IEE REVIEW

Electrical ionisation and breakdown of gases in acrossed magnetic field

A.E.D. Heylen, Ph.D., D.SdEng.), C.Eng., M.I.E.E.

Indexing terms: Breakdown and gas discharge, Ionisation, Magnetic fields

Abstract: A unified approach to the subject of electrical ionisation and breakdown of gases in the presence ofa crossed magnetic field is presented by invoking the equivalent reduced electric field (e.r.e.f.) concept whichis a generalisation of the previously accepted equivalent gas density (pressure) principle. The e.r.e.f. conceptis derived from basic electron trajectories in a simplified way using a geometrical approach rather than from afictitious, but later shown to be measurable, equivalent gas density; this method, it is hoped, should appeal toengineers. The ej.e.f. approach immediately leads to an almost satisfactory interpretation of breakdowncharacteristics but in order to obtain better agreement with experiment, the influence of a crossed magneticfield on the primary and secondary prebreakdown ionisation processes is considered. Although coaxialsystems provide unrestricted electron motion in the E X B direction, and are usually preferred, it is shown thatparallel plate electrodes yield as good or even better agreement with theory. For the case of restricted motiondue to wall losses which occurs below the Paschen minimum, the usual equations have to be suitablymodified. The influence of the crossed magnetic field on the salient minimum breakdown voltage is thendealt with, and by considering in more detail the ionisation-current build-up, the sideways Townsendpropagation in the E X B direction is considered. An almost chronological survey of the literature traces thedevelopment of the subject from its earliest beginning, and a number of applications are given. The presentstate of the art is in a concluding section, and likely future developments are outlined.

List of symbols

BNB/NB/pEEIN,(E/N)B,(E/N)0

E/p

Femv

Vy

t

= magnetic field strength, T= gas number density, cm"3

= reduced magnetic field, T cm3

= reduced magnetic field, T torr"1

= electric field, V cm"1

= reduced electric field in general,in a magnetic field and in itsabsence, respectively

= equivalent reduced electric field(e.r.e.f.) Vcm2

= reduced electric field, V cm"1

torr"1

= force,N= charge of electron, C= mass of electron, Kg= resultant velocity of electron in

a crossed electric and magneticfield, m s"1

= magnetic electron deflectionangle, degrees

= force on electron in E direction,N

= force on electron normal toplane containing E and B, N

= velocity of electron normal toplane containing E and B, m s-1_

= velocity of electron in the Edirection, m s"1

= time,s= constants in equations, Section 3= (e/m)B, electron cyclotron fre-

quency, rads"1

Paper 648A, Commissioned IEE ReviewDr. Heylen is with the Department of Electrical & Electronic Engin-eering, University of Leeds, Leeds LS2 9JT, West Yorkshire, England

IEEPROC, Vol. 127, Pt. A, No. 4, MA Y1980

Ky

d

7> 7//> TO

V VH V

E/cocB, cycloidal radius, cmtransverse electron drift velocityin E direction in the presenceand absence of B respectively,ms"1

perpendicular electron driftvelocity normal to planecontaining E and B, and in thepresence and absence of B,respectively, m s"1

free time, i.e. time betweensuccessive collisions, stotal number of free timesaverage distance travelled byelectrons in E direction during r,mconstant electron moleculecollision frequency in generaland at TV = 3-32 x 1016 cm"3

(1 torr), respectively, s"1

primary ionisation coefficient,cm"1

gap distance over which electronsattain equilibrium energy, cmconstants in Townsend'sexpression for ainterelectrode gap distance, cmsecondary ionisation coefficientin general, in a magnetic fieldand in its absence, respectively;dimensionlessionisation potential of gas, eVvoltage, breakdown voltage in amagnetic field and in its absence,respectively, Velectron mean energy, eVionisations per volt in a magneticfield and in its absence,respectively, V"1

221

0413-702X/80/040221 +24 $01-50/0

(Nd)e

in> \'s)min

- > •

V

XX'

F(e)de

v(e)

ionisations per volt at e.r.e.f.,V"1

equivalent gas density times gapspacing, cm"2

electron-molecule collision cross-section atA^= 3-32 x 1016 cm"3

(1 torr), cm"1

thyratron tube radius, cmexperimental and calculated Ndvalue corresponding to Paschenminimum, respectively, cm"2

experimental and calculatedminimum breakdown voltage,respectively, Vexperimental and calculatedreduced electric field corre-sponding to Paschen minimumrespectively, V cm2

critical reduced electric field,Vcm2

sideways Townsend propagationvelocity, m s"1

positive ion drift velocity, m s"1

cycloidal distance, mdistance travelled in E directionby electrons during X, mI/NoQ = electron mean freepath at W=3-32x 1016 cm"3

(1 torr),melectron energy distributionfunction, dimensionlessenergy dependent electron col-lision frequency at N = 3-32 x

(1 tones'1

1 Preamble

The subject of the influence of a magnetic field, appliedperpendicularly to the electric field, on the ionisation andbreakdown mechanism in gases is hardly, if at all, treated instandard texts or reference books, not even Meek andCraggs's latest book.66 This is possibly because of theapparent complexity of the processes involved which hasprevented a clear understanding of the subject until 1958.Even so, after then, claims and counterclaims about theapplicability of the equivalent reduced electric fieldconcept, or what amounts to almost the same, theequivalent gas density concept, have abounded and it is notuntil very recently that the subject has been firmly put ona sure footing.

It is thus thought timely and appropriate to bringtogether, in one review paper, a unified approach to thesubject of the topics scattered about in the literature and topresent as concisely and as lucidly as possible the under-lying principles involved of what happens when a magneticfield is applied perpendicularly to an electric field in aTownsend discharge. It will be shown that an appreciationof this leads to an understanding of a number of applicationswithin this field and in related areas.

2 Introduction

The first thing to appreciate is that a magnetic field onlyinfluences the ionisation and breakdown voltage of gaseswhen the gas density is low, corresponding to the regionaround the Paschen minimum and below, despite the fact

that in a solid the Hall effect is appreciable. At gas densitiesnear to atmospheric pressure and above, the effect isnegligible and can only be detected by special measuringtechniques. This is because, as we shall show, the basicparameter which influences the breakdown process is B/Nwhere B is the magnetic field (in Teslas) and N is the gasnumber density (in cm"3). Thus, as the gas number densityincreases, the reduced magnetic field BIN decreases untilat atmospheric pressure and above, its influence is difficultto detect. Another way of looking at this dependence onBIN is to consider the electron mean free path. This isinversely proportional to the gas number density so that,the larger N, the smaller is the electron mean free path andhence the smaller is the bending influence due to themagnetic field. The reduced magnetic field B/N, acharacteristic of the magnetic flux, is analogous in itsfundamental importance to the reduced electric field E/N,which is the basic parameter when a high voltage alone isapplied to a gas. It will be shown that, in the presenceof a crossed magnetic field, the equivalent reduced electricfield (E/N)e, applies.

The second point is that a magnetic field primarilyinfluences the electrons in a gas discharge. This is becausethe force, F (newtons), on a charged particle travelling at avelocity v (m s"1) in a magnetic field, is given by

F = e•v xB (1)

As the electrons have a small mass compared with the ions,their velocity is much higher under the influence of anelectric field and consequently the perpendicular force onthe electrons is much greater and it is just the electronswhich are responsible for ionising the gas. A secondary,though smaller, consideration is that the mean free path ofelectrons is also greater than that of ions and thus theperpendicular deflection experienced before anothercollision occurs is greater for electrons.

3 Magnetic electron deflection angle 6

Similar to the Hall effect in semiconductors, the ionisingelectrons in a gas are deflected through an angle 6 to theelectric field due to the crossed magnetic field. We willexamine which basic parameters determine this angle 6 andto do so we briefly recapitulate the basic equationspertaining in a vacuum before we go on to consider themodification of these equations by the collisions with gasmolecules suffered by the electrons.

3.1 Deflection of electrons in a crossed electric andmagnetic field in a vacuum

Consider the configuration shown in Fig. 1 with an electricfield along the .y-axis and a magnetic field in the z-direction.Suppose the electron starts at the origin with zero velocity;then it will move in a cycloid in a plane containing the xy-axes. The forces Fy and Fx in the y- and x-direction,respectively, are given by

- _ - dvvFv = eE + e - vx x B = m —-y x dt

and

Fv = - -x B = m

dvx

dt

(la)

(2b)

222 IEEPROC, Vol. 127, Pt. A, No. 4, MAY 1980

where m is the mass of the electron, and e = — \e\ for anelectron. Differentiating eqn. 2a with respect to time andmaking use of eqn. 2b yields

'eB

and

dt m

The standard solution to this second-order differentialequation is

vy = a cos <jjct + b sin coct

where the cyclotron angular frequency, coc = eBjm. Theinitial conditions are that (vy )t=0 = 0 = a and (dvy/dt)t=0 =eE/m from eqn. 2a. Differentiation of vy with a = 0 yields

dVy

it cos

and thus eE/m = (e/m) Bb or b = E/B. Substitution there-fore yields

Vy = — sin ioct (3a)B

From eqn. 2b with eqn. 3a we get by integration,remembering that (vx)t=0 = 0,

vx = — (1 — cosco-0

For the cycloid shown in Fig. 2,

vy = cjcr sin cocf

where

0- = co r̂

Fig. 1 /><afft a/id velocity of electron leaving cathode (y = 0) atzero velocity under influence of electric field E, (y-direction) andcrossed magnetic field B (z-direction)

Fig. 2 Cycloidal path of electron for configuration as in Fig. I

IEE PROC, Vol. 127, Pt. A, No. 4, MA Y 1980

vx =cos

Eqns. 3a and 3b are analogous to these and thus theelectrons indeed move in cycloidal paths. Comparisonshows that

and thus the cycloidal radius is given by

Er —

To find the distances covered in the y- and x-directions intime t, we integrate eqns. 3a and 3b, and with suitable initialconditions this yields

m Ey = - To 0 - cos

and

E / mx = — \t — sin coj

B \ eB c

(4?)

(4b)

3.2 Deflection of electrons in a crossed electric andmagnetic field in a gas

In a gas the electrons make incessant collisions with gasmolecules and drift in the electric-field direction with atransverse velocity v%, which is much less than theirrandom velocity. With a crossed magnetic field applied, thistransverse drift velocity is reduced to v% for the geometryshown in Fig. 3 and at the same time the electrons acquire aperpendicular drift velocity vf, so that the resultant driftvelocity under the combined electric and magnetic field isat an angle 6 to the electric field in the xy-plane where

tan0 = ^4- (5)

In between collisions, the electrons move as in a vacuum sothat eqns. 4a and 4b apply but have to be averaged over the

Fig. 3 Perpendicular drift velocity v^ and transverse drift velocityVf of electron swarm in crossed electric (y-direction) and magnetic(into paper) field, making angle 6 with electric-field direction

223

distribution of free times, i.e. the time between collisions,which is given by

dND = ^ e~t/TdtT

(6)

where dND is the number of free times between t and t + dtof Njj free times and r is the average free time betweencollisions. Thus, the average distance travelled in the ydirection y during an average free time is given by

ydND

From eqn. 4a and eqn. 6 this becomes

Therefore

_ r°° m£ l _t/, r°° mh cos cy = —5 - e t/Tdt — —z

Jo eB T Jo ei? T

00 mE 1 _f / r°° miT cos u>ct _t/

or

Thus

rt/T\ _ mE r°°

eB2^ Jocos cj^t e~t/Tdt*

mE mE \ r2e t/T I cossin

or

y =mE mE T

eB2 eB2r

and thus

e r'

Now the transverse drift velocity vf is given by y/r andthus

It is more usual to employ the collision frequency v, wherev= 1/T, and writing v = v0 N where v0 is the collisionfrequency at standard gas density

e E v0T mN vl +

(7a)

By a similar averaging process it may be shown that theperpendicular electron drift velocity vf is given, from eqns.4b and 6, after integration, by

n _ e E <JOC/N1 ~ mNvl+ (coc/N)2 Ob)

*This is a standard integral.70

224

and thus' the deflection angle 6 is given by tan = vf/v% or

e 1 Btan 9 = -

m v0 N(8)

So we observe that the tangent of the magnetic electrondeflection angle 6 is directly proportional to the reducedmagnetic field B/N, as pointed out in the Introduction, andis greater, the smaller the collision frequency v0. Also thesmaller the mass, the greater 6 and thus in ordinarymagnetic fields, the angle 6 is only significant for electronsand small to negligible for ions, as pointed out in theIntroduction.

