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UofO- Geology 619. Electron Beam MicroAnalysis- Theory and Application Electron Probe MicroAnalysis - (EPMA). X-ray Production: Ionization and Absorption. Schrödinger Model:. EM Spectrum Lines Produced by Electron Shell Ionization. Time. K shell. (=photoelectron). 1. L shell.

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UofO- Geology 619

Electron Beam MicroAnalysis- Theory and ApplicationElectron Probe MicroAnalysis -(EPMA)

X-ray Production:

Ionization and Absorption


K shell



L shell

Blue Lines indicate subsequent times: 1 to 2, then 3 where there are 2 alternate outcomes



Inner-shell ionization:Production of X-ray or Auger e-

Incident electron knocks inner shell (K here) electron out of its orbit (time=1). This is an unstable configuration, and an electron from a higher energy orbital (L here) ‘falls in’ to fill the void (time=2). There is an excess of energy present and this is released internally as a photon. The photon has 2 ways to exit the atom (time=3), either by ejecting another outer shell electron as an Auger electron (L here, thus a KLL transition), or as X-ray (KL transition).

(Goldstein et al, 1992, p 120)

X-ray Lines - K, L, M

Ka X-ray is produced due to removal of K shell electron, with L shell electron taking its place. Kb occurs in the case where K shell electron is replaced by electron from the M shell.

La X-ray is produced due to removal of L shell electron, replaced by M shell electron.

Ma X-ray is produced due to removal of M shell electron, replaced by N shell electron.

Ionization of Electron Shells

(Originally Woldseth, 1973, reprinted in Goldstein et al, 1992, p 125)

X-ray Lines with initial + final levels

(Reed, 1993)

Nomenclature of X-rays

  • X-rays are described, from the traditional Siegbahn notation (e.g. Ka1) to the the IUPAC (K-L3).

  • (International Union of Pure and Applied Chemistry).

  • This table is from their 1991 recommendation.

(Reed, 1993)

One slide Schrödinger Model of the Atom

  • n = principal quantum number and indicates the electron shell or orbit (n=1=K, n=2=L, n=3=M, n=4=N) of the Bohr model. Number of electrons per shell = 2n2

  • l = orbital quantum number of each shell, or orbital angular momentum, values from 0 to n –1

  • Electrons have spin denoted by the letter s, angular momentum axis spin, restricted to +/- ½ due to magnetic coupling between spin and orbital angular momentum, the total angular momentum is described by j = l + s

  • In a magnetic field the angular momentum takes on specific directions denoted by the quantum number m <= ABS(j) or m = -l… -2, -1, 0, 1, 2 … +l

  • Rules for Allowable Combinations of Quantum Numbers:

    • The three quantum numbers (n, l, and m) that describe an orbital must be integers.

    • "n" cannot be zero. "n" = 1, 2, 3, 4...

    • "l" can be any integer between zero and (n-1), e.g. If n = 4, l can be 0, 1, 2, or 3.

    • "m" can be any integer between -l and +l. e.g. If l = 2, m can be -2, -1, 0, 1, or 2.

    • "s" is arbitrarily assigned as +1/2 or –1/2, but for any one subshell (n, l, m combination), there can only be one of each. (1 photon = 1 unit of angular momentum and must be conserved, that is no ½ units, hence “forbidden transitions)

No two electrons in an atom can have the same exact set of quantum numbers and therefore the sameenergy. (Of course if they did, we couldn’t observably differentiate them but that’s how the model works.)

Fluorescence yield

Fluorescence yield (w) is fraction of ionizations that yield characteristic X-ray versus Auger yield (a) within a particular family of X-rays. w + a =1

Absorption Edge Energy

Edge or Critical ionization energy: minimum energy required to remove an electron from a particular shell. Also known as critical excitation energy, X-ray absorption energy, or absorption edge energy. It is higher than the associated characteristic (line) X-ray energy; the characteristic energy is value measured by our X-ray detector.

Example: Pt (Z=78) X-ray line energies and associated critical excitation (absorption edge) energies, in keV


  • Overvoltage is the ratio of accelerating (gun) voltage to critical excitation energy for particular line*. U = E0/Ec Maximum efficiency (cross-section) is at 2-3x critical excitation energy.

  • Example of Overvoltage for Pt: for efficient excitation of this line, would be (minimally) thisß accelerating voltage

  • La -- 23 keV

  • Ma -- 4 keV

Example: Pt (Z=78) X-ray line energies and associated critical excitation (absorption edge) energies, in keV

* recall: E0=gun accelerating voltage; Ec=critical excitation energy

Duane-Hunt Limit

Continuum X-rays

HV beam electrons can decelerate in the Coulombic field of the atom (+ field of nucleus screened by surrounding e-). The loss in energy as the electron brakes is emitted as a photon, the bremsstrahlung (“braking radiation”). The energy emitted in this random process varies up from 0 eV to the maximum, E0.

On an EDS plot of X-ray intensity vs energy, the continuum intensity decreases as energy increases. The high energy value where the continuum goes to zero is known as the Duane-Hunt limit.

Continuum and Atomic Number

At a given energy (or l), the intensity of the continuum increases directly with Z (atomic number) of the material. This is of critical importance for minor or trace element analysis, and also lends itself to a timesaving technique (Mean Atomic Number,“MAN”).

