Numerical solutions of the Boltzmann transport equation

Computer Physics Communications 19 (1980) 377—393 © North-Holland Publishing Company

NUMERICAL SOLUTIONS OF THE BOLTZMANN TRANSPORT EQUATION

*

S.D. ROCKWOOD and A.E. GREENE University of California, Los Alamos Scientific Laboratory, Los Alamos, NM 87545, USA Received 14 August 1979; in revised form 21 April 1980

PROGRAM SUMMARY Title of program: NOMAD

Keywords: Boltzmann transport equation, electron energy distribution, gas discharge lasers, atomic and molecular transport data

Catalogue number: ABVC Program obtainable from: CPC Program Library, Queen’s University of Belfast, N. Ireland (see application form in this issue)

Nature of physical problem The purpose ofNOMAD is to calculate the distribution function of electrons accelerated by a dc electric field in a mixture of atomic or molecular gases. This distribution function is then convolved with elastic and inelastic cross sections to provide energy loss rates. The code is most frequently used to provide pumping rates for gas discharge lasers.

Computer for which the program is designed and others on which it is operable: Computer: CDC 7600; Installation: Los Alamos Scientific Laboratory

Method of solution NOMAD solves the linear Boltzmann equation in a flux divergent form. It converts the one-dimensional electron energy axis into a discrete energy grid by finite differencing the respective electron energy gain and loss terms. The result is a finite set of coupled, linear differential equations which define the number density of electrons at each energy as a function of time. This matrix of densities is then evolved forward in time using an implicit back-substitution algorithm until a predetermined convergence criterion defining a steady state is achieved. Once the steady state energy distribution of electron number densities is determined, the elastic and inelastic energy gain and loss terms are used to calculate rates for the gas mixture, heavy particle number density, and electric field strength in question.

Operating system: LTSS Programming language used: FORTRAN IV High speed storage required: 38356 words No. of bits in a word: 60 Overlay structure: none No. of magnetic tapes required: none Other peripherals used: line printer, disk file input and output, large core memory

Restrictions on the complexity of the problem . . The version of NOMAD published here .is intended for use in situations where the ionization level is not so high that . . electron—electron (e—e) collisions will have an important

No. of cards in combined program and test deck: 1509 Card punching code: none

impact on the electron energy distribution. Modifications to . . . . handle e—e collisions are discussed in ref. [11. These modifications add significantly to the running time of the program. It should be noted that NOMAD, following the work of Holstein [21,assumes that the two-term expansion of the Boltzmann equation in terms of spherical harmonics is adequate. It is beyond the scope of this work to discuss the ramifications of this limitation.

CPC library subprograms used: none

*

Work performed under auspices of the US Department of Energy, 377

S.D. Rockwood, A.E. Greene / The Boltzmann transport equation

378

NOMAD assumes the computer core has been preset to zero.

for five different E/N values typically requires 6 s on a CDC 7600.

Typical running time A run without superelastic collision, calculating distributions

References [1] S.D. Rockwood, Phys. Rev. 8A (1973) 2348. [2] T. Holstein, Phys. Rev. 70 (1946) 367.

LONG WRITE-UP 1. Introduction NOMAD is a computer code written in FORTRAN IV which solves the Boltzmann transport equation in energy space. The resulting solution gives the distribution function of electrons accelerated by a dc electric field and in a mixture of atomic or molecular gases. From the distribution function the transport parameters: drift velocity, average energy, diffusion coefficient and mobility are calculated for comparison with experimental data. In addition, rates of energy loss through elastic and inelastic scattering are evaluated and displayed for use in discharge kinetic models, The fundamental approach encoded in NOMAD was initially developed by Rockwood and Canavan and Proctor at Air Force Weapons Laboratory in 1972. This initial development was directed at determining electron energy distributions, transport coefficients and pumping rates for gas discharge lasers [1]. Since its development, the code has been used extensively at AFWL and LASL and has been distributed to many academic and research institutions. In addition, with some modification, this same formalism has been used to predict gas breakdown due to laser radiation (i.e., ac fields) [21, to back out elastic and inelastic cross sections from experimental electron transport data [3], to simulate the cooling of electrons under conditions found in planetary atmospheres [4], and to simulate energy deposition in intense neutron sources [51. The following sections of this report will describe briefly the numerical method and tests of its accuracy, present a logic diagram, and conclude with an example which can be used as a standard test case,

tropic. The resulting equation for the time rate of change of the isotropic component of the velocity distribution is then converted from velocity to energy variables and projected onto a discrete energy grid. The details of this derivation are presented in ref. [3] and in appendix A. The result is a set of coupled, linear differential equations of the form C~fl~ (1) =

where ~k is the number of electrons per unit volume per unit energy in the interval (k+i Ek). The matrix elements Ckl are constants which give the rate of transfer of electrons from energy e~to energy 6k due to elastic and inelastic scattering. The matrix C has a tridiagonal component which arises from elastic scattering and acceleration of electrons by the electric field. Explicit expressions for these matrix elements are containedin appendix A. Elements above the principal diagonal in C describe inelastic scattering where an electron drops from energy c 1 to ~k’ > k, by exciting an atomic or molecular state of energy Ci Ck. Elements below the principal diagonal describe superelastic scattering, e.g. Cik, 1
—

Cki

=

~II~ N5u51(c1) v(e,),

(2)

S~J

where N5 is the number density of species s, cr51(e,) is the cross section for species s at electron energy e~to undergo an excitationf with energy change Ek Cj, and v(e1) is the velocity of electrons with energy ,. The solution of eq. (1) proceeds by making the finite difference approximation flk(t +h) flk(t) hCklnI(t +h), (3) —

2. Mathematical techniques Following the work of Holstein [6], the velocity distribution function is assumed to be nearly iso-

—

=

~I

S.D. Rockwood, A.E. Greene / The Boltzmann transport equation

where h is a time step. Notice that this algorithm is implicit by evaluation of the right-hand side at time t+h. Solution of eq. (3) for n1(t n,(t +h)

~

=

+ h)

379

sider two coupled equations: i~

=—an~+bn2

(7a)

yields n2 =an~—bn2

(I— hC)j~nk(t),

(7b)

