TPP-2M and IMFPs


 

 

TPP-2M IMFPs for the Thermo K-Alpha XPS

Values of electron inelastic mean free paths (IMFPs) in solids are of fundamental physical importance for describing the inelastic scattering of electrons in different materials. The IMFPs also define the surface sensitivity of Auger electron spectroscopy (AES) and X-ray photoelectron spectroscopy (XPS) and are used in quantitative applications of these techniques.

Tanuma, Penn, and Powell have previously calculated IMFPs for many solid materials over a wide energy range. Initially, we reported IMFPs for 50 eV to 2,000 eV electrons for 27 elemental solids 15 inorganic compounds and 14 organic compounds. These IMFPs were calculated from experimental optical data with the non-relativistic full Penn algorithm (FPA) for electron energies under 200 eV and the single-pole approximation or simple Penn algorithm (SPA) for higher energies. We analyzed these calculated IMFPs with the Bethe equation for inelastic scattering of electrons in matter to develop an IMFP predictive formula (designated TPP-2M). The TPP-2M equation could be used to estimate IMFPs in other materials, again for energies between 50 eV and 2,000 eV, the energy range of interest for many AES and XPS experiments.

 

TPP-2M  Theoretically Calculated IMFPs   versus   KE^0.6 IMFP Calculation


 

 


 

 


 

 

 

 


 

TPP-2M IMFPs 

 

 

 


 

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7047655/

TPP-2M are theoretically Calculated Inelastic Mean Free Path (IMFP) values
used for Quantification

The following publication was originally
published in final edited form as:
Surf Interface Anal. 2018; 51(4): 427–457.

doi: 10.1002/sia.6598

PMCID: PMC7047655
NIHMSID: NIHMS1530804
PMID: 32116395

Calculations of electron inelastic mean free paths. XII. Data for 42 inorganic compounds over the 50 eV to 200 keV range with the full Penn algorithm.

Use of this article is for research only. Copies may not be made or sold.

Abstract

We have calculated inelastic mean free paths (IMFPs) for 42 inorganic compounds (AgBr, AgCl, AgI, Al2O3, AlAs, AlN, AlSb, cubic BN, hexagonal BN, CdS, CdSe, CdTe, GaAs, GaN, GaP, GaSb, GaSe, InAs, InP, InSb, KBr, KCl, MgF2, MgO, NaCl, NbC0.712, NbC0.844, NbC0.93, PbS, PbSe, PbTe, SiC, SiO2, SnTe, TiC0.7, TiC0.95, VC0.76, VC0.86, Y3Al5O12, ZnS, ZnSe, and ZnTe) for electron energies from 50 eV to 200 keV. These calculations were made with energy-loss functions (ELFs) obtained from measured optical constants for 15 compounds while calculated ELFs were utilized for the other 27 compounds. Checks based on ELF sum rules showed that the calculated ELFs were superior to the measured ELFs that we had used previously. Our calculated IMFPs could be fitted to a modified form of the relativistic Bethe equation for inelastic scattering of electrons in matter for energies from 50 eV to 200 keV. The average root-mean-square (RMS) deviation in these fits was 0.60 %. The IMFPs were also compared with a relativistic version of our predictive Tanuma-Powell-Penn (TPP-2M) equation. The average RMS deviation in these comparisons was 10.7 % for energies between 50 eV and 200 keV. This average RMS deviation is almost the same as that found in a similar comparison for a group of 41 elemental solids (11.9 %) although relatively large deviations were found for cubic BN (65.6 %) and hexagonal BN (34.3%). If these two compounds are excluded in the comparisons, the average RMS deviation becomes 8.7 %. We found generally satisfactory agreement between our calculated IMFPs and values from other calculations and from experiments.

1. Introduction

Values of electron inelastic mean free paths (IMFPs) in solids are of fundamental physical importance for describing the inelastic scattering of electrons in different materials. The IMFPs also define the surface sensitivity of Auger electron spectroscopy (AES) and X-ray photoelectron spectroscopy (XPS) and are used in quantitative applications of these techniques.[,]

We have previously calculated IMFPs for many solid materials over a wide energy range.[,,,,,] Initially, we reported IMFPs for 50 eV to 2,000 eV electrons for 27 elemental solids,[,] 15 inorganic compounds,[] and 14 organic compounds.[] These IMFPs were calculated from experimental optical data with the non-relativistic full Penn algorithm (FPA)[] for electron energies under 200 eV and the single-pole approximation or simple Penn algorithm (SPA)[] for higher energies. We analyzed these calculated IMFPs with the Bethe equation for inelastic scattering of electrons in matter[] to develop an IMFP predictive formula (designated TPP-2M).[] The TPP-2M equation could be used to estimate IMFPs in other materials, again for energies between 50 eV and 2,000 eV, the energy range of interest for many AES and XPS experiments.

In recent years, there has been growing interest in XPS and related experiments performed with X-rays of much higher energies for both scientific and industrial purposes. We then published IMFPs for 41 elemental solids for electron energies up to 30 keV.[] Since there is also a need for IMFPs in transmission electron microscopy (TEM), we also calculated IMFPs in 41 elemental solids for energies up to 200 keV with a relativistic version of the FPA.[] In addition, we developed a relativistic version of the TPP-2M equation that provides reasonable IMFP estimates for energies between 50 eV and 200 keV. The root-mean-square (RMS) deviation between the estimated IMFPs from the TPP-2M equation and the directly calculated values was 11.9 % for the group of 41 elemental solids.[] This RMS deviation was similar to that found (10.2 %) in a similar comparison for our original group of 27 elemental solids for the 50 eV to 2 keV energy range.[]

In this paper, we extend our previous calculations[] of IMFPs for 15 inorganic compounds and electron energies between 50 eV and 2,000 eV. We report here new calculations of IMFPs with the relativistic FPA for 42 inorganic compounds (AgBr, AgCl, AgI, Al2O3, AlAs, AlN, AlSb, cubic BN (c-BN), hexagonal BN (h-BN), CdS, CdSe, CdTe, GaAs, GaN, GaP, GaSb, GaSe, InAs, InP, InSb, KBr, KCl, MgF2, MgO, NaCl, NbC0.712, NbC0.844, NbC0.93, PbS, PbSe, PbTe, SiC, SiO2, SnTe, TiC0.7, TiC0.95, VC0.76, VC0.86, Y3Al5O12, ZnS, ZnSe, and ZnTe) for energies between 50 eV and 200 keV. These inorganic compounds were chosen because they had experimental ELF data below the bandgap energy and the errors in the two sum rules for evaluating ELFs were less than 10 %. Two inorganic compounds that we considered previously [], LiF and Si3N4, were omitted in this study because their ELFs gave large errors in the sum rules discussed in Section 3.2. The relativistic TPP-2M formula[] must therefore be a more reliable means for determining IMFPs for these two compounds.

The IMFP calculations with the FPA require knowledge of the energy-loss functions (ELFs) of each material, typically for photon energies (or, equivalently, electron energy losses) between 1 eV and 200 keV. Our previous IMFP calculations were based on ELFs obtained from experimental optical data or from electron energy-loss spectroscopy (EELS). However, optical constants and optical ELFs as well as EELS ELFs are lacking for most semiconductor materials for energy losses larger than about 10 eV. We therefore calculated ELFs for the 27 compound semiconductors in our group of 42 inorganic compounds. We utilized the WIEN2k[] and FEFF[] codes to compute the imaginary part of the complex dielectric constant, typically for photon energies from several eV to 1 MeV, from which we calculated the real part and then the ELF. We note here that Werner et al.[] successfully used the WIEN2k code to calculate optical constants and ELFs for 16 elemental metals and one semimetal that agreed satisfactorily with corresponding data obtained from an analysis of reflection EELS data.

We also report comparisons of our calculated IMFPs for the 42 inorganic compounds with IMFPs calculated by Reich et al.[], Akkerman et al.[], Pandya et al.[], Kwei and Li[], Kwei et al.[], Boutboul et al.[], Ashley and Anderson[], Dapor and Miotello[], and Dapor[], and with IMFPs measured by lakoubovskii et al.[], Egerton[], Meltzman et al.[], Gurban et al.[], Pi et al.[], Krawczyk et al.[], Chung et al.[], Wang et al.[], MacCartney et al.[], Lee et al.[], and Bideux et al.[], Zommer et al.[], and Jung et al.[]. We also make comparisons between the measured IMFPs and IMFPs from the relativistic TPP-2M equation.

2. IMFP Calculations with the Relativistic Full Penn Algorithm

We utilized the relativistic FPA to calculate IMFPs for energies in the 50 eV to 200 keV range in order to be consistent with our previous IMFP calculations for elemental solids.[] The IMFPs were calculated at equal energy intervals on a logarithmic scale corresponding to increments of 10 % from 10 eV to 1 MeV. We will present IMFPs for energies between 10 eV and 50 eV and between 200 keV and 1 MeV in Figures but these results are shown only to illustrate trends.

The relativistic differential cross section (DCS) for inelastic scattering can be expressed as the sum of a longitudinal DCS and a transverse DCS.[] Since the transverse DCS can be neglected for electron energies less than about 0.5 MeV,[] the relativistic DCS can be written using Hartree atomic units (me=e==1, where me is the electron rest mass, e is the elementary charge, and  is the reduced Planck constant), as

d2σdωdQ=d2σLdωdQ+d2σTdωdQd2σLdωdQ=1v21+Q/c2Q(1+Q/2c2)1πNIm(1ε(Q,ω))
(1)

where ω is the energy loss, c is the speed of light, Q is the recoil energy defined by[]

Q(Q+2c2)=(cq)2,
(2)

q is the momentum transfer, N is the number of molecules per unit volume, v is the electron velocity, and Im[1/ε(Q,ω)] is the ELF expressed as a function of energy loss ω and recoil energy Q. Using Eqn (2) and dQ=q/[1+(Q/c2)]dq, the ELF in Eqn (1) can be conveniently written as a function of ω and momentum transfer q.

