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Contribution of inverse neutral bremsstrahlung to the absorption coefficient of heated air
J. Quant.
Spec~rosc.
Radiar.
Trons/er.
Vol.
1. pp. 423427.
Pergamon
Press Ltd..
1967. Printed
m Great
Britam
CONTRIBUTION OF INVERSE NEUTRAL BREMSSTRAHLUNG TO THE ABSORPTION COEFFICIENT OF HEATED AIR* R. C. MJOLSNE~Sand H. M. RUPPEL University
of California,
Los Alamos
Scientific
Laboratory,
Los Alamos,
New Mexico
(Received 2 November 1966) Abstract-The contributions to the spectral absorption coefficient of inverse bremsstrahhmg from electrons in the fields of neutral oxygen and nitrogen atoms and nitrogen molecules are tabulated for air temperature of 400&12 000°K and for photon energies of 0.6-3.0 eV. The atomic and total contributions are tabulated separately, since the molecular contributions may well be evaluated significantly more accurately in the future. For brevity, values are given at only four air densities. OUTLINE
OF
THEORETICAL
CONSIDERATIONS
AIR heated to between 4000 and 12 000°K and for photon energies of less than 3 eV, the inverse bremsstrahlung from electron collisions with neutral species is a significant contributor to the spectral absorption coefficient. Over an appreciable part of this range it is dominant. However, the cross section for this process has not been accurately known, and previous work has typically included this process”’ through use of an effective 2 approximation’2’ to KRAMERS’(~) formula for hydrogenic bremsstrahlung or has left this have made it possible to treat process for later evaluation. (4) Recent calculations’5*6’ electron-atom bremsstrahlung quite accurately. The calculations are less accurate for electron-molecule bremsstrahlung. The cross section per unit photon energy for electron-neutral particle bremsstrahlung is given by FOR
d -----d d(W
= &(E,,
E)?,
;
(1)
where hv is the photon energy, a, is the Bohr radius, E, and E are the initial and final electron energies respectively (E, > E), and Co is a dimensionless constant of the order of (e2/hc)“. From the above one may obtain the spectral absorption coefficient for inverse bremsstrahlung by use of the principle of detailed balance. For a neutral species of density ‘N and electrons of density N, with temperature T the expression for the linear absorption coefficient takes the form a:
NN,(e2/ao)5’2 (kT)3’2(hv)3
mdE[E+hv]&(E,,E)e-E’kT,
s
0
* Work performed
under the auspices
of the U.S. Atomic 423
Energy
Commission.
(2)
424
R. C. MJOLSNESS and H. M. RUPPEL
where E, = E +hv during the integration over E. The generalization to several neutral species is straightforward. Thus, the calculation of the absorptiou coefficient due to inverse bremsstrahlung reduces to the calculation of X0. For atomic oxygen and nitrogen, the results of accurate calculations’5’ may be fairly well approximated by the linear relation Xo(Eo, E) = o.
i9
1 E,
(3)
where o. = 0.71 x 10e6 for oxygen and co = 080 x lop6 for nitrogen. expression for the absorption coefficient ’ (e2/uo)3/2(kT)3/2 (W3
1 +tfi [
h\* 1
a;NN,
The corresponding
(4)
