NRL_FORMULARY_09-1
.pdf
41 |
IONOSPHERIC PARAMETERS23
The following tables give average nighttime values. Where two numbers are entered, the first refers to the lower and the second to the upper portion of the layer.
Quantity |
E Region |
F Region |
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Altitude (km) |
90–160 |
160–500 |
Number density (m−3) |
1.5 × 1010–3.0 × 1010 |
5 × 1010–2 × 1011 |
Height-integrated number |
9 × 1014 |
4.5 × 1015 |
density (m−2) |
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Ion-neutral collision |
2 × 103–102 |
0.5–0.05 |
frequency (sec−1) |
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Ion gyro-/collision |
0.09–2.0 |
4.6 × 102–5.0 × 103 |
frequency ratio κi |
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2.2 × 10−3–2 × 10−4 |
Ion Pederson factor |
0.09–0.5 |
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κi/(1 + κi 2) |
8 × 10−4–0.8 |
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Ion Hall factor |
1.0 |
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κi2/(1 + κi 2) |
1.5 × 104–9.0 × 102 |
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Electron-neutral collision |
80–10 |
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frequency |
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Electron gyro-/collision |
4.1 × 102–6.9 × 103 |
7.8 × 104–6.2 × 105 |
frequency ratio κe |
2.7 × 10−3–1.5 × 10−4 |
10−5–1.5 × 10−6 |
Electron Pedersen factor |
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κe /(1 + κe 2) |
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Electron Hall factor |
1.0 |
1.0 |
κe 2/(1 + κe 2) |
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Mean molecular weight |
28–26 |
22–16 |
Ion gyrofrequency (sec−1) |
180–190 |
230–300 |
Neutral di usion |
30–5 × 103 |
105 |
coe cient (m2 sec−1) |
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The terrestrial magnetic field in the lower ionosphere at equatorial lattitudes is approximately B0 = 0.35×10−4 tesla. The earth’s radius is RE = 6371 km.
42
SOLAR PHYSICS PARAMETERS24
Parameter |
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Symbol |
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Value |
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Total mass |
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M |
1.99 × 1033 |
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g |
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Radius |
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R |
6.96 × 1010 |
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cm |
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Surface gravity |
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g |
2.74 × 104 |
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cm s−2 |
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Escape speed |
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v∞ |
6.18 × 107 |
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cm s−1 |
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Upward mass flux in spicules |
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— |
1.6 × 10−9 |
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g cm−2 s−1 |
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Vertically integrated atmospheric density |
— |
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4.28 |
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g cm−2 |
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Sunspot magnetic field strength |
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Bmax |
2500–3500 |
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G |
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Surface e ective temperature |
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T0 |
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5770 |
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K |
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Radiant power |
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L |
3.83 × 1033 |
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erg s−1 |
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Radiant flux density |
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F |
6.28 × 1010 |
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erg cm−2s−1 |
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Optical depth at 500 nm, measured |
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τ5 |
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0.99 |
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— |
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from photosphere |
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1.50 × 1013 |
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Astronomical unit (radius of earth’s orbit) |
AU |
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cm |
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Solar constant (intensity at 1 AU) |
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f |
1.36 × 106 |
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erg cm−2 s−1 |
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Chromosphere and Corona25 |
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Parameter (Units) |
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Quiet |
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Coronal |
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Active |
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Sun |
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Hole |
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Region |
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Chromospheric radiation losses |
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(erg cm−2 s−1) |
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2 × 106 |
