- Email: [email protected]
Enhanced photon ionization cross-section of shock-heated materials
High Energy Density Physics 32 (2019) 14–17
Contents lists available at ScienceDirect
High Energy Density Physics journal homepage: www.elsevier.com/locate/hedp
Short Communication
Enhanced photon ionization cross-section of shock-heated materials
T
Carlos A. Iglesias Lawrence Livermore National Laboratories, P.O. Box 808, Livermore, CA 94550, USA
ARTICLE INFO
ABSTRACT
Keywords: Photon ionization Ion-sphere models
A recent paper [1] reported an enhancement of the boron K-shell photon ionization cross-section per electron at high photon energies along the principal Hugoniot. It is shown that the enhancement is due to known properties of isolated ions rather than subtle plasma effects. In addition, approximations and corrections to the photon absorption cross-section are examined.
1. Introduction In a recent paper the average atom (AA) formalism was applied to shock-heated boron [1]. The results, relevant to ongoing high-pressure experiments [2], showed an enhanced absorption cross-section per electron for 9 keV photons along the principal Hugoniot. Since the results come from the AA model, it is natural to surmise that subtle plasma effects are responsible for the enhancement [1]. In that case, opacity models that exclude plasma effects in the cross-section, for example models relying on isolated ion calculations [3,4], would not capture the enhancement. Some time ago it was shown that the K-shell photon ionization cross-section is unchanged, except for the ionization energy threshold, when removing bound electrons from outer shells (L-shell and above) along an isonuclear sequence [5,6]. Removing one K-shell electron, however, yields an enhanced photon ionization cross-section per electron. The results were explained in terms of Slater inner and outer screening concepts that only produce a significant change to the 1 s electron wavefunctions when the first K-shell electron is removed. It is emphasized that those calculations were performed for isolated ions using Hartree-Slater wavefunctions. The present goal is to show that the same isolated ion physics, rather than subtle plasma effects, explains the shock-heated boron cross-sections. In addition, corrections and approximations to the photon absorption relevant to the high-pressure experiments are examined. 2. Photon ionization The nonrelativistic photon ionization cross-section per electron of a bound s-wave electron with principal quantum number n is given by [7] ns (
)=
4 2ao2 3
ns P (1) p
2
(2.1)
where α is the fine structure constant and ℏω is the photon energy (Rydberg energy units are assumed throughout). The reduced matrix element is defined by
n
P (t )
= ( 1)
(2 + 1)(2
+ 1)
t 0 0 0
drPn (r ) r tP (r ) o
(2.2) where r 1Pi (r ) is the radial wavefunction for the ith level with energy ɛi and radii are in units of ao, the Bohr radius. The continuum level has normalization
P (r
)=
1 q
cos qr
( + 1) 2
+
(2.3)
with energy
=
+
ns
= q2
(2.4)
and ϕ the phase shift. 2.1. Isolated ions The 1 s photon ionization cross-sections were computed for the B2+, B , and B4+ ions (respectively electronic configurations 1s22s, 1s2, and 1s) over the range of photon energies 350–9000 eV. The wavefunctions come from numerical solutions to the Schrodinger equation using parametric potentials fitted to experimental energies for isolated ions [8]. The ratio of the cross sections for the B2+ and B3+ ions is unity within 1% over the entire spectral range. The ratio of the cross-sections per electron for the B4+ and B3+ ions is plotted in Fig. 1 showing agreement (∼18%) with the enhancement reported by [1] at = 9 keV . Note that the results in Fig. 7 of [1] show the increased cross-section per electron occurring simultaneously with the start of the K-shell ionization consistent with the isolated ion physics result. 3+
E-mail address: [email protected]. https://doi.org/10.1016/j.hedp.2019.04.003 Received 19 March 2019; Received in revised form 9 April 2019; Accepted 9 April 2019 Available online 10 April 2019 1574-1818/ © 2019 Elsevier B.V. All rights reserved.