yields values within 20 per cent of the numbers tabulated below, the accuracy increasing at low photon energy. From equation (4) it is readily seen that both the temperature dependence and the frequency dependence of the free-free electron-atom absorption coefficient differ from that of the Kramers formula. However, the new calculations will have little effect on the Rosseland mean absorption coefficient which will differ by only a few percent from values computed with TAYLOR’S”’ fit to the Kramers formula. The contribution from nitrogen molecules was calculated from HOLSTEIN’S formula.‘6’ which yields for C,
(5) c~=(j$) 1:)‘(&)1;)1’2 {(y - (EoW2) o.,(E.)+(+oE)1/20,..(E.) where the elastic scattering’and momentum transfer cross sections, cr,, and crmom, are evaluated at the midpoint energy, E, = (E,+ E)/2. Data for the elastic scattering and momentum transfer cross sections were taken from NORMAND and ENGELHARDT.@) Since the elastic scattering cross section is not well known below roughly 1 eV, two quite different extrapolations to zero energy are used, yielding negligible differences in the absorption coefficient. In the tables below the absorption coefficient due to atomic oxygen and nitrogen and molecular nitrogen are given. Other neutral species should have little effect on the absorption coefficient. The atomic and total contributions are listed separately, since the molecular numbers may well be subject to significant improvement. For brevity, values have been tabulated at only four values of the air density ratio p/pa, where p. = 1.2931 x lo- 3 g cme3. Values at other densities may be obtained when the populations of the various species are known, since the absorption coefficient per electron-neutral depends only on temperature and photon energy. The absorption coefficients tabulated below differ appreciably from the recent calculations of KIVEL.(~’ For the atomic contributions the exchange forces are correctly treated in the present work by use of integrodifferential equations. The previous work approximates their effect through local potentials. Both calculations account for polarization effects through a central potential containing a cut-off parameter. A much larger value for the parameter was obtained by Kivel to compensate for the approximations in his treatment of exchange. In the present calculation the value of the parameter was determined
Contribution of inverse neutral bremsstrahlung to the absorption coefficient of heated air
425
by fitting to elastic scattering data. However, photodetachment data could equally well have been used, since in the present model these data are consistent. The two calculations of molecular contributions are quite similar. Different approaches are used to account for finite photon energies, and different averages over electron energies are taken. While the present method seems better founded theoretically, it is by no means certain that the resulting numerical values are more accurate. Clearly, more precise information on this process would be desirable. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