We write the DCS as

d2σdωdq2πNv2Im(1ε(q,ω))1q,
(3)

where Im[1/ε(q,ω)] is the ELF. We will use the Penn algorithm[] to evaluate Eqn (3). The dependence of the ELF on ω can be obtained from the measured ELF for each material (typically from optical experiments) while the dependence of the ELF on q can be obtained from the Lindhard model dielectric function.[] In this paper, we will show that we can also use calculated ELFs for materials for which there are no measured ELFs over key energy ranges.

The ELF in Eqn (3) can be expressed using the full Penn algorithm as

Im[1ε(q,ω)]=0dωpg(ωp)Im[1εL(q,ω;ωp)]
(4)

where εL denotes the Lindhard model dielectric function of a free-electron gas with plasmon energy ωp(=4πn−−−√)n is the electron density, g(ωp) is a coefficient introduced to satisfy the condition Im[1/ε(q=0,ω)]=Im[1/ε(ω)], and Im[1/ε(ω)] is the measured or calculated ELF. The coefficient g(ωp) is then given by

g(ω)=2πωIm[1ε(ω)]
(5)

 

The ELF from the FPA in Eqn (4) can be described as the sum of two contributions, one associated with the plasmon pole and the other with single-electron excitations:

Im[1ε(q,ω)]=Im[1ε(q,ω)]pl+Im[1ε(q,ω)]se
(6)

Details of the evaluations of the plasmon-pole and single-electron contributions were given in a previous paper [].

We calculated IMFPs from Eqns (3) to (6) using the approach of Boutboul et al.[] to include the effect of the bandgap energy in semiconductors and insulators. The IMFP, λ, at electron energy T(>Eg+Ev), which is measured from the bottom of the valence band for semiconductors and insulators, can be expressed as []:

λ(T)1=(1+T/c2)2(1+T/(2c2)1πTD1qIm[1ε(ω,q)]dqdω
(7)

where T=TEg and Eg is the bandgap energy. The integration domain D is determined from the maximum and minimum energy losses and the largest and smallest kinematically-allowed momentum transfers for a given energy T and ω:

D={(ω,q):Egω(TEV),qqq+},
(8)

where Ev is the width of the valence band for semiconductors and insulators, and q±T(2+T/c2)−−−−−−−−−−−√±(Tω)[2+Tω/c2]−−−−−−−−−−−−−−−−−−√.

Table 1 shows the material-property data used in our IMFP calculations and in our analyses of the ELFs and IMFPs. We show values of the molecular weight M, bulk density ρ (g cm−3), number of valence electrons per molecule (Nv), free-electron plasmon energy (Ep), bandgap energy (Eg), and valence-band width (Ev) for each compound. The bandgap energy is also an important material parameter in the use of our TPP-2M predictive equation for IMFPs []. We chose the median of the Eg values in the literature [,,,,,,] for most of our compounds; however, only single values of Eg were available for NaCl [] and Y3Al5O12 []. The Ev values were obtained from the WIEN2k calculations for 27 compound semiconductors and values for the other eight insulators were estimated from the literature: Al2O3 [], KBr [], KCl [], MgF2 [], MgO [], NaCl, [], SiO2 [], Y3Al5O12 []. For the niobium, titanium, and vanadium carbides, the Ev values correspond to the Fermi energies that were estimated from the literature [].

Table 1.

Values of material parameters used in the IMFP calculations and in the analysis of IMFP results for our 42 inorganic compounds.

Compound M ρ (g cm−3) Nv Ep (eV) Eg(eV) Ev (eV)
AgBr 187.772 6.48 18 22.71 2.68 4.27
AgCl 143.321 5.59 18 24.14 3.25 4.26
h-AgI 234.773 5.72 18 19.08 2.92 3.23
Al2O3 101.961 3.97 24 27.86 8.63 8.0
AlAs 101.903 3.73 8 15.59 2.16 5.27
h-AlN 40.988 3.26 8 22.99 6 6.03
AlSb 148.742 4.28 8 13.83 1.62 5.32
c-BN 24.818 3.49 8 30.56 7.2 8.51
h-BN 24.818 2.3 8 24.81 5 8.77
h-CdS 144.476 4.8 18 22.28 2.46 4.42
h-CdSe 191.371 5.66 18 21.03 1.7 4.53
CdTe 240.011 5.85 18 19.09 1.51 4.62
GaAs 144.645 5.32 8 15.63 1.47 7.07
h-GaN 83.7297 6.09 8 21.98 3.4 7.09
GaP 100.697 4.13 8 16.51 2.26 6.67
GaSb 191.483 5.61 8 13.95 0.73 7.02
h-GaSe 148.683 5.07 9 15.96 1.98 7.73
InAs 189.74 5.67 8 14.09 0.36 6.12
InP 145.792 4.79 8 14.77 1.38 5.98
InSb 236.578 5.78 8 12.74 0.18 6.18
KBr 119.002 2.75 8 12.39 7.26 2.6
KCl 74.548 1.98 8 13.28 7.4 2.7
MgF2 62.302 3.177 16 26.03 10.95 5.5
MgO 40.304 3.576 8 24.28 7.69 6.3
NaCl 58.443 2.165 8 15.69 9 4.1
NbC0.712 101.458 7.746 7.848 22.31 0 7.4*
NbC0.844 103.044 7.769 8.376 22.90 0 7.4*
NbC0.93 104.077 7.781 8.72 23.27 0 7.4*
PbS 239.265 7.62 10 16.26 0.42 5.21
PbSe 286.16 8.29 10 15.51 0.29 5.00
PbTe 334.8 8.27 10 14.32 0.32 4.54
SiC 40.096 3.22 8 23.10 2.31 6.95
SiO2 60.008 2.19 16 22.02 9.1 10.0
SnTe 246.31 6.47 10 14.77 0.19 8.25
TiC0.7 56.288 4.627 6.8 21.54 0 5.7*
TiC0.95 59.290 4.843 7.8 23.00 0 5.7*
VC0.76 60.070 5.582 8.04 24.91 0 7.5*
VC0.86 61.271 5.605 8.44 25.32 0 7.5*
Y3Al5O12 593.618 4.554 96 24.73 6.5 6.5
ZnS 97.445 4.09 18 25.05 3.81 5.37
ZnSe 144.34 5.26 18 23.34 2.68 5.42
ZnTe 192.98 5.64 18 20.9 2.25 5.42
*Fermi energy (eV).

3. Optical energy-loss functions

We have shown that IMFPs can be calculated with the FPA and SPA from ELFs obtained from experimental optical data (i.e., optical ELFs) for many materials of interest.[] The ELF is thus the critical material-dependent parameter in our IMFP calculations. Ideally, optical ELFs should be determined from measured optical constants. In many cases, we utilized optical constants from Handbooks[] although we have occasionally used optical constants derived from analyses of electron energy-loss experiments.[] Our choices from sets of optical data were guided by evaluations of the ELFs using the two important sum rules described in the ‘Evaluations of energy-loss functions’ subsection.

Unfortunately, experimental optical constants are generally not available for many inorganic compounds, especially for the 20 eV to 50 eV energy range that is particularly important for the IMFP calculations. As an alternative, we have made first-principles calculations of optical constants using the WIEN2k[] and FEFF[] codes, as described in the next subsection. These calculations were made for 27 inorganic compounds (c-AgBr, c-AgCl, h– AgI, c– AlAs, h-AlN, c- AlSb, c– BN, h-BN, CdS, h-CdSe, c-CdTe, c– GaAs, h-GaN, c- GaP, c- GaSb, h– GaSe, c– InAs, c– InP, c– InSb, c– PbS, c– PbSe, c– PbTe, c– SiC, c– SnTe, c– ZnS, c– ZnS, and c– ZnTe). These compounds were selected because optical data for photon energies less than 0.1 eV were needed for evaluation of one of the sum rules and were mostly available in the literature.[] Details of the first-principle calculations of ELFs will be published elsewhere.

Table 2 shows the sources of optical data used in our IMFP calculations. For 15 of the compounds (Al2O3, KBr, KCl, MgF2, MgO, NaCl, NbC0.712, NbC0.844, NbC0.93, SiO2, TiC0.7, TiC0.95, VC0.76, VC0.86, and Y3Al5O12), optical data from literature sources were available for photon energies up to at least 50 eV.

Table 2.

Sources of optical data used in the IMFP calculations for the 42 inorganic compounds.