High Energy Density Physics 32 (2019) 14–17
C.A. Iglesias
line applies for a hydrogenic system with nuclear charge Z. Unfortunately, comparisons with the exact results for hydrogenic boron [10] reveal that the Born approximation overestimates the K-shell = 9 keV . cross-section by ∼64% at 2.3. Higher energy bound levels Although the Born approximation is not very accurate, it can still provide estimates for scaling purposes. For example, at high photon energies the 1 s and 2 s hydrogenic cross-sections require continuum wavefunctions with energies differing by O(Z2/q2) due to the different bound electron energies. The ratio of the cross-sections per electron is estimated using
2s P (1) p
Born
2Zq (2q2 Z 2) (4q2 + Z 2 ) 4
= 512qZ 2
(2.3.1)
together with Eq. (2.2.1) to get Fig. 1. Ratio of K-shell photon ionization cross-section per electron as a function of photon energy.
2s 1s
E2 E1
q
drr 2P1s (r ) j1 (kr )
=
2
ao
2
2
ao
2
16Z 2q (Z 2
+
q 2 )3
Born
1s P (2) d 1s P (1) p
5 = 36
o
Zq
1 Z2 1+O 2 8 q
=
(2.3.2)
The dipole approximation is valid when the photon wavelength is much larger than the size of the ion becoming less accurate with increasing photon energies. An estimate for the ratio of hydrogenic Kshell quadrupole (E2) to dipole (E1) [7] using the Born cross-sections for hydrogenic ions yields
It is tempting to use the Born approximation to obtain the necessary high-energy cross-sections. At high energies, the continuum wavefunction is approximately given by a plane wave and the radial dipole integral for the K-shell electron is approximated by
=
2
2.4. Higher multipoles
2.2. Born approximation
Born
Born
Therefore, for heavier elements than boron where the 2 s level may still be bound along the principal Hugoniot, it is expected to increase the cross-section per electron by ∼10%. It is emphasized that the Lshell photon ionization per electron is enhanced with the removal of an L-shell electron with increasing ionization [5,6] and cold, isolated absorption data does not capture the effect.
The reason for the enhancement in Fig. 1 is provided in Fig. 2 where P1s(r) for the same three ions are plotted. The figure shows the wavefunctions for B2+ and B3+ to be indistinguishable while that for B4+ displays a collapse as reported in earlier work [5,6]. On the other hand, the continuum wavefunctions are almost identical for the three ion stages at high energies. Thus, due to the oscillations of the continuum wavefunctions, the overlap integrals are determined at small radii where the B4+ ion has a larger 1 s electron probability density.
1s P (1) p
2s P (1) p 1s P (1) p
q q2 + Z 2
Born
2
Born 2
= 0.014
(2.4.1)
where λ is the photon wavelength,
(2.2.1)
where jℓ(x) is the spherical Bessel function of order ℓ [9] and the second
1s P (2) d
Born
=
q
drr 3P1s (r ) j2 (kr ) o
=
Zq
96Z 2q2 (Z 2 + q2) 4
(2.4.2)
and the numerical value is for boron and a 9 keV photon. 3. Test of AA model Johnson and Nilsen [1] correctly noted that the boron photon ionization cross-section along the Hugoniot is not given by the product of the K-shell occupation times the cold, solid density result [11]. The present interpretation suggests the following expression for the boron total photon absorption, K(
) = cH
H(
) + 2cHe
He (
)
(3.1)
where σH and σHe are the isolated K-shell cross-sections per electron for the ground configurations of hydrogenic and helium-like ions, respectively, cH and cHe are respectively the fractional number of ions with a one and two K-shell electrons, and the factor of 2 accounts for the two K-shell electrons in the He-like species. The result for σHe can be obtained from cold data [11]. The hydrogenic isolated ion results [10] provides σH.