A typical order of magnitude calculation is B = 1, TV =3-32 x 1016 (corresponding to 1 torr at 20°C), elm =1-76 x 1011 C/kg and v0 for nitrogen equal to 8 x 109 s"1

at 1 torr (see Dargan and Heylen39) yieldingvQ = 2 - 4 X 10~7

S"1 at unit density. This gives, fromeqn. 8, 6 = 87° which is a very substantial deflection nearto the Paschen minimum. At 10 x and 100 x this density,the angle drops to 65° and 12°, respectively, and atatmospheric pressure, 6 is a mere 2°. This illustrates whatwe pointed out in the Introduction, that a magnetic fieldonly influences the electron swarm substantially at gasnumber densities around and below the Paschen minimum.

It should be noted that the above is an approximatederivation as we have omitted to consider the electronenergy distribution, but from an engineering point of viewthis derivation is adequate because if we consider a constantcollision frequency, then the correct formulae reduce tothose above. This is considered in Appendix 18.1 where theHall angle for solids is also briefly treated in Appendix 18.2.

4 Breakdown in a crossed electric and magnetic field

At high electric fields near to the breakdown voltage, theelectrons ionise the gas molecules on collision and electronavalanches are formed. It is clear from the previous Sectionthat the avalanches will be inclined at an angle 6 to theelectric field E. Thus in a crossed magnetic field, theelectric field along which the electrons travel is reducedfrom E to E cos 6, while on the other hand the path lengthin the electric field direction is increased by cos"1 d. A wellknown formula10 for the primary ionisation coefficient aparticularly suitable near the Paschen minimum is

BTNOL/N = AT exp

In a crossed magnetic field this formula changes to

(9)

cos 6 expBTN

E cos 9(10)

due to the two effects described above. Combining thisequation with the breakdown criterion

1 = 7 [exp {a(d — 5)}— 1] (11)

where y is the secondary ionisation coefficient, d is the gapdistance and 5 is that portion of the gap distance overwhich the electrons acquire the equilibrium energy and isgiven by 5 =* VJE, in which Vt is the ionisation potential ofthe gas, we obtain an expression for the breakdown voltage

IEEPROC, Vol. 127, Pt. A, No. 4, MAY 1980

in the presence of a crossed magnetic field, rememberingthat E = Vs/d, as follows:

BTNd sec 6(12)

in which from eqn. 8

sec0 =e_\_B_m vn N,

2W2

(13)

We see that the breakdown voltage now not only dependson the product Nd alone, but that Paschen's law isextended to Vs =f(Nd, B/N) provided 5 is small comparedwith d and j B varies mildly with B. This is the case asillustrated in Figs. 4 and 5. In Fig. 4.the breakdown voltageis shown at two values of Nd for different gap distances asa function of the magnetic field alone. It is seen that thebreakdown characteristic is different for each value of N, dand B. However, plotting the breakdown voltage as afunction of B/N collapses the characteristics on to the samegraph for each Nd value, thus illustrating the extension ofPaschen's law in a crossed magnetic field, as shown in Fig. 5.

It is seen from eqn. 13 that for zero magnetic field,sec 0 = 1 and thus eqn. 12 becomes in the absence of amagnetic field

VB =BTNd

\n{N(d-b)}+\nIn (1 + l/7o).

(14)

1800 r

1700 -

0 0-1 0-2 0-3 0-4 0 5 0 6 0 7 0 8800

Fig. 4 Breakdown voltage V8 in nitrogen against crossed magneticfield39

At Nd = 2 X 10 l s cm"2 (6 torrcm):o d = 0-3cmx 0-6 cm• 0-9 cmAt Nd= 3 X 10'scm~2 (9 torrcm):A d = 0-3cmv 0-6 cm

IEEPROC, Vol. 127, Pt. A, No. 4, MAY 1980

Comparison of eqn. 14 with eqn. 12 shows that with amagnetic field the value oiNd is simply increased to Nd sec0 or the gas number density N is increased to N sec 0. This'is known as the equivalent gas number density concept(previously known as the equivalent pressure concept19

because of the relation between density and pressure). Thusthe effect of a magnetic field is effectively to increase thegas number density. Thus if the initial Nd value is aboveand to the right of the Paschen minimum, the breakdownvoltage will increase with increase in BIN; below thePaschen minimum, the opposite takes place whereas at thePaschen minimum no change in breakdown voltage occurs.

Paschen characteristics for various values of BIN areshown typically for nitrogen in Fig. 6. It is seen that, asmost of the Nd values are to the right of the Paschenminimum, the breakdown voltage for a given Nd valueincreases with increase in BIN in conformity with thetheory. The few Vs values below the Paschen minimum areseen to decreases initially with increase in B/N. By applyingthe equivalent gas number density concept, outlined above,the breakdown voltages are shown plotted in Fig. 7 as afunction of (Nd)e =Nd sec 0. It is seen that all thecharacteristics collapse onto a single Paschen curve withina maximum deviation of only + 10% in voltage and —15%in (Nde). This evidently satisfactory state of affairs wouldalmost close the subject but it can be shown71 that whilstsimilar agreement is reached for argon, the deviation inhydrogen amounts to —30% in voltage and in ethane thedeviation is —36-5% and is as large as 536% in (Nd)e. So wehave to dig a bit deeper to find out more accurately thechange in V$ with BIN and actually to measure the value ofthe all important electron collision frequency whichdetermines the deflection angle according to eqn. 8.

1800

1700

1600

1500

U00

1300

1200

1100

1000

900

800

B/p.T torr"1

002 004 006 008 010 0 12 0 U

AAy

06 12 18 24 30B/NxiO18. Tern3

3 6 U2

Fig. 5 Breakdown voltage V8 in nitrogen against reduced magneticfield B/N in nitrogen,39 data as for Fig. 4

Note collapse of graphs in Fig. 4 onto two graphs only in the Figurecorresponding to a given Nd

225

It should be pointed out that the collapse onto a singlePaschen curve, shown in Fig. 7, does not depend on theapplicability of eqn. 9 but only on the equivalent gasnumber density principle, but, if eqn. 9 is used, then thevalue of the secondary ionisation coefficient and its changewith the magnetic field, which is mainly responsible for theaforesaid deviations, can be calcualted as shown in Section9.

The Paschen curves shown in Fig. 6 are actually takenfrom experimental plots showing the increase in breakdownvoltage with H/N for various Nd values. Two suchexamples, for hydrogen and argon, are given in Figs. 8 and9, respectively. It is seen that in both graphs the Vs — B/Ncharacteristics bend over at large B/N and low Nd, and thiswill be referred to later. No such bending over occurs in thecharacteristic for ethane gas. This leads us to the equivalentreduced electric field concept (e.r.e.f.).

5 Equivalent reduced electric field, [EIN )e

It has been shown that when a crossed magnetic field isapplied to an electron swarm, the electrons drift at an angle

1300

6 A 4

NdxiO'^crrf235 40 45

Fig. 6 Paschen characteristics in nitrogen for various reducedmagnetic fields B/N X 10'* Tern3

o B/N = ox 0-6• 1-2+ 8-4

1300

= 1-82-43-6

• B/N - 4-8* 6 0• 7-2

(pdL. torr cm10 11 12 13

Fig. 7 Paschen characteristics of Fig. 6 with same data plotted asfunction of equivalent gas number density times gap distance(Nd)e=Ndsecd

Line through points for B/N = 0

226

Haydon

20012 24 36 48 60 72 84 96

B/Nx10l.8Tcm3

Fig. 8 Breakdown voltage for hydrogen against reduced magneticfield61 for various Nd X 10~11 cm'2 values

Values in brackets are pd in torr cmtheory

o o o o experiment

650r

600-

20012 24 36 48 60

B/N x1018. Tern3

7 2 8 4

Fig. 9 Breakdown voltage for argon against reduced magneticfield61 for various Nd X 10~xn cm'2 values

Values in brackets are pd in torr cmtheory

o o o o experiment

IEEPROC, Vol. 127, Pt. A, No. 4, MA Y1980

0 to the electric field. This means that the electric fieldexperienced by the electrons is E cos 8. As the basicindependent variable governing all transport quantities isthe reduced electric field E/N (see for instance eqns. la, 1band 9), this in turn is reduced to E/N cos 0 and thus themean energy of the electrons e is also reduced by themagnetic field. Now the dependence of e on E/N is acomplicated one which is usually derived experimentallyand compared with involved theory; also, e, the dependentvariable of E/N, determines most of the electron transportquantities such as drift velocity, diffusion and ionisation.Thus the application of a magnetic field would mean thatall these quantities would have to be determined for a givenE/N at each value of B/N, which would be very lengthy andlaborious. A method to obviate this is to postulate thepresence of an equivalent reduced electric field (E/N)e

which is the reduced electric field in the absence of amagnetic field required to keep the mean electron energythe same as in the presence of a magnetic field. This e.r.e.f.concept is established in the following manner.

In a magnetic field, the reduced electric field alongwhich the electrons travel is E/N cos 0 corresponding to agiven mean energy e. This reduced electric field is equivalentto (E/N)e without a magnetic field to keep e the same.Thus all the basic fomulas for the transport quantitiesremain unaltered and from the above

or

(E/N)e = E/N cos 8

(E/N)e = E/N (1 + I - x - x j\m v0 N

2\-l/2

Thus we see that when H = 0, (E/N)e = E/N, the reducedelectric field at zero magnetic field.

Another way of looking at it is to write eqn. 15b as

E/N = (E/N)e1 J . I e l B

1 + |— x — x —m v0 N

2\ 1/2

(16)

(15a)

Thus when a magnetic field is applied, the working E/Nhasto increase according to the eqn. 16 in order to keep econstant.

The relationship between E/N and {E/N)e is shown inFig. 10 for hydrogen, which was obtained by working outthe integrals in Appendix 18.1 for greater accuracy. At lowB/N, E/N = (E/N)e, whereas at high B/N, E/N = (E/N)e

(e/m x \/v0 x B/N) and thus a plot of E/N against B/Nslants at 45° as shown. In between these two extremes ofweak and strong reduced magnetic field is the intermediateregion, the range of which varies with the mean electronenergy on account of the dependence of the electroncollision frequency v(e) on electron energy. Similar graphsfor oxygen, nitrogen and air have been obtained.43

10

o 10

10

E/Nx1017.vcm2

10T I 1—1 I I I 1 1 1 1 1—Tyr-T

10'

CO

6 8 10'

Fig. 10 Calculated dependence of equivalent reduced electric field (E/N)e for hydrogen on reduced magnetic fields*0 B/N for variousconstant average electron energies e, in eV

Horizontal arrows show the limits of medium B/N region. Inclined arrows represent experimental data of Bernstein24 for strong magnetic fieldcondition

IEEPROC, Vol. 127, Pt. A, No. 4, MA Y1980227

6 Application of e.r.e.f. concept

The e.r.e.f. concept is a generalisation of the equivalent gasnumber density concept and eqn. 15a can be written as

(17a)

(176)

Thus for the same voltage, from eqn. 17a

(Nd)e = (Nd)secd

which is the equivalent gas number density times gapdistance which was shown to apply in Section 4. If now thegap distance is also kept constant then from equation 11b

NP = TV sec (17c)

which is indeed the equivalent gas number density concept,as outlined in Section 4 and propounded by Blevin andHaydon.19 We shall see in Section 8 that the e.r.e.f. conceptprovides an easy method of obtaining the all-importantelectron collision frequency which, through eqn. 8,determines the electron magnetic deflection angle, fromsimple breakdown voltage determinations.

