Detailed Theoretical Investigations on the L-Shell Absorption of Open-M-Shell Germanium Plasmas:Effect of Autoionization Resonance Broadening

doi:10.4236/jmp.2012.330204

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Journal of Modern Physics, 2012, 3, 1670-1677

http://dx.doi.org/10.4236/jmp.2012.330204 Published Online October 2012 (http://www.SciRP.org/journal/jmp)

Detailed Theoretical Investigations on the L-Shell

Absorption of Open-M-Shell Germanium Plasmas:

Effect of Autoionization Resonance Broadening

Wenjun Xiang, Jiaolong Zeng, Yongsheng Fu, Cheng Gao

Department of Physics, College of Science, National University of Defense Technology, Changsha, China

Email: [email protected]

Received August 23, 2012; revised September 25, 2012; accepted October 2, 2012

ABSTRACT

Radiative opacity of open-M-shell germanium plasmas in the L-shell photon energy region were investigated in detail

by using a fully relativistic detailed level accounting approach. Among other physical effects such as relativistic and the

interaction between fine-structure levels belonging to the same non-relativistic configuration and different configura-

tions, particular attention is paid on the effect of autoionization resonance broadening on the L-shell absorption. The

results show that for plasmas at present and past typical experimental conditions, line width due to autoionization reso-

nance broadening dominate among all the physical broadening mechanisms including electron impact and Doppler

broadenings. Such an effect is most pronounced for ions with just a few 2p-nd transition lines such as , while it is

not so pronounced for complex ions such as

Ge 



, where there are so many 2p-nd lines that line overlapping partly

conceal the effect of autoionization resonance broadening. After taking the effect of autoionization resonance broaden-

ing into account, detailed comparisons are made with available experimental spectra at different physical conditions of

different plasma temperatures and densities. The L-shell absorption is sensitive to the plasma temperature, especially in

the 2p-3d excitation energy region. The potential of utlizing the relative shape and intensity of the 2p-3d spin-orbit

splitting as temperature diagnostics is investigated.

Keywords: Autoionization Resonance Broadening; Germanium; Absorption; Radiative Opacity

1. Introduction

Germanium is one of the absorption and switching mate-

rial in indirectly driven inertial confinement fusion and

therefore its radiative property is of great interest both

experimentally and theoretically [1-9]. The radiative opa-

city of open-M-shell germanium plasmas are of particu-

lar experimental interest, especially in the region of 2p-

3d and 2p-4d transition arrays. Foster et al. [1] measured

the L-shell absorption spectrum of a germanium plasma,

which was generated by radiation heating using thermal

X radiation from a laser-produced gold plasma. Tracer

elements of Al and Mg were used to characterize the

temperature (76 eV) and density (0.05 g/cm3) of the

plasma. Perry et al. [2] experimentally measured the X-

ray absorption of germanium plasmas at a lower tempe-

rature of about 38 eV. Renaudin et al. [3] measured the

absorption spectra of germanium plasma in the 0.8 - 1.1

nm range. The germanium sample was radiatively heated

by a gold hohlraum. Recently, Loisel et al. [4] experi-

mentally measured the radiative opacity of medium Z

element plasmas including germanium in the X-ray re-

gion (0.8 - 1.8 nm) at temperatures between 15 and 25

eV and densities between 2 and 10 mg/cm3.

The experimental observations provide valuable data

to validate different approximations made in the opacity

calculations. In order to interpret these experiments, va-

rious theoretical methods were used to analyze the ob-

served spectra. Foster et al. [1] used a detailed configura-

tion accounting (DCA) model which includes an appro-

ximate treatment of term widths using the unresolved

transition array (UTA) approach. Perry et al. [2] em-

ployed a Super Transition Array (STA) approach [10] to

describe the state of plasmas, which can be thought as a

hybrid approach between an average atom description

and a full DCA scheme. In these statistical methods, the

line width caused by the statistical approach are gener-

ally dominant and therefore a Gaussian line shape is used.

