Jacobian transformed and detailed balance approximations for photon induced scattering

Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 150–157

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Jacobian transformed and detailed balance approximations for photon induced scattering B.R. Wienke a,n, K.G. Budge a, J.H. Chang a, J.A. Dahl a, A.L. Hungerford b a b

Computing and Computational Sciences Division, Los Alamos National Laboratory, Los Alamos, NM 87545, United States Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, United States

a r t i c l e in f o

abstract

Article history: Received 21 August 2009 Received in revised form 27 September 2011 Accepted 30 September 2011 Available online 10 October 2011

Photon emission and scattering are enhanced by the number of photons in the final state, and the photon transport equation reflects this in scattering–emission kernels and source terms. This is often a complication in both theoretical and numerical analyzes, requiring approximations and assumptions about background and material temperatures, incident and exiting photon energies, local thermodynamic equilibrium, plus other related aspects of photon scattering and emission. We review earlier schemes parameterizing photon scattering–emission processes, and suggest two alternative schemes. One links the product of photon and electron distributions in the final state to the product in the initial state by Jacobian transformation of kinematical variables (energy and angle), and the other links integrands of scattering kernels in a detailed balance requirement for overall (integrated) induced effects. Compton and inverse Compton differential scattering cross sections are detailed in appropriate limits, numerical integrations are performed over the induced scattering kernel, and for tabulation induced scattering terms are incorporated into effective cross sections for comparisons and numerical estimates. Relativistic electron distributions are assumed for calculations. Both Wien and Planckian distributions are contrasted for impact on induced scattering as LTE limit points. We find that both transformed and balanced approximations suggest larger induced scattering effects at high photon energies and low electron temperatures, and smaller effects in the opposite limits, compared to previous analyzes, with 10–20% increases in effective cross sections. We also note that both approximations can be simply implemented within existing transport modules or opacity processors as an additional term in the effective scattering cross section. Applications and comparisons include effective cross sections, kernel approximations, and impacts on radiative transport solutions in 1D geometry. The additional computing time for processing opacities (cross sections) within these approximations is negligible as induced terms are merely added (multipliers) to cross sections at the end of the processing cycle. Published by Elsevier Ltd.

Keywords: Induced scattering Photon detailed balance Numerical approximations Effective cross sections

1. Introduction Radiative equations with approximate scattering kernels are employed across many applications [1–16]. Although

n

Corresponding author. E-mail address: [email protected] (B.R. Wienke).

0022-4073/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.jqsrt.2011.09.018

the radiative equations are classical in origin and scope, quantum effects [9,13] impact the radiative equations through induced scattering–emission processes, important effects as the number of photons in the final state increases. These processes enter the transfer equation through the socalled induced terms. While they merely enhance photon processes by the numbers of photons in the final state, induced terms are quadratic in field intensities, rendering

B.R. Wienke et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 150–157

formal and numerical solutions of the radiative equations difficult. Approximation schemes are usually necessary to provide quick and accurate solutions [17–19]. We take a closer look at induced terms, earlier approximate representations, and underlying assumptions for photon transport applications. In particular, we examine three schemes, the mixed transport expansion approximation (MTA) proposed many years ago, an alternative, the transformed distribution approximation (JTA), and the detailed balance approximation (DBA). Contrasts are given, and asymptotic limits for electron temperatures and photon energies are obtained. 2. Kernel dynamics and thermodynamic equilibrium The radiative equation is a classical equation of transfer, based on simple physics concepts. However, there is more to a complete representation. The radiative equation is frame invariant under Lorentz and Galilean transformations. Quantum mechanics also requires that the scattering and emission sources are enhanced by the number of photons in the final state. This modification to the radiative transfer equation is easily written down, with the stimulated emission terms simple multipliers of the source and scattering integrands:   Z Z n 1 @I c2 I þ X  =I þ sa I ¼ dn dX 0 m0s I0 1þ c @t n 2hn3 Z 

dn0

Z

    c2 I0 c2 I dX0 ms I 1 þ þ S 1þ 03 3 2hn 2hn

ð1Þ

Differential scattering coefficients, ms , and total scattering coefficients, ss , are simply connected for outscatter: @ss @n0 @O0 ZZ ss ¼ ms dn0 dO0