(4)

k

where I is the identity matrix, The numerical solution thus proceeds by calculating the matrix (I hC) given the input cross section data, then finding the inverse (I hC)_i and taking initial conditions for the distribution function ff(t), determined ff(r + h) by eq. (4). If the time step h is made small compared to the eigenvalues of C then one obtains the correct temporal evolution of the distribution function from its initial condition to its steady state. However, in most cases, only the steady state solution is interesting, thus h of is taken be very largewithout compared to the eigenvalues C. Thistocan be done sacrificing stability because as mentioned above we have chosen an implicit algorithm, If one considers n(t + mh) as the mth iteration, then eq. (4) may be written as —

which are representative of eq. (1) when only elastic scattering is included. By inspection one observes that the steady state solution is n2 = (a/b)n1 The matrix I hC required by this algorithm is: —

—

(1 + ha —hb \ —ha 1 + hb)

I—hC=

from which

Ii + hb (I

—

hCY

D

~

=

(I- hC)~n~’,

k

so that numerically one used n°kto find to find

(4a)

4, then 4

(8)

DJ

where D = 1 + h(a + b) is the determinant of (I Applying the algorithm of eq. (4a) yields m

ni nr

hb 1D+ ha

haD

1

+(~)~,

m+i—(1+hb)

D n~ (ha)rn(l+ha) D D

m+ 1

hC).

(9a)

m

_______

—

—

n 2

4, etc., iterating until

(9b)

.

Takirfg the ratio of eq. (9b) to (9a) gives in ~k ~

m—1

rn—i

II~k

<~

(5)

for all k where ~ is some small number. In practice, since one only desires that the distribution achieves a self-similar form, it is better to perform the convergence test on the normalized distribution =

~kI

~

~,

(6)

and thus eq. (5) is used in NOMAD with ~k replaced by ~k• Use of the normalized distribution for convergence is particularly important under conditions where the total number of electrons is not constant (i.e. there is a net At gain this or point lossitdue is instructive to ionization to study or attachment). the convergence properties of eq. (4) by a simple example. Con-

~

_________________

ha ÷ (1 + ha)(nT/nr) (10) l+hb+hb(nT/nr) Now in the limit that ha and hb are large compared to 1, eq. (10) has the correct limit ~2

nT~/n~” a/b independent of the value for nT/nr. Notice, therefore, that for elastic scattering the algorithm is unconditionally stable and should converge to the correct steady state solution on the first iteration. In practice, it takes 4 or 5 iterations as will be discussed below. To test the applicability of the convergence test given by eq. (5) insert eq. (9a) into eq. I

m+1

nj

in —~i j

nm i

I=

(5)

to obtain

/1 +hb\ /hb\InT D +

(,

)

—

1 (11)

S.D. Rockwood, A.E. Greene / The Boltzmann transport equation

380

again in the limit of ha, hb >> 1 this yields

The average energy and characteristic energy are obtained by definition as

m

j3(m)= (———~(1+~-~—~_1 . (12) \a+bJ\ nr/ Now from eq. (10) after one iteration one will haven~/n~ a/b,thuseq.(l2)yieldsjl(l)Osothat eq. (5) should be satisfied for arbitrarily small ~. It should be noted in passing that J3(m) also approached 0 ash 0. Thus eq. (5) is appropriately applied only to the case of overrelaxation when ah >> 1. —~

3. Numerical calculation of transport data Appendix A describes the derivation of the coefficients ak, bk which give, respectively, the rate at which electrons move from energy cell k to k ÷1 and k to k 1 due to interaction with the applied electric field. It is thus possible to calculate the rate at which electrons gain energy from the applied field by: P (a,~ bk)nk~sC. (13) —

~

—

~

()=

(19)

Cj~fl~s/flo

k

and = eD/p

(20)

where e is the electric charge,D is given by eq. (16) and p by eq. (15). The output of NOMAD uses practical units with (e> and Ek in eV, Vd in cm/s, p in cm2 /Vs and inelastic rates in cm3/s. Energy balance is not assumed in the derivation of the numerics and is thus used as an accuracy test. The rate of energy input per electron and per neutral should equal the rate of inelastic and elastic losses. For a properly zoned problem energy balance in NOMAD is generally good to 1 part in iO~, 4. Test cases To test the accuracy of the solutions produced by NOMAD two test problems were used. In the first

/C

The drift velocity is then obtained from

case only elastic scattering was included with a cross ~14~

—

VD P/Eno, where E is the applied electric field and n —

‘

~‘

0 = ~k~k is the total electron density. From the computed drift velocity the mobility is obtained by definition I~ Di1E (15) = V The diffusion coefficient is calculated as in ref. [8] ‘

section which was independent of energy. The distribution function for this case has been derived analytically by Dryvesteyn [7]. The second test problem entails a comparison of calculated and experimental transport coefficients for pure N2. This was chosen to test by inelastic conditions collisions. where Detailed the shape numerical of f(c) output is dominated for an N 2 run at one E/N is provided in the test run out-

D=

(~) ~jI~

1 12\h/2 3m

—

~

C/çf/CL~iC/

k

q5a5(c/C),

(16)

put. To obtain an accurate numerical representation of . the true distribution function requires: 1) that ~e 0 and 2) that f(max) 0, where f is the normalized dis6max = KL~eis the tribution asenergy defined by eq. (17) and maximum obtainable with K zones along the energy axis. Condition (2) is necessary so that inte.

-~

where q5 is the concentration of the species Ns, UNS is the cross sectionmass for and elastic scattering of species s, m is the electron fk is the normalized distri. bution function defined by

f~= nk/~~’2 &no)

.

(17)

grals of products quantities the distribution functionofinvarious the limits e = 0 towith °° can be adequately replaced by sums over f(e/C) which trun-

(18)

cate at ~max’ 4.1. Elastic scattering

All inelastic excitation rates are given by R~=

~ k

,

cr~(c/C)v(c/C)n/CLxe

-+

no

where a~is the cross section for excitation of level i in species s.

The analytic results for various transport parameters for the pure elastic scattering case are presented

S.D. Rockwood, A.E. Greene / The Boltzmann transport equation 3C

‘I I

I

I

y —10

20

—

—e

LOG f(

MAX)

—6

_:

E/N C=1o

381

—2

—4

X = 00545 III

I

Ui -J

C.)