Renaudin et al. [3] took into account of the term structure

using UTA statistical methods and a detailed line ac-

counting approach in pure jj coupling with orbital relaxa-

tion effect being considered. In their work, a Gaussian

shape is used, which assumed that Doppler broadening

W. J. XIANG ET AL. 1671

dominates all other broadening mechanisms.

In a recent work, Blenski et al. [11] carried out theo-

retical interpretation of the X-ray photo-absorption ex-

periments measured at LULI2000 facility [4]. Their ana-

lysis was performed using the statistical superconfigura-

tion code SCO, two line-by-line opacity codes based on

the HULLAC and FAC packages and a hybrid statistical-

detailed code SCORCG. The authors pointed out that in

the detailed line-by-line treatment the autoionization (AI)

resonance broadening appeared to be the most important

in all broadening mechanisms. Yet no detailed investiga-

tions were found in the literature on this effect partly due

to the very challenging computations of the detailed line

spectrum with the complex open-M-shell Ge ions.

With the development of computational ability, com-

putation of plasma opacities is developing toward more

accurate direction including DLA, mixed DLA and UTA

and improvement of UTA [12-14]. In this work, we in-

vestigated the radiative opacity of Ge plasmas by using a

detailed level accounting (DLA) approach. In such a

DLA method, we do not need to introduce additional line

broadening caused by the statistical method, and there-

fore we include only true physical broadening mecha-

nisms such as Doppler, electron impact and AI resonance

broadenings. Among other physical effects on the opac-

ity, particular attention was paid on the effect of AI re-

sonance broadening of L-shell excitation region. To the

best of our knowledge, no detailed systematic investiga-

tions were found in the literature on this physical effect

on the L-shell absorption of open-M-shell germanium

plasmas, although many researches (for example, see

[15-21]) highlighted the importance of AI process to de-

termine the population balance and radiative properties in

non-local thermodynamic equilibrium (non-LTE) plas-

mas with a wide range of temperature and density.

2. Theoretical Method

The total radiative opacity for a plasma at a temperature

and mass density



is the sum of the bound-bound,

bound-free, free-free and scattering processes. The con-

tribution of bound-bound process for radiation of energy



can be obtained from the cross sections of bound-

bound lines:



il ll

bb il





















(1)

where il is the population density of level of ioni-

zation stage and ll





l

 is the cross section for

photoexcitation from level to l of ionization state

and can be expressed in terms of the absorption oscil-

lator strength

ill





iill ll

hfSh









hcm

(2)

where is the Planck constant, is the speed of light

in vacuum, e is the electron charge, e is the rest

mass of electron, and







is the line shape function.

In the DLA approach, the line shape function takes the

Voigt profile:

 

ln 2,

πd

ShHa (3)









,Ha



is the Voigt function: where

 













ln 2ld









ln 2d







where d



nd l



are the Doppler and Lorentzian half

width at half maximum (HWHM), respectively. The

Doppler HWHM is related to the temperature of the

plasma and transition energy



[22]:



3.858 10

dkT Ah



  (4)



where

is atomic weight of the ion in gram and the

units of , 0

kT h



and d



are eV. The Lorentzian

HWHM (l



) is contributed by the electron impact

broadening (e



), AI resonance broadening (a



) and

natural life broadening. In general, line width due to

natural life are much smaller than the electron impact and

AI resonance broadening. Yet the line width caused by

AI resonance broadening is in many cases the dominant

broadening mechanism and therefore should be taken

into account. The accurate determination of the AI reso-

nance broadening is complicated because of the complex

atomic structure for open-M-shell Ge ions and thus is not

carefully considered in previous work. In this work, we

systematically investigated the effect of AI resonance

broadening on the L-shell absorption of open-M-shell Ge

plasmas. For the line width of electron impact broaden-

ing, we use a semiempirical method [23,24]



1/2

8π2π1.1

0.9

6π3

jjjj

jif

NkT z

nnll



 

 

 

 

 









(5)

where i (i

l) and

n (



) are the effective principal

(orbital angular momentum) quantum numbers of the

lower and upper energy levels of the transition, respec-

tively.

