Radiative and Thermodynamic Properties of High Energy Density Systems using Screened Hydrogenic Model with l –Splitting

A. Ali, G. Shabbir Naz, M. Saleem Shahzad, Rukhsana Kouser , M. H. Nasim

Department of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad, 45650, Pakistan

Overview

Hydrodynamic simulations of High Energy Density Systems(HEDS) requires sufficiently accurate, smooth and fast atomicstructure formulism for computation of equation of state (EOS)and Opacity data. For this purpose we have developed a computercode SHOPEOS (Screened Hydrogenic model for Opacity andEquation Of State) proposed by Faussurier et all [1]. SHOPEOSis based on average atom model for High Energy Density plasmaand calculates screened charges and level energies used forcalculation of average charge state. In addition to continuumlowering and pressure ionization effects for HEDS, we have alsoimplemented the exchange, correlation and level interactioneffects. We have benchmarked our results for average charge stateand wave functions against reported atomic data for comparison.

Problem Statement

HEDS are to be probed using Radiation driven Hydrodynamicsimulations.

Sufficiently accurate, smooth and fast atomic structure data forcomputation of equation of state (EOS) and Opacity data.

For EOS and Opacity data generation Detail Level (DL)calculations gives accurate results but not a practical solution.

EOS and Opacity data should cover HEDS regime . EOS and Opacity data should cover LTE and Non-LTE regimes. EOS and Opacity data could be inline with Hydro codes. EOS and Opacity data for Mixture of elements can be produced.

Average Atom (AA) is a fictitious atom model which gives fast ,smooth, sufficiently accurate and thermodynamically consistentelectron structure data for material at some density and temperature.

AA is computationally much cheaper than (DL) . Can be made inline with Hydro codes . High energy density effects are incorporated. Currently we have LTE regime for single material.

Screened hydrogenic model with l-splitting

𝒁𝒁𝒌𝒌 = 𝒁𝒁 − �𝒌𝒌′

𝒌𝒌𝒎𝒎𝒎𝒎𝒎𝒎

𝝈𝝈𝒌𝒌𝒌𝒌′(𝟏𝟏 −𝜹𝜹𝒌𝒌𝒌𝒌′𝑫𝑫𝒌𝒌𝟎𝟎 )𝑷𝑷𝒌𝒌′

𝜺𝜺𝒌𝒌 = −𝑹𝑹𝑹𝑹𝑹𝑹 −𝒁𝒁𝒌𝒌𝟐𝟐

𝒏𝒏𝒌𝒌𝟐𝟐+ 𝟐𝟐�𝝈𝝈𝒌𝒌𝒌𝒌/

𝑷𝑷𝒌𝒌/𝒁𝒁𝒌𝒌/

𝒏𝒏𝒌𝒌/𝟐𝟐

∆𝑰𝑰 =𝒁𝒁𝑹𝑹𝒘𝒘𝒘𝒘

𝟑𝟑.𝟔𝟔 −𝒏𝒏𝒌𝒌𝟐𝟐

𝒁𝒁𝒌𝒌𝟐𝟐𝑹𝑹𝒘𝒘𝒘𝒘𝟐𝟐𝟓𝟓𝒏𝒏𝒌𝒌𝟐𝟐 − 𝟑𝟑𝒍𝒍𝒌𝒌 𝒍𝒍𝒌𝒌 + 𝟏𝟏 − 𝟏𝟏

𝑷𝑷𝒌𝒌 =𝑫𝑫𝒌𝒌

𝟏𝟏 + 𝒆𝒆𝜷𝜷(𝝐𝝐𝒌𝒌+∆𝑰𝑰−𝝁𝝁)