Compound Photon energy range (eV) Source of data
AgBr 0.005283 to 0.02254 Ref.[]
2.6 to 30 Calculation with WIEN2k
30.40 to 1000000 Calculation with FEFF
AgCl 0.01 to 0.023 Ref.[]
3.0 to 30 Calculation with WIEN2k
30.40 to 1000000 Calculation with FEFF
AgI 0.004138 to 0.02857 Ref.[]
1.4 to 30 Calculation with WIEN2k
30.40 to 1000000 Calculation with FEFF
Al2O3 0.0496 to 100.0 Ref. []
101.94 to 30000 Ref. []
30156.09 to 988553.1 Ref. []
AlAs 0.01 to 0.1 Ref. []
2.1 to 50 Calculation with WIEN2k
50.5 to 1000000 Calculation with FEFF
AlN 0.01 to 1.0 Ref. []
4.3 to 70 Calculation with WIEN2k
70.5 to 10413.56 Calculation with FEFF
10570.17 to 1000000 Ref. []
AlSb 0.01 to 0.3 Ref. []
1.70 to 50 Calculation with WIEN2k
50.5 to 1000000 Calculation with FEFF
c-BN 0.01 to 0.14 Ref. []
8.90 to 80.0 Calculation with WIEN2k
80.5 to 9821.20 Calculation with FEFF
10413.56 to 1000000 Ref. []
h-BN 0.01 to 1 Ref. []
4.7 to 80.0 Calculation with WIEN2k
80.5 to 9821.20 Calculation with FEFF
10413.56 to 1000000 Ref. []
CdS 0.01 to 0.06 Ref. []
1.2 to 30 Calculation with WIEN2k
30.4 to 1000000 Calculation with FEFF
CdSe 0.01 to 0.1 Ref. []
0.7 to 30 Calculation with WIEN2k
30.4 to 1000000 Calculation with FEFF
CdTe 0.004 to 0.1 Ref. []
0.8 to 30 Calculation with WIEN2k
30.4 to 1000000 Calculation with FEFF
GaAs 0.01 to 0.4 Ref. []
0.6 to 50 Calculation with WIEN2k
50.5 to 1000000 Calculation with FEFF
GaN 0.01 to 0.5 Ref. []
1.9 to 50 Calculation with WIEN2k
50.5 to 1000000 Calculation with FEFF
GaP 0.01 to 0.1 Ref. []
1.8 to 50 Calculation with WIEN2k
50.5 to 1000000 Calculation with FEFF
GaSb 0.01 to 0.3 Ref. []
0.4 to 50 Calculation with WIEN2k
50.5 to 1000000 Calculation with FEFF
GaSe 0.0031 to 0.3105 Ref. []
0.9 to 46 Calculation with WIEN2k
46.4 to 1000000 Calculation with FEFF
InAs 0.01 to 0.1 Ref. []
0.2 to 30 Calculation with WIEN2k
30.40 to 1000000 Calculation with FEFF
InP 0.01 to 0.3 Ref. []
0.7 to 30 Calculation with WIEN2k
30.40 to 1000000 Calculation with FEFF
InSb 0.001 to 0.04 Ref. []
0.1 to 30 Calculation with WIEN2k
30.40 to 1000000 Calculation with FEFF
KBr 0.00003682 to 40.0 Ref. []
41.85834 to 1070165 Ref. []
KCl 0.007439 to 43 Ref. []
44.4673 to 30000.0 Ref. []
30309.08 to 944060.9 Ref. []
MgF2 0.0006 to 83 Ref. []
84.164 to 977240 Ref. []
MgO 0.002 to 75.0 Ref. []
76.01 to 30000.0 Ref. []
30156.09 to 988553.13 Ref. []
NaCl 0.00004053 to 26 Ref. []
27.032 to 1011600 Ref. []
NbC0.712 0.02 to 79 Ref. []
80.177 to 30000 Ref. []
30070.87 to 1120601 Ref. []
NbC0.844 0.02 to 79 Ref. []
80.177 to 30000 Ref. []
30070.87 to 1120601 Ref. []
NbC0.93 0.02 to 79 Ref. []
80.177 to 30000 Ref. []
30070.87 to 1120601 Ref. []
PbS 0.001 to 0.035 Ref. []
0.4 to 50 Calculation with WIEN2k
50.5 to 1000000 Calculation with FEFF
PbSe 0.001 to 0.007 Ref. []
0.3 to 50 Calculation with WIEN2k
50.5 to 1000000 Calculation with FEFF
PbTe 0.001 to 5.4 Ref. []
5.5 to 50 Calculation with WIEN2k
50.5 to 1000000 Calculation with FEFF
SiC 0.01 to 0.4 Ref. []
4.6 to 50 Calculation with WIEN2k
80.5 to 9821.20 Calculation with FEFF
10143.01 to 1000000 Ref. []
SiO2 0.002 to 2000 Ref. []
2025.3 to 30000 Ref. []
30156.09 to 1000000 Ref. []
SnTe 0.1 to 36.8 Calculation with WIEN2k
37.2 to 1000000 Calculation with FEFF
TiC0.7 0.02 to 79.6 Ref. []
80.177 to 30000 Ref. []
30070.87 to 1120600 Ref. []
TiC0.95 0.02 to 79.6 Ref. []
80.177 to 30000 Ref. []
30070.87 to 1120600 Ref. []
VC0.76 0.02 to 79 Ref. []
80.177 to 30000 Ref. []
30071 to 1120600 Ref. []
VC0.86 0.02 to 79 Ref. []
80.177 to 30000 Ref. []
30071 to 1120600 Ref. []
Y3Al5O12 0.0031 to 49.45 Ref. []
50.935 to 30000 Ref. []
30156 to 1022000 Ref. []
ZnS 0.01 to 0.07 Ref. []
2.1 to 50 Calculation with WIEN2k
50.5 to 1000000 Calculation with FEFF
ZnSe 0.01 to 0.0744 Ref. []
1.4 to 50 Calculation with WIEN2k
50.5 to 1000000 Calculation with FEFF
ZnTe 0.01 to 0.04 Ref. []
1.3 to 38 Calculation with WIEN2k
38.4 to 1000000 Calculation with FEFF

3.1. Optical energy-loss functions from first-principles calculations

Crystal information for the inorganic compounds used in the WIEN2k[] and FEFF[] calculations is shown in Table 3.[] The WIEN2k code performs calculations of the electronic structure of solids using density functional theory (DFT). We used WIEN2k Version 08.02. The generalized gradient approximation (GGA) as parameterized by Perdew, Bruke, and Ernzerhof was adopted for the exchange-correlation potential []. We calculated the imaginary part of the dielectric function, ε2, from the optically-allowed transition amplitudes for photon energies between 0.1 eV and 136 eV. For hexagonal crystals, we calculated the optical constants under the condition that the electric field was perpendicular to the c-axis of the crystal.

Table 3.

Crystal information used in the FEFF and WIEN2k calculations for the indicated compound semiconductors. The cell parameters were obtained from the AtomWork database[].

Material Space Group Cell parameter (nm)
AgBr F m −3 m a = 0. 5775
AgCl F m −3 m a = 0.5543
AgI P 63 m c a = 0.45856
c = 0.749
γ = 120°
AlAs F −4 3 m a = 0.56605
AlN P 63 m c a = 0.311
c = 0.498
γ = 120°
AlSb F −4 3 m a = 0.6135
c-BN F −4 3 m a =0.36159
h-BN P 63 /mmc a = 0.2.5045
c =0.6606
γ = 120°
CdS P 63 m c a = 0.4142
c =0.6724
γ = 120°
CdSe P 63 m c a = 0.4299
c = 0.701
γ = 120°
CdTe F −4 3 m a = 0.6482
GaAs F −4 3 m a = 0.56532
GaN P 63 m c a =0.31891
c = 0.51855
γ = 120°
GaP F −4 3 m a = 0.54508
GaSb F −4 3 m a = 0.60959
GaSe P 63 /mmc a = 0.375
c = 1.5995
γ = 120°
InAs F −4 3 m a = 0.60577
InP F −4 3 m a = 0.58687
InSb F −4 3 m a = 0.64794
PbS F m −3 m a = 0.59315
PbSe F m −3 m a = 0.61213
PbTe F m −3 m a = 0.64541
SiC F −4 3 m a = 0.43581
SnTe F m −3 m a = 0.6323
ZnS F −4 3 m a = 0.54102
ZnSe F −4 3 m a = 0.56692
ZnTe F −4 3 m a = 0.61026

The FEFF 8.2 code is an automated program for calculating X-ray absorption spectra based on an ab initio all-electron, real-space relativistic Green’s function formalism[]. This code gives photoabsorption cross sections for all elements in a compound. Only crystal-structure information is needed to obtain the absorption spectra. We can then easily calculate ε2 These calculations were made for photon energies between 10 eV and 1 MeV.

Our calculations of ε2 with the FEFF and WIEN2k codes were made independently, and we found the values of ε2 from each code to be reasonably consistent for energies between 30 and 80 eV, as will be shown shortly. We made a dataset of ε2 values calculated by the FEFF and WIEN2K codes for each compound for photon energies from 0.1 eV to 1 MeV, and Table 2 shows the energy range utilized for the ε2 values from each code. The real part of the dielectric function, ε1, was calculated by use of the Kramers-Kronig relation[]:

ε1(ω)=1+2πP0ωε2(ω)(ω)2ω2dω,
(9)

where P is the Cauchy principal value. The actual integration range of Eqn (9) was between 0.1 eV and 1 MeV. We then calculated the optical ELF, Im[1/ε(ω)], from the data sets of ε1 and ε2.[]

We now give brief comments on the calculated ELFs for two compound semiconductors, GaAs and InSb, to assess the accuracy of our first-principles calculations of the EL ε2Fs. Figure 1 shows the plots of ε2 for GaAs and InSb calculated from the WIEN2k and FEFF codes for energies from 0.1 eV to 100 eV together with ε2 values calculated from the atomic scattering factors of Henke et al. []. We see that there is good agreement in Fig. 1(a) among the ε2 eV. For energies over 60 eV, the ε2 data for GaAs for energies between 30 eV and 60 values from WIEN2k are smaller than those from FEFF and from Henke et al. On the other hand, the ε2 values from FEFF are in good agreement with those from Henke et al. for energies between 30 eV and 100 eV. This agreement occurs because the FEFF code was developed mainly to calculate inner-shell X-ray absorption spectra, and it is thus expected to be particularly useful for energy losses greater than about 100 eV. In contrast, the WIEN2k was developed mainly for calculating the electronic states of the outer shells, and is particularly useful for energy losses less than about 50 eV.[]

An external file that holds a picture, illustration, etc. Object name is nihms-1530804-f0001.jpg

Plots of the imaginary part of the complex dielectric function, ε2, as a function of photon energy and of the energy-loss functions (ELFs) as a function of energy loss for GaAs and InSb. (a) ε2 for GaAs. The solid and dotted lines show calculated ε2 values from the WIEN2k and FEFF codes, respectively. The solid circles indicate experimental data from Henke et al.[]. (b) ε2 for InSb. See caption to (a). (c) ELFs for GaAs. The solid line shows the optical ELF calculated from the WIEN2k code. The dot-dashed line is the ELF for GaAs measured by Jin et al.[] and the dotted line is the ELF determined by Chen et al.[]. The solid circles show the ELF from the optical data tabulated by Palik.[] (d) ELFs for InSb. The solid line is the optical ELF from the WIEN2k code. The short-dashed line is the ELF measured by Festenberg et al.[] The solid circles show the ELF from the optical data tabulated by Palik.[]

For InSb, as shown in Fig. 1(b), the ε2 data from WIEN2k are in good agreement with those from FEFF for energies between 30 eV and 50 eV. We also see that the ε2 data of Henke et al. are slightly smaller than the WIEN2k and FEFF values for the same energy range. For energies over 60 eV, the ε2 results from FEFF are larger than those from WIEN2k, as found for GaAs.