ms ¼

ð2Þ

and similarly for the inscatter differential and total cross sections: @s0s m0s ¼ @n@O

s0s ¼

ZZ

m0s dn dO

ð3Þ

Denoting incoming electron and photon energies, e0 and o0 , and outgoing electron and photon energies, e and o, energy differential cross sections are linked by an energy conservation d-function: @ss @ss ¼ dðe0 þ o0 eoÞ @o0 @O0 @O0 @s0s @s0s ¼ dðe þ oe0 o0 Þ @o@O @O

Z

f ðp0 Þ

@s0s 02 0 p dp @n@O

ð6Þ

Below pair production thresholds, the differential scattering cross sections are the Compton and inverse Compton expressions for photon–electron scattering [7,12,14]. In most applications, the induced contributions are neglected. But we focus on approximations for numerical implementation, even though effects are small. At low energy or high temperature, effects of stimulated emission become more important, as seen in the exponential term of the Planckian distribution, that is, the equilibrium distribution of photons at temperature, T:   2hn3 1 B¼ 2 ð7Þ expðhn=kTÞ1 c Origins of the induced terms can be seen in the Bose– Einstein statistics satisfied by photons. If p represents the probability of single photon emission or scattering, and n represents the number of photons in the final state, the actual probability induced by photons in the final state, P, is

P ¼ pð1 þ nÞ

ð8Þ

For electrons, neutrons, and neutrinos, obeying Fermi–Dirac statistics, the exclusion principle [13] holds and a minus sign appears in the above, with n ¼ 0; 1, only,

P ¼ pð1nÞ

ð9Þ

For photons, the number, n, can be related to the intensity, I, via the uncertainty principle in the following way. Denoting the distribution function, f, in phase space, the number of photons at a point, (n, X,r,t), is the integral over the phase space element, D, Z ð10Þ n ¼ f ðn, X,r,tÞ dn dX dr D

The phase space element is bounded by the uncertainty principle (assuming two polarization states for the photon): 3

D ¼ dq dr ¼ q2 dq dX dr ¼

h 2

ð11Þ

with h Planck’s constant. The intensity, I, is related to the distribution function, f, I ¼ hcnf

ð12Þ

and the photon momentum, q, depends on the frequency, n, in the usual fashion: q¼

hn c

ð13Þ

Employing the above relationships, it is seen: ð4Þ

For an isotropic distribution, f, of scattering electrons, we have, Z @s ms ¼ f ðpÞ 0 s 0 p2 dp @n @O

m0s ¼

with normalization: Z Z 0 f ðpÞp2 dp ¼ f ðp0 Þp02 dp ¼ 1

151

ð5Þ

n¼

c2 4 h 3

n

ID ¼

c2 I 2hn3

so that,   c2 P ¼ p 1þ I 3 2hn

ð14Þ

ð15Þ

Additionally, with elastic scattering for fermions and bosons, where energies and frequencies do not change, the induced terms all vanish.

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The source, S, usually denotes photons arising from spontaneous atomic emission, and is treated by a local thermodynamic equilibrium (LTE) assumption [1–4,14], first introducing a corrected absorption coefficient, ka :   c2 B sa ¼ ka 1þ 2hn3 S ¼ ka B

ð16Þ

whereby, at complete thermodynamic equilibrium:

ka ¼ sa ½1expðhn=kTÞ B¼

  2hn3 1 2 expðhn=kTÞ1 c

ð17Þ

The exponential term multiplying ka measures the effective decrease in absorption due to induced emission, and thus corrected absorption coefficient for T the absolute temperature, Above, B is the Planck blackbody distribution and at equilibrium, the radiation field, I, must equal the Planck field, B. Consequently, the number of photons inscattered must then equal the number of photons outscattered, yielding a detailed balance condition for the scattering kernels:   0 0     c2 B ms B c 2 B0 ms B 1þ ¼ 1þ ð18Þ 0 hn hn 2hn3 2hn03 Further, at complete thermodynamic equilibrium:

ka ðBIÞ ¼ 0

ð19Þ

But even if local thermodynamic equilibrium is not invoked, the form of the radiative transfer equation is written using the foregoing: 1 @I þ X  =Ika ðBIÞ c @t   Z Z n c2 I ¼ dn dX 0 m0s I0 1 þ n 2hn3     Z Z 2 0 c I c2 I  dn0 dX0 ms I 1 þ þ S 1þ 2hn3 2hn03 ¼ XþC

