In the SEM and electron microprobe, the sample is bombarded by a focused beam of electrons. Most incident electrons, rather than penetrating the sample in a linear fashion, interact with specimen atoms and are scattered. They follow complicated twisting paths through the sample material, losing energy as they interact. The scattering events are of two types either elastic or inelastic. In elastic scattering, the electron trajectory changes, but its kinetic energy and velocity remain essentially constant. This is because of the large difference between the mass of the electron and atomic nuclei. Alternatively, in inelastic scattering, the trajectory of the incident electron may be only slightly perturbed, but energy is lost through interaction with the orbital electrons of the atoms in the specimen. Inelastic interactions produce a number of effects:
- Secondary electrons (SE)
- Backscattered electrons (BSE)
- Cathodoluminescence (CL)
- Continuum x-ray radiation (bremsstrahlung)
- Characteristic x-ray radiation
- Phonons (heat)
Interaction Effects. Some the interaction effects due to electron bombardment emerge from the sample. Some, such as sample heating (not shown) stay within the sample. The lines within the interaction volume delineate regions where the effect indicated predominates (see figure below). For example, only x-rays emerge from the sample from the deeper parts of the volume.
The volume of material analyzed by an electron microprobe (or SEM equipped with X-ray spectrometers) depends upon many factors. The interaction volume is limited by energy loss through inelastic interactions and electron loss or backscattering through essentially elastic interactions. The probability of beam electrons interacting (scattering) with an atom or ion in the sample is specified by the interaction cross-section. However, interaction cross-sections are very small, corresponding to very low probabilities of interaction with a given atom. Cross-sections are measured in barns (1 barn = 10-24 cm2 = 10-4 pm2). However, there are a very large number of atoms in a small volume of a material, making the total probability of interaction unity. A single electron may be scattered many times.
The depth of electron penetration of an electron beam and the volume of sample with which it interacts are a function of its angle of incidence, the magnitude of its current, the accelerating voltage, and the average atomic number (Z) of the sample. The resulting excitation volume is a hemispherical to jug-shaped region with the neck of the jug at the specimen surface. The analyst must remember that the interaction volume penetrates a significant depth into the sample.
Schematic shapes of electron interaction volumes. The accelerating voltage and density play the largest roles in determining the depth of electron interaction.
Electron penetration generally ranges from 1-5 µm with the beam incident perpendicular to the sample. The depth of electron penetration, x, is approximately (Potts, 1987, p. 336):
where, E0 = accelerating voltage (keV) and ρ = density (g/cm3). For example, bombarding a material with a density of 2.5 g/cm3, about the minimum density for silicate minerals, with Eo = 15 keV, gives x = 2.3 µm. The width of the excited volume, y, can be approximated by (Potts, 1987, p. 337):
Both of these are empirical expressions. A theoretical expression for the range of an electron, r, the straight line distance between where an electron enters and its final resting place, for a given Eo is (Kanaya and Okayama, 1972):
where, A = average atomic weight (g/mole), E0 = accelerating voltage (keV), Z = average atomic number, ρ = density (g/cm3)
The volume of electron interaction may be modeled from first principles using a Monte Carlo method, where the paths of a series of incident electrons are modeled probabilistically with equations for elastic and inelastic scattering determining the scattering angles, mean free-paths, and the rate of energy-loss. Each electron trajectory is simulated in stepwise fashion.
Monte Carlo Simulation of Electron Paths. This simulation is of 15 KeV electrons in fayalite (Fe2SiO4). Distances are given in nanometers (1000 nm = 1 µm). Paths of backscattered electrons are in red; those of absorbed electrons in blue. One should remember that this slice through a three-dimensional volume. This model was run using the Casino software described at http://www.gel.usherbrooke.ca/casino/What.html.
Monte Carlo results may be used to models not just the electron paths, but distribution of the resulting effects and implanted energy. This documents the strong dependence of depth of interaction on atomic number and density.
Monte Carlo Simulation of Energy Distributions. Energy distributions assuming 15 keV electrons and the materials indicated. The contour lines correspond to energy amounts of 5% (pale blue), 10% (red), 25% (green), 50% (yellow), 75% (dark blue), 90% (purple). The models were run using the Casino software described at http://www.gel.usherbrooke.ca/casino/What.html.
