2592:
plasmas as continuous media. An N-body calculation reveals that the bare
Coulomb acceleration of a particle by another one is modified by a contribution mediated by all other particles, a signature of Debye shielding (see section 8 of ). When starting from random particle positions, the typical time-scale for shielding to set in is the time for a thermal particle to cross a Debye length, i.e. the inverse of the plasma frequency. Therefore in a weakly collisional plasma, collisions play an essential role by bringing a cooperative self-organization process: Debye shielding. This shielding is important to get a finite diffusion coefficient in the calculation of Coulomb scattering (
2580:
With repulsion, this number is smaller. Therefore, according to Gauss theorem, the apparent charge of the first electron is smaller than in the absence of repulsion. The larger the sphere radius, the larger is the number of deflected electrons, and the smaller the apparent charge: this is Debye shielding. Since the global deflection of particles includes the contributions of many other ones, the density of the electrons does not change, at variance with the shielding at work next to a
1851:
1410:
1627:
2579:
For a weakly collisional plasma, Debye shielding can be introduced in a very intuitive way by taking into account the granular character of such a plasma. Let us imagine a sphere about one of its electrons, and compare the number of electrons crossing this sphere with and without
Coulomb repulsion.
2591:
This intuitive picture leads to an effective calculation of Debye shielding (see section II.A.2 of ). The assumption of a
Boltzmann distribution is not necessary in this calculation: it works for whatever particle distribution function. The calculation also avoids approximating weakly collisional
1620:
2160:
3970:
When doping profiles exceed the Debye length, majority carriers no longer behave according to the distribution of the dopants. Instead, a measure of the profile of the doping gradients provides an "effective" profile that better matches the profile of the majority carrier density.
1215:
695:
2796:
2042:
sets the scale for variations in the potential and in the concentrations of charged species. All charged species contribute to the DebyeâHĂŒckel length in the same way, regardless of the sign of their charges. For an electrically neutral system, the
Poisson equation becomes
2013:
2474:
1118:
3520:
3321:
3002:
3649:
2300:
3927:
1481:
2046:
3795:
3012:
In space plasmas where the electron density is relatively low, the Debye length may reach macroscopic values, such as in the magnetosphere, solar wind, interstellar medium and intergalactic medium. See the table here below:
561:
2639:
4226:
Escande, D. F., BĂ©nisti, D., Elskens, Y., Zarzoso, D., & Doveil, F. (2018). Basic microscopic plasma physics from N-body mechanics, A tribute to Pierre-Simon de
Laplace, Reviews of Modern Plasma Physics, 2, 1-68
1846:{\displaystyle \varepsilon \nabla ^{2}\Phi (\mathbf {r} )=\left(\sum _{j=1}^{N}{\frac {n_{j}^{0}\,q_{j}^{2}}{k_{\rm {B}}T}}\right)\,\Phi (\mathbf {r} )-\,\sum _{j=1}^{N}n_{j}^{0}q_{j}-\rho _{\rm {ext}}(\mathbf {r} )}
1884:
738:
2344:
996:
1472:
308:
3418:
2215:
1405:{\displaystyle \varepsilon \nabla ^{2}\Phi (\mathbf {r} )=-\,\sum _{j=1}^{N}q_{j}n_{j}^{0}\,\exp \left(-{\frac {q_{j}\,\Phi (\mathbf {r} )}{k_{\rm {B}}T}}\right)-\rho _{\rm {ext}}(\mathbf {r} ).}
890:
3230:
166:
4434:
Guo, Lingjie; Effendi
Leobandung; Stephen Y. Chou (199). "A room-temperature silicon single-electron metalâoxideâsemiconductor memory with nanoscale floating-gate and ultranarrow channel".
2908:
3563:
2220:
959:
838:
552:
465:
3678:
2632:
2339:
69:
3844:
804:
765:
518:
2522:
2040:
1875:
487:
3834:
The Debye length has become increasingly significant in the modeling of solid state devices as improvements in lithographic technologies have enabled smaller geometries.
3730:
1414:
Solutions to this nonlinear equation are known for some simple systems. Solutions for more general systems may be obtained in the high-temperature (weak coupling) limit,
1147:
1183:
3712:
338:
424:
2905:
Even in quasineutral cold plasma, where ion contribution virtually seems to be larger due to lower ion temperature, the ion term is actually often dropped, giving
3405:
2569:
2546:
1203:
987:
921:
397:
377:
226:
206:
186:
1208:
Identifying the instantaneous concentrations and potential in the
Poisson equation with their mean-field counterparts in the Boltzmann distribution yields the
4289:
International
Standard ISO 13099-1, 2012, "Colloidal systems â Methods for Zeta potential determination- Part 1: Electroacoustic and Electrokinetic phenomena"
3826:
There is a method of estimating an approximate value of the Debye length in liquids using conductivity, which is described in ISO Standard, and the book.
2302:
The bare
Coulomb potential is exponentially screened by the medium, over a distance of the Debye length: this is called Debye screening or shielding (
1615:{\displaystyle \exp \left(-{\frac {q_{j}\,\Phi (\mathbf {r} )}{k_{\rm {B}}T}}\right)\approx 1-{\frac {q_{j}\,\Phi (\mathbf {r} )}{k_{\rm {B}}T}}.}
2599:
In a non-isothermic plasma, the temperatures for electrons and heavy species may differ while the background medium may be treated as the vacuum
2155:{\displaystyle \nabla ^{2}\Phi (\mathbf {r} )=\lambda _{\rm {D}}^{-2}\Phi (\mathbf {r} )-{\frac {\rho _{\rm {ext}}(\mathbf {r} )}{\varepsilon }}}
4469:
Tiwari, Sandip; Farhan Rana; Kevin Chan; Leathen Shi; Hussein Hanafi (1996). "Single charge and confinement effects in nano-crystal memories".
700:
690:{\displaystyle \varepsilon \nabla ^{2}\Phi (\mathbf {r} )=-\,\sum _{j=1}^{N}q_{j}\,n_{j}(\mathbf {r} )-\rho _{\rm {ext}}(\mathbf {r} ),}
489:. According to the so-called "primitive model", these charges are distributed in a continuous medium that is characterized only by its
4103:
2791:{\displaystyle \lambda _{\rm {D}}={\sqrt {\frac {\varepsilon _{0}k_{\rm {B}}/q_{e}^{2}}{n_{e}/T_{e}+\sum _{j}z_{j}^{2}n_{j}/T_{j}}}}}
2588:). Ions bring a similar contribution to shielding, because of the attractive Coulombian deflection of charges with opposite signs.
2015:
that commonly is referred to as the DebyeâHĂŒckel length. As the only characteristic length scale in the DebyeâHĂŒckel equation,
1857:: The second term on the right-hand side vanishes for systems that are electrically neutral. The term in parentheses divided by
1417:
231:
2008:{\displaystyle \lambda _{\rm {D}}=\left({\frac {\varepsilon \,k_{\rm {B}}T}{\sum _{j=1}^{N}n_{j}^{0}\,q_{j}^{2}}}\right)^{1/2}}
344:
and the ThomasâFermi wave vector. They are of interest in describing the behaviour of electrons in metals at room temperature.
