Analytical expressions for phase shifts and cross sections for low energy electron-atom scattering in noble gases

Nuclear Instruments and Methods in Physics Research A242 (1986) 327-337 North-Holland, Amsterdam

327

ANALYTICAL E X P R E S S I O N S F O R P H A S E S H I F T S AND C R O S S S E C T I O N S F O R L O W E N E R G Y E L E C I ' R O N - A T O M SCA'ITERING IN NOBLE GASES A.D. S T A U F F E R Physics Department, York University, Toronto, Canada M3J 1P3

T.H.V.T. DIAS and C.A.N. C O N D E Departamento de Fisica, Universidade de Coimbra, 3000 Coimbra, Portugal

Received 23 April 1985

Analytical expressions which reproduce recent reliable data on low energy (0 to - 2 0 eV) electron-noble gas atom elastic scattering phase shifts are presented, allowing an accurate calculation of elastic integral and differential cross sections at any chosen energy through the use of the appropriate formulae. Fittings to the integral elastic cross sections thus obtained and to inelastic and total (elastic+ inelastic) cross sections are also presented within this energy range. These expressions are of interest in Boltzmann and Monte Carlo calculations of electron drift phenomena in noble gases at low values of E / p and are applicable to certain types of nuclear radiation detectors.

1. Introduction

Most nuclear radiation detectors using gases as the detection medium are based on phenomena associated with the transport of electrons through the gas, usually under the influence of an electric field [1-3]. The main physical properties involved are the drift velocity for electrons, the diffusion coefficients, the electroluminescence intensity, the charge multiplication factor and the recombination and negative ion formation coefficients. The choice of the gas or the gas mixture used in these instruments is dictated by the performance to be achieved, but often one is led to the conclusion that the most convenient gas is one of the heavier noble gases: argon, krypton or xenon. To model the operation of these devices, a detailed understanding of the transport of electrons in noble gases under an electric field is needed. This modelling is usually carried out via Boltzmann equation analysis or by Monte Carlo simulation methods. A Boltzmann analysis, although often used, is difficult to carry out for nonspaciaUy uniform distributions or nonequilibrium processes. The conceptual simplicity of Monte Carlo methods is hindered by the large amounts of computer time required but even so it has been frequently used not only for modelling but also for testing the accuracy of Boltzmann equation solutions. For either of these techniques a detailed knowledge of the electron atom scattering cross sections is required. Even when these quantities are well known the data are available in the literature only at certain fixed energies. If extensive calculations are to be carried out for modelling a gas nuclear radiation detector using these techniques accurate values for the elastic (both integral and differential) and inelastic cross sections are required at a large number of arbitrary energies. In order to facilitate these calculations we present here simple analytical expressions for the energy dependence of electron-atom scattering phase shifts and elastic, inelastic and total cross sections for the heavier noble gases: argon, krypton and xenon. These expressions permit a rapid evaluation of the quantities required for use in model calculations as was the case in a recent study of electroluminescence and electron drift velocity in noble gases at low electron energy by a Monte Carlo method [4] where they were used. Accurate elastic cross sections are particularly required in order to get reliable results as was borne out in ref. [4]. For the inelastic data our formulae are inferred from the rather meagre data available. 0168-9002/86/$03.50 © Elsevier Science Publishers B.V. (North-HollandPhysics Publishing Division)

A.D. Stauffer et al. / Analytical expressions for phase shifts

328

Analytical expressions have previously been given for the momentum transfer cross sections in ref. [5]. An accurate calculation of momentum transfer cross sections for the noble gases can also be obtained from the phase shifts given in this paper.

2. Basic data and formulae

At present there is a great deal of accurate data both experimental and theoretical for the elastic scattering of electrons from the heavier noble gases (refs. [6-9] and references therein). The situation for inelastic cross sections is much less satisfactory, however. The existing data [10-15] permit only a rough approximation to the partial and total cross sections for these processes. The ionization cross sections are well known [16]. In the case of low energy elastic scattering reliable phase shifts 8t(k ) are available [7-9]. Here l represents the angular momentum and k the linear momentum of the scattered electron (~ = 13.6025k 2 with c in eV and k in amu). From these phase shifts, differential, integral and momentum transfer elastic cross sections can be calculated via the well known nonrelativistic formulae [17] do dO

(O, k)

1 = ~ )-'.e 's'tk, sin t

8t(k)(21+

0) 2,

1)?t(cos

(1) (2)

Oe,(k) = ~ 2 • (21 + 1) sinZSt(k), /

and °m(k) = ~ 2 Y . ( l + 1) sinZ[6t(k) - S t + l ( k ) ] , t

(3)

where Pz(cOs 0) are the usual Legendre polynomials and 0 is the scattering angle. The cross sections are in units of a02 where a 0 is the Bohr radius. For the heaviest gases relativistic effects become important and in these cases phase shifts 8t±(k) are obtained from a relativistic treatment of the scattering process [18,19]. Here the + refer to the spin-up and spin-down cases. The expressions for the cross sections now become [20] (P/(cos 0) are the associated Legendre polynomials) do