Fig. 2. Radial wavefunctions for the K-shell electron in several charge states of boron as a function of radius. 15
High Energy Density Physics 32 (2019) 14–17
C.A. Iglesias
contribution from the He-like ion to the total cross-section at the highest temperature. On the other hand, at the higher temperatures the AA from [1] yields the hydrogenic cross section (within 0.4% of the exact formula [10]) and applies it to the total K-shell occupation. Thus, the AA approach does not capture the He-like cross-section, which is smaller by 18% per electron. The consequence is a small over estimate of the cross-section. 4. Conclusion It was shown that the enhancement of the K-shell photon ionization cross-section per electron in the AA treatment of shock-heated boron [1] follows from known isolated ion phenomena rather subtle plasma effects. The role of the plasma is simply to produce higher ionization with increasing temperature. Consequently, opacity models that account for the changing bound level populations with plasma conditions will describe the enhanced cross-section per electron even if excluding plasma effects on the photon ionization at the high energies of interest to the high-pressure experiments [2]. A simple approximation was proposed to test the average atom calculations for boron along the principal Hugoniot. The test indicates that the interchange of the averaging procedure inherent in the AA atom approach leads to errors of <2%. Clearly the ion concentrations from the constrained Saha–Boltzmann model are not unique since they depend on the choice for the assumed continuum lowering model (see Appendix). Nevertheless, the discrepancy between the two cross-section approximations is due to the explicit treatment of the He-like ion in the discrete charge state approximation absent in the AA model. That is, the precise He-like concentration may be uncertain, but it is expected to be significant except at very high temperatures. Consequently, the numerical details of the comparisons are not definitive but should be representative. Of course, there remains the open question of whether AA models accurately reproduce the K-shell occupations at extreme matter conditions. The high-pressure experiments require high precision material density in order to determine an accurate equation of state. Therefore, corrections and approximations to the photon ionization were considered at the photon energy of interest. For example, the Born approximation was found insufficient to compute the K-shell photon absorption. The presence of bound L-shell electrons in samples containing heavier elements enhance the cross-section by ∼10% per electron. In addition, the L-shell electrons require explicit treatment of the various ion charge states present in the plasma. Finally, higher multipole contributions were found to increase the cross-section by a 1–2%.
Fig. 3. Relative difference between photon ionization cross-sections from the AA and detail ion charge formula as a function of temperature (left axes). Also, the contribution fraction from individual charge states to the total cross-section in Eq. (3.1) (right axes).
The AA formalism also provides an approximation for the crosssection. The AA model, however, interchanges the correct procedure of averaging the cross-sections of physically realizable ion charge states with integer electron occupations (as in Eq. (3.1)) by the cross-section in the average potential generated by average non-integer subshell occupations. To test the AA against Eq. (3.1) requires a calculation of the individual ion concentrations not available from an AA approach. For the test, the concentrations are obtained from a Saha–Boltzmann model (see Appendix) constrained to reproduce the AA K-shell occupations from [1]. The results from Eq. (3.1) using the concentrations from the constrained Saha–Boltzmann model, which reproduced the AA total K-shell occupation within 1 part in 106, agree to better than 2% with the AA calculations. The relative difference,
Diff% =
AA (
) K(
K(
)
)
× 100%
(3.2)
where σAA is the AA result from [1], is displayed in Fig. 3. The comparison uses the nonrelativistic exact nonrelativistic formula [10] for hydrogenic boron in Eq. (3.1). For the He-like system the cross-section in Eq. (3.1) is taken from the AA calculations at the lowest temperature rather than the cold, solid density data for the He-like system [11] mitigating possible small numerical uncertainties in the AA cross-sections in [1]. The remaining discrepancies, less than 2%, are due to the reversal of the averaging procedure by the AA formalism. These differences are most easily explained at the highest temperature. The fractional contribution from the individual ions to the total cross-section in Eq. (3.1) are displayed in Fig. 3. The figure shows ∼4%
Acknowledgments Thanks are due to Daniel Aberg for valuable discussions, Walter J. Johnson for comments on the manuscript, and Joseph Nilsen for the AA results. The work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
Appendix. Constrained Saha–Boltzmann model An ionization balance model is necessary to apply Eq. (3.1). To test the Average Atom (AA) model interchange of the averaging procedures, a Saha–Boltzmann model is used but constrained to reproduce the AA K-shell occupation from [1]. Along the Hugoniot, the AA calculations only has bound K-shell levels and all other subshells are delocalized. Thus, the discrete charge state plasma consists of ions with one and two bound K-shell electrons, bare nuclei, and free electrons. The ratio of two adjacent charge state ground configuration concentrations in thermodynamic equilibrium is given by [12]
cj cj + 1
=
ne 2 2 2 mT
3/2
gj gj + 1
e
j /T
(A.1)
where gj is the degeneracy of the ground configuration for species j, χj > 0 is the energy required to ionize an electron in the K-shell of species j and 16
High Energy Density Physics 32 (2019) 14–17
C.A. Iglesias
create an ion of species j + 1, m is the electron mass, and T is the plasma temperature in energy units. The average free electron number density includes all electrons not in bound states, (A.2)
ne = Z n i where ni is the total ion number density,
cj Z nj
Zn =
(A.3)
j
with Zje the net charge of ion species j and e the fundamental electric charge The binding energy includes plasma effects and assumes the form, j
=
o j
(Zj + 1)
(A.4)
with jo > 0 the isolated ion binding energy for species j and Ω a dimensionless parameter to be determined. There are two equations for the three ion charge states in the plasma represented in Eq. (A.1). Particle conservation requires
cj = 1
(A.5)
j
and provides a third equation. Finally, the constrain to match the AA K-shell occupation provides a fourth equation,
cH + 2cHe = n1s
(A.6)
AA
where cH and cHe are respectively the fractional number of ions with a one and two K-shell electrons and 〈n1s〉AA is the AA occupation of the K-shell. The four unknowns, the three ion concentrations and Ω, together with the four equations are solved iteratively given ne and T where the former is obtained from the AA calculations.