2165:
4311:
4148:
4117:
4061:
847:
2469:{\displaystyle \lambda _{\rm {D}}=\left(4\pi \,\lambda _{\rm {B}}\,\sum _{j=1}^{N}n_{j}^{0}\,z_{j}^{2}\right)^{-1/2},}
359:
The Debye length arises naturally in the thermodynamic description of large systems of mobile charges. In a system of
119:
4361:
4336:
4244:
4086:
3975:
1113:{\displaystyle n_{j}(\mathbf {r} )=n_{j}^{0}\,\exp \left(-{\frac {q_{j}\,\Phi (\mathbf {r} )}{k_{\rm {B}}T}}\right),}
4513:
3515:{\displaystyle \kappa ^{-1}={\sqrt {\frac {\varepsilon _{\rm {r}}\varepsilon _{0}RT}{2\times 10^{3}F^{2}C_{0}}}}}
1209:
4548:
4553:
3992:
3359:
490:
17:
3316:{\displaystyle \kappa ^{-1}={\sqrt {\frac {\varepsilon _{\rm {r}}\varepsilon _{0}k_{\rm {B}}T}{2e^{2}I}}}}
1854:
3004:
although this is only valid when the mobility of ions is negligible compared to the process's timescale.
926:
812:
526:
432:
4377:
Stern, Eric; Robin Wagner; Fred J. Sigworth; Ronald
Breaker; Tarek M. Fahmy; Mark A. Reed (2007-11-01).
3654:
2604:
2315:
45:
4558:
2818:
774:
341:
743:
496:
4563:
4033:
2997:{\displaystyle \lambda _{\rm {D}}={\sqrt {\frac {\varepsilon _{0}k_{\rm {B}}T_{e}}{n_{e}q_{e}^{2}}}}}
893:
93:
3644:{\displaystyle \kappa ^{-1}={\frac {1}{\sqrt {8\pi \lambda _{\rm {B}}N_{\rm {A}}\times 10^{-24}I}}}}
88:
and how far its electrostatic effect persists. With each Debye length the charges are increasingly
4007:
2479:
2303:
314:. Expressions in MKS units will be given below. The analogous quantities at very low temperatures (
89:
4257:
2018:
1860:
470:
1123:
1156:
990:
3687:
4304:
Characterization of liquids, nano- and micro- particulates and porous bodies using Ultrasound
555:
4134:
2295:{\displaystyle \Phi (\mathbf {r} )={\frac {Q}{4\pi \varepsilon r}}e^{-r/\lambda _{\rm {D}}}}
317:
4573:
4543:
4478:
4443:
4390:
4184:
4031:[The theory of electrolytes. I. Freezing point depression and related phenomenon].
1878:
901:
402:
85:
3922:{\displaystyle L_{\rm {D}}={\sqrt {\frac {\varepsilon k_{\rm {B}}T}{q^{2}N_{\rm {dop}}}}}}
8:
4217:
Meyer-Vernet N (1993) Aspects of Debye shielding. American journal of physics 61, 249-257
3819:
3553:
3346:
4482:
4447:
4394:
4188:
3727:
At room temperature (20 °C or 70 °F), one can consider in water the relation:
100:
is a volume whose radius is the Debye length. Debye length is an important parameter in
4411:
4378:
4200:
4174:
4140:
4029:"Zur Theorie der Elektrolyte. I. Gefrierpunktserniedrigung und verwandte Erscheinungen"
3997:
3390:
3372:
2831:
2554:
2531:
1188:
1150:
972:
962:
906:
521:
382:
362:
211:
191:
171:
4379:"Importance of the Debye Screening Length on Nanowire Field Effect Transistor Sensors"
4264:
4165:
Brydges, David C.; Martin, Ph. A. (1999). "Coulomb Systems at Low Density: A Review".
4416:
4357:
4332:
4307:
4240:
4144:
4113:
4082:
4057:
3408:
2846:
2593:
2549:
966:
768:
4204:
3790:{\displaystyle \kappa ^{-1}(\mathrm {nm} )={\frac {0.304}{\sqrt {I(\mathrm {M} )}}}}
4486:
4451:
4406:
4398:
4192:
4002:
3540:
1475:
841:
31:
2162:
To illustrate Debye screening, the potential produced by an external point charge
4568:
4526:
4376:
4109:
3987:
3815:
3681:
3333:
3224:, the Debye length for a monovalent electrolyte is usually denoted with symbol
2581:
2310:
961:, may be considered to be thermodynamic (ensemble) averages and the associated
311:
101:
81:
4196:
4537:
4028:
3838:
2525:
427:
4420:
3530:
2585:
105:
4179:
3217:
348:
347:
The Debye length is named after the Dutch-American physicist and chemist
113:
35:
4054:
Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices
806:
is a charge density external (logically, not spatially) to the medium.
4468:
4402:
4490:
4455:
3805:
897:
1624:
This approximation yields the linearized PoissonâBoltzmann equation
4433:
3221:
733:{\displaystyle \varepsilon \equiv \varepsilon _{r}\varepsilon _{0}}
520:. This distribution of charges within this medium gives rise to an
109:
3714:
derives from transforming unit volume from cubic dm to cubic nm.
3087:
3382:
4101:
2865:
are the temperatures of the electrons and ions, respectively,
3966:
is the net density of dopants (either donors or acceptors).
1881:
leads to the definition of the characteristic length scale
4132:
1467:{\displaystyle q_{j}\,\Phi (\mathbf {r} )\ll k_{\rm {B}}T}
303:{\displaystyle k_{\rm {D}}^{2}=4\pi nq^{2}/(k_{\rm {B}}T)}
4507:
4327:
Russel, W. B.; Saville, D. A.; Schowalter, W. R. (1989).
2888:
2309:
The DebyeâHĂŒckel length may be expressed in terms of the
2210:{\displaystyle \rho _{\rm {ext}}=Q\delta (\mathbf {r} )}
4326:
809:
The mobile charges not only contribute in establishing
92:
and the electric potential decreases in magnitude by 1/
27:
Measure of electrostatic effect and how far it persists
3847:
3733:
3690:
3657:
3566:
3421:
3393:
3233:
2911:
2642:
2607:
2557:
2534:
2482:
2347:
2318:
2223:
2168:
2049:
2021:
1887:
1863:
1630:
1484:
1420:
1218:
1191:
1159:
1126:
999:
975:
929:
909:
850:
815:
777:
746:
703:
564:
529:
499:
473:
435:
405:
385:
365:
320:
234:
214:
194:
174:
122:
48:
1877:, has the units of an inverse length squared and by
969:. With these assumptions, the concentration of the
3921:
3789:
3717:For deionized water at room temperature, at pH=7,
3706:
3672:
3643:
3514:
3399:
3315:
2996:
2790:
2626:
2563:
2540:
2516:
2468:
2333:
2294:
2209:
2154:
2034:
2007:
1869:
1845:
1614:
1466:
1404:
1197:
1177:
1141:
1112:
981:
953:
915:
885:{\displaystyle -q_{j}\,\nabla \Phi (\mathbf {r} )}
884:
832:
798:
759:
732:
689:
546:
512:
481:
459:
418:
391:
371:
332:
302:
220:
200:
180:
160:
63:
116:). The corresponding Debye screening wave vector
4535:
1185:is the mean concentration of charges of species
161:{\displaystyle k_{\rm {D}}=1/\lambda _{\rm {D}}}
923:, then the concentrations of discrete charges,
4041:(9). Translated by Braus, Michael J.: 185â206.
4523:Fundamentals of Interface and Colloid Science
4351:
4258:"Chapter 20: The Particle Kinetics of Plasma"
4164:
3211:
3415:or, for a symmetric monovalent electrolyte,
840:but also move in response to the associated
351:(1884â1966), a Nobel laureate in Chemistry.
4301:
4026:
892:. If we further assume the system to be in
4255:
4410:
4297:
4295:
4178:
3978:may be required instead of Debye length.
2425:
2388:
2375:
1968:
1914:
1764:
1746:
1705:
1574:
1512:
1431:
1327:
1299:
1252:
1067:
1039:
864:
630:
598:
4105:Electrodynamics of particles and plasmas
4056:. New York: Cambridge University Press.