The contribution of bound-free opacity can be

W. J. XIANG ET AL.

1672

obtained from the photoionization cross sections per ion:



il il













(6)

where il is the total population density of all possible

levels of ionization stage and i





i is the photo-

ionization cross section per ion and it can be calcu-

lated from the photoionization cross sections





il h





from level in ion i

 

EkT

iil











(7)

where il

is the statistical weight for level of ion i,

il is the energy of level of ion above the ground

state and

Eli

is the partition function for ion . i



ehL

 









Ge 

In the X-ray region, the bound-bound and bound-free

contributions to the opacity are generally dominant and

thus only simple approximations are used to describe the

free-free (Kramers cross section) and scattering (Thomp-

son scattering cross section) contribution to the opacity.

The population density of different charge states i

is determined by ionization equilibrium equation in LTE.

For a particular level of each charge state, the population

density obeys the Boltzmann distribution. The ionization

potential depression is considered by using the Debye-

Huckel model [25].

The fraction of radiation transmitted with respect to

some incident source of arbitrary intensity is given by

Fh (8)

where is the path length traversed by the light source

through the plasma. To compare directly with experiment,

one should include the effects of instrumental broaden-

ing.

3. Results and Discussion

For the experimental conditions [1-4], the charge states

which contribute to the L-shell absorption range from

(the ground configuration





269

33 3ds p

Ge Ne



) to

(the ground configuration



e3Ns

 

ππ

). For these

ionization stages of Ge, the upper levels of 2p-nd transi-

tions are autoionized ones, and therefore it is necessary

to include the effect of AI resonance broadening on the

L-shell absorption. These charge states have complex

atomic structure with an open-M-shell characteristics and

it is a challenging work to accurately determine the ato-

mic data including energy levels, oscillator strengths,

photoionization cross sections and line width due to au-

toionized levels.











In order to take into account of the configuration in-

teraction (CI), an atomic state is approximated by a linear

combination of configuration state functions (CSFs) with

same symmetry

(9)

where c is the number of CSFs and i



denotes

the representation of the atomic state in this basis. The

CSFs are anti-symmetrized products of a common set of

orthogonal orbitals which are optimized on the basis of

the relativistic Hamiltonian. Once the CI wavefunctions

have been obtained, the oscillator strengths can be calcu-

lated

iijij

gf 

P (10)

where

EE E E, i and



are, respectively, the

energies of initial and final levels and

is the statisti-

cal weight of the lower state, i.e. . is 21

gJP

dipole transition operator with

p



Pr in the length

formalism and

E







in the velocity formalism,

where is the total number of bound electrons. The

AI rates are obtained in the relativistic distorted-wave

approximation, which reads in the first order perturbation

theory (in atomic units)

2,;

ijjT Ti

ijij

AJM



 





(11)

where i



is the autoionizing state,



is the final

state which has one less electron than i





is the

relativistic angular quantum number of the free electron.

The total angular quantum number of the coupled final

state must be equal to that of i



, i.e., Ti



and



. The AI resonance width of level can be

obtained by summing all final levels

iij



(12)



where 2πh





. All atomic data required in the calcu-

lation of opacity were obtained out using the Flexible

Atomic Code (FAC) developed by Gu [26].

We first investigate the effect of AI resonance broad-

ening on the opacity of Ge plasma. Extensive calcula-

tions were carried out and the results show that the AI

resonance HWHM for the inner shell 2p excited levels

range from 0.05 eV to 0.4 eV for charge states from



. For Ge14



, the ground configuration





e3 3Nsp

52 6

2333dpsp

only has one fine-structure level. The 2p

excited configuration is split into 12

fine-structure levels, where



23dp

3232 1,



23dp

3252 1,

and





23dp

32 1 the total angular momentum is 1.





23dp

3232 1 means one hole in orbital 2p, one electron

in orbital 3d and full electron orbitals have been omitted.

W. J. XIANG ET AL. 1673

For the 2p-3d transition array, there are three dipole al-

lowed transitions: 32 32

23dp,32 52, and 23dp

12 32

3d2p

2 3dp14

Ge 

. The radiative transition probability of the

first transition is much smaller than the last two lines.