Figure 1 also shows ELFs of GaAs and InSb calculated from the WIEN2k code together with ELFs obtained from the transmission-EELS experiments of von Festenberg,[] as analyzed by Chen et al.,[] and the reflection-EELS experiments of Jin et al.[] The GaAs ELF from WIEN2k for energy losses up to 30 eV is in excellent agreement with the ELF determined by Jin et al. except at around 20 eV, as shown in Fig. 1(c). It is also in good agreement with the ELF obtained by Chen et al.,[] especially for the large peak at around 16 eV that corresponds to bulk-plasmon excitation, although there are some differences at around 10 eV and over 20 eV. However, the peak found by Chen et al. at around 10 eV is not seen in the ELF of Jin et al. The plasmon peak is at 15.6 eV in the WIEN2k ELF, in good agreement with the peak positions in the ELFs of Jin et al. (15.4 eV) and Chen et al. (15.7 eV), and with the tabulation of Egerton[] (15.8 eV).

On the other hand, the GaAs ELF from WIEN2k is very different from the ELF shown in Fig. 1(c) that was calculated from experimental optical constants tabulated by Palik[] and which we used in our previous IMFP calculations for GaAs[]. For photon energies between 6 eV and 23 eV, we utilized optical constants that were obtained from near-normal-incidence reflectance measurements in a vacuum spectrometer[]. The GaAs sample used by Philipp and Ehrenreich[] was prepared by chemical etching to remove distorted layers produced by mechanical polishing but was exposed to air before mounting in their sample chamber. The sample was also exposed to “the poor vacuum” of their monochromator during their measurements; at the time of this work, reflectance measurements in the ultraviolet spectral region were typically made in high-vacuum chambers. As noted by Palik[], the GaAs sample of Ehrenreich and Philipp probably had a ≈ 2 nm thick native oxide on its surface. It is therefore not surprising now that the GaAs ELF in Fig. 1(c) from the Palik optical data differs considerably from the other ELFs. We note particularly that the plasmon peak from the Palik data is at 14.0 eV and is much weaker than for the other ELFs in Fig. 1(c).

The ELF for InSb from the WIEN2k code is shown in Fig. 1(d) and is in good agreement with the ELF determined by von Festenberg[] from transmission-EELS experiments. The bulk-plasmon peak is at 13.2 eV in the WIEN2k ELF which is also in good agreement with the peak position in the von Festenberg ELF (12.9 eV). As for InSb, the ELF from the optical data of Palik[] (also used in our previous IMFP calculations for InSb[]) shows a plasmon peak at 12.0 eV that is much weaker than the plasmon peak found in the ELFs from WIEN2k and von Festenberg[]. The optical constants tabulated by Palik for InSb were also obtained from the reflectivity measurements of Philipp and Ehrenreich[] for photon energies between 6 eV and 24 eV. Their InSb sample was prepared and handled in the same way as their GaAs sample, and it is again likely that the InSb sample had a surface oxide.

We conclude that the ELFs of GaAs and InSb obtained from WIEN2k are superior to the ELFs obtained from published optical data[]. We also point out that small differences in peak positions and peak heights in ELFs from different sources are expected to have insignificant effects on calculated IMFPs since these IMFPs are derived from an integration of the ELF using Eqn (7). Further evaluations of the ELFs from WIEN2k and FEFF are given in the following subsection.

3.2. Evaluations of energy-loss functions

We checked the internal consistency of our ELF data for each compound with the oscillator-strength or f-sum rule and a limiting form of the Kramers-Kronig integral (or KK-sum rule).[,] The f-sum can be evaluated as the total effective number of electrons per molecule, Zeff, contributing to the inelastic scattering:

Zeff=(2/π2Ω2p)ΔEmaxEgΔEIm[1/ε(ΔE)]d(ΔE)
(10)

where ΔE=ω,Ωp=(4πnae2/m)1/2na=Naρ/M is the density of atoms, Na is Avogadro’s number, ρ is the mass density, and M is the molecular weight. The maximum energy loss in Eqn (10), ΔEmax, was chosen to be 1 MeV. Ideally, the value of Zeff should be equal to (or otherwise close to) the total number of electrons per molecule, Z. The KK-sum can be expressed as:

Peff=(2/π)ΔEmax0ΔE1Im[1/ε(ΔE)]d(ΔE)+n2(0)
(11)

where n(0) is the limiting value of the refractive index at low photon energies.

In the limit ΔEmax → ∞,, Zeff → Z and Peff → 1. We determined Z eff and Peff from Eqns (10) and (11) for each compound as a function of ΔEmax up to a maximum value of 1 MeV. Table 4 lists the errors in the f-sum and KK-sum rules for each inorganic compound, that is, the differences between the computed values of Zeff and Peff and the expected values (the total number of electrons per molecule and unity, respectively). The average RMS errors in the f-sum and KK-sum rules were 4.1 % and 3.5 %, respectively, for our sets of ELF data. These average RMS errors are comparable to those found in our ELF data sets for a group of 41 elemental solids[], 4.2 % for the f-sum error and 7.7 % for the KK-sum error. Over 80 % of our compounds had sum-rule errors less than 5 % for both sum rules. In our previous analyses of optical ELFs, obtained from experimental optical data for 15 inorganic compounds, the average sum-rule errors were 8 % and 24 % for the f-sum and KK-sum rules, respectively[].

Table 4.

List of 42 inorganic compounds with values of the number of electrons per molecule, ZZeff from Eqn (10), errors in the f-sum rule, values of Peff from Eqn (11), and errors in the KK-sum rule. Values of Zeff and Peff were determined with ΔEmax = 1 MeV.

Compound Z Zeff Error in f-sum rule (%) Peff Error in KK-sum rule (%)
AgBr 82 80.10 −2.3 1.023 2.3
AgCl 64 62.48 −2.4 0.977 −2.3
AgI 100 98.40 −1.6 1.035 3.5
Al2O3 50 49.39 −1.2 0.922 −7.8
AlAs 46 46.22 0.5 1.012 1.2
AlN 20 20.25 1.3 0.937 −6.3
AlSb 64 62.75 −1.9 1.018 1.8
c-BN 12 11.96 −0.3 0.953 −4.7
h-BN 12 12.18 1.5 0.997 −0.3
CdS 64 62.84 −1.8 1.005 0.5
CdSe 82 80.35 −2.0 1.058 5.8
CdTe 100 100.07 0.1 1.036 3.6
GaAs 64 62.76 −1.9 1.014 1.4
GaN 38 37.70 −0.8 1.033 3.3
GaP 46 45.04 −2.1 1.076 7.6
GaSb 82 79.95 −2.5 1.014 1.4
GaSe 65 63.98 −1.6 1.003 0.3
InAs 82 80.18 −2.2 1.031 3.1
InP 64 62.45 −2.4 1.001 0.1
InSb 100 97.09 −2.9 1.018 1.8
KBr 54 54.07 0.1 0.943 −5.7
KCl 36 35.33 −1.9 0.984 −1.6
MgF2 30 32.76 9.2 1.038 3.8
MgO 20 20.22 1.1 1.003 0.3
NaCl 28 27.01 −3.6 0.917 −8.3
NbC0.712 45.272 41.05 −9.3 0.996 −0.4
NbC0.844 46.064 41.55 −9.8 0.996 −0.4
NbC0.93 46.58 41.87 −10.1 0.996 −0.4
PbS 98 95.38 −2.7 0.993 −0.7
PbSe 116 112.70 −2.8 0.989 −1.1
PbTe 134 130.47 −2.6 1.012 1.2
SiC 20 19.91 −0.4 1.070 7.0
SiO2 30 28.14 −6.2 1.045 4.5
SnTe 102 100.20 −1.8 0.999 −0.1
TiC0.7 26.2 26.23 0.1 1.005 0.5
TiC0.95 27.7 27.66 −0.1 1.006 0.6
VC0.76 27.56 24.83 −9.9 1.011 1.1
VC0.86 28.16 25.68 −8.8 1.011 1.1
Y3Al5O12 278 280.24 0.8 0.962 −3.8
ZnS 46 44.83 −2.6 1.022 2.2
ZnSe 64 63.13 −1.4 0.992 −0.8
ZnTe 82 82.28 0.4 1.032 3.2

For the example case of GaAs in Fig. 1(c), we previously found large sum-rule errors (−13 % for the f-sum rule and −37 % for the KK-sum rule) for the GaAs ELF obtained from the Palik[] optical data for photon energies between 1.5 eV and 100 eV and from Henke et al.[] for higher energies.[] In contrast, the sum-rule errors for the GaAs ELF from the WIEN2k and FEFF calculations were −1.9 % for the f-sum rule and 1.4 % for the KK-sum rule. This large reduction in the sum-rule errors is due mainly to the more reliable ELF from WIEN2k for photon energies less than 30 eV in Fig. 1(c). As noted in the previous subsection, the ELF from WIEN2k generally agrees well with the ELFs from the EELS experiments of von Festenberg[] (analyzed by Chen et al.[]) and of Jin et al[] The large improvement in the results of the KK-sum rule is due to the fact that this evaluation emphasizes the ELF for relatively small energy losses [Eqn (11)] whereas the f-sum rule [Eqn (10)] emphasizes the ELF for relatively large energy losses. As we can see from Fig. 1(c), there is a substantial difference in the GaAs ELF from the Palik data and the more recent ELFs from WIEN2k and EELS data. We reach similar conclusions for the example case of InSb in Fig. 1(d).