4520:
4126:
3951:is the absolute temperature in kelvins,
989:-th charge species is described by the
14:
4536:
4292:
4285:
4283:
4157:
4102:PC Clemmow & JP Dougherty (1969).
4095:
4051:
3336:of the electrolyte in number/m units,
4302:Dukhin, A. S.; Goetz, P. J. (2017).
4045:
3829:
3684:of the medium in nm, and the factor
3552:is the electrolyte concentration in
4280:
4133:RA Robinson &RH Stokes (2002).
954:{\displaystyle n_{j}(\mathbf {r} )}
833:{\displaystyle \Phi (\mathbf {r} )}
547:{\displaystyle \Phi (\mathbf {r} )}
460:{\displaystyle n_{j}(\mathbf {r} )}
24:
4508:Goldston & Rutherford (1997).
4501:
4076:
4070:
3909:
3906:
3903:
3876:
3854:
3777:
3754:
3751:
3673:{\displaystyle \lambda _{\rm {B}}}
3664:
3613:
3601:
3448:
3282:
3260:
2947:
2918:
2678:
2649:
2627:{\displaystyle \varepsilon _{r}=1}
2382:
2354:
2334:{\displaystyle \lambda _{\rm {B}}}
2325:
2284:
2224:
2181:
2178:
2175:
2129:
2126:
2123:
2097:
2083:
2060:
2051:
1921:
1894:
1826:
1823:
1820:
1747:
1729:
1644:
1635:
1597:
1575:
1535:
1513:
1455:
1432:
1382:
1379:
1376:
1350:
1328:
1232:
1223:
1133:
1090:
1068:
868:
865:
816:
790:
787:
784:
667:
664:
661:
578:
569:
530:
379:different species of charges, the
354:
288:
241:
152:
129:
64:{\displaystyle \lambda _{\rm {D}}}
55:
25:
4585:
4354:Intermolecular and Surface Forces
4266:Applications of Classical Physics
3207:
3007:
2883:is the density of atomic species
799:{\displaystyle \rho _{\rm {ext}}}
84:'s net electrostatic effect in a
4238:Principles of plasma diagnostics
4079:Electrokinetics in Microfluidics
2231:
2200:
2139:
2104:
2067:
1836:
1754:
1651:
1582:
1520:
1439:
1392:
1335:
1239:
1075:
1014:
944:
875:
823:
760:{\displaystyle \varepsilon _{0}}
677:
645:
585:
537:
513:{\displaystyle \varepsilon _{r}}
475:
450:
4514:Institute of Physics Publishing
4462:
4427:
4370:
4345:
4320:
4249:
4112:. pp. § 7.6.7, p. 236 ff.
4027:Debye, P.; HĂŒckel, E. (2019) .
3381:is the absolute temperature in
2528:that relates the charge on the
4510:Introduction to Plasma Physics
4331:. Cambridge University Press.
4230:
4220:
4211:
4167:Journal of Statistical Physics
4020:
3781:
3773:
3758:
3747:
2574:
2235:
2227:
2204:
2196:
2143:
2135:
2108:
2100:
2071:
2063:
1840:
1832:
1758:
1750:
1655:
1647:
1586:
1578:
1524:
1516:
1443:
1435:
1396:
1388:
1339:
1331:
1243:
1235:
1079:
1071:
1018:
1010:
948:
940:
879:
871:
827:
819:
681:
673:
649:
641:
589:
581:
541:
533:
454:
446:
324:
297:
279:
13:
1:
4013:
3976:ThomasâFermi screening length
3957:is the elementary charge, and
2517:{\displaystyle z_{j}=q_{j}/e}
78:DebyeâHĂŒckel screening length
3189:
3172:
3155:
3138:
3121:
3104:
3085:
3068:
2874:is the density of electrons,
2035:{\displaystyle \lambda _{D}}
1870:{\displaystyle \varepsilon }
491:relative static permittivity
482:{\displaystyle \mathbf {r} }
7:
3981:
3936:is the dielectric constant,
1853:which also is known as the
1142:{\displaystyle k_{\rm {B}}}
399:-th species carries charge
10:
4590:
4352:Israelachvili, J. (1985).
3974:In the context of solids,
3945:is the Boltzmann constant,
3347:permittivity of free space
3212:In an electrolyte solution
2819:permittivity of free space
1210:PoissonâBoltzmann equation
4034:Physikalische Zeitschrift
2548:-th ionic species to the
1178:{\displaystyle n_{j}^{0}}
894:thermodynamic equilibrium
168:for particles of density
3993:DebyeâFalkenhagen effect
3707:{\displaystyle 10^{-24}}
2636:and the Debye length is
4471:Applied Physics Letters
4436:Applied Physics Letters
4197:10.1023/A:1004600603161
3923:
3791:
3708:
3674:
3645:
3516:
3401:
3317:
2998:
2792:
2628:
2565:
2542:
2518:
2470:
2409:
2335:
2296:
2211:
2156:
2036:
2009:
1952:
1871:
1847:
1785:
1686:
1616:
1468:
1406:
1273:
1199:
1179:
1143:
1114:
991:Boltzmann distribution
983:
965:to be a thermodynamic
955:
917:
886:
834:
800:
761:
734:
691:
619:
548:
514:
483:
461:
420:
393:
373:
334:
333:{\displaystyle T\to 0}
304:
222:
202:
182:
162:
65:
4329:Colloidal Dispersions
4136:Electrolyte solutions
4052:Kirby, B. J. (2010).
3924:
3792:
3709:
3675:
3646:
3517:
3402:
3318:
3191:Intergalactic medium
2999:
2847:charge of an electron
2793:
2629:
2566:
2543:
2519:
2471:
2389:
2336:
2297:
2212:
2157:
2037:
2010:
1932:
1872:
1855:DebyeâHĂŒckel equation
1848:
1765:
1666:
1617:
1469:
1407:
1253:
1200:
1180:
1144:
1115:
984:
956:
918:
887:
835:
801:
762:
735:
692:
599:
549:
515:
484:
462:
421:
419:{\displaystyle q_{j}}
394:
374:
335:
305:
223:
203:
183:
163:
90:electrically screened
80:), is a measure of a
66:
4549:Electronics concepts
3845:
3837:The Debye length of
3731:
3688:
3655:
3564:
3419:
3391:
3231:
3222:colloidal suspension
3174:Interstellar medium
3035:Electron temperature
2909:
2808:is the Debye length,
2640:
2605:
2555:
2532:
2480:
2345:
2316:
2221:
2166:
2047:
2019:
1885:
1879:dimensional analysis
1861:
1628:
1482:
1418:
1216:
1189:
1157:
1124:
997:
973:
927:
907:
902:absolute temperature
848:
813:
775:
744:
701:
562:
527:
497:
471:
433:
403:
383:
363:
318:
232:
212:
192:
172:
120:
46:
4554:Colloidal chemistry
4483:1996ApPhL..69.1232T
4448:1997ApPhL..70..850G
4395:2007NanoL...7.3405S
4256:Kip Thorne (2012).
4189:1999JSP....96.1163B
4108:. Redwood City CA:
3556:units (M or mol/L).
3360:dielectric constant
3208:
2989:
2758:
2703:
2440:
2424:
2096:
1983:
1967:
1800:
1720:
1704:
1298:
1174:
1038:
342:ThomasâFermi length
340:) are known as the
251:
4356:. Academic Press.
4173:(5/6): 1163â1330.
4141:Dover Publications
4081:. Academic Press.