Press, Berkeley, 1981. [8] F.J. Rogers, B.G. Wilson, C.A. Iglesias, Phys. Rev. A38 (1988) 5007. [9] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions 55 NBS, Applied Mathematics Series, 1972. [10] H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Two-Electrons Atoms, Plenum Publishing Co., New York, 1977. [11] R.B.I. Henke, E.M. Gullikson, J.C. Davis, At. Data Nucl. Data Tables 54 (1993) 181 http://henke.lbl.gov/optical_constants. [12] D. Mihalas, Stellar Atmospheres, W.H. Freeman and Co., San Francisco, 1978.
References [1] [2] [3] [4] [5] [6] [7]
W.R. Johnson, J. Nilsen, HEDP 31 (2019) 83. D. Swift, A. Kritcher, J. Hawreliak, et al., Rev. Sci. Instrum. 89 (2018) 053505. N.R. Badnell, M.A. Bautista, K. Butler, F. Delahaye, et al., MNRAS 360 (2005) 458. J. Colgan, D.P. Kilcrease, N.H. Magee, M.E. Sherril, et al., ApJ 817 (2016) 116. D.W. Missavage, S.T. Manson, G.R. Daum, Phys. Rev. A15 (1977) 1001. R.F. Reilman, S.T. Manson, Phys. Rev. A18 (1978) 2124. R.D. Cowan, The Theory of Atomic Structure and Spectra, University of California
17
Contents lists available at ScienceDirect
High Energy Density Physics journal homepage: www.elsevier.com/locate/hedp
Short Communication
Enhanced photon ionization cross-section of shock-heated materials
T
Carlos A. Iglesias Lawrence Livermore National Laboratories, P.O. Box 808, Livermore, CA 94550, USA
ARTICLE INFO
ABSTRACT
Keywords: Photon ionization Ion-sphere models
A recent paper [1] reported an enhancement of the boron K-shell photon ionization cross-section per electron at high photon energies along the principal Hugoniot. It is shown that the enhancement is due to known properties of isolated ions rather than subtle plasma effects. In addition, approximations and corrections to the photon absorption cross-section are examined.