ð20Þ

with scattering kernel, X, given by, X¼

Z

dn

Z

dX

n

n

0



m0s I0 1 þ

 Z   Z c2 I c2 I0  dn0 dX0 ms I 1 þ 3 03 2hn 2hn

ð21Þ and induced source, C,   c2 I C ¼ S 1þ 3 2hn

only, in schemes that allow the induced terms to be included as part of the effective cross sections. 3. Effective cross sections Photon scattering off an ensemble of moving electrons [11–14] is an important ingredient in transport calculations, so called Compton and inverse Compton scattering in the basic interaction picture. A relativistic Maxwell– Boltzmann distribution [10] at temperature, T, takes the normalized isotropic form:   1 f ðpÞ ¼ exp½ðp2 þ m2 Þ1=2 =kT ð23Þ m2 kTK 2 ðm=kTÞ with momentum, p, electron rest mass, m, and temperature, T, in the same energy units, and for K2 the modified Bessel function (order 2). That is to say that mc2, cp, and kT are energy quantities here, with c suppressed for notational and textual simplicity, a convention often employed in relativistic dynamics and analysis. Effective total and differential cross sections for radiation transport require folding over any electron distribution, f, in same fashion as standard in transport analysis. The effective differential cross section (inverse mean free path in transport terms) for photon–electron scattering has the form: Z @s ms ¼ 0 s 0 ¼ dðe0 þ o0 eoÞf ðpÞLðo0m , e0m , om , em Þ d3 p @o @O

ss ¼

ZZ

ms do0 dO0

ð24Þ

for incident photon energy, o, background electron energy, e, final photon energy, o0 , and final electron energy, e0 . Using four vector photon and electron invariants [12,13], that is, the energy–momentum four vectors, e0m and em for the final and initial electrons and o0m and om for the final and initial photons, we have, recalling that photons have equal energy and momentum apart from the factor c2:

o0m þ e0m ¼ om þ em em em ¼ m2 ¼ ðo0m þ e0m om Þðo0m þ e0m om Þ

ð25Þ

and with cos a0 ¼ cos a cos y þ sin a sin y cos f

ð26Þ

where a and a are the angles between incident electron and incident and final photons, y is the angle between incident and final photons, and f is the azimuthal orientation of the scattered photon. Taking the scalar product of incident electron and photon four vectors, and using four vector conservation, we also obtain, 0

ð22Þ

Above, I is generally not the Planckian blackbody function. However, the LTE assumption is widespread in radiation hydrodynamics analysis because of the vast simplification it introduces, that is, equilibrium thermodynamics reduces the analysis. The form is useful for making further approximations, outside of LTE. It is the scattering kernel, X, that introduces complexities into solution schemes of the radiative equation, iteratively or directly inverted, and is our focus in the following. Particularly, the focus will be approximations to the induced parts of the scattering kernels, that is, the ð1 þ c2 I=2hn3 Þ and ð1 þ c2 I0 =2hn03 Þ terms

e0m o0m ¼ ðom þ em o0m Þðom þ em e0m Þ eopocos a ¼ o0 oð1cos yÞ þ eo0 po0 cos a0

ð27Þ

with the same angular definitions. In the above, in the electron rest and photon rest frames, respectively,

e0m e0m ¼ em em ¼ m2 , o0m o0m ¼ om om ¼ 0

ð28Þ

B.R. Wienke et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 150–157

and, additionally, the set closes with, 0

om

em

153

4. Scattering kernel approximations

0m

¼ om e

0m

0m

0

om o ¼ oo ð1cos yÞ ¼ em e m

2

ð29Þ

The Lorentz invariant interaction term takes four vector form (59)–(61): 2 3 !2   nr 20 o0 2 4 m2 om em o0m em 5 m2 L¼  þ 1 1þ 0 m þ 2 o om em o0m em om e om em  