Eqn. 156 can be written, by squaring, rearranging andmultiplying throughout by elm as

1 _ L [ ^ | IE) = 1 £ u° E

m u0 \Nje X \ N ) . ~ mXNX ul+ (ujNfN

and from eqn. la, with v? corresponding to B, and thusCJC = 0, this yields

(18)

This application of the e.r.e.f. concept shows that a plot ofthe transverse electron drift velocity v% against E/N shouldyield a straight line with slope —45° for various reducedmagnetic fields as (vj<)e and (E/N)e remain constantcorresponding to a given mean electron energy e. Such aplot is shown in Fig. 11 for hydrogen evaluated from theintegrals given in Appendix 18.1. Similarly, the perpen-dicular electron drift velocity v± and deflection angle tan 6may be obtained as shown in Figs. 12 and 13. Similargraphs for oxygen, nitrogen and air have been obtained.43

E/Nx1017.V cm2

A 6 8 10"

Fig. 11 Calculated transverse electron drift velocity V? in hydrogen as function of reduced electric field"0 E/N for various constant averageelectron energies e in e V, and reduced magnetic field B/N where B/N = B/p (T/torr) X 3 X 10''7 Tcm3

Inclined arrows represent experimental data of Bernstein24 for strong magnetic field condition

228 IEEPROC, Vol. 127, Pt. A, No. 4, MAY 1980

7 Ionisation in a crossed magnetic field

The ionisation per volt in the presence of a crossedmagnetic field is given by

1/21

ml vf(F/N)F(e)de (19a)

In which No Qt is the ionisation cross-section and F(e)de isthe electron distribution function. At the equivalentreduced electric field

,9 = 2J1/2

1

m (v°T)e(E!N)eN0QiemF(e)de(\9b)

As the electron energy is the same, the term under theintegral sign is the same in both equations, provided nochange in the distribution function occurs due to themagnetic field. As the e.r.e.f. condition (eqn. 18) holds, wesee that

rfe (20)

and this gives a first method of determining the electroncollision frequency.

An example of ionisation in a crossed magnetic field fornitrogen is given in Fig. 14. It is seen that with increase in

B/N (or B/p), the slope of the conductivity curves, which isproportional to the primary ionisation coefficient,decreases. Note the gradual reduction in the initial photo-electric current /0, due to electron recapture by thecathode, with increase in BjN.

A collection of ionisation curves rj, is shown in Fig. 15.At low F/N, the magnetic field reduces ionisation and thusincreases the breakdown voltage as has already beenpointed out, whereas at high E/N (low Nd) the ionisation isincreased and consequently the breakdown voltage isreduced. At a constant ionisation per volt in the presenceand absence of a magnetic field, the e.r.e.f. concept appliesand consequently eqn. 16 is valid. Thus any horizontal linedrawn on Fig. 15 cuts the 17° curve at (F/N)e and a r?B

curve at EfN. Therefore for one value of (E/N)e, variousvalues of F/N at different B/N values are obtained and bythe application of eqn. 16, v0 may be determined for agiven (E[N)e; this process is repeated for various horizontallines. This is a valuable though cumbersome way ofobtaining the all-important electron collision frequency andvalues so obtained and by other methods and theory fornitrogen are shown compared in Fig. 16. It is seen that withincrease in (F/N)e, v0 increases slowly.

The satisfactory agreement with experiment of theinfluence of a crossed magnetic field on the primaryionisation coefficient oc/N, given by eqn. 10, is illustrated

E/Nx1017.V cm2

1 1—1—1 1 1 1 11 1 1—1 j i n n 1 1—1—1 1 1 111 1 1 1 1 r T 111

10- . V m " Torr"p

6 8 105

Fig. 12 Calculated perpendicular electron drift velocity vf in hydrogen as function of reduced electric fieldw E/N for various constantaverage electron energies, e, in eV, and reduced magnetic field B/N where B/N = B/p (T/torr) X 3 X 10'xl Tern3

Horizontal arrows represent experimental data of Bernstein24 for strong magnetic field condition

IEEPROC, Vol. 127, Pt. A, No. 4, MA Y1980 229

for nitrogen in Fig. 17, where the collision frequency usedis that determined by the method given in the next Section.

Comparing eqn. 10 with eqn. 9, we see that a crossedmagnetic field increases both the constants AT and BT bysec 0. It is useful to examine whether, for a given E/N, theionisation coefficient a/N is increased or decreased by acrossed magnetic field. Differentiating eqn. 10 with respectto 0 yields the fractional change in a/N namely

da/N/de I BTsecd= tan 0 1 —

a/N E/N(21)

Thus a/N increases or decreases with a crossed magneticfield depending on whether E/N is greater or smaller thanBT sec 0, or for small magnetic fields BT, the Townsendionisation constant.

8 Determination of the electron collision frequency

In the presence of a crossed magnetic field, the breakdowncriterion according to eqn. 11 is given by

VBVS = l n d + T B 1 )

This corresponds to the breakdown voltage at the equivalentreduced magnetic field

V°e Vs = l n O + y ; 1 )

Assuming y to change according to the e.r.e.f. concept, andholding the breakdown voltage constant in the presence ofa magnetic field by either changing the gas number densityor the gap distance, the equality expressed by eqn. 20 holds

and the e.r.e.f. given by eqn. 17a applies. Thus byrearranging eqn. 17a and making use of eqn. 13, we get

(Nd)2 =-[-—] (Nd)2 + (Nd) (22)

Therefore a plot of (Nd)2 against (NBd)2 should yield astraight line of slope —(ejmv0)

2 from which vQ can befound. This has been called the equivalent gas numberdensity times gap distance method39 and a typical plotkeeping the gas number density constant is shown in Fig.18 for nitrogen. The values of u0 so deduced are shown inFig. 16 and good agreement with those obtained by themuch more cumbersome ionisation method is observed. Itwill be noted that the graphs curve over at large (Bd)2

indicating that the collision frequency increases withincrease in B even though the e.r.e.f. concept applies. Thisis wholly in agreement with the calculated v0 values derivedfrom the formulas given in Appendix 18.1. In commercialvalves it may not be possible to vary the gap distance, andso a change in the gas number density to keep Vs constantwould be a practical alternative.

A complementary but more cumbersome method wouldbe to hold the prebreakdown current rather than thebreakdown voltage constant for various magnetic fieldvalues. A similar graph could be drawn and the value of vQ

deduced. This would have the advantage that the ionisationcurrent could be set to such a value that the secondaryionisation coefficient does not contribute to the currentand any variation of y with B/N would not matter. Thedisadvantage, however, would be that the initial photo-

E/Nx1017Vcm2

U 6 8 105

Fig. 13 Calculated dependence of tan 6 = V^/Vx in hydrogen on reduced electric field*0 E/N for various constant average electron energiese in e V, and reduced magnetic field B/N where B/N = B/p (T/torr) X 3 X 10~'7 Tern3

Inclined arrows show experimental data of Bernstein24 for strong magnetic field condition

230 IEEPROC, Vol. 127, Pt. A, No. 4, MAY 1980

electric current /0 would have to remain constant withincrease in magnetic field, which is not the case, as shownin Fig. 14. This latter disadvantage is eliminated in thebreakdown method as the current is self-sustaining and thusindependent of/0.

9 Influence of a crossed magnetic field on thesecondary ionisation coefficient

It has so far been assumed that the magnetic field has noinfluence on the secondary ionisation coefficient 7. It iswell known that in breakdown, secondary ionisation is duemainly to positive ions falling on the cathode therebyliberating further electrons and is also due to photonsfalling on the cathode releasing additional electrons. Bothprocesses are dependent on the electrode material. Near the

B/p=C

0 2 OGd, cm

0 8 10

Fig. 14 Ionisation current as function of gap distance d in nitro-gen™ for various values of B/N

B/N = B/p (T/torr) X 3 X 10'17 Tern3 for given reduced electricfield E/N= 3 X 10"15 Vcm2 (E/p = lOOVcm"1 torr"1)

E/Nx1015Vcm2

12 18

200 A00 600 800E/p.V crn'torr"'

1000

Fig. 15 Ionisation coefficient TJ in nitrogen as function of reducedelectric field"6 E/N for various reduced magnetic fields B/N X10lB Tern3

• B/N = 0X 0-9a 1-5

a B/N =2-1o 3-0+ 4-5

Paschen minimum where the effect of the reduced magneticfield is mostly felt, secondary ionisation by positive ions isgenerally the predominant process so that this will beconsidered first.

As ions are heavy compared with electrons, their,cyclotron frequency is very much smaller, so much so that

1312

(E/N)x10*V cm2

7U 36 6 0 7 2

o ji

80 120 160 200(E/p) .Vcm-'torr-1 at 18°C

240

Fig. 16 Calculated and experimental electron collision frequencyin nitrogen as function of equivalent reduced electric field** (EN)e

for various reduced magnetic fields B/N = B/p (Tesla/torr) X i X10'11 Tern3

Experimental:0 Bagnall and Haydon,33 conductivity methodX Fletcher and Haydon,36 conductivity method1 Heylen and Dargan,48 conductivity method

T Dargan and Heylen,39 equivalent Nd method

v0 increases with increase in B/N for given (E/N)e as shown. Calcu-lated values • shown by arrow and giving corresponding B/p.

T 1-0-

OK)

U 6 8 10 12 Up/E. torr cm 'V ' x 103

Fig. 17 Comparison between experimental and calculated primaryionisation coefficient a/N in nitrogen versus recriprocal of reducedelectric field™ N/E for various B/N X 10ls Tern3

• B/N=0 a B/N =2-1x 0-9 o 3 0A 1-5 + 4-5Solid lines represent calculated values

IEEPROC, Vol. 127, Pt. A, No. 4, MA Y 1980 231

the magnetic field has a negligible influence on themovement of the ions. Thus the e.r.e.f. concept does notapply to ions and the 7 coefficient will be given, not by(E/N)e but by the working E/N. However, the electrons,released by these ions falling on the cathode, will move incycloidal paths and it is possible at high magnetic-fieldvalues that some of the released electrons will be recapturedby the cathode, thus leading to a reduction in the 7 process.The fraction recaptured has been estimated by Somerville12

as

exp - 8 -m (E/N)N0Q

{BIN)7 (23)

on the basis of an electron executing a complete cycloidgiven by 8 (m/e) (E/B2) compared with the mean free pathl/N0Q. This applies when electrons leave the surface withzero velocity and has been considered in detail by Heylen67

who also took into account the general case when electronsleave the surface with some velocity which, as expected,reduces their chance of recapture.

For photons falling on the cathode, the e.r.e.f. principleapplied as photons are produced by the original ionisingelectrons and thus the number of photons falling on thecathode at a given E/N will depend on B/N.

It is significant that no directly measured values of thesecondary ionisation coefficient in a crossed magnetic fieldexist. Indirectly, however, these can be obtained frombreakdown characteristics and the use of the primaryionisation coefficient, according to, from eqn. 12

InJ_\_ ATN(d -8) sec

BTNd sec 07B

exp

(24)

A plot of yB as a function of E/N so obtained for H2 andAr shows a confused67 variation with H/N. However, forhydrogen the increase in 7 with increase in E/N for zeromagnetic field indicates that positive ions are responsiblefor secondary electron emission. Applying to these valuesSomerville's formula (eqn. 23) for recapture yields thecurve on the right-hand side of Fig. 19. Subtracting thesepositive ion secondary-ionisation contributions from theoriginal curves leaves us with the net photon electric effectto which, as indicated, the e.r.e.f. concept applies. Thusplotting these values as a function of (E/N)e yields the

2-33(70)

0-100 140 280 420 560 700 840 980 1120 1260 1400

Fig. 18 Plot ofd* against (BdJ2 in nitrogen to determine electroncollision frequency from slope39

Numbers are gas density, AT X 10"18 cm"3; those in brackets are p intorr

232

remarkably consistent photoelectric coefficients shown onthe left-hand side of Fig. 19. Thus it is seen that at low(E/N)e, corresponding to large B/N, the 7 coefficientincreases rapidly with decrease in (E/N)e and thismechanism is responsible for the flattening off of thebreakdown characteristic shown in Fig. 8.