X~O.0363 10

—

E/N

X= 0. 0272

0 ~ —6

io’~

Ui

U

I

IIII~Il I

I

I

hull I

I

iü~

0 C

12h —8

-

Fig. 1. The number of computational cycles required to converge under the criteria ~ = i0~ displayed as a function of the iteration step size Ci2h. —ic I

-10

in appendix B. Test calculations were performed for various values of the relaxation parameter h, using 200 zones along the energy axis so that C is a 200 X 200 matrix, The results displayed in fig. 1 give the number of cycles for convergence as a function of the dimensionless variable C~2h,where C52 is the first row, second column element of C. The choice of this particular element is arbitrary, but sufficient, because in this test case all the elements of C are proportional to one another, For values of C~2h>2 X iO~convergence as defined by eq. (5) with ~ = l0~is accomplished in 4 to 5 cycles. Further tests varied ~ from lO~to lO~and indicated only changes of ±1cycle in the convergence. From the brief analysis performed with the 2-component example above, eq. (7), one expects the convergence to be rapid. The results of fig. 1 fulfil this expectation. The 4 or 5 cycles are in the inversion of the I C7~matrix. To display the probably required to damp out small errors introduced results of the test calculation in comparison with the analytic results of appendix B, let x = .~e/(e)and

-8

I

I

-6

-4

—2

Fig. 2. The percentage error in the drift velocity as a function of the truncation value in the distribution function, parametric in the computational energy zoning.

values ofx. It is observed that fory <—5 there is no measurable variation in any of these parameters for constant x. This is quite reasonable since all of the transport coefficients measure of the distribution function andproperties neglectingofI the partbulk in

y -10

I

—8

2~

LOG ~(~uAX)

—6 • x00545

—4

—2

“fl—....

X = 0.0363 ___________________

X ii

—

2 0 Ui

—4

y = log[f(cmax)]. These two dimensionless parameters then measure, respectively, the fineness of the zoning, ~e, and the error introduced by truncation of the distribution function at Cmax. Figs. 2—4 display the percentage error in drift velocity, VD, average energy (c) and diffusion coefficient D resulting from variation ofy for constant

-

-10

I

I

-8

I

I

-6

,

I

-4

-2

Fig. 3. The percentage error in average energy as a function of the truncation value in the distribution function, parametric in the computational energy zoning.

S.D. Rockwood, A.E. Greene / The Boltzmann transport equation

382 y

LOG f(EMAXI -6 -4

-8

-10 21 I

I

I

I

I

I

X

—2 .1

.0

0.05

I

=

t~E/
I

0,15

I

X=O.0545 -

•

X

=

X

=

0.0362

p

O.O

27i~~,

____________________________

C

it 0

~

>

-2

-1

0

-

0 U

Ui

-4

—2-

I

-10

I

I

-8

-6

I

I

-4

I

I

Fig. 4. The percentage error in the diffusion coefficient as a function of the truncation value of the distribution function, parametric in the computational energy zoning.

iO~of the electrons is insignificant for their determination. It is noted that V0 from the numerical calculations is always less than the theoretical value. This is to be expected since V0 is determined by calculating energy input as explained in section 3 and truncation of f(e) at Emax will always “throw away” a small amount of the power input. Asy is increased all transport parameters begin to fall below their theoretical values. For (e> and D this may produce solutions of accidentally high accuracy since in the limit of y _oo these two parameters err on the high side. Having thus determined that y <—5 yields no further improvement in VD, (e) and D for constantx, variations of x were made withy <—5. Figs. 5—7 display the results of these calculations. Notice that V0 is most accurately obtained with the error being less than 1 part in iO~if x <0.04. To achieve 0.5%

I

.,

-2

0

examined. Because of storage limitations it was not possible to go to smaller values of x and still keep y <—5. The behavior of the error in (e)asx -+0 is interesting since it must either break sharply from linearity to reach 0 asx 0 or else it proceeds to a finite value of ~—0.5%.In either case the question is mostly academic since the solutions for all variables -~

0.10

I

0.15

Fig. 5. The percentage error in the drift velocity as a function of the computational energy zoning,

are much more accurate than the experimentally known cross sections which must be used as input data. It can thus be stated that for the case of elastic scattering NOMAD (e> and D which are will onlyproduce limited solutions by zoningfor if V0, f(e) < l0~and to have all of these parameters accurate to 1% requiresx <0.035. 4.2. Inelastic scattering To test NOMAD under conditions that are dominated by inelastic scattering we have made runs I

—~

accuracy and D both only increase requiredx linearly <0.13. with xThe over errors the range in (e>

I

0.05

I

6

~

-

0

2

-

/ 00

/

/ /

I 0.05

x

I 0.10

I 0,15

=

Fig. 6. The percentage error in the average energy as a function of the computational energy zoning.

S.D. Rockwood, A.E. Greene / The Boltzmann transport equation I

3

-

~ 2

-

I

0

/

/

I

•

/

2

-

.7

0.10 X

(106 V0 cm/s)

(eV) D/~

(eV)

5.0 10.0 15.0

1.09 1.78 2.43

0.786 0.984 1.084

0.846 0.959 1.006

20.0 30.0 40.0

3.06 4.21 5.26.

1.146 1.237 1.312

1.033 1.079 1.134

(10’-i7 V cm

~/‘

0.05

2)

E/N

/ I

Tablel Calculated transport coefficients for N

/ -

383

0,15

I~E/

Fig. 7. The percentage error in the diffusion coefficient as a function of the computational energy zoning.

Here we have assumed that the “correct” values are those calculated with the smallest x listed. Unlike the situation in the pure elastic scattering case, there seem to be no obvious systematic trends, This result is not too surprising since we are now dealing with cross sections that are complicated functions of energy. In this case a fortuitous choice of ~c, such as

to compare with experimental results for pure N 2 gas. The input data used for the pure N2 runs are from Engelhart et al. [8] and Rapp and Englander-Golden [9] as cited by Kieffer [10]. The runs made for N2 encompass a range2.ofWe E/N fromthat 5,0 forE/Nvalues X l0~~to found 4.0 X l0~6 cm cm2 electron energy losses due below 5.0 X V iO—’~V to rotational excitation of the N 2 molecules begin to play an important role in f(e). In the absence of accurate cross sections for this process we have elected to restrict our attention to higher F/N values where f(e) is primarily determined by vibrational excitation processes. Values calculated by NOMAD for V0,D/p, and (e> are presented in table 1. The calculated VD and Dip values are compared-to experimental values as citedbyHuxleyandCrompton [11] infig.8.Inthe 2 the range ofE/N differences arefrom approximately 5 X 10i7 to 5 and 1 X 3% l0~6 forVD/p cmand 1~D~ respectively. These errors may be due to the neglect of rotational excitation even at these E/N values. Above E/N 1.2 X lO_16 Vcm2 only V 0 data are available for comparison and in this range our calculated values are well within the errors cited for the scattering, experimental data. Analogous to the study of elastic table 2 shows the errors incurred in calculating (e>, V 0 and D/p when the energy zoning parameter, x = ~e/(e>, is allowed to become too large.

