Lines from a Crookes-Tube

Home , Crookes tube, Degree of ionization

Lecture 6 X-ray sources based on Crookes tubes and Crookes-like capillary plasma tubes

Outline • Plasma as 4th state of matter. Plasma of charges. Kinds of plasmas. Crookes X-tube in the frame of Townsend model. K,L, M… lines from a Crookes-tube. K,L… lines and bremsstrahlung in 1st regime. K,L, … lines and bremsstrahlung in 2nd regime. • Electron binding energies [eV] for atoms in their natural forms (relevant to K, L,M,… lines ). A ltlot of x-ray emiiission lines from multip le-chdharged ions (MCIs) . • Crookes tube vs capillary Crookes-like discharge: Generation of high-intensity x-rays.

• Ionization energies Eionization. [eV] of atoms and multiple charged ions (MCI). X-ray spectra using balance equations for the ion stages and energy levels of uniform plasma. • Three plasma-physics models of Crooke plasma and capillary Crookes-like plasma. Microscopic model by using the Dirac delta distribution function of a point-like particle. Microscopic model like a kinetic (Klimontovich) model of Crookes and Crookes-like plasmas. • Kinetic model by using Vlasov equation Kinetic model by using Vlasov and Maxwell equations for Crookes or Crookes-like plasma.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 1 Outline ctd.

• MHD for capillary Crookes-like plasma with 2 temperatures (Te ,Ti) and 2 fluids (ne,Ve , ni,Vi).MHDfor Crookes or Crookes-like plasma. Continuity equation for the conservation of mass or particles in MHD of Crookes or Crookes-like plasma. The force equation of conservation of momentum in MHD of Crookes or Crookes-like plasma. The equation of conservation of energy in MHD of Crookes or CkCrookes-likeplasma. BlBalance equations for the energy lllevelsand ion stages for Crookes or Crookes-like plasma. • Comparison of the x-ray (XUV) radiation from a Crookes tube with a Crookes- like capillary discharge. X-ray (XUV) resonant line emission from capillary Crookes-like Oxygen discharge. Intensity of x-ray (XUV) vs Oxygen pressure of • Capillary Crookes-like discharge. X-ray (XUV) resonant line emission from the capillary Crookes-like Xenon discharge. • The use of Van Cittert-Zernike theorem for x-rays generated by Crookes-like plasma. • Understanding capillary Crookes-like plasma sources requires theory, computations and experiments. • Problems as home assignments • References

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 projekt 2 Plasma as 4th state of matter

T(K)T (K) T(T (K) Æ T(T (eV) = kBT 11600 K ~ 1eV T ? Plasma of electrical charges c Degree of ionization Tc x = N /N/ N Gas ch 0 Liquid x = 0 – gas 0 < x < 1 – non-perfect plasma Solid-state material x = 1 – perfect plasma (plasma) 0 x ? Fig. 1 Plasma as 4th stttate of matt er 1. For th e l oca l thermo-dildynamical equilibrium (LTE) with the Boltzmann statistics N = N A e-(Em/kBT) T ~ 0.25 eV m 0 c Saha equation 2 2 2 3/2 -(Ui/k T) x /(1-x )=2A(2πm/h ) [kBT/P]e B Æ Tc

2F2. For a general case x by: balance equation of ionization and recombination

Æ Tc

Fig. 2 Degree of ionization x of hydrogen plasma vs temperature T (Figure 2 from Abonyi Iván, A negyedik halmazállapot. Bevezető a plazmafizikába, Gondolat kiadó, Budapest (1971)) 3 TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project Kinds of plasmas

Basic Iono- plasma Space sphere between parameters: starss Flame Alkali vapor High- Te [K] plasmas pressure arcs kBTe [eV] Low- Glow pressure n [cm-3] disch. arcs e Space Shock between tubes ωp [rad/sec] planets High- pressure Fusion Theta ωp = arcs 2 experiments pinch (4πne e / LIPs, m)1/2 Cap illary

plasmas, λD = Z-pinches (kBT/ 2 1/2 Fusion 4πne e /m) reactors

ND = 3 (4π/3)ne λD

Fig. 3 Plasmas of charges. Kinds of plasmas (Figure from Abonyi Iván, A negyedik halmazállapot. Bevezető a plazmafizikába, Gondolat kiadó, Budapest (1971)).4 TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project Crookes X-tube in the frame of Townsend model Let us consider Crookes tubes in the context of transition from incoherent to coherent x-ray sources