Figure 2 shows comparisons of the f-sum and KK-sum errors for the 15 inorganic compounds in our previous work[] and the corresponding values for the same compounds in the present work. We see that the new values cluster much closer to the origin with sum-rule errors generally less than 10 %. The present ELF data are clearly superior to the previous ELF data, especially in the KK-sum rule results.

An external file that holds a picture, illustration, etc. Object name is nihms-1530804-f0002.jpg

Plots of KK-sum rule errors versus f-sum rule errors for our group of 42 inorganic compounds (solid circles) and for the 15 compounds in our previous work (solid squares) [].

4. Results

4.1. Calculated IMFPs from the relativistic full Penn algorithm

Table 5 shows our calculated IMFPs for the 42 inorganic compounds as a function of the electron kinetic energy E(=TEgEv) with respect to the bottom of the conduction band between 50 eV and 200 keV. Plots of IMFPs as a function of electron energy are shown as solid circles in Figs. 3 to to10.10. IMFPs are included in these plots for energies less than 50 eV and over 200 keV to illustrate trends. The IMFPs for energies less than 50 eV, however, are not considered as reliable as those at higher energies[,] while the IMFPs for energies larger than 200 keV must be slightly larger than the correct values because we neglected the contribution of the transverse DCS shown in Eqn (1).[]

An external file that holds a picture, illustration, etc. Object name is nihms-1530804-f0003.jpg

Plots of our calculated inelastic mean free paths as a function of electron kinetic energy for AgBr, AgCl, AgI, Al2O3, and AlAs. The solid circles show calculated IMFPs from the relativistic full Penn algorithm (Table 5). The solid lines show fits to these IMFPs with the relativistic modified Bethe equation [Eqns (12) and (13)] and the derived parameters are listed in Table 6. The long-dashed lines indicate IMFPs calculated from the relativistic TPP-2M equation [Eqns (12)(13) and (15)].

An external file that holds a picture, illustration, etc. Object name is nihms-1530804-f0010.jpg

Plots of calculated electron inelastic mean free paths as a function of electron kinetic energy for VC0.76, VC0.86, Y3Al5O12; YAG, ZnS, ZnSe, and ZnTe. See caption to Fig. 3.

Table 5.

Calculated IMFPs for the 42 inorganic compounds as a function of electron energy E with respect to the bottom of the conduction band.

Inelastic mean free path (nm)


E(eV) AgBr AgCl AgI Al2O3 AlAs AlN AlSb c-BN h-BN CdS CdSe
54.6 0.533 0.562 0.524 0.757 0.401 0.533 0.416 0.718 0.597 0.503 0.489
60.3 0.524 0.550 0.524 0.711 0.410 0.513 0.430 0.652 0.550 0.504 0.492
66.7 0.520 0.543 0.529 0.677 0.424 0.502 0.446 0.597 0.522 0.509 0.498
73.7 0.520 0.539 0.537 0.654 0.440 0.499 0.464 0.555 0.510 0.516 0.507
81.5 0.523 0.540 0.547 0.640 0.459 0.501 0.485 0.521 0.504 0.526 0.518
90.0 0.528 0.543 0.560 0.634 0.481 0.510 0.507 0.498 0.504 0.538 0.531
99.5 0.536 0.548 0.575 0.634 0.506 0.523 0.532 0.491 0.512 0.552 0.547
109.9 0.546 0.557 0.593 0.640 0.533 0.540 0.559 0.491 0.525 0.568 0.565
121.5 0.560 0.569 0.614 0.651 0.563 0.560 0.588 0.497 0.542 0.587 0.585
134.3 0.578 0.584 0.638 0.667 0.597 0.585 0.619 0.508 0.563 0.608 0.608
148.4 0.599 0.604 0.665 0.688 0.633 0.613 0.652 0.524 0.588 0.632 0.634
164.0 0.625 0.628 0.694 0.713 0.672 0.644 0.687 0.543 0.616 0.659 0.663
181.3 0.655 0.657 0.723 0.743 0.715 0.679 0.724 0.567 0.649 0.691 0.696
200.3 0.690 0.690 0.752 0.776 0.760 0.717 0.764 0.595 0.685 0.728 0.734
221.4 0.728 0.728 0.788 0.814 0.810 0.758 0.808 0.626 0.725 0.769 0.776
244.7 0.772 0.770 0.830 0.856 0.862 0.803 0.856 0.661 0.770 0.816 0.823
270.4 0.819 0.818 0.878 0.903 0.919 0.851 0.910 0.699 0.819 0.867 0.874
298.9 0.872 0.871 0.933 0.954 0.980 0.903 0.969 0.742 0.873 0.925 0.931
330.3 0.930 0.929 0.993 1.01 1.05 0.960 1.03 0.789 0.932 0.988 0.993
365.0 0.992 0.992 1.06 1.07 1.12 1.02 1.11 0.841 0.996 1.06 1.06
403.4 1.06 1.06 1.13 1.14 1.20 1.09 1.18 0.898 1.07 1.13 1.13
445.9 1.13 1.14 1.22 1.21 1.28 1.16 1.27 0.961 1.14 1.21 1.21
492.7 1.22 1.22 1.30 1.29 1.37 1.24 1.36 1.03 1.23 1.30 1.30
544.6 1.30 1.31 1.40 1.38 1.47 1.33 1.47 1.10 1.32 1.40 1.40
601.8 1.40 1.41 1.51 1.47 1.58 1.43 1.58 1.18 1.42 1.51 1.50
665.1 1.50 1.52 1.62 1.58 1.70 1.53 1.70 1.27 1.53 1.62 1.62
735.1 1.62 1.64 1.75 1.69 1.83 1.65 1.83 1.37 1.65 1.75 1.74
812.4 1.74 1.76 1.89 1.81 1.97 1.77 1.98 1.47 1.78 1.88 1.87
897.8 1.88 1.90 2.03 1.95 2.12 1.91 2.13 1.59 1.92 2.03 2.02
992.3 2.02 2.05 2.20 2.10 2.29 2.06 2.31 1.71 2.07 2.19 2.18
1096.6 2.18 2.21 2.37 2.26 2.47 2.22 2.49 1.85 2.24 2.37 2.35
1212.0 2.35 2.39 2.57 2.44 2.67 2.40 2.70 2.00 2.42 2.56 2.54
1339.4 2.54 2.58 2.77 2.63 2.89 2.59 2.92 2.16 2.62 2.77 2.74
1480.3 2.75 2.79 3.00 2.84 3.12 2.81 3.16 2.33 2.83 3.00 2.97
1636.0 2.97 3.02 3.25 3.07 3.38 3.03 3.42 2.52 3.07 3.24 3.21
1808.0 3.21 3.27 3.52 3.32 3.66 3.28 3.71 2.73 3.32 3.51 3.47
1998.2 3.48 3.53 3.81 3.59 3.97 3.56 4.02 2.95 3.60 3.80 3.76
2208.3 3.76 3.83 4.13 3.88 4.30 3.85 4.36 3.20 3.91 4.12 4.07
2440.6 4.07 4.14 4.47 4.20 4.66 4.18 4.73 3.47 4.24 4.47 4.41
2697.3 4.41 4.49 4.85 4.55 5.06 4.53 5.14 3.76 4.60 4.84 4.78
2981.0 4.78 4.87 5.26 4.94 5.49 4.91 5.57 4.08 4.99 5.25 5.19
3294.5 5.19 5.28 5.70 5.35 5.96 5.33 6.05 4.42 5.42 5.70 5.63
3641.0 5.63 5.73 6.19 5.80 6.47 5.78 6.57 4.80 5.88 6.18 6.11
4023.9 6.10 6.21 6.71 6.29 7.03 6.28 7.13 5.21 6.39 6.71 6.63
4447.1 6.62 6.74 7.29 6.83 7.64 6.82 7.75 5.65 6.94 7.29 7.19
4914.8 7.19 7.32 7.91 7.41 8.30 7.41 8.42 6.14 7.54 7.91 7.81
5431.7 7.80 7.95 8.59 8.04 9.02 8.04 9.15 6.67 8.19 8.60 8.48
6002.9 8.48 8.63 9.33 8.73 9.80 8.74 9.95 7.24 8.91 9.34 9.21
6634.2 9.20 9.38 10.14 9.48 10.7 9.50 10.8 7.87 9.68 10.1 10.0
7332.0 10.0 10.2 11.01 10.3 11.6 10.3 11.8 8.55 10.5 11.0 10.9
8103.1 10.9 11.1 11.96 11.2 12.6 11.2 12.8 9.29 11.4 12.0 11.8
8955.3 11.8 12.0 13.00 12.2 13.7 12.2 13.9 10.1 12.4 13.0 12.8
9897.1 12.8 13.1 14.13 13.2 14.9 13.2 15.1 11.0 13.5 14.2 14.0
10938.0 13.9 14.2 15.35 14.3 16.2 14.4 16.4 11.9 14.7 15.4 15.2
12088.4 15.1 15.4 16.68 15.6 17.6 15.6 17.8 13.0 16.0 16.7 16.5
13359.7 16.4 16.8 18.12 16.9 19.1 17.0 19.4 14.1 17.4 18.2 17.9
14764.8 17.8 18.2 19.69 18.4 20.8 18.5 21.1 15.3 18.9 19.7 19.4
16317.6 19.4 19.8 21.38 19.9 22.6 20.1 22.9 16.6 20.5 21.4 21.1
18033.7 21.0 21.5 23.22 21.7 24.5 21.8 24.9 18.0 22.3 23.3 22.9
19930.4 22.8 23.3 25.21 23.5 26.6 23.7 27.0 19.6 24.2 25.3 24.9
22026.5 24.8 25.3 27.36 25.5 28.9 25.7 29.4 21.3 26.3 27.5 27.0
24343.0 26.9 27.4 29.69 27.7 31.4 27.9 31.9 23.1 28.5 29.8 29.3
26903.2 29.2 29.8 32.20 30.0 34.1 30.2 34.6 25.0 31.0 32.3 31.8
29732.6 31.6 32.3 34.90 32.5 36.9 32.8 37.5 27.1 33.6 35.0 34.5
32859.6 34.2 35.0 37.81 35.2 40.0 35.5 40.6 29.4 36.4 38.0 37.4
36315.5 37.1 37.8 40.94 38.1 43.4 38.4 44.0 31.8 39.4 41.1 40.5
40134.8 40.1 41.0 44.30 41.2 46.9 41.6 47.6 34.4 42.7 44.5 43.8
44355.9 43.4 44.3 47.90 44.6 50.8 45.0 51.5 37.2 46.1 48.1 47.3
49020.8 46.8 47.8 51.75 48.1 54.9 48.6 55.7 40.2 49.9 52.0 51.1
54176.4 50.6 51.6 55.86 51.9 59.2 52.5 60.1 43.4 53.8 56.1 55.2
59874.1 54.5 55.7 60.24 56.0 63.9 56.6 64.8 46.8 58.1 60.5 59.5
66171.2 58.7 60.0 64.88 60.3 68.8 60.9 69.9 50.4 62.5 65.2 64.1
73130.4 63.2 64.5 69.80 64.9 74.1 65.6 75.2 54.2 67.3 70.1 69.0
80821.6 67.8 69.3 74.99 69.7 79.6 70.4 80.8 58.3 72.3 75.3 74.1
89321.7 72.8 74.3 80.46 74.7 85.4 75.6 86.7 62.5 77.6 80.8 79.5
98715.8 78.0 79.6 86.19 80.0 91.5 81.0 92.9 66.9 83.2 86.6 85.2
109097.8 83.4 85.2 92.19 85.6 97.9 86.6 99.4 71.6 89.0 92.6 91.1
120571.7 89.0 90.9 98.42 91.4 105 92.5 106 76.4 95.0 98.9 97.3
133252.4 94.9 96.9 105 97.3 111 98.6 113 81.5 101 105 104
147266.6 101 103 112 104 119 105 120 86.6 108 112 110
162754.8 107 109 118 110 126 111 128 92.0 114 119 117
179871.9 113 116 125 116 133 118 135 97.4 121 126 124
198789.2 120 122 133 123 141 125 143 103 128 133 131
Inelastic mean free path (nm)