0.3 eV which is quite close to the threshold energy of the first N2 vibrational state, can produce results that are fortuitously more accurate than one would initially have anticipated. We can say from the results shown in table under mostabout circumstances theto user should try 2tothat keepx below 0.1 in order get all transport data correct to <1%. The user may, however, find it necessary to violate this restriction if he is seeking specific information. For example, in a laser mixture which contains a polyatomic molecule and a rare gas, if the user wishes to know the impor-

I

6

-

~-

-

2

-

I

I

I

.-

1.4 1,2

--

1.0 0.8

-

0.6

-

0.4

-

...-°

~

-

_=~

D/~IL~’

— ‘~ ‘~

I

..-~

‘ VD

o-——oCalculation

ExperIment 00

I I

I 17 V’cm2)

0.2 I

10 0

E/N Fig. 8. Comparison of the (xlO’ calculated and experimental drift velocity (left scale) and characteristic energy DIM (right scale) as a function of E/N,

S.D. Rockwood, A.E. Greene / The Boltzmann transport equation

384

Briefly the function of each of the routines within NOMAD is as follows:

Table 2 Zoning errors in inelastic calculations 2 Error (%) E/N = 5.0 X 10i7 V cm _____________________ ______________

X

NOMAD

Reads input data, prints input data, sets

SETUP

initial conditions, Normal program exit is STOP 77 from NOMAD. Computes energy mesh, excitation rates,

V 0

DIM

tel

0.012

0.0

0.0

0.060 0.090 0.120 0.180

0.0 0.0 0.0 0.0

+0.1 0.0 +0.6 +0.2

+0:3 +0.5 +1.0 +1.4

+8.2

~

______________________

ATURPL BTURPL CALCOE

2 E/N = 1.0 X 10—16 V cm 0.01 0.0 0.03 0.0 0.05 0.0 0.075 0.0 0.10 —1.0 0.15 0.0 0.20 —3.9 0.30 +0.6

0,0 0.0 +0.3 0.0 +1.5 +0.2 +5.2 +0.4

0.0 0.0 +0.3 +0.3 +1.0

+i.o

+4.6 +3.1

tance of the rare gas in the formation of secondary electrons he will have to extend the energy range of the calculation far enough to adequately sample the ionization cross sections of the rare gas. It may not be possible to do this, keep x below 0.1, and stay within the core restrictions of the computer.

5. Program set-up and execution Fig. 9 shows a flow chart of the NOMAD program. The specific ordering of the input data and the formats are described in tables 3 and 4. Cross section data outside the data set given are automatically set to zero. Table 5 provides a description of the important symbols in the program. The user should be aware that the version of NOMAD presented here automatically includes elastic scattering of electrons from ions a cross section 2 eV2/2. Thewith ion density is assumed to be equal to the electron density which is ~ei = 2 X l0~~ cm an input to the code. Electron—ion elastic scattering becomes important only for large fractional ionizations and low electron energies.

and arraysA(k), B(k). Performs linear interpolation for 2-dimensional data arrays. Performs linear interpolation for 3-dimenCalculates sional data elements arrays. of the coefficient matrix D. If superelastic option is chosen, D is treated as full 2d matrix. If superelastic option is not used, then D is assumed to be an upper Hessenberg

matrix. Directs the solution of the linear equations when D is a full matrix. Note that the convergence criterion in this case is ~ ~ l0~ because of increased algebraic operations needed with a full matrix. SOLV2 Directs the solution of the linear equations when D is an upper Hessenberg matrix. Note that the convergence criterion here is set at ~ ~ 10’~. TSPORT Computes transport data V 0, Ek, (C>, momentum transfer, inelastic collision, ionization and attachment frequencies, excitation rates and energy balance. MATTRI Triangulates the upper Hessenberg matrix computed by CALCOE and CALM2. SOLN’VE Uses back substitution to obtain solution vector V(I) for input vector Y(1) and coefficient matrix triangulated in MATTRI. MATTRI and SOLNVE are a linear equation solver for equations of an upper Hessenberg form. MATSLV Solves the general linear set of equations used when D computed in CALCOE does not have upper Hessenberg form. Part I triangulates the input part 2 solves resulting equations bymatrix, back substitution. Comment cards explain most of this routine. CALM 1 Takes coefficient matrix computed by CALCOE and converts to working matrix SOLV1

S.D. Rockwood, A.E. Greene / The Boltzmann transport equation

NOMAD

____

[ATURPL

JSOLV2

1~

SIJPERELASTIC

L..~jCALM? JMATTRI

I I

_________

Yes TIPORT ~4

No

~

I

_(,,~ONlERGE~,~—_l

385

IBTt~1.

I

Yes

ICALCOE

14”14_ø{CALM1

1

________

_________

I

MATSLV

t~o

‘—.—(.CON~,~RGEl

Yes P-.~TSPORT ~

Fig. 9. Schematic flow diagram for the computer code.

Table 3 Master control cards, the first three control cards dictate the modes of operation required of NOMAD. The order of these cards is the same for all calculations

Card

Format

Field

Parameter

Range

1

14

1 2 3 4

no. of energy mesh points no. of distribution function to be calculated no.ofgasspecies super-elastic option

~250 ~20

1 2 3 4

energy zoning (eV) concentration of species 1 concentration of species 2 concentration of species 3 initial temperature (eV) of electron distribution

arbitrary

1 2 3 4 5 6

background gas temperature (K) neutral number density (#1/cm3)

arbitrary arbitrary

electron number density (#1/cm3)

arbitrary

mass of species 1 (atomic units) mass of species 2 mass of species 3

arbitrary but ~ 0. If the concentration of species I = 0 set this number to 1

7F10.4

1 2 M

E/N

for first calculation in V cm2 E/N for 2nd calculation E/N for Mth calculation where M is the value of card 1,field2

arbitrary

14

1

number of vibrational levels in species 1

0

2

F10.4

5 3

4,

(5) (6)

A.

F 10.4

~,

0, 1

0 ~p ~ 1 0 ~p ~ 1 0 ~p ~ 1 arbitrary

~p ~

8

S.D. Rockwood, A.E. Greene / The Boltzrnann transport equation

386

Table 3 (ôontinued) Card

Format

Field

Parameter

Range

If the value of this card is zero go to E B.

