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APPLICATION 0F THE cI.ASsIcAL THEORY 0F ELECTRONS IN GASES T0 MULTIWIRE PROPORTIONAL AND DRIFT GHAMBERS
V. Palladino
B. Sadoulat
April 1974
Prepared for the U.S. Atomic Energy Commission under Contract W-7405-ENG-48
Exit
{h`: " -... /\"¢·\ Quoavi OCR Output Printed in the United States of America Available from National Technical Information Service U.S. Department of Commerce 5285 Port Royal Road Springfield, Virginia 22151 Price: Printed Copy $5.45; Microfiche $1.45 OCR Output -]_¤
APPLICATION OF THE CLASSICAL THEORY OF ELECTRONS
IN GASES TO MULTIWIRE PROPORTIONAL AND DRIFT CHAMBERS
V. Palladino
CERN, Geneva, Switzerland
B. Sadoulet Lawrence Berkeley Laboratory, University of California Berkeley, California 94720
April 1974
Present address: University of California, Santa Cruz, California
Part of this work was done at CERN. OCR Output -11
CONTENTS
Ab stract ...... iv
Intrgducti gn ......
Akldgcnowements ......
Section 1. Transport Coefficients Characterizing the Behavior of an Electron Swarm in Electric and Magnetic Fields
1.1 Drift of Electrons in Electric Field ......
1.2 Effect of the Magnetic Field ......
1.3 Life History of an Electron in a Multiwire Proportional ounter ......
1. 3. 1 Production of Electrons .... 1. 3. 2 Drift Region ...... 1. 3. 3 Avalanche Region ·...... 10
Section 2. Simplified Theory of Electrons in Gases
21. rtDifVli eocty ...... 12
2. 2 Diffusi on ...... 15
2. 3 Effect of Magnetic Field ...... 16
2.4 Energy Conservation: A Simplified Theory 18 2.4.1 Constant Cross Section ...·
2.4. 2 Application to Pure Argon · . 20 2.4. 3 Non-Uniform Electron Density 22
Section 3. Classical Theory of Electrons in Gases
3. 1 Boltzmann Equation ------·· 23
3. 2 Druyvestein Distribution ,,,,,,,,, 26
3. 3 General Case ...... 27 3. 3. 1 Approximate Methods .,,,. 28 3. 3. 2 Rigorous Numerical Methods . 29 3. 3. 3 Application to Argon ..... 30
3.4 Special Cases: Low Electron Temperature or High 32 MgtiFildanec es ...... 3. 4.1 Movement of Gas Molecules ...... 32 3. 4.2 High Magnetic Fields ...... 33 OCR Output -111
3. 5 Fit t0 Realistic Gas Mixtures .... 36 3. 5. 1 Argon-Isobutane Mixtures 37
3. 5. 2 Methane ...... 37
3. 6 Intuitive Interpretation ...... 42 3. 6. 1 Approximate Theory . . . 42
3.6.2 Pure G ases ...... 42
3. 6. 3 Mi xtures ...... 44
Section 4. Application to Drift Chambers
4. 1 Minimum Diffusion ...... 45
4. 2 Low Drift Velocities ...... 48
4. 3 Behavior in Magnetic Field ......
4. 4 Constancy and Stability ofthe Drift Velocity 52
Section 5. Multiplication
5.1 Definitions and Experimental Behavior . . . 55
5. 2 Multiplication Theory in a Constant Field ·· 56 5. 2.1 H01stein‘ s Theory ...... 56
5. 2. 2 Numerical Evaluation ...... 58
5. 3 Multiplication in a Multiwire Proportional Counter . 59 5. 3. 1 Validity of the Equilibrium Assumption 5. 3. 2 The Empirical Formulae for Gain in Proportional Counters ··-· 61
5. 4 Parasitic Phenomena --·-······ 63
5. 4. 1 Loss of Electrons ...... 63 5. 4. 2 Space Charge Effects ..... 63 5. 4. 3 Breakdown Phenomena .... as 5. 4. 4 An Example: The "Magic Gas" 70
Ref erences ...... 71
Appendix: The Korff Model for Proportional Counters . . 74 OCR Output -]_\/•·•
APPLICATION OF THE CLASSICAL THEORY OF
ELECTRONS IN GASES TO
MULTIWIRE PROPORTIONAL AND DRIFT CHAMBERS
ABSTRACT
In this paper we review the classical theory of electrons in gases and some of its results which may be useful in the design of multiwire proportional and drift chambers. We discuss especially how low diffusions and constant drift velocity may be obtained in certain gas mixtures, and the multiplication process. OCR Output -4
INTRODUCTION
Among the various particle detection devices multiwire proportional countersl and drift z-5 chambersappear to be very promising. They have been the object of numerous recent exper imental studies that have resulted in many cooking recipes, most of which are,however, of doubtful fundamental significance. ln fact, very few high energy physicists working in this field seem to be aware -- at least we were not —- of the existence of a rather classical and successful theory of the behavior of electrons in gases. After the pioneering work of Townsend" in the twenties, Druyvestein' , Morse, Allis, and Lamaru and others were able to construct a successful theory just before World War II. The development of radar and gaseous switching tubes triggered some /10 ii new theoretical developments from Holstein’ , Margenau, Golant, etc. after the war. The iz theory was then good enough to be used by Phelpsand co-workers in 1962 not only for the qualitative explanation of the observed behavior of the drift velocity and the diffusion, but to extract from measurement of these quantities detailed information on the various elastic and excitation cross sections of interest for atomic physicists.
ln this paper we give a summary of this theory for high energy and nuclear physicists, since we think that in spite of the complexity of the gas mixtures we are using, some knowledge of the basic phenomena may be useful in understanding their properties. The first section of our work deals with the definition of the so called transport coefficients (drift velocity, diffusion co efficient, characteristic energy) describing the behavior of electrons in gases under the action of external fields. We will give some experimental results concerning these coefficients for the gases usually used in multiwire proportional chambers.
In the second section we develop a simplified theory for the calculation of these coefficients; although it uses only intuitive arguments and rough approximations, it shows how the different contributions appear. The action of a magnetic field is (which is important because detectors are often located inside strong magnetic fields, as in the new generation of large magnetic spectro meters for particle physics) is also considered.
The third section introduces a more rigorous theory of gaseous kinetics, starting from general transport equations,which enables computation of the energy distribution of the free electrons in the gas and therefore the various transport coefficients. As an example we show explicitly that the numerical solution of these equations describes reasonably well the behavior of electrons in pure argon. We then apply this theory to gas mixtures commonly used, namely argon-isobutane and argon-methane mixtures. We try to give an intuitive interpretation of the behavior of drift velocities.
These results allow us to comment in the fourth section on the desirable properties of gases to be used in drift chambers and the way to achieve them. We discuss especially the diffusion, the behavior in magnetic fields and the constancy of drift velocities.
The fifth section deals with gaseous amplification phenomena. We first show that the classical theory developed in Section 3 can describe quantitatively the multiplication processes in homogeneous fields. Considering then the case of multiwire proportional counters, where the electric field is strongly varying, we show through computer simulation that we can use the results obtained for homogeneous fields. This allows us to understand the various empirical formulae OCR Output -2
proposed for the amplification in proportional counters. We finally comment on various parasitic phenomena and the way they are eliminated in modern proportional counters.
None of our results are entirely original. Section 2 is a generalization of Townsend's approach in "Electrons in Gases". U The proof of the capability of the theory developed in Section 3 to accurately describe experimental data was already given by Golant, Phelps and co-workers. However, we have developed a program for computing the transport coefficients and will send it to any interested physicist. Sections 4 and 5 are more original, at least in the attempt to explain phenomena of great importance for design and operation of multiwire proportional and drift chambers.
For readers uninterested in mathematical details, we would recommend the reading of Sections 1, 2, 3.5 and 3.6, 4, 5.3.2, and 5.4.
ACKNOWLEDGEMENTS
The authors wish to thank G. Charpark for many discussions and suggestions about this work. They are also greatly indebted to F. Bourgeois, S. Derenzo, D. Nygren, S. Parker, and F. Sauli for fruitful discussions on various subjects presented here. This work was done in part under the auspices of the U. S. Atomic Energy Commission. OCR Output -3
SECTION 1
TRANSPORT COEFFICIENTS CHARACTERIZING THE BEHAVIOR OF
AN ELECTRON SWARM IN ELECTRIC AND MAGNETIC FIELDS
1. 1 Drift of Electrons in Electric Field
The ionization of a gas by an ionizing radiation, as is well known, produces free electrons (and ions). If the ionized gas is not subject to any electric field, these electrons will move randomly with an average agitation energy that is given by the Maxwell formula 3/2 kT (about 3.7 10-LeV at 18°C). If an electric field is applied, the electrons will continue to have a thermal non—directional velocity v (and energy €), but they will exhibit a general overall drift along the field direction, with a mean drift velocity w = (vv). Of course one gets
’>) w = {V}x];/1/‘x =vF(r,v,t) dv 1.1(1) where F(i·,v,>> t) is the velocity distribution of the electrons at the point r and at time t; in presence of external fields F is not the Maxwellian distribution. The mobility is defined as it = w/E where E is the applied electric field.