3998:Plasma oscillation
3919:
3787:
3704:
3670:
3641:
3512:
3397:
3373:Boltzmann constant
3313:
2994:
2975:
2832:Boltzmann constant
2788:
2744:
2743:
2689:
2624:
2561:
2538:
2514:
2466:
2426:
2410:
2331:
2292:
2207:
2152:
2077:
2032:
2005:
1969:
1953:
1867:
1843:
1786:
1706:
1690:
1612:
1464:
1402:
1284:
1195:
1175:
1160:
1151:Boltzmann constant
1139:
1110:
1024:
979:
963:electric potential
951:
913:
882:
830:
796:
757:
730:
687:
556:Poisson's equation
544:
522:electric potential
510:
479:
457:
416:
389:
369:
330:
300:
235:
218:
198:
178:
158:
61:
4559:Plasma parameters
4403:10.1021/nl071792z
4389:(11): 3405â3409.
4313:978-0-444-63908-0
4236:I. H. Hutchinson
4150:978-0-486-42225-1
4119:978-0-201-47986-7
4063:978-0-521-11903-0
3917:
3916:
3830:In semiconductors
3785:
3784:
3639:
3638:
3510:
3509:
3409:elementary charge
3400:{\displaystyle e}
3311:
3310:
3209:
3206:
2992:
2991:
2786:
2785:
2734:
2594:Coulomb collision
2564:{\displaystyle e}
2550:elementary charge
2541:{\displaystyle j}
2260:
2150:
1985:
1739:
1607:
1545:
1478:the exponential:
1360:
1198:{\displaystyle j}
1100:
982:{\displaystyle j}
916:{\displaystyle T}
769:electric constant
392:{\displaystyle j}
372:{\displaystyle N}
221:{\displaystyle T}
208:at a temperature
201:{\displaystyle q}
181:{\displaystyle n}
16:(Redirected from
4581:
4564:Electrochemistry
4530:
4521:Lyklema (1993).
4517:
4512:. Philadelphia:
4495:
4494:
4491:10.1063/1.117421
4466:
4460:
4459:
4456:10.1063/1.118236
4431:
4425:
4424:
4414:
4374:
4368:
4367:
4349:
4343:
4342:
4324:
4318:
4317:
4299:
4290:
4287:
4278:
4277:
4275:
4273:
4262:
4253:
4247:
4234:
4228:
4224:
4218:
4215:
4209:
4208:
4182:
4180:cond-mat/9904122
4161:
4155:
4154:
4130:
4124:
4123:
4099:
4093:
4092:
4074:
4068:
4067:
4049:
4043:
4042:
4024:
4008:Screening effect
4003:Shielding effect
3928:
3926:
3925:
3920:
3918:
3915:
3914:
3913:
3912:
3896:
3895:
3885:
3881:
3880:
3879:
3865:
3864:
3859:
3858:
3857:
3804:is expressed in
3796:
3794:
3793:
3788:
3786:
3780:
3769:
3765:
3757:
3746:
3745:
3713:
3711:
3710:
3705:
3703:
3702:
3679:
3677:
3676:
3671:
3669:
3668:
3667:
3650:
3648:
3647:
3642:
3640:
3634:
3633:
3618:
3617:
3616:
3606:
3605:
3604:
3588:
3584:
3579:
3578:
3541:Faraday constant
3521:
3519:
3518:
3513:
3511:
3508:
3507:
3506:
3497:
3496:
3487:
3486:
3470:
3463:
3462:
3453:
3452:
3451:
3440:
3439:
3434:
3433:
3406:
3404:
3403:
3398:
3322:
3320:
3319:
3314:
3312:
3309:
3305:
3304:
3291:
3287:
3286:
3285:
3275:
3274:
3265:
3264:
3263:
3252:
3251:
3246:
3245:
3065:
3052:
3042:
3032:
3016:
3015:
3003:
3001:
3000:
2995:
2993:
2990:
2988:
2983:
2974:
2973:
2963:
2962:
2961:
2952:
2951:
2950:
2940:
2939:
2929:
2928:
2923:
2922:
2921:
2887:, with positive
2797:
2795:
2794:
2789:
2787:
2784:
2783:
2782:
2773:
2768:
2767:
2757:
2752:
2742:
2730:
2729:
2720:
2715:
2714:
2704:
2702:
2697:
2688:
2683:
2682:
2681:
2671:
2670:
2660:
2659:
2654:
2653:
2652:
2635:
2633:
2631:
2630:
2625:
2617:
2616:
2570:
2568:
2567:
2562:
2547:
2545:
2544:
2539:
2523:
2521:
2520:
2515:
2510:
2505:
2504:
2492:
2491:
2475:
2473:
2472:
2467:
2462:
2461:
2457:
2445:
2441:
2439:
2434:
2423:
2418:
2408:
2403:
2387:
2386:
2385:
2359:
2358:
2357:
2340:
2338:
2337:
2332:
2330:
2329:
2328:
2304:Screening effect
2301:
2299:
2298:
2293:
2291:
2290:
2289:
2288:
2287:
2277:
2261:
2259:
2242:
2234:
2216:
2214:
2213:
2208:
2203:
2186:
2185:
2184:
2161:
2159:
2158:
2153:
2151:
2146:
2142:
2134:
2133:
2132:
2115:
2107:
2095:
2087:
2086:
2070:
2059:
2058:
2041:
2039:
2038:
2033:
2031:
2030:
2014:
2012:
2011:
2006:
2004:
2003:
1999:
1990:
1986:
1984:
1982:
1977:
1966:
1961:
1951:
1946:
1930:
1926:
1925:
1924:
1909:
1899:
1898:
1897:
1876:
1874:
1873:
1868:
1852:
1850:
1849:
1844:
1839:
1831:
1830:
1829:
1810:
1809:
1799:
1794:
1784:
1779:
1757:
1745:
1741:
1740:
1738:
1734:
1733:
1732:
1721:
1719:
1714:
1703:
1698:
1688:
1685:
1680:
1654:
1643:
1642:
1621:
1619:
1618:
1613:
1608:
1606:
1602:
1601:
1600:
1589:
1585:
1573:
1572:
1562:
1551:
1547:
1546:
1544:
1540:
1539:
1538:
1527:
1523:
1511:
1510:
1500:
1476:Taylor expanding
1473:
1471:
1470:
1465:
1460:
1459:
1458:
1442:
1430:
1429:
1411:
1409:
1408:
1403:
1395:
1387:
1386:
1385:
1366:
1362:
1361:
1359:
1355:
1354:
1353:
1342:
1338:
1326:
1325:
1315:
1297:
1292:
1283:
1282:
1272:
1267:
1242:
1231:
1230:
1204:
1202:
1201:
1196:
1184:
1182:
1181:
1176:
1173:
1168:
1148:
1146:
1145:
1140:
1138:
1137:
1136:
1119:
1117:
1116:
1111:
1106:
1102:
1101:
1099:
1095:
1094:
1093:
1082:
1078:
1066:
1065:
1055:
1037:
1032:
1017:
1009:
1008:
988:
986:
985:
980:
960:
958:
957:
952:
947:
939:
938:
922:
920:
919:
914:
891:
889:
888:
883:
878:
863:
862:
839:
837:
836:
831:
826:
805:
803:
802:
797:
795:
794:
793:
766:
764:
763:
758:
756:
755:
739:
737:
736:
731:
729:
728:
719:
718:
696:
694:
693:
688:
680:
672:
671:
670:
648:
640:
639:
629:
628:
618:
613:
588:
577:
576:
553:
551:
550:
545:
540:
519:
517:
516:
511:
509:
508:
488:
486:
485:
480:
478:
466:
464:
463:
458:
453:
445:
444:
425:
423:
422:
417:
415:
414:
398:
396:
395:
390:
378:
376:
375:
370:
339:
337:
336:
331:
309:
307:
306:
301:
293:
292:
291:
278:
273:
272:
250:
245:
244:
227:
225:
224:
219:
207:
205:
204:
199:
187:
185:
184:
179:
167:
165:
164:
159:
157:
156:
155:
145:
134:
133:
132:
70:
68:
67:
62:
60:
59:
58:
21:
4589:
4588:
4584:
4583:
4582:
4580:
4579:
4578:
4534:
4533:
4504:
4502:Further reading
4499:
4498:
4467:
4463:
4432:
4428:
4375:
4371:
4364:
4350:
4346:
4339:
4325:
4321:
4314:
4300:
4293:
4288:
4281:
4271:
4269:
4260:
4254:
4250:
4235:
4231:
4225:
4221:
4216:
4212:
4162:
4158:
4151:
4139:. Mineola, NY:
4131:
4127:
4120:
4100:
4096:
4089:
4077:Li, D. (2004).