R. P. MAIN and E. BAUER,JQSRT6, 1 (1966) or H. A. BETHE,Los Alamos Report LA-3064. R. L. TAYLOR,J. Chem. Phys. 39,2354 (1963). W. FINKELBURG and TH. PETERS,Hundbuch der Physik Vol. XXVIII, Springer, Berlin (1957). D. R. CHURCHILL,B. H. ARMSTRONG and K. G. MUELLER,Absorption Coefficients of Heated Air, Vol. I and II, Air Force Weapons Laboratory Report, AFWL-TR-65-132 (1965). R. C. MJ~LSNESS and H. M. RUPPEL,Phys. Reo., to be published. T. HOLSTEIN,Westinghouse Scientific Paper 65-lE2-Gases-P2; see also M. ASHKIN,Phys. Rev. 141,4l (1966). C. E. NORMAND,Phys,. Rev. 35, 1217 (1930). A. G. ENGELHARDT, A. V. PHELP~and C. G. RISK, Phys. Rev. 135, Al566 (1964). B. KIVEL,AVCO Research Report 249. JQSRT. to be published. TABLE 1. ATOMICAND TOTALCONTRIBUTIONS TO THESPECTRAL ABSORPTION COE~FITIENT
Photon energy (eV)
10-Z
1.0
Atomic (cm- ‘)
Total
0.6 0.8 1.0 1.2 I.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
3.21-l 1.566’ 8.96-s 5.88-s 4.16-’ 3.11-8 2.42 - ’ 1.95-a 1,62-’ 1.37-8 1.17-s 1.01-s 8.78-9
6.65 - 6 3.46-6 2.15-” 15O-6 1.14-6 9.13-’ 7.62-l 6.533’ 5.69-’ 5.033’ 4.51-’ 4.10-’ 3.766’
0.6 0.8
1.57-5 7.52-b 4.3116 2.80-’ 1.96-6 1.46-6 l.13-6 9-05 - ’ 7.46 - ’ 6.27 - ’ 5.333’ 4.588’ 3.97-l
1.67-4 8.4115 5.11-S 3.49- 5 2.58-5 2.02-S 1.65-5 1,38-5 1.18-’ 1.03-5 9.096 8.11-6 7.3t-6
1.0
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
Atomic
Total
T = 4000°K 5.87- ” 4.56-9 2.85-” 2.36-9 1.46-9 1.o2-9 76-” 7.65- i” 5.699” 4.43- ’ ’ 3.57-11 2.96- ’ ’ ;I;:’
::
10-b
10-A
;:;;:
:;
Atomic
6.63 - l4 ;:;;:
::
;:s’;:
::
1.45-“+ 1.20- I4 1.02- I4 8.71- ”
Total
Atomic
Total
1,555’* 8.02- l3
1.24- ” 6.02- ” 3.47- ”
4.37- I6
;f:‘:: 2.599i3
;f;‘:: 1.22117
5.44 is 9.06 14
4.69- I8
8.28 - I4 T = 5000°K 1.03-8 7.06s 4.94-9 3.55-a 2.833’ 2.15-8 1.84-9 1.47-a 1.29-9 1.08-a ;:;: :: 598-I 4.933’0 ;::;‘:;
8.44-9 6.87-9 5.76-9 4.92-9 4.28-9 3.77-q 3.36-9 3.02-9
5.58- ” ;::I
4.90- i5 2.35- ” 1.35- 15 ;:;y g::
2-92- ” 2.42- l6 ;:y;:
:;
::
3.38- ”
426
R. C. MJOLSNE~~ and H. M. RUPPEL
TABLE 2. ATOMIC AND TOTALCONTRIBUTIONS TO THESPECTRALABSORPTION COEFFICIENT
F
1.0
10-z
10-4
Atomic (cm- ‘)
Total
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
1.51-4 7.14-5 4.06- 5 2.61-’ 1.82- ’ 1,34-* 1.04-5 8.27-6 6.78- 6 f1.67-~ 4.80-’ 4.11-6 3.55-e
l.29-3 6.36-4 3.78-4 2.53-4 1.83X4 1.41-4 1.13-4 9.31-5 7.85-5 6.73-s 5,85-5 5,15Y5 4.57-5
T = 6000°K 1.07-’ 4.18_’ 5.09-8 2,05-’ 2.90-’ 1.21-’ 1.87-’ 8.06-” 1.30- B 5.81-* 9.64-9 4.44-a 7.46- 9 3.54-S 5,97-9 2.91-* 4.91-9 2.45-a 4.11-9 2.09-‘3 3.50-9 l.82-8 3.00-9 1.59-e 2.59-9 1.41-13
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
8,32-4 3.91-4 2.21-4 l4-4 9.78-S 7.18-5 5.51-5 4.38-5 3.57-5 2.97- 5 2-51-* 2.14-5 1.84-5
5.31-3 2.57-3 1.50-a 9.88-4 7.04-4 5.32-’ 4.19-4 3.41-4 2.84-“ 2.4O-4 2.06-4 1.80-“ 1.5sm4
T = 7000°K 8.51-’ l.52-6 4~Ow’ 7.25- ’ 2.26-’ 4.17-l 1.45s’ 2.71-’ 1.01-’ 19-’ 7.40-S 1.42-’ 5.70-s 1.11-7 4.54-a 8.96-8 3.71-e 7.40-S 3.10-a 6.23-8 2.62- * 5.31-B 2.24- ’ 4.59-8 1.92-* 3.99- 8
Atomic
Total
lo-6
Atomic
Total
Atomic
Total
1.01-‘0
1.17-‘O 5.60- ’ ’ 3,21-” 2.09- ‘I
9.92- I4
9.93- I4
;:;;I
::
;:g:::
1.47-” 1.10-” 8,55_‘2
;:;;:
::
8.98- I5
1.74- I4 l.21-‘4 9.0(- ‘5
;:;;::: 1.76-” 1.23-l’ 9.11-12 7.06- I2 5.66- ” ;:;;I
::
;:;‘:I
::
2.48-”
;:;;I
::
;::;I
:f;:
::
::
;:;;I
::
4.61-I5 3.88-l” 3.30- I5 2.84-15 2.45- ‘5
3.09- ‘2
2.45- I5
9.65- I0
X.78_ ‘3 4.13-13 2.34-13 1.50- ” 1.04-13
;:;:I::
7.65- I4 5.89- I4
7.65- I4 5.89- I4
;:;;I
;:;;I
::
;:;;I
::
::
;:;;I::
;:;;I
::
8,30-” 6.39-” 5.09-L’
8.43-l’
4.17-I’ 3.48- ’ ’ 2.94- ’ ’ 2.51-” 2.16-l’
::
4.82- ”
;:;;I
2.54-l”
;:;;I
4.60- I5 3.87_ I5
4:;‘;:: : 4.24- ’ ’ 3.53- ’ ’ 2.99-”
::
3.20- I4 ;:;;I:: 1.99- I4
8.78- I3
;:;I
::
II
Spec~rosc.