Let us consider, for the sake of simplicity, a region where no ion or electron is formed and none disappears by attachment or recombination. The motion of the electrons has to be deter mined by a diffusion equation. If p is the spatial density of particles, the conservation of the number of particles yields
pdV = N or Bp/Bt + div (pv) = O
The diffusion coefficient D is defined by
—+ 1 Vp = -; (PV) and the diffusion equation is
Bp/Bt = DV°p 1. 1(2)
D describes the mean spread of a swarm of electrons
gig Z 2 dv : 3D ’ Bt 8t N where we have used Eq. 1. 1(Z) and integrated by parts. So we get
I- : _ Bt
Similarly the quantity V 2D/w gives a measure for the lateral and longitudinal spread that the electrons undergo while traveling a unit distance in the gas, since
zz z agxz agx) at zo agyg ag}; = — = -- = = 1. 1(3) 5Bx Bt Bx W Bt Bt OCR Output -4
Note that in the presence 0f electric or magnetic fields the diffusion coefficients are not necessarily the same for all directions of space (see Section 2. 4. 3). The mean energy of electrons is of course
(e) : [ eF(i·,e,t)dv*> where 6 : g mV. 2 Another similar parameter, which is directly measurable, is the so called characteristic energy, defined as
€, = E2 1. i(4)
E/P300 (V cnf' torr") E/P300 (V cnf' ton") '°`° '°" 'O" '°" "° I0., 10** 10** 10·* 1o·' 1·0 IO mo I0. 9;} {JI
—`—-— ;`l0' A 105 Ar Z _ He {Q V'°’ LS K 310*H/E / __ 1 l0" O" E, *200·1<
.- ws 0·0l 1: I0,
.0• 0001
{0t1?,0;. mr". ,9-.. ,0-.. ,0--. |O¤ _“ IO" I0 "‘ IO" 10·"‘ 10·" I0 “ 10 E/N (\/Cm!) xg1_141—357g E/N (V cm?) x¤u41·s¤¤¤ (¤> E/N (V Cm;) V 10‘·" 10"" 10**** 10"’ IO" [Y7"YT;T;T'T`T`|""T'—`F ·w·rj—1——r··1 E/N tv cmg) ·20 -—l'3 -l8 -I7 I0 10 I0 IO 10, Kr
5 IO + Xc
T7°K E s O ‘° / 5 · v*3,,,+w :· ,0• . >500’K Ne ;f c '0°l` 77·1<
s00·1< ° (Ol
10* 10** 10** 10-2 10-' 1·0 E/P300 (V cm' ton"') I0} *-1 (C) _4 _] |0_z I0_l LO *°L"’·”"*’ I0 I0 -1 -1 xsu4·r—ss·n E/P300(V Cm TOYT )
Fig. 1. Experimental behavior of drift velocity and some characteristic energies in noble gases (Ar, I-Ie, Ne, Kr, Xe). In Ar the `lmagnetic drift velocity, WM" is also shown. OCR Output -5
where e is the charge of the electron, p is the mobility, and D is again the diffusion coefficient.
Note that B x,_ _ 2 — S/€E BX k 1.115)
Fig. 1 gives, as an example, the experimental behavior of w and eh for the usual noble gases. It is seen that the behavior of these quantities is by no means simple. Note that they are universal functions of E/N, where N is the number of molecules per unit volume, or E/P30O,Wh€y€ p3O0 is the atmosphe1‘iCp1‘€SS¤I‘€ at 3000 K. Drift velocities are typically 106cm/s. It must be noticed that they are very much lower than the "thermal" random velocity. Mean energy and character istic energy are in fact usually in the region of few electron volts, that is, we get a thermal velocity of the order of 108 cm/sec (v = N/1.6><1O_19><2€/___, where m is the mass of the electron in kilograms and 6 is in eV). The diffusion coefficient has the dimension of cm°/sec. For diatomic and poliatomic molecules the characteristic energies for E/P300 S, 1 are much lower (Fig. 2 gives w and 6k for N2, HZ, and CO2). Another striking experimental fact is that the E/P300 (V/cm ton) me- no-· n·o 10 10*
,yI. A > X:T */f" at 2 \ I 1·0 ° ’'Q4 10*
5 6 E 10 I 0 g · 10* 10-¤ T;
: 5 ·; IO k CO2 ·· 1 E> f Iwgy |9—¤ 6
ml -1 l0 10* 10* max no-¤· 1o—·· w—¤ no-·· no—·· (vcmz) XIL74'T'3§76 9 LJ l95°K ·I . 1 ¤ il ui"? E/P500 (V cnfl ton" ) I0. ll1*’ 10** IO" l0 I0 ml _ In-19 w-It 10-17 4 10-16 m-15 E/N, V·cm (b) xs1.141-ssrn
10 __
Fig. 2. Experimental behavior of drift : W S ° * velocities and characteristic ,k 4] 2 ID" °’ in energies in more complex-mole —’” W cules than monoatomic gases (H2, CO2, N2) showing in partic '°" ular smaller values of ck at low and intermediate electric fields.
W" ll) ·• 10 ·- HV" |0’" in-·· 10*** (c) E/N (V cm?) x¤¤.1•1-aan
ig pFor theexperimental method of measurement we refer the reader to Loeband Massey and ilBurhop . OCR Output
OCR Output-7
1. 2 Effect of the Magnetic Field
along a path which is different from the electric field lines. We. define wu as the drift velocity V component along the electric field lines and wl as the drift velocity component along the normal to the electric field and magnetic field. If the electric field is along Ox, and the magnetic field along Oz, w=u {V) xf= vx F(r,v,t))’, dv w=l {V) Yf= vy F(¥,v,t)>, dv
The magnetic drift velocity is defined as Wl E W _ M ‘ Tv] E
Notice that the lateral displacement after a drift length is
Ay = WM? x
Experimentally WM is usually close to w (see Fig. 1 for pure argon).
1. 3 Life History of an Electron in a Multiwire Proportional Counter
In order to clarify the following discussions, let us follow the life history of an electron and give some relevant formulas for completeness.
}15 1. 3. 1 Production of Electrons
When a high energy particle of velocity [ic and charge unity goes through a proportional 16 chamber, it deposits in the chamber a mean amount of energy given by the Bethe—Bloch formula.
2 Zcz z
an 2 NZ Cmep2e e — = -2vrp mrT 7 ln-- --e-2B - <°>(B)| 1.3[1) dz TE max {3 1 1-p
where p is the density of the gas, A and Z its atomic weight and atomic number, mn is the electron mass, In is the classical electron radius, N the Avogadro number, eis the max maximum energy allowed for the electrons to stop in the chamber (typically 22 keV for a 1--cm thick chamber filled with argon), I is the mean ionization potential which is given roughly by:
I = 13 eV X Z
6(B) is a function of the velocity which accounts for the density effect responsible for the Fermi plateau. However the mean number of electrons is not given by Eq. 1.3(1). One should distinguish between (a) the primary electrons, namely those directly produced by the high energy particles 17 (Table I, taken from Korfr'gives their number, S, per cm per atmosphere. This number iS important because it gives the basic inefficiency of a chamber of thickness E, assuming a Poisson law ti = e-`H.) and (b) the secondary electrons some of these primary electrons are very energetic OCR Output -8
and will ionize the gas. Moreover they also rapidly lose their energy through elastic and exci tation collisions. One introduces the mean energy
(No. of primary ion (E) Total no. of ion pairs)per cm P pairs per cm
Hydrogen 33 ~1O
Helium 5.9 or 6.5 27.8 ~1Z
Argon 29.4 25.4 ~9O Methane (CHA) 16. 27.3 ~50
Neon 12. 27.4 ~5O
Xenon 44. 20.8 ~300
§ |O3 Xe ISO C4H|0 CO2 ¤ 3 Q. IO Ar Ne He ; i0'
0 noi no2 io¤'· i04 T kinetic energy (MeV)
xsi.14s—ziv0A
Fig. 4. Number of ion-electron pairs per centimeter deposited by pions in different gases as a function of the energy of the pions.