1. Introduction In a recent paper the average atom (AA) formalism was applied to shock-heated boron [1]. The results, relevant to ongoing high-pressure experiments [2], showed an enhanced absorption cross-section per electron for 9 keV photons along the principal Hugoniot. Since the results come from the AA model, it is natural to surmise that subtle plasma effects are responsible for the enhancement [1]. In that case, opacity models that exclude plasma effects in the cross-section, for example models relying on isolated ion calculations [3,4], would not capture the enhancement. Some time ago it was shown that the K-shell photon ionization cross-section is unchanged, except for the ionization energy threshold, when removing bound electrons from outer shells (L-shell and above) along an isonuclear sequence [5,6]. Removing one K-shell electron, however, yields an enhanced photon ionization cross-section per electron. The results were explained in terms of Slater inner and outer screening concepts that only produce a significant change to the 1 s electron wavefunctions when the first K-shell electron is removed. It is emphasized that those calculations were performed for isolated ions using Hartree-Slater wavefunctions. The present goal is to show that the same isolated ion physics, rather than subtle plasma effects, explains the shock-heated boron cross-sections. In addition, corrections and approximations to the photon absorption relevant to the high-pressure experiments are examined. 2. Photon ionization The nonrelativistic photon ionization cross-section per electron of a bound s-wave electron with principal quantum number n is given by [7] ns (
)=
4 2ao2 3
ns P (1) p
2
(2.1)
where α is the fine structure constant and ℏω is the photon energy (Rydberg energy units are assumed throughout). The reduced matrix element is defined by
n
P (t )
= ( 1)
(2 + 1)(2
+ 1)
t 0 0 0
drPn (r ) r tP (r ) o
(2.2) where r 1Pi (r ) is the radial wavefunction for the ith level with energy ɛi and radii are in units of ao, the Bohr radius. The continuum level has normalization
P (r
)=
1 q
cos qr
( + 1) 2
+
(2.3)
with energy
=
+
ns
= q2
(2.4)
and ϕ the phase shift. 2.1. Isolated ions The 1 s photon ionization cross-sections were computed for the B2+, B , and B4+ ions (respectively electronic configurations 1s22s, 1s2, and 1s) over the range of photon energies 350–9000 eV. The wavefunctions come from numerical solutions to the Schrodinger equation using parametric potentials fitted to experimental energies for isolated ions [8]. The ratio of the cross sections for the B2+ and B3+ ions is unity within 1% over the entire spectral range. The ratio of the cross-sections per electron for the B4+ and B3+ ions is plotted in Fig. 1 showing agreement (∼18%) with the enhancement reported by [1] at = 9 keV . Note that the results in Fig. 7 of [1] show the increased cross-section per electron occurring simultaneously with the start of the K-shell ionization consistent with the isolated ion physics result. 3+
E-mail address: [email protected]. https://doi.org/10.1016/j.hedp.2019.04.003 Received 19 March 2019; Received in revised form 9 April 2019; Accepted 9 April 2019 Available online 10 April 2019 1574-1818/ © 2019 Elsevier B.V. All rights reserved.
High Energy Density Physics 32 (2019) 14–17
C.A. Iglesias
line applies for a hydrogenic system with nuclear charge Z. Unfortunately, comparisons with the exact results for hydrogenic boron [10] reveal that the Born approximation overestimates the K-shell = 9 keV . cross-section by ∼64% at 2.3. Higher energy bound levels Although the Born approximation is not very accurate, it can still provide estimates for scaling purposes. For example, at high photon energies the 1 s and 2 s hydrogenic cross-sections require continuum wavefunctions with energies differing by O(Z2/q2) due to the different bound electron energies. The ratio of the cross-sections per electron is estimated using
2s P (1) p
Born
2Zq (2q2 Z 2) (4q2 + Z 2 ) 4
= 512qZ 2
(2.3.1)
together with Eq. (2.2.1) to get Fig. 1. Ratio of K-shell photon ionization cross-section per electron as a function of photon energy.
2s 1s
E2 E1
q
drr 2P1s (r ) j1 (kr )
=
2
ao
2
2
ao
2
16Z 2q (Z 2
+
q 2 )3
Born
1s P (2) d 1s P (1) p
5 = 36
o
Zq
1 Z2 1+O 2 8 q
=
(2.3.2)
The dipole approximation is valid when the photon wavelength is much larger than the size of the ion becoming less accurate with increasing photon energies. An estimate for the ratio of hydrogenic Kshell quadrupole (E2) to dipole (E1) [7] using the Born cross-sections for hydrogenic ions yields
It is tempting to use the Born approximation to obtain the necessary high-energy cross-sections. At high energies, the continuum wavefunction is approximately given by a plane wave and the radial dipole integral for the K-shell electron is approximated by
=
2
2.4. Higher multipoles
2.2. Born approximation
Born
Born
Therefore, for heavier elements than boron where the 2 s level may still be bound along the principal Hugoniot, it is expected to increase the cross-section per electron by ∼10%. It is emphasized that the Lshell photon ionization per electron is enhanced with the removal of an L-shell electron with increasing ionization [5,6] and cold, isolated absorption data does not capture the effect.