In contrast to hydrogen, Ar shows that with increase inE/N, the 7 coefficient decreases. This suggests that photo-electric ionisation is solely responsible for the secondaryionisation coefficient. Applying the e.r.e.f. conceptcollapses all the original curves, save the one at the lowestNd values, onto a single curve as shown in Fig. 20, thusvindicating the e.r.e.f. principle. This may seem surprisingas it might have been expected that the 7 curves, as forpositive ions, are subject to electron recapture. However itis suggested67 that electrons liberated by the energeticphotons are released with a considerable escape velocitybecause the excitation potential of a gas is usually muchlarger than the work function of a metal; this substantiallyreduces electron recapture. In nitrogen, the photoelectriccontribution is small and in ethane it is nonexistent so thatbreakdown characteristics for these gases do not show anyflattening-off effect.

It can be concluded that the original deviations, shownin Fig. 7, in accordance with the e.r.e.f. principle werewholly due to the variation of 7 with magnetic field onaccount of recapture and onset of photoelectric secondaryionisation.

E/P,v cm"1 torr"1

6 8 102 2 6 8

161

i 6

c *•o

"5•iE.O2

•S 8

o 6•o

§ 4

6 8 10° 2 U 6 8 1O1 2

E/Nx1O15.Vcm2

Fig. 19 Secondary ionisation coefficient for hydrogen against re-duced electric field61 E/N for various B/N XlO18 Tern3

o B/N = 0 AB/N= 1-8x 0*6 v 2-4• 1-2 • 3-6+ 8-4Abscissa below E/N = 1-8 X 10~ ' s V cm 2 in equivalent reducedelectric field (E/N)e

IEEPROC, Vol. 127, Pt. A, No. 4, MAY 1980

i B/N = 4-86 07-2

10 Breakdown in a crossed magnetic field well below thePaschen minimum with wall losses

The breakdown characteristic around the Paschen minimumwas considered in Section 4 and it was found that belowthe Paschen minimum the breakdown voltage shoulddecrease with increase in magnetic field. This is true to acertain extent but it has been found37' ** that well belowthe Paschen minimum the breakdown voltage increasesdramatically with increase in even modest magnetic fields,reaching factors of ten times or more. Because of itsapplication as switching tubes, this is now considered. Atypical characteristic obtained in a uniform electric andcrossed magnetic field for hydrogen is shown in Fig. 21.How can this huge increase in breakdown strength beexplained when actually a decrease is expected?

One important aspect is that, in this gas number densityregion, the gas present is very tenuous and thus N is verylow. This means that the reduced magnetic field parameterBIN, even for modest B, is very high with the consequencethat, by eqn. 8, the deflection angle 0 is quite large, in theregion 40° to 80°. Because of this, the electron avalanche isso much deflected that, instead of reaching the anode, ithits the sidewalls of the discharge vessel. This was allowedfor in the design, and conducting rings were placed alongthe wall in order to conduct the captured electrons harm-lessly away. Thus it is clear that the full development ofthe usual avalanche build-up leading, by means of thesecondary mechanism, to breakdown, is severely curtailedand that electron loss mechanisms to the walls take place.

A typical discharge tube in this application is shown inFig. 22. For an angular deflection 6, the electron avalanche

E/PeVcm"1 torr"1

E/Nexi015,Vcm2

Fig. 20 Secondary ionisation coefficient for argon against equiva-lent reduced electric fields (E/N)e for various reduced magneticfields" B/NX1018 Tern3

o B/N = 0X 0-6• 1-2+ 8-4Upper graphlower graph

a B/N = 1-8v 2-4D 3-6

for all values of Nd

• B/N = 4* 6• 7

except Nd

•8•0•2

= 0-3 X 1017 shown in

10'

en 3a

o 3a 10

gas pressure, p, mtorr

JO1 2 3 5 102 2 3 5 10:

T 1 1—I I I

3 6 1015 3 6 1016 3

gas number density , N , cm*3

Fig. 21 Breakdown characteristic in hydrogen for various magneticfields53 B X 103

oB = 0x 1-8A 3-6

• 5 = 5 - 4a 7-2

anode

Fig. 22 Cross-section of experimental tube53

Arrow shows direction of electron avalanche, tilted at angle 6 andintercepted by outer wall at distance de from cathode

IEEPROC, Vol. 127, Pt. A, No. 4, MAY 1980 233

will strike the side walls at a height de from the cathode.It is now postulated that this is the effective gap distance sothat

de = ro/tand (25)

This expression for the gap distance is then substituted ineqn. 12 but only in the denominator, not in the numeratoras the electric field is still given by E = V/d, Thus we obtain

Vf =BrNd sec 6

In (Nr0 cosec 0) + Inl/7 B)

(26)

and from this equation it should be possible to predict thebreakdown voltage in the presence of wall losses.Comparison with eqn. 12 shows that although Vf shouldincrease substantially with sec 0 in the numerator, in eqn.12, sec 0 also appears in the denominator and thus Vf doesnot increase much and even decreases below the Paschenminimum. However, in eqn. 26 cosec 0 now replaces sec 0in the denominator and at large angles of 6, as is the case,cosec 0 m 1, so that the full effect of the term sec 0 in thenumerator is now felt.

There is however one difficulty which prevents thecalculation of Vf from eqn. 26 and that is the ignorance ofthe term yB. Previously (Section 4) we have circumventedthis difficulty by calculating y from the breakdown value atzero magnetic field from eqn. 12 with sec 0 = 1 andassumed that y hardly varies with B/N. However, as will beshown further on, y varies quite markedly andunexpectedly with E/N and moreover the calculated valuesof Vf become increasingly sensitive to the value of 7chosen as one goes further and further below the Paschenminimum. In addition there is the possibility that thebreakdown process involves the wall rings, made in this caseof copper, rather than the stainless-steel cathode, and thus adifferent 7 value would prevail. To overcome all thesedifficulties, the experimental values of Vf were taken fromFig. 21 and with the aid of the constants Ar and BT in eqn.26, available from the literature, the corresponding valuesof 7 were calculated; these are shown in Fig. 23.

It is seen that 7 increases markedly with increase in E/Nindicating that positive ion impact on the cathode is thedominant process (curve 1). With moderate magnetic fieldsand increase in 7 takes place (curve 2) and this was ascribed53

to the copper rings now being involved in the breakdownprocess rather than the stainless-steel cathode. At thehighest B fields used (curve 3) 7 decreases with increase inB at high E/N showing that some recapture of the emittedsecondary electrons takes place. In contrast, at low E/N, 7is increased and shows the characteristic hump due tophotoelectric secondary ionisation. It is well known that 7due to this process increases with decrease in E/N, asoutlined in Section 9, and because the e.r.e.f. conceptapplies, secondary ionisation by photons takes place at ahigher E/N value than normal. As the magnitude and trendin the curves of Fig. 23 seem reasonable and amenable tostraightforward interpretation, it can be concluded that thebreakdown criterion embodied in eqn. 26 spplies at low gasnumber densities well below the Paschen minimum, wherewall losses are active.

Similar conclusions have been reached for the othergases, He, Ar and N2 tested.53 The outstanding feature isthe large increase in breakdown voltage obtainable withvery modest values of B. This is illustrated in Fig. 24. It is

234

noted that at a given gas number density, helium is the gaswhich gives the largest increase in Vf followed by H2, Arand N2 in that order. As the dominating influence is theterm sec 0 in eqn. 26 which decides the increase in Vf it isclear, through eqn. 8, that vQ is the most important gasquantity which determines Vf and it is not surprising thatthe gases with increasingly lower vQ have increasingly largerVf / Vf, helium being the best gas to use having the lowestelectron collision-frequency value.

reduced electric field. E/p .V cm^torr"

3 6 1Cfu 3 6 10reduced electric field , E/N, V cm2

Fig. 23 Secondary ionisation coefficient 7 as function of reducedelectric field" E/N for various magnetic fields B X 103

gas number density . N , cm

1010 6 10 3 6

gas pressure, p. torr10

Fig. 24 Comparison of breakdown ratio with and withoutmagnetic field as function of gas number density" N for magneticfield of 72x1 Q-3 T

Gases:o heliumX hydrogen

A argon• nitrogen

IEEPROC, Vol. 127, Pt. A, No. 4, MA Y1980

Table 1. Characteristic values at and near the Paschen minimum

Gas

N2

H2

C2H6

Ar

Column

Gas constants

AT

cm"1 torr"1

12tt>

ir(b)b

21-5(C)

13-6(b)

1

BT

Vcm"1 torr"1

335(a)

130(5)

321 ( c )

235(b)

2

Calculated

(V \c •l"s'min

V

443

421

746

22-7

3

cm"2

1-32

3-24

2-32

00965

4

y at*{Nd) cm"2

2-42 X 10"3

(12)2-61 X 10"3

(12)104 X 10"8

(4-2)1 61(6)5

Experimental

V

400

245

475

205

6

* </W)£in*

cm"2

0-9

0-9

0-3

1-5

7

IF/N\E- *v c / M "rmn

Vcm2

445

272

1583

136-5

8

(E/N)k*

Vcm2

370

132

360

237

9

Columns 1 and 2: values from literature (a) Heylen and Dargan(b) Meek and Craggs72

(c) Heylen68

Columns 3 and 4: according to eqns. 27 and 28, respectivelyColumns 5, 6, 7, 8 and 9: experimental according to Dargan and Heylen:

*/Vat3-32X 10l 6cm"3 (1 torr)

11 Influence of a crossed magnetic field on theminimum breakdown voltage

As the Paschen minimum breakdown voltage (Vs)min,which occurs at a given (Nd)min value, is a landmark in thePaschen characteristic and occurs right in the middle of itfor crossed magnetic fields, it is of interest to examine whatinfluence a crossed magnetic field has on this salientfeature.

11.2 7 is a function of E/N

Repeating the above procedure by taking into account 7 =f(E/N), we have, in the absence of a magnetic field, fromeqn.14

InAT(Ndfmin\ln(l + l/7)J

(NdYrdy

= 1 +ld(Nd)

7ln(l + 1/7)(30)

//. / Assuming 7 independent of E/N

The minimum quantities can be found by differentiatingeqn. 12 with respect to Nd, neglecting 5, and equating tozero. This yields the calculated values

(27)AT sec 6

and thus the experimental values

E =min

(31)

1 +7ln(l

and

and

(28)

Thus with a crossed magnetic field, (Nd)min is decreasedgradually with increase in B/N from its value in the absenceof a crossed magnetic field. However, it is noted that (ys)

cmi

remains unchanged. Another useful quantity is (E/N)\where

mmmin

(EIN)\'min{Vs)\min

(Nd)cm

= BT sec 6 (29)

This value is given by the slope of a straight line from theorigin on a Paschen characteristic to the point correspondingto (Nd)min and (Vs)min and the slope increases withincrease in B/N from its value equal to BT at B/N = 0.

Now, it has been observed that, in actual breakdowncharacteristics with a crossed magnetic field, the minimumbreakdown is not constant but can be increased or decreased,by a magnetic field, compared with its value at zeromagnetic field, and to explain this, we have to take intoaccount the variation of 7 with E/N which occurs in mostgases in practice.