m2 eðep cos aÞ

 ð30Þ

or, simplifying   " nr 20 o0 2 m4 ð1cos yÞ2 L¼ 2 o ðep cos aÞ2 ðep cos a0 Þ2 2m2 ð1cos yÞ oðep cos aÞ þ  ðep cos aÞðep cos a0 Þ o0 ðep cos a0 Þ   o0 ðep cos a0 Þ m2 þ oðep cos aÞ eðep cos a0 Þ

4.1. Mixed transport approximation (MTA)

ð31Þ

with r0 the classical electron radius ð2:8  1013 cmÞ, and n the electron (spatial) number density. The d function permits one integration over do0 trivially, but remaining 3 integrations over d p and dO0 are formidable. Numerical techniques are requisite to generate the total effective cross, ss . Turning to the effective differential cross section, a number of interesting cases result for various forms of the background electron distribution, f. Integrating over the d function, the effective differential cross section presents a form that is convenient for numerical approximations, recovers standard Klein–Nishina, Thomson, and Bjorken results [11], and also extends them with temperature dependency. First, introducing the dimensionless parameters, k, k1 , and k2 ,

km ¼ e k1 m ¼ ep cos a ¼ kmðk2 1Þ1=2 m cos a k2 m ¼ ep cos a0 ¼ kmðk2 1Þ1=2 m cos a0

ð32Þ

the Compton-like expression results from the earlier four vector identity, 0

k1 mo ¼ o oð1cos yÞ þ k2 mo

0

An early scheme (MTA) is described first, with a simple extension (JTA) suggested next. The detailed balance scheme (DBA) is proposed, differences noted, and comparisons are given. In the limit of equal incident and scattered photon energies, induced scattering terms in all schemes vanish as required. Approximations in the induced part of the kernel are denoted, H and H0 , so that the inscatter minus outscatter kernel takes the form:   Z Z n c2 H X ¼ dn dX 0 m0s I0 1 þ n 2hn3   Z Z 2 0 c H  dn0 dX0 ms I 1 þ ð34Þ 2hn03

ð33Þ

The equation above permits upscatter and downscatter, unlike the pure Compton case where only photon downscatter is kinematically possible. For k1 4 k2 , upscatter occurs, while for k1 r k2 , only downscatter is possible. For k ¼ k1 ¼ k2 ¼ 1 that is, e ¼ m and p¼0, the Compton scattering law is recovered. Useful forms beyond the stationary Klein–Nishina expression have been detailed (60)–(62) and employed in both deterministic and Monte Carlo transport applications [20]. They can be employed here within the induced kernel approximations for further simplification of numerical quadratures. The numerical approach is to bootstrap induced terms into the full transport equation, updating intensities at each step. The iterative process continues until the full transport equation is converged after the last bootstrap of induced terms.

This scheme was proposed by Sampson [17] expanding the radiative flux in low order. LTE was assumed, so that the photon intensities are nearly Planckian, according to, 0

0

H0 ¼ B0 l X  =B0 þ lX  =l X  =B0 0

H ¼ BlX  =B þ l X  =lX  =B

ð35Þ

with mean free paths

l0 ¼

l¼

1

k0a þ s0s 1

ka þ ss

ð36Þ

Electron distributions in the final and initial state were assumed to be linked, f ðp0 Þ ¼ f ðpÞ exp½hðnn0 Þ=kT

