Microanalysis – Basic Principles

When highly accelerated electrons strike any kind of matter, x-rays are produced. These x-rays are analyzed by various spectrographic techniques –x-ray spectroscopy. The different techniques used to measure the x-ray spectra have led to a proliferation of acronyms. We will focus in two techniques, wave dispersive spectroscopy (WDS), also known as x-ray fluorescence (XRF) and energy dispersive spectroscopy (EDS).We have seen that the interaction of highly accelerated electrons striking any kind of matter results in two types of interactions, elastic and inelastic. Part of the inelastic interaction results in the production of two types of x-ray radiation, namely the continuum radiation and the characteristic radiation.

Continuum Radiation - BremsstrahlungContinuous radiation or bremsstrahlung (“braking radiation”) are generated when the highly accelerated incident beam interacts with the matter and it is slowed down by the electric field (Coulomb field) of the atomic nuclei which results on a partially or complete conversion of the kinetic energy into electromagnetic radiation.

The energy of the electromagnetic radiation can take any value from zero to the energy of the incident beam, thus, the wavelength can take any value from infinity to a minimum given by the energy of the incident beam, resulting in a continuous spectrum.

The minimum wavelength given by the conversion of all the kinetic energy of the incident electron into a photon (and the electron stopped) is called the short-wavelength limit (swl).

Where h is the Planck constant, c is the light velocity.

Continuous radiation is particular to particle (electrons & photons) bombardment. Bombarding the sample with x-ray as in x-ray fluorescence analysis, does not produce the continuum.

Each incident electron actually undergoes many interactions to produce the continuum.

The production of photons with energy close to Eo is low. The probability of producing photons increases as the energy decreases. At lower energies the probability is high, but these photons could be absorbed by the material.

The intensity of the continuum radiation emitted by a decelerating particle is:

nmE

2398.1=λ

The proportion (p) of the total energy of the electron beam incident upon a sample that is emitted in the continuum is approximately:

This fraction is very small! A sample of iron (Z = 26) bombarded by 15 keV electrons produces only 0.04% of its energy as continuum X-rays, the rest is dissipated as heat in the sample and, to a very much lesser extent, as characteristic X-rays.

The intensity of the x-ray continuum (IC) is a function of atomic number (Z) and the energy of interest (E). It may be modeled using Kramer's equation, derived from classical theory in 1923:

Continuum X-ray emission. The EDS spectrum at right was accumulated on a small carbon inclusion within meteoritic metal.

Effect of changing the intensity of the electron beam (mA), accelerating potential (keV), and target atomic number (Z) on the continuous spectrum

The peak in the continuum ( λmax) is located at approximately 1.5 times λswl. Increasing Eo shifts λmax towards λswl, λswl moves to shorter wavelengths, and the overall X-ray output across the continuum increases. Thus, increasing the Eo does not improve the peak to background ratio. Increasing the incident current also increases the total X-ray output, but λmax and λswl remain the same.

Inner Shell IonizationAn electron beam with sufficient energy can interact with an atom and cause the ejection of a tightly bound inner shell electron, leaving the atom in an ionized and highly energetic excited state.

Two types of electron transition occur during the subsequent de-excitation of the atom. The emission of characteristic x-rays and/or the emission of Auger electron.

Characteristic X-Rays: The energy difference between electron shells is a specific or characteristic value for each element. If this element is present, its characteristic energy (EDS) or wavelength (WDS) can be detected and the intensity at its characteristic energy will be proportional to its concentration in the material.

Auger Emission: The difference in shell energies can be absorbed or transmitted to another outer shell, ejecting it from the atom as an electron with a specific kinetic energy.

Fluorescence Yield (ω):The partition of the de-excitation process between x-rays and Auger branches is described by the fluorescence yield (ω).

sionizationshellKofNumberproducedphotonsKofNumber

K ________

=ω

Low atomic number material (ωk~0.005). The x-ray process is not favored.High atomic number material (ωk~0.5). The x-ray process dominates.

Characteristic X-RaysThe vacancy left by the ejection of a tightly bound inner shell electron by the electron beam, is immediately (10-15sec) filled by an electron of a higher orbital and the difference in energy between the orbitals is irradiated in the form of x-rays. The energy of the x-rays is equal to the difference in energy of the electron levels, thus, as the electronic structure of each atom is unique, this x-ray energy can be used to identify the atom. The name assigned to a given characteristic x-ray depends on the electron shells involved in the transition (Kα, Kβ, Lα, Lβ, Mα, etc).