1 ] (0, k) - 4 - ~ I y]~ [(l + 1)(eZin'+ - 1) + l(e 2i~7

-

-

1)] P, (cos 0)

2

I

+4-4@~ (e2iSi-e:i'V)P/(cos O) 2' °e,(k) =-~2

(4)

Y~.(I + 1) sin281+ + I sin28t- ,

K-

(5)

I

4~r

( l + 1)(1+ 2) sinZ(St+ _ 8t+a )

° ' ( k ) = k--7E

2/+3

I

+ l(l+ _

_

(t + 1)

1) sin2(8 t- _ 8L1) 4 sin2(87 2l+ 1 ( 2 l + 1)(2/+ 3)

- -

~/~-1)"

(6)

It is obvious that a knowledge of the energy dependence of the phase shifts will allow a calculation of the elastic differential, integral and momentum transfer cross sections via eqs. (1)-(6). However, the energy dependence of these latter quantities will not be simple. Thus we also present simple analytical fits to the integral elastic cross sections to enable a rapid calculation of these quantities when needed.

A.D. Stauffer et at / Analytical expressions for phase shifts

329

Finally, it should be noted that although the summations in eqs. (1)-(6) are infinite in range, in practice only a finite number of terms are retained as described in sects. 3.2 and 3.3.

3. Analytical formulae 3.1. Elastic phase shifts Effective range theory [21,22] provides analytical expressions for the nonrelativistic phase shifts as a function of the momentum k. For 1 >/2 [22] tan 8,(k) = k2fla, + k 4( fl2b, + ac, ) + O(k6),

(7)

qr

a, = ( 2 1 - 1)(21+ 1)(2l + 3 ) '

(8)

b,=

(9)

~r(15(2l+ 1) 4 - 140(21+ 1)2+ 128) (2•- 3 ) { ( 2 l - 1)(21+ 1)(2l+ 3)}3(2l+ 5 ) '

3~r c,= ( 2 l - 3)(21- 1)(2l+ 1 ) ( 2 l + 3 ) ( 2 l + 5 ) '

(10)

and fl and a are the dipole and quadrupole polarizabilities of the atom, respectively. However, the range of validity of eq. (7) may be restricted to values of k very near to zero especially for small values of 1. For larger values of 1 the explicit terms of eq. (7) give accurate values of Bt(k) over a wide range of values for k. In these cases eq. (7) is also a good approximation to the relativistic phase shifts since for larger 1 the difference between the spin-up and spin-down values become negligible. For small l we have used two analytical forms for 8l(k). For the lowest 1 values we have

St(k) = N,(k ) / D t ( k ),

(11)

where Nt(k) = [ hOlk + h°2k2 + . . . + ho,k"

(1=0),

(12)

t ht2 k2 + h13k 3 + . . . +hi, k"

(l*0),

(13)

D,(k) =

1+

duk

+ d12 k 2 + d13 k 3 I n k + d14k4 + • • • dim k m

1+

dnk

+ dl2 k2 + d13 k3 + dr4 k 4 + • • •

dl,,k"

(1=1),

(14)

( l * 1),

(15)

the number of coefficients m and n in these polynomials varying from case to case. For somewhat larger values of / we use a form suggested by eq. (7), viz.

St(k) = kEfla, + k4( fl2bt + act) +Pt6 k6 + p n k 7,

(16)

with at, b I and c t as in eqs. (8), (9) and (10). Note that in expressions (11) and (16) we are giving forms for St(k), rather then tan 8t(k ) as suggested by effective range theory. When 8/ is small tan 8/ -- ~t and when 8t becomes larger we avoid the infinite values that can occur for tan 8 t. The values of the coefficients h . , d . and p . are obtained by fitting the above expressions to the results of refs. [7,8,19] for Ar, Kr and Xe respectively. These theoretical calculations are the most extensive set of results available and have been shown to be quite accurate especially at low energies. The fittings reported here agree with the original data to within 1% over the range of energies indicated. In using eqs. (7) or (16) we use the values of fl from ref. [23] and put a = 0 to be consistent with the original calculations. We give the values of the coefficients h . and d . (14 3) and coefficients p . (4 ~<1~< 6) for argon, krypton and xenon in tables l a - l d . For / >1 7 eq. (7) may always be used.

hoi

0.150575E1 0.671099E1 - 0.103246E3 0.119870E2

i

1 2 3 4 5 6 7

0.122028E2 0.429028E2 0.547674E1

doi

0.225686E1 - 0.805571E1 0.213514E1

hi i

1

hOi

0.310290E1 0.995427E1 -0.959457E2 -0.957331E2 - 0.148661E2

i

1 2 3 4 5 6 7

0.936060E1 0.561505E2 0.348326E2 0.268920E2

doi

0.341894E1 -0.119992E2 0.135713E1

hll

1

0

ho~

0.703886E1 0.699360E2 - 0.382597E3

I

i

1 2 3 4 5 6 7

0.140574E2 0.134270E3 0.840150E1 0.277726E2 - 0.763291E1

doi

0.565255E1 - 0.217485E2 0.279415E1

hl~

1

(c) S T , x e n o n (fl = 27.06 ainu, k < 1.2 a m u )