ð37Þ

which is a restrictive statement about the scattering process in general, really only holding for small values of nn0 , assuming a Wien photon distribution. The relationship connecting electron–photon phase space number densities is usually posed: Z Z 3 3 3 3 f ðn0 Þf ðp0 Þ d n0 d p0 ¼ f ðnÞf ðpÞ d n d p ð38Þ with f ðnÞ and f ðn0 Þ the Planckian blackbody distributions, within which case the kinematical constraints are retained:  3  3  @ ðnÞ @ ðpÞ f ðnÞf ðpÞ ¼ Jf ðnÞf ðpÞ ð39Þ f ðn0 Þf ðp0 Þ ¼ 3 0 @ ðn Þ @3 ðp0 Þ with Jacobian transformation, J, of conserved energy– momentum linking the distributions. In the original MTA, J¼1, as a simplification. In the following scheme, the kinematical properties of J will be maintained in calculations. 4.2. Jacobian transformed approximation (JTA) Assuming LTE provides a simplification to the induced scattering kernel: H 0 ¼ B0 H¼B

ð40Þ

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B.R. Wienke et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 150–157

but also serves as a convenient numerical calibration point for all schemes. Kernels with effective cross sections integrated over Planckian, B0 or B, serve as physical limit points. Away from low energy and high temperature limits, the Wien distribution W, can be used: H0 ¼ W 0

Rearranging terms, we can rewrite  0 n m0s I0 ¼ ms I x

n

x¼

1 þ c2 H0 =2hn03 1þ c2 H=2hn3

or H¼W

ð41Þ

as behavior of the Wien distribution, W, parallels the Planckian distribution, B. This is seen by noting the maxima in both as a function of energy:   @B ¼0 @n hn=kT ¼ 3   @W ¼0 @n hn=kT ¼ 3½1expðhn=kTÞ

ð42Þ

are roughly equal at high energies and low temperatures. In some cases, kernel integrals over W can be performed analytically. Employing the full Jacobian, J, linking initial and final electron and photon distributions, the complete transformation is applied to the product distributions: f ðpÞHðnÞ ¼ Jf ðp0 ÞH0 ðn0 Þ

ð43Þ

negating direct integration of many terms in the MTA effective induced cross section. This extends the MTA, but no other assumptions are made. Test functions, H and H0 , are normalized over the range of photon energy. In particular, the product Jacobian is written J¼

@ðp,cos a, ZÞ @ðn,cos y, zÞ @ðp0 ,cos a0 , Z0 Þ @ðn0 ,cos y0 , z0 Þ

ð44Þ

with variables defined previously, and angles linked:

ZZ0 ¼ f z0 z ¼ f0 cos a0 ¼ cos a cos y þ sin asin ycos f 0

0

0

cos a ¼ cos a0 cos y þ sin a0 sin y cos f

ð45Þ

Spherically symmetric scattering reduces the complexity of the 3  3 electron and photon Jacobian determinants.

ms I ¼ m0s I0

n

n0

ð47Þ

x0

1 þc2 H=2hn3 1 þc2 H0 =2hn03

x0 ¼

where I and I0 are field intensities, not necessarily Planckian LTE distributions, B and B0 . The relationship states incoming and outgoing photon intensities differ by a factor of x, a convenient representation for approximations. In general, x is itself a complex function of intensities, but serves as a simple starting point. Within such framework, we can stairstep the level of thermodynamic equilibrium, LTE to non-LTE. 5. Results and conclusions First, numerical comparisons of effective cross sections and approximate kernels are given. Application to radiation transport and comparisons follows. 5.1. Cross sections and kernels For simplicity, only the outscatter effective kernels are computed under the MTA, JTA, and DBA at LTE. The full Compton cross section is employed for a relativistic Maxwellian electron scattering distribution. The nested 4D integrations are performed numerically in all cases. The effective Compton kernel is standard in radiative transport applications, providing a baseline for Compton cross sections integrated over background Maxwellian electron distributions with induced terms included in the scattering kernel, that is, recalling the total Compton cross section, SC , ZZ SC ¼ ss ¼ do0 dX0 ms ð49Þ with

ms ¼

@ss ¼ @o0 @O0

Z

dðe0 þ o0 eoÞf ðpÞLðo0m , e0m , om , em Þ d3 p ð50Þ

4.3. Detailed balance approximation (DBA) Electron phase space balance yields additional constraints. As before, photon scattering dynamics are upscatter and downscatter for high and low temperature electron backgrounds. The detailed balance condition for the induced scattering kernel is the starting point for this approximation scheme. Both LTE and non-LTE assumptions can be made after the inscattering and outscattering kernels are linked. Accordingly, we formally equate the integrands of the scattering kernel:   0 0     c2 H ms I c2 H0 ms I 1þ ¼ 1þ ð46Þ 0 3 03 hn hn 2hn 2hn