Two factors control which effects can be detected from the interaction volume. Firstly, some effects are only produced from certain parts of the interaction volume. Beam electrons lose energy during interactions within the sample. Electron energy generally decreases with depth; thus, if a certain amount of energy is required to produce an effect, it will not be possible to produce it from the deeper portions of the volume. Secondly, the degree to which an effect, once produced, can be observed is controlled by how strongly it is diminished by absorption and scattering in the sample. For example, secondary electrons are produced throughout the interaction volume, but have very low energies and can only escape from a thin layer near the sample's surface. Similarly, soft x-rays, which are absorbed more easily than hard x-rays, will escape more readily from the upper portions of the interaction volume. Absorption is an very important phenomenon and is discussed in detail below.
Generalized illustration of interaction volumes for various electron-specimen interactions. Auger electrons (not shown) emerge from a very thin region of the sample surface (maximum depth about 50 Å). The outline of the volume of characteristic x-ray production is defined by the case where the energy of an electron, E, is just sufficient to produce x-rays requiring energy, Ec.The critical energy, Ec, varies with the x-ray of interest. Secondary fluorescence will be discussed below.
Of all the effects produced from the interaction volume, characteristic X-rays provide the greatest amount of information, reflecting as they do the composition of the material. X-rays represent an energetic portion of electromagnetic spectrum with λ from ~1 nm to 1000 nm (0.1 to 100 Å). X-rays with λ from ~50 to 100 Å are termed "soft" X-rays; those with shorter wavelengths are called "hard" X-rays and overlap with γ-rays.
The Electromagnetic Spectrum.
X-rays are specified either by their energies (in electron volts) or wavelengths (in Å). It is very useful to be able to convert readily between these units. For all electromagnetic radiation:
where h = Planck's constant (6.6260 x 10-34 Joule-second) and ν = frequency in cycles/second (Hertz).
For electromagnetic radiation,
where c = speed of light (2.99782 x 108 m/sec) and λ = wavelength (m)
and plugging in values gives...
where λ = wavelength (m) and E = energy (J). Conversion to nanometers and electron volts (1 eV = 1.6021 x 10-19 Joule) yields the Duane-Hunt equation:
where, λ is in nanometers (nm) and E is in electron volts (eV). If a conversion from Ångstroms to eV is desired, the appropriate conversion factor is 12398.4.
Continuum X-rays are produced when incident beam electrons are slowed to varying degrees by the strong electromagnetic field of atomic nuclei in the sample. All degrees of electron braking are possible and, thus, the resulting X-rays have a continuous range of all energies. Each incident electron potentially can undergo many such interactions in a solid. Continuum radiation is also called bremsstrahlung, German for "braking radiation".
Production of Continuum Radiation. A beta minus particle (β-) is simply an electron. Images' sources: (left) http://www.rstp.uwaterloo.ca/manual/interaction/graphic/brem_mov.gif, (right) http://microanalyst.mikroanalytik.de/Images/Infos22.jpg
The highest energy X-ray that can be produced by electrostatic braking has an energy equivalent to the accelerating voltage of the electron beam. This cutoff is sometimes described as the short wavelength limit and may be written as:
The wavelength of maximum continuum intensity (λmax) occurs at approximately 1.5 λswl. Increasing the accelerating voltage shifts λmax toward λswl, while λswl moves to shorter wavelength and the overall X-ray output of the continuum increases. Increasing the beam current, increases the overall X-ray output of the continuum, but λswl and λmax remain the same. Continuum intensity is a function of three variables: atomic number (Z), beam current (ib) and the accelerating voltage (E0). The relationship between intensity, I, and energy of interest, E, is given by Kramer's equation, which is derived from classical theory:
This relationship yields a exponential curve with I = 0 at E = E0 that increases to I → ∞ at E = 0. However, the observed X-ray continuum drops off at very low energies (long wavelengths), because very soft X-rays are easily absorbed. Increasing either the accelerating voltage, average atomic number of the sample, or the beam current produces a higher continuum.