IEEPROC, Vol. 127, Pt. A, No. 4, MA Y 1980

(ElN)Emin = (32)

Thus if secondary ionisation is due to positive ions, thiscoefficient decreases with decrease in E/N and, as in generalE/N decreases with increase in Nd, the secondary ionisationcoefficient decreases with increase in Nd. Thereforedy/d(Nd) is negative and according to eqn. 30, (Nd)min

decreases below its value given by eqn. 27. Similarly in eqn.31, although the denominator is less than unity, thenumerator is even less and (Vs)min is smaller than its valuegiven by eqn. 28. Finally it is seen that (E/N^in accordingto eqn. 32 is greater than BT as given by eqn. 29 in theabsence of a magnetic field. The opposite changes occur inthe case of secondary ionisation by photons, as for thisprocess, 7 increases with increase in Nd.

The above described effects are illustrated in Table 1.Assuming 7 constant and equal to a value given in column 5well away from the Paschen minimum, the calculatedquantities according to eqns. 27 and 28 are shown incolumns 3 and 4, respectively; it is clear that (V8)

cminl

(Nd)cmin= BT. The experimental values are shown in

columns 6, 7 and 8. For the first three gases (,Vs)min and

235

xin are less than those given in columns 3 and 4, thusshowing that in these gases the secondary ionisation processnear the Paschen minimum is due to positive ions inaccordance with Section 9. For Ar gas, the opposite is thecase and thus in this gas secondary ionisation is due tophotons, also in accordance with Section 9. In the sameway, as expected, (E/N)min is greater than the gas constantBT for N2, H2 and C2 H6 and less than BT for Ar. It is notedthat the changes are {east in N2 and greatest in C2 H6 thusindicating that y changes little with E/N in N2 and a lot inC2H6, again in agreement with the contents of Section 9.

It has been observed by Dargan and Heylen39 that acrossed magnetic field lowers the (Vs)min of N2, H2 andC2H6 whereas that of Ar is increased. This can be explainedas follows. Eqn. 21 shows that primary ionisation in acrossed magnetic field is increased if E/N is greater than BT

thus yielding a lower breakdown minimum. This occurs asshown in Table 1 (compare column 8 with 2) for N2, H2

and C2H6 whereas the opposite occurs for Ar. Moreover,the greater (E/N)^in is with respect to BT the bigger thelowering in (Vs)min which is the case for C2H6. This is truefor a weak magnetic field. For a strong magnetic field(E/N)^nin has to be greater than BT sec 0 by eqn 21 andthus as sec 0 increases, this condition will be more difficultto satisfy at larger B/N and thus strong magnetic fields nolonger reduce (Vs)min below its value in the absence of B.

12 Critical value, (E//V)*

Because a crossed magnetic field alters the Paschen curve asoutlined above, intersections between the Paschen curves ina crossed magnetic field will occur with the one at zeromagnetic field, thus yielding the same value of V* at agiven (Nd)k in the presence or absence of a magnetic field.Thus at this value of (Vs/Nd)k = (E/N)k, called the criticalvalue, the magnetic field has no influence on the break-down value. It is of interest to find (E/N)k as this value isbound up with the accuracy with which fundamentalparameters can be determined in this region of E/N.

Rewriting eqns. 12 and 14 slightly with E/N = VjNd wehave

BT sec 0

In {Nd sec 0) + In

and

In (Nd) + Inl/7o)

As (E/N)8 = (E/N)° = (E/N)k we have, assuming yB to belittle influenced by the magnetic field,

InATNd

[In ( 1 + 1 / 7 )

and substituting

In (sec 0)sec 0 - 1

(sec 0)

Thus for small 0

(E/N)k ^ BT

evenify=f(E/N)

236

(33)

Thus the critical value of E/N occurs at BT and experimentalvalues of (E/N)k are given in Table 1, column 9, whichshould be compared with column 2. It is seen thatexperiment is in good agreement with theory. So, iffundamental quantities, such as the electron collisionfrequency, are determined in the region of (E/N)k, theerror in the derived value will be large as the magnetic fieldhas been shown to have little or no influence on theionisation process. This is borne out by the error analysesof Haydon and Robertson28 and Heylen and Dargan.48

13 Discharge propagation in the Ex B direction

So far we have restricted our view and concentrated on thefirst crossing of the gap by the ionising electron avalanchein a crossed magnetic field, although the breakdownequations of Section 4 take into account the fact that theelectrons and secondaries cross the gap many times beforea breakdown occurs. A detailed picture of what happensduring the breakdown build-up is shown in Fig. 25.

Consider an electron or bunch of electrons to be releasedat the point 0 on the cathode surface. The ionisingelectrons travel in the electric-field direction with a velocityV8 and at the same time are deflected in the perpendicular(E x B) direction with a velocity Kj8 . As they reach thepoint A (or A', A"" or A'", which corresponds to theanode), the resultant positive ions, 90% of which in aTownsend discharge are created within less than 10% of thegap distance near the anode, drift towards the cathode andare hardly deflected at all in the E x B direction. Theystrike the cathode therefore directly beneath the spot onthe anode where the electrons entered and the cycle isrepeated over and over again as shown in Fig. 25. It isinteresting to note from the Figure that the propagationvelocity of the discharge, which is the resultant velocitywith which the electrons and ions move in the perpendiculardirection, is independent of the gap separation. Neglectingthe velocity of the ions compared with the much largerelectron drift velocity, it has been shown32 that thepropagation velocity ~v is given by

v = v+ tan (34)

where v+ is the ion drift velocity. Thus the dischargevelocity is given by the slower ion component in thedischarge and the larger electron deflection angle.

It has been noted that in some cases a time delay occursbetween the instant of impact on the cathode of the

cathode

Fig. 25 Propagation pattern of Townsend discharge in crossedmagnetic field32

IEEPROC, Vol. 127, Pt. A, No. 4, MA Y1980

positive ions and the release of secondary electrons, thetime being of the order of the ion transit time r+ (i.e. timefor ions to cross the interelectrode gap) and this wouldsignificantly slow down the velocity of propagation. If thistime delay, called the site time rs, is taken into account,eqn. 34 is modified as follows:

v = v+ tan T+

+(35)

and the velocity is slowed down by the ratio shown. Ofcourse, now the propagation velocity is gap dependentthrough the term T+.

When secondary ionisation by photon bombardmenttakes place, then

(36)= v i = v T tan

As v% ^ v+, it should be possible from experiment todistinguish which mechanism is mainly responsible forsecondary ionisation.

Watts58 has shown that in ethane gas, where second-ary ionisation only occurs through positive ions, v+ =3-74 x 1012 (E/N)1'2 and tan 0 = 4-77 x 1017 BIN. Thussubstituting into eqn. 34, we get

v = 1-79 x 1030 (B/N) (E/N)1/2 (37)

Experimental measurements of v were made between ski-shaped electrodes with two probes in the cathode somedistance apart to record on an oscilloscope the pulseinduced in each successive probe as the Townsend dischargetravelled over it. A typical result is shown in Fig. 26, wherev is plotted against the independent variable (BIN) (EjN)1'2

as indicated by eqn. 37. It is seen that agreement betweentheory and experiment is excellent. The characteristicshown is smooth but usually steps appear in the graph.62

14 Survey of work carried out in a crossed magneticfield

The survey of work carried out in a crossed magnetic fieldcan be conveniently grouped into four sections. The firstsection deals with all the work from the beginning right upto 1958 when the significant breakthrough was made by

0-82 164 i, 2-46 3-28B / N ( E / N ) 2 x 1 0 TV;2cmA

Fig. 26 Propagation velocity If of Townsend discharge as functionof reduced electric and magnetic field58

Gap distance d = 0-6cm, and gas density N = 1-8 X 10~15 cm"3

(p = 0-54 torr) in ethane gastheory

o o o o experiment

Blevin and Haydon who put the interpretation of resultson an accurate footing through the use of the equivalentgas number density (gas pressure) concept. A few publi-cations after this date from independent sources are alsoincluded in this Section for completeness. The secondsection deals with the pioneering work by Haydon and hisco-workers at Armidale, Australia. The third section re-ports the later work of Heylen and his colleagues at Leeds,England and the final section covers the recent work byIndians, notably that by Govinda Raju and his associates atBangalore, India.

14.1 Initial investigations up to about 1958

J.J. Thomson1 in 1893 was the first to investigate the in-fluence of a magnetic field on the breakdown of a gas andhe found that a crossed magnetic field made the establish-ment of a discharge more difficult. Townsend3 already in1913 explained the influence of a crossed magnetic fieldto be equivalent to an increase in gas density; with hisassociates, Townsend2'6'9 experimented extensively witha crossed magnetic field but often his E/N values were be-low the region where ionisation occurs. In the interveningyears, several workers explored the field but because of theuse of a nonuniform geometry, they established complicateddependencies of the breakdown voltage on the magneticfield. Several uniform-field experiments were subsequentlyperformed but no quantitative data of interest were forth-coming until the first comprehensive work of Meyer5'6 atZurich, Switzerland. He investigated air using circular brassplates of 2-1 cm o.d. with a gap of 1 to 5 mm surroundedby hard rubber and ebonite, i.d. 8-1 mm, and found thatabove and below the Paschen minimum, the breakdownvoltage is increased and that the increase is the same if Bdis kept concept. This is a variant on our preferred indepen-dent parameter B/N, which we use because of its analogyE/N. Meyer explained the increase in Vs to be due to theshorter distance travelled by electrons in the electric-fielddirection between collisions with an applied magneticfield and this therefore confirmed Townsend's equivalentgas density (or pressure) concept. Meyer's pupil, Wehrli,7

eliminated the influence of the sidewalls by using a coaxialelectrode arrangement with outer radius up to 3-2 cm andgap up to 0-5 cm in air. He observed the expected loweringof the breakdown voltage with increase in magnetic fieldbelow the Paschen minimum and established the criticalelectric-field value (E/N)k, dealt with in Section 12, atwhich the magnetic field has no influence on Vs. He alsofound that in a nonuniform field, the magnetic field hasless influence. Wehrli7 was the first to treat the equivalentgas density Ne quantitatively and established that

= AM 1 -\oe/mB'

8EN(38)

This he obtained by considering the length of a cycloidalpath compared with the distance travelled by the electronsin the electric-field direction. Thus the resultant velocity ofthe electrons is given by v = (z/2 + Vy)xn and from eqns. 3aand 3b

IEEPROC, Vol. 127, Pt. A, No. 4, MA Y1980 237

Integrating this expression yields the cycloidal distance X as

4EmL w,r , ( 3 9 )

whereas the distance in the electric field direction X' is givenby eqn. 4a, namely

m EX' = r (1 - C O S C J C 0

e Bor

2mE

eB2i 2 wC*

1 — cos2—-- (40))

Substitution of cos cjcf/2 from eqn. 39 into eqn. 40 yieldsexpr. 38 derived by Wehrli if we remember that gas densityis inversely proportional to free path. Eqn. 38 should becompared with the modern eqn. 17c.

This work was followed by that of Townsend and Gill,9

Valle,11 Somerville12 and Haeffer13'14 and lately by theArmidale group, Blevin and Haydon.18 The earlier workersexamined the problem through the study of individual elec-tron trajectories as outlined above, assuming that the freepath is constant for all electrons and that the electron-gasmolecule collisions are all inelastic. The agreement betweentheory and experiment was far from satisfactory. A moregeneral approach was attempted by Somerville12 andHaeffer13'14 by taking into account the distribution of freepaths about the mean as Townsend and Gill9 had doneearlier; agreement was better but still not satisfactory.Redhead17 developed Haeffer's work at low gas densitieswhere the electrons are not in energy equilibrium with thegas,, which has applications in gas density measurements.