F10.4

1

excitation energy of the first vibrational level (eV)

arbitrary

C.

14

1

number of data points to be read in for the cross section of this level

~36

D.

F 10.4

1 energy at which cross section is defined (eV) 2) 2 cross section at above energy (A return to B and repeat for all vibrational levels of this species as given in A.

arbitrary arbitrary

E.

14

number of electronic levels in species 1

0

1

~ p ~6

if the value of this card is zero go to I F.

F 10.4

1

excitation energy of the first electronic level (eV)

arbitrary

G.

14

1

number of data points to be read in for the cross section of this level

~20

H.

F 10.4

1 2

energy at which cross section is defined (eV) cross section at above energy (A2)

arbitrary arbitrary

return to F and repeat for all electronic levels of this species as given in E. 1.

14

1

does the species attach? 1 = yes, 0 = no

0, 1

if no, i.e. p = 0, go to L. J.

14

1

number of data points for the attachment cross section

~20

K.

F10.4

1 2

energy at which cross section is defined (eV) value of attachment cross section at above energy (A2)

arbitrary arbitrary

L.

F 10.4

1

ionization potential of species 1 (eV)

arbitrary

M.

14

1

number of data points for ionization cross section

~20

N.

F10.4

1

energy at which ionization cross section is defined (eV) cross section for ionization at above energy (A2)

arbitrary

number of data points for momentum transfer cross section energy at which cross section is defined value of cross section at above energy (A2)

~60

2 0.

14

1

P.

F 10.4

1 2

arbitrary

go back to A and repeat for species 2 up to number defined in card 1, field 3. If superelastic collisions are included, i.e. card 1, field 5 = 1, go to table 4. Otherwise this concludes the in~iitdata. *

From here on the card number depends on the number of data points and will not be explicitly counted.

S.D. Rockwood, A.E. Greene / The Boltzmann transport equation

387

Table 4 Superelastic collisions, NOMAD permits superelastic collisions with species in the first 8 inelastic levels. In table 3 these were formally called vibrational excitations; however, the designation is arbitrary and one may read in data for any inelastic processes, i.e. rotational excitation, electron excitation, if superelastic interactions with these processes are desired. When employing the superelastic option, the following cards are required after the cross section data read in table 3 Card

Format

1

8F10.4

Field

Parameter

Range *

1

ground state number density of species 1

2

first excited state number density of species 1

rn

last excited state number density of species 1

repeat for all species *

The total ofall these number densities should equal the number read in card 3, field 2.

Table 5 List of symbols used in NOMAD Symbol

Purpose

Units

Symbol

Purpose

A(i)

rate of excitation of electrons from energy bin ito i + 1 due to the electric field

S~

COEF(i)

temporary storage for elements of the I — Ch matrix

CSA

binary rate of attachment in final distribution

cm

temporary for an attachment location cross section

cm

AAR

CSE

average electron energy in the

eV

. temporary location for an electronic excitation cross

cm 2

AEE AIR

binary rate of ionization in final distribution

AK

an intermediate term in calculating the A(i) and B(i) rates

AMAS(i)

mass of heavy species i

amu

AMUN

electron mobility

cm2/V

ANA

total number density of all heavy species

cm’3

ANAl

inverse of ANA

cm3

ATE

energy used to thermalize electrons formed by ionization. It is assumed that all secondary electrons are born with zero energy

eV cm3/s

AVR

temporary storage for excitation rates

cm3/s

AVRS

temporary storage for superelastic deexcitation rates

cm3/s

B(i)

CONC(i)

3/S

Units

.

final distribution

2

section

cm3/s

CSI

temporary location for an ionization cross section

cm2

CSV

temporary location for a vibrational excitation cross section

cm2

D(i)

contains the elements of the matrix I — Ch, computed in CALCOE

DE

width ofeach bin or zone on the energy axis

eV

DEIN

1/DE

eV~

DlF

intermediate calculation step in determining B(i) values time step element [h in eq. (3)1

DT .

DTG(i)

array of time steps to provide a different time step for each E/N if necessary

E(i)

energy of midpoint of energy bin i

eV

rate of de-excitation of dcctrons from energy bin ito — 1 due to the electric field

EAB

temporary storage of energy of each bin used in cross section interpolation

eV

fractional concentration of species i

EBN

E/N value, units changed from

V(stat) cm2

V cm2 to V(stat) cm2

S.D. Rockwood, A.E. Greene / The Boltzrnann transport equation

388 Table 5 (continued) Symbol

Purpose

Units

EBNOLD

when calculating rates for several F/N values EBNOLD holds previous E/N values to update terms that scale linearly with

V(stat) cm

2

E/N EBYN(i)

array of E/N values input by user

V cm2

ECAL

temporary storage of actual energy loss by an electron due to an inelastic collision

eV

cross section for electron impact excitation of species i to electronic state! due Roan electron with energy EEX(i,

cm2

ECS-

(i, J~k)

Symbol

Purpose

Units

EMC(i,j)

momentum transfer cross seetion for an electron with energy EEM(i,f) to transfer momentum to species i

cm2

ENOT

time test used to determine if code is consuming too much CP time (an indication some-

s

EOC(i,/)

difference between the actual and integerized energy losses (see ECAL and ESTAR) divided by the actual energy loss

EE(i,j)

energy of electronic state! of species i

eV

energies corresponding to dissociative attachment cross sections (see EOC)

eV

energies corresponding to ionization cross sections (see EIC)

eV

energies corresponding to momentum transfer cross sections (see EMC)

eV

EEV(i,/, k)

energies corresponding to vibrational excitation cross sections (see VCS)

eV

EEX -

energies corresponding to electronic state excitation cross sections (see ECS)

eV

ionization threshold of species i cross section for electron at energy EEI(i,!) to ionize species i

eV

rate of excitation of electrons trom energy bin ito i + 1 due to elastic scattering

s—i

rate of deexcitation of electrons from energy bin ito i — 1 due to elastic scattering

~

EEC(iJ)

EEI(i, /)

EEM(i,!)