Continuum X-rays. (left) Effects of changing accelerating voltage and beam current. (right) Continuum observed from graphite enclosed in Fe-Ni metal. The spectrum was accumulated using an EDS system, using a 15 keV accelerating voltage. Low channel numbers correspond to low energies. The small peak near channel 622 is due to secondary fluorescence producing characteristic X-rays in the surrounding Fe-Ni metal.
It should be noted that the continuum is nonlinear. The continuum radiation limits minimum detectable amount of an element and its curvature may present difficulties in insuring that the backgrounds of characteristic X-ray peaks are correctly determined. Backgrounds are usually measured on each side of a peak of interest assuming a linear background between them. However, if the continuum is nonlinear and the backgrounds are measured too far from peak assuming linear background, an incorrect peak intensity will be determined. The peak intensity will be under measured if there is a slight upward curvature in the continuum; conversely, intensity will be overestimated if there is a downward curvature. These problems are only significant when analyzing for trace elements.
Only a very small fraction of the incident energy from the electron beam is emitted as continuum X-rays. The fraction of beam energy, p, may be approximated by:
where Z = the average atomic number, and E0 is the accelerating voltage (energy) of the incident electrons. For example, 15 keV electrons bombarding an iron sample (Z = 26) yield only 0.04% of their energy as continuum X-rays. Most energy is dissipated as heat in sample. The low efficiency of continuum production makes it necessary to cool the X-ray tubes used in XRF and X-ray diffraction (XRD) analysis.
Continuum radiation is peculiar to particle bombardment (electrons or protons), resulting as it does from electrostatic charges; bombardment of a sample with X-rays, as in X-ray fluorescence (XRF) analysis will not produce continuum radiation, yielding much lower background counts. Consequently, element detection limits are much lower in XRF analysis than electron beam analysis. Particle mass also plays a role in determining the magnitude of the continuum radiation. The intensity of continuum radiation, I, emitted by a decelerating particle is:
where a = the acceleration of the particle, f = the decelerating force, and m = the mass of the particle. The intensity of continuum produced by electron bombardment is greater than that produced by comparably accelerated protons because of the mass difference. Similar nuclear coulombic forces (f) are involved, but protons are 1836 times more massive than electrons. As a consequence, proton-induced X-ray emission (PIXE) has markedly lower backgrounds than those produced during electron bombardment, allowing detection of trace elements.
Spectrum observed from proton bombardment. PIXE spectrum of a spin-valve type recording head, which has many thin layers composed of similar mass elements. Note the accelerating voltage of 2.3 MeV and the essentially flat continuum background! Image source: http://www.almaden.ibm.com/st/scientific_services/materials_analysis/ib_surface/pixe/PIXEData.gif
The beam electrons knock a small fraction of electrons out of inner shells orbitals, in a process called inner-shell ionization. An atom remains ionized for only ~10-14 second before inner-shell vacancies are filled by outer-shell electrons, emitting a characteristic X-ray. Each atom interacts with an incident electron every ~10-12 seconds, thus, it is possible for a given atom to be repeatedly ionized. Approximately 0.1% of the beam electrons cause K-shell ionizations.
This process requires critical threshold energy to occur; beam electrons with lower energies cannot displace inner-shell electrons. This threshold energy is called the excitation potential or critical ionization potential (Ec). Excitation potentials decrease with distance from the nuclei reflecting the diminishing electrostatic forces holding the orbital electrons. Excitation potentials are always larger than the energy of the associated X-ray. Incident electrons that are not energetic enough to excite Kα radiation may still have sufficient energy can excite Lα or Mα radiation. For example, Ec for Fe-Kα is 7.11 keV; whereas, Ec for Fe-Lα is only 0.71 keV. X-ray analysts use Lα (and β) lines for elements with atomic numbers from about 30 to 70 and Mα lines for elements with higher atomic numbers.
Production of Characteristic X-rays. Image source: (left) http://nobelprize.org/educational_games/physics/x-rays/what-3.html; (right) http://ehs.unc.edu/training/self_study/xray/9.shtml
The spectrum of characteristic X-ray peaks produced during neutralization of inner-shell vacancies is superimposed on the continuum X-ray spectrum.