The reason for the enhancement in Fig. 1 is provided in Fig. 2 where P1s(r) for the same three ions are plotted. The figure shows the wavefunctions for B2+ and B3+ to be indistinguishable while that for B4+ displays a collapse as reported in earlier work [5,6]. On the other hand, the continuum wavefunctions are almost identical for the three ion stages at high energies. Thus, due to the oscillations of the continuum wavefunctions, the overlap integrals are determined at small radii where the B4+ ion has a larger 1 s electron probability density.
1s P (1) p
2s P (1) p 1s P (1) p
q q2 + Z 2
Born
2
Born 2
= 0.014
(2.4.1)
where λ is the photon wavelength,
(2.2.1)
where jℓ(x) is the spherical Bessel function of order ℓ [9] and the second
1s P (2) d
Born
=
q
drr 3P1s (r ) j2 (kr ) o
=
Zq
96Z 2q2 (Z 2 + q2) 4
(2.4.2)
and the numerical value is for boron and a 9 keV photon. 3. Test of AA model Johnson and Nilsen [1] correctly noted that the boron photon ionization cross-section along the Hugoniot is not given by the product of the K-shell occupation times the cold, solid density result [11]. The present interpretation suggests the following expression for the boron total photon absorption, K(
) = cH
H(
) + 2cHe
He (
)
(3.1)
where σH and σHe are the isolated K-shell cross-sections per electron for the ground configurations of hydrogenic and helium-like ions, respectively, cH and cHe are respectively the fractional number of ions with a one and two K-shell electrons, and the factor of 2 accounts for the two K-shell electrons in the He-like species. The result for σHe can be obtained from cold data [11]. The hydrogenic isolated ion results [10] provides σH.
Fig. 2. Radial wavefunctions for the K-shell electron in several charge states of boron as a function of radius. 15
High Energy Density Physics 32 (2019) 14–17
C.A. Iglesias
contribution from the He-like ion to the total cross-section at the highest temperature. On the other hand, at the higher temperatures the AA from [1] yields the hydrogenic cross section (within 0.4% of the exact formula [10]) and applies it to the total K-shell occupation. Thus, the AA approach does not capture the He-like cross-section, which is smaller by 18% per electron. The consequence is a small over estimate of the cross-section. 4. Conclusion It was shown that the enhancement of the K-shell photon ionization cross-section per electron in the AA treatment of shock-heated boron [1] follows from known isolated ion phenomena rather subtle plasma effects. The role of the plasma is simply to produce higher ionization with increasing temperature. Consequently, opacity models that account for the changing bound level populations with plasma conditions will describe the enhanced cross-section per electron even if excluding plasma effects on the photon ionization at the high energies of interest to the high-pressure experiments [2]. A simple approximation was proposed to test the average atom calculations for boron along the principal Hugoniot. The test indicates that the interchange of the averaging procedure inherent in the AA atom approach leads to errors of <2%. Clearly the ion concentrations from the constrained Saha–Boltzmann model are not unique since they depend on the choice for the assumed continuum lowering model (see Appendix). Nevertheless, the discrepancy between the two cross-section approximations is due to the explicit treatment of the He-like ion in the discrete charge state approximation absent in the AA model. That is, the precise He-like concentration may be uncertain, but it is expected to be significant except at very high temperatures. Consequently, the numerical details of the comparisons are not definitive but should be representative. Of course, there remains the open question of whether AA models accurately reproduce the K-shell occupations at extreme matter conditions. The high-pressure experiments require high precision material density in order to determine an accurate equation of state. Therefore, corrections and approximations to the photon ionization were considered at the photon energy of interest. For example, the Born approximation was found insufficient to compute the K-shell photon absorption. The presence of bound L-shell electrons in samples containing heavier elements enhance the cross-section by ∼10% per electron. In addition, the L-shell electrons require explicit treatment of the various ion charge states present in the plasma. Finally, higher multipole contributions were found to increase the cross-section by a 1–2%.
Fig. 3. Relative difference between photon ionization cross-sections from the AA and detail ion charge formula as a function of temperature (left axes). Also, the contribution fraction from individual charge states to the total cross-section in Eq. (3.1) (right axes).