E(eV) CdTe CdTe GaN GaP GaSb GaSe InAs InAs InSb KBr KCl
54.6 0.489 0.415 0.531 0.408 0.428 0.428 0.446 0.445 0.452 0.838 0.785
60.3 0.497 0.422 0.516 0.414 0.439 0.435 0.454 0.451 0.463 0.812 0.755
66.7 0.507 0.433 0.507 0.424 0.453 0.445 0.464 0.461 0.477 0.793 0.732
73.7 0.519 0.447 0.505 0.437 0.469 0.458 0.478 0.473 0.492 0.783 0.721
81.5 0.533 0.464 0.508 0.453 0.487 0.474 0.493 0.488 0.510 0.782 0.722
90.0 0.549 0.483 0.515 0.471 0.507 0.493 0.510 0.504 0.528 0.788 0.733
99.5 0.567 0.505 0.525 0.492 0.529 0.515 0.529 0.522 0.549 0.805 0.753
109.9 0.587 0.529 0.538 0.516 0.552 0.539 0.550 0.542 0.570 0.831 0.780
121.5 0.608 0.555 0.555 0.542 0.578 0.566 0.572 0.564 0.593 0.866 0.814
134.3 0.630 0.584 0.574 0.571 0.606 0.596 0.597 0.587 0.618 0.907 0.854
148.4 0.651 0.616 0.595 0.603 0.635 0.629 0.623 0.613 0.643 0.955 0.899
164.0 0.672 0.651 0.620 0.638 0.665 0.665 0.652 0.641 0.670 1.01 0.951
181.3 0.696 0.688 0.647 0.676 0.697 0.704 0.684 0.673 0.699 1.07 1.01
200.3 0.725 0.729 0.677 0.717 0.732 0.746 0.720 0.709 0.732 1.13 1.07
221.4 0.760 0.773 0.711 0.762 0.771 0.791 0.760 0.750 0.769 1.21 1.14
244.7 0.801 0.820 0.748 0.811 0.815 0.841 0.805 0.795 0.811 1.29 1.22
270.4 0.848 0.871 0.789 0.864 0.863 0.894 0.855 0.846 0.859 1.37 1.30
298.9 0.901 0.927 0.834 0.922 0.917 0.953 0.909 0.902 0.913 1.47 1.39
330.3 0.960 0.987 0.883 0.985 0.976 1.02 0.969 0.964 0.973 1.57 1.49
365.0 1.02 1.05 0.937 1.05 1.04 1.08 1.03 1.03 1.04 1.68 1.60
403.4 1.10 1.12 0.996 1.13 1.11 1.16 1.11 1.11 1.11 1.80 1.72
445.9 1.17 1.20 1.06 1.21 1.19 1.24 1.18 1.19 1.19 1.93 1.85
492.7 1.26 1.28 1.13 1.29 1.28 1.32 1.27 1.27 1.28 2.07 1.99
544.6 1.35 1.37 1.21 1.39 1.37 1.42 1.36 1.37 1.38 2.23 2.14
601.8 1.46 1.47 1.29 1.49 1.47 1.52 1.46 1.47 1.48 2.39 2.31
665.1 1.57 1.58 1.39 1.60 1.58 1.63 1.57 1.58 1.59 2.58 2.49
735.1 1.69 1.70 1.49 1.72 1.70 1.76 1.69 1.71 1.72 2.77 2.68
812.4 1.82 1.83 1.60 1.85 1.84 1.89 1.82 1.84 1.85 2.99 2.90
897.8 1.96 1.97 1.72 2.00 1.98 2.03 1.97 1.99 2.00 3.22 3.13
992.3 2.12 2.12 1.85 2.15 2.14 2.19 2.12 2.14 2.16 3.47 3.38
1096.6 2.29 2.28 1.99 2.32 2.31 2.36 2.29 2.32 2.33 3.75 3.65
1212.0 2.48 2.47 2.14 2.51 2.49 2.55 2.48 2.50 2.52 4.05 3.95
1339.4 2.68 2.66 2.31 2.71 2.69 2.75 2.68 2.71 2.73 4.38 4.27
1480.3 2.90 2.88 2.50 2.93 2.91 2.98 2.89 2.93 2.96 4.73 4.62
1636.0 3.14 3.11 2.70 3.17 3.15 3.22 3.13 3.17 3.20 5.12 5.00
1808.0 3.40 3.37 2.92 3.44 3.41 3.48 3.39 3.44 3.47 5.54 5.42
1998.2 3.68 3.65 3.15 3.72 3.70 3.77 3.67 3.72 3.76 6.00 5.87
2208.3 3.99 3.95 3.41 4.04 4.01 4.08 3.98 4.04 4.07 6.50 6.37
2440.6 4.32 4.28 3.70 4.38 4.34 4.43 4.31 4.38 4.41 7.04 6.90
2697.3 4.68 4.64 4.00 4.75 4.71 4.80 4.67 4.75 4.78 7.63 7.49
2981.0 5.08 5.03 4.34 5.15 5.11 5.20 5.07 5.15 5.19 8.28 8.13
3294.5 5.51 5.46 4.70 5.59 5.54 5.65 5.49 5.59 5.63 8.98 8.83
3641.0 5.98 5.92 5.10 6.07 6.01 6.13 5.96 6.06 6.11 9.75 9.59
4023.9 6.49 6.43 5.53 6.59 6.53 6.65 6.47 6.58 6.63 10.6 10.4
4447.1 7.04 6.98 6.00 7.16 7.09 7.22 7.03 7.15 7.20 11.5 11.3
4914.8 7.64 7.58 6.51 7.78 7.70 7.84 7.63 7.76 7.81 12.5 12.3
5431.7 8.30 8.23 7.07 8.45 8.36 8.52 8.29 8.44 8.49 13.6 13.4
6002.9 9.02 8.94 7.68 9.18 9.08 9.26 9.00 9.17 9.22 14.7 14.5
6634.2 9.80 9.72 8.33 9.98 9.87 10.1 9.78 9.96 10.0 16.0 15.8
7332.0 10.6 10.6 9.05 10.8 10.7 10.9 10.6 10.8 10.9 17.4 17.2
8103.1 11.6 11.5 9.83 11.8 11.7 11.9 11.5 11.8 11.8 18.9 18.7
8955.3 12.6 12.5 10.7 12.8 12.7 12.9 12.5 12.8 12.9 20.6 20.3
9897.1 13.7 13.5 11.6 13.9 13.8 14.0 13.6 13.9 14.0 22.3 22.1
10938.0 14.8 14.7 12.6 15.1 15.0 15.2 14.8 15.1 15.2 24.3 24.0
12088.4 16.1 16.0 13.7 16.5 16.3 16.6 16.1 16.4 16.5 26.4 26.1
13359.7 17.5 17.4 14.9 17.9 17.7 18.0 17.5 17.9 17.9 28.7 28.3
14764.8 19.0 18.9 16.1 19.4 19.2 19.6 19.0 19.4 19.5 31.2 30.8
16317.6 20.7 20.5 17.5 21.1 20.8 21.2 20.6 21.1 21.2 33.8 33.5
18033.7 22.4 22.3 19.0 22.9 22.6 23.1 22.4 22.9 23.0 36.8 36.4
19930.4 24.4 24.2 20.6 24.9 24.6 25.0 24.3 24.9 25.0 39.9 39.5
22026.5 26.5 26.3 22.4 27.1 26.7 27.2 26.4 27.0 27.1 43.3 42.9
24343.0 28.7 28.5 24.3 29.4 29.0 29.5 28.7 29.3 29.4 47.0 46.5
26903.2 31.1 30.9 26.3 31.9 31.4 32.0 31.1 31.8 31.9 51.0 50.5
29732.6 33.7 33.5 28.5 34.5 34.0 34.7 33.7 34.5 34.6 55.3 54.8
32859.6 36.6 36.3 30.9 37.4 36.9 37.6 36.5 37.3 37.5 59.9 59.4
36315.5 39.6 39.3 33.5 40.5 40.0 40.7 39.6 40.4 40.6 64.9 64.3
40134.8 42.8 42.5 36.2 43.9 43.2 44.0 42.8 43.8 43.9 70.2 69.6
44355.9 46.3 46.0 39.1 47.5 46.8 47.6 46.3 47.3 47.5 75.9 75.3
49020.8 50.1 49.7 42.3 51.3 50.5 51.5 50.0 51.1 51.3 82.0 81.3
54176.4 54.0 53.6 45.6 55.4 54.5 55.6 54.0 55.2 55.4 88.6 87.8
59874.1 58.3 57.8 49.2 59.7 58.8 59.9 58.2 59.5 59.7 95.5 94.7
66171.2 62.7 62.3 53.0 64.3 63.3 64.5 62.7 64.1 64.3 103 102
73130.4 67.5 67.0 57.0 69.2 68.2 69.4 67.5 69.0 69.2 111 110
80821.6 72.5 72.0 61.2 74.4 73.2 74.6 72.5 74.2 74.4 119 118
89321.7 77.8 77.3 65.6 79.8 78.6 80.0 77.8 79.6 79.8 128 127
98715.8 83.4 82.8 70.3 85.5 84.2 85.8 83.3 85.3 85.5 137 136
109097.8 89.2 88.5 75.2 91.5 90.0 91.7 89.1 91.2 91.5 146 145
120571.7 95.2 94.5 80.2 97.7 96.1 97.9 95.2 97.4 97.7 156 155
133252.4 101 101 85.5 104 102 104 101 104 104 166 165
147266.6 108 107 90.9 111 109 111 108 110 111 177 176
162754.8 115 114 96.5 118 116 118 115 117 118 188 187
179871.9 121 120 102 125 123 125 121 124 124 199 198
198789.2 128 127 108 132 129 132 128 131 132 210 209
Inelastic mean free path (nm)