ð48Þ

as given in the previous section. We also take the electron number density, n¼1. To compare with earlier analyzes, results are normalized to the Thomson cross section, ST ,

ST ¼

8p 2 r ¼ 0:6568 barns 3 0

ð51Þ

denoting the ratio, z,

z¼

S ST

ð52Þ

as well as the Compton cross section, SC , by the ratio, Z,

Z¼

S SC

ð53Þ

B.R. Wienke et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 150–157

for values of the effective cross section, S, under approximations of the induced term. Induced terms are included in the scattering kernel by writing   Z Z c2 S ¼ dn0 dO0 ms 1 þ H0 ð54Þ 03 2hn with H0 given in the MTA, JTA, and DBA schemes detailed. Table 1 lists Compton total cross sections, SC , for various temperatures and photon energies [11,12]. Because of the additive effects of photons in the final state, we have, across all energies and temperatures (see Tables 2 and 3):

Table 3 Wien MTA, JTA, DBA induced scattering ratios ðzÞ.

hn=kT 0.001

0.025

2.000

S Z SC but, against a moving electron background (see Table 1)

SC r ST

8.000

20.000

with, in general, both the effective induced and Compton cross sections decreasing with temperature. Both effects are well known, reflected in Tables 1–3. Said another way,

and

z

with both ratios and cross sections decreasing with temperature. Incoming photon energies, o, and electron temperatures, T, are in keV, and cross sections, SC , are in barns. Table 1 Compton Maxwellian electron cross sections ðSC Þ. 1

10

100

500

1000

5000

0.6624 0.6401 0.4921 0.2886 0.2107 0.0826

0.6623 0.6393 0.4879 0.2846 0.2074 0.0804

0.6610 0.6281 0.4482 0.2499 0.1789 0.0673

0.6528 0.5771 0.3367 0.1673 0.1123 0.0382

0.6438 0.5280 0.2683 0.1201 0.0794 0.0252

0.5839 0.3581 0.1241 0.0442 0.0268 0.0075

oðkeVÞ 1 10 100 500 1000 5000

2

20

50

125

Wien Wien Wien Wien Wien Wien Wien Wien Wien Wien Wien Wien Wien Wien Wien

1.0079 1.5440 1.5367 1.0047 1.4233 1.4218 0.9829 1.0689 1.0687 0.9150 0.9537 0.9535 0.8052 0.8803 0.8804

1.0860 1.0632 1.0592 1.0472 1.0298 1.0321 0.8405 0.8772 0.8770 0.5231 0.6518 0.6517 0.3321 0.4794 0.4795

1.2462 1.0443 1.0376 1.1132 0.9761 0.9779 0.6604 0.7254 0.7254 0.3276 0.4634 0.4622 0.1991 0.3322 0.3322

1.8281 1.0718 1.0689 1.2094 0.9010 0.9026 0.4263 0.5240 0.5240 0.1844 0.2981 0.2981 0.1064 0.2016 0.2016

MTA JTA DBA MTA JTA DBA MTA JTA DBA MTA JTA DBA MTA JTA DBA

5.1.1. MTA The mixed expansion takes the form:

Z1

T (keV)

T (keV)

Specifics of the MTA, JTA, and DBA used in numerical comparisons follow.

Z Z1 Z

155

0

0

H0 ¼ B0 l X  =B0 þ lX  =l X  =B0 where the last terms drops out of the integral, and   @B0 1 hn0 1 B0 rT 0 rB0 ¼ 0 rT 0 ¼ 0 @T T kT 0 1expðhn0 =kT 0 Þ

ð55Þ

ð56Þ

according to the original analysis of Sampson [17] using Rosseland mean opacities (not reproduced here). Again, T ¼ T 0 . Some overlapping results of the MTA are quoted, however, in Table 3, and contrasted with predictions of the JTA and DBA. 5.1.2. JTA As a calibration point, we simply consider two cases: H 0 ¼ B0 H0 ¼ W 0