1. 3. 2 Drift Region
These approximately 100 electrons begin to drift in the electric field of the gap towards the sense wires. The electric field in a multiwire proportional counter has been given by G. A. ig Erskine. The complex potential w at an arbitrary point z = x+iy arising from a single wire with linear charge density qn, at position zn with lm zn = O
2qO Sin h [(·n/4L) (2. - zO)] w(z,z0) = —-j In 4"TE0 cos h[(·n·/4L) (z — zO)] OCR Output -9_
For a set of n wires at positions zi and potentials Vi , the resulting potential at z (if the wire 19 radius is negligib1e)is
wiz) =i Li=1 qw(z,zi) where the q; are determined by
Re [w(z)] = V ii
The case where there are no ground planes (cathode made out of wires) is treated by letting L go to infinity. When the arrangement of wires is periodic and the wires are in the middle plane between two grounded cathode planes, the potential at position x,y is given a closed form *9** I V = ZtfI' —Z- (0log4 Lsin+ s 4gs sinh4**52 z Di s where the origin is at one wire, S is the spacing between wires, and L/S?. 5 has been assumed. The charge density q is given by 4·rreOVO L Z]rrr 3 isL -1 .4).Og s where Va is the applied voltage and rn is the radius of the wires. Figure 5 shows the mechanical
MULTIWIRE PROPORTIONAL CHAMBER
V0=5kV
2mm 0 O 0 ` O O O 0 O 0 0 6 O 0 0 Sense wires (Radius 20;:.) (ground patential) 8`mm
V¤=5kV Fig. 5. Mechanical structure of DRIFT CHAMBERS . . . a multiwire proportional Chargak etal. chamber and of two types "¤ AS¤·>¤¤*¤¤¤¤*¤¤**¤'—-· of aria chambers (char 0 0 0 0 0 D 0 24 mm pack et al. and Valenta "` Sense wire (24,u.) Field wire-Pf? Bmm Bt all )
O 0 O 0 0 O
Wclenta eral. Vc 7.5mm {4.•(20p.) 0Bmm _ Sense wire Field wire $6*)* WW (ground VF potential)
xsuas-siss OCR Output -10 construction of usual multiwire proportional chambersl and drift chambers.2—5 Figure 6 and 7 show the corresponding fields for the dimensions and applied potentials shown. Section 2, 3, and 4 will be devoted to the study of the behavior of the electrons in this region.
IO4 Chorpok at ul. mic In ¤ Mwpc vo · 1700 v V.- • 4700 V IOS ` 3 EIO
2 'O Q l0 0 5 10 15 20 25 Ey · Becm direction ` . _ P0s1t10r1 (mm)
IO4.
@3 Wolento et ol. Vr _ 2,50 V VF · 2750 V Ex\NormoI to beam direction > 3 V IO
I02 \O;.i IOO;.t O.! cm 1 cm XSL 745-3ISS 2 'O 0 1.5 5.0 4.5 6.0 7.5 Position (mm) KSL 745·3|67
Fig. 6. Shape of the electric field in a multi wire proportional chamber along the Fig. 7. (a) Shape of the electric field in a incoming particle direction (E") and drlft Chamber of the type blllll by along the normal to the beam (Ev). Cllarpack st al' The field increases as 1/r in the (bl Shape Ol the electric fleld in 3* region Close to the Wire. drift chamber of the type built by Valenta et al.
1. 3. 3 Avalanche Region
Finally when the electrons arrive at some 100 ii from the wire, their energy is high enough to ionize the gas (see Section 5). The movement of negative charges towards the wire and of positive charge away from the wire produces a pulse on the wire. Because the electrons are in the majority produced on the last mean free path, they do not contribute very much to the pulse, and the positive ions which have to move down along the entire potential will be responsible for 2O most of the induced charge` . The time build up of the charge has been given by Wilkinsonand zo the pulse shape by Mitra. Assuming that a resistor R is attached to the wire and that the total (parasitic) capacity in parallel to the wire is C, the difference of potential across the resistance is OCR Output -11
t + tO - Q q ` RC , **0 _t(0) where q is the charge per unit length of wire, V is the difference of potential across the chamber, Q is the total charge of the electrons in the avalanche, and tn is given by
4¤€Od 0 16q(-L where d is the diameter of the wire, and p the mobility of the positive ions (typically 1.6 cmé/s/V for Argon). E; is the exponential integral function. OCR Output -12
SECTION Z
SIMPLIFIED THEORY OF ELECTRONS IN GASES
Before presenting a rigorous theory of the behavior of electrons in gases, we would like to give an elementary derivation of the rigorous expressions of the transport coefficients in terms l of the thermal velocity v, the collision length I with the molecules of the gas, and the fraction of energy A lost at each collision.
2. 1 Drift Velocity
Let us first start with the drift velocity in an electric field. Consider an electron which has undergone a collision with a molecule of the gas at the time 0 and which has emerged at an angle Gm with the field direction, and a velocity vn (Fig. 8a). If the field is along the x direction the path covered along x direction by this electron in a time 6t is
6x =% é(w+ z VO ces 00 6r Z.1(1)
We will evaluate the drift velocity w along the field direction first in the following three approximations (relaxed later): a) the differential cross section for electron-molecule scattering is isotropic. b) the time 6t between two collisions is independent of the first collision emission angle Bn (which is wrong, as we will see). c) the mean free path f is constant over the whole energy spectrum.
After averaging over 06we fget O(cos 9 d cosG = O)
e<6x>cos9,, · 1/2 ETE (bt) 2
We now have to make the average over the distribution g(6t) of the collision time St, to get the mean path along the x direction between two collisions. If E is the mean free path and v the random velocity, then T :2/v is the mean free time 1 -6t *r g(5t) = ; e /
The mean of (6t)" is L Z .21(Gt >coll.2 /VZ ’
where v, in a first approximation, is not different from vn. Therefore we get
E I2 _ e<§x>cos9n _ Tn- 3 coll.
Since we have v/I collisions in a second, the drift velocity of an electron which has an energy 6 I 1/2 m vb is W =
We have still to average over the different electrons that have a different velocity v to get
eE Z _ W - -5 (;)V 2.1(2)
Remark: It is very easy to construct wrong arguments; for instance, because v = F-- IE'5t + v cos 9 one could write dow x m O O ’ n
__ _ _
But that is wrong because the time between collisions is not a constant.
We now go to a more exact evaluation of w. We retain the hypothesis (a) but we drop hypotheses (b) and (c). Now 2 =!(€) and, as we will see, 6t= 6t(9n). We have again x = %% t+vOcos90t2
The real path of the electron is drawn in Fig. 8b. The path covered is given by
= v;+v5+vZ,2 where VX = Eng t+vO cos60 vi + vi = vg sinGOz
Now let us expand these expressions in Taylor series and disregard the terms at the orders greater than one in E because, for not too high fields, Av/vn << 1. We get
ds : v 1+ v eEcos9 ( 7,0t + . at 0 0 o }
By integrating one obtains s Z vOt (QEEmvO1+---? v0 cos0O t)+ . and, always to the first order in E
_ t(s)s eE—T1-———ZvOcos9OT+. (0 2mvO s )O 2.i(3)
So we get v(s) = S; : v01 +Z v0 cos8O + . (2mvO O for the dependence of the random velocity on the path covered. Now the number of collisions in the element of path ds is given by OCR Output -14
_dSds _(V_V)£_ 1 _ S9 gd f(v) _ !(v) O dv 2 _ !(v) — Z VO CO Os 2 dv S I 0 1 ds eE I(vn) O mvn (v0) where we have expanded [!(v)]-1. Taking into account that, for events with mean free path !(s), the probability of collision between s and s+ds is
- Sdz/1(Z) g(s) ds Z % e `{ 2. 1(4) we get
2 - g(s) 1ds E (E=i- Lm"0 2- v0 cosdl 90 E -EWds§ ) (g§exp+ 0v0 0 cos GO 2-mz E)HV 0
Expanding the exponential factor and retaining only the first term in E we arrive at
N g(s)ds-e...5/20 ds T-; 1c0s9OHT[-(T-E-; eE dl s 1 O O 2 s ) I 20 O
The mean path is then
°° n -s Using [A s e ds = nf and taking the average over cos Gm we obtain
_ Z eE lk 1 eE I dl
W - Q EE L + 1 8E di * 3 m VO szaavivzv
We should now introduce v0 as a function of the mean random velocity v of the considered electron. But the two values differ by terms in g , introducing correction of second order only. So we get at last
_;W·s "¤T
where the averages are to be performed over the energy distributions of the electrons. Note that for constant collision frequency (v/Z = cgnstant) W : EE m v
Remarks: (1) A simpler (and wrong) argument that might have been used is the following: at the time t the velocity of an electron along the field direction is vx = % t+vO cos9O
From our function t(s) [eq. 2. 1(3)] and the distribution g(s) [eq. 2. 1(4)] . we derive
_ _ (vX}COu —l vX(s) g(s) ds — eEK -51 + v0 cos 80
Averaging now over cos Gn in the isotropy hypothesis, we get again the old result
I m I (V)eE /v Xm
This is wrong and shows the importance of dropping the hypothesis (b), that there was no depen dence of <'>t from 90. In this last simplified derivation we have in fact forgotten that at the time t, the distribution in cos 90 is no more uniform because the electrons that started with cos 6O> O are now less numerous than the electrons that started with cos 6O< O, because for them the time between collision is shorter and a greater part of them has already undergone another collision. (2) Our expressions are rigorous (to the first order in eE/rn) and are those obtained in the general theory (Section 3). lf the hypotheses (1) that the differential cross section for the electron— molecule scattering is isotropic is dropped, one should make the difference between the mean free path \F_(v, cos 6) for deflection at an angle 8 and the "momentum" mean free path f(v) defined as
1 + . 1 [1-cos6 dcosG f(v) —_!1)(€(v,cosG) 2
Then the quantity to use in computing w (and so on) can be shown to be JZ and not Xe (see Section 3).