𝒁𝒁 = 𝐙𝐙 − �𝑷𝑷𝒌𝒌 =𝟒𝟒𝝅𝝅

𝑨𝑨𝑵𝑵𝑨𝑨 𝝆𝝆

𝒎𝒎𝒌𝒌𝒎𝒎𝟐𝟐𝝅𝝅ℏ𝟐𝟐

⁄𝟑𝟑 𝟐𝟐

𝒇𝒇𝟏𝟏𝟐𝟐

𝝁𝝁𝒌𝒌𝒎𝒎

𝒇𝒇𝜶𝜶𝝁𝝁𝒌𝒌𝒎𝒎

= �𝟎𝟎

∞

𝑹𝑹𝒎𝒎𝒎𝒎𝜶𝜶

𝟏𝟏 + 𝒆𝒆𝒎𝒎−𝝁𝝁𝒌𝒌𝒎𝒎

0.00 0.05 0.10 0.15 0.20 0.25-4

-2

0

2

4

6

√r/r0

1s 2s 3s 4s

Rnl (r)

𝑹𝑹𝒌𝒌 𝒓𝒓 =𝒁𝒁𝒌𝒌𝒏𝒏

𝒏𝒏 − 𝒍𝒍 − 𝟏𝟏 !𝒏𝒏 + 𝒍𝒍 ! 𝒆𝒆−

𝟐𝟐𝒁𝒁𝒌𝒌𝒓𝒓𝒏𝒏

𝟐𝟐𝒁𝒁𝒌𝒌𝒓𝒓𝒏𝒏

𝒍𝒍+𝟏𝟏

𝑳𝑳𝒏𝒏−𝒍𝒍−𝟏𝟏𝟐𝟐𝒍𝒍+𝟏𝟏 𝟐𝟐𝒁𝒁𝒌𝒌𝒓𝒓𝒏𝒏

Radiative Properties

Thermodynamic Properties

Radial Wave Function for Gold at Temperature 1 KeV and Density 0.1 g/cm3

10-3 10-2 10-1 100 101 102 1030123456789

1011121314

Ave

rage

Cha

rge

Stat

e

Density (g/cm3)

Temp (eV) 5.00 20.00 50.00 100.00 200.00 400.00 600.00 800.00 900.00 1000.00

Average Charge State For Aluminum

Average Charge State For Aluminum

We get average behavior not detail structures in all variables. Screening constant fit has omitted 1st ionization potential data. Inherently there are oscillations around 1st ionization potential. Fermi Dirac distribution is very sharp below 3eV which results

in oscillations in convergence. The system is highly non linear coupled algebraic equations.

which is problematic at near full or empty shell. Phase transition are not modeled .

