SIGMAK and SIGMAL Instructions

SIGMAK and SIGMAL instructions

(courtesy of Rick Noecker, Lehigh University)

SIGMAK and SIGMAL can be used to calculate partial ionization cross-sections and total K and L shell cross-sections, which are essential for EELS quantification and XEDS microanalysis, respectively. In general, they provide good agreement with more sophisticated calculations for incident electron energies ≤ 400 keV.

These executable programs must be recorded on a medium that allows files to be written in the same directory from which the programs are being executed. As such, you can not run these programs from a CD-ROM, but rather must copy them to a form of recordable media because the programs generate an output file upon their execution. The program crashes if this file can not be written. The following will provide information on how to calculate partial ionization cross-sections using the SIGMAK3 and SIGMAL3 programs.

Total K and L shell cross-sections (relevant to x-ray emission spectroscopy) can be calculated using these programs by setting BETA = 3142 mrad (π in mrad = 180°), and EINC≈EK/10 (for SIGMAK). The total cross-section is then equal to SIGMA that corresponds to the largest value of E(EV) in the output file. The aforementioned input and output information is explained below.

The newest version of SIGMAK utilizes an inner shell screening constant, s = 0.50. 

For more information on these programs, to include the source code, reference Electron Energy-Loss Spectroscopy in the Electron Microscope, 3rd ed. (Plenum, 1996) by R.F. Egerton.

SIGMAK

When SIGMAK3 is executed in a windows environment it opens a command prompt window with the following command line:

Z, EK, EINC, E0, BETA =

where:

Z is the atomic number of the element of interest

EK is the ionization energy of the K-shell in eV

EINC is the energy increment of output data in eV – for EELS partial cross-sections use EINC 1/3 or 1/4 of the required energy window (delta)

E0 is the incident beam energy in keV (must be less than 400 keV)

BETA is the collection semi-angle (in milliradians) contributing to the cross-section

Executing SIGMAK3 will generate a SIGMAK3.OUT file in the same directory as SIGMAK3. This output file will contain the results of the SIGMAK3 calculations.

Example:

Calculate the partial ionization cross-section for Oxygen K given a 120 keV incident beam energy, 100 mrad scattering angle and 80 eV energy window.

Z, EK, EINC, E0, BETA =

8, 532, 20, 120, 100

Which produces the below output where:

E(EV) is the energy window in eV

DSBYDE is the energy differential cross section (dσ/dE) integrated up to BETA at an energy loss given by the first column (E(EV)) – units are cm2/eV

DELTA is the energy window for which SIGMA is calculated

SIGMA is DSBYDE integrated up to energy loss E, otherwise known as the partial ionization cross section – units are cm2

F(0) is the (dimensionless) optical (dipole) oscillator strength integrated up to E

 

0Z,EK,EINC,E0,BETA =

E(EV) DSBYDE DELTA SIGMA f0

532.0 0.388E-22 0.0 0.000E+00 0.000

552.0 0.344E-22 20.0 0.731E-21 0.101

572.0 0.307E-22 40.0 0.138E-20 0.192

592.0 0.275E-22 60.0 0.196E-20 0.276

612.0 0.247E-22 80.0 0.248E-20 0.353

632.0 0.223E-22 100.0 0.295E-20 0.424

672.0 0.184E-22 140.0 0.376E-20 0.549

752.0 0.128E-22 220.0 0.499E-20 0.750

912.0 0.698E-23 380.0 0.652E-20 1.024

1232.0 0.269E-23 700.0 0.792E-20 1.319

1872.0 0.662E-24 1340.0 0.881E-20 1.556

3152.0 0.792E-25 2620.0 0.913E-20 1.693

5712.0 0.443E-26 5180.0 0.919E-20 1.751

10832.0 0.165E-27 10300.0 0.919E-20 1.771

The partial ionization cross section is then the value of SIGMA at a DELTA of 80, which is 0.248E-20.

SIGMAL

SIGMAL can be used to calculate partial ionization cross-sections, with relative accuracy, for Z=13 to 36. When SIGMAL3 is executed in a windows environment it opens a command prompt window with the following command line:

SIGMAL3: VALUES OF Z, E0 AND BETA ARE:

Where Z, E0 and BETA are given above. Note that SIGMAL does not allow input of the energy increment or edge energy. Also, the minimum DELTA in the output file is 50 eV.

Executing SIGMAL3 will generate a SIGMAL3.OUT file in the same directory as SIGMAL3. This output file will contain the results of the SIGMAL calculations.

Example:

Calculate the partial ionization cross-section for Nickel L given a 120 keV incident beam energy, 100 mrad scattering angle and 80 eV energy window.

SIGMAL3: VALUES OF Z, E0 AND BETA ARE: 28, 120, 100

This produces the same output as SIGMAK, but with the DELTA starting at 50 eV

0SIGMAL3: VALUES OF Z, E0 AND BETA ARE:

E(eV) DSBYDE DELTA SIGMA f0

905.0 0.203E-22 50.0 0.170E-20 0.475

915.0 0.204E-22 60.0 0.190E-20 0.533

925.0 0.203E-22 70.0 0.211E-20 0.591

935.0 0.202E-22 80.0 0.231E-20 0.649

945.0 0.200E-22 90.0 0.251E-20 0.707

955.0 0.198E-22 100.0 0.271E-20 0.764

975.0 0.192E-22 120.0 0.310E-20 0.878

1015.0 0.220E-22 160.0 0.392E-20 1.120

1095.0 0.188E-22 240.0 0.555E-20 1.606

1255.0 0.135E-22 400.0 0.810E-20 2.422

1575.0 0.737E-23 720.0 0.113E-19 3.575

2215.0 0.271E-23 1360.0 0.142E-19 4.852

3495.0 0.573E-24 2640.0 0.159E-19 5.910

6055.0 0.584E-25 5200.0 0.164E-19 6.565

11175.0 0.338E-26 10320.0 0.165E-19 6.889

21415.0 0.138E-27 20560.0 0.165E-19 7.032

The partial ionization cross section is then the value of SIGMA at a DELTA of 80, which is 0.231E-20.