ð57Þ 0

0

Though both functions, B and W , have similar limits, the induced terms, c2 B0 =2hn03 and c2 W 0 =2hn03 , differ in the low energy and high temperature limits:

Table 2 Induced scattering ratios in the JTA and DBA schemes ðZÞ. T(keV)

1

10

100

1000

Planck JTA Planck DBA Wien JTA Wien DBA Planck JTA Planck DBA Wien JTA Wien DBA Planck JTA Planck DBA Wien JTA Wien DBA Planck JTA Planck DBA Wien JTA Wien DBA

1.4166 1.4168 1.4755 1.4774 1.0010 1.0010 1.0007 1.0006 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.2558 1.2822 1.0921 1.1095 1.0159 1.0160 1.0377 1.0377 1.0001 1.0003 1.0001 1.0001 1.0000 1.0000 1.0000 1.0000

1.1268 1.3242 1.0099 1.1099 1.0128 1.0252 1.0084 1.0408 1.0002 1.0009 1.0032 1.0014 1.0001 1.0000 1.0000 1.0002

1.0258 1.6783 1.0010 1.3732 1.0029 1.1147 1.0110 1.2900 1.0053 1.0031 1.0061 1.0017 1.0001 1.0004 1.0002 1.0003

c2 B0 1 ¼ expðhn0 =kTÞ1 2hn03

oðkeVÞ 1

10

100

1000

c2 W 0 ¼ expðhn0 =kTÞ 2hn03

ð58Þ

apart from any normalization factors. In those limits, the former ðB0 Þ is unbounded, while the latter ðW 0 Þ remains bounded. The effective cross section will then be larger in the Planckian case than in the Wien case. Exact transformation of the distributions was given for the JTA: f ðpÞHðnÞ ¼ Jf ðp0 ÞH0 ðn0 Þ whereby we take, H0 ðn0 Þ ¼ J

f ðpÞ HðnÞ f ðp0 Þ

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B.R. Wienke et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 150–157

with test functions, H and H0 , Wien or Planckian for numerical integrations, and the product Jacobian, J, given previously, J¼

@ðp,cos a, ZÞ @ðn,cos y, zÞ @ðp0 ,cos a0 , Z0 Þ @ðn0 ,cos y0 , z0 Þ

ð59Þ

As will be seen (Table 3), the finer granularity of the JTA versus the MTA affects induced cross sections in the high energy and low temperature regimes. 5.1.3. DBA Recall, requiring equality of the approximate scattering kernel integrands, that is, across both sides with H and H0 replacing I and I0 in the induced terms:   0 0     c2 H ms I c2 H0 ms I 1þ ¼ 1 þ hn hn0 2hn3 2hn03

m0s I0 ¼ ms I

 0

n x n

ð60Þ

which in all LTE cases requires

x¼

1þ c2 B0 =2hn03 1expðhn=kTÞ ¼ 1expðhn0 =kTÞ 1þ c2 B=2hn3

ð61Þ

as seen. This is a fairly complex requirement and constraint involving differential cross sections. To render approximation, we make a kernels simplification by using total cross sections, as the kernels are pointwise and complex. Extending the test functions, H and H0 , over the full scattering kernel, that is, writing   0 0     c2 H ms H c2 H0 ms H 1þ ¼ 1þ 0 3 03 hn hn 2hn 2hn 0 0 sH

m

 0

n ¼ ms H x n

ð62Þ

and taking H¼ B and H0 ¼ B0 , the DBA further simplifies employing the stationary total cross sections, k0 and k, for Klein–Nishina scattering: h k in0  H0 ¼ B 0 x ð63Þ

k

n

normalized over energy, temperature, and frequency regions of interest. The DBA used here represents a detailed balance correction to the JTA used above. To compare approximations, Table 2 contrasts induced effects in the JTA and MTA schemes for both Planckian and Wien test functions, H0 . It is seen that the JTA is always larger than the DBA, particularly as the ratio of photon energy over electron energy decreases for low energies and temperatures (where induced effects play more strongly). This is true in both the Planckian and Wien cases. What is tabulated is the ratio, Z, of induced cross section, S, divided by Compton cross section, SC :

