emission

59 Appendix III: DT neutron polarization Denoting with n + and n the fractions of neutrons polarized parallel and antiparallel to B 0 and setting f = 1, the neutron polarization at θ = 90 is: n + n = (3/4)(d t d + t + )+(1/6)d 0 (t + t )+(1/12)(d + t d t + ) (3/4)a +(1/6)b +(1/12)c roughly independent of energy within the range of fusion interest