4075:
4071:
4064:
4050:
4046:
4025:
4021:
4016:
3984:
3965:
3944:
3902:
3901:
3897:
3891:
3887:
3886:
3875:
3874:
3870:
3866:
3863:
3853:
3852:
3848:
3846:
3843:
3842:
3832:
3776:
3764:
3750:
3738:
3734:
3732:
3729:
3728:
3723:
3695:
3691:
3689:
3686:
3685:
3663:
3662:
3658:
3656:
3653:
3652:
3626:
3622:
3612:
3611:
3607:
3600:
3599:
3595:
3583:
3571:
3567:
3565:
3562:
3561:
3560:Alternatively,
3551:
3502:
3498:
3492:
3488:
3482:
3478:
3471:
3458:
3454:
3447:
3446:
3442:
3441:
3438:
3426:
3422:
3420:
3417:
3416:
3392:
3389:
3388:
3370:
3357:
3344:
3300:
3296:
3292:
3281:
3280:
3276:
3270:
3266:
3259:
3258:
3254:
3253:
3250:
3238:
3234:
3232:
3229:
3228:
3214:
3063:
3057:
3056:
3047:
3046:
3037:
3036:
3030:
3024:
3023:
3010:
2984:
2979:
2969:
2965:
2964:
2957:
2953:
2946:
2945:
2941:
2935:
2931:
2930:
2927:
2917:
2916:
2912:
2910:
2907:
2906:
2900:
2896:
2881:
2872:
2863:
2856:
2844:
2829:
2816:
2807:
2778:
2774:
2769:
2763:
2759:
2753:
2748:
2738:
2725:
2721:
2716:
2710:
2706:
2705:
2698:
2693:
2684:
2677:
2676:
2672:
2666:
2662:
2661:
2658:
2648:
2647:
2643:
2641:
2638:
2637:
2612:
2608:
2606:
2603:
2602:
2600:
2577:
2556:
2553:
2552:
2533:
2530:
2529:
2524:is the integer
2506:
2500:
2496:
2487:
2483:
2481:
2478:
2477:
2453:
2446:
2435:
2430:
2419:
2414:
2404:
2393:
2381:
2380:
2376:
2368:
2364:
2363:
2353:
2352:
2348:
2346:
2343:
2342:
2324:
2323:
2319:
2317:
2314:
2313:
2283:
2282:
2278:
2273:
2266:
2262:
2246:
2241:
2230:
2222:
2219:
2218:
2199:
2174:
2173:
2169:
2167:
2164:
2163:
2138:
2122:
2121:
2117:
2116:
2114:
2103:
2088:
2082:
2081:
2066:
2054:
2050:
2048:
2045:
2044:
2026:
2022:
2020:
2017:
2016:
1995:
1991:
1978:
1973:
1962:
1957:
1947:
1936:
1931:
1920:
1919:
1915:
1910:
1908:
1904:
1903:
1893:
1892:
1888:
1886:
1883:
1882:
1862:
1859:
1858:
1835:
1819:
1818:
1814:
1805:
1801:
1795:
1790:
1780:
1769:
1753:
1728:
1727:
1723:
1722:
1715:
1710:
1699:
1694:
1689:
1687:
1681:
1670:
1665:
1661:
1650:
1638:
1634:
1629:
1626:
1625:
1596:
1595:
1591:
1590:
1581:
1568:
1564:
1563:
1561:
1534:
1533:
1529:
1528:
1519:
1506:
1502:
1501:
1499:
1495:
1491:
1483:
1480:
1479:
1454:
1453:
1449:
1438:
1425:
1421:
1419:
1416:
1415:
1391:
1375:
1374:
1370:
1349:
1348:
1344:
1343:
1334:
1321:
1317:
1316:
1314:
1310:
1306:
1293:
1288:
1278:
1274:
1268:
1257:
1238:
1226:
1222:
1217:
1214:
1213:
1190:
1187:
1186:
1169:
1164:
1158:
1155:
1154:
1132:
1131:
1127:
1125:
1122:
1121:
1089:
1088:
1084:
1083:
1074:
1061:
1057:
1056:
1054:
1050:
1046:
1033:
1028:
1013:
1004:
1000:
998:
995:
994:
974:
971:
970:
943:
934:
930:
928:
925:
924:
908:
905:
904:
874:
858:
854:
849:
846:
845:
822:
814:
811:
810:
783:
782:
778:
776:
773:
772:
751:
747:
745:
742:
741:
724:
720:
714:
710:
702:
699:
698:
676:
660:
659:
655:
644:
635:
631:
624:
620:
614:
603:
584:
572:
568:
563:
560:
559:
554:that satisfies
536:
528:
525:
524:
504:
500:
498:
495:
494:
474:
472:
469:
468:
449:
440:
436:
434:
431:
430:
410:
406:
404:
401:
400:
384:
381:
380:
364:
361:
360:
357:
355:Physical origin
319:
316:
315:
287:
286:
282:
274:
268:
264:
246:
240:
239:
233:
230:
229:
213:
210:
209:
193:
190:
189:
173:
170:
169:
151:
150:
146:
141:
128:
127:
123:
121:
118:
117:
54:
53:
49:
47:
44:
43:
28:
23:
22:
15:
12:
11:
5:
4587:
4577:
4576:
4571:
4566:
4561:
4556:
4551:
4546:
4532:
4531:
4527:Academic Press
4518:
4503:
4500:
4497:
4496:
4461:
4426:
4369:
4362:
4344:
4337:
4319:
4312:
4291:
4279:
4248:
4229:
4219:
4210:
4156:
4149:
4143:. p. 76.