EDS X-ray Spectrum. The characteristic peaks of gold (Au) and copper (Cu) are superimposed upon the background of the continuum. Although the peaks have distinct quantized energies, their detection is a statistical process, which broadens the peaks. Image source: http://www.trincoll.edu/~alehman/_images/Engr232_02/Au_latex_042302.gif
The Bohr model of the atom provides a convenient way to understand the nature of characteristic X-rays. One may consider an atom to be a positively charged nucleus surrounded by shells of negatively charged electrons, labeled K, L, M, N… (starting from innermost, most strongly bound shell). Each electron in an atom is uniquely defined by several quantum numbers.
|n||principle||positive integers (1, 2, 3…)||electron shell (1=K, 2=L, 3=M…)||Principle binding energy|
|l||azimuthal||integers from 0 to (n-1)||electron cloud shape (0=sphere, 1=dumbbell…)||Orbital angular momentum. Chemists follow optical spectroscopy conventions using letters rather than numbers: sharp (l = 0), principal, (l = 1), diffuse (l = 2), and fundamental (l = 3)|
|m||magnetic||-l to +l (including 0)||electron orientation in magnetic field||Not significant in absence of an external magnetic field|
|s||spin||± ½||electron spin direction||Clockwise or counterclockwise (up or down)|
|j||inner precession||l ± ½, ( j ≠ 0 - ½)||total angular momentum vector||Determines which transitions are permitted between electron shells. For s orbitals (l = 0), j can only be +½ (vector sum always positive)|
The Pauli Exclusion Principle was proposed by Wolfgang Pauli to explain the arrangement of electrons in atoms. It has been generalized to assert that no two fermions of the same type (electrons are fermions) can exist in the same state, i.e., have the same quantum numbers, at the same place and time. Thus, the maximum number of electrons per level is 2n2. The correspondence between first four electron shell names and their quantum numbers is given in the following table. Note that the j values increase systematically within a given shell.
Characteristic X-rays are produced by transitions between electron shells. However, not all transitions are possible. Permissible transitions are specified by what are termed quantum selection rules:
- The change in n must be ≥ 1 (Δn ≠ 0)
- The change in l can only be ±1
- The change in j can only be ±1 or 0
Consider the element scandium, Sc, which has an electronic configuration of 1s22s22p63s23p63d24s2. We can represent this configuration in n-l space:
Now we can restate the first two rules as: (1) no transitions between shells in same row, and (2) no transitions between shells in the same column or that skip columns (e.g., from l = 3 to l = 1).
The transitions permitted according to these two rules are:
Finally, we must apply Rule 3 (Δj=0,±1). There are four possible cases to consider for the p → d transitions (±½ spin in the p orbital → ±½ spin in the d orbital).
Note that the second transition in the list is not possible because Δj = 2. There are only two potential transitions to consider when the s orbitals are involved because j cannot be negative: ±½ spin (non-s orbital) → +½ spin (s orbital). Both are possible.
All the other transition possibilities can be evaluated in the same manner...
In 1923, Manne Siegbahn (1886-1978) introduced a notation for identifying X-rays in his book “The Spectroscopy of X-rays.” His nomenclature, now called the Siegbahn notation, was based on the relative intensity of lines from different series, but provided no information about specific origin of X-ray lines. Beta (β) and gamma (γ) lines were discriminated based on intensities and some inconsistencies resulted. Siegbahn did such good enough job labeling the lines, that, after the fact, generalizations can be made about the relationship between the names of the spectral lines and the processes that produced them:
- X-ray lines are labeled according to the element that they represent (Si, Ca, Fe, etc.)
- Lines are further labeled (K, L, M...) based on the electron shell that was ionized and subsequently filled.
- Greek letters are added based on the "distance" of shell that provided the neutralizing electron (α indicates that the electron came from the next shell out, β indicates two shells out, γ indicates three shells, etc.)
- If needed, sublines are identified by numerical subscripts.