The AA formalism also provides an approximation for the crosssection. The AA model, however, interchanges the correct procedure of averaging the cross-sections of physically realizable ion charge states with integer electron occupations (as in Eq. (3.1)) by the cross-section in the average potential generated by average non-integer subshell occupations. To test the AA against Eq. (3.1) requires a calculation of the individual ion concentrations not available from an AA approach. For the test, the concentrations are obtained from a Saha–Boltzmann model (see Appendix) constrained to reproduce the AA K-shell occupations from [1]. The results from Eq. (3.1) using the concentrations from the constrained Saha–Boltzmann model, which reproduced the AA total K-shell occupation within 1 part in 106, agree to better than 2% with the AA calculations. The relative difference,
Diff% =
AA (
) K(
K(
)
)
× 100%
(3.2)
where σAA is the AA result from [1], is displayed in Fig. 3. The comparison uses the nonrelativistic exact nonrelativistic formula [10] for hydrogenic boron in Eq. (3.1). For the He-like system the cross-section in Eq. (3.1) is taken from the AA calculations at the lowest temperature rather than the cold, solid density data for the He-like system [11] mitigating possible small numerical uncertainties in the AA cross-sections in [1]. The remaining discrepancies, less than 2%, are due to the reversal of the averaging procedure by the AA formalism. These differences are most easily explained at the highest temperature. The fractional contribution from the individual ions to the total cross-section in Eq. (3.1) are displayed in Fig. 3. The figure shows ∼4%
Acknowledgments Thanks are due to Daniel Aberg for valuable discussions, Walter J. Johnson for comments on the manuscript, and Joseph Nilsen for the AA results. The work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
Appendix. Constrained Saha–Boltzmann model An ionization balance model is necessary to apply Eq. (3.1). To test the Average Atom (AA) model interchange of the averaging procedures, a Saha–Boltzmann model is used but constrained to reproduce the AA K-shell occupation from [1]. Along the Hugoniot, the AA calculations only has bound K-shell levels and all other subshells are delocalized. Thus, the discrete charge state plasma consists of ions with one and two bound K-shell electrons, bare nuclei, and free electrons. The ratio of two adjacent charge state ground configuration concentrations in thermodynamic equilibrium is given by [12]
cj cj + 1
=
ne 2 2 2 mT
3/2
gj gj + 1
e
j /T
(A.1)
where gj is the degeneracy of the ground configuration for species j, χj > 0 is the energy required to ionize an electron in the K-shell of species j and 16
High Energy Density Physics 32 (2019) 14–17
C.A. Iglesias
create an ion of species j + 1, m is the electron mass, and T is the plasma temperature in energy units. The average free electron number density includes all electrons not in bound states, (A.2)
ne = Z n i where ni is the total ion number density,
cj Z nj
Zn =
(A.3)
j
with Zje the net charge of ion species j and e the fundamental electric charge The binding energy includes plasma effects and assumes the form, j
=
o j
(Zj + 1)
(A.4)
with jo > 0 the isolated ion binding energy for species j and Ω a dimensionless parameter to be determined. There are two equations for the three ion charge states in the plasma represented in Eq. (A.1). Particle conservation requires
cj = 1
(A.5)
j
and provides a third equation. Finally, the constrain to match the AA K-shell occupation provides a fourth equation,
cH + 2cHe = n1s
(A.6)
AA
where cH and cHe are respectively the fractional number of ions with a one and two K-shell electrons and 〈n1s〉AA is the AA occupation of the K-shell. The four unknowns, the three ion concentrations and Ω, together with the four equations are solved iteratively given ne and T where the former is obtained from the AA calculations.
Press, Berkeley, 1981. [8] F.J. Rogers, B.G. Wilson, C.A. Iglesias, Phys. Rev. A38 (1988) 5007. [9] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions 55 NBS, Applied Mathematics Series, 1972. [10] H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and Two-Electrons Atoms, Plenum Publishing Co., New York, 1977. [11] R.B.I. Henke, E.M. Gullikson, J.C. Davis, At. Data Nucl. Data Tables 54 (1993) 181 http://henke.lbl.gov/optical_constants. [12] D. Mihalas, Stellar Atmospheres, W.H. Freeman and Co., San Francisco, 1978.
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