1) An electron in the K shell is ejected from the atom by an external primary excitation x-ray, creating a vacancy. 2) An electron from the L or M shell "jumps in" to fill the vacancy. In the process, it emits a characteristic x-ray unique to this element and in turn, produces a vacancy in the L or M shell.3) When a vacancy is created in the Lshell by either the primary excitation x-ray or by the previous event, an electron from the M or N shell "jumps in" to occupy the vacancy. In this process, it emits a characteristic x-ray unique to this element and in turn, produces a vacancy in the M or N shell.

Naming TransitionsIn theory, characteristic X-rays are named according to the shell being filled and the number of shells changed by the electron is indicated by α (1 shell), β (2 shells), or γ (3 shells). However, spectrographic nomenclature was developed before atomic electronic structure was well-understood, producing many inconsistencies.

Critical Ionization EnergyBecause the energy of a shell and sub-shell are sharply defined, the minimum energy necessary to remove an electron from a specific shell has a sharply defined value as well. This energy is called the critical ionization or excitation energy(Ec), also known as the excitation potential or x-ray absorption edge. It is higher than the associated characteristic (line) X-ray energy.

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

Line EdgeKα1 K-L3 66.83 78.38Lβ3 L1-M3 11.23 13.88Lβ1 L2-M4 11.07 13.27Lα1 L3-M5 9.442 11.56

M1-N3 2.780 3.296Mζ M2-N4 2.695 3.026Mγ1 M3-N5 2.331 2.645Mβ1 M4-N6 2.127 2.202Mα1M5-N7 2.051 2.133

OvervoltageOvervoltage is the ratio of accelerating (gun) voltage to critical excitation energy for particular line*.. 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) this ß accelerating voltage• Lα -- 23 keV• Mα – 4.2 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

Line EdgeKα1 K-L3 66.83 78.38Lβ3 L1-M3 11.23 13.88Lβ1 L2-M4 11.07 13.27Lα1 L3-M5 9.442 11.56

M1-N3 2.780 3.296Mζ M2-N4 2.695 3.026Mγ1 M3-N5 2.331 2.645Mβ1 M4-N6 2.127 2.202Mα1M5-N7 2.051 2.133

C

O

EEU =

Moseley’s LawHenry Moseley (1913 – Manchester University) discovered that the wavelength (energy) of an X-ray depended upon the nuclear charge of an atom. He photographed the X-ray spectrum of 10 elements that occupied consecutive places in the periodic table. Moseley's Law describes the relationship between atomic number and wavelength of a spectral line. Moseley's determination of this relationship provided a simple test of the order of the elements according to Z. It showed where elements were missing from the periodic table and led to the discovery of some of these elements. The constant σ is equal to 1 for the K-lines and 7.4 for the L-lines.

Element Kabs Kα1 Kβ1 L-IIIabs Lα1 Lβ1 M-Vabs Mα1 Mβ9 F 0.687 0.677

11 Na 1.072 1.041 1.06712 Mg 1.305 1.253 1.29513 Al 1.559 1.486 1.55314 Si 1.838 1.740 1.82915 P 2.142 2.013 2.13616 S 2.472 2.307 2.46417 Cl 2.822 2.62218 Ar 3.202 2.957 3.19019 K 3.607 3.313 3.58920 Ca 4.038 3.691 4.012 0.346 0.341 0.34521 Sc 4.496 4.090 4.460 0.403 0.395 0.40022 Ti 4.965 4.510 4.931 0.454 0.452 0.45823 V 5.465 4.951 5.426 0.513 0.511 0.51924 Cr 5.989 5.414 5.946 0.574 0.573 0.58325 Mn 6.540 5.898 6.489 0.641 0.637 0.64926 Fe 7.112 6.403 7.057 0.709 0.705 0.71827 Co 7.709 6.929 7.648 0.779 0.776 0.79128 Ni 8.333 7.477 8.263 0.855 0.851 0.86929 Cu 8.979 8.046 8.904 0.932 0.930 0.95030 Zn 9.659 8.637 9.570 1.021 1.012 1.03438 Sr 16.105 14.163 15.833 1.940 1.806 1.87140 Zr 17.998 15.772 17.665 2.223 2.042 2.12456 Ba 37.441 32.188 36.372 5.247 4.465 4.82757 La 38.925 33.436 37.795 5.483 4.650 5.041 0.85458 Ce 40.449 34.714 39.251 5.724 4.839 5.261 0.90260 Nd 43.571 37.355 42.264 6.208 5.229 5.721 0.99672 Hf 65.351 55.781 63.222 9.561 7.898 9.021 1.69782 Pb 88.006 74.965 84.922 13.035 10.550 12.612 2.484 2.345 2.44290 Th 109.646 93.334 105.591 16.300 12.967 16.199 3.332 2.996 3.14592 U 115.036 98.422 111.281 17.167 13.612 17.217 3.552 3.170 3.336

ExampleEstimate the energy of Kα X-rays off of a silver (Ag) target (Z=47).