Bernstein24>2S conducted electron-drift and diffusionmeasurements in hydrogen and deuterium which are in goodagreement with the calculations of Heylen and Bunting.40

He also determined the Townsend primary ionisation co-efficient in hydrogen in a crossed magnetic field anddeduced a collision frequency of 3-1 x 109 s"1 which isabout three-fifths of the accepted averaged collision fre-quency. This discrepancy results from changes in theelectron-energy distribution with magnetic field. Phelpsand his associates26'29'31 calculated collision frequenciesfor hydrogen and nitrogen which agree with those in theliterature.

14.2 Work of Haydon and co-workers

A new approach to the equivalent gas number density(pressure) concept was made by the Armidale group. Blevinand Haydon18 considered the transport properties of theelectron swarm; their approach is the same as that ofTownsend and Gill9 except that the latter authors did notallow for the change in the direction of the drifting elec-trons which takes place when a crossed magnetic field isapplied. This new approach, further developed by Heylenand Bunting,40 has been used in this review and givesexcellent agreement with experiment, as shown in Fig. 27compared with the previous latest theories of Somerville12

and Haeffer.13>14 Blevin and Haydon19 also considered theinfluence of the magnetic field on the minimum breakdownthrough its influence on the secondary ionisation coefficientas outlined in Sections 11 and 12.

This analytical work was followed by a series of papers,under the direction of Haydon, describing experimental

verification of the equivalent gas density concept. In thisconcept, as we have seen in Section 5, the important par-ameter is the electron-molecule collision frequency and theaim of the experimental work is to determine this collisionfrequency from the measurements of the variation ofTownsend's primary ionisation coefficient with magneticfield. Using 5-83 cm diameter parallel circular plates and amagnetic field up to 0-2 T, Haydon and Robertson28 de-duced a constant collision frequency of 2-5 x 109 s"1 atN = 3-32 x 1016 cm"3 (1 torr) for hydrogen. This result isbriefly described in a previous communication (Haydon andRobertson),20 at which time Haydon21 also confirmed theequivalent gas density approach to hydrogen. With the sameapparatus, Bagnall and Haydon33 obtained for nitrogen aconstant v0 =8-3 x 109 s"1 but found that this value in-creased with increasing (E/N)e. Because of possible perpen-dicular loss of electrons, Haydon was not satisfied with theuse of parallel plates and so Fletcher and Haydon36 used acoaxial electrode arrangement of diameter 9-9 cm and gap3-4mm, so that the electric field nonuniformly was lessthan 3%, with a magnetic field up to 1 T. This providesunrestricted motion to the electrons but the collision fre-quency values for hydrogen were found to be similar toprevious values. Again they found the collision frequencyto be a function of (E/N)e only and to decrease for hydro-gen and increase for nitrogen with increase in (E/N)e.

The problem was once more examined from a theoreticalpoint of view by Haydon, Mclntosh and Simpson;44 theyshowed that the Townsend primary ionisation coefficient issensitively dependent on the change in the electron energy

1 0

0 8

0 6

&

&

0 4

0 2

0 0 5 10 15 20 2-5 30B/E x lO^.TV'cm

Fig. 27 Ratio of ionisation coefficient with and without crossedmagnetic field as function of ratio of magnetic to electric fieldli

o experimental18

theory18

theory, Somerville,12 Haeffer13'14

238 IEEPROC, Vol. 127, Pt. A, No. 4, MA Y1980

distribution with magnetic field. Fletcher and Walsh49 con-cluded from their work with cylindrical electrodes, diameter3-0 cm, gap 1 mm giving a maximum nonuniformity of theelectric field of 6% and with a magnetic field up to 0-5 T,that their results, obtained from a study of formative time-lags in hydrogen, cannot be interpreted in conformity withthe equivalent gas density concept. It is strange that theseauthors ascribe to photon action an increasing secondaryionisation coefficient with increasing E/N. Blevin andStock51 also reported on prebreakdown current growth atvery low gas densities such that the electron mean freepathis much greater than the dimensions of the vessel. Thiswork has a bearing on that of Redhead17 and Haeffer.13'14

14.3 Publications by Heylen and colleagues

The main aim of Heylen's work using a crossed magneticfield is to find the perpendicular propagation velocity of aTownsend discharge under these conditions so as to shedlight on the mechanisms operating in the cathode fallregion of an arc discharge in the presence of a crossed mag-netic field and to this end a mechanism involving secondaryionisation, illustrated in Fig. 25, was put forward,30 whichcalculation showed to yield good agreement betweenTownsend and arc movement. Encouraged by this, Heylen32

calculated the electron perpendicular and transverse elec-tron drift velocities and in particular the all important mag-netic electron deflection angle 9, as a function of thereduced electric field for various electron energies inhydrogen. The calculated values for tan 6 at low electronenergies and low B/N agree within only 33% with the lowervalues measured by Hallls but agree to within 15% with thelarger calculated values of Frost and Phelps26 who usedcross-sections within this same agreement. At high B/N, the

calculations of Engelhardt and Phelps29 agree with theexperimental values of Bernstein24>2S and coincide withthose of Heylen.32 The following year the calculationswere extended to N2, O2, air and C2H6 by Heylen andDargan.35 At low B/N, their tan 0 values are some 30%larger than those measured by Townsend and Bailey6 andthis discrepancy was accounted for by Engelhardt, Phelpsand Risk.31

Extensive measurements using a uniform electric fieldbetween ski-shaped electrodes up to a separation of 1 -2 cmin a uniform magnetic field up to 0-75 T, were reported byDargan and Heylen39 for breakdown voltages in N2, H2, Arand C2 H6. The new equivalent Nd method (see Section 8)for deriving the electron collision frequency was presentedand values so derived agree well with those obtained by theArmidale group. Heylen and Bunting40'43 generalised theequivalent gas density concept proposed by Blevin andHaydon18 by presenting the equivalent reduced electricfield (e.r.e.f.) concept, outlined in Section 5, and this al-lowed them to extend their calculations to large B/N andfor moderate electron energies (2 to 5 eV) agreement withexperimental data of Bernstein24>2s for hydrogen is com-plete.

The first ever recorded measurements of the Townsendperpendicular propagation velocity in a crossed magneticfield were presented by Watts and Heylen45 in 1972. It wasshown that the mechanisms involved conform to the earlyproposed theory32 and more detailed inspection andanalysis revealed the stepped nature of the movement.62

Electron collision frequency values in nitrogen derivedfrom conductivity measurements by Heylen and Dargan48

are shown to be in good agreement with those obtained from

the equivalent Nd method39 and the latter method, basedas it is on a null principle, is shown, using an error analysis,to be capable of much greater accuracy than conventionalmethods, although this has been unsuccessfully queried byHaydon,41 who concluded that experimental collision fre-quency values do not give a satisfactory interpretation ofbreakdown values using the equivalent gas density concept.Due to an error in averaging, the calculated collision fre-quency values, especially in N2, obtained by Haydon34 donot agree with the above.

In two successive papers, Heylen and Lister53*61 pro-posed a method of taking wall losses into account formagnetically controlled thyratrons, as outlined in Section10, for the region well below the Paschen minimum. Usingestablished theory, they obtained sensible values for thesecondary ionisation coefficients in He, H2, Ar and N2. Thereason why He is the best gas to use is correctly explained.As the equilibrium distance 6 was neglected in these calcu-lations, higher y values would have been obtained if thishad been taken account of in eqn. 12.

The earlier breakdown data39 were successfully inter-preted by Heylen67 using the e.r.e.f. concept and takinginto account electron recapture by the cathode. As fore-shadowed by Blevin and Haydon,19 the y mechanism in Aris due to photons to which the e.r.e.f. concept applies. ForH2 and to a lesser extent N2, 7 is due to photons andpositive ions which can be distinguished. As expected,C2H6 yields very low 7. Thus the subject is shown to becapable of successful interpretation using the establishedequivalent reduced concepts and it shown possible usingdata obtained in the absence of a magnetic field to predictaccurately breakdown values in its presence. The poor agree-ment between theory and experiment for ethane gas68 isshown to be due to the electron collision frequency in thisgas not having a unique value as originally measured39 andalso to be gas density dependent in agreement with the dataof Watts.58

14.4 Indian work

Workers from India are making a valuable contribution. Senand Gosh22 measured breakdown voltages in dry air in aglass vessel with the distance between the glass electrodesvarying from 9 to 26-5 cm at low gas densities with a crossedmagnetic field up to 0-2 T. They concluded that Wehrli'sexpression, eqn. 38, but also Blevin and Haydon's one, eqn.17c for the equivalent gas density, are of limited applica-bility and there is a need to incorporate the variation of 7with H, alongside the equivalent gas density concept, toexplain the observed results better. It should be noted thatthese workers used an a.c. voltage and that the electricfield was probably far from uniform. In a second paper,23

there authors show that 7 decreases hyperbolically withincrease in magnetic field in accordance with their proposedtheory. It should be pointed out that the breakdown vol-tages (not given) measured by these authors were reportedto increase always with increase in magnetic field even atthe lowest gas densities used which are presumably wellbelow the Paschen minimum and therefore wall lossescannot be ruled out as was also the case in Allen's37'46

experiments. This is commented upon by Bhiday et a/.38

and Blevin and Haydon37 who both point out that indeedSen and Gosh's measurements were carried out in a restric-ted E x B drift direction and that the values of the electroncollision frequencies are not correctly deduced and thuscannot be relied upon. In their preliminary paper, Bhiday

IEEPROC, Vol. 127, Pt. A, No. 4, MA Y1980 239

etal.38 present experimental primary ionisation coefficientsfor dry air and show that these agree with the equivalentgas density concept where the collision frequency is derivedfrom electron drift-velocity data taken from the literature.Not surprisingly, the collision frequency values are close tothose for nitrogen and increase, as in N2, with increase inE/N. In a full paper, Bhiday et al.42 present a/N values fordry air obtained in a coaxial system of diameter 15 cmwith a gap up to 6 mm so that the electric field is uniformto within 4%. Unlike in Bernstein's25 case, the gap wasvaried by interchanging the anode, thus keeping /0 moreconstant; the magnetic field could be increased to 0-1 T.At low E/N, good agreement is obtained with theory but atlarge E/N experimental a/N values are below those ob-tained from theory, thus showing that actual values ofv0 may be less than those obtained from drift-velocity data.

A major contribution is being made by Govinda Rajuand co-workers. Gurumurthy and Govinda Raju52 reportedclassical measurements of a/N in a crossed magnetic fieldup to 0-31 T in oxygen and dry air and Govinda Raju andRajapandians4 made similar measurements in nitrogen usinga coaxial arrangement with outer diameter 4-85 cm andinner diameter 4-60 cm so that the electric field was con-stant to within ± 2\%. All the derived collision-frequencyvalues, except those for air at the highest (E/N)e, increasewith increase in (E/N)e and are not very different from oneanother. Most but not all of the v0 values in N2 agree withthose of Heylen and Dargan48 and Bagnal and Haydon33

and those for air are in good agreement with the data ofBhiday et al.42 Also Gurumurthy and Govinda Raju59

measured minimum breakdown voltages in N2 and O2

which are in agreement with theory assuming y does notchange with magnetic field, but for air y is thought to varymore with B. Rajapandian and Govinda Raju also measuredbreakdown voltages in a nonuniform field coaxial geometryfor dry air,47 nitrogen5S and oxygen.75 In air and oxygen,agreement with the theory developed for a nonuniformfield is good, but for nitrogen agreement with theory is notsatisfactory and this is attributed more to the uncertaintyin the v0 value than nonuniformity of the electric field. Inboth papers 7 is deduced not to vary much with B. In 1977,Govinda Raju and Gurumurthy59'60 measured breakdownvoltages in N2 and air in a coaxial system with changeablegap so that the electric field could gradually be changedfrom being uniform to becoming nonuniform. Again theequivalent gas density concept is shown to apply and withnonuniform fields a polarity effect sets in. In 1978, GovindaRaju and Gurumurthy64 extended their earlier calcu-lations56 in nitrogen to the E/N region where ionisation andbreakdown occurs. The calculated values of a/N agree verywell indeed with the experimental values of Heylen andDargan.48 The cross-sections used and method of compu-tation are similar to those of Engelhardt and Phelps.29 Graphsof calculated transverse and perpendicular electron driftvelocities are also given. For a constant E/N, the transversedrift velocity declines steadily whilst the perpendiculardrift velocity increases, reaches a peak and then decreasesfaster with increasing E/N. This variation is analogous tothat shown in Figs. 11 and 12 although it must be bornein mind that there the graphs are given for constant elec-tron mean energies and thus constant (E/N)e. Electronenergy distribution functions are also given for a fixedE/N and with increase in B/N, the high-energy electronpopulation diminishes as expected. This work shows greatpromise and it will be interesting to see what these authors

can make of the gas hydrogen where the electron collisionfrequency required to fit the equivalent gas density conceptis three-fifths of that found by averaging the basic collisioncross-section, as shown by many authors, even for weakmagnetic fields.