0

1

0a) K r y p t o n ( f l = 16.46 a m u , k ~ 1.2 a m u )

0

l

Table 1 C o e f f i c i e n t s o f 6 l ( k ) fittings (0 < 1 < 6) a) (a) A r g o n (fl = 10.755 ainu, k _~ 1.3 a m u )

- 0.218277E0 0.998044E1 0.345797E0

dl~

- 0.155652E0 0.719842E1 0.420052E0

dl i

--0.802810E - 2 0.425572E1 - 0.677844E0

dli

0.800069E0 -0.230394E1 0.439708E1 -0.177411E1

h2i

0.497005E0 -0.136592E1 0.337916E1 -0.531442E1 0.493903E1 -0.166390E1

h2i

0.319699E0 -0.471566E0 0.662807E0

h2i

-0.294671E1 0.271655E1

d2i

-0.243304E1 0.175754E1

d2i

- 0.145269E1 0.201048E0 0.584853E0

d21

-

0.274239E0 0.216850E0 0.258845E0

h3i

- 0.342904E0 0.584499E0

0.164194E0

h 3i

-0.183263E0 0.315322E0

0.107419E0

h3~

3

- 0.613335E0 0.175085E0

d3~

- 0.220459E1 0.355981E1 - 0.695569E0

d 3i

-0.177524E1 0.293544E1 -0.497051E0

d3i

0.324336E- 2 0.867549E - 3

0.115713E- 1 0.632895E - 2

-

Psi

5

0.754620E- 3

Psi

5

0.382582E- 3

Psi

5

P6i

6

P6~

6

P6i

6

0.378435E - 1 0.280246E - 2 0.281535E - 3 0.241583E - 1 - 0.780738E - 3 0.489967E - 4

P4i

4

-

P4i

4

-

P4i

4

0.168879E1 - 0 . 1 8 6 6 1 3 E 0 - 0 . 2 0 3 3 4 3 E 1 0.875782E0

") 0.150575E1 = 0.1505'75 × 10 s.

6 7

4 5

3

0.560248E0

hli

doi

i hoi

1 -0.201190E0 0.163957E1 2 -0.310805E1 -0.752074E0

1

10

(f) N e o n (fl = 2.669 a m u , k < 1.4 a m u )

6 7

0.802815E - 1

h2i

- 2

- 1 - 1 - 1

-0.938189E0 -0.852250E- 1 0.118350E0 -0.193714E - 1

-0.139339E0 0.113390E1

dlt

2

-0.802140E0 0.276586E0 0.102218E1 0.396706E 0.741701E - 1 0.159467E0 - 0 . 3 8 6 3 1 6 E 0.953195E - 2 0.437796E - 0.891220E

h2i

dli

hit

i hot

1 -0.115754E1 0.129511E1 2 -0.269704E1 -0.146378E1 3 0.382044E1 - 0 . 3 6 3 0 3 7 E 0 4 -0.522851E0 5

2

1

doi

h21

0.950567E1 0.803605E0 0.720979E0 - 0 . 2 3 1 5 8 9 E 1 0.432197E1 - 0.172023E1

-0.344694E0

dlt

2

l 0

(e) H e l i u m ( B = 1.322 ainu, k < 1.49 a m u )

6 7

2 3 4 5

0.563812E1 -0.184394E2 0.151314E1

hli

dot

i hot

1

1

10

(d) 8 7 , x e n o n (fl = 27.06 ainu, k < 1.2 ainu)

T a b l e 1 (continued)

0.666633E0

- 0.966691E0

d2t

0.707432E0

- 0.921308E0

d2i

- 0.391452E - 2

0.658760E - 2

P3i

3

0.156586E - 2 - 0.102726E - 2

P3i

3

0.174509E0

0.273965E0 -0.221426E0 0.239560E0

0.264906E1

d3i -0.649407E0

h3i

-0.291455E1

d2i

3

0.216312E - 3

P4i

4

0.205123E - 4

P5t

5

P6t

6

P6i

6

3 0.198219E- 3

0.853256E - 5

Psi

5

P6t

6

0.247173E - 2 0.128418E - 3

Psi

5

1 -0.630014E-

0.447793E - 4

P4i

4

-0.211047E-

0.337899E - 1

P4t

4

~x

e,

A.D. Stauffer et al. / Analytical expressions for phase shifts

332

For completeness we include the coefficients h , and dti (l <~2) and Pti (3 ~< 1 ~< 6) for helium and neon in tables le, f. These were obtained by fitting the phase shifts of refs. [24,25]. 3.2. Elastic differential cross sections Using the expressions for 8~(k) given in sect. 3.1 the elastic differential cross sections can be calculated for arbitrary angle and energy using eqs. (1) or (4). For accurate values at all angles, terms up to 1 = 50 should be included.