E(eV) MgF2 MgO NaCl NbC0.712 NbC0.844 NbC0.93 PbS PbSe PbTe SiC SiO2
54.6 1.06 0.705 0.819 0.466 0.466 0.466 0.449 0.442 0.439 0.451 0.831
60.3 0.985 0.671 0.808 0.455 0.454 0.454 0.452 0.446 0.446 0.433 0.801
66.7 0.933 0.654 0.815 0.451 0.449 0.449 0.458 0.453 0.456 0.426 0.782
73.7 0.898 0.646 0.830 0.451 0.449 0.448 0.467 0.463 0.469 0.425 0.773
81.5 0.874 0.644 0.854 0.454 0.452 0.451 0.478 0.474 0.483 0.430 0.772
90.0 0.860 0.647 0.883 0.458 0.456 0.455 0.491 0.488 0.498 0.440 0.778
99.5 0.853 0.655 0.917 0.465 0.463 0.462 0.503 0.501 0.514 0.453 0.791
109.9 0.853 0.667 0.956 0.473 0.470 0.469 0.517 0.517 0.531 0.470 0.808
121.5 0.859 0.684 0.999 0.483 0.480 0.479 0.534 0.535 0.551 0.490 0.831
134.3 0.871 0.704 1.05 0.497 0.494 0.493 0.553 0.555 0.572 0.513 0.859
148.4 0.888 0.727 1.10 0.515 0.512 0.511 0.575 0.578 0.594 0.539 0.892
164.0 0.909 0.753 1.15 0.537 0.534 0.533 0.599 0.604 0.618 0.569 0.930
181.3 0.935 0.783 1.22 0.563 0.560 0.559 0.628 0.634 0.642 0.602 0.973
200.3 0.966 0.817 1.28 0.593 0.590 0.589 0.660 0.667 0.670 0.638 1.02
221.4 1.00 0.854 1.36 0.626 0.623 0.622 0.697 0.704 0.702 0.678 1.08
244.7 1.04 0.896 1.44 0.663 0.660 0.659 0.738 0.746 0.739 0.722 1.14
270.4 1.09 0.943 1.53 0.705 0.701 0.700 0.783 0.791 0.782 0.769 1.21
298.9 1.14 0.994 1.62 0.750 0.747 0.745 0.834 0.842 0.830 0.821 1.28
330.3 1.20 1.05 1.73 0.801 0.797 0.795 0.890 0.897 0.883 0.877 1.36
365.0 1.27 1.11 1.85 0.856 0.851 0.850 0.951 0.957 0.942 0.937 1.45
403.4 1.35 1.18 1.98 0.916 0.911 0.909 1.02 1.02 1.01 1.00 1.55
445.9 1.43 1.26 2.12 0.982 0.977 0.975 1.09 1.09 1.08 1.07 1.66
492.7 1.52 1.34 2.27 1.05 1.05 1.05 1.17 1.17 1.16 1.15 1.78
544.6 1.62 1.43 2.43 1.13 1.13 1.12 1.26 1.26 1.24 1.23 1.90
601.8 1.73 1.53 2.61 1.22 1.21 1.21 1.35 1.35 1.34 1.33 2.04
665.1 1.85 1.64 2.81 1.31 1.30 1.30 1.45 1.45 1.44 1.42 2.19
735.1 1.98 1.76 3.03 1.41 1.40 1.40 1.56 1.56 1.55 1.53 2.36
812.4 2.13 1.89 3.26 1.52 1.51 1.51 1.69 1.68 1.67 1.65 2.54
897.8 2.29 2.04 3.51 1.64 1.63 1.62 1.82 1.81 1.80 1.78 2.73
992.3 2.46 2.19 3.78 1.76 1.76 1.75 1.96 1.95 1.94 1.92 2.95
1096.6 2.65 2.36 4.08 1.90 1.89 1.89 2.11 2.11 2.10 2.07 3.18
1212.0 2.86 2.55 4.40 2.05 2.04 2.04 2.28 2.27 2.26 2.24 3.44
1339.4 3.08 2.75 4.76 2.22 2.21 2.20 2.47 2.45 2.45 2.42 3.71
1480.3 3.33 2.97 5.14 2.40 2.38 2.38 2.66 2.65 2.65 2.62 4.02
1636.0 3.59 3.21 5.56 2.59 2.57 2.57 2.88 2.87 2.86 2.83 4.35
1808.0 3.88 3.47 6.02 2.80 2.78 2.78 3.12 3.10 3.10 3.07 4.70
1998.2 4.20 3.76 6.52 3.03 3.01 3.01 3.37 3.35 3.35 3.32 5.09
2208.3 4.55 4.07 7.06 3.28 3.26 3.25 3.65 3.63 3.63 3.60 5.52
2440.6 4.92 4.41 7.66 3.55 3.53 3.52 3.96 3.93 3.93 3.90 5.98
2697.3 5.33 4.77 8.30 3.85 3.82 3.82 4.29 4.26 4.26 4.23 6.49
2981.0 5.77 5.18 9.01 4.17 4.15 4.14 4.65 4.62 4.62 4.59 7.04
3294.5 6.26 5.61 9.77 4.52 4.49 4.48 5.04 5.01 5.01 4.98 7.63
3641.0 6.78 6.09 10.6 4.90 4.87 4.86 5.47 5.43 5.43 5.41 8.28
4023.9 7.36 6.60 11.5 5.32 5.29 5.28 5.93 5.89 5.89 5.88 8.99
4447.1 7.98 7.17 12.5 5.77 5.74 5.73 6.44 6.39 6.40 6.38 9.77
4914.8 8.66 7.78 13.6 6.26 6.23 6.22 6.99 6.94 6.94 6.94 10.6
5431.7 9.40 8.44 14.8 6.80 6.76 6.75 7.59 7.53 7.54 7.54 11.5
6002.9 10.2 9.17 16.0 7.39 7.35 7.33 8.25 8.18 8.19 8.19 12.5
6634.2 11.1 9.96 17.4 8.02 7.98 7.96 8.96 8.89 8.89 8.90 13.6
7332.0 12.0 10.8 18.9 8.71 8.67 8.65 9.73 9.65 9.66 9.68 14.8
8103.1 13.1 11.7 20.6 9.47 9.42 9.39 10.6 10.5 10.5 10.5 16.1
8955.3 14.2 12.8 22.4 10.3 10.2 10.2 11.5 11.4 11.4 11.4 17.5
9897.1 15.4 13.9 24.3 11.2 11.1 11.1 12.5 12.4 12.4 12.4 19.0
10938.0 16.7 15.1 26.4 12.1 12.1 12.0 13.6 13.4 13.5 13.5 20.6
12088.4 18.2 16.4 28.7 13.2 13.1 13.1 14.7 14.6 14.6 14.7 22.4
13359.7 19.7 17.8 31.2 14.3 14.3 14.2 16.0 15.9 15.9 16.0 24.4
14764.8 21.4 19.3 33.9 15.6 15.5 15.4 17.4 17.2 17.3 17.3 26.5
16317.6 23.2 21.0 36.8 16.9 16.8 16.8 18.9 18.7 18.7 18.8 28.7
18033.7 25.2 22.8 40.0 18.4 18.3 18.2 20.5 20.3 20.4 20.5 31.2
19930.4 27.4 24.7 43.4 19.9 19.8 19.8 22.3 22.1 22.1 22.2 33.9
22026.5 29.7 26.8 47.2 21.6 21.5 21.5 24.2 23.9 24.0 24.1 36.8
24343.0 32.2 29.1 51.2 23.5 23.3 23.3 26.2 26.0 26.0 26.2 39.9
26903.2 34.9 31.5 55.5 25.4 25.3 25.2 28.4 28.2 28.2 28.4 43.3
29732.6 37.8 34.2 60.2 27.6 27.4 27.4 30.8 30.5 30.6 30.8 46.9
32859.6 41.0 37.0 65.2 29.9 29.7 29.6 33.4 33.1 33.1 33.4 50.9
36315.5 44.3 40.0 70.6 32.3 32.2 32.1 36.1 35.8 35.9 36.2 55.1
40134.8 48.0 43.3 76.4 35.0 34.8 34.7 39.1 38.7 38.8 39.1 59.6
44355.9 51.8 46.8 82.6 37.8 37.6 37.5 42.3 41.9 41.9 42.3 64.4
49020.8 56.0 50.6 89.3 40.9 40.6 40.6 45.7 45.2 45.3 45.7 69.6
54176.4 60.4 54.6 96.4 44.1 43.9 43.8 49.3 48.8 48.9 49.4 75.2
59874.1 65.1 58.9 104 47.6 47.3 47.2 53.2 52.7 52.7 53.3 81.0
66171.2 70.1 63.4 112 51.2 50.9 50.8 57.3 56.7 56.8 57.4 87.3
73130.4 75.4 68.2 120 55.1 54.8 54.7 61.6 61.0 61.1 61.7 93.9
80821.6 81.0 73.2 129 59.2 58.9 58.7 66.2 65.5 65.6 66.3 101
89321.7 86.9 78.6 139 63.5 63.2 63.0 71.0 70.3 70.4 71.2 108
98715.8 93.0 84.2 149 68.0 67.7 67.5 76.1 75.3 75.4 76.3 116
109097.8 99.5 90.0 159 72.8 72.4 72.2 81.4 80.5 80.7 81.6 124
120571.7 106 96.1 170 77.7 77.3 77.1 86.9 86.0 86.1 87.1 132
133252.4 113 102 181 82.8 82.3 82.1 92.6 91.6 91.8 92.8 141
147266.6 120 109 193 88.0 87.6 87.4 98.5 97.5 97.6 98.8 150
162754.8 128 116 205 93.4 92.9 92.7 104 103 104 105 159
179871.9 135 122 217 99.0 98.4 98.2 111 110 110 111 169
198789.2 143 129 229 105 104 104 117 116 116 117 178
Inelastic mean free path (nm)