2. Z Diffusion
Let us now compute D in a similar way. After a time t, one electron will move by R = v6t and the dispersion between two collisions is 2 _ 2 2 _ 2 1* _ ,2 (R )COu - V (st ) - 2v V2 - 2 cos 9 ...... ag RQ _ where we integrated over cos On too. Because of the v/B collisions in the unit time, — Zvi Bt which yields, after the usual ave rage over the energy distribution,
D = UV)/3 OCR Output -46
2. 3 Effect of Magnetic Field
A similar computation can be made when the electrons are drifting in an electric field E normal to a magnetic field B. Let us take (Fig. 8c) the field E along the x direction, B along the z direction. Now there is no more symmetry in qa, because of the presence of B and averages over ¢ are to be made as well. The dynamical equations for the motion of an electron in the fields are
d°'x _ eE eB dy {_ dtZ ` _nT Tn- HT
d"y _eB dx _dtZ ` YT HT
dbz. 2 O dtz
which implies ..-d3X -()dt s;§2 _ m dtQ
Solving this equation in dx/dt and taking in account that v; = v0 sin 90 cos dpO and v; Z v0 sin90 sincpo we get the velocity components
x,E x,E
B,z V; \(OYxv ) xv,
@0 % OE`? ~. Y ‘ OY `po \\ I _ xn E,x
XRJNT-5573
Fig. 8. Various axis systems helpful in the reading of Sections 2 and 3 of this report. OCR Output -47
dx/dt = v0 sin 60 cos apo cos + (v0 sin9O sinqno + E/B) sin[E1T1? t) dy/dt = v0 $11190 cos q>O sin + (v0 sin 90 sin¢0 + E/B) cos - E/B
dz/dt = v0 cos 90
By integrating we now can get the three components of the displacement
x : x0 + v0 s1n9O c0sq>O -33 s1n{%¤¥)-1 (v0 s1n9O sin¢0 + E/B) ig (cos(%12)1 -1)*: z x0 + v0 cos ¢0 sin9O t + v0 sin9O sin¢O + E/B -ii;-E
y = y0 + v0 s1n90 cos ¢O gg) (c0s(§—£i) - 1) + (v0 sin90 sin¢0 + E/B) Er-1% s1n z = zO+vO c0s9Ot We assume as above that the differential cross section of collision is isotropic and take explicitly into account the effects of the lengthening of the trajectory and the increase of velocity. To the fl1"St OI'd€1° in €B/YH, €E/YD, Ong ggts s = v0t<1+ EE-T v0 sin9O cosq>O t + é fi;-[ t2 v0 sin9O sinqpo) + . Emv O mv O t = ;1 - -v0 sin9O cos¢0€·- —%-EZ -;nifE(§ v0 sin9O sinq>O) + . <6;% O Zmv O O mv v O O v(s) = vO1+ -2 sin90 cos ¢0s + -2;% ssin9Oz sinq>O + . (2mv O Zmv O and the probability of collision between s and s+ds is . g(s)ds4-1 = i —=expdzf(z'= We find, at last, after long but straightforward computation g eE M 4 eE 1 dll ``` +` <6X?9,¢,CO1i ‘ 3 n~T:23EI v so 6 --1sE2§1’-;£>1$¤‘df So we obtain, after the usual average over energy distribution _ _ 2 x?;; _ _1eEeB£" 2e§IeB£d£ W·"·‘* i D keeps the same expressions, as might be shown, UV}/3 Remarks: (1) We have everywhere expanded sinuses and cosinuses to the first order in (eB/m)t. That is not correct if the collision time is long; in this case the particle would turn over an appreciable angle and our approximation would be wrong. We would have, in this case, to integrate expressions like the following ones an ds 1 ws cos e -— = —-—-i 2 1+u»2l2/v -sin eS/( 9;- = aé of/v 1+w I /v where w = eB/rn. We then have to divide the expressions we got before for w, w, (The M diffusion coefficient transverse to the magnetic field) 2 2 2 1 + wziz : 1 + if V2 2mE An order of magnitude of the connection is obtained by noticing that I Z 3 wm V 2 eE Therefore, the connection factor is roughly K. 1 +§- §——w2 2 EZ For electric fields greater than 10 kV/cm and magnetic fields smaller than 2 tesla, these corrections are usually small (for w< 10‘ cm/s). We will treat this question quantitatively in Sections 3. 4. 2 and 4. 3. (2) In the general case where the differential cross section is not isotropic, IZ should be the momentum transfer mean free path. 2. 4 Energy Conservation: A Simplified Theory In order to compute the various transport coefficients one has to know the energy dis tribution of electrons. That will be done in Section 3. We present here, however, a very OCR Output -49 simple argument based on the energy conservation that allows a rough determination of these coefficients and a qualitative discussion. An electron, continuously accelerated by a constant -M field E, reaches very quickly (in ~ 10sec) a stable drift velocity, exactly as any body accelerated in a viscous medium does. If we call A(€) the mean fractional energy loss in a collision at energy E , the equilibriuni between the energy gained from the field acceleration and the energy lost in atomic collisions can be written eEw = (AE v/E) If there are excitation collisions for which the mean free path is fh(€) and the excitation energy is E - v V EEW - (Ae I) +€ eh (EET) 2. 4(1) This is equivalent` to the hypothesis of a frictional force lv=: wv. with eEw:<-_§•;;’>, Z I and WH :/ (1/3)<=E A/Y \1 +(1/4)g2B2(A/y)2 W, Z/(4/42)fBE(A/Y)Z22 A 1 +(i/4>¤ZB2Z/ 2. 4. 1 Constant Cross Section In order to make a rough estimation of transport coefficients, let us assume that the momentum transfer mean free path is independent of v and that the distribution in energy is narrow (this is not too true, in general). Then J eEw Z % Am -<—1;—z ic) cf F. Bourgeois, J. P. Dufey; they adopt a completely classical approach (without considering the random motion). From g = -kvthey’ get er: 1 _ PLBE 1 +-2-- ` k (1 +-*7- ) which does not differ too much from our expression, if R = 3 OCR Output -20 d “` - ,/i ya2 SM3 Wm ’ I 2 \/ 3 SE. V 3A m L 2 t E2 <€> V 3A e 2 1 eE2 J9D '='\/———·· 3A m 2. 4 2 ( ) eD €.,= T = <<> W E L 2*2 \/A iz i 2x/E m N 1 eE 2 __ 3 WM · z K v · 1 W We see that the shorter the mean free path is the lower are the drift velocity, the mean thermal velocity, and the diffusion coefficient. On the other hand with a great mean frac tional energy loss per collision we get a high drift velocity, a small thermal velocity and diffusion. 2, 4. 2 Application to Pure Argon As an example of the problems involved in the application of the foregoing simplified theory to real life cases, let us take the example of argon. Figure 9 gives the experimental elastic cross section. IO_I5l_ Argon 2 M 10**6 IO·|7|_i..|.;.L.1.q IO 10 E XBL?45·3I66 Fig. 9. Momentum transfer cross sections for electrons in argon. Notice the Ramsauer dip at 0.3 eV of electron energy. The dotted line shows the fit we have made for practi cal computer application. OCR Output -31 Let us first note that in our approximation 0f narrow energy distribution (from 2. 1(5)): W ~ E E A + 2. SE d' — 3 m v 3 m HV 1 I dl _ — E v eE E- + H-E which is zero for a cross section rising linearly with energy. This happens in argon from 1 to 10 eV and in that range the approximate equation 2. 4(1) in 6 1we 2 !(e) TE d1(e) evq£€,§1r;!h€ Ehv - E — + = A e > l——+— has no solution. For practical application we are therefore obliged to use an even cruder approximation wherein we neglect dl/dv w = é veE £(€)/E Aev v E = + - e w —(—Y£ E eh £—J 6 The result of this crude model for argon is given in Fig. 10. The general behavior of eb and w is reproduced only qualitatively, and wm = 3/4 w in contradiction with experiment. This shows the necessity of taking into account the distribution of energy, as will be done in the following section. E (V/cm)(p, 2760mm)34 .1 I IO IO IO IO 108 ME *0 107 cog ug I experimental wuE! -¤¤°_4[• ; IOGE g’ experimental wv_| 3 0 D, 2 ww . IOSE @8 _ W .O| 104}; . .OOI 020 Idle 1616 2 E/N (V cm ) xal.v4v-a5vl Fig. 10. Results of the crude model of Section 2. 4. 2 applied to argon compared to experimental results. OCR Output ..22 2_. 4. 3 Non-uniform Electron Density When the electron density is non-uniform, because of equation 1. 1(Z) there are diffusion currents which contribute to the energy balance. For an electric field along O? , equation 2. 