Table 2 C o e f f i c i e n t s f o r t h e fitting o f oeu(c) b y a series o f N c u b i c splines (a) A r g o n ( N = 1 8 , ~ < 20 eV) p

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

c~ [eV] 1.65D- 5 3.00D-4 1.00D- 3 4.80D-3 1.01D-2 5.00D-2 1.45D-1 2.00D-1 2.50D- 1 3.00D- 1 3.50D-1 4.00D-1 6.00D-1 1.25D0 3.50D0 5.50D0 9.25D0 14.50D0 19.00D0

q,l [ ~ra 02/eV]

q~2 [ ~ra02/eV]

-0.3144620811D5 -0.1166604511D5 -0.2069580066D4 -0.1201380112D4 -0.1118286385D3 -0.2125524382D2 -0.1624537315D2 -0.1120026305D2 -0.8189885701D1 -0.6800194152D1 -0.6489337690D1 -0.1419820347D1 -0.7301506475D0 -0.8718626375D0 -0.3543993911D1 - 0.3252682114D1 - 0.6960753123D1 - 0.6191480659D1

0.2954450691D5 0.1127432807D5 0.1704597279D4 0.1056707458D4 0.6587738787D2 0.8476022949D1 0.1008665299D2 0.8189885701D1 0.6800194152D1 0.6489337690D1 0.6842455087D1 0.2936111442D1 0.2997270859D1 0.3113736206D1 0.6770843794D1 0.8783754271D1 0.5282074526D1 0.5035888889D1

qv3 [ ~ra 2 / e V 3 ] 0.1805193286D10 0.2829021641D7 0.2123675855D7 0.1373648957D6 0.2054309839D4 0.1548655137D3 0.1817225290D3 0.1080457198D3 0.7192233908D2 0.4426492391D2 0.3101796528D2 0.1472213954D1 -0.4464481506D-2 0.3433468634D-1 0.6678905143D-1 - 0.7120741486D - 1 0.3406344825D - 2 0.0000000000D0

qv4 [ ~ra 02/e V 3 ]

Validity [eV]

0.0000000000D0 -0.7311032809D9 -0.5211355654D6 -0.1522635519D7 -0.1824646484D5 -0.8628101323D3 -0.2674949782D3 -0.1998947819D3 -0.1080457198D3 -0.7192233908D2 -0.4426492391D2 -0.7754491321D1 -0.4529889090D0 0.1289739102D-2 -0.3862652213D-1 - 0.3562082743D - 1 0.5086243919D - 1 - 0.3974068963D- 2

0.0 to c 2 c 2 to c 3 c 3 to c 4 c 4 to c 5 c 5 to c 6 c 6 to c 7 C 7 t o C8 c s to c 9 c 9 to c10 c10 to c H e l l t o C12 c12 to c~3 on3 to c14 on4 to c15 c~5 to c16 c16 to E17 c17 to cls cls to 20.0

q~4 [ ~ra 2 / e V 3 ]

Validity [eV]

0.0000000000D0 - 0.7696360830D8 0.2128414127D7 -0.9705573731D6 -0.2707349926D5 -0.8103499526IM -0.2137762180D4 - 0.9347512540D3 -0.3776797196D3 -0.1540892746D3 -0.4686988817D2 -0.2060143591D2 -0.5391813721D1 -0.50225995731)0 -0.8413043810D-1 -0.12209369641)0 0.9306931678D - 1 -0.2754688908D- 2

0.0 to c 2 c 2 to c 3 c 3 to E4 c 4 to c 5 c 5 to c 6 c 6 to c 7 c 7 to c 8 c a to c 9 c 9 to Clo c10 to cll ell to c12 c12 to c13 c13 to c14 cl4 to c15 c15 to on6 c16 to c17 c17 to cls cls to 12.0

(b) K r y p t o n ( N = 18, c < 18 eV) c~ [eV] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1.00D-4 1.00D- 3 4.00D-3 9.00D-3 2.00D-2 6.00D- 2 1.10D-1 1.60D- 1 2.30D-1 3.00D-1 4.00D-1 5.50D-1 7.00D-1 1.00D0 2.00D0 4.00D0 6.00D0 11.20D0 17.30D0

q,1 [ ~ra o2/eV]

qJ,2 [ ~'a o2/eV]

-0.4135822222D5 - 0.1078232753D5 -0.6134690353D4 -0.2299862558D4 -0.4940924012D3 -0.2344872512D3 -0.1520175946D3 - 0.7033829028D2 -0.4349079795D2 -0.1863710725D2 -0.6582094183D1 -0.3969801025D1 -0.1705403432D1 -0.7603400427D0 -0.2283478248D1 -0.8109225214D1 - 0.8630940480D1 -0.5882416059D1