Radiar.
Trons/er.
Vol.
1. pp. 423427.
Pergamon
Press Ltd..
1967. Printed
m Great
Britam
CONTRIBUTION OF INVERSE NEUTRAL BREMSSTRAHLUNG TO THE ABSORPTION COEFFICIENT OF HEATED AIR* R. C. MJOLSNE~Sand H. M. RUPPEL University
of California,
Los Alamos
Scientific
Laboratory,
Los Alamos,
New Mexico
(Received 2 November 1966) Abstract-The contributions to the spectral absorption coefficient of inverse bremsstrahhmg from electrons in the fields of neutral oxygen and nitrogen atoms and nitrogen molecules are tabulated for air temperature of 400&12 000°K and for photon energies of 0.6-3.0 eV. The atomic and total contributions are tabulated separately, since the molecular contributions may well be evaluated significantly more accurately in the future. For brevity, values are given at only four air densities. OUTLINE
OF
THEORETICAL
CONSIDERATIONS
AIR heated to between 4000 and 12 000°K and for photon energies of less than 3 eV, the inverse bremsstrahlung from electron collisions with neutral species is a significant contributor to the spectral absorption coefficient. Over an appreciable part of this range it is dominant. However, the cross section for this process has not been accurately known, and previous work has typically included this process”’ through use of an effective 2 approximation’2’ to KRAMERS’(~) formula for hydrogenic bremsstrahlung or has left this have made it possible to treat process for later evaluation. (4) Recent calculations’5*6’ electron-atom bremsstrahlung quite accurately. The calculations are less accurate for electron-molecule bremsstrahlung. The cross section per unit photon energy for electron-neutral particle bremsstrahlung is given by FOR
d -----d d(W
= &(E,,
E)?,
;
(1)
where hv is the photon energy, a, is the Bohr radius, E, and E are the initial and final electron energies respectively (E, > E), and Co is a dimensionless constant of the order of (e2/hc)“. From the above one may obtain the spectral absorption coefficient for inverse bremsstrahlung by use of the principle of detailed balance. For a neutral species of density ‘N and electrons of density N, with temperature T the expression for the linear absorption coefficient takes the form a:
NN,(e2/ao)5’2 (kT)3’2(hv)3
mdE[E+hv]&(E,,E)e-E’kT,
s
0
* Work performed
under the auspices
of the U.S. Atomic 423
Energy
Commission.
(2)
424
R. C. MJOLSNESS and H. M. RUPPEL
where E, = E +hv during the integration over E. The generalization to several neutral species is straightforward. Thus, the calculation of the absorptiou coefficient due to inverse bremsstrahlung reduces to the calculation of X0. For atomic oxygen and nitrogen, the results of accurate calculations’5’ may be fairly well approximated by the linear relation Xo(Eo, E) = o.
i9
1 E,
(3)
where o. = 0.71 x 10e6 for oxygen and co = 080 x lop6 for nitrogen. expression for the absorption coefficient ’ (e2/uo)3/2(kT)3/2 (W3
1 +tfi [
h\* 1
a;NN,
The corresponding
(4)