X-rays from Crookes tubes: - Glow discharge at Electron beam P ~ 7×10−4 -4×10−5 Torr, U > 5kV Cathode air + AtAston NtiNegative Faraday Positive Anode Anode Anode dark glow space column glow dark space space - + C A HV power Supply X-ray Cathode glow Cathode dark space Crookes x-ray tube (electron beam = cathode Fig. 4 Fig. 5 X-rays from a Crookes tube (glow discharge) beam) Quantitative description of Crookes X-ray tube in Townsend model of ionization 1. Townsend model of glow plasma. Paschen Curve: Ionization and plasma current: α =(1/= (1/λ)exp [ -Ui / eEλ) Ub=Apd / (ln(Pd)+B) Experiment: αd U – ionization potential I/I0=e i e- electron charge Townsend equation U I /I = eαd /[1 - γ(eαd-1)] E – eltifildlectric field b 0 <λ> – mean free path α- 1-st Townsend avalanche ionization coefficient by anion (in volume) γ- Townsend 2-nd ionization coefficient by ion (on the cathode surface) Pd d- distance between electrodes, P - gas pressure

I0 – initial current TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 5 K,L, M… lines from a Crookes-tube

Eb is the electron binding energy

Continuum) Shells: K, L, M, N, … Townsend model of glow plasma Einf (n=inf.) Electron avalanche Æ E 4 N (n=4) self-sustained avalanche E3 M (n=3) (glow discharge) Lα Lβ + E L (n=2) b E2 Emission models of X-rays (waves): Photon Kα 1. E (r, t) ~ Σi ei aTi (t - ri/c + ϕi) / ri Kα Kβ Kγ 2. Atomic (quantum) physics

E1 K (n=1) Fig. 6 Ionization of an atom byyq the avalanche electron and subsequent radiation of K-line photon

1st regime: The mean free path (λ) in a Crookes tube The mean free path in a Crookes tube is given by <λ> >d, where d is the distance between the λ~ 1/p ~ n-1/3 electrodes. Then, Crookes tube Æ vacuum tube. 2 if mv /2 ~ eΔϕ > Eb. In such a case, the Crookes tube Rutherford model gives 2 2 4 generates the K,L,M photons with energies hωK,L,M,.. λ = 9k BT / πne 2nd regime:. The mean free path (λ) in a Crookes v tube is given by <λ> Eb. In such a case, the Crookes tube generates the K,L,M photons with energies hωK,L,M, +eZ

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 6 K,L… lines and bremsstrahlung in 1st regime I (arb. u.) Eb is the electron binding energy 2 Kα Continuum) Shells: K, L, M, N, … Einf (n=inf.) E 4 N (n=4) Δλ< 0.01 nm E3 M (n=3) 1

Lα Lβ L (n=2) λmin~2πh/eΔϕ Eb E 2 Photon K α bremsstrahlung Kα Kβ Kγ 0 24 6810 E1 K (n=1) λ (nm) Fig. 7 Ionization of an anode atom by the avalanche electron and subsequent radiation of a Kα-line photon 1st regime (characteristic lines by anode atoms): 2 2 4 Since <λ> >d and <λ>Rutherford =9k BT / πne ,

4 2 1/2 then T > (d πne /9k B) (1) and eΔϕ > Eb (2)

BthlBremsstrahlung st 1 regime v by an anode atom (bremsstrahlung E (r, t) ~ e a (t - r / c) / r -e T by the anode atoms): +eZ

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 7 K,L, … lines and bremsstrahlung in 2nd regime I (arb. u.) Eb is the electron binding energy 2 Kα Bremsstrahlung Continuum) Shells: K, L, M, N, … + recombination Einf (n=inf.) E 4 N (n=4) Δλ< 0.01 nm E3 M (n=3) 1 L L α β λ ~2π /eΔϕ E L (n=2) min h b E2 Photon Kα Kα Kβ Kγ 0 24 6810 E1 K (n=1) λ (nm) Fig. 8 Ionization of a gas atom by the avalanche electron and subsequent radiation of a Kα-line photon 2nd regime (characteristic lines by gas atoms): 2 2 4 Since <λ> Rutherford =9k BT / πne ,

4 2 1/2 then T>(dT > (d πne /9k B) (3) and eΔϕ (λ/d) > Eb (4)

Bremsstrahlung Bremsstrahlung radiation intensity ([eV/cm-3]): st v by an anodtde atom -27 2 1/2 1 regime IB ~1.5x10 Ne Ni Z (Te) Recombination radiation intensity ([eV/cm-3]) (bremsstrahlung -e -27 4 -1/2 I R~5x10 Ne Ni Z (Te) by the gas atoms): +eZ (from L.A. Arcimovics, R.Z. Szaggyejev, Plasmafizika fizikusnak, Akadémiai kiadó. Bpp(. (1985))

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 8 Electron binding energies [eV] for atoms in their natural forms (relevant to KLMK, L,M,… lines ) 1

Tab. 1 Electron binding energies (eV) for the atoms in their natural forms.