Thus, Ca-Kα is a calcium X-ray produced when an electron fills the K shell from the L shell; Ca-Kβ indicates the electron came from the M shell. Siegbahn got Nobel Prize in Physics in 1924 for this work.
Manne Siegbahn and Siegbahn Notation. The diagram shows both notations used for the electron shells and examples of transitions. the Image sources: (left) http://nobelprize.org/nobel_prizes/physics/laureates/1924/siegbahn.jpg, (right) http://xdb.lbl.gov/Section1/Sec_1-2.html
The table below summarizes all significant characteristic X-rays and gives the line intensities relative to the a lines of each series. Note that the intensities of most lines are very low, making them of no use for analytical work. Analysts use the α lines almost exclusively, with β lines used for analyses of the rare-earth elements where there are significant line overlaps on the α lines.
Recognizing that there were inconsistencies in the Siegbahn notation, the International Union of Pure and Applied Chemistry proposed a new system for identifying X-ray lines. The IUPAC system labels X-rays using:
- A level symbol for final state (K, L, M ...), i.e., the shell being filled;
- A level symbol for initial state (K, L, M ...), i.e., the source of the neutralizing electron; and
- Arabic numerals rather than Roman numerals for identifying subshells (L2 and L3 instead of LII and LIII, etc.)
For example, the Kα1 line, which is produced by the transition from the LIII to the K shell, is labeled K-L3. When a single spectral line results from two transitions, this is indicated by duplicating the symbol for the source shells. For example, the Kβ2 corresponds to K-N2N3. When X-ray lines are unresolved, subscripts are used to indicate this. For example, the recommended IUPAC notation for the Kα1,2 X-ray lines, which are unresolved in the light elements, are labeled K-L2,3 (Kα2 is K-L2 and Kα1 is K-L3).
The IUPAC notation is consistent with notation used for Auger electron spectroscopy; however, it is cumbersome and the X-ray analytical community largely uses the Siegbahn notation.
The relationship between atomic number and characteristic X-ray wavelength was established by Henry G.J. Moseley in 1914:
where, K and σ are constants for given X-ray spectral line. Sigma (σ) is called the shielding constant and compensates for the effect of electron shells within the ionized shell; it is approximately 1 for K-lines, but 7.4 for the more shielded L-lines. Moseley derived his formula empirically by plotting the square root of X-ray frequencies against a line representing atomic number. However, it was almost immediately noted in 1914 that his formulae could be explained in terms of the 1913 Bohr model of the atom, if certain reasonable extra assumptions about atomic structure in other elements were made.
Moseley's Data. Moseley's data redrafted for clarity. Image source: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/moseley.gif
Moseley's Law may be rewritten to relate atomic number and X-ray frequency:
where, K and σ are constants for given X-ray spectral line (but not the same constants as in the previous equation). For the Kα and Lα lines, these equations are:
X-ray line intensity is function of the transition probability and rate of ionization. The probability of ionization depends on the target-atom ionization cross-section, Q. Typical capture cross-sections for K shell are about 10-20 cm2 (~1/100 atomic diameter); there is a very low probability of K-shell ionization! The capture cross-section depends on the overvoltage, U, which is the ratio accelerating voltage, E0, to the critical ionization potential, Ec. For the K-shell:
where, ns = number of electrons; bs and Cs = constants; E0 = accelerating voltage; Ec = critical ionization potential for shell of interest (keV); and U = overvoltage (E0/Ec).
Effect of Overvoltage on Capture Cross-section. Note that the optimum overvoltage for the K-lines is about 2.5. Higher overvoltages also increase the chance of producing satellite peaks.
The intensity of a given spectral line, I, may be described by the following empirical relationship:
where, C = constant; ib = beam current; Ec = critical excitation potential of the line of interest; E0 = accelerating voltage; and p = 1.7 for E0 < 1.7Ec (smaller for higher values of Ec). The only ways an analyst can increase a specific line's intensity is to optimize the over voltage (as above) or increase the beam current, ib.