Level Screened Charge

Level energy

Continuum Lowering

Occupation number

Average Charge State

Fermi Dirac Integral

Density-Temperature Diagram for HEDS

Exchange and Correlation Effect

𝑬𝑬𝒄𝒄 = 𝟎𝟎.𝟎𝟎𝟔𝟔𝟐𝟐𝟏𝟏𝟎𝟎𝟏𝟏𝟒𝟒�𝐥𝐥𝐥𝐥𝒎𝒎𝟐𝟐

𝒎𝒎+ 𝟐𝟐𝟐𝟐

𝟒𝟒𝒄𝒄−𝟐𝟐𝟐𝟐𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐥𝐥𝟐𝟐

+ 𝟐𝟐𝒎𝒎 + 𝒄𝒄

𝟒𝟒𝒄𝒄 − 𝟐𝟐𝟐𝟐

𝟐𝟐𝒎𝒎 + 𝟐𝟐

�−𝟐𝟐𝒎𝒎𝟎𝟎

𝒎𝒎𝟎𝟎𝟐𝟐 + 𝟐𝟐𝒎𝒎𝟎𝟎 + 𝒄𝒄𝐥𝐥𝐥𝐥

𝒎𝒎 − 𝒎𝒎𝟎𝟎 𝟐𝟐

𝒎𝒎𝟐𝟐 + 𝟐𝟐𝒎𝒎 + 𝒄𝒄 +)𝟐𝟐(𝟐𝟐 + 𝟐𝟐𝒎𝒎𝟎𝟎

𝟒𝟒𝒄𝒄 − 𝟐𝟐𝟐𝟐𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐚𝐥𝐥

𝟒𝟒𝒄𝒄 − 𝟐𝟐𝟐𝟐

𝟐𝟐𝒎𝒎 + 𝟐𝟐

𝑬𝑬𝒎𝒎 = −𝟎𝟎.𝟗𝟗𝟏𝟏𝟔𝟔�𝒁𝒁𝑹𝑹𝒘𝒘𝒘𝒘

Exchange Energy Ryd

Correlation Energy

b= 3.72744 , c=12.9352 , x0 = -0.10498, x = (Rws)1/2Where

Ryd

Ryd

Limitation and Approximation

Model for Solution

Introduction : High Energy-Density SystemsSystems having Energy per unit Volume > 1011 J/cm3

Cores of Giant Planets

Laser Fusion Targets

Conclusion

HEDS can be probed by using radiation driven hydrodynamic simulations These simulations require sufficiently accurate, fast and thermodynamically consistent data of EOS and

Opacity.We have used screened hydrogenic model with l-splitting for electronic structure calculations. Average charge state is compared against published data. Radial Wave functions for Gold are compared with the results presented in Nikiforv [2] . Radiative Opacity for Aluminum and Carbon is compared with LEDCOP [3] data. Electron Pressure and Internal energy for Aluminum is also obtained and compared against results by More[4].

References [1]. G. Faussurier and C. Blancard and P. Renaudin, High Energy Density Physics, 4 (2008)114 –123.[2]. Nikiforov AF, Novikov VG and Uvarov VB. ‘Quantum-Statistical Models Of Hot Dense Matter: Methods for computation opacity and equation of state’. Birkhäuser Verlag, 2005.[3]. www.nist.gov[4]. R.M. More. ‘JQSRT 27-3 (1982) 345-357’.[5]. Balazs and F. Rozsnyai ,High Energy Density Physics 8 (2012) 88-100.[6]. Gennady Miloshevsky and Ahmed Hassanein, Phys Rev E 92, 033109 (2015).

Physical Feeling

Pressure ≥ 1 Mbar

Fields ≥ 300 Tesla

Radiation ≥ 1. 1016 W/cm2

Radiative properties like opacity is obtained from level charges , level energies and average charge state. The frequency dependent absorption coefficient is obtained from four contributions

1E-3 0.01 0.1 1 1010-3

10-2

10-1

100

101

102

103

104

105

106

107

108

Line BroadiningsNatural and Doppler

AluminumTemperature = 100eVDensity = 0.1 g/cm3

bound - bound bound - free free - free total

Opac

ity

(cm

2 /g)

hν (KeV)

1E-3 0.01 0.1 1 10101

102

103

104

105

106

107

108

Line BroadiningsNatural and Doppler

CarbonTemperature = 10eVDensity = 0.01 g/cm3

Opa

city

(cm

2 /g)

hν (KeV)

SHOPEOS LEDCOP

1E-3 0.01 0.1 1 10101

102

103

104

105

106

107

108

Line BroadiningsNatural and Doppler

Opac

ity

(cm

2 /g)

hν (KeV)

SHOPEOS LEDCOP

AluminumTemperature = 100eVDensity = 0.1 g/cm3

1 /( ) [ ]( )Bh K Tbb bf ff scatk k k k e kνν −= + + − +

( ) ( ) ( ) ( ) ( ), , , , , , , , , ,SHML SHML QEOS QEOSc e c iP T Z P T Z P T Z P T Z P T Zρ ρ ρ ρ ρ= − + +

( ) ( ) ( ) ( ) ( ), , , , , , , , , ,SHML SHML QEOS QEOSc e c iE T Z E T Z E T Z E T Z E T Zρ ρ ρ ρ ρ= − + +

00

0

1 12

P PE Eρ ρ

+− = −

Degeneracy Lowering and Continuum Lowering

Continuum is lowered by ∆𝑰𝑰(𝒌𝒌)