The binding energies can be used for quantitative modeling of X-tubes with the Cu anode and Ar gas

Fig. 9 Transmissions that give rise to the characteristic emission lines in atoms of the anode or gas.

(Fig. 9 and Table 1 from http://xdb.lbl.gov) 9 TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project A lot of x-ray emission lines from multiple- charged ions (MCIs) Li Li+1 Li+2 Continuum) (n=inf. ) Continuum) (n=4) (n=inf.) (n=4) (n=3) Continuum) (n= inf. ) (n=3) (n=4) e (n=2) (n=3) e (n=2) Photon e (n=2) Photon hω3 Photon hω2 hω1

(n=1) (n=1) (()n=1) hω3 > hω2 > hω1 +3e +3e +3e Faraday-like screenifing of a positive charge +3e

Fig. 10 Multiple-charged ions (MCIs) radiating a lot of x-ray emission lines. 10 TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project Crookes tube vs Crookes-like capillary disch arge: GtifGeneration of high -itintensit y x-rays

−4 −5 The intensity of x-rays from the Crookes (Pair~ 7×10 -4×10 Torr, U > 5kV) glow-discharge plasma is weak.

The intensity of x-rays from Crookes-like (Pgas ~1 Torr, U > 5kV) powerful capillary discharge is strong.

Capillary X-rays X-rays

A -+K Plasma

Fig. 11 X-rays from Crookes-like (glow-like) powerful capillary discharge.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 11 Ionization energies [eV] of atoms and multiple charged ions (MCIs)

Hydrogen H 1 13.5984 Nickel Ni 28 7.6398 18.16884 35.19 54.9 76.06 Helium He 2 24.5874 54.417760 Copper Cu 29 7.7264 20.2924 36.841 57.38 79.8 Lithium Li 3 5.3917 75.6400 122.45429 Zinc Zn 30 9.3942 17.96439 39.723 59.4 82.6 Beryllium Be 4 9.3227 18.21114 153.89661 217.71865 Gallium Ga 31 5.9993 20.51514 30.71 64 87 Boron B 5 8. 2980 25. 1548 37. 93064 259. 37521 340. 22580 Germanium Ge 32 7.8994 15.93461 34.2241 45.7131 93.5 Carbon C 6 11.2603 24.3833 47.8878 64.4939 392.087 Arsenic As 33 9.7886 18.5892 28.351 50.13 62.63 Nitrogen N 7 14.5341 29.6013 47.44924 77.4735 97.8902 Selenium Se 34 9.7524 21.19 30.8204 42.9450 68.3 Oxygen O 8 13.6181 35.1211 54.9355 77.41353 113.8990 Bromine Br 35 11.8138 21.591 36. 47.3 59.7 Fluorine F 9 17.4228 34.9708 62.7084 87.1398 114.2428 Krypton Kr 36 13.9996 24.35984 36.950 52.5 64.7 Neon Ne 10 21.5646 40.96296 63.45 97.12 126.21 Rubidium Rb 37 4.1771 27.2895 40 52.6 71.0 Sodium Na 11 5.1391 47.2864 71.6200 98.91 138.40 Strontium Sr 38 5.6949 11.0301 42.89 57 71.6 Magnesium Mg 12 7.6462 15.03527 80.1437 109.2655 141.27 Yttrium Y 39 6.2171 12.22 20.52 60.597 77.0 Aluminum Al 13 5.9858 18.82855 28.44765 119.992 153.825 Zirconium Zr 40 6.6339 13.1 22.99 34.34 80.348 Silicon Si 14 8.1517 16.34584 33.49302 45.14181 166.767 Niobium Nb 41 6.7589 14.0 25.04 38.3 50.55 Phosphorus P 15 10.4867 19.7695 30.2027 51.4439 65.0251 Molybdenum Mo 42 7.0924 16.16 27.13 46.4 54.49 Sulfur S 16 10.3600 23.33788 34.79 47.222 72.5945 Technetium Tc 43 7.28 15.26 29.54 46 55 Chlorine C l 17 12.9676 23.8136 39.61 53.4652 67.8 Ruthenium Ru 44 7.3605 16.76 28.47 50 60 Argon Ar 18 15.7596 27.62965 40.74 59.81 75.02 Rhodium Rh 45 7. 4589 18. 08 31. 06 48 65 Potassium K 19 4.3407 31.63 45.806 60.91 82.66 Palladium Pd 46 8.3369 19.43 32.93 53 62 Calcium Ca 20 6.1132 11.87172 50.9131 67.27 84.50 Silver Ag 47 7.5762 21.47746 34.83 56 68 Scandium Sc 21 6.5615 12.79977 24.75666 73.4894 91.65 Cadmium Cd 48 8.9938 16.90831 37.48 59 72 Titanium Ti 22 6.8281 13.5755 27.4917 43.2672 99.30 Indium In 49 5.7864 18.8703 28.03 54.4 77 Vanadium V 23 6.7462 14.618 29.311 46.709 65.2817 Tin Sn 50 7.3439 14.6322 30.50260 40.73502 72.28 Chrom ium C r 24 6. 7665 16. 4857 30. 96 49. 16 69. 46 Antimony Sb 51 8. 6084 16. 63 25. 3 44. 2 56 Manganese Mn 25 7.4340 15.6400 33.668 51.2 72.4 Tellurium Te 52 9.0096 18.6 27.96 37.41 58.75 Iron Fe 26 7.9024 16.1877 30.652 54.8 75.0 Iodine I 53 10.4513 19.1313 33 42 66 Cobalt CO 27 7.8810 17.084 33.50 51.3 79.5 Xenon Xe 54 12.1298 20.9750 32.1230 46 57 Table 2 Ionization energies [eV] of atoms and multiple charged ions (from http://dept.astro.lsa.umich.edu/~cowley/ionen.htm) Ionization ”bottlenecks” limit the degree of ionization of atoms and MCIs in a Crookes plasma if