(i,/,k) EI(i) EIC(i,j)

ESTAR

temporary storage of integerized energy loss by an electron due to an inelastic collision (see also ECAL and EDIF)

eV

ET

initial value for the energy axis, code cannot take ET equal to zero, usually set to the energy bin width

eV

EV(i,j)

energy threshold for vibrational excitation of species i to vibrational state!

eV

FREL

intermediate step in elastic scattering calculation

FREQU FREQM

inelastic collision frequency momentum transfer collision

cm3/s cm3/s

GASTE

frequency gas temperature

K

ELA(e)

ELB(i)

HW IG

identical to DE DO loop index usually running over the number of heavy species

cm2

cm2

attach to species i

j, k) EDIF

thing else is wrong) cross section for an electron with energy EEC(i,/) to

MG

total number of heavy species (<3)

MDSE

indicator that turns on superelastic collisions, 0 = no, 1 = yes

N NAT(i)

number of bins on the energy axis indicator of whether or not

NB6

species i attaches, 0 = no, 1 = yes N/6 — used for print out

NE(i,j)

number of energy bins an electron will lose if it excites species ito electronic state!

eV

S.D. Rockwood, A.E. Greene

/ The Boltzrnann transport equation

389

Table 5 (continued) Symbol

Purpose

QMOD(A, B)

aused defined function which is in determining the numher of bins that will be lost by an electron in inelastic collisions

QT

temporary storage for the result of the QMOD function

R(i)

temporary storage for the sum of all coefficients that together make up the total loss of electrons from bin i

RE(i,j, k)

Units

Symbol

Purpose

Units

SV

rate of energy loss the electron gas due to from vibrational excitation

eV cm

TEO

starting estimate for temperature of the electron gas

eV

TEST

the convergence criterion for the population in each bin

3/s

from iteration to iteration TIME1

computer starting time returned by SECOND

NEL(i)

number of electronic states considered for species i

NI(i)

number of energy bins an dcc-

rate for electrons in energy bin k to excite species ito electronic statej

~

REN

sum of all electrons or total electron number density

cm3

RES(i)

square root of energy bin i

eV~2

NPRINT

number of iterations between prints

RES3

temporary storage for the cube of the RES values

eV312

NQ(i,!)

RI(i,!)

rate for electrons in energy bin Ito ionize species i

s—i

number of cross sections to be read in for species i, energy loss process!

NSPEC

number of heavy species

RV(i,j,k)

rate for electrons in energy bin

s

tron will lose if it ionizes species i

5~

included (~3)

k to excite species ito vibrational level! S2

intermediate term used to calculate the elastic scattering rates

SA

electron gas due to from attachrate of energy loss the ment

SCALE

intermediate term used to calculate the elastic scattering rate

SE

rate of energy loss from the electron gas due to electronic state excitation

eV cm3/s

SEXS

rate of energy returned to the electron gas by superelastic collisions

eV cm3/s

SI

rate of energy loss from the electron gas due to ionization

eV cm3/s

SL

energy input by the field (rate)

eV cm3/s

SQ

energy given up by electrons in inelastic collisions (rate)

eV cm3/s

SU

superelastic correction to the vibrational energy loss rate

eV cm3/s

SUM

intermediate calculation step in determining A(i) values

3/s

NUM

temporary storage for bin numbers, used when calculatmg coefficients in the 0 matrix

NV(i,!)

electron of number will energy lose ifbins it excites an species i vibrational levelj

NVL(i)

number of vibrational levels included for species i

TIME2

temporary storage for cornputer time during the calculation returned by SECOND

s

TOTEN

total number density of electrons

cm3

TRA(i)

total rate of loss of electrons from bin i due to attachment

s~

TRE(i)

total rate of loss of electrons

s~

eV cm

from bin i due to electronic state excitations TRI(i)

total rate of loss of electrons

s’’1

TRV(i)

from bin i due to ionizations total rate of loss of electrons

~

from bin i due to vibrational excitations

S.D. Rockwood, A.E. Greene / The Boltzmann transport equation

390

mined by

Table 5 (continued) Symbol

Purpose

Units

VCS(i,j,k)

cross sections impact excitation for electron of species i

cm

~

—

—

~L~i+ ~ IV~[Rs,(C+ ;1)~(~ + C;1)

2 + RJ(C

—

C:!) n(C

—

e 1)N~/N3° + R’5(C + C:i)n(C + :i)

to level! due to an electron with energy EEV(i,!, k) VEL

+ ~(C)

temporary storage for velocity of electrons in a given energy bin

cm/s

drift velocity of the converged distribution

cm/s

X(i)

in number energydensity bin z of electrons

cm’

XX(i)

“normalized” number density of electrons inenergy bin

cm ~ eV’312

number density of species i in vibrational state /, used only if the superelastic option is used

cm3

W

R’5(C)n(C) dC

Cs1

[R51(e)+R~J(C)+R~(C)]n(C)].

-

-

.

—

—

V

—

(2C’~2

~

q

a

(A.2)

5(C)

N\m/

B = I dt*D (dt is implicit time step), then calls first part of MATSLV to tnangulate B. Takes coefficient matrix computed by CALCOE and computes B = I dt ~D (with implicit assumption that D and B have upper Hessenberg form), then calls MATTRI. —

For a test case we have used the runs that produced the N2 results. Part the test run output for E/N 2 isofprovided in table 6. This run = 1.0 Xa~250 6point V cmenergy grid with a zoning of 0.01 eV. used The gas temperature was set at 300 K and the N 2 and electron number densities were set, at Note 3, arbitrarily respectively. 2.67 this X lO~and 2.67only X 10~ cm’ to 2.5 eV so that that energy grid extends the electronic states and ionization threshold are not reached.

Appendix A The time evolution of the number density n(, t) dC of electrons with energy between e and e + dC in a mixture of gases of total number density N is deter-

s

where ci.~is the mole fraction of species s, 05(C) the cross section for momentum transfer from electrons at energy C to molecules N5, and e and m are the electron charge and mass, The second term on the right-hand side of eq. (A.1) is the flux of electrons along the energy axis driven by elastic collisions. ~ J’A~ \ an

—

CALM2

(A.l)

. the flux electrons energy space driven by theis The first of term on the in right-hand side of eq. (A.1) applied field E, 2Ne2 (Eli’!)2 C (n an 3m(z.’/iV) ~2C ac

3 eV~ —

Y(i,j)

f

~ei =

—

C)

2 3

2mN(2C/m)~

—

kTC

~

C q~u~(C)/M 5,

(A.3)

-

where M5 Both is thefluxes mass Jf of and species and T isa current the gas ternperature. ~el scontain term proportional to n(C) and a diffusion term proportional to the gradient dn/dC. The final terms onand the superelastic right-hand side of eq. which (A.l) describe the inelastic collisions give rise to nonlocal interactions in energy space. The quantity R 51(c) = a51(e)u(c) is the rate at which dcctrons with velocity v() produce excitation from the ground state of species s to excited state / losing energy CJ while R~(e)= [(e + C1)IC] a~(c+ C;1) v(C) is the rate at which electrons at C suffer superelastic collisions with molecules in state N~and gain energy e.~.R~(e)is the ionization rate for species s and the term multiplied by 6(C) indicates that all secondary electrons are produced at zero energy.