At high overvoltages more than one electron may be ejected simultaneously from an atom. This causes the overall structure of the electron shells to change, resulting in the production of X-rays with slightly lower energies than those produced during single electron ionization. These multi-ionization X-rays appear as small satellite peaks near the haracteristic X-ray peaks. Another important source of satellite peaks is the Auger process described below.
Satellite peaks are generally not important, except when there is a very large overvoltage. An accelerating voltage, E0, of 15 keV is about optimum for exciting Fe-Kα X-rays (U = 15 / 7.112 = 2.1), but results in an eight-fold overvoltage for Si-Kα (U = 15 /1.838 = 8.2). Production of satellite peaks may decrease peak-to-background ratios.
There are small, but detectable, changes in the wavelength (energy) of an X-ray peak produced from an pure element compared with that produced from a compound of the element. This occurs because the configuration of the inner shells of an atom is influenced by the outer valence electrons. The orbitals of the valence electrons reflect (1) the manner in which an element is bonded (which is related to its oxidation state) and (2) the coordination of a atom inside a crystal. Peak shifts most significant when the X-ray-producing inner shells are less shielded from the valence electrons. This is the case for the outermost transitions (M- and N-lines) in most elements and in low-Z elements (B through F for Kα, Al through Cl for Kβ).
The shifts are most apparent comparing metals with oxides and halides (figure below). Consequently, it is not advisable to use a metallic standard material for analysis of oxide compounds. In the case of energy-dispersive analysis, the peak shifts are undetectable and metals can be used as reference standards for quantification.
Mg Wavelength Shifts. After ...
As an detailed example, consider Al as a metal and as an oxide, Al2O3 (figure below). In Al metal, X-rays are produced by transitions from the 3s and 3p levels into the 1s level; the 3s and 3p electrons are not involved in chemical bonding. The result is a relatively narrow X-ray peak. In contrast, in Al2O3, the valence electrons are combined with oxygen 2s and 2p electrons and span wider range of energies. As a result, the Kβ emission becomes broader, asymmetric, and separates into Kβ and a satellite Kβ' peak.
Al Wavelength Shifts. After Figure 2.7 from Williams, 1986, p. 26.
People have attempted to use observed wavelength shifts to determine oxidation state. For example, iron in minerals exhibits two common valences, 2+ and 3+. In theory, there should be a detectable energy change between Fe-Lα and Fe-Lβ, however, this is very difficult to observe. The only apparently workable method uses the shape of the Fe-Lβ peak to evaluate oxidation state. A readily observable shift occurs for the S-Kβ peak when comparing sulfides (S2-) with sulfates (S6+)
The coordination of an atom or ion in a mineral will also have an affect on the electron cloud configuration. In essence, while equal numbers of protons pull on the electron clouds, variable numbers of coordinating anions pull out. Higher coordination (more anions pulling out) results in the electron shells being pulled farther from nucleus, yielding higher energy transitions. As with bonding, coordination effects are most easily observed in light elements. For example, the shift in Al-Kα lines relative to line from Al metal for kaolinite (AlVI) is about twice that observed for feldspar (AlIV).
Luminescence is the emission of visible light from a solid when it is excited by some form of energy. Luminescence can be subdivided into two subtypes: fluorescence, which ceases immediately after withdrawal of the exciting source (persisting <10-8 seconds) and phosphorescence, which persists for some time after removal of excitation (persisting >10-8 seconds). However, the distinction between these is somewhat arbitrary and confusing. Alternatively, perhaps more usefully, luminescence may be subdivided based upon the type of exciting energy. Luminescence produced by incident X-rays or γ-rays is radioluminescence, that produced by visible and UV light is photoluminescence, that produced by heat is thermoluminescence (TL) and that generated by incident electrons is cathodoluminescence (CL).
Cathodoluminescence was first studied by William Crookes in 1879. He discovered that scheelite, CaWO4, placed opposite the cathode in a vacuum discharge tube yielded spectacular fluorescence. He concluded that some particle was traveling from cathode to the target causing the fluorescence well before J. J. Thomson discovered the electron in 1897. Crookes also studied luminescence from many other minerals (diamond, ruby, sapphire, zircon, etc.).
Cathodoluminescence in Carbonate