1392:'s Radiation Laboratory. They quoted each other’s results in their respective papers. The first reference deals with the mean-field part of the interaction by using perturbation theory in electric field amplitude. Within the same approximations, a more elegant derivation of the collisional transport coefficients was provided, by using the Balescu–Lenard equation (see Sec. 8.4 of and Secs. 7.3 and 7.4 of ). The second reference uses the Rutherford picture of two-body collisions. The calculation of the first reference is correct for impact parameters much larger than the interparticle distance, while those of the second one work in the opposite case. Both calculations are extended to the full range of impact parameters by introducing each a single ad hoc cutoff, and not two as in the above simplified mathematical treatment, but the transport coefficients depend only logarithmically thereon; both results agree and yield the above expression for the diffusion constant.
795:
443:, an electron will have many such encounters simultaneously, with various impact parameters (distance to the ion) and directions. The cumulative effect can be described as a diffusion of the perpendicular momentum. The corresponding diffusion constant is found by integrating the squares of the individual changes in momentum. The rate of collisions with impact parameter between
420:. Fast particles are "slippery" and thus dominate many transport processes. The efficiency of velocity-matched interactions is also the reason that fusion products tend to heat the electrons rather than (as would be desirable) the ions. If an electric field is present, the faster electrons feel less drag and become even faster in a "run-away" process.
552:
1372:
of a full
Rutherford deflection. Therefore, the above perturbative theory can also be done by using this full deflection. This makes the calculation correct up to the smallest impact parameters where this full deflection must be used. (ii) The effect of Debye shielding for large impact parameters can
800:
Obviously the integral diverges toward both small and large impact parameters. The divergence at small impact parameters is clearly unphysical since under the assumptions used here, the final perpendicular momentum cannot take on a value higher than the initial momentum. Setting the above estimate
74:
In a plasma, a
Coulomb collision rarely results in a large deflection. The cumulative effect of the many small angle collisions, however, is often larger than the effect of the few large angle collisions that occur, so it is instructive to consider the collision dynamics in the limit of small
57:
where the typical kinetic energy of the particles is too large to produce a significant deviation from the initial trajectories of the colliding particles, and the cumulative effect of many collisions is considered instead. The importance of
Coulomb collisions was first pointed out by
790:{\displaystyle D_{v\perp }=\int \left({\frac {Ze^{2}}{4\pi \epsilon _{0}}}\right)^{2}\,{\frac {1}{v^{2}b^{2}}}\,nv(2\pi b\,{\mathrm {d} }b)=\left({\frac {Ze^{2}}{4\pi \epsilon _{0}}}\right)^{2}\,{\frac {2\pi n}{v}}\,\int {\frac {{\mathrm {d} }b}{b}}}
380:
1163:
962:
1367:
An N-body treatment accounting for all impact parameters can be performed by taking into account a few simple facts. The main two ones are: (i) The above change in perpendicular velocity is the lowest order approximation in
253:
839:
1075:
by the tendency of electrons to cluster in the neighborhood of the ion and other ions to avoid it. The upper cut-off to the impact parameter should thus be approximately equal to the
544:
1534:
498:
289:
1042:
1000:
1229:
126:
1357:
1331:
1085:
1305:.) The limits of the impact parameter integral are not sharp, but are uncertain by factors on the order of unity, leading to theoretical uncertainties on the order of
1249:
418:
1543:
1453:
1199:
281:
152:
1297:
1002:
as an estimate of the cross section for large-angle collisions. Under some conditions there is a more stringent lower limit due to quantum mechanics, namely the
862:
99:
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870:
461:
441:
192:
172:
1485:
Escande DF, Elskens Y, Doveil F (2015) Uniform derivation of
Coulomb collisional transport thanks to Debye shielding. Journal of Plasma Physics 81, 305810101
1251:. It is the factor by which small-angle collisions are more effective than large-angle collisions. The Coulomb logarithm was introduced independently by
1380:). This cancels the above divergence at large impact parameters. The above Coulomb logarithm turns out to be modified by a constant of order unity.