4(1) becomes (omitting for simplicity inelastic collisions) E .. l QE. : A X e w eE DL D ax ( 6 I} zi where DT is the diffusion along the Ox axis. Parker and Lowkehave shown that in that case DT is different from the diffusion coefficient D along the two other axes. What is happening is in fact easy to understand: consider an electron pulse in an electric field in a gas with constant cross section. In the leading edge Bp/6x< O and because 2 1) 1 Bp v E _ _ ...... : A _ e(3eEV +|eEDLpaXI E1 € has to increase and w to decrease. In the trailing edge, the opposite happens and we have therefore a "bunching" effect, decreasing the value of the longitudinal diffusion coefficient. Obviously, depending on the behavior of the cross section, the effect may be opposite. We thank D. Nygren for having brought this fact to our attention. OCR Output -23.. SECTION 3 CLASSICAL THEORY OF ELECTRONS IN GASES The above approximations, if they give an intuitive picture of the phenomena, are too rough to describe quantitatively the experimental behavior of transport coefficients. The history 13 of the various attempts for a rigorous theory is very intricate and we refer the reader to Loeb for an historical account. The real breakthrough was made by Morse, Allis and Lamaru in io 12 1935 and their theory was improved by Mazgenau, Holstein’ and Phelps.This classical theory is based on the Boltzmann transport equation which expresses the conservation of the number of electrons in the absence of ionization. 3. 1 Boltzmann Equation One introduces a distribution function f(v,r,’’ t) such that f(v,r,.>. t)dvdi? gives the number of electrons in the element of six-dimensional volume around the point livin, phase space. We take the x axis along the electric field and consider stationary states. f(v,1?,t)’ is then only a function of x,v, vv=vcos 9 for obvious symmetry reasons and we will consider the density of probability vLf(x, v, cos 9)dx dv dcos 9 d¢ . Since dx/dt = vv, dv/dt = eE cos 9/m, d cos 9/dt = eE sinr'9/mv, the Boltzmann transport equation may be written Bf Bf eE Bf eE . Z Bf Bf _ _ E ‘ VX52 * TH °°S"·5v * aw Sm gmt; ·5;)cOu. · O Mm where Bf/3t)HCO is the difference between the gain and the loss of electrons because of collisions. We will assume that only two kinds of collisions are important: a) Elastic collisions. If dcr/df? = q!a (M\J,v) is the cross section for elastic diffusion at an angle QJ of an electron of velocity v in our gas, the difference between the gain and loss per unit time of electrons in the six-volume element v'“dv dS2 is _ r v v s 1 1 _ v Bf/BijglasticL[ - N, M9 , QD VI, V3 , X) V qc! » V) {(9; ¢» V» X)V qa! (Lp: V) I dg I NVL, If(9`.¢'.v`.><)(%)4 qc2(=P'.v') - f(9.¢.v.><)qe2 (·1¤,v)|dQ' where f(x,v>) has become, by simple change of variables f(v, 6,¢,x), QJ is the angle between the direction 6',q>' before the impact and the direction 9,¢ after the impact, v' is the velocity before the impact, N is the number of diffusion centers per unit volume, and dQ' is the solid angle of diffusion (Fig. 8d). Introducing the aximuthal angle v in the axes vnx(6xXv=),’> (O?cXvw),vw dQ' : dcos•·]J dv 3. 1(2) cos9 = cos 9` coslb + sin Gl sinkb cosv and v = v' I;ing]1 — - (1 -cos¢) Zm = I 1 - [—’l M OCR -Output W 6 6 ( cos ) -24 Note that the mean fractional energy loss is A = Eé-l . The term (v'/v)J in the first equation is the Jacobian v’dv'/vdv.2 z Because v' is very close to v, one can approximate the functions of v' by a Taylor expansion around v, obtaining Bf lat I |elastic —) I Nv] [f(9$7 .<1> wax) —f(9,¢.v,><)] q!a (~)»,v) d9 —rNv IIn B, 7; yM §-£_l'+(1-cos9)q£(4»,v)v€ f(9,q>,v,x)dQ 3.i(3) We will assume here that the electron temperature, that is, their energy, is high enough to allow to discard the energy transferred to the electrons by the molecules of the gas (see Section 3. 4.1). b) Excitative collisions. Their contribution is 3. i(4) at lexcitation=Z Nj h [f(9»¢'.v’.><)(!—)v'YZ' (¢`.v`) —f(9,<1>.v,x)vq (dmv) d9' = V qh 11 ] =ZN[f(9'.¢'.h -£2_ v'.x)(%—)L<1h(¢'.v') - f(9.¢, wx) q._(¢» v)] df? ,. . . where 1/Zmv2 - 1/Zmv2 = 1/Zmvh2 = eh is the excitation energy of the h-th level and qL((|J,v) the excitation cross section for that level. The Jacobian is now v'/v since dv'/dv = v/v'. Because of the large values of Eh no Taylor expansion is now possible. Here again if the temperature of the electrons is high enough one can neglect the collisions of second kind where excited molecules give back their excitation energy to the electrons. We neglect for the time being ionization collisions, which we will treat in Section 5. Let us assume also that the distribution of electrons is uniform. It will be independent of x. In order to solve this Boltzmann equation we expand f(v, vv) = f(v, cos G) in Legendre polynomials. It can be shown a posteriori that the first two terms are usually sufficient and we write f(v,cos 9) = fn(v) + cos6 fl (v) cos 6 . -. Taking thedcos averaes 9 gr — and f? dcos 9 we get the two equations -m ,a . 3, - 2-. 2f + 3———- 4LO ——- fIi I - fv' Z : ;|O (v BV) v ()1MBV (v )+Z()v QEh (v)/ vOh0h (v) v(v)/(v) and af O I - HW vfi/e — O 3. OCR Output -25 (Ie)`1 = 21rNI (1 -c0skIJ) sinklu q(¢,v)€l dkI»· (zh)‘* : xm] sm kp qh(¢,v) dw a = eE/rn We have followed here the standard practice t0 neglect in the second equation the terms 1 m 3 4 cos 9 cos 9` f4(v')(v'/V) €l;,,(¢`, vl) dcos 9 dcos G' édu - Zf(v)hf1h q(*|»".v') dc0s\IJ du which are usually small (the first one because of the factor m/M, the second one since for moderate energies q1_<< q__” ). If one works with the function F(€,cos 9) = Fn(€)+ F4 (€)cos9 ll 61 V A such that € 1'I`13.X f FO(€)d€ = 1 the foregoing equations become "2/m"€h 2*2wp L...... LQ - §€VF0F (gig )- 3 86 1 M 86?:hh 26 Eh !(€+€) 0 h ~/267mF£hi€$ 0 (g)|g O ,1(6) 8 2eE VF1 _ €E §("F0’·mF¤* T · 0 _1(7> or by eliminating F4 , 3, 2 3(es) m 3 ge4 6(FQ/V) as Mam ae3 evr tele) 6 ME) + + .1(8> + Z JZ7m\/6+e] F 4€+€ )_ {Ze/m Fg(€) I : O h gh 6+644 0 h gh 6 The various transport coefficients are F H = /vc0s8 F(v,cos6)d€@ = Y-gde = -E $5 efemdé 2 3 3 m 66 ,1(9) 2eE le 2eE € 3,4 _2eE eE dl which is exactly the expression obtained in Section 2 (Eq. 2. 1(5)). The diffusion coefficient D is from its definition D = fg3- F (e)0 de .1(10) OCR Output -36. One can show (see Section 3. 4. 2) that, if the magnetic field is small enough =/vsin9F(v,cos9)d6-@@:IF22 2 3m 1 d6= l3 eE eB I"m Z eE eB m52EI dz j -- F0(6)d6 + K ?—/-G1O av F(6)d6 3.i(11) which is again exactly the same result as in Section 2 (apart from the minus sign due to our axis convention). The conservation of energy written in the same section has, of course, its exact analog. Performing on Eq. 3. 1(8) the integration _[6d6, we get v E -- 2_ _Q..deEr/* ZB(F ...... Q_/v) d e2m 3 m6vF 1e(6) (6) 86 6 M 6 l6 +2h eh ma F0(6) de e 3. M12) or, eEw : elastic losses per unit time + inelastic losses per unit time,which is Eq. 2. 4(1). Remarks: We have assumed that f is independent of x. This is not right for an isolated pulse of electrons as we have seen in Section 2. 4. 3, since the distribution of velocity depends on the position in the pulse. zi Relatively recently (1967) J. H. Parker and S. J. Lowkehave developed a method of solution of the complete Boltzmann equation where the Bf/Bx terms are not rejected. The result is quite complicated and we will not describe it here. 3. 2 Druyvestein Distribution If !e(6) is constant and there is no excitation, our Eq. 3. 1(B) becomes 2 (eE)3 3(F/v) Zm 3 6vF_ O ````€I€ + é () § m §éas M BG ze 3·2(1) OCR Output By integrating on d6, we get (¢E)° 8(F/v) 2m 5 - 2"€1"+ Fc C ?r7¢?M fe where C is a constant. From that we get easily, because of the constancy of the fe $ Z k exp {2/E(eE!e)21C [g il|if +;*.*1“.°3..2A 2(f E) Efeexp (E2/3i$(sEi€2>) de The second term gives a non-convergent contribution (J F(6)d6 = °°); so C=O and, finally -37 F- F(e)de = —;-£ ex-% 5/d 2 (3 4 [yl.- E1 il / )L 3Aep<--2) e 2AeE! e) 3. Z(2) , where A = 2m/M is the usual fraction of energy loss per collision. We can see that this distri bution, found by Druyvestein' in 1930, decreases faster at high energies than a Maxwellian one 2 1 -e/kT F d = — -— d (e)`}§ e V? -1€kT e e 3.2 (3) With Druyvestein's distribution we can compute the transport parameters. Y _ “· no2 E [ 31 EE dll v§ 1 E ;av5Z1( ) dl Hfe e _I;vavl(`n e e _ <>‘<>· _—‘F(€)°)€‘ 4/2 ·=·> A 1 2 -1/2 di)A/O _ 2 eE 3(F/v) EE€ F(€)(!+€H-EE,$ d€-E/·€[B—T—·— eEMZm vf 6 ff:F(€) de °° or-1 -t _ from where we can get (because [ t e dt —- l”`(cx) ) 2 1`(3/2)11374) J V? ,J— eE!e sv? W = -·- rn A —. 