yields values within 20 per cent of the numbers tabulated below, the accuracy increasing at low photon energy. From equation (4) it is readily seen that both the temperature dependence and the frequency dependence of the free-free electron-atom absorption coefficient differ from that of the Kramers formula. However, the new calculations will have little effect on the Rosseland mean absorption coefficient which will differ by only a few percent from values computed with TAYLOR’S”’ fit to the Kramers formula. The contribution from nitrogen molecules was calculated from HOLSTEIN’S formula.‘6’ which yields for C,
(5) c~=(j$) 1:)‘(&)1;)1’2 {(y - (EoW2) o.,(E.)+(+oE)1/20,..(E.) where the elastic scattering’and momentum transfer cross sections, cr,, and crmom, are evaluated at the midpoint energy, E, = (E,+ E)/2. Data for the elastic scattering and momentum transfer cross sections were taken from NORMAND and ENGELHARDT.@) Since the elastic scattering cross section is not well known below roughly 1 eV, two quite different extrapolations to zero energy are used, yielding negligible differences in the absorption coefficient. In the tables below the absorption coefficient due to atomic oxygen and nitrogen and molecular nitrogen are given. Other neutral species should have little effect on the absorption coefficient. The atomic and total contributions are listed separately, since the molecular numbers may well be subject to significant improvement. For brevity, values have been tabulated at only four values of the air density ratio p/pa, where p. = 1.2931 x lo- 3 g cme3. Values at other densities may be obtained when the populations of the various species are known, since the absorption coefficient per electron-neutral depends only on temperature and photon energy. The absorption coefficients tabulated below differ appreciably from the recent calculations of KIVEL.(~’ For the atomic contributions the exchange forces are correctly treated in the present work by use of integrodifferential equations. The previous work approximates their effect through local potentials. Both calculations account for polarization effects through a central potential containing a cut-off parameter. A much larger value for the parameter was obtained by Kivel to compensate for the approximations in his treatment of exchange. In the present calculation the value of the parameter was determined
Contribution of inverse neutral bremsstrahlung to the absorption coefficient of heated air
425
by fitting to elastic scattering data. However, photodetachment data could equally well have been used, since in the present model these data are consistent. The two calculations of molecular contributions are quite similar. Different approaches are used to account for finite photon energies, and different averages over electron energies are taken. While the present method seems better founded theoretically, it is by no means certain that the resulting numerical values are more accurate. Clearly, more precise information on this process would be desirable. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.
R. P. MAIN and E. BAUER,JQSRT6, 1 (1966) or H. A. BETHE,Los Alamos Report LA-3064. R. L. TAYLOR,J. Chem. Phys. 39,2354 (1963). W. FINKELBURG and TH. PETERS,Hundbuch der Physik Vol. XXVIII, Springer, Berlin (1957). D. R. CHURCHILL,B. H. ARMSTRONG and K. G. MUELLER,Absorption Coefficients of Heated Air, Vol. I and II, Air Force Weapons Laboratory Report, AFWL-TR-65-132 (1965). R. C. MJ~LSNESS and H. M. RUPPEL,Phys. Reo., to be published. T. HOLSTEIN,Westinghouse Scientific Paper 65-lE2-Gases-P2; see also M. ASHKIN,Phys. Rev. 141,4l (1966). C. E. NORMAND,Phys,. Rev. 35, 1217 (1930). A. G. ENGELHARDT, A. V. PHELP~and C. G. RISK, Phys. Rev. 135, Al566 (1964). B. KIVEL,AVCO Research Report 249. JQSRT. to be published. TABLE 1. ATOMICAND TOTALCONTRIBUTIONS TO THESPECTRAL ABSORPTION COE~FITIENT