0.3804219826D5 0.1016772621D5 0.5264679344D4 0.1942305842D4 0.3022255009D3 0.1520175946D3 0.1016143706D3 0.4349079795D2 0.2774708936D2 0.1075195168D2 0.3969801025D1 0.4138701716D1 0.4057988679D1 0.5071739124D1 0.8109225214D1 0.1686522089D2 0.6933390499D1 0.4754508197D1

q~3 [ ~rao2 / e V 3 ] 0.2565453610D9 - 0.3547356879D7 0.2135226221D7 0.9844908822D5 0.1012937441D5 0.2137762180D4 0.1308651756D4 0.3776797196D3 0.2201275352D3 0.7030483225D2 0.2060143591D2 0.1078362744D2 0.1674199858D1 0.1682608762D0 0.1220936964D0 -0.2419802236D0 0.3231461988D - 2 0.0000000000D0

A.D. Stauffer et al. / Analytical expressions for phase shifts

333

Table 2 (continued) (c) X e n o n ( N = 1 9 , c ~ 1 5 eV) 1,

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

c~ [eV] 5.00D- 5 4.00D-4 1.20D-3 4.50D- 3 1.20D-2 2.60D-2 5.00D-2 1.00D-1 1.50D-1 2.50D-1 4.00D- 1 5.50D- 1 7.50D- 1 1.00D0 1.40D0 2.20D0 4.00D0 6.30D0 9.00DO 14.00D0

q,1 [ ~ a o2 / e V ]

q~2 [~rao2 / e V l

q~3 [~ra2/eV3 ]

- 0.5567711429D6 -0.2320165923D6 -0.5385599701D5 - 0.2032706568D5 -0.9022769341D4 -0.3913356823D4 -0.1274312271D4 -0.7847610367D3 -0.2387801656D3 -0.7923736666D2 - 0.3592416779D2 - 0.1266372734D2 - 0.5946266498D1 -0.4673083233D1 -0.5390500992D1 -0.6193573395D1 -0.2576388041D2 - 0.2215275845D2 -0.9425031942D1

0.5358415437D6 0.2239621029D6 0.4753537828D5 0.1715912961D5 0.6950849268D4 0.2858834550D4 0.7847610367D3 0.5271155828D3 0.1298004667D3 0.3592416779D2 0.1832596217D2 0.8034453199D1 0.8536433270D1 0.1256075050D2 0.1861741379D2 0.3065698297D2 0.2578115637D2 0.1830162328D2 0.8559840000D1

0.1066261677Dll 0.1762063672D9 0.2948636981D8 0.2265666136D7 0.6374672940D6 0.1060381079D6 0.1546038533D5 0.6607366889D4 0.8755533336D3 0.1904814315D3 0.8234242213D2 0.2673867004D2 0.1086666767D2 0.3707809401D1 0.1800720636D1 -0.1104161410D1 -0.1121278590D0 0.4787467097D - 1 0.0000000000D0

q~4 [ ~ a o2//eV 3 ]

Validity [eV]

0.00000000~D0 -0.4664894835D10 -0.4271669507D8 - 0.1297400272D8 -0.1213749716D7 -0.3718559215D6 -0.5089829179D5 -0.1546038533D5 -0.3303683445D4 -0.5837022224D3 - 0.1904814315D3 - 0.6175681660D2 - 0.2139093603D2 -0.6791667296D1 -0.1853904701D1 -0.8003202828D0 0.8641263205D0 0.9551632432D - 1 -0.2585232233D- 1

0.0 to c 2 c 2 to c a c s to c 4 c 4 to c 5 c 5 to c 6 c 6 to c 7 c 7 to c s c a to c 9 [9 to [1o c10 to q l cll to c12 ~12 to Cl3 c13 to c14 c14 to cls c~5 to c16 c1~ to c17 c17 to c18 c18 to c19 c19 to 15.0

Table 3 C o e f f i c i e n t s f o r the fittings o f inelastic c r o s s s e c t i o n s (a) A r g o n fo [~ra 2 ] ls5 ls4 ls3 ls2 P.S. Ion.