3. Z 4 ( ) Similarly we get (E) =/‘€F(€)d€ = V2/3A eE2e Fo/4> {2 1 °° 1 M? } 2 D = .. - ~/‘ F d = - - m 3 L E ME) (6) E 3 (374)J 3A eE£3 2 leB f“"g“3 eB ml`(5/4) O J2./x e ‘ : .. ..-. E .... F md : I(3/4)i. .-——...Y Ie WBT (E) E ` W: M W[H3/2>]i- gz-; : 1,067w 3,z(5) These expressions for the transport coefficients do not differ but for numerical factors of the order of 1 from those we got in Section 2. 3. 3 General Case We have already seen that in real life (Fig. 9 for argon) that cross sections are not constant. The solution of (Eq. 3. 2(1)) is then OCR Output -28 3/ie F(€) = CV?E gf 7exp O [eE ———-d€fe(€)] 3.3(1) where A = ilig- and C is fixed by fFd6 = 1 . However, we have neglected the excitation in the above equation, Its inclusion transforms the differential equation in a difference —- differential equation: .4 F 2 E (eE) €£€(€) 8( Q/v) + 2m 3 6vFQ + B6 3 m B6 M 3€ 2€(€) M2/m*¤!€+€h F (6+E) \12?m '\(?FQ(€) |_ - _ ' .4 l- I O h g 6 h h(e+eh) h 3 32 ° ( ) 3. 3. 1 Approximate Methods Most of the authors (Refs. ii, 22 for argon) have tried to get rid of the third term and to come back to an ordinary differential equation. Let us consider two extreme cases: a) eh Large In noble gas the excitation is large and one may safely assume that Nl 2/m '\J 6+Eh F(€+€h) << Nl 2 7m rpg F(€) !h(€+€h) fi i6 due to the fact that F(€+6h) << F(€) , (except maybe in the neighborhood of 6:0). We are then left with 3.3(3) 2 eEL 8 BF/v Zm 3€ €vF Z1) ZE _ g g e£(€)—&——+—M§(7;)-(big FO(€) - O ii Golanthas used this method very cleverly with approximations to fg and fh especially designed to allow analytical integration, and his result is not far from those obtained by Z3 Engherardt and Phelpsor the present authors with rigorous methods. b) Eh Small In the organic vapor, on the other hand, Eh is usually small and one may replace the two last terms by E _§_ vF(e) h h 86 Phle) Inserting in the equation this gives 2 eE L 6(FQ/v) A(€) _ el(€) + vF— O 3. $(4) OCR Output 5 €86 Tm O -g9 where we have replaced iz-EE-+ 2 Eh 1 '1 b A 1 , H h v <<>/ <<> . · where U6) = + IS1 the1 1 7Tmean e free path and A(€) is the mean energy loss by collision. We may note that the presence in the first term of fe instead of I comes from the fact that we have neglected the angular dependence of excitation collision. If these are important they have to be taken into account and we would obtain approximately 23 eE ‘ m866(F /v) me) ..(TN _ 1 ..O_ : ()O 6 +-VF0 The solution 3. 3(1) is then replace 3A(€) E de F(€)6 =:/· CV? T 9 l¢E1(€)lexp — 3.3(5) 3. 3. 2 Rigorous Numerical Methods These approximations were useful before the availability of large computers. Nowadays it is possible to solve Eq. 3. 3(2) directly. The method follows directly from the observation first z'4 made by Shermanthat if you know F between 6 and €+<—ih (6h being the last important excitation energy) and F'(€), you can solve the equation between 6-61,1 and G; it is just an ordinary differential equation. Sherman has developed such a method of backward prolongation 1223 of the elastic solution valid at infinity. Phelps and co—workers’have used this method extensively for a large number of gases and mixtures. The present authors have developed a similar method; instead of starting from equation 3. 3. (Z) we have started from a system of equations easily deduced from Eqs. 3. 1(6) and 3.1(7); F BFD/v 2 _ 1 QE eE gev N/s+eh F (s) 2El..<€>. 3 as. E a ._<> h - 0 as QE 6 as 1 + 1 FO(€) + Eh gE UsF I (6%1 (ca) l e e (€+€) h hh 1 + 3Ae - _ ik (em.? F (6) ‘ 3. 3(6) where v Z '\}2€;m and A = Zm/M. We take as starting value, a reasonably high energy 6 (about 4 eb), choose FO(€O) = 0, and F1(€O) = 1, since we expect the asymmetry to be large at high 6 (in fact it is infinite in our approximation) and solve backward the system (with a modified 25 is version of the Lawrence Berkeley Laboratory library routine DIFSUB). Our program is available upon request, from B. Sadoulet, Lawrence Berkeley Laboratory, Berkeley, California 94720 USA. OCR Output -30 3. 3. 3 Application to Argon On Fig. 9 we have plotted (dotted line) the momentum transfer cross section we have used: > _ -N! E 11.5 eV U — 1.52><1O 15/ (6/11.5) Z cm 15 1.15 < e < 11.5 O : 1.52><1O`6/11,5 cm2 .3 <€< 1.15 O = 1.46><1O`U + 1.9 1O`(616 -.3)cm 2 2 1265 2 6 < .3 c : 1.46><10"' + 1.24 10‘(.3 - E)cm The excitation cross section has been taken as a = 9><10'*' ~/6/11.5 (6/11.5 - 1) Cm and eh = 11.5 eV The resulting transport coefficient are plotted on Fig. 11. The agreement with the data is ii excellent over five orders of magnitudes of E, a result already obtained by Golantfor the 23 high energy region and by Engherardt and Phelpsin a more detailed analysis. The cross sections used by these authors are essentially identical to ours. A remark may be useful for application of such calculation to drift for chambers operating in a magnetic field: The "magnetic velocity" WM is very sensitive to the low energy part of the spectrum and therefore to the hypothesis made for the losses in excitation collision; the dashed curve in Fig. 11 shows the behavior of wm when instead of assuming an energy loss equal to Eh, we assume a total energy loss. In that context the deviation observed in our calculation (and the one of Engherardt and Phelps) between theory and experiment is not too worrying. E (V/can) (p=76O235 nlm) .1 I IO IO IO IO IO IO8 1O O 1/ goo i! 107 lzi 1 |Q6 .1 iw IOS? wd W {O4): • .OO| IGZO ‘618 I61G 1614 E/N (v cm?) xm. 745-sues Fig. 11. Results of rigorous approach to the argon transport coefficients compared to experimental results. OCR Output -34 Another warning may be given. Figure 12 gives the distribution function and the asymmetry as functions of 6 for various fields. At high fields (where we have taken into account the ioni zation)(see Section 5) the asymmetry is very large and our approximation of F(€, cos9) by first two terms of the Legendre expansion FO(€) + cos9 F1(€) breaks down. The method described in the last section can easily be extended to a mixture of two gases 1 and 2 in proportion 6 and 1-6. It is only necessary to write 1 Z 6 L J, (4-6) 1 11 12 3. 3(7) A ZA +é (1-)6; 62 1 I 1 Excitation [¤niz¤ti0¤ -¤ 10 Enercvl Enemy. 80 kV/crn 80/rl// C ¢ 25 kV/cm 1/ Q , IG 0.25 kV/cm 162 2.5 kV/cm 162 163 0.8 kV/cm 1 JL O O IO 10 6 (eV) ‘ (G) E (eV) (b) XBL T45·3\64 XEL 745*3liB Fig. 12. (a) Distribution function Fn(€) for argon at different electric fields. (b) Asymmetry, that is F1(€)/F(€)O for argon in different electric fields. lt is very large at very high fields where the approximation used in the computation (only two te rms in the Legendre expansion) breaks down. OCR Output -32 3. 4 Special Cases: Low Electron Temperature or High Magnetic Fields The above theory neglects several effects: the movement of gas molecules, the action of high magnetic fields and ionization. This last effect at high electric field is important and we will discuss it at length in Section 5. We will review here the modifications to Eqs. 3. 3(2) or 3. 3(fn) due to the first two effects. 3. 4. 1 Movement of Gas Molecules lf the electron temperature is low enough (6**kT ~ 2.5 10-° eV at ’l8°C) we should take into account the movement of gas molecules. Moreover, if these are excitation energies of the order of kT, the thermally excited molecules can give back to electrons their energy (collision of the second kind). These two effects may be taken into account by the addition to Eq. 3. 3(2) of the iz two termsz 2Am d (ecge kT dF/v)e gh+2 V2/m ($614)we-6b FK E) N/2/m gh V6 F(6)`| e 3 M1) H? Hh h where fh is the mean free path for collision of second kind and is given by detailed balance argument: (6-Eh) E W = exp (- 6h/kT) UE) An a posteriori justification for the modification 3. 4(i) comes from the fact that the energy loss due to elastic and inelastic collisions goes then to zero for a Maxwell distribution F ~ y-6 8-6/kT In order to see the effect of the temperature let us assume that fe is constant and that there is no inelastic collision. Equation 3. 