L

K

n=2

n=1

photon

EL = −13.6eV(47 −1)2 1

22= −7.2keV

EK = −13.6eV(47 −1)2 1

12= −28.8keV

E(Kα ) =EL −EK = 21.6 keV

(vs. 21.7 keV Expt)

Not bad!

Careful! the formula En =

(−13.6)Z2

n2

assumed a single electron bound to just a positive nucleus.

λinte

nsity

Ka

30

Atomic Spectra: X-Ray Spectra

K

L

M

N

O

Kα Kβ Kχ Kδ Kε

Lα Lβ Lχ Lδ

Mα Mβ Mχ

Nα Nβ

Energy levels of a heavy atom showing the origin of x-ray spectra

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

Wavelength Energy (keV)

Z Name Kα1 Kα2 Kβ1 Kα11 Na 11.909 11.909 11.617 1.041 1.04114 Si 7.125 7.128 6.768 1.739 1.83826 Fe 1.936 1.940 1.757 6.398 7.057

0.109 97.143

Kβ

92 U 0.126 0.131 111.786

The fact that characteristic x-ray photons are described by either their energy or their wavelength enables us to isolate and count x-rays for a desired element either by discrimination based on energy or by wavelength. This fact produces two principal types of x-ray detection systems:WDS: wavelength dispersive detection system in which x-rays from different elements are recognized and separated from one another by their wavelength using Bragg diffraction.EDS: energy dispersive detection system in which x-rays from different elements are recognized and separated from one another by their characteristic energy using a solid state detector and multichannel analyzer.

Some examples:

• Remembering that the energy of an electron in the hydrogen atom is given by

,......,,neVn

.En 3216132 =

−=

the energy of the innermost electron in the complex atom is approximately given by

eVn

)Z(.En 2

21613 −−=

The energy of the Kα line is given by

eV)Z(.)Z(.EEhchfE 222

212 1210

11

211613 −=⎟

⎠⎞

⎜⎝⎛ −−−=−=

λ==Δ

α

The wavelength λα is given by

CCZ)Z(C −=−=⎟⎟⎠

⎞⎜⎜⎝

⎛λα

11 21

Note: here h = 4.135x10-15eV s

( )

(Equation of a straight line) (C = 2.86x103 m-1)

)(2398.1nm

keVEλ

=

ExampleThe wavelength of the Kα line from iron is 193 pm. (a) What is the energy difference between the two states of the iron atom that give rise to this transition?(b) What is the atomic number of iron?

(a)

(b)

eVx.Jx.x

)x)(x.(hcE 31512

8341044610031

10193103106266

===λ

=Δ −−

−

eVx.eV)Z(.E 32 104461210 =−=Δ 26=Z

Natural Width of Characteristic X-Ray Lines

Energy levels are so sharply defined that the energy width of a characteristic x-ray is a small fraction of the line energy.

For example, elements such as Ca (Z=20) have an x-ray line-width at half-height intensity of only 2eV. Note that the width of the measured peak is primary governed by the resolution of the spectrometer.

Ionization Cross Section

The probability that an energetic electron will ionize an atom depends on the atom, the line to be excited, and the over-voltage. For high over-voltages, the probability (ionization cross section) can be quite small; consequently, few atoms are ionized, thus few X-rays or Auger electrons are produced.

Auger emission is the dominant de-excitation process for low atomic number elements such as boron and carbon.

Electron Auger In many cases the photon emitted in an X-ray transition is absorbed by another electron within the same atom, which is therefore ejected as a result of an internal photo-electric effect.

This process of the internal conversion of X-rays into photo-electrons is called the Auger effectand the emitted photo-electrons are called auger electrons.

Auger electrons are emitted with specific kinetic energies T depending on the electronic levels involved in the process.E.g.: T=EK-EL1-EL3

Schematic diagram of various two-electron de-excitation processes.

The KL1L1 Auger transition corresponds to an initial K hole which is filled with L1 electron and simultaneously the other L1 electron is ejected to the vacuum.The LM1M1 Auger transition is the corresponding process with an initial 2s vacancy.