Guharay and Sen Gupta57 extended Dargan andHeylen's39 work for H2, air and Ar to lower gas densities.Agreement with theory is good with regard to breakdownvoltages for H2 and air, though for Ar some discrepanciesare reported because again they assume y constant. Thispaper is similar in nature to that of Haydon.41 In 1978,Guharay et al.65 presented collision-frequency values forthe same gases obtained using the equivalent Nd method,but unlike Dargan and Heylen,39 they varied the gas densityinstead of the gap distance. Although they used an electrodegap of 5 cm with circular electrodes of only 5 cm diameter,they claim a nonuniform field geometry of only ± 5%. Alltheir vQ values are much higher than those reported byother workers and in a comparison with theory for hydro-gen they claim wall losses to be unimportant. We wouldthink that their nonuniform field geometry is the cause oftheir too high v0 values. Also it must be noted that theirequivalent Nd graphs are far from straight lines.

15 Applications

15.1 Prediction of breakdown voltages in a crossedmagnetic field

It is of interest to the designer to be able to predict break-down characteristics in a crossed magnetic field withouthaving to carry out the measurements. As a starting pointit is assumed that the breakdown characteristic is onlyavailable in the absence of a magnetic field. Then with theaid of eqn. 14, the 7 coefficient can be calculated andplotted as a function of E/N. The shape of the graph wouldsuggest which mechanism is responsible for secondaryionisation. If we take Ar for instance, then from Fig. 20,photon secondary ionisation is seen to be predominant, towhich the e.r.e.f. concept is applicable. Using this infor-mation, calculated breakdown values can be obtained inthe presence of a crossed magnetic field and these areshown to agree in Fig. 9 with experiment to within 3%which is within the experimental error with which the actualvalues were obtained, except at the lowest Nd value usedwhere a choice of a slightly smaller value of electron col-lision frequency would also have given perfect agreement.

On the other hand for hydrogen, Fig. 19 suggests posi-tive ion action to be predominant in the absence of a mag-netic field and using these values at the correct E/N yieldsthe calculated graphs shown in Fig. 8 which agree withexperiments to within 6% except again at the lowest Ndvalues where photon ionisation comes into play. All thisassumes of course that the electron collision-frequencyvalue is available. This can either be calculated from basiccollision data according to the formulas given in Appendix18.1, as has been done by Heylen and Bunting,40'43 or canbe obtained from experimental electron drift velocities asshown in Section 15.3 below.

75.2 Magnetically controlled thyratron as a high voltageswitch

Because of the large losses of ionising electrons to the walls,quite modest magnetic fields, applied orthogonally to adischarge tube shown for instance in Fig. 22, can increase

240 IEEPROC, Vol. 127, Pt. A, No. 4, MAY 1980

the hold-off voltage of a thyratron by a factor of ten ormore (see Fig. 24), thus making it suitable as a high-voltageswitch. The tube has to be of necessity fairly large, at leastaxially so that low breakdown voltages are reached in theabsence of a crossed magnetic field. A compact tube withimproved performance has been described in Reference 61.

15.3 Determination of the electron drift velocity

The application of a crossed magnetic field to an electronswarm moving under the influence of a uniform electricfield enabled Townsend2 as early as 1912 to derive the fasttransverse electron drift velocity using a steady-state d.c.method without recourse to pulse techniques or the use ofwideband oscilloscopes unavailable at that time. In a weakmagnetic field, i.e. vQ > <JOJN, eqn. la becomes

e 1 EvT =

m v0 N(41)

and using the expression for tan 6 given by eqn. 8, we ob-tained Townsend's equation

=E

= ~ tanB

(42)

Townsend6 ingeniously measured 0 by using a split elec-trode and thus determined ify. Naturally, of course, thesame procedure gives the electron collision frequency v0 byeqn. 8, which figures in the expression for vT (eqn. 41), soany method which allows the basic electron collision fre-quency to be determined yields also the electron driftvelocity.

15.4 Use of the electron collision frequency in microwavebreakdown

It is possible to calculate the breakdown voltage at low gasdensities of any microwave cavity having a specific diffusionlength by using d.c. ionisation data such as given by eqn. 9,making use of the effective d.c. field principle which is thatsteady electric field which transfers energy to the electronsat the same rate as the actual applied microwave field EirrKS

as given by16

F — F (43)

where in this case co == 2nf in which / is the applied fre-quency. It is observed that in this expression a knowledgeof VQ is necessary which with advantage can be obtainedfrom the data presented.76

15.5 Movement of a cathode-fall controlled arc in acrossed magnetic field

Seeker and Guile,73 measuring the velocity of magneticallydriven arcs in' air at atmospheric pressure, showed, that, fora magnetic field of 0-05 T, the arc velocity could varybetween 1-4 and 2-6 ms"1. If eqn. 34 is evaluated for thesame conditions of pressure and magnetic field, the calcu-lated Townsend propagation velocity58 is 19m s"1, whichis in excellent agreement with the experimental values forthe arc. The implication of this good agreement, andothers,77 still have to be worked out and should shed lighton the mechanisms operating in the arc cathode-fall region.For instance, it is no coincidence that Prinzler74 has as-cribed high-frequency noise emitted by cold cathode arcs to

be due to the presence of electron avalanches, at the sametime as the good agreement between arc and spark move-ment was obtained.30

15.6 Garnitron78 (glow annular magnetic interrupter tube)

This high-power switch tube, developed by Lutz andHofmann for high-voltage d.c. interruption, functions atlow gas pressure (005 torr), well below the Paschen mini-mum, similar to the tube by Allen and co-workers37'46 out-lined in Section 15.2, but operates in the inverse way inthat, unlike the Allen tube, it has a high hold-off voltage inthe absence of a crossed magnetic field, as one would expectfor its Nd value, the electrode gap separation being only1 cm. The tube is made to fire by the application of even aweak crossed magnetic field (~0-01T), but B/N is suf-ficiently high so that the (Nd)e value corresponds to thePaschen minimum. By using a spherical vessel of 24 indiameter and an electrode area of 500 cm2, the designershave been successful in eliminating wall losses and thusachieved the low breakdown voltage of 500 V in the pres-ence of the crossed magnetic field. The tube has a claimedrating of 2kA, lOOkV with a recovery rate in excess of2 kV/jtis.

15.7 Ion thrusters for space electric propulsion

A preliminary survey of ion thrusters has been made byDwarakanath and Govinda Raju,79 and although it is re-ported that axial magnetic fields are normally used in thesedevices, the authors propose to explore the use of a crossedmagnetic field to achieve improved specification.

16 Conclusion: present state of the art and likely futuredevelopments

It has been shown in the preceding Sections that agreementbetween experiment and theory is excellent, although itcould be argued that perhaps a somewhat circular argumenthas been used; that is, theory and experiment go step bystep, hand in hand. This agreement has been reached forparallel-plate electrodes from which some authors haveshrunk away because apparently in their view, such a systemdoes not provide unrestricted freedom to the movement ofthe ionising electrons in the E x B direction and theseauthors preferred coaxial electrode arrangements for thatreason. However, it has been shown that, provided the elec-tric field at the edge of a parallel plate configuration,though nonuniform, is everywhere less than in the centreuniform section, then this is no hindrance to good agree-ment. Its advantage indeed is that the gap distance caneasily and very accurately be varied. Many authors assumefor simplicity that the magnetic field has no influence onthe secondary ionisation coefficient; this is contrary towhat is said in Section 9 and this neglect has often led toquite erroneous values of the electron collision frequencybeing deduced. A further application of the use of a crossedmagnetic field is that gas density regions can be investigatedwhich are, without such a field, inaccessible because abreakdown would intervene. Also a crossed magnetic fieldallows, with the use of some theory, the secondary ion-isation contributions to be distinguished and separated. Asingle electron collision-frequency39 value for each gas isadequate67 except for ethane.68

Future development is likely to include strong electro-negative gases and a start has been made with air and oxy-gen. This will have applications in that the electron collision

IEEPROC, Vol. 127, Pt. A, No. 4, MA Y1980 241

frequency of these highly insulating gases will be foundwhich has various applications, not least in the field ofmicrowave breakdown. It will be interesting to see howGovinda Raju's theoretical work will deal with hydrogengas for which the experimental collision frequency valuesare well below the calculated ones. In this context, it wouldbe useful if theory could be more closely allied to thee.r.e.f. concept which has so far been shown to be thecorner stone of theory and the lynch-pin between theoryand experiment. A closer link should be forged betweenmicrowave and crossed magnetic field breakdown, as thebasic equations are similar in nature and a satisfactorycorrelation between arcs and sparks should be established.This embryonic state of affairs has been outlined in thepreceding Section.

17 References

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2 TOWNSEND, J.S.: 'The diffusion and mobility of ions in a mag-netic field', Proc. Royal Soc, 1912, A86, pp. 571-577

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4 MEYER, E.: 'liber die beeinflussung des funkenpotentials durchein transversales magnetfeld', Ann. Phys., 1919, 58, pp. 297-332

5 MEYER, E.: 'Zur deutung des einflusses eines transversalenmagnetfeldes auf das funkenpotential', ibid, 1922, 67, pp. 1-12

6 TOWNSEND, J.S., and BAILEY, V.A.: 'The motion of electronsin gases', Phil. Mag., 1921, 42, pp. 873-891

7 WEHRLI, M.: 'Funkenpotentiale in transversalen magnetfelde',Ann. Phys., 1922, 69, pp. 285-314

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40 HEYLEN, A.E.D., and BUNTING, K.A.: 'Electron drift velocitiesin a moderate and in a strong crossed magnetic field', Int. J.Electron., 1969, 27, pp. 1-12

41 HAYDON, S.C: 'Critical comparison of methods for the evalu-ation of electron-molecule collision frequencies in crossed E andH fields',/Voc. IEE, 1970, 117, (2), pp. 473-479

42 BHIDAY, M.R., PAITHANKAR, A.S., and SHARDA, B.L.:'Measurement of Townsend's first ionisation coefficient for air intransverse electric and magnetic fields', /. Phys. D: Appl. Phys.,1970, 3, pp. 943-950

43 BUNTING, K.A., and HEYLEN, A.E.D.: 'Moderate and strongmagnetic field electron drift velocities in nitrogen, oxygen andair', Int. J. Electron., 1971, 31, pp. 9-18

44 HAYDON, S.C, McINTOSH, A.I., and SIMPSON, A.A.: 'Theeffect of magnetic fields on electron energy distribution func-tions and derived collision frequencies', /. Phys. D: Appl. Phys.,1971, 4, pp. 1257-1265

45 WATTS, M.P., and HEYLEN, A.E.D.: 'Townsend-Huxleypropagation in the E X B direction', IEE Conf. Publ. 90, 1972,pp. 163-165

46 ALLEN, N.L., and LISTER, N.T.: 'A thyratron employingorthogonal electric and magnetic fields', IEE Conf. Publ. 90,1972, pp. 159-160