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2 × 106 |
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Low chromosphere |
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> |
107 |
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Middle chromosphere |
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6 |
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6 |
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7 |
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2 × 10 |
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2 × 10 |
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10 |
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Upper chromosphere |
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3 × 105 |
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3 × 105 |
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2 × 106 |
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Total |
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6 |
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6 |
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> |
7 |
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4 × 10 |
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4 × 10 |
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2 × 10 |
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Transition layer pressure (dyne cm−2) |
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0.2 |
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0.07 |
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2 |
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Coronal temperature (K) at 1.1 R |
1.1–1.6 × 106 |
106 |
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2.5 × 106 |
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Coronal energy losses (erg cm−2 s−1) |
2 × 105 |
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6 × 104 |
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Conduction |
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105–107 |
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Radiation |
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< |
105 |
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104 |
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5 × 106 |
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Solar Wind |
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5 × 104 |
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7 × 105 |
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< 105 |
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Total |
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5 |
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5 |
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7 |
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3 × 10 |
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8 × 10 |
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10 |
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Solar wind mass loss (g cm |
−2 −1 |
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< |
−11 |
−10 |
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−11 |
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s |
2 × 10 |
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2 × 10 |
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< 4 × 10 |
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43
THERMONUCLEAR FUSION26
Natural abundance of isotopes:
hydrogen nD /nH = 1.5 × 10−4
helium nHe3 /nHe4 = 1.3 × 10−6 lithium nLi6 /nLi7 = 0.08
Mass ratios: |
me/mD |
= 2.72 |
× 10−4 |
= 1/3670 |
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(me/mD )1/2= 1.65 × 10−2 = 1/60.6 |
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me/mT |
= 1.82 |
× 10−4 |
= 1/5496 |
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(me/mT )1/2 = 1.35 × 10−2 = 1/74.1 |
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Absorbed radiation dose is measured in rads: 1 rad = 102 erg g−1. The curie (abbreviated Ci) is a measure of radioactivity: 1 curie = 3.7×1010 counts sec−1.
Fusion reactions (branching ratios are correct for energies near the cross section peaks; a negative yield means the reaction is endothermic):27
(1a) D + D |
50% |
T(1.01 MeV) + p(3.02 MeV) |
(1b) |
−−−−→ |
He3(0.82 MeV) + n(2.45 MeV) |
−−−−→ |
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50% |
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(2)D + T −−−−→He4(3.5 MeV) + n(14.1 MeV)
(3)D + He3−−−−→He4(3.6 MeV) + p(14.7 MeV)
(4)T + T −−−−→He4 + 2n + 11.3 MeV
(5a) He3 + T |
51% |
He4 |
+ p + n + 12.1 MeV |
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−−−−→ |
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(5b) |
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He4 |
(4.8 MeV) + D(9.5 MeV) |
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−−−−→ |
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43% |
He5 |
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(5c) |
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(2.4 MeV) + p(11.9 MeV) |
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−−−−→ |
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6% |
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(6)p + Li6 −−−−→He4(1.7 MeV) + He3(2.3 MeV)
(7a) p + Li7 |
20% |
2 He4 + 17.3 MeV |
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(7b) |
−−−−→ |
Be7 + n |
− |
1.6 MeV |
−−−−→ |
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80% |
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(8)D + Li6 −−−−→2He4 + 22.4 MeV
(9)p + B11 −−−−→3 He4 + 8.7 MeV
(10)n + Li6 −−−−→He4(2.1 MeV) + T(2.7 MeV)
The total cross section in barns (1 barn = 10−24 cm2) as a function of E, the energy in keV of the incident particle [the first ion on the left side of Eqs.