2 p /2m ~ eΔϕ (λ/d) < Eionization (5)

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 12 X-ray spectra by using balance equations for ion stages and energy levels of uniform plasma Balance equations for the ion stages assuming the known

values of Te (t),Ti(t), ne(t) and ni(T) dn Z ion = n ⋅{n Z −1I Z −1 + n Z +1(R Z +1 + R Z +1 + R Z +1 )− n Z I Z − n Z (R Z + R Z + R Z )} (6) dt e ion ion rr dr cr ion ion rr dr cr BlBalance equa tions f or th e energy l evel s assumithing the known τ values of Te (t),Ti(t), ne(t) and ni(T) d(nZ ) ⎡ ⎤ ion j = −(nZ ) (A + Pd )+ Pe + (nZ ) (A + Pd )+ (nZ ) Pe (7) ion j ⎢∑ βτji ji ∑ ji ⎥ ∑ ion i ji ji ∑ ion i ij dt ⎣ ij<>j i ⎦ ij>

S How can we find Te (r,t),Ti(r,t) ne(r,t) and ni(r,T) ? 13 TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project Three plasma-physics models of a Crooke plasma and a Crookes-like capillary plasma 1st : Microscopic model 2nd: Kinetic model 3rd : MHD model for all particles for the spatio- for the spatio- r temporal evolution temporalr evolutionr i (t), of the distribution of r ne ( ,t),nion ( ,t), v r r d i (t) function r r i (t) = Te ( ,t),Tion ( ,t), dt v r v f (), ,t v (r ,t), v (r ,t) a d (t) q E v B e e i i i i (t) = = [ + × ] + + dt mi Maxwell equations Maxwells equations