1442:
Chandrasekhar, S. (1943). Dynamical friction. I. General considerations: the coefficient of dynamical friction. Astrophysical
Journal, 97, 255–262.
1503:
Rosenbluth, M. N., MacDonald, W. M. and Judd, D. L. 1957 Fokker-Planck equation for an inverse-square force. Phys. Rev. 107, 1–6.
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1462:
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17:
197:
804:
1388:
In the 1950s, transport due to collisions in non-magnetized plasmas was simultaneously studied by two groups at
503:
1494:
Gasiorowicz, S., Neuman, M. and
Riddell, R. J. Jr. 1956 Dynamics of ionized media. Phys. Rev. 101, 922–934
375:{\displaystyle \Delta m_{\text{e}}v_{\perp }\approx {\frac {Ze^{2}}{4\pi \epsilon _{0}}}\,{\frac {1}{vb}}}
1256:
466:
1562:
1521:
Hazeltine, R. D. and
Waelbroeck, F. L. 2004 The Framework of Plasma Physics. Boulder: Westview Press
1009:
1201:
thus yields the logarithm of the ratio of the upper and lower cut-offs. This number is known as the
1374:
1158:{\displaystyle \lambda _{\text{D}}={\sqrt {\frac {\epsilon _{0}kT_{\text{e}}}{n_{\text{e}}e^{2}}}}}
1072:
970:
1512:
Balescu, R. 1997 Statistical
Dynamics: Matter Out of Equilibrium. London: Imperial College Press.
1208:
104:
1336:
63:
1308:
283:. The product of these expressions divided by the mass is the change in perpendicular velocity:
1234:
1401:
388:
1301:
1362:
957:{\displaystyle b_{0}={\frac {Ze^{2}}{4\pi \epsilon _{0}}}\,{\frac {1}{m_{\text{e}}v^{2}}}}
8:
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1176:
258:
131:
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844:
81:
1262:
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426:
177:
157:
43:
35:
62:
in 1936, who also derived the corresponding kinetic equation which is known as the
54:
1547:
1065:
50:
39:
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1424:
Landau, L.D. (1936). "Kinetic equation for the case of coulomb interaction".
1333:. For this reason it is often justified to simply take the convenient choice
1377:
1076:
1003:
1363:
Mathematical treatment for plasmas accounting for all impact parameters
1252:
59:
47:
255:
at the closest approach and the duration of the encounter is about
1359:. The analysis here yields the scalings and orders of magnitude.
1259:
in 1943. For many plasmas of interest it takes on values between
864:, we find the lower cut-off to the impact parameter to be about
69:
1461:. The Office of Naval Research. pp. 31 ff. Archived from
1373:
be accommodated by using a Debye-shielded
Coulomb potential (
46:, the resulting trajectories of the colliding particles is a
38:
between two charged particles interacting through their own
1299:. (For convenient formulas, see pages 34 and 35 of the
27:
Binary elastic collision between two charged particles
1339:
1311:
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1071:
At large impact parameters, the charge of the ion is
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385:Note that the deflection angle is proportional to
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93:
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423:In passing through a field of ions with density
248:{\displaystyle Ze^{2}/4\pi \epsilon _{0}b^{2}}
834:{\displaystyle \Delta m_{\text{e}}v_{\perp }}
70:Simplified mathematical treatment for plasmas
924:
764:
745:
676:
657:
627:
356:
546:, so the diffusion constant is given by
539:{\displaystyle nv(2\pi b\mathrm {d} b)}
14:
1555:
1423:
78:We can consider an electron of charge
53:. This type of collision is common in
1451:
1419:
1417:
1168:
154:and much larger mass at a distance
128:passing a stationary ion of charge
24:
1390:University of California, Berkeley
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493:{\displaystyle (b+\mathrm {d} b)}
1037:{\displaystyle h/m_{\text{e}}v}
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13:
1:
1541:NRL Plasma Formulary 2013 ed.