The effect of AI resonance width on the absorption in the

2p-3d region can clearly be seen from Figure 1, which

shows the absorption cross section for the transition array

of of . In order to

obtain the results, was assumed to be embedded

in an LTE plasma at a temperature of 60 eV and a den-

sity of 0.01 g/cm3. In Figure 1(b), the electron impact

broadening and Doppler broadening mechanisms are

considered, while the AI resonance broadening is not

included. In Figure 1(a), three broadening mechanisms

are all taken into account. One can see that when AI

resonance broadening is included, the peak absorption

cross section is one order of magnitude smaller, which is

a reflection of the effect of AI resonance broadening on

the absorption. In the above plasma condition, the

HWHM due to electron impact and Doppler broadening

is evaluated to be 0.003 eV and 0.041 eV, respectively.

The HWHM due to AI resonance broadening is calcu-

lated to be 0.247 eV and 0.263 eV for 2p excited levels

62 6

233psp52 6

33sp

14



3252 1 and 23pd





23dp

32 1, respectively. Among

the three broadening mechanisms, the AI resonance

broadening is the dominant broadening mechanism. The

AI resonance width is larger than the electron impact and

Doppler width by more than one order of magnitude.

The effect of AI resonance broadening on the absorp-

tion of is evident from the inspection of Figure 1.

As there are only two strong absorption lines and the

2p-3d spin-orbit splitting (~30 eV) is much larger than

Ge 

Figure 1. Absorption cross section for the transition array

2p63s23p6 − 2p53s23p63d of Ge14+: (a) including and (b) not

including the eﬀect of AI resonance broadening.

the line width caused by all broadening mechanisms, two

2p-3d transition lines are separate each other with an

interval of ~30 eV. With the decrease of the number of

3p electrons, especially near the half full 3p orbital, the

number of 2p-3d lines becomes rapidly great and the

absorption approaches unresolved quasi-continuum bands.

Figure 2 shows the absorption cross section for the tran-

sition array of of

62 452 4

233 2333dpsp psp16



which was assumed to be embedded in an LTE plasma at

a temperature of 90 eV and a density of 0.015 g/cm3.

Similar to 14



, the AI resonance width is also larger

than the electron impact and Doppler width by more than

one order of magnitude. Yet the peak absorption cross

section without considering the AI resonance broadening

is just 2.43 times larger than that which includes this ef-

fect. This is due to many lines overlapped together re-

sulting in less pronounced for the effect of AI resonance

broadening than in 14



, where lines do not overlap.

The two strong absorption bands with an interval of

spin-orbit splitting of ~30 eV in Figure 2 originate from

32 52

23dp



and 12 32

23dp



transitions for the lower

and higher photon energy range, respectively. Such a

feature differ obviously from that of Ge . With the

further decrease of ionization stage, the 2p-3d absorption

shows a different feature from Ge and

14

1416



, as

illustrated in Figure 3, which shows the absorption cross

section for the transition array of

626 5526 6

2333d2333dpsp psp99

Ge of Ge .



was

assumed to be embedded in an LTE plasma at a tem-

perature of 38 eV and a density of 0.015 g/cm3. For



and Ge16



, 32 52

23dp



absorption is weaker

than 9



. Fur-

12 32

23dp, while this is reversed for



Figure 2. Absorption cross section for the transition array

2p63s23p4 − 2p53s23p43d of Ge16+: (a) including and (b) not

including the eﬀect of AI resonance broadening.

W. J. XIANG ET AL.

1674

Figure 3. Absorption cross section for the transition array

2p63s23p63d5 − 2p53s23p63d6 of Ge9+: (a) including and (b)

not including the eﬀect of AI resonance broadening.

thermore, the AI width for many upper levels of

32 52

23dp transitions is much smaller than those of

12 32

As an illustrative example, Figure 4 shows the effect

of AI resonance broadening on the radiative opacity of

Ge plasma at a temperature of 38 eV and a density of

0.015 g/cm3 with plot (a) including and (b) not including

such an effect. Due to many overlapped lines, it was

found that the effect of AI resonance broadening reduce

the peak value of opacity by only 25% for 2p-3d transi-

tion arrays. For 2p-4d and higher transition arrays, this

effect is small as line width caused by the electron im-

pact broadening is comparable or even larger than that of

AI resonance broadening.