st The 1 model is based on the use of the Lorentz force ma=q(E+vxB). Non-collective motion nd The 2 model is based on the use of a concept of the distribution EK >> Ep, v r v r (1/2)k T~ e2/< function determined by (10) B λ>, ∫∫ f ( , ,t)d d = N <λ>~ n-1/3 where N is the particle number, which determines the particle density Collective motion v v E ~ E , as K p f ( )d = n (1/2)k T~ e2/< ∫ (11) B λ>, <λ>~ n-1/3 For the time-independent, homogeny case we have the Maxwell distribution n 2 f (v) = e−mv / 2kBT (12) (2π )3 v3/ 2 TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 14 Microscopic model by using the Dirac delta distribution function of a point -like particle In Electrodynamics, an electron in a point r is described by the point-like charge density δ r r r r ρ = q n( − )δ= qδ ( − ) (13) i v r i n i r r i v v which yields δ (14) f ( , ,t)= ∑ ( − i (t))δ ( − i (t)) i Then the time evolution of the distribution function is determinedδ as ∂ v r n ∂ r r v v ∂ v v r r ⎡ ⎤ ⎡ ⎤ (15) f ( , ,t)= ∑ ⎢ ( − i (t))⎥ ( − i (t))⎢ ( − i (t))⎥δ ( − i (t))= ∂t i ⎣∂t ⎦ ⎣∂t ⎦ r v n ⎡− ∂ ∂ r r ⎤ v v ⎡− ∂ ∂ v v ⎤ r r i r i v (16) = ∑ ⎢ δ ( − i (t))⎥δ ( − i (t))⎢ δ ( − i (t))⎥δ ( − i (t))= i ⎣ ∂t ∂ ⎦ ⎣ ∂t ∂ ⎦ n ⎡ v ∂ r r ⎤ v v ⎡ a ∂ v v ⎤ r r r v (17) = ∑ ⎢− i δ ( − i (t))⎥δ ( − i (t))⎢− i δ ( − i (t))⎥δ ( − i (t)) i ⎣ ∂ ⎦ ⎣ ∂ ⎦ Then we get the microscopic description of plasma by the Klimontovich equation ∂ v ∂ a ∂ f + r f + v f = (18) ∂t ∂ ∂ v E v B ∂ qi (20) = f + ∇f + [ + × ]∇v f = 0 ∂t mi TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 15 Microscopic model as a kinetic (Klimontovich) modldel of CkCrookes and dCk Crookes-like pl asmas It should be stressed that the microscopic description of plasmas and the Klimontovich equation is applicable also to other kinds of plasmas ((ynot only to a Crookes p lasma or a Crookes-like capillary plasma).

The relation between the microscopic description of plasma and the Klimontovich equation is demonstrated by the following equations

v r n r r v v (21) f ( , ,t)= ∑ δ ( − i (t))δ ( − i (t)) i ∂ v q v B ∂ f + ∇f + i []E + i × v f = 0 (22) ∂t mi ∂

d r (t) v (t) = i (()23) i dt d v (t) q a i i E v i B (24) i (t) = = []+ × dt mi

TÁMOP-4.1.1.C-12/1/KONVTÁMOP-4.1.1.C-12/1/KONV-2012-0005-2012-0005 project project 16 Kinetic model using Vlasov equation

Let us assume that the distribution function of the plasma can be presented as a superposition of the slowly varying part and a rapidly fluctuating part: v r v r ~ v r f ( , ,t)= f ( , ,t)+ f ( , ,t) (25) Similarly, we suppose that fields E and B can be presented as the sums of the slowly and rapidly varying components: E v r E v r ~E v r ( , ,t)= ( , ,t)+ ( , ,t) (26) B v r B v r ~B v r ( , ,t)= ( , ,t)(+ , ,t ) (27)

Substituting Eqs. (25)-(27) into the Klimontovich equation (20) and the subsequent spatial averaging yields the collisional Vlasov equation v E v B E v B ∂ qi qi (28) f + ∇f + []+ × ∇v f = − [ + × ]∇v f ∂t mi mi The use Eq. (28) for electron and ions yields the collision-less Vlasov equation

v E v B ∂ qi (29) f + ∇f + []+ × ∇v f = 0 ∂t mi

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 17 Kinetic model using Vlasov and Maxwell equations for Crookes or Crookes-like plasma Thus a kinetic model, which uses Vlasov equation and Maxwell equations for a Crookes plasma or a Crookes-like capillary plasma, v E v B ∂ qi is given by the equation f + ∇f + [ + × ]∇v f = 0 (30) ∂t mi coupled with Maxwell equations. In theB equations, the sums are over all kinds of charged particles. E ∂ ∇× = − (31) ∂Dt H ∂ v v r v (32) ∇× = + q f ( , ,t)d ∑ j ∫ j ∂t j D v r v (33) ∇ = q f (), ,t d ∑ j ∫ j j B (34) ∇ = 0

D E (35) = ε 0 B H (36) = μ0 It should be stressed that the microscopic kinetic model of plasma is applicable also to other kinds of plasmas (not only to a Crookes plasma or a Crookes-like capillary plasma).