S.D. Rockwood, A.E. Greene / The Boltzmann transport equation

Eq. (A.l) is converted to a set of K-coupled ordinary differential equations by finite differencing the electron energy axis into K cells of width ~C, ~1C

—

L~5C

—

Ns[Rs/k+ms!flk+ms!

+~

+R;jkm.flkms.N~/Ns +

5jflkmsjN~/No

-

2 F

~3m (N) E

\2

~rflk+,

+

Jei(k) = vic[

kTC~

4

.

m

(flk+i

(A.4)

~—

nk+i —

+)

(kT

+ ~k

2

+flk

44

z4/N

(~) 2

+

(Ck +

Ck + 4) 2~C 2 +_!~_-~-~i 2kT 2Ne21E~2 1 ‘1

bk

+1

—

~C

)j~

2

4/NU~CI (C~ -~) \

1~ kT

2/cT

k~~Ck).

is interpreted as the rate at which electrons at energy Ck are promoted to energy Ck+1, while bk is the rate for demotion from Ck to Ck_1. Notice that all the rates ak, bk, RSJk are constants which only need to be evaluated once per calculation. Eq. (A.6) may be expressed in matrix notation as ak

(A.5a)

Ck

— nk\l 1~C

+

~t~)

—

t’k

—

i4/N

2~C \

with

1

—

(RSJk +R~/k -+R~k)nk]~

Jj~(k)=

1

2

+ R~km

~DR~mflm

61k 5,1 +R~s.k+m.flk+m.+

—

2Ne

ak =

+R~k+msink+msi

R~n~ (R~k+R~1k+R~k)nk]~(A.6)

~

with

[J~(k)_J~(k)] [J~(k)_J~j(k)] =

391

(A.Sb)

nk =

~I~Cklnl

[see eq.(1) in main text].

and +

‘=~

N =

Appendix B

+ \i/2

~-1 m i

~I~qsasie,j,

+‘

Analytic solutions for the case of elastic scattering

.~

2mN(~~-~

~

q 5a5(4)

~ where = k&, Ck = J’~’(k)=J’1(k—1),

M5

with a

ao F (5/4) ____ e 4(E/IV) R~ F(3/4) (3m*)~~ where m~ = 2m/M, and e is the electron charge. =

___

(C>—v’~ —

4

m, -

5,C1/L~C.

The first two terms. on the right-hand side of eq. (A.4) conserve the total number of electrons n 0 = physical exactly even in finite-difference form if the boundary conditionsJ~—J~= 0 are employed. The entire equation is then required to conserve particles exactly by setting to zero rates for which k + m 51 > K and K m51 < 1. Substitute eqs. (A.Sa) and (A.5b) in (A.4) and of perform the indicated coefficients nk, nk÷ito obtain: algebra, grouping

4(1r/m)~2 3F(3/4) e (6m*)h/

ii

2

2(~L~.)L~

2 D = (m) 2 1/2/

~

2

‘~i/4

1 3F(3/4)

~ 1/2 (E/N~i/ a~ i a~N

Taking M = 1 amu yields the numerical values (C) =

18.36 (E/N —JeV, 2cm/s,

U~ /

—

VD

= 5.749 X 10 6 (E/N’~~’ U

=

ak_ink_i +bk+lnk.fi

+

2I~Ns[R~k+m.flk+m.

—

(ak 4-bk)nk

0 / 2/s, E/N 1/2 1 cm D=8,024X I07(_-__) a~ a 2 N is0N in cm’~’3and a 2, where F/N is in eV cm 0 is in cm ,

5/

392

S.D. Rockwood, A.E. Greene

References [1] S.D. Rockwood, J.E. Brau, G.H. Canavan and W.A. Proctor, IEEE JQE 9(1973)120. [2] P.E. Nielsen, G.H. Canavan and S.D. Rockwood, Proc. IEEE Lett. 59 (1971) 709. [3] S.D. Rockwood, Phys. Rev. 8A (1973) 2348. [4] M.A. Morrison and A.E. Greene, J. Geophys. Res. 83 (1978) 1172. [5] A.E. Greene, Los Alamos Laboratory Report LA-6821MS (1977).

/ The Boltzmann transport equation [6] T. Holstein, Phys. Rev. 70 (1946) 367. [7] M.J. Dryvesteyn and F.M. Penning, Rev. Mod. Phys. 12 (1940) 87. [8] A.G. Engelhardt, A.V. Phelps and C.G. Risk, Phys. Rev. 135 (19,64) A1566; 131 (1963) 2115. [9] D. Rapp and P. Englander-Golden, J. Chem. Phys. 43 (1965) 1464. [10] L.T. Kieffer, JILA Information Center Report 13 (1973). [ll]L.G.H. Huxley and R.W. Crompton, The diffusion and drift of electrons in gases (John Wiley, New York, 1974).

S.D. Rockwood, A.E. Greene / The Boltzmann transport equation

393

TEST RUN OUTPUT 6/N r i.00216 CYCLE 6001,868 CYCLE 6U”668 CYCL6 ,429,691,

I 2 3

7 8 9 ii ii 12 15 I’ iS 16 17 18 IS 20 21 22 23 20 25 26 27 26 29 30 31

32 33 311 35 36 37 35 39 00 41

0.i2E111 6.03E01 8.026—01 8.Ui6—U1 8.012—Ui 8.016—UI 8.006—01 8.059—01 8.006—01 6.01)2—Ui 6.006—01 6.002—01 6.006—01 7.996—01 7.996—UI 7.996—01 7.996—01 7.996—01 7.9901—01 7.866—01

