Download 1. In a low temperature plasma device called a magnetron, B is

1. In a low temperature plasma device called a magnetron, B is typically 300 gauss, the potential
difference V is 500 V over 2 mm in the region of interest and the E is perpendicular to the B.
Estimate the drift velocity of the electrons.
2. (a) In a magnetic mirror where the magnetic field is B0, the trajectory makes an angle θ0 with the
magnetic field line. Show that reflection occurs where the magnetic field is B=
(b) Suppose now Bmax is the maximum value of the magnetic field.
Show that if
√
then there is no reflection. In a magnetic
mirror device, this would mean the particle was not trapped - it would be lost. This
angle defines a cone in velocity space - the loss cone
3. A plasma discharge has a pressure of 2 mbar, temperature 400K and is 1% ionized. At what
wavelength is the plasma opaque ?
6
4. Calculate vrms for protons and electrons at 10 K.
19
−3
5. Consider a plasma in a tokamak where ne = ni = 10 m , T = 100 eV and volume of plasma = 1
−3
m . How much would the energy in this plasma raise the temperature of 200 ml of water?
6. Collisions – Explain why electrons and ions do not simply recombine to extinguish the plasma?
Consider collisions between electrons and ions and the fact that energy and momentum must be
conserved.
Hint: The probability of a particular collision occurring is proportional to the product of the
number densities of each of the particles involved in the collision.
7. A hydrogen plasma of suitable density may be 50% ionized at a temperature of 1eV. How could
this be the case when 1eV is much less than the 13.6 eV ionization potential of hydrogen?
8. Normally, self-absorption is observed only from the ground level of atoms in light- emitting
plasmas. I what other cases can self absorption occur?
9.
Why are laboratory plasmas practically electrically neutral?
10. Helium has the ground state 1s2 1S0 . Which of the following excited states have the shortest
lifetime: 1s2p 1P1 or 1s2p 3P1. Why?
11. Make a sketch of a 2-level atom and indicate the relevant coefficients for collision and radiation
processes. Write down the conditions for equilibrium if a: Collisions are ignored, b: if collisions
dominate. Which case corresponds best to local thermal equilibrium?
12. Describe in words the significance of the Debye-radius.
13. What are the S-, L- och J-quantum numbers for the following levels:
1
S0, 2S1/2, 1P1, 3P2, 3F4, 5D1, 1D2, 6F9/2
14. In a neutral hydrogen gas there is a temperature when the population in the n=3 state is 1% of the
population in the n=2 state. Calculate this temperature.
15. A low-temperature plasma has an electron density of 109 cm-3. At what frequencies will the
plasma be opaque to electromagnetic radiation? What happens with the radiation of higher
frequency?
16. In practice all known elements are in the gaseous phase at a temperature of 5000K.
a. What is the corresponding mean energy of free atoms at this temperature?
b. What is the vrms of Hg-atom with this energy?
c. What is the vrms of electrons at the same energy, and what fraction of the speed of light
is this?
17. What is the degree of ionization in
a. a Tokamak plasma
b. a fluorescent tube plasma
c. the solar corona?
18. Mercury has an ionization potential of 10.5 eV. Calculate the fraction of ions at a temperature of
0,5eV and a total pressure of 1 mbar.
19. What is roughly the electron temperature in a fluorescent tube plasma? What is the ion
temperature? Why is this difference maintained over time?
20. Sketch an energy level diagram of a two-level atom including the relevant coefficients for
radiative and collisional processes. Indicate which of the processes dominate in
a. Corona balance
b. Thermodynamic equilibrium
21. The three lowest excited levels in Mercury are 6s6p 3P0, 6s6p 3P1 and 6s6p 3P2 . From which of
these do you expect transitions to the ground state, 6s2 1S0 ?. Why?
Solution to all your problems:
1:
B = 300 gauss = 0.03 T
E= 200/0.002 V/m
Vdrift = (E x B)/B2 ≈ 0,33 x E7 m/s
2.
At start position (“0”): v0perp = vtot * sinθ, B = B0, (v0tot)2 = (v0perp)2 + (v0parallell)2
At reflection: vperp = vtot, , B = Br
½ m v2perp/B is a preserved quantity =>
 (v0perp)2/B0 = (vtot*sinθ)2/B0 = (vperp)2/BR = (vtot)2/BR
 Br = B0 / (sinθ)2
If B never reaches this condition (sinθ too small) then vperp never reaches zero and
there is a remaining non-zero vparallell => the particle is lost from the trap.
3.
p = 2mbar = 2 hP = 200P. pV = nkT, ne = 0.01*p/kT ≈ 3,6 * 1020 m-3
ω2p = (e2 * ne)/(2*ε0*m) => ωp ≈ 760 GHz, λ = c*2*π/ω ≈ 2,5 mm
4.
vrms = SQRT(3RT/M) e: vrms = 6,7*106 m/s p: vrms = 1,6*105 m/s
5.
ions: E= N* 3/2 kT = n*V*3/2 kT = 240J, Etot = 480J, 1cal = 4,19J, m=200g => ∆T = 0,6K
7.
See Spectrophysics 9.6.1 and the discussion therein.
9.
See Spectrophysics 10.1 .
10.
In LS-coupling (i.e. for most light elements), the selection rule ∆s = 0 is valid.
11.
See Spectrophysics section 9.4 .
12.
See Spectrophysics section 10.2 .
13.
1
S0 :
S1/2 :
1
P1 :
3
P2 :
3
F4 :
5
D1 :
1
D2 :
6
F9/2 :
2
14.
S
0
½
0
1
1
2
½
5/2
L
0
0
1
1
3
2
2
3
J
0
1/2
1
2
4
1
2
9/2
n3/n2 = g3/g2 * exp(-∆E/kT)
n3/n2 = 0,01, n = 2: 2s1/2 2p1/2 2p3/2 . Stat w. = 2J+1 gives g2 = 2+2+4 = 8
n = 3: 3s1/2 3p1/2 3p3/2 3d3/2 3d5/2 Stat w. = 2J+1 gives g3 = 2+2+4+4+6 = 18
∆E = 13.6*(1/32 – 1/22) eV ≈ -1.89 eV. Note: E = 0 for the ground state in section 9.5.1
in Spectrophysics! (Theoretical) atomic physicists set E=0 at the ionization limit!
Beware!
kT = ln((n3*g2)/(n2*g3))/∆E => T ≈ 4000 K.
15.
See 3. Above! ν ≈ 0,2 GHz
16.
E: 3/2 kT, vrms = SQRT(3RT/M) = SQRT(3kT/m)
17.
a: 100%, b: 1%, c: 100%
18.
IP = 10,5eV, ntot = 1,25 x 1021 m-3 , Qi/Qa = 1
kT = 2eV:
100% joniserat
kT = 1eV:
10% joniserat
kT = 0.5eV: 3.5% joniserat
19.
Te : 1eV < - > Ti: 40oC . See handout (Waymouth).
20.
See Spectrophysics section 9.4 .
21.
The selection rule ∆J = 0 or ± 1 but 0 -> 0 forbidden is valid also when In LS-coupling
breaks down.