A [ ~rao2/eVl

f2 [~ao2/eV 2]

0.5321E-3 0.2037E- 3 0.8463E - 4 0.7999E- 3 0.3025E- 2 - 0.9458E0 0.1894E0

-0.1543E-4 - 0.4394E- 0.2335E - 0.3438E- 0.7096E0.3259E-

fo [ ~ra 02]

fl [~ra2/eV]

f2 [~a02/eV 2]

0.7693E- 1 -0.2611E0 - 0.4320E- 1 - 0.2225E0 - 0.5039E0 0.1597E1 -0.3578E1

- 0.3812E- 1 0.2603E-1 0.5307E- 2 0.2090E- 1 0.4032E- 1 - 0.3913E0 0.2490E0

fo

/1

f2

Range

[ era 2 ]

[ ~a o2/eV]

[ ~a o2 / e V 2 ]

[eV]

-0.4087E- 0.1775E- 0.6713E - 0.5562E- 0.2844E0.6810E1 - 0.3075E1

2 2 3 2 1

A [~rao2/eV 3 ]

5 5 4 4 1

0.5510E- 6

Range [eV] 11.55 11.62 11.72 11.83 14.0 15.76 17.5

to to to to to to to

20.0 20.0 20.0 20.0 20.0 17.5 20.0

(b) K r y p t o n

ls5 ls4 ls3 ls2 P.S. Ion.

0.4252E- 2

f3 [ ~ra o2 / e V 3 ]

Range [eV]

- 0.1200E- 3

9.92 10.03 10.57 10.64 12.5 14.00 16.0

- 0.1153E- 3

0.1981E- 1

(c) X e n o n

ls5 ls4 ls3 ls2 P.S. Ion.

-

0.9455E0.2247E 0.2354E0.2062E 0.3087E0.2121E1

3 4 3 4 4

0.1137E0.2663E 0.2491E0.2155E 0.2858E0.2643E-

3 3 2 3 3 1

0.1226E- 1

8.32 8.44 9.45 9.57 10.8 12.13

to to to to to to

15.0 15.0 15.0 15.0 15.0 15.0

to to to to to to to

18.0 18.0 18.0 18.0 18.0 16.0 18.0

A.D. Stauffer et al. / Analytical expressionsfor phase shifts

334

If a large number of such cross section values are required we suggest tabulating them for a range of energies and angles and interpolating to find the desired results. 3.3. Elastic integral cross sections

The integral elastic cross sections were calculated from eq. (2) for Ar and Kr and from eq. (5) for Xe. It suffices to include only terms up to l = 6 in these equations. To facilitate rapid calculations we have fitted each resulting cross section thus obtained by a series of N cubic splines Q~(c) of the form (~ = 1, 2 . . . . . N ) Q,(c) = q,1(c - c,) + q,2(c - c,+i) + q,3(c - c,) 3 + q,4(c - c,+i) 3, Table 4 Coefficients for the fitting of

at(¢ ) by a series o f N (a) A r g o n ( N = 18, ¢ ~ 20 eV) ~'

cubic splines Validity

¢, [eV]

q~l " [,rra2/eV]

q,2 [ era 02 / e V ]

qpa [ ~ a o2 / e V 3 ]

q J,4 [ ¢rao2 / e V 3 ]

[eV]

1 2 3

1.65D-5 3.00D-4 1.00D-3

-0.3144620811D5 -0.1166604511D5 -0.2069580066D4

0.2954450691D5 0.1127432807D5 0.1704597279D4

0.1g05193286D10 0.2829021641D7 0.2123675855D7

0.0000000000D0 - 0.7311032809D9 -0.5211355654D6

0.0 to ¢2 c 2 to ¢3 c 3 to ~4

4 5 6

4.80D-3 1.01D-2 5.00D-2

-0.1201380112D4 -0.1118286385D3 -0.2125524382D2

0.1056707458D4 0.6587738787D2 0.8476022950D1

0.1373648957D6 0.2054309839D4 0.1548655137D3

-0.1522635519D7 -0.1824646484D5 -0.8628101323D3

c 4 to ¢5 ~5 to c 6 ¢6 to ¢7

7 8 9 10 11 12 13 14 15 16 17

1.45D-1 2.00D- 1 2.50D-1 3.00D - 1 3.50D- 1 4.00D- 1 6.00D- 1 1.25D0 3.50D0 5.50D0 9.25D0

-0.1624537315D2 -0.1120026304D2 -0.8189885706D1 - 0.6800194134D1 -0.6489337759D1 - 0.1419819315D1 -0.7301588079D0 - 0.8717923314D0 - 0.3544147754D1 - 0.3251588004D1 - 0.6965235123D1

0.1008665298D2 0.8189885706D1 0.6800194134D1 0.6489337759D1 0.6842454829D1 0.2936113953D1 0.2997250548D1 0.3113909279D1 0.6770260269D1 0.8786955700D1 0.5267801777D1

0.1817225294D3 0.1080457178D3 0.7192234649D2 0.4426489626D2 0.3101806848D2 0.1472151182D1 -0.4416408959D0.3430049907D 0.6693493281D - 0.7143507203D 0.5094158618D -

-0.2674949781D3 -0.1998947824D3 -0.1080457178D3 - 0.7192234649D2 -0.4426489626D2 - 0.7754517121D1 -0.4529695944D0 0.1275851477D - 0.3858806145D - 0.3569863083D 0.5102505145D - -

c 7 to ¢8 c 8 to ~9 c 9 to ~10 c10 to ¢11 cll to ¢12 c12 to ¢13 el3 to ¢14 c14 to ¢15 ¢15 to ¢16 c16 to ¢17 Cl7 to ¢18