C0NCF~.T6AT1O5S i.OOE000 0. 0. I 2 3 60688L1720 71151619L-IIUN 6(5) 160—3/2) ~l? 7.916—01 83 7.672—01 12” 6.596—01 ‘IS 7.416—01 811 7.662—sI 125 2.532—01 40 7.906—01 85 7.656—si i26 6.076—01 OS 7.906—01 89 7.602—01 127 o.0OE—01 06 7.8016—01 87 7~63E—01 128 #.3”6—0l ‘7 7.8016—01 78 7.625—SI 129 6.276—01 0*’ 7.886—UI 79 7.~i9—01 130 6.206—01 1101 7.882—al 90 7.606—01 ill 8.139—01 50 7,596—al SI 7.596—lI 132 6.062—01 51 7.876—Si 92 7,582—01 133 5.985—nI 52 7.872—01 93 7.579—0! 130 5,906—sI 53 7.862—01 90 7.551—al 135 5.836—01 50 7.962—01 95 7.506—01 136 5.752—01 55 7.856—nI 901 7.532—01 137 5.876—01 58 7.856—01 97 7.516—01 138 5.566.01 57 7.806—01 98 7.506—01 135 5.505—01 8 7.805—01 59 7.495—Ui lOS 5.025—0! 59 7.892—01 100 7.066—01 101 5.835—ui 80 7.938—01 101 7,005—01 10? 5.202—01 61 7.926—uI 102 7.021—01 103 5.l66—Ui

7.98t—oI 7.986—UI 7.986’.UI 7.076—UI 7.976—UI 1.87L—UI 7.97t—UI 7.966—01 7.5001”UI 7.966—01 7.856—01 7.956—UI 7.950—01 7.946—UI 7.942—01 7.906—UI 7.936—UI 7.932—UI 7,922—UI 7.926—UI 7.9101—UI -

62 63 0,0

85

so 67 9.8 9.9 70 71 72 73 70

75 701 77 7* 79 50 UI 110

I0~.17AIlU6 U. IrsIZ’1IUS 7’OIUUESCY

7.812—01 7.016—01 7,005—01 7.906—01 7.796—)11 7.786—01 7.786—Ill 7.772—01 7.772—01 7.766—01 7.756—01 7.756—01 7.706—00 7.736—01 7.726—01 7.726—01 7.716—01 7.706—UI 7.0192—01 7.,0F~0l 7.676—Ill

lOl IOU 105 1001 107 107 109 110 III 112 113 II’ 110 1101 Ill 118 II~ IOC 121 122 123

7.006—SI 7.396—lI 7.362—0! 7.336—01 7.30t—UI 7.279—UI 7.206—01 7.216—UI 7.186—01 7.109—01 7.116—01 7.072—07 7.039—01 01.096—01 6.556’~U! 6.0106—01 8.859—01 01.602—01 8.756—01 0.702—01 01.0,46—UI

IOU 105 1001 107 10$ lOS 151 15! IS? 153 154 ISO I5*~ 157 158 59 160 161 101? 163 160

5.076—01 0.982—01 0.856—01 0.905U1 4.712—Ill 4.629—01 0,536—01 0.009—01 4.306—01 0.256—01 4.15601 0.065—UI 3.Sbf—OI 3.856—01 3.751—01 3.602—01 3.546—Ill 3.036—01 3.325—Ill 3.206—01 3.09901

165 166 i67 168 109 1711 i71 172 173 170 179 176 177 178 179 180 181 Ill? 163 170

2.486—0! 2.875—SI ?.76E—Oi 2.652—01 2.506—5! 2.036—01 2.322—01 2.216—01 2.106—01 1.992—01 1.662—01 1,776—01 1.662—01 1.962—01 1.052—01 l.SUF,—OI 1,202—01 1.111201 1,002—01 9,005—02

206 207 208 209 210 21I 212 213 210 215 2l6 217 2I5 219 220 221 222 223 220 225

“.006—03 3.396—03 2.806—03 2.332—03 I .90E—03 1.616—03 1.322—03 1.092—03 8.YiE.OlI 7.272—00 5.912—00 0.792—00 3.672—00 3.126—00 2.516—00 2.026—04 l.63E00 1.322—00 1.062—00 8.585—05

185 186 187 18*) 189 190 191 192 193 90 195 I91, l97 198 195 200 201 202 203 204 205

8.54502 7,856—0? 6.896—02 6.162—0? 5.476—02 4.806—52 0.276~02 3.746—02 3.276—02 2.742—02 2.4,2—02 2.116—02 1.816—02 1.556—02 1.326—02 1.126—02 9.046—03 7.976—03 8.736—03 5.6016—03 4.766—03

276 227 729 2201 2311 231 232 233 234 235 2301 237 238

6.952—05 5.635—05 4.565—05 3.706—05 3.015—05 2.052—05 2.002—05 1.632—05 1.336—oS 1.092—05 7.915—06 7.SIE—OI, 6.006—08 4.041—08 4.069—06 3.352—06 2.776—06 2.306—08 1.522—06 1.626—06 I.386—06

10)002 90580Y LOSS 00120 150—C53/SECI 016041i0’.0 1.795426—10 FL6C—E0CITOTiON 0. IC’I3/SECI

0.

07TUC85~.T FU5II068CY

2s~ 800 201 242 243 244 205 248

-

STTACO,I’rol 0.

ICY’3,’56C) 0.

EL6LT806

S6CON0008

ELECTRONS

0.

8076

IC83/S5CI

0.

580,19

09191 06206110 I.782046,06IC01/SOCI 0/708 5.P’35079—0lIEo) 0056005 969800 9.589375—0I(E0l OSENTU’ 10063290 FOElI, 9.983941—0101L53,SECI ISFLUS01C CULL. FOES. 1.810066—lu

261058 8ALANC6 C8EL# INPUT l.780932—IOIIU—C015/S50) FL#STIC LOSS 2.5S1091—12I6V—C73/SECI VI86ATIONAL 60010*1106 063860163 i(C#3/O5CI IN6LAS1IL RAIlS SUP5#ELA,571C OAIES i 1 3,o53301—IC, 0. I 2 6.032016—il 0. 322—lI U. 1 3 0.5I5I99—i? 4 2.755 0. I 5 8.7620105~13 0. 1 6 9.910976—IS 0. 1 7 7.474UUE—l~ 0. 1 80. 0. ELECT8OSIC 65611*1106 CF SPECIS3 ICCY3/SECI I 1 U. 1 2 0. 1 3 0. I 4 U. I 5 0. I 6 0.

INELOSTIC LOSS

1.755026—10

G*IS—LUSSS 2.555995—?2IEV—C83/SEC)