4125:
4118:
4110:Addison-Wesley
4094:
4087:
4069:
4062:
4044:
4018:
4017:
4015:
4012:
4011:
4010:
4005:
4000:
3995:
3990:
3988:Bjerrum length
3983:
3980:
3968:
3967:
3963:
3958:
3952:
3946:
3942:
3937:
3911:
3908:
3905:
3900:
3894:
3890:
3884:
3878:
3873:
3869:
3862:
3856:
3851:
3839:semiconductors
3831:
3828:
3824:
3823:
3816:ionic strength
3809:
3783:
3779:
3775:
3772:
3768:
3763:
3760:
3756:
3753:
3749:
3744:
3741:
3737:
3721:
3701:
3698:
3694:
3682:Bjerrum length
3666:
3661:
3637:
3632:
3629:
3625:
3621:
3615:
3610:
3603:
3598:
3594:
3591:
3587:
3582:
3577:
3574:
3570:
3558:
3557:
3549:
3544:
3534:
3505:
3501:
3495:
3491:
3485:
3481:
3477:
3474:
3469:
3466:
3461:
3457:
3450:
3445:
3437:
3432:
3429:
3425:
3413:
3412:
3396:
3386:
3376:
3368:
3363:
3355:
3350:
3342:
3337:
3334:ionic strength
3308:
3303:
3299:
3295:
3290:
3284:
3279:
3273:
3269:
3262:
3257:
3249:
3244:
3241:
3237:
3213:
3210:
3205:
3204:
3201:
3198:
3195:
3192:
3188:
3187:
3184:
3181:
3178:
3175:
3171:
3170:
3167:
3164:
3161:
3158:
3154:
3153:
3150:
3147:
3144:
3141:
3140:Magnetosphere
3137:
3136:
3133:
3130:
3127:
3124:
3120:
3119:
3116:
3113:
3110:
3107:
3106:Gas discharge
3103:
3102:
3099:
3096:
3093:
3090:
3084:
3083:
3080:
3077:
3074:
3071:
3067:
3066:
3061:
3053:
3045:Magnetic field
3043:
3033:
3028:
3020:
3009:
3008:Typical values
3006:
2987:
2982:
2978:
2972:
2968:
2960:
2956:
2949:
2944:
2938:
2934:
2926:
2920:
2915:
2903:
2902:
2898:
2894:
2879:
2875:
2870:
2866:
2861:
2854:
2850:
2840:
2835:
2827:
2822:
2814:
2809:
2805:
2781:
2777:
2772:
2766:
2762:
2756:
2751:
2747:
2741:
2737:
2733:
2728:
2724:
2719:
2713:
2709:
2701:
2696:
2692:
2687:
2680:
2675:
2669:
2665:
2657:
2651:
2646:
2623:
2620:
2615:
2611:
2582:Langmuir probe
2576:
2573:
2560:
2537:
2513:
2509:
2503:
2499:
2495:
2490:
2486:
2465:
2460:
2456:
2452:
2449:
2444:
2438:
2433:
2429:
2422:
2417:
2413:
2407:
2402:
2399:
2396:
2392:
2384:
2379:
2374:
2371:
2367:
2362:
2356:
2351:
2327:
2322:
2311:Bjerrum length
2286:
2281:
2276:
2272:
2269:
2265:
2258:
2255:
2252:
2249:
2245:
2240:
2237:
2233:
2229:
2226:
2206:
2202:
2198:
2195:
2192:
2189:
2183:
2180:
2177:
2172:
2149:
2145:
2141:
2137:
2131:
2128:
2125:
2120:
2113:
2110:
2106:
2102:
2099:
2094:
2091:
2085:
2080:
2076:
2073:
2069:
2065:
2062:
2057:
2053:
2029:
2025:
2002:
1998:
1994:
1989:
1981:
1976:
1972:
1965:
1960:
1956:
1950:
1945:
1942:
1939:
1935:
1929:
1923:
1918:
1913:
1907:
1902:
1896:
1891:
1866:
1842:
1838:
1834:
1828:
1825:
1822:
1817:
1813:
1808:
1804:
1798:
1793:
1789:
1783:
1778:
1775:
1772:
1768:
1763:
1760:
1756:
1752:
1749:
1744:
1737:
1731:
1726:
1718:
1713:
1709:
1702:
1697:
1693:
1684:
1679:
1676:
1673:
1669:
1664:
1660:
1657:
1653:
1649:
1646:
1641:
1637:
1633:
1611:
1605:
1599:
1594:
1588:
1584:
1580:
1577:
1571:
1567:
1560:
1557:
1554:
1550:
1543:
1537:
1532:
1526:
1522:
1518:
1515:
1509:
1505:
1498:
1494:
1490:
1487:
1463:
1457:
1452:
1448:
1445:
1441:
1437:
1434:
1428:
1424:
1401:
1398:
1394:
1390:
1384:
1381:
1378:
1373:
1369:
1365:
1358:
1352:
1347:
1341:
1337:
1333:
1330:
1324:
1320:
1313:
1309:
1305:
1302:
1296:
1291:
1287:
1281:
1277:
1271:
1266:
1263:
1260:
1256:
1251:
1248:
1245:
1241:
1237:
1234:
1229:
1225:
1221:
1194:
1172:
1167:
1163:
1135:
1130:
1109:
1105:
1098:
1092:
1087:
1081:
1077:
1073:
1070:
1064:
1060:
1053:
1049:
1045:
1042:
1036:
1031:
1027:
1023:
1020:
1016:
1012:
1007:
1003:
978:
950:
946:
942:
937:
933:
912:
881:
877:
873:
870:
867:
861:
857:
853:
829:
825:
821:
818:
792:
789:
786:
781:
754:
750:
727:
723:
717:
713:
709:
706:
686:
683:
679:
675:
669:
666:
663:
658:
654:
651:
647:
643:
638:
634:
627:
623:
617:
612:
609:
606:
602:
597:
594:
591:
587:
583:
580:
575:
571:
567:
543:
539:
535:
532:
507:
503:
477:
456:
452:
448:
443:
439:
413:
409:
388:
368:
356:
353:
329:
326:
323:
312:Gaussian units
299:
296:
290:
285:
281:
277:
271:
267:
263:
260:
257:
254:
249:
243:
238:
217:
197:
177:
154:
149:
144:
140:
137:
131:
126:
102:plasma physics
82:charge carrier
57:
52:
26:
9:
6:
4:
3:
2:
4586:
4575:
4572:
4570:
4567:
4565:
4562:
4560:
4557:
4555:
4552:
4550:
4547:
4545:
4542:
4541:
4539:
4528:
4524:
4519:
4515:
4511:
4506:
4505:
4492:
4488:
4484:
4480:
4476:
4472:
4465:
4457:
4453:
4449:
4445:
4441:
4437:
4430:
4422:
4418:
4413:
4408:
4404:
4400:
4396:
4392:
4388:
4384:
4380:
4373:
4365:
4363:0-12-375181-0
4359:
4355:
4348:
4340:
4338:0-521-42600-6
4334:
4330:
4323:
4315:
4309:
4305:
4298:
4296:
4286:
4284:
4268:
4267:
4259:
4252:
4246:
4245:0-521-38583-0
4242:
4239:
4233:
4223:
4214:
4206:
4202:
4198:
4194:
4190:
4186:
4181:
4176:
4172:
4168:
4160:
4152:
4146:
4142:
4138:
4137:
4129:
4121:
4115:
4111:
4107:
4106:
4098:
4090:
4088:0-12-088444-5
4084:
4080:
4073:
4065:
4059:
4055:
4048:
4040:
4036:
4035:
4030:
4023:
4019:
4009:
4006:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3985:
3979:
3977:
3972:
3962:
3959:
3956:
3953:
3950:
3947:
3941:
3938:
3935:
3932:
3931:
3930:
3898:
3892:
3888:
3882:
3871:
3867:
3860:
3849:
3840:
3835:
3827:
3821:
3818:expressed in
3817:
3813:
3810:
3807:
3803:
3800:
3799:
3798:
3770:
3766:
3761:
3742:
3739:
3735:
3725:
3720:
3715:
3699:
3696:
3692:
3683:
3659:
3635:
3630:
3627:
3623:
3619:
3608:
3596:
3592:
3589:
3585:
3580:
3575:
3572:
3568:
3555:
3548:
3545:
3542:
3538:
3535:
3532:
3528:
3525:
3524:
3523:
3503:
3499:
3493:
3489:
3483:
3479:
3475:
3472:
3467:
3464:
3459:
3455:
3443:
3435:
3430:
3427:
3423:
3410:
3394:
3387:
3384:
3380:
3377:
3374:
3367:
3364:
3361:
3354:
3351:
3348:
3341:
3338:
3335:
3331:
3328:
3327:
3326:
3323:
3306:
3301:
3297:
3293:
3288:
3277:
3271:
3267:
3255:
3247:
3242:
3239:
3235:
3227:
3223:
3219:
3202:
3199:
3196:
3193:
3190:
3185:
3182:
3179:
3176:
3173:
3168:
3165:
3162:
3159:
3156:
3151:
3148:
3145:
3142:
3139:
3134:
3131:
3128:
3125:
3122:
3117:
3114:
3111:
3108:
3105:
3100:
3097:
3094:
3091:
3089:
3086:
3081:
3078:
3075:
3072:
3069:
3060:
3054:
3050:
3044:
3040:
3034:
3027:
3021:
3018:
3017:
3014:
3005:
2985:
2980:
2976:
2970:
2966:
2958:
2954:
2942:
2936:
2932:
2924:
2913:
2901:
2890:
2886:
2882:
2876:
2873:
2867:
2864:
2857:
2851:
2848:
2843:
2839:
2836:
2833:
2826:
2823:
2820:
2813:
2810:
2804:
2801:
2800:
2799:
2779:
2775:
2770:
2764:
2760:
2754:
2749:
2745:
2739:
2735:
2731:
2726:
2722:
2717:
2711:
2707:
2699:
2694:
2690:
2685:
2673:
2667:
2663:
2655:
2644:
2621:
2618:
2613:
2609:
2597:
2595:
2589:
2587:
2583:
2572:
2558:
2551:
2535:
2527:
2526:charge number
2511:
2507:
2501:
2497:
2493:
2488:
2484:
2463:
2458:
2454:
2450:
2447:
2442:
2436:
2431:
2427:
2420:
2415:
2411:
2405:
2400:
2397:
2394:
2390:
2377:
2372:
2369:
2365:
2360:
2349:
2320:
2312:
2307:
2305:
2279:
2274:
2270:
2267:
2263:
2256:
2253:
2250:
2247:
2243:
2238:
2193:
2190:
2187:
2170:
2147:
2118:
2111:
2092:
2089:
2078:
2074:
2055:
2027:
2023:
2000:
1996:
1992:
1987:
1979:
1974:
1970:
1963:
1958:
1954:
1948:
1943:
1940:
1937:
1933:
1927:
1916:
1911:
1905:
1900:
1889:
1880:
1864:
1856:
1815:
1811:
1806:
1802:
1796:
1791:
1787:
1781:
1776:
1773:
1770:
1766:
1761:
1742:
1735:
1724:
1716:
1711:
1707:
1700:
1695:
1691:
1682:
1677:
1674:
1671:
1667:
1662:
1658:
1639:
1631:
1622:
1609:
1603:
1592:
1569:
1565:
1558:
1555:
1552:
1548:
1541:
1530:
1507:
1503:
1496:
1492:
1488:
1485:
1477:
1461:
1450:
1446:
1426:
1422:
1412:
1399:
1371:
1367:
1363:
1356:
1345:
1322:
1318:
1311:
1307:
1303:
1300:
1294:
1289:
1285:
1279:
1275:
1269:
1264:
1261:
1258:
1254:
1249:
1246:
1227:
1219:
1211:
1206:
1192:
1170:
1165:
1161:
1152:
1128:
1107:
1103:
1096:
1085:
1062:
1058:
1051:
1047:
1043:
1040:
1034:
1029:
1025:
1021:
1005:
1001:
992:
976:
968:
964:
935:
931:
910:
903:
899:
895:
859:
855:
851:
843:
842:Coulomb force
807:
779:
770:
752:
748:
725:
721:
715:
711:
707:
704:
684:
656:
652:
636:
632:
625:
621:
615:
610:
607:
604:
600:
595:
592:
573:
565:
557:
523:
505:
501:
492:
441:
437:
429:
428:concentration
411:
407:
386:
366:
352:
350:
345:
343:
327:
321:
313:
294:
283:
275:
269:
265:
261:
258:
255:
252:
247:
236:
215:
195:
175:
147:
142:
138:
135:
124:
115:
111:
107:
103:
99:
95:
91:
87:
83:
79:
75:
71:
50:
41:
37:
33:
19:
4522:
4509:
4474:
4470:
4464:
4439:
4435:
4429:
4386:
4383:Nano Letters
4382:
4372:
4353:
4347:
4328:
4322:
4306:. Elsevier.
4303:
4272:September 7,
4270:. Retrieved
4265:
4251:
4237:
4232:
4222:
4213:
4170:
4166:
4159:
4135:
4128:
4104:
4097:
4078:
4072:
4053:
4047:
4038:
4032:
4022:
3973:
3969:
3960:
3954:
3948:
3939:
3933:
3836:
3833:
3825:
3822:(M or mol/L)
3811:
3801:
3726:
3718:
3716:
3559:
3546:
3536:
3531:gas constant
3526:
3414:
3378:
3365:
3352:
3339:
3329:
3324:
3225:
3215:
3058:
3055:Debye length
3048:
3038:
3025:
3011:
2904:
2892:
2884:
2877:
2868:
2859:
2852:
2841:
2837:
2824:
2811:
2802:
2598:
2590:
2586:Debye sheath
2578:
2308:
1623:
1413:
1207:
808:
467:at position
358:
346:
228:is given by
106:electrolytes
98:Debye sphere
97:
77:
74:Debye radius
73:
42:
40:Debye length
39:
36:electrolytes
29:
18:Debye sphere
4574:Peter Debye
4544:Electricity
4477:(9): 1232.
3218:electrolyte
3157:Solar wind
3123:Ionosphere
3070:Solar core
2575:In a plasma
1153:and where
349:Peter Debye
114:DLVO theory
4538:Categories
4442:(7): 850.
4014:References
3841:is given:
3806:nanometres
967:mean field
3868:ε
3740:−
3736:κ
3697:−
3660:λ
3628:−
3620:×
3597:λ
3593:π
3573:−
3569:κ
3476:×
3456:ε
3444:ε
3428:−
3424:κ
3268:ε
3256:ε
3240:−
3236:κ
2933:ε
2914:λ
2736:∑
2664:ε
2645:λ
2610:ε
2448:−
2391:∑
2378:λ
2373:π
2350:λ
2321:λ
2280:λ
2268:−
2254:ε
2251:π
2225:Φ
2194:δ
2171:ρ
2148:ε
2119:ρ
2112:−
2098:Φ
2090:−
2079:λ
2061:Φ
2052:∇
2024:λ
1934:∑
1912:ε
1890:λ
1865:ε
1816:ρ
1812:−
1767:∑
1762:−
1748:Φ
1668:∑
1645:Φ
1636:∇
1632:ε
1576:Φ
1559:−
1553:≈
1514:Φ
1497:−
1489:
1447:≪
1433:Φ
1372:ρ
1368:−
1329:Φ
1312:−
1304:
1255:∑
1250:−
1233:Φ
1224:∇
1220:ε
1069:Φ
1052:−
1044:
898:heat bath
869:Φ
866:∇
852:−
817:Φ
780:ρ
749:ε
722:ε
712:ε
708:≡
705:ε
657:ρ
653:−
601:∑
596:−
579:Φ
570:∇
566:ε
531:Φ
502:ε
325:→
259:π
188:, charge
148:λ
51:λ
4421:17914853
4205:54979869
3982:See also
426:and has
110:colloids
86:solution
4479:Bibcode
4444:Bibcode
4412:2713684
4391:Bibcode
4185:Bibcode
3814:is the
3724:â 1ÎŒm.