Isolated Atom

AA𝑫𝑫𝒌𝒌 =𝑫𝑫𝒌𝒌𝟎𝟎

𝟏𝟏 + 𝒎𝒎𝒛𝒛𝒎𝒎𝒓𝒓𝒌𝒌𝟎𝟎

𝑹𝑹𝑾𝑾𝑾𝑾

𝟐𝟐𝒛𝒛𝒎𝒎

𝒓𝒓𝒌𝒌𝟎𝟎 = 𝒎𝒎𝟎𝟎𝟑𝟑𝒏𝒏𝒌𝒌𝟐𝟐 − 𝒍𝒍𝒌𝒌 𝒍𝒍𝒌𝒌 + 𝟏𝟏

𝟐𝟐𝒁𝒁𝒌𝒌𝟎𝟎

𝑹𝑹𝒘𝒘𝒘𝒘 =𝟒𝟒𝝅𝝅𝝆𝝆𝑵𝑵𝑨𝑨

𝟑𝟑𝑨𝑨

𝟏𝟏/𝟑𝟑

𝑟𝑟𝑘𝑘0

RW

S

For rk0 < RWSAtoms are not pressure

ionized

For rk0 ≥ RWS Atoms are pressure

ionized

EOS calculations includes Pressure , Energy and Thermal conductivities etc. are obtained from electrons and ions contribution . For Electronic part we used SHML and QEOS is adapted for ion part. The results are presented for Aluminum and shock Hugoniot is also compared with experimental data.

Pressure

Energy

Shock Hugoniot

1E-3 0.01 0.1 1 10 100 1000 10000101

102

103

104

105

106

107

108

109

1010

1011

1012

1013

Pres

sure

(J/c

c)

Density (g/cc) 1E-3 0.01 0.1 1 10 100 1000 10000

105

106

107

108

Ener

gy (J

/g)

Density (g/cc)

1E-3 0.01 0.1 1 10 100 1000101

102

103

104

105

106

107

108

109

1010

1011

1012

Total

Pre

ssur

e (J/

cc)

Density (g/cc)1E-3 0.01 0.1 1 10 100 1000 10000

105

106

107

108

109

1010

Total

Ene

rgy (

J/g)

Density (g/cc)

P (Mbar)

Present Calculation

Shock Hugoniot for Aluminum

Pressure and Energy against density for Aluminum using SHML

Total Pressure and Total Energy against density for Aluminum using SHML+QEOS

𝑃𝑃𝑘𝑘 =𝐷𝐷𝑘𝑘

1 + 𝑒𝑒𝛽𝛽(𝜖𝜖𝑘𝑘+∆𝐼𝐼−𝜇𝜇)

�̅�𝑍 = 4𝜋𝜋

𝐴𝐴𝑁𝑁𝐴𝐴 𝜌𝜌

𝑚𝑚𝑘𝑘𝑚𝑚2𝜋𝜋ℏ2

⁄3 2𝑓𝑓12( 𝜇𝜇𝑘𝑘𝑚𝑚

)

𝜇𝜇 = 𝑓𝑓1/2 𝑍𝑍−1

�̅�𝑍 = 𝑍𝑍 −�𝑃𝑃𝑘𝑘

Self Consistent Numerical Scheme

Hydrodynamic or Radiation Hydrodynamic simulation for region of interest

Atomic structure Modeling

Ionic Part Electronic Part

EOS or Opacity modeling for the region of interest

SHMLQEOS

10 20 30 40 50 60 70 80 90 1001

2

3

4

5

6

7

8

9

Aluminum ρ= 2.7 g/cm3

Aver

age

Char

ge S

tate

Temperature (eV)

SHOPEOS QEOS REODP Sesame UBCAM

Thanks to Ganday [6] for Providing the data

}

Comparison of 𝒁𝒁 for Aluminum

Fig Taken From [5]

10 eV <Temp < 1 KeV