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 18 MHD for a Crookes-like capillary plasma with

2 temperatures (Te , Ti) and 2 fluids (ne,Ve , ni,Vi) (37) dne/dt+ div (neVe) = 0 Te , dni/dt+ div (niVi) = 0 (38) Ti, ne,

mene deVeα/dt = dPeα/dxα –ene(Eα +(+ (Ve x B)α +R+ Rα (39) Ve, ni,

mini diViα/dt = dPiα/dxα +Zeni(Eα + (Vi x B)α -Rα (40) Vi

(3/2)ne deTe/dt = - Pe div Ve –div Qe + We (41) For Crookes plasma or Crookes-like (3/2)ni diTi/dt = - Pi div Vi –div Qi + Wi (42) capillary plasma, pliitilasma viscosity is (43) Pe = neTe , Pi = niTi neglected. (For details, see S.I. (44) de/dt = d / dt+Ve (d/dr) Braginskii, Transport processes in a plasma, d /dt = d / dt+V (d/dr) Reviews of Plasma i i (45) Physics, 1, AP, NY (1965)). + Maxwell equations (46)

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 19 MHD for Crookes or Crookes-like plasma It should be stressed that MHD of plasmas is applicable also to other kinds of plasmas (not only to a Crookes plasma or a Crookes-like capillary plasma). The equations of magnetohydrodynamics (MHD) of a Crookes plasma or a Crookes-like capillary plasma are derived by using the equalities

⎡ ∂ V q E V B ⎤ V 1⋅ f + ∇f + i []+ × ∇ f d = 0 ∫ ⎢ v ⎥ ⎣∂t mi ⎦ (47)

V ⎡ ∂ V q E V B ⎤ V ⋅ f + ∇f + i []+ × ∇ f d = 0 ∫ ⎢ v ⎥ (48) ⎣∂t mi ⎦ V2 m ⎡ ∂ V q V B ⎤ V ⋅ f + ∇f + i []E + × ∇ f d = 0 ∫ ⎢ v ⎥ (49) 2 ⎣∂t mi ⎦

v V obtained from the Vlasov equation (29) with the notation ≡ . The equation (47) y ie lds the continu ity equati ons for el ect rons (37) and i ons (38) . Th e equa tion (48) yields the conservation momentum equations for electrons (39) and ions (40). The conservation energy equations for electrons (41) and (42) are obtained from Eq. (49).

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 20 Continuity equation for the conservation of mass or particles in MHD of Crookes or Crookes- like plasma The calculation of Eq. (47) yields

⎛ ⎡ ∂ ⎤ V⎞ V V ⎧ ⎡ q E V B ⎤⎫ ⎜ f d ⎟ + [][]∇f d + i []+ × ∇ f = 0 (50) ⎜ ∫ ⎢ ⎥ ⎟ ∫ ⎨∫ ⎢ v ⎥⎬ ⎝ ⎣∂t ⎦ ⎠ ⎩ ⎣mi ⎦⎭

⎛ ∂n ⎞ V ⎜ ⎟ + []div(n⋅ ) +{}0 = 0 (()51) ⎝ ∂t ⎠ ∂n V + div(n⋅ ) = 0 ∂t (52)

The continuity equation (52) for the electron and ion fluids yields respectively (53) dne/dt+ div (neVe) = 0

dni/dt+ div (niVi) = 0 (54)

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 21 The force equation for conservation of momentum iMHDfin MHD of a CkCrookes or C Ckrookes- like plasma For a Crookes-like, viscosity-less capillary plasma, the calculation of Eq. (48) yields

⎛ V⎡ ∂ ⎤ V⎞ V V V ⎧ V⎡ q E V B ⎤⎫ ⎜ m f d ⎟ + [ m [ ∇f ]d ]+ m i [ + × ]∇ f = 0 ⎜ ∫ ⎢ ⎥ ⎟ ∫ ⎨∫ ⎢ v ⎥⎬ (55) ⎝ ⎣∂t ⎦ ⎠ ⎩ ⎣mi ⎦⎭

mn dVα / dt = dPα / dxα –en (Eα + (V x B)α + Rα (56)

The force equation of conservation of momentum (55) for the electron and ion fluids yields respectively

mene deVeα/dt = dPeα/dxα –ene(Eα + (Ve x B)α + Rα (57)

mini diViα/dt = dPiα/dxα –Zeni(Eα + (Vi x B)α -Rα (58)