47 RAJAPAND1AN, S., and GOVINDA RAJU, G.R.: 'Sparkingpotentials for dry air in crossed electric and magnetic fields',IEE Con. Publ. 90, 1972, pp. 169-170

48 HEYLEN, A.E.D., and DARGAN, C.L.: 'Electron-moleculecollision frequencies in a crossed electric and magnetic field',Int. J. Electron., 1973, 35, pp. 433-451

242IEE PROC, Vol. 127, Pt. A, No. 4, MA Y1980

49 FLETCHER, J., and WALSH, T.I.: 'Secondary ionisation mech-anisms in an E X B discharge in hydrogen', Austral. J. Phys.,1973,26, pp. 761-769

50 STOCK, H.M.P.: 'Prebreakdown current growth in crossedelectric and magnetic fields at low gas pressures, I. Effects ofnonuniform applied fields',/. Phys. D., 1973, 6, pp. 988-999

51 BLEVIN, H.A., and STOCK, H.M.P.: 'Prebreakdown currentgrowth in crossed electric and magnetic fields at low gas pres-sures. II. Spatial growth of current', ibid., 1973, 6, pp. 1467-1476

52 GURUMURTHY, G.R., and GOVINDA RAJU, G.R.: Town-send's first ionisation coefficients and sparking potentials incrossed electric and magnetic fields', IEEE Trans., 1975, PS-3,pp. 131-143

53 HEYLEN, A.E.D.: 'Criterion for Townsend breakdown in acrossed electric and magnetic field with wall losses', Int. J.Electron., 1976,41, pp. 209-220

54 GOVINDA RAJU, G.R., and RAJAPANDIAN, S.: 'Townsend'sfirst ionisation coefficient in crossed electric and magneticfields in nitrogen', ibid., 1976, 40, pp. 65-79

55 GOVINDA RAJU, G.R., and RAJAPANDIAN, S.: 'Prebreak-down current and sparking potentials in a non-uniform electricfield with crossed magnetic field', IEEE Trans., 1976, EL-11,pp. 1-8

56 GOVINDA RAJU, G.R., and GURUMURTHY, G.R.: 'Electrontransport coefficients in N2 in crossed fields at low E/p', ibid.,1976, PS4, pp. 241-245

57 GUHARAY, S.K., and SEN GUPTA, S.N.: 'Effect of a trans-verse magnetic field on the sparking characteristics of gases atlow pressure', Indian J. Phys., 1976, 50, pp. 9-17

58 WATTS, M.P.: 'Pulsed Townsend discharge in a d.c. crossedelectric and magnetic field', Ph.D. thesis, University of Leeds,1977

59 GURUMURTHY, G.R., and GOVINDA RAJU, G.R.: 'Minimumbreakdown potentials in E x B fields', Proceedings of the 13thinternational conference on ionisation phenomena in gases,1977, pp. 361-362

60 GOVINDA RAJU, G.R., and GURUMURTHY, G.R.: 'Electricalbreakdown of gases between coaxial cylindrical electrodes incrossed electric and magnetic fields', IEEE Trans., 1977, EL-12,pp. 325-334

61 HEYLEN, A.E.D., and LISTER, N.T.: 'Breakdown criterion fora magnetically controlled thyratron', Int. J. Electron., 1978,44, pp. 79-84

62 WATTS, M.P., and HEYLEN, A.E.D.: 'Stepped sideways Town-send discharge movement in a crossed magnetic field', Proc. IEE,1978, 125,(6), pp. 563-564

63 HEYLEN, A.E.D.: 'Electric strength of a moving gas and of agas subjected to a crossed electric and magnetic field', IEEConf. Publication, 165, 1978, pp. 262-264

64 GOVINDA RAJU, G.R., and GURUMURTHY, G.R.: 'Electronenergy distributions and transport coefficients in nitrogen inE X B fields', Int. J. Electron., 1978, 44, pp. 355-365

65 SEN GUPTA, D., GUHARAY, S.K., GHOSHROY, D.N. andSEN GUPTA, S.N.: 'Electron-neutral collision frequency frombreakdown measurements in crossed electric and magnetic fields',Pramana, 1978, 11, pp. 661-671

66 MEEK, J.M. and CRAGGS, J.D.: 'Electrical breakdown of gases'(Wiley, 1978)

67 HEYLEN, A.E.D.: 'Interpretation and prediction of gaseouselectrical breakdown characteristics in a crossed magnetic field',Proc. IEE, 1979, 126, (2), pp. 215-220

68 HEYLEN, A.E.D.: 'Electron-molecule collision frequencies frombreakdown data in a crossed magnetic field for ethane gas',Proceedings of the 14th International conference in ionised gases,1979 (to be published)

69 HUXLEY, L.G.H., CROMPTON, R.W.,and ELFORD.M.T.: 'Useof the parameter E/N', Brit. J. Appl. Phys., 1966, 17, pp. 1237-1238

70 DWIGHT, H.B.: Tables of integrals and other mathematicaldata', (McMillan, 1957), p. 128, No. 576.1

71 HEYLEN, A.E.D.: 'Paschen characteristics of gases in a crossedmagnetic field', 2nd International symposium on gaseous di-electrics, (to be published, March 1980), Knoxville, Tennessee,USA

72 MEEK, J.M., and CRAGGS, J.D.: 'Electrical breakdown ofgases' (Clarendon Press, 1953) p. 60

73 SECKER, P.E., and GUILE, A.E.: 'Arc movement in a trans-verse magnetic field at atmospheric pressure', Proc. IEE, 1958,106A, pp. 311-320

74 PRINZLER, H.: 'Hochfrequentes Rauschen von Elektronenlawinen und sein Zusammenhang mit dem Mechanismus desKalt Kathodenbogens', Z. Naturforsch., 1965, 20A, pp. 876-883

75 GOVINDA RAJU, G.R., and RAJAPANDIAN, S.: 'Cross-fieldgas breakdown in a coaxial cylinder geometry', Int. J. Electron.,1979,46, pp. 393-400

76 HEYLEN, A.E.D.: 'Microwave breakdown voltages from d.c.data' (to be published)

77 HEYLEN, A.E.D., and GUILE, A.E.: 'Magnetic arc movementfrom a multi-collisional viewpoint', IEE Vlth Int. Conference onGas Discharges and their Applications, Heriot-Watt University,Sept. 1980 (accepted for publication)

78 LUTZ, M.A., and HOFMANN, G.A.: The GAMITRON - ahigh power cross-field switch tube for H.V.D.C. interruption',IEEE Trans. (Plasma Science), 1974, 2, pp. 11-24

79 DWARAKANATH, K., and GOVINDA RAJU, G.R.: 'Ionthrusters for space electric propulsion', Indian Space ResearchOrganisation Report, 1978, pp. 1-25

18 Appendix

18.1 Precise formulae for electron transport data

According to Allis,16 the transverse (see Fig. 3) electrondrift velocity in the presence of a crossed magnetic field isgiven by

B - _ 1 L EC v(^e*n d fF(e>UVT 3 m X N Jo He)}2 + (uc/N)2 * de [ein j

(44)

where e is the electron energy and F(e)de is the electronenergy distribution function. Similarly the perpendicularelectron drift velocity is given by

B 2 e Ee_ Er~ (UclNr*m X N Jo {^(e)}2 + (ujN)2 X

d F(e)de \em

(45)

Normally in a gas the electrons are spread out according tosome distribution function and assuming a Maxwellian one,as given by

271/2

.1/2 3e(46)

where e is the mean energy, then substituting eqn. 46 intoeqns. 44 and 45, respectively, we obtain

e—m

E \

and

If now v(e) is assumed constant, independent of electronenergy, then eqns. 47 and 48 reduce to eqns. la and 1bgiven in Section 3.2 from which the expression for 6, eqn.8, follows. Basically there is no need to specify a Maxwelliandistribution48 but other distributions will give a slightlydifferent averaged constant electron collision frequency.

IEE PROC, Vol. 127, Pt. A, No. 4, MAY 1980 243

18.2 Hall angle in solids

Because of the electron transport theorem in solids, theabove transport equations also apply in metals and semi-conductors. In a metal, the electron energy distribution isthat of Fermi for which

d_\F(e)de All = - 1

thus the Hall angle is given by

tan0 =u>JN _ e B

v(e) m v

(49)

(50)

where v is the electron collision frequency at the Fermienergy.

For semiconductors, the Maxwellian energy distributionapplies. As the energy of the electrons extends over a widerrange than in metals, the collision frequency cannot beassumed constant. It is usual to consider the collision cross-section constant (Druyvesteyn) and to make the approxi-mation v2 > co2. i.e. for a weak magnetic field. If this isdone we get, after integration from eqns. 47 and 48 andtheir ratio

/37T\ e Btan d = (—

8/ m v(51)

where v is the electron collision frequency at the meanenergy e.

18.3 Electric and magnetic units

The c.g.s. unit for magnetic field is the gauss, which is thesame as the older unit, the oerstedt. This was suitable inthe early days when only weak magnetic fields could begenerated. The SI unit for magnetic field is the Tesla where

1 tesla = 1 weber/m2 = 104 gauss = 104 oerstedt

and these days of magnetic fields of one tesla can readily begenerated, so the tesla is a suitable unit.

The c.g.s. unit for gas pressure p is the torr, and in gas

discharge work this is also a convenient unit as gas pressuresnear Paschen minimum usually occur near to 1 torr. Now itis shown in this review that the important parameterdetermining the behaviour of a gas in a crossed magneticfield is the reduced magnetic field B/p, so that as a result ofthe above definition, B/p is in units of gauss/torr or themixed unit tesla/torr. However, the SI unit system doesnot employ the torr, but rather the pascal, which in gas dis-charge work is not a convenient unit. Usually, the gas num-ber density TV is used where

N = 3-32 x 1016 p(torr)cm~3 at 20°C

As this unit introduces very large factors of ten, thetownsend was proposed69 for that other fundamentalparameter, the reduced electric field E/p, which is usuallygiven in Vcm"1 torr"1, where

1 townsend = 10~17Vcm2

so that, as at 20°C

E/N = E/p x 3-0 x 10"17 Vcm2

the correspondence is

— (townsend) = 3-0 (E/p)

In a similar way, we would like to propose a Huxley unit, inhonour of his distinguished work in gas discharges, for thereduced magnetic field, where

1 Huxley = 10"17Tcm3

so that, as at 20°C

- = - x 3-0 x l(T17Tcm3

N p

we have

B/N (Huxley) = 3 (B/p)

As a safeguard, the units used in this review are the con-ventional SI ones where the relation between E/N, E/p andB/N, B/p are as given above.

I Albert E.D. Heylen graduated and ob-tained a Ph.D (Eng)., at the Electrical

I Engineering Department of QueenMary College, London, after studyingthe classics in Belgium. He spent five

(years as an ICI and BICC ResearchFellow at Queen Mary College, re-

[ searching into the basic ionisationand breakdown processes of hydro-carbon gases, before being appointed

'a lecturer in 1961 at the Departmentof Electrical and Electronic Engineering, University ofLeeds, England, where he established and is in charge of theHigh Voltage Laboratory. He became involved in developing

very low electric strength gas mixtures for nuclear countersand, under the influence of Professor A.E. Guile, in thearea of the review paper, a good deal of the support workbeing funded by the Office of Aerospace Research, USA.

Dr. Heylen has published over 65 papers on high voltageionisation and sparking phenomena in gases and obtaineda D.Sc. (Eng.) in 1971 from the University of London. Heis particularly interested in reconciling theory with experi-ment and in the field of direct energy conversion in whichhe published a paper on thermo-electricity. In recognitionof his pioneering work, the IEE, of which he became amember in 1965, awarded him a Student Premium andlately on Ayrton Premium.

244 IEEPROC, Vol. 127, Pt. A, No. 4, MA Y1980