(1)–(5)], assuming the target ion at rest, can be fitted by28a
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(A |
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−1 |
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σT (E) = |
A5 + |
4 − A3E1/2 |
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A2 |
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E exp(A1E− |
) − 1 |
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44
where the Duane coe cients Aj for the principal fusion reactions are as follows:
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D–D |
D–D |
D–T |
D–He3 |
T–T |
T–He3 |
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(1a) |
(1b) |
(2) |
(3) |
(4) |
(5a–c) |
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A1 |
46.097 |
47.88 |
45.95 |
89.27 |
38.39 |
123.1 |
A2 |
372 |
482 |
50200 |
25900 |
448 |
11250 |
A3 |
4.36 × 10−4 |
3.08 × 10−4 |
1.368 × 10−2 |
3.98 × 10−3 |
1.02 × 10−3 |
0 |
A4 |
1.220 |
1.177 |
1.076 |
1.297 |
2.09 |
0 |
A5 |
0 |
0 |
409 |
647 |
0 |
0 |
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Reaction rates σv (in cm3 sec−1), averaged over Maxwellian distributions:
Temperature |
D–D |
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D–T |
D–He3 |
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T–T |
T–He3 |
(keV) |
(1a + 1b) |
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(2) |
(3) |
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(4) |
(5a–c) |
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1.0 |
1.5 × 10−22 |
5.5 |
× 10−21 |
10−26 |
3.3 |
× 10−22 |
10−28 |
2.0 |
5.4 × 10−21 |
2.6 × 10−19 |
1.4 × 10−23 |
7.1 × 10−21 |
10−25 |
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5.0 |
1.8 × 10−19 |
1.3 × 10−17 |
6.7 × 10−21 |
1.4 × 10−19 |
2.1 × 10−22 |
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10.0 |
1.2 × 10−18 |
1.1 × 10−16 |
2.3 × 10−19 |
7.2 × 10−19 |
1.2 × 10−20 |
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20.0 |
5.2 × 10−18 |
4.2 × 10−16 |
3.8 × 10−18 |
2.5 × 10−18 |
2.6 × 10−19 |
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50.0 |
2.1 × 10−17 |
8.7 × 10−16 |
5.4 × 10−17 |
8.7 × 10−18 |
5.3 × 10−18 |
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100.0 |
4.5 × 10−17 |
8.5 × 10−16 |
1.6 × 10−16 |
1.9 × 10−17 |
2.7 × 10−17 |
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200.0 |
8.8 × 10−17 |
6.3 × 10−16 |
2.4 × 10−16 |
4.2 × 10−17 |
9.2 × 10−17 |
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500.0 |
1.8 × 10−16 |
3.7 × 10−16 |
2.3 × 10−16 |
8.4 × 10−17 |
2.9 × 10−16 |
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1000.0 |
2.2 × 10−16 |
2.7 × 10−16 |
1.8 × 10−16 |
8.0 × 10−17 |
5.2 × 10−16 |
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For low energies (T < 25 keV) the data may be represented by (σv)DD = 2.33 × 10−14T −2/3 exp(−18.76T −1/3) cm3 sec−1;
(σv)DT = 3.68 × 10−12T −2/3 exp(−19.94T −1/3) cm3 sec−1,
where T is measured in keV.
A three-parameter model has also been developed for fusion cross-sections of light nuclei.28b
The power density released in the form of charged particles is
PDD = 3.3 × 10−13nD 2(σv)DD watt cm−3 (including the subsequent D–T reaction);
PDT = 5.6 × 10−13nD nT (σv)DT watt cm−3;
PDHe3 = 2.9 × 10−12nD nHe3 (σv)DHe3 watt cm−3.
45
RELATIVISTIC ELECTRON BEAMS
Here γ = (1 − β2)−1/2 is the relativistic scaling factor; quantities in analytic formulas are expressed in SI or cgs units, as indicated; in numerical formulas, I is in amperes (A), B is in gauss (G), electron linear density N is in cm−1, and temperature, voltage and energy are in MeV; βz = vz /c; k is Boltzmann’s constant.
Relativistic electron gyroradius:
re = mc2 (γ2 − 1)1/2 (cgs) = 1.70 × 103(γ2 − 1)1/2B−1 cm. eB
Relativistic electron energy:
W = mc2γ = 0.511γ MeV.
Bennett pinch condition:
I2 = 2N k(Te + Ti)c2 (cgs) = 3.20 × 10−4N (Te + Ti) A2.
Alfv´en-Lawson limit:
IA = (mc3/e)βz γ (cgs) = (4πmc/µ0e)βz γ (SI) = 1.70 × 104βz γ A.
The ratio of net current to IA is
I = ν .
IA γ
Here ν = N re is the Budker number, where re = e2/mc2 = 2.82 × 10−13 cm is the classical electron radius. Beam electron number density is
nb = 2.08 × 108Jβ−1 cm−3,
where J is the current density in A cm−2. For a uniform beam of radius a (in cm),
nb = 6.63 × 107Ia−2β−1 cm−3,
and
2re = ν .
aγ
46
Child’s law: (non-relativistic) space-charge-limited current density between parallel plates with voltage drop V (in MV) and separation d (in cm) is
J = 2.34 × 103V 3/2d−2 A cm−2.