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 22 The equation of conservation of energy in MHD of Crookes or Crookes-like plasma

For a Crookes-like, viscosity-less plasma, the calculation of Eq. (49) yields V V V ⎛ m 2 ⎡ ∂ ⎤ V⎞ ⎡ m 2 V V⎤ ⎧ m 2 ⎡ q E V B ⎤⎫ ⎜ f d ⎟ + []∇f d + i []+ × ∇ f = 0 (59) ⎜ ∫ ⎢ ⎥ ⎟ ⎢∫ ⎥ ⎨∫ ⎢ v ⎥⎬ ⎝ 2 ⎣∂t ⎦ ⎠ ⎣ 2 ⎦ ⎩ 2 ⎣mi ⎦⎭

(3/2)n dT/dt = - P div V –div Q + W (60)

The equation of conservation of energy (60) for the electron and ion fluids yields respectively

(61) (3/2)ne deTe/dt = - Pe div Ve –div Qe + We

(()3/2)ni diTi/dt = - Pi div Vi – div Qi + Wi (()62)

It should be stressed again that the MHD equations (37)-(46) of plasmas are applicable also to other kinds of plasmas (not only to Crookes or Crookes -like capillary plasmas) .

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 23 Balance equations for energy levels and ion stages for Crookes or Crookes-like plasma Balance equations for the ion stages

(with Te(r,t), Ti(r,t), ne(r,t), Ve(r,t), n(r,t)i and Vi(r,t) calculated by MHD) dn Z ion = n ⋅{n Z −1I Z −1 + n Z +1(R Z +1 + R Z +1 + R Z +1 )− n Z I Z − n Z (R Z + R Z + R Z )}(63) dt e ion ion rr dr cr ion ion rr dr cr

BlBalance equa tions f or th e energy l evel s τ (with Te(r,t), Ti(r,t), ne(r,t), Ve(r,t), n(r,t)i and Vi(r,t) calculated by MHD) d(nZ ) ⎡ ⎤ ion j = −(nZ ) (A + Pd )+ Pe + (nZ ) (A + Pd )+ (nZ ) Pe (42) ion j ⎢∑ ji βτ ji ∑ ji ⎥ ∑ ion i ji ji ∑ ion i ij dt ⎣ ij<>j i ⎦ ij>

24 TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project Comparison of the x-ray (XUV) radiation from Crookes tube with Crookes-like capillary discharge

CktbCrookes tube: It was f ound ithin the pas t that th e eltlectron b eam produces low-intensity x-rays that make foggy marks on nearby unexposed photographic plates.

Crookes-like capillary discharge: High-intensity (~10 W at ~ 5kH)5 kHz) o f x-rays (XUV) is eas ily ac hieve d. The XUV nano-photolithograpy requires intensities more than about 50 W.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 25 The x-ray (XUV) resonant line emission from the Crookes-like Oxygen capillary discharge

Fig. 12 X-ray (XUV) resonant line emission from the Crookes-like Oxygen capillary discharge (from http://www.google.com/patents/US6031241) TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 26 Intensity of x-ray (XUV) radiation vs Oxygen pressure of Crookes-like capillary discharge

Fig. 13 Intensity of x-ray (XUV) radiation vs Oxigen presuure of the Crookes-like discharge (Imax=6kA) (from http://www.google.com/patents/US6031241) TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 27 X-ray (XUV) resonant line emission from Crookes-like Xenon capillary discharge

Fig. 14 X-ray (XUV) resonant line emission from the Crookes-like Xenon capillary discharge (from http://www.google.com/patents/US6031241) TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 28 The use of Van Cittert-Zernike theorem for x-rays generatdbCted by Crook es-like pl asma Output aperture of an x-ray source based on a Crookes-like plasma d 2R Fig. 15 Schema for the use of theorem in Θ case of an x-ray source based on a Crookes- like plasma (2-Dimensional. circular, radius R) Z source composed from incoherent, uncorrelated emitters

The Crookes-like plasma produces partially coherent or fully transversally coherent x-rays (waves) in the region.