MAN plot (Z-bar = average Z = MAN)

Continuum intensity around the Si Ka peak, varying with Z: Mo (42), Ti (22), B (5). X axis is sin theta position units.

General observations

  • Above atomic number 10, the K family splits into Ka and Kb pairs

  • Above atomic number 20, the L family becomes generally observable at about 0.2 keV

  • Above atomic number 50, the M family of lines begin to appear

  • When an electron beam has sufficient energy to excite the shell edge (absorption edge), all lines of lower energy for that element will also be excited.

  • This is an important fact to consider when trying to positively identify the presence of an element in a spectra. Therefore if K lines are present, then L and M lines must also be present at their appropriate energies.

  • X-ray units: A, keV, sin q, mm

    l = hc/E0 where h=Plancks constant, c=speed of light

    l = 12.398/E0 where is l is in Å and E0 in keV

    also, the 2 main EMPs plot up X-ray positions thusly:

    Cameca: n l = 2d sin q so for n=1 and a given 2d, an X-ray line can be given as a sin value (or 105 times sin q)

    JEOL: distance (L, in mm) between the sample (beam spot) and the diffracting crystal, i.e. L= l R/d, where R is Rowland circle radius (X-ray focusing locus of points) and d is interlayer spacing of crystal.

    Electron interaction volumes

    Effect of beam interaction (damage) in plastic (polymethylmethacrylate), from Everhart et al., 1972. All specimens received same beam dosage, but were etched for progressively longer times, showing in (a) strongest electron energies, to (g) the region of least energetic electrons. Note teardrop shape in (g). Same scale for all.

    (Goldstein et al, 1992, p 80)

    Ranges and interaction volumes

    It is useful to have an understanding of the distance traveled by the beam electrons, or the depth of X-ray generation, i.e. specific ranges. For example: If you had a 1 um thick layer of compound AB atop substrate BC, is EPMA of AB possible?

    Electron and X-ray Ranges

    Several researchers have developed physical/mathematical expressions to approximate electron and X-ray ranges. Two common ones are given below.

    Electron range. Kanaya and Okayama (1972) developed an expression for the depth of electron penetration:

    RKO=(0.0276 A E01.67)/(r Z0.89)

    X-ray range. Anderson and Hasler (1966) give the depth of X-ray production as:

    RAH=(0.064)(E01.68 - Ec1.68)/ r

    where Ec is the absorption edge (critical excitation) energy.

    Contact Lens simulation using a 5 um beam on carbon coated (20 nm) silicon rubber of nominal composition.

    Ranges and Interaction Volumes

    5 keV, 5 um beam diameter:

    Electron energy is color coded and red indicated electrons that were backscattered out of the sample.

    10 keV, 5 um beam diameter:


    Secondary electrons



    electrons 1-2µm


    X-rays 2-5um

    20 keV, 5 um beam diameter:

    Ionization cross section (in units of ionizations/e (20 nm) silicon rubber of nominal composition. -/(atom/cm2)

    - Bethe (1930)

    where: ns is the number of electron in a shell of subshell (e.g., 2 for K shell)

    bs and cs are constants for a particular shell

    Ec is the critical excitation energy

    U is the overvoltage

    To calculate the number photons generated we need to modify the previous expression to units of photons/e- or nx

    where: Q is the ionization cross section from above

    w is the florescent yield (explain later)

    NO is the Avogadro’s number

    A is atomic weight

    r is density

    t is thickness of the thin foil (which experimentally) minimizes the effect of elastic scattering

    For thick targets the thickness is replaced by the infinitesimal path increment ds and so the bulk x-ray yield, Ic, is calculated by the integration along the Bethe range (RB) to the point where the electron energy falls below the critical excitation energy for the x-ray of interest

    where: dE/ds is the rate of energy loss from inelastic scattering (the Bethe range expression we discussed last week)

    “Harper’s Index” of EPMA infinitesimal path increment ds

    1 nA of beam electrons = 10-9 coulomb/sec

    1 electron’s charge = 1.6x 10-19 coulomb

    ergo, 1 nA = 1010 electrons/sec

    Probability that an electron will cause an ionization: 1 in 1000 to 1 in 10,000

    ergo, 1 nA of electrons in one second will yield 106 ionizations/sec

    Probability that ionization will yield characteristic X-ray (not Auger electron):

    1 in 10 to 4 in 10.

    ergo, our 1 nA of electrons in 1 second will yield 105 xrays.

    Probability of detection: for EDS, solid angle < 0.01 (1 in 100). WDS, <.001

    ergo 103 X-rays/sec detected by EDS, and 102 by WDS. These are for pure

    elements. For EDS, 10 wt%, 102 X-rays; 1 wt% 10 X-rays; 0.1 wt % 1 X-ray/sec.

    ergo, counting statistics are very important, and we need to get as high count rates

    as possible within good operating practices.

    From Lehigh Microscopy Summer School

    Phi-Rho-z Distribution: infinitesimal path increment ds

    Fig 1: from Bastin and Heijligers, 1992, Present and future of light element analysis with electron beam instruments. Microbeam Analysis, 1, 61-73.

    X-ray Absorption infinitesimal path increment ds

    Mass absorption coefficients infinitesimal path increment ds