2(’l) becomes now, after one integration 2 3 Ze 86 or writing 6(`L = (eE!e)L/3A (6,3 + kT6)Qg£;#’) = 6F/v Therefore F/v = 6 exp 6 6 T/4 -2---- d6 0 60 + kT6 62/(KT) kT6 0F : which for large kT converges to the Maxwell distribution Eq. 3. 2(3). Then from Eqs. 2. ’l(5) and 2. 2(i) OCR Output -33 W;/ifi E%Fd5:._;-...... -.. @*5 r3 V 3'\/?'\lmkT D:[g L-d€:z~121~/kr 35 '\}m and 6D - - Ek · T · RT It should be noted that 6k is then fixed and equal to kT, and that the mobility is constant. It is for the same reason that at low field, the mobility is constant for ions: A is very big and there fore their temperature is close to kT. It is difficult to incorporate collisions of the second kind in the backward prolongation method 24 we have described in Section 3. 3. 2. Sherman and Gibson have developed other methods. In our program, in order to minimize modifications, we have used an approximation based on the following remarks: a) For low E of the order of kT, F converges towards a Maxwell distribution, and to firstorder F ¤¤ V? (1 + ae) e` E/kT with _ 1 a - ----T€7m.. (F/~/?+kT7.i?)d F/V? kT e b) For complex molecules chg E for es kT, since at that energy the vibrational cross sections are important. Therefore “€'€h F(€'€h) _ ¤/?F(€) 2 L 8-e/kr ae z gi B-6/kr az ` HZ kl M65 lh - ~lH@ e -e/kT)/ d(F/·~/?) (1-e \r+~lEl? mi. and we have used the last expression, which for large 6 converges towards —T-YR E (E) These modifications have even for pure isobutane (see Section 3. 5) very small effects and just prevent eb from decreasing below kT. The crudeness of our approximation has presumably very small effects on the final result. 3. 4. 2 High Magnetic Fields We have already mentioned in Section 2 that in a large magnetic field, particles turn over an appreciable angle between collisions and that this increases the apparent density of the medium. zé The way to do it quantitatively with this phenomena has been given by Allis OCR Output -34 If the electric field is along Oz and the magnetic field is along Oy, the Boltzmann equation 3. 1(1) is replaced by V 2£+dv Bf + dcos8 Bf +d¢ Bf Bf _ O - _ X 3x HT gv Ht dcosg HE EQ? ECON where 9 and ¢ are the usual polar and aximuthal angles and dv _ eE - - - cos9 ETn wwrp;dcos9 _ eE . 29 _Q Singm COS - · Sm ¢ d¢ _ _eB cosG Sin — HF TH sing The angles after a collision are given by Eq. 3. 1(Z) supplemented by sin9 cosqn = cosrp sin9' sin¢' - sin cos (v -¢) sin9 sinq> = cos¢ sin9` sin¢” + sin sin (v-cp) Proceeding as before, but now expanding f(v, cos8,¢) as f(v, cos6,q>) = fn(v) + cos9 f4(v) + sin9 cosdp f7(v) + sin8 sinq> f2(v) we get equations similar to 3. 1(5): - eE`3mVZ 8v°f1 1 m B 3v4 - v'VZ , . .M —-+—-—-—(vf)+Z|—vf(v)/£(v) gv O h v0 h l) — v O)/h]f Bf SE i J, Q { - V1 Z O m 3v m Z I e E2 { - 2 m 1 ge f3 = O Therefore eB [ _ fz ‘‘ "5F£ fi from which one deduces transforming to FO, F1, FZ, F3 eE 8 2.m 3 O -_ ¤m *¤+sh6VF}27}? Nm- e 2 OOCR ·? Output5-%- (V F1) · V g 'ivl FO(€+€h)W Z/mw F(€) -35 2 2 2 vF 3 eE8€0mv e B I 1 E— - t O2meT 2 e (vF) ZF+(1+--6->le O SF/v 2 O_ v _ cE el -—·- Wn-/‘:F1d · *5 H/...-Q.-Q Bf de , 44-Lg gms cB _ V _ wi -/? Fzde -3%/1€F1de jg V 6 aB 1;e= /423 - -eE- 1+ E B—-.? Ié ds Zme where we find again the factor EQBLIE. 1 +i-— 2m€ which we have already encountered in Section 2. 3. Similarly, the diffusion transverse to the magnetic field is [ v F DL ‘ I 1 + 22;. d6 Zm€ The effect on the transport coefficients for a field of 1 tesla is exemplified for pure argon in Fig. 13. The bumps observed for ek and WM come from the fact that the diffusion D and wl are less affected than wu, which is dramatically reduced. tgs IOO 108 IO 107 ¤°°° rm · A f' 4 Q, Fig. 13. Calculated variations of 196 the transport coefficients for pure argon in a mag 105 €¢,,,.•°° I OI · . . netic field of 1 Tesla §• (10 kG). • I \·W 104 .OOI -20 -|8 -|6 IO IO 2 IO E/N (V cm ) xm. Ms-suse *wher€ we have kept the definition eb = eD/k where D is the diffusion parallel to the magnetic field. OCR Output -36 3. 5 Fit to Realistic Gas Mixtures Unfortunately the gas used in multiwire proportional counters or drift chambers are more complex than pure argon: argon-CO, argon-isobutane, argon-methane or pure organic vapor such as methane or ethylene. The fact that some of these gases are mixtures can be dealt with very easily: the various cross sections are just the appropriate weighted average of the cross sections for each component (Eq. 3. 3(7)). On the other hand, the fact that we are using complex molecules introduces important modifications: in addition to electronic excitation, we now have excitation of the rotational and vibrational modes of the molecules. Figure 14 gives the excitation spectrum of 23 CO.) as deduced by Hake and Phelpsand shows two general effects which are very important for understanding the behavior of electrons in complex molecular gases: a) The vibrational excitation cross sections QV are very important compared to the momentum transfer elastic cross section QM in the energy range O.1 to 1 eV. In other words, the mean fractional energy loss per collision A is large, and from Eq. 2. 4(Z) the drift velocities will be large and the mean energies low. So low in fact that they are of the order of the thermal energy l b) The important vibrational energies are limited to a few tenths of an electron volt. This leads for higher fields to an effective A :6/6 where eis the highest important vibrational max max energy. This will generate as we will see, constant or even decreasing drift velocities. 1u"“ Ramsauer!. \ Kollathi ) ·· \{ Q; ,—¤mm•e0,n .| 40 ( H `= 2 ll Q O .,% ;. `~ l 0x/Nl_¤ _ /-s· \ \·< N -15 E ‘° v, ‘· WX =·/ \ é 6 l ". \h,-, 0. 20% y [ (BH4 ¢ 51% /<*,H6 E, ,0-16 °» //·/°¤—l __ _ _ on :34a —l7 10 m 2 lo-2 [Q-I ] 10 10340 ` Electron Energy, av (G) xm.141 - asvo Fig. 14. (a) Excitation spectrum of CO (QM is the momentum transfer cross section, Q the vibra- 2** NN (.0 " mo tional cross section, Q the _ W *"° * . . electronic cross section, and (O md Q. the ionization cross section. (b) Total cross sections for _ _ - 0.0 {.4) 3-an 31) 4-0 ..-0 Q.-ll M! electrons with various com V F?l—··**¤‘¤¤ *‘*'**°*‘>‘(~""l‘*l xsuav-ssss OCR Output plex molecules -37 Of course when speaking of realistic mixtures we encounter the usual problem that we have not enough data on cross sections to predict through the use of the above theory the various trans port coefficients. We have therefore attempted to deduce these cross sections from a rough fit to the drift velocities in various mixtures of argon and the gas considered. We have done this for isobutane using data kindly given to us before publication by Charpak and Sauli° and for zg methane where drift velocity measurementsexist for the pure gas and one mixture with argon (10% CHA, 90% Ar). In both cases, we can compare the predicted eb with either the eb which can be estimated from the measurement by Charpak, Sauli and Duinker“ of the position accuracy in a drift chamber filled with 25% isobutane and 75% argon or the eb measured by Cochran and zg Foresterfor pure methane. 3. 5. 1 Argon — lsobutane Mixtures 30 The total cross section for isobutane is only known above 1 eV and not very accurately (Fig. 14b). The data on drift velocities seems to be incompatible with the simple extrapolation of the measured cross section to zero (over estimation of w at low field). As a first guess we took, therefore, a flat total cross section below 1 eV. We also took a flat excitation cross 31 section and from infrared spectrawe fixed the upper limit to vibrational energies at 0.36 eV. We then fitted by hand the data at the lower and the upper concentrations (7% and 38% isobutane) varying only the value of the total cross section below 1 eV and the excitation cross section. Reasonable values seem to be O = tot 1.1X 10-15 cm2 for €$ 1 eV {i.i>< 10‘*’ (8. -€) + 4.8 10'U (E - 1.)]