The saturated parapotential current (magnetically self-limited flow along equipotentials in pinched diodes and transmission lines) is29
Ip = 8.5 × 103Gγ ln γ + (γ2 − 1)1/2 A,
where G is a geometrical factor depending on the diode structure:
w |
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G = |
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2πd |
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G = ln |
R2 |
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−1 |
R1 |
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G = Rc d0
for parallel plane cathode and anode of width w, separation d;
for cylinders of radii R1 (inner) and R2 (outer);
for conical cathode of radius Rc , maximum separation d0 (at r = Rc ) from plane anode.
For β → 0 (γ → 1), both IA and Ip vanish.
The condition for a longitudinal magnetic field Bz to suppress filamentation in a beam of current density J (in A cm−2) is
Bz > 47βz (γJ)1/2 G.
Voltage registered by Rogowski coil of minor cross-sectional area A, n turns, major radius a, inductance L, external resistance R and capacitance C (all in SI):
externally integrated |
V |
= (1/RC)(nAµ0I/2πa); |
self-integrating |
V |
= (R/L)(nAµ0I/2πa) = RI/n. |
X-ray production, target with average atomic number Z (V < 5 MeV):
η ≡ x-ray power/beam power = 7 × 10−4ZV.
X-ray dose at 1 meter generated by an e-beam depositing total charge Q coulombs while V ≥ 0.84Vmax in material with charge state Z:
D = 150Vmax2.8QZ1/2 rads.
47
BEAM INSTABILITIES30
Name |
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Saturation Mechanism |
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Electron- |
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Electron trapping until |
Vd > Vej , j = 1, 2 |
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electron |
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Vej Vd |
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Buneman |
Vd > (M/m) |
1/3 |
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Electron trapping until |
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Ve Vd |
Beam-plasma |
Vb > (np/nb ) |
1/3 ¯ |
Trapping of beam electrons |
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Weak beam- |
Vb < (np/nb ) |
1/3 ¯ |
Quasilinear or nonlinear |
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plasma |
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Beam-plasma |
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Quasilinear or nonlinear |
Ve > Vb |
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Ion acoustic |
Te Ti, Vd Cs |
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ation, nonlinear scattering, |
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or resonance broadening. |
Anisotropic |
Te > 2Tek |
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Isotropization |
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temperature |
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Ion cyclotron |
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Ion heating |
Vd > 20Vi (for |
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Te ≈ Ti) |
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Beam-cyclotron |
Vd > Cs |
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Resonance broadening |
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Modified two- |
Vd < (1 + β)1/2VA, |
Trapping |
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stream (hydro) |
Vd > Cs |
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Ion-ion (equal |
U < 2(1 + β)1/2VA |
Ion trapping |
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beams) |
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Ion-ion (equal |
U < 2Cs |
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Ion trapping |
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For nomenclature, see p. 50.
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Parameters of Most Unstable Mode |
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Electron- |
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plasma |
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e |
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ωe |
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Ion acoustic |
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re |
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Modified two- |
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0.9ΩH |
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stream |
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Ion-ion (equal |
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0.4ΩH |
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U |
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beams) |
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Ion-ion (equal |
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0.4ωi |
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1.2 |
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U |
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beams) |
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For nomenclature, see p. 50. |
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49
In the preceding tables, subscripts e, i, d, b, p stand for “electron,” “ion,” “drift,” “beam,” and “plasma,” respectively. Thermal velocities are denoted by a bar. In addition, the following are used:
m |
electron mass |
re , ri |
gyroradius |
M |
ion mass |
β |
plasma/magnetic energy |
V |
velocity |
|
density ratio |
T |
temperature |
VA |
Alfv´en speed |
ne , ni |
number density |
Ωe , Ωi |
gyrofrequency |
n |
harmonic number |
ΩH |
hybrid gyrofrequency, |
Cs = (Te/M )1/2 |
ion sound speed |
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ΩH 2 = Ωe Ωi |
ωe , ωi |
plasma frequency |
U |
relative drift velocity of |
λD |
Debye length |
|
two ion species |
50