23dp.

After investigating the effect of AI resonance broad-

ening on the opacity of Ge plasma, we turn to the com-

parison of our theoretical results with the experimental

data. Figure 5 shows the calculated transmission of Ge

plasma at temperatures of 58, 60 and 62 eV, while the

mass density is fixed to be 0.01 g/cm3 in solid lines. The

dashed lines show the experimental data measured by

Renaudin et al. [3]. Areal density used in the calculations

is equal to 0.11 mg/cm2. It can be seen that the transmis-

sion is sensitive to the temperature of the plasmas, in

particular in 2p-3d region. At the temperature of 58 eV,

there is a close agreement for absorption line position

between theory and experiment at the higher photon en-

ergy wing of the 2p-3d (1300 - 1340 eV), while such a

good agreement turned to the lower photon energy region

(1220 - 1260 eV) at the temperature of 62 eV. This shows

that there is small temperature or density gradient in the

experiment. In the whole shown photon energy range,

Figure 4. The eﬀect of AI resonance broadening on the ra-

diative opacity of Ge plasma at a temperature of 38 eV and

a density of 0.015 g/cm3: (a) including and (b) not including

such an eﬀect.

Figure 5. Transmission of Ge plasma at temperatures of (a)

58; (b) 60 and (c) 62 eV with the mass density being 0.01

g/cm3 in solid lines. The dashed lines refer to the experi-

mental data of Renaudin et al. [3].

there is a better agreement between theory and experi-

ment at the temperature of 60 eV, which is consistent

with the experimental determination of plasma tempera-

ture from element tracer of Mg and radiative-hydrody-

namic simulations [3]. At the temperature of 60 eV, a

more complete comparison between experiment and dif-

W. J. XIANG ET AL. 1675

ferent theoretical results is shown in Figure 6. Two addi-

tional theoretical results are compared in Figure 6 which

were obtained by average atom calculations including

UTA with and without orbital relaxation (OR) treatment

[3]. Renaudin et al. [3] also carried out DLA calculations,

yet it is difficult to extract the data from the original fig-

ure.

At the experimental condition of Renaudin et al. [3],

the 2p-3d absorption does not show an interval of spin-

orbit splitting. 32 52 and 23dp1232 are

smoothly merged together, expanding an extended pho-

ton energy region from 1220 to 1320 eV. Similar conclu-

sion applies for the experimental spectra of Foster et al.

[1]. To save space, we do not show detailed comparison

between theory and experiment under the physical condi-

tion of Foster et al. [1]. With the decrease of plasma

temperature, the separation of 2p-3d spin-orbit splitting

begins to appear evidently. Such a trend can easily be

seen from Figures 7 and 8, which shows the transmis-

sion at temperatures near 38 eV and 24 eV.

23dp

In Figure 7, the spectrally resolved transmission of Ge

plasma are shown in solid lines at temperatures of 36, 38

and 40 eV with the mass density being fixed to be 0.012

g/cm3. The dashed lines shows the experimental data

measured by Perry et al. [2] with the temperature being

determined to be 38 ± 2 eV and density 0.012 ± 0.003

g/cm3. In plot (b), the STA prediction is also given for

Ge plasma at the temperature of 38 eV. Areal density

used in the calculations is equal to 0.111 mg/cm2. In this

case, the 2p-3d spin-orbital splitting is very sensitive to

the temperature of the plasmas. With just 2 eV difference

in temperature from (a) to (c), the 2p-3d spin-orbital

Figure 6. Comparison of different theoretical (DLA and

UTA) and experimental spectra of transmission of Ge pla-

sma at a temperature of 60 eV and a density of 0.01.