Photon energy (eV)
10-Z
1.0
Atomic (cm- ‘)
Total
0.6 0.8 1.0 1.2 I.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
3.21-l 1.566’ 8.96-s 5.88-s 4.16-’ 3.11-8 2.42 - ’ 1.95-a 1,62-’ 1.37-8 1.17-s 1.01-s 8.78-9
6.65 - 6 3.46-6 2.15-” 15O-6 1.14-6 9.13-’ 7.62-l 6.533’ 5.69-’ 5.033’ 4.51-’ 4.10-’ 3.766’
0.6 0.8
1.57-5 7.52-b 4.3116 2.80-’ 1.96-6 1.46-6 l.13-6 9-05 - ’ 7.46 - ’ 6.27 - ’ 5.333’ 4.588’ 3.97-l
1.67-4 8.4115 5.11-S 3.49- 5 2.58-5 2.02-S 1.65-5 1,38-5 1.18-’ 1.03-5 9.096 8.11-6 7.3t-6
1.0
1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
Atomic
Total
T = 4000°K 5.87- ” 4.56-9 2.85-” 2.36-9 1.46-9 1.o2-9 76-” 7.65- i” 5.699” 4.43- ’ ’ 3.57-11 2.96- ’ ’ ;I;:’
::
10-b
10-A
;:;;:
:;
Atomic
6.63 - l4 ;:;;:
::
;:s’;:
::
1.45-“+ 1.20- I4 1.02- I4 8.71- ”
Total
Atomic
Total
1,555’* 8.02- l3
1.24- ” 6.02- ” 3.47- ”
4.37- I6
;f:‘:: 2.599i3
;f;‘:: 1.22117
5.44 is 9.06 14
4.69- I8
8.28 - I4 T = 5000°K 1.03-8 7.06s 4.94-9 3.55-a 2.833’ 2.15-8 1.84-9 1.47-a 1.29-9 1.08-a ;:;: :: 598-I 4.933’0 ;::;‘:;
8.44-9 6.87-9 5.76-9 4.92-9 4.28-9 3.77-q 3.36-9 3.02-9
5.58- ” ;::I
4.90- i5 2.35- ” 1.35- 15 ;:;y g::
2-92- ” 2.42- l6 ;:y;:
:;
::
3.38- ”
426
R. C. MJOLSNE~~ and H. M. RUPPEL
TABLE 2. ATOMIC AND TOTALCONTRIBUTIONS TO THESPECTRALABSORPTION COEFFICIENT
F
1.0
10-z
10-4
Atomic (cm- ‘)
Total
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
1.51-4 7.14-5 4.06- 5 2.61-’ 1.82- ’ 1,34-* 1.04-5 8.27-6 6.78- 6 f1.67-~ 4.80-’ 4.11-6 3.55-e
l.29-3 6.36-4 3.78-4 2.53-4 1.83X4 1.41-4 1.13-4 9.31-5 7.85-5 6.73-s 5,85-5 5,15Y5 4.57-5
T = 6000°K 1.07-’ 4.18_’ 5.09-8 2,05-’ 2.90-’ 1.21-’ 1.87-’ 8.06-” 1.30- B 5.81-* 9.64-9 4.44-a 7.46- 9 3.54-S 5,97-9 2.91-* 4.91-9 2.45-a 4.11-9 2.09-‘3 3.50-9 l.82-8 3.00-9 1.59-e 2.59-9 1.41-13
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
8,32-4 3.91-4 2.21-4 l4-4 9.78-S 7.18-5 5.51-5 4.38-5 3.57-5 2.97- 5 2-51-* 2.14-5 1.84-5
5.31-3 2.57-3 1.50-a 9.88-4 7.04-4 5.32-’ 4.19-4 3.41-4 2.84-“ 2.4O-4 2.06-4 1.80-“ 1.5sm4
T = 7000°K 8.51-’ l.52-6 4~Ow’ 7.25- ’ 2.26-’ 4.17-l 1.45s’ 2.71-’ 1.01-’ 19-’ 7.40-S 1.42-’ 5.70-s 1.11-7 4.54-a 8.96-8 3.71-e 7.40-S 3.10-a 6.23-8 2.62- * 5.31-B 2.24- ’ 4.59-8 1.92-* 3.99- 8
Atomic
Total
lo-6
Atomic
Total
Atomic
Total
1.01-‘0
1.17-‘O 5.60- ’ ’ 3,21-” 2.09- ‘I
9.92- I4
9.93- I4
;:;;I
::
;:g:::
1.47-” 1.10-” 8,55_‘2
;:;;:
::
8.98- I5
1.74- I4 l.21-‘4 9.0(- ‘5
;:;;::: 1.76-” 1.23-l’ 9.11-12 7.06- I2 5.66- ” ;:;;I
::
;:;‘:I
::
2.48-”
;:;;I
::
;::;I
:f;:
::
::
;:;;I
::
4.61-I5 3.88-l” 3.30- I5 2.84-15 2.45- ‘5
3.09- ‘2
2.45- I5
9.65- I0
X.78_ ‘3 4.13-13 2.34-13 1.50- ” 1.04-13
;:;:I::
7.65- I4 5.89- I4
7.65- I4 5.89- I4
;:;;I
;:;;I
::
;:;;I
::
::
;:;;I::
;:;;I
::
8,30-” 6.39-” 5.09-L’
8.43-l’
4.17-I’ 3.48- ’ ’ 2.94- ’ ’ 2.51-” 2.16-l’
::
4.82- ”
;:;;I
2.54-l”
;:;;I
4.60- I5 3.87_ I5
4:;‘;:: : 4.24- ’ ’ 3.53- ’ ’ 2.99-”
::
3.20- I4 ;:;;I:: 1.99- I4
8.78- I3
;:;I
::
II