1407:
995:{\displaystyle \pi b_{0}^{2}}
194:. The perpendicular force is
1224:{\displaystyle \ln \Lambda }
1205:and is designated by either
121:{\displaystyle m_{\text{e}}}
7:
1395:
1352:{\displaystyle \lambda =10}
10:
1584:
1383:
1326:{\displaystyle 1/\lambda }
1257:Subrahmanyan Chandrasekhar
1244:{\displaystyle \lambda }
413:{\displaystyle 1/v^{2}}
64:Landau kinetic equation
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1535:Effects of Ionization
1402:Rutherford scattering
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1455:NRL Plasma formulary
1426:Phys. Z. Sowjetunion
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1452:Huba, J.D. (2016).
1194:{\displaystyle 1/b}
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276:{\displaystyle b/v}
147:{\displaystyle +Ze}
1546:2016-12-23 at the
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1292:{\displaystyle 15}
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94:{\displaystyle -e}
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44:inverse-square law
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1203:Coulomb logarithm
1169:Coulomb logarithm
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1057:{\displaystyle h}
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456:{\displaystyle b}
436:{\displaystyle n}
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187:{\displaystyle v}
167:{\displaystyle b}
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36:elastic collision
32:Coulomb collision
18:Coulomb logarithm
16:(Redirected from
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1563:Plasma phenomena
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1470:. Retrieved
1463:the original
1454:
1447:
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1387:
1378:Debye length
1369:
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1255:in 1936 and
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1077:Debye length
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73:
34:is a binary
31:
29:
1568:Scattering
1557:Categories
1472:2017-10-19
1432:: 154–164.
1408:References
1253:Lev Landau
60:Lev Landau
48:hyperbolic
1341:λ
1321:λ
1239:λ
1219:Λ
1216:
1108:ϵ
1091:λ
975:π
913:ϵ
909:π
841:equal to
827:⊥
809:Δ
766:∫
753:π
724:ϵ
720:π
671:π
606:ϵ
602:π
573:∫
565:⊥
520:π
345:ϵ
341:π
317:≈
312:⊥
294:Δ
227:ϵ
223:π
101:and mass
86:−
1544:Archived
1396:See also
1073:shielded
1384:History
1064:is the
55:plasmas
1044:where
1466:(PDF)
1459:(PDF)
1279:and
801:for
463:and
1231:or
500:is
1559::
1430:10
1428:.
1416:^
1368:1/
1347:10
1287:15
1213:ln
1079::
1068:.
66:.
30:A
1475:.
1370:b
1344:=
1317:/
1313:1
1267:5
1189:b
1185:/
1181:1
1147:2
1143:e
1137:e
1133:n
1125:e
1121:T
1117:k
1112:0
1100:=
1095:D
1052:h
1032:v
1027:e
1023:m
1018:/
1014:h
988:2
983:0
979:b
947:2
943:v
937:e
933:m
928:1
917:0
906:4
899:2
895:e
891:Z
885:=
880:0
876:b
852:v
849:m
823:v
817:e
813:m
783:b
779:b
774:d
760:v
756:n
750:2
741:2
736:)
728:0
717:4
710:2
706:e
702:Z
696:(
691:=
688:)
685:b
680:d
674:b
668:2
665:(
662:v
659:n
650:2
646:b
640:2
636:v
631:1
623:2
618:)
610:0
599:4
592:2
588:e
584:Z
578:(
570:=
562:v
558:D
534:)
531:b
527:d
523:b
517:2
514:(
511:v
508:n
488:)
485:b
481:d
477:+
474:b
471:(
451:b
431:n
406:2
402:v
397:/
393:1
367:b
364:v
360:1
349:0
338:4
331:2
327:e
323:Z
308:v
302:e
298:m
271:v
267:/
263:b
241:2
237:b
231:0
220:4
216:/
210:2
206:e
202:Z
182:v
162:b
142:e
139:Z
136:+
114:e
110:m
89:e
20:)
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