-- 0.6189228280D1

0.5274066667D1

0 . ~ D 0

- 0.5943185054D- 2

cls to 20.0

q,4 [ ~ a o~ / e V ~]

Validity

18 19

14.50D0 19.00D0

2 1 1 1 2

2 1 1 1

(b) K r y p t o n ( N ffi 18, ¢ < 18 eV) u

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

¢, [eV]

q,1 [ ~,a o2 / e V ]

q~2 [,~a~/eV]

q~3 [,,ao~ / e V 3 ]

1.00D-4

- 0.4135822222D5

0.3804219826D5

0.2565453610D9

1.00D - 3 4.00D--3 9.00D - 3 2.00D-2 6.00D-2 1.10D-1 1.60D-1 2.30D - 1 3.00D- 1 4.00D- 1 5.50D- 1 7.00D-1 1.00D0 2.00D0 4.00D0 6.00D0 11.20D0 17.30D0

- 0.1078232753D5 -0.6134690353D4 - 0.2299862558D4 -0.4940924012D3 -0.2344872512D3 -0.1520175946D3 -0.7033829028D2 - 0.4349079796D2 -0.1863710715D2 -0.6582094668D1 -0.3969799511D1 -0.1705414581D1 -0.7602336041D0 - 0.2284009431D1 - 0.8107738103D1 - 0.8645025360D1 -0.5834798578D1

0.1016772621D5 0.5264679344D4 0.1942305842D4 0.3022255009D3 0.1520175946D3 0.1016143706D3 0.4349079796D2 0.2774708929D2 0.1075195200D2 0.3969799511D1 0.4138707290D1 0.4057956748D1 0.5072004715D1 0.8107738103D1 0.1687063816D2 0.6895873326D1 0.4973213115D1

- 0.3547356879D7 0.2135226221D7 0.9844908822D5 0.1012937441D5 0.2137762179D4 0.1308651757D4 0.3776797157D3 0.2201275496D3 0.7030479994D2 0.2060150324D2 0.1078337969D2 0.1674554653D1 0.1679952846D0 0.1224654742D0 - 0.2433345390D0 0.5034981681D- 2 0.0000000000D0

0 . ~ D 0 - 0.7696360830D8 0.2128414127D7 - 0.9705573731D6 -0.2707349926D5 -0.8103499526D4 -0.2137762179D4 -0.9347512552D3 - 0.3776797157D3 - 0.1540892847D3 -0.4686986663D2 -0.2060150324D2 -0.5391689846D1 -0.5023663959D0 - 0.8399764228D- 1 - 0.1224654742D0 0.9359020733D- 1 -0.4292115532D- 2

[eV] 0.0 to c 2 c 2 t o E3 c 3 to c 4 c 4 to e 5 c 5 to c 6 c 6 to c 7 c 7 to ¢8 ¢8 to c 9 c 9 to c10 on0 to ¢11 ell to ~12 c12 to ¢13 ~13 to ¢14 c14 to ¢15 ¢15 to ¢16 ¢16 to ¢1T ~17 to ¢1s ¢1s to 18.0

A.D. Stauffer et a L / Analytical expressions for phase shifts

335

T a b l e 4 (continued) (c) X e n o n ( N = 19, c < 15 eV) ~, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

c~ [eV]

q~l [ era02/eV]

q,2 [ ~ra02/eV]

-0.5567711429D6 - 0.2320165923D6 -0.5385599701D5 -0.2032706568D5 -0.9022769341D4 -0.3913356823D4 -0.1274312271D4 -0.7847610367D3 -0.2387801655D3 -0.7923736671D2 -0.3592416765D2 -0.1266372803D2 -0.5946263625D1 -0.4673098667D1 -0.5390406413D1 -0.6194177075D1 -0.2576177286D2 - 0.22160871MID2 - 0.9362219372D1

0.53584154371)6 0.22396210291)6 0.4753537828D5 0.1715912961D5 0.6950849268D4 0.28588345501)4 0.7847610367D3 0.5271155828D3 0.1298004667D3 0.3592416765D2 0.1832596269D2 0.8034450900D1 0.8536442917D1 0.1256070321D2 0.1861768209D2 0.3065533358D2 0.2578806656D2 0.1827638654D2 0.8772240000D1

5.00D-5 4.00D - 4 1.20D-3 4.50D-3 1.20D-2 2.60D-2 5.00D-2 1.00D-1 1.50D-1 2.50D-1 4.00D-1 5.50D-1 7.50D-1 1.001)0 1.401)0 2.20D0 4.00D0 6.30D0 9.00D0 14.00D0

q J,3 [ ~ra 0~ / e V 3 ] 0.1066261677Dll 0.1762063672D9 0.2948636981D8 0.2265666136D7 0.6374672940D6 0.1060381079D6 0.1546038533D5 0.6607366891D4 0.8755533305D3 0.1904814378D3 0.8234239889D2 0.2673872750D2 0.1086651333D2 0.3708104961D1 0.1800301414D1 -0.1103652340D1 -0.1134341325D0 0.5286523168D - 1 0.000000(0~D0

q,4 [ era 02/eV 3 ]