/7. em for 1 S € S 8 eV (4.8X1O-/'\/6/8.)15 cmd for €>8eV O exc 8X 10-17 cm2 for E S 0.36 eV h Figures 15 and 16 show our results for w, WM and ek. The agreement for drift velocities is fair, taking into account the crudeness of our model and the experimental difficulties of absolute normalization. As will be shown in Section 4.1, we predict rather well the position accuracy obtained by Charpak et al in their drift chamber. From our fit we can predict the drift velocity in pure isobutane. On the small range of E/p 27 where it is measured,we agree quite well with experiment (see Fig. 22). 3. 5. Z Methane In order to have more confidence in the mechanisms proposed to explain the data, we have played the same game for methane. With the same adjustment by hand of the total and excitation cross sections: i" Above we assume a straight line joining the point at 1 eV to the known maximum at 8 eV. OCR Output -38 4r5v"$§~<> 4}- L7 /’ Ar 75% lsob. 25% Ar 93% Isnb. 7% 4l* //7, K ; 4I- %’ ` ; Ar 70% lsob. 30% 2} ’ -E‘-$.*2:m::s°' * Ar 86.5% lsob. l3.5°/. V v1“(lheory) A w Experimental (3l%) » \ ; 4r- rj —-.:_ 4|— ,l/ Ar B1 °/° {sob. 19 °/¤ Ar 62% 1»¤¤.38·/. Zkp D Experimental w 2}- { ——-· w (theory) V wuhheoryl O 500 1000 l5OO 2000 2500 0 500 1000 1500 2000 2500 (q) E (V/cm, a1 300° K ond l atm.) E (V/cm, ul 300° K and 1 alm.) XBL 745-3lGI XBL747'3|EO Fig. 15. Computer w and wm for different concentrations of argon isobutane mixtures. Comparison is done with Sauli et al. measurements. OCR Output -39 GK in severol Ar-Isobutone mixtures provided by the theory 1·°'\ ‘°y we . tw} 1¤\¤ N" o°‘° \y¤‘¤ ·.=··‘* 6*+/ ,~·°‘ q\ I @·t Wy O.| wty zve} KT = 0.025 0,0l IOO 5OO IOOO 2000 E (V/cm, ot 300°K, totm) x¤¤.v4¤—ar¤s Fig. 16. Computed €L_ for different concentrations of argon isobutane mixtures. OCR Output -40.. -16 2 Umt = I 1.4><1Ocmfor ES 1 [1.4><10'*" (8. -e) + 2.8 10-13 (6-1.)]/7. cm for 1 S E S 8 eV (z.8><10‘1°/~/678.) for E > 8 eV -17 2 USXC : 5 X10cmfor 6hS 0.36 eV we obtained the drift velocities shown in Fig. 17. The data are from Borner et al.?Cottrel8 28 28 28 Walker,and Wagner et al,The experimental points of Hurst et a1for pure CHA seem too high and are usually not quoted in the literature, 1O A Argon 90% CH4 10% 8|—§>;‘— Experiment ’§ I rt ~ i gl \ b -..- IO % CH b 6I—:f\ --+--12% cn Kt} \ ‘u} 4 \ I \ ,,_` " \0" \ *O 3 2 C0ttrel;W¤|ker (1965) IO —~`_ _ Bortner et ¤I.(1957) "\\ Q 8F Wx / \-Wagner et cl. (l967) Pure CH4 1000 2000 BOOO E (V/cm, cnt 500° K ond 1 cutm) xet 745—3n58 Fig. 17. Computer w for pure methane and 10% methane in argon. 3.I`gOY1. OCR Output -44 The agreement is worse than in isobutane. In particular our assumption of flat cross section below 1 eV does not allow us to describe very well the sharp use of drift velocity for pure methane below 900 V/cm, and we predict too sharp a peak for the mixture 10% CHA, 90% argon. We may note that in this region of concentration the drift velocity predicted by our program depends very much on the exact percentage of methane as shown in Fig. 17 (dashed—dotted curve). There are also problems in our analysis due to the fact that F4 is of the same order of magnitude as F] and that therefore the two first terms of the Legendre expansion of the distribution function F(€, cos9) are not sufficient. We may however, compared our 6,, with the measured values of Cochran and Forester (Fig. 18) which agree for pure CHA with Cottrel and Walker. The agreement is rather good. We think therefore that in spite of its problems, our model is able to describe reasonably well the qualitative features of the data both for isobutane and methane mixtures with argon: the saturation of the drift velocity, the dramatic decrease of drift velocity at low concentration, and CH4 , CO2 IO `%` f (06/ °y\<°°/ C»° / CH4 theory X C,\’\°· / I coz coci .x><>¢ C02 w Thermal limit 300°K IOL. I_. IOO 5OO IOOO 3000 E (V/cm, 01 300°|·< , I 01m) xsuas-sis? Fig. 18. Computed Gk for CH4. Experimental data for methane, CO,, and pure argon are also shown. OCR Output -42 the small values of €l__. We have not attempted to distinguish between the two coefficients of 28 of diffusion D and DT (Section 2. 4. 3): experimentally (Wagner et al.) for CI-I4, DL is not very 21 different from D, presumably because of the small variation of the collision frequency. 3. 6 Intuitive Interpretation The strange behavior of the drift velocities encountered in the preceding sections are some what difficult to understand intuitively. We therefore would like to conclude this section by the discussion of their origin. 3. 6. 1 Approximate Theory Let us go back to our Eq. 3. $(5) which is an approximation to a general theory of electron in gases Fm . Cir. HPe s) 0&Is2 [grim] 2 The drift velocity is then W 2 '~(27m A(€) 63/2 F(€)d€ eE £(€) Let us assurne that ./\(e) = xm 6 £(€) Z IDE-(¤’~€)H n We have then 3)\ €n+2n+2 F(€)de =% C\}€—- Z0 exp(m+ 2n+2)(eE)-ji d€ From dimensionality arguments (F(€)d€ has no dimension): C z E4/(m+ 2¤+z) e¤ (6) = E2/(m+2“+2) 4.1(3) k and W z Em+1/(m+2n+2) The same result may be deduced from the simplified theory of Section 2. 4. It is seen then Eb is always a rising function of E, but may rise very slowly in agreement with intuition if the cross section or A are fast rising functions of 6 (m and n large). On the other hand, w will rise slowly if the cross section rises rapidly or if A decreases with 6. Eventually for m< -1, w will_de_c_rease. We will now study how in practice we have a rising cross section or decreasing A. 3. 6. 2 Pure Gases In order to exemplify these effects let us first look at argon. The cross section has been given in Fig. 9. The cross section below 0.3 eV is decreasing. Therefore, from 4. 1(3) for OCR Output -43 6bS 0.3 eV we expect the strong rise of Eb and w seen experimentally (Fig. 1). Then the cross section rises sharply and we expect w and €L_ to be nearly constant with 6b rising twice as fast as w. This will continue until an appreciable proportion of the tail of the energy distribution is beyond the excitation potential of 11.5 eV. From the plots of Fig. 1 we see that this happens around E/N = 3. 10-V/cm.17 2A then is effectively increasing very much, producing a sharp rise in w and a leveling off of 6 This qualitative discussion may also be applied to the other heavy noble gases whe re the Rarnsauer dip in the cross section occurs (Fig. 19), producing basically the same structure (Fig. 1) Xe (ul Kr Ar O 2 4 6 8 IO Electron veloclty (,/volts l lb) 6 l-"· He Ne O 2 4 6 8 IO Electron veloclty (.,/volts ) xBL.,4.,-3568 Fig. 19. Cross sections for argon and other noble gases showing the Ramsauer dip below 1 eV and cross sections for the other gases He and Ne. Units are Wag ¤ 10cm OCR -15ZOutput -44 or to CO2 where the cross section (Fig. 14) is falling off rapidly at small 6, giving at relatively 16 2 high fields (E/N > 10-V/cm) a very fast rising w and 6,f (Fig. 2). In the last case for small fields, 6k is limited by ambient temperature and, as shown in Section 3. 4. 1, w is then propor tional to the field (constant mobility). However, the rising cross sections cannot alone explain the saturation of the drift velocity in organic vapors. The upper limit to the important vibrational energies is the responsible mechanism: it leads for high 6 to an effective fractional energy loss A = 6...... /6 where eis the highest vibrational energy. Therefore mz-1 and w is approximately constant. max 3. 6. 3 Mixtures Another mechanism may occur in mixtures especially with the heavy noble gases (argon, krypton, xenon) where there is a Ramsauer minimum in the cross section. I-et us add to argon a hypothetical gas of constant cross section and constant A in such a proportion that the Ramsauer dip is filled partially. For small enough 6, the added gas will dominate, and I and A are constant. eb and w rise with the field until the cross section of argon becomes comparable to the cross section of the added gas. From that point on Z and A begin to decrease. This results in the leveling off of the drift velocity, which then becomes constant or even decreases. If the proportion of the additional gas is increased, the Ramsauer dip is filled completely and such a behavior no longer occurs. This is the mechanism responsible for the behavior of mixtures of argon-CO, (Fig. 3), argon—isobutane (Fig. 15) and argon-methane (Fig. 17).