Z¼

S SC

ð66Þ

Z Z 1:0 with corresponding Compton cross sections, SC , listed in Table 1. Induced effects diminish as Z-1:0. Units in Table 2 are the same as Table 1, and Z is the dimensionless ratio of induced cross section, S, over Compton cross section, SC . Differences between the Wien and Planckian entries in Table 2 mainly result from the differences in the low energy behavior of both distributions. In the Planckian case, hn

lim B0 ¼

0 =kT-0

2kT n02 c2

ð67Þ

while, in the Wien case, lim W 0 ¼

hn0 =kT-0

2hn03 c2 ð1 þ hn0 =kTÞ

ð68Þ

The induced contributions for small values of the ratio, hn0 =kT, will then be larger in the Planckian case. Table 2 exhibits some other trends which can be summarized: 1. induced scattering effects in both the JTA and DBA vanish as the ratio of incident photon energy over temperature approaches 10, that is,

Z-1 as

hn -10 kT

that is,

b0 ¼

k0 ¼

n0

2. induced scattering effects consistently increase within the DBA as the ratio of incident photon energy over temperature decreases, for instance:

m " # 8p 2 1 b02 r0 þ1:2 0 0 3 1 þ2b 1 þ 2b

ð64Þ

and,

b¼

n m

" # 8p 2 1 b2 r k¼ þ1:2 3 0 1 þ 2b 1þ 2b

ð65Þ

where one reasonably expects the ratios of differential and total cross sections to scale in the same proportions:

ms k  m0s k0 with differences due to the moving electron background small. As before in JTA cases, the test function can be

Z-1:67 as

hn -0:001 kT

but not within the JTA for both Planck and Wien cases; 3. over the range of energies and temperatures in Table 2, the Planckian JTA and DBA estimates of Z remain closer numerically than in the Wien case. Table 3 contrasts estimates of z for the Wien JTA and DBA with earlier reported [17] ratios for the Wien MTA. The ratio of induced cross section divided by Thomson cross section, z, is employed instead of Z because the effective Compton cross section, SC , is not given in the Wien MTA. Except for very high temperatures at low photon energy, both the JTA and DBA approximations yield higher induced scattering effects than the MTA, with differences increasing as the ratio of incident photon energy over

B.R. Wienke et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 113 (2012) 150–157

157

opacities are 0.6624 cm2/g and 1.4166  0.6624 cm2/g in corresponding cases without and with induced scattering. Discrete ordinates calculations are performed in S4 angular quadratures using the LANL code SERRANO. In this simple case, decrease in radiation temperature also tracks roughly with Z as a maximum [1,14,17]. As background and incident radiation temperatures increase, induced scattering effects diminish accordingly (Table 2). With material coupling, where absorption and emission play dominant roles, the induced scattering effects are even smaller. A complete comparison of coupled material-induced scattering effects is planned for another communication focusing on transport applications in thick and thin materials. References Fig. 1. Radiation slab transport with and without induced scattering.

temperature, hn=kT, increases. In all cases, induced effects are enhanced at low photon energies and high background temperatures. The JTA and MTA are closest in formulation, with the JTA containing the MTA as a subset if the Jacobian, J, is unity in the Wien case, but the effects of transformation of product initial distributions to final product distributions are not negligible as seen in Table 3. The JTA and DBA, on the other hand, track more closely over the ranges in Table 3. Both the JTA and DBA suggest higher induced scattering cross sections in the 10–20% range compared to the MTA. 5.2. Radiative transfer in 1D slabs As application to radiative transfer in a slab, with and without induced scattering, consider an isotropic distribution of photons at 1.0 keV incident on a nominal slab of unit density, 1 g/cm3, at 1.0 keV. Only Compton scattering in the gray opacity approximation is depicted in Fig. 1. The slab is 100 cm and the radiation temperature is shown after 10 ns. We take Z ¼ 1:4166 in both the Planck JTA and DBA schemes as the largest case for comparison. Electron

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