Validity [eV]

0.00000000~D0 - 0.4664894835D10 -0.4271669507D8 -0.1297400272D8 -0.1213749716D7 -0.3718559215D6 -0.5089829179D5 -0.1546038533D5 -0.3303683446D4 -0.5837022204D3 -0.1904814378D3 -0.6175679917D2 -0.2139098200D2 -0.6791570829D1 -0.1854052480D1 -0.8001339619D0 0.8637279183D0 0.9662907585D - 1 - 0.2854722511D - 1

0.0 to e2 • 2 to c E3 to ~4 e 4 to ~5 e 5 to ~6 c 6 to c 7 c 7 to c~ c 8 to ~9 q[9 to c10 ~1o to ~ln ~H to el2 ~12 to c13 ~13 to c14 ~ 4 to ~15 E1s to c16 c16 to c17 el7 to c18 E1a to c19 E19 to 15.0

the pth spline Q,(c) thus representing Oel(C) between the boundaries (knots) c, and ~,+1 (except Q1 which is valid from zero to c 2, and QN which is valid from c N to the maximum energy specified in the tables). The four coefficients q,; and the knots E,, ~,+1 of each spline Q, are given in tables 2a, b, c for Ar, Kr and Xe respectively. These cubic spline fittings reproduce the calculated data [7,8,19] within 1% for the range given in each table. 3.4. Inelastic cross sections In this section we present some simple analytical expressions of the form

r(,) = El,,', i

for near threshold partial excitation and ionization cross sections in Ar, Kr and Xe. For the Ar and Kr first four excited states (ls5, ls4, ls3 and ls2) we based our fittings on the partial excitation data published in refs. [10,11]. For xenon where similar data was not available we assumed linear variation and relative magnitudes similar to the Kr case. To take into account the excitation of the remaining levels, we considered a fifth excited pseudostate. Its threshold (14.0 eV for Ar, 12.5 eV for Kr, 10.8 eV for Xe) was taken by averaging over the structure of the excitation functions in refs. [13,14]. Its excitation cross section was taken as a smoothly increasing function from the corresponding threshold, which together with the previously mentioned four levels would join the total excitation cross sections of ref. [12]. At present there is little information on partial excitation for the individual levels and so the cross sections we present here are thus not very accurate, but may be very useful, tentative as they are. We should emphasize that along the range of E / p in which we used them ( E / p < 5 V c m - 1 Torr-1) the uncertainty of the inelastic cross sections did not significantly affect our results, as the inelastic contribution to the total cross section amounts only to a few percent in that range. The ionization cross sections fits we present are based on the well established data from ref. [16].

A.D. Stauffer et al. / Analytical expressions for phase shifts

336

The coefficients f~ for the inelastic cross sections mentioned in this section are shown in tables 3a, b, c for Ar, Kr and Xe, where P.S. designates the pseudostate. 3.5. Total cross sections

We present in tables 4a, b, c the coefficients of the cubic spline fittings (cf. sect. 3.3) for the total scattering cross section ot(c ) obtained by adding the integral elastic and inelastic contributions we have described. Again these fittings reproduce the calculated data within 1%.

4. Conclusions

The modelling of gas nuclear radiation detectors requires the detailed knowledge of the energy dependence of electron impact elastic, inelastic and total scattering cross sections. The values of these cross sections may be required at a very large number of energies during extensive calculations as in a Monte Carlo simulation. We present in this paper simple analytical expressions which reproduce recent reliable data on electron-noble gas a t o m elastic scattering phase shifts, thus allowing an accurate calculation of elastic scattering integral and differential cross sections at any energy by the use of the appropriate phase shifts formulae (eqs. (1)-(6)). But as a repeated computation of cross sections using those formulae is not a rapid process, in order to enable a quicker evaluation we also present fittings to the integral elastic cross section which reproduce within 1% the cross sections obtained from those formulae. Also presented are simple expressions which tentatively fit the rather scarce data on near threshold inelastic cross sections, and fittings to the total scattering cross sections obtained by adding the described elastic and inelastic data within the low energy range considered. Most of these expressions were used in a previous study of electroluminescence and drift velocity of electrons in noble gases by a Monte Carlo method at low E / p ( E / p __<5 V cm-1 Torr-1) [4].

Acknowledgments We acknowledge support from INIC (Lisboa) and NATO (Research Grant 133/80). We thank Prof. L. Alte da Veiga and Prof. Margarida R. Costa for the use of The PDP-11/34 computer donated by the Deutsche Gesellschaft f'ur Technische Zusammenarbeit.

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