Figure 7. Transmission of Ge plasma at temperatures of (a)

36; (b) 38 and (c) 40 eV with the mass density being 0.012

g/cm3 in solid lines. The dashed lines refer to the experi-

mental data of Perry et al. [2].

splitting is much more obvious at 36 eV than at 40 eV,

even the shape of absorption structures is much different

at the three temperatures. For example, look at the ab-

sorption feature near photon energy 1240 eV. The ab-

sorption structure for the spin-orbital splitting is sharper

at 36 eV than at 38 and 40 eV. For the wide and rela-

tively flat absorption bottom at 38 and 40 eV, the varia-

tion trend with photon energy is reversed. This is an in-

dication that the 2p-3d absorption has the potential of

temperature diagnosis, especially in temperature region

where absorption is very sensitive to the temperature. As

there are enormous numbers of absorption lines, the in-

dividual lines merged to become quasi-continuum ab-

sorption bands, DLA results tend to agree with UTA and

STA ones. Note, however, that at some particular tem-

perature, where there are not so many absorption lines to

merge together, DLA results will have distinct difference

from those of UTA and STA.

The 2p-3d spin-orbit splitting becomes more evident

with the further decrease of temperature. In Figure 8, the

spectrally resolved transmission of Ge plasma are shown

in solid lines at temperatures of 16, 18, 20, 22 and 24 eV

and the mass density being fixed to be 0.015 g/cm3. The

dashed lines shows the experimental data [4,11]. In plot

(e), the SCO and SCORCG predictions [11,27] are given

for Ge plasma at temperature of 24 eV. Areal density

W. J. XIANG ET AL.

1676

Figure 8. Transmission of Ge plasma at temperatures of (a)

16; (b) 18; (c) 20; (d) 22 and (e) 24 eV with the mass density

being 0.015 g/cm3 in solid lines. The dashed lines refer to the

experimental data [11,13]. In plot (f), the dotted and dot-

dashed lines refer to results of SCO and SCORCG codes,

respectively.

used in the calculations is equal to 0.08 mg/cm2. At these

temperatures, the shape and intensity of the 2p-3d spin-

orbital splitting is more sensitive to the temperature of

the plasmas than in physical conditions of Figures 5 and

7. With just 2 eV difference in the temperature, the shape

and relative intensity of the 2p-3d spin-orbital splitting is

noticeably different from case to case for plot (a)-(e), and

therefore the potential of temperature diagnosis at these

temperatures is better than in Figures 5 and 7. Reason-

able agreement is found between our work and those of

SCO and SCORCG predictions at 24 eV, although

stronger absorption is predicted by SCO and SCORCG

codes for the 1232 transition lines near the pho-

ton energy 1240 eV. Compared with the experiment, all

theoretical results deviate from experimental spectrum, in

particular in the 2p-3d region of 1190 - 1260 eV. Such a

deviation is due to the strong gradients of temperature

and density in the experimental plasma.

23dp

In conclusion, spectrally resolved L-shell absorption

spectra of open-M-shell germanium plasmas were inves-

tigated by using a detailed line-by-line method. In the

DLA approach, spectral line profile plays an important

role on the radiative opacity of plasmas. In this work, we

focus on the effect of AI resonance broadening on the

L-shell absorption of germanium plasmas. It was found

that for plasmas at typical present and past experimental

conditions, line width due to AI resonance broadening is

an order of magnitude larger than that due to electron

impact and Doppler broadenings. For germanium ion

with near closed atomic structure such as



, the

effect of AI resonance broadening is most pronounced

with the peak absorption cross section of one particular

2p-3d absorption line being an order of magnitude

smaller than that of not including this effect. For germa-

nium ion with near half filled 3d or 3p electron such as



and Ge16



, such an effect is not so pronounced

due to many 2p-nd transition lines merged together to

form quasi-continuum bands. Detailed comparisons are

carried out with available experimental spectra at differ-

ent physical conditions of different plasma temperatures

and densities. The results show that absorption spectra

are sensitive to the temperature of plasmas and show the

potential of temperature diagnostics by using the 2p-3d

transition arrays. The relative shape and intensity of the

2p-3d spin-orbit splitting is very sensitive to the tem-

perature and therefore should be an ideal tool of tem-

perature diagnostics.

4. Acknowledgements

This work was supported by the National Natural Sci-

ence Foundation of China under Grant Nos. 11274382,

11274383, and 11204376.

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