E(eV) SnTe TiC0.7 TiC0.95 VC0.76 VC0.86 Y3Al5O12 ZnS ZnSe ZnTe
54.6 0.427 0.486 0.492 0.463 0.471 0.807 0.460 0.449 0.460
60.3 0.437 0.476 0.475 0.453 0.458 0.767 0.461 0.452 0.468
66.7 0.449 0.471 0.465 0.449 0.452 0.736 0.467 0.459 0.480
73.7 0.463 0.470 0.462 0.450 0.452 0.713 0.476 0.470 0.494
81.5 0.480 0.474 0.462 0.456 0.455 0.692 0.490 0.484 0.511
90.0 0.498 0.479 0.466 0.464 0.462 0.674 0.507 0.501 0.531
99.5 0.518 0.487 0.471 0.475 0.472 0.662 0.526 0.521 0.552
109.9 0.539 0.494 0.478 0.487 0.483 0.659 0.548 0.544 0.575
121.5 0.561 0.502 0.484 0.501 0.496 0.663 0.573 0.570 0.599
134.3 0.585 0.511 0.491 0.516 0.510 0.674 0.601 0.599 0.624
148.4 0.609 0.524 0.504 0.534 0.527 0.692 0.632 0.631 0.648
164 0.633 0.543 0.521 0.556 0.548 0.715 0.666 0.665 0.672
181.3 0.658 0.566 0.543 0.582 0.573 0.743 0.704 0.703 0.698
200.3 0.685 0.594 0.569 0.612 0.602 0.776 0.744 0.744 0.729
221.4 0.718 0.625 0.599 0.645 0.634 0.813 0.789 0.788 0.765
244.7 0.755 0.660 0.632 0.682 0.670 0.856 0.838 0.836 0.806
270.4 0.798 0.699 0.670 0.723 0.710 0.902 0.892 0.889 0.853
298.9 0.847 0.743 0.711 0.769 0.754 0.955 0.950 0.945 0.905
330.3 0.902 0.791 0.757 0.818 0.802 1.01 1.01 1.01 0.962
365.0 0.962 0.844 0.807 0.872 0.855 1.07 1.08 1.07 1.03
403.4 1.03 0.902 0.862 0.932 0.913 1.14 1.16 1.14 1.10
445.9 1.10 0.965 0.922 0.997 0.977 1.22 1.24 1.22 1.17
492.7 1.18 1.03 0.988 1.07 1.05 1.30 1.33 1.31 1.25
544.6 1.27 1.11 1.06 1.15 1.12 1.39 1.43 1.40 1.35
601.8 1.36 1.19 1.14 1.23 1.21 1.49 1.53 1.50 1.44
665.1 1.47 1.28 1.22 1.32 1.30 1.60 1.64 1.61 1.55
735.1 1.58 1.38 1.32 1.42 1.39 1.72 1.77 1.73 1.67
812.4 1.71 1.49 1.42 1.53 1.50 1.84 1.90 1.86 1.80
897.8 1.84 1.60 1.53 1.65 1.62 1.98 2.05 2.00 1.94
992.3 1.99 1.73 1.65 1.79 1.75 2.13 2.21 2.16 2.09
1096.6 2.15 1.87 1.78 1.93 1.89 2.30 2.38 2.33 2.26
1212.0 2.32 2.02 1.93 2.08 2.04 2.48 2.57 2.51 2.44
1339.4 2.51 2.18 2.08 2.25 2.20 2.67 2.78 2.71 2.64
1480.3 2.71 2.36 2.25 2.43 2.38 2.89 3.00 2.93 2.85
1636.0 2.94 2.56 2.44 2.63 2.58 3.12 3.25 3.17 3.09
1808.0 3.18 2.77 2.64 2.85 2.79 3.37 3.52 3.43 3.34
1998.2 3.45 3.00 2.86 3.09 3.02 3.65 3.81 3.71 3.62
2208.3 3.73 3.25 3.10 3.34 3.27 3.94 4.13 4.02 3.92
2440.6 4.05 3.52 3.35 3.62 3.54 4.27 4.47 4.36 4.25
2697.3 4.39 3.81 3.64 3.93 3.84 4.62 4.85 4.72 4.61
2981.0 4.76 4.14 3.94 4.26 4.16 5.01 5.26 5.12 5.00
3294.5 5.16 4.49 4.28 4.62 4.52 5.43 5.71 5.55 5.42
3641.0 5.60 4.87 4.64 5.01 4.90 5.89 6.20 6.03 5.88
4023.9 6.08 5.28 5.04 5.44 5.32 6.39 6.73 6.54 6.38
4447.1 6.59 5.74 5.47 5.90 5.77 6.93 7.31 7.10 6.93
4914.8 7.16 6.23 5.94 6.41 6.27 7.52 7.94 7.71 7.52
5431.7 7.78 6.77 6.45 6.96 6.81 8.16 8.63 8.38 8.17
6002.9 8.45 7.35 7.00 7.56 7.39 8.86 9.37 9.10 8.87
6634.2 9.18 7.99 7.61 8.22 8.03 9.62 10.2 9.89 9.64
7332.0 9.97 8.68 8.27 8.93 8.73 10.5 11.1 10.7 10.5
8103.1 10.8 9.43 8.98 9.70 9.48 11.4 12.0 11.7 11.4
8955.3 11.8 10.3 9.76 10.5 10.3 12.3 13.1 12.7 12.4
9897.1 12.8 11.1 10.6 11.5 11.2 13.4 14.2 13.8 13.4
10938.0 13.9 12.1 11.5 12.5 12.2 14.5 15.4 15.0 14.6
12088.4 15.1 13.2 12.5 13.5 13.2 15.8 16.8 16.3 15.9
13359.7 16.4 14.3 13.6 14.7 14.4 17.2 18.2 17.7 17.2
14764.8 17.8 15.5 14.8 16.0 15.6 18.6 19.8 19.2 18.7
16317.6 19.4 16.9 16.1 17.3 17.0 20.2 21.5 20.9 20.3
18033.7 21.0 18.3 17.4 18.8 18.4 22.0 23.4 22.7 22.1
19930.4 22.8 19.9 18.9 20.5 20.0 23.8 25.4 24.6 24.0
22026.5 24.8 21.6 20.6 22.2 21.7 25.9 27.6 26.7 26.0
24343.0 26.9 23.4 22.3 24.1 23.5 28.0 29.9 29.0 28.2
26903.2 29.2 25.4 24.2 26.1 25.5 30.4 32.4 31.4 30.6
29732.6 31.6 27.5 26.2 28.3 27.7 33.0 35.2 34.1 33.2
32859.6 34.3 29.8 28.4 30.7 30.0 35.7 38.1 36.9 35.9
36315.5 37.1 32.3 30.8 33.2 32.5 38.6 41.3 40.0 38.9
40134.8 40.2 35.0 33.3 36.0 35.1 41.8 44.7 43.2 42.1
44355.9 43.4 37.8 36.0 38.9 38.0 45.2 48.3 46.8 45.5
49020.8 46.9 40.9 38.9 42.0 41.0 48.8 52.2 50.5 49.2
54176.4 50.6 44.1 42.0 45.4 44.3 52.7 56.3 54.5 53.1
59874.1 54.6 47.6 45.3 48.9 47.8 56.8 60.7 58.8 57.3
66171.2 58.8 51.2 48.8 52.7 51.5 61.1 65.4 63.3 61.7
73130.4 63.3 55.1 52.5 56.7 55.4 65.7 70.4 68.1 66.4
80821.6 68.0 59.2 56.4 60.9 59.5 70.6 75.6 73.2 71.3
89321.7 73.0 63.5 60.5 65.3 63.8 75.7 81.2 78.5 76.5
98715.8 78.2 68.1 64.8 70.0 68.4 81.1 87.0 84.1 81.9
109097.8 83.6 72.8 69.3 74.9 73.1 86.8 93.0 90.0 87.6
120571.7 89.3 77.7 74.0 79.9 78.1 92.6 99.3 96.1 93.6
133252.4 95.1 82.8 78.8 85.2 83.2 98.7 106 102 99.7
147266.6 101 88.1 83.8 90.6 88.5 105 113 109 106
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