3680:is the
3539:is the
3529:is the
3407:is the
3383:kelvins
3371:is the
3358:is the
3345:is the
3332:is the
3088:Tokamak
3022:Density
3019:Plasma
2891:charge
2845:is the
2830:is the
2817:is the
1149:is the
896:with a
767:is the
32:plasmas
4569:Length
4525:. NY:
4419:
4409:
4360:
4335:
4310:
4243:
4203:
4147:
4116:
4085:
4060:
3929:where
3797:where
3651:where
3522:where
3325:where
3216:In an
2798:where
2476:where
1120:where
771:, and
697:where
108:, and
38:, the
4261:(PDF)
4201:S2CID
4175:arXiv
3820:molar
3767:0.304
3554:molar
3220:or a
2889:ionic
1474:, by
4417:PMID
4358:ISBN
4333:ISBN
4308:ISBN
4274:2017
4241:ISBN
4163:See
4145:ISBN
4114:ISBN
4083:ISBN
4058:ISBN
3808:(nm)
2858:and
2306:).
96:. A
34:and
4487:doi
4452:doi
4407:PMC
4399:doi
4193:doi
3964:dop
3203:10
3197:10
3186:10
3183:10
3180:10
3177:10
3169:10
3166:10
3163:10
3160:10
3152:10
3149:10
3146:10
3143:10
3135:10
3132:10
3129:10
3126:10
3118:10
3112:10
3109:10
3101:10
3098:10
3095:10
3092:10
3082:10
3076:10
3073:10
3064:(m)
3051:(T)
3041:(K)
3031:(m)
2596:).
2341:as
2217:is
1486:exp
1301:exp
1041:exp
900:at
310:in
76:or
30:In
4540::
4485:.
4475:69
4473:.
4450:.
4440:70
4438:.
4415:.
4405:.
4397:.
4385:.
4381:.
4294:^
4282:^
4263:.
4199:.
4191:.
4183:.
4171:96
4169:.
4039:24
4037:.
3700:24
3693:10
3631:24
3624:10
3480:10
3200:â
3194:1
3115:â
3079:â
2634:),
2571:.
1212::
1205:.
993:,
844:,
740:,
558::
493:,
104:,
4529:.
4516:.
4493:.
4489::
4481::
4458:.
4454::
4446::
4423:.
4401::
4393::
4387:7
4366:.
4341:.
4316:.
4276:.
4207:.
4195::
4187::
4177::
4153:.
4122:.
4091:.
4066:.
3961:N
3955:q
3949:T
3943:B
3940:k
3934:Δ
3910:p
3907:o
3904:d
3899:N
3893:2
3889:q
3883:T
3877:B
3872:k
3861:=
3855:D
3850:L
3812:I
3802:Îș
3782:)
3778:M
3774:(
3771:I
3762:=
3759:)
3755:m
3752:n
3748:(
3743:1
3722:B
3719:λ
3665:B
3636:I
3614:A
3609:N
3602:B
3590:8
3586:1
3581:=
3576:1
3550:0
3547:C
3543:,
3537:F
3533:,
3527:R
3504:0
3500:C
3494:2
3490:F
3484:3
3473:2
3468:T
3465:R
3460:0
3449:r
3436:=
3431:1
3411:,
3395:e
3385:,
3379:T
3375:,
3369:B
3366:k
3362:,
3356:r
3353:Δ
3349:,
3343:0
3340:Δ
3330:I
3307:I
3302:2
3298:e
3294:2
3289:T
3283:B
3278:k
3272:0
3261:r
3248:=
3243:1
3226:Îș
3062:D
3059:λ
3049:B
3039:T
3029:e
3026:n
2986:2
2981:e
2977:q
2971:e
2967:n
2959:e
2955:T
2948:B
2943:k
2937:0
2925:=
2919:D
2899:e
2897:q
2895:j
2893:z
2885:j
2880:j
2878:n
2871:e
2869:n
2862:i
2860:T
2855:e
2853:T
2849:,
2842:e
2838:q
2834:,
2828:B
2825:k
2821:,
2815:0
2812:Δ
2806:D
2803:λ
2780:j
2776:T
2771:/
2765:j
2761:n
2755:2
2750:j
2746:z
2740:j
2732:+
2727:e
2723:T
2718:/
2712:e
2708:n
2700:2
2695:e
2691:q
2686:/
2679:B
2674:k
2668:0
2656:=
2650:D
2622:1
2619:=
2614:r
2601:(
2584:(
2559:e
2536:j
2512:e
2508:/
2502:j
2498:q
2494:=
2489:j
2485:z
2464:,
2459:2
2455:/
2451:1
2443:)
2437:2
2432:j
2428:z
2421:0
2416:j
2412:n
2406:N
2401:1
2398:=
2395:j
2383:B
2370:4
2366:(
2361:=
2355:D
2326:B
2285:D
2275:/
2271:r
2264:e
2257:r
2248:4
2244:Q
2239:=
2236:)
2232:r
2228:(
2205:)
2201:r
2197:(
2191:Q
2188:=
2182:t
2179:x
2176:e
2144:)
2140:r
2136:(
2130:t
2127:x
2124:e
2109:)
2105:r
2101:(
2093:2
2084:D
2075:=
2072:)
2068:r
2064:(
2056:2
2028:D
2001:2
1997:/
1993:1
1988:)
1980:2
1975:j
1971:q
1964:0
1959:j
1955:n
1949:N
1944:1
1941:=
1938:j
1928:T
1922:B
1917:k
1906:(
1901:=
1895:D
1841:)
1837:r
1833:(
1827:t
1824:x
1821:e
1807:j
1803:q
1797:0
1792:j
1788:n
1782:N
1777:1
1774:=
1771:j
1759:)
1755:r
1751:(
1743:)
1736:T
1730:B
1725:k
1717:2
1712:j
1708:q
1701:0
1696:j
1692:n
1683:N
1678:1
1675:=
1672:j
1663:(
1659:=
1656:)
1652:r
1648:(
1640:2
1610:.
1604:T
1598:B
1593:k
1587:)
1583:r
1579:(
1570:j
1566:q
1556:1
1549:)
1542:T
1536:B
1531:k
1525:)
1521:r
1517:(
1508:j
1504:q
1493:(
1462:T
1456:B
1451:k
1444:)
1440:r
1436:(
1427:j
1423:q
1400:.
1397:)
1393:r
1389:(
1383:t
1380:x
1377:e
1364:)
1357:T
1351:B
1346:k
1340:)
1336:r
1332:(
1323:j
1319:q
1308:(
1295:0
1290:j
1286:n
1280:j
1276:q
1270:N
1265:1
1262:=
1259:j
1247:=
1244:)
1240:r
1236:(
1228:2
1193:j
1171:0
1166:j
1162:n
1134:B
1129:k
1108:,
1104:)
1097:T
1091:B
1086:k
1080:)
1076:r
1072:(
1063:j
1059:q
1048:(
1035:0
1030:j
1026:n
1022:=
1019:)
1015:r
1011:(
1006:j
1002:n
977:j
949:)
945:r
941:(
936:j
932:n
911:T
880:)
876:r
872:(
860:j
856:q
828:)
824:r
820:(
791:t
788:x
785:e
753:0
726:0
716:r
685:,
682:)
678:r
674:(
668:t
665:x
662:e
650:)
646:r
642:(
637:j
633:n
626:j
622:q
616:N
611:1
608:=
605:j
593:=
590:)
586:r
582:(
574:2
542:)
538:r
534:(
506:r
476:r
455:)
451:r
447:(
442:j
438:n
412:j
408:q
387:j
367:N
328:0
322:T
298:)
295:T
289:B
284:k
280:(
276:/
270:2
266:q
262:n
256:4
253:=
248:2
242:D
237:k
216:T
196:q
176:n
153:D
143:/
139:1
136:=
130:D
125:k
112:(
94:e
72:(
56:D
20:)
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