For a circular (radius R) surface of incoherent 2J1( k RΘ) (uncorrelated) emitters of a Crookes-like-ray source, we have μ = (69) 12 k RΘ

Thus the use of the Van Cittert-Zernike theorem for a Crookes-like x-ray source yields

Transverse coherence

- Incoherent radiation: 2R >> <λ>Z / d (70) - Partially coherent radiation: 2R ~<~ <λ>Z / d (71) - Coherent radiation: 2R << <λ>Z / d (72)

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 29 Understanding Crookes-like capillary plasma sources requithires theory, comput ttidations and experiments

Theory Computations

Experiment For an example: T. Donkó, P. Hartmann, K. Kutasi, On the reliability of low-pressure DC glow discharge modelling, XXVIIthe ICPIG, Eindhoven , the Netherlands, 18-22 July (2005)

Why can the 25-year theoretical and experimental experience of the University of Pecs in capillary plasmas be useful for R&D of x-ray (XUV) sources for photolithography?

Fig. 16 Understanding Crookes-like plasma x-ray sources requires theory, computations and experiments. 30 TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project Problems as home assignments (A)

1. Explain the transition of matter in the 1st state (solid-state) to matter in the 4th state. 2. Describe kinds of plasmas and basic processes in plasmas. 3. Calculate the electron plasma frequency, the Debye distance and Larmor (cyclotron) frequency for a LIP of the electron density 1015 cm-3, kinetic temperature 50 eV and magnetic flux density 2 Tesla. 4. Show the connection of ionization and the Townsend model with the operation of a Crookes X-tube. 5. Describe the two types of emission of x-rays by Crookes-tube glow plasma. 6. Give a qualitative explanation of the continuum radiation by using Maxwell equations and the line emission by using atomic physics. 7. How does ionization ”bottlenecks” limit the number of ionization states in a Crookes Ar-plasma? 8. Compare the microscopic model vs kinetic descriptions of plasma of a Crookes x- ray tube. 9. Show the transition of the Vlasov kinetic equation to the collisionless Maxwell- Vlasov equations for a Crookes glow-plasma. 10. Describe the basic points of the magnetohydrodynamic (MHD, fluid) model of a Crookes glow-plasma. 11. What are the two approaches in the MHD model of a Crookes glow-plasma. 12. Derive the continuity equation for mass (particles) conservation for MHD of a Crookes glow-plasma. 13. Derive the force equation for MHD of a Crookes glow-plasma.

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 31 Problems as home assignments (B)

14. Derive the conservation of energy equation for MHD of a Crookes glow- plasma. 15. Summar ize MHD equa tions for a Croo kes g low-plasma.

16. Give an example of 2 temperature (Te and Ti), 2 fluids (Ne,Ve and Ni,Vi) MHD equations for a Crookes plasma. 17. Present balance equations for the energy levels and ion stages for a CkCrookes g low-plGiltifthtithtilasma. Give explanations of the terms in the equations. 18. Why does understanding Crookes glow-plasma x-ray sources require theory, computations and experiments?

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 32 References

1. Abonyi Iván, A negyedik halmazállapot. Bevezető a plazmafizikába, Gondolat kiadó, Budapest (1971) 2. L.A. Arcimovics, R.Z. Szaggyejev, Plasmafizika fizikusnak, Akadémiai kiadó, Bp. (1985)) 3. S.I. Braginskii, Transport processes in a plasma, Reviews of Plasma Physics, 1. AP. NY (1965). 4. David Attwood, Soft-X-rays and Extreme Ultraviolet Radiation, Cambridge University Press, 2000; David Attwood, Soft-X-rays and Extreme Ultraviolet Radiation (www.coe. berkeley.edu).

For additional information see: 1. R. C. Elton, X-ray lasers, Academic Press, 1990. 2. David Attwood, Soft-X-rays and Extreme Ultraviolet Radiation, Cambridge University Press, 2000; David Attwood, Soft-X-rays and Extreme Ultraviolet Radiation (www.coe. berkeley.edu). 3. J.J. Rocca, Review article. Table-top soft x-ray lasers, Rev. Sci. Instr. 70, 3799 (1999) 4. H. Daido, Review of soft x-ray laser researches and developments, Rep. Prog. Phys. 65, 1513 (2002) 5. A.V. Vinogradov, J.J. Rocca, Repetitively pulsed X-ray laser operating on the 3p-3s transition of the Ne-like argon in a capillary discharge, Kvant. Electron., 33, 7 (2003) 6. S. Suckewer, P. Jaegle, X-Ray laser: past, present, and future, Laser Phys. Lett. 6, 411 (2009).

TÁMOP-4.1.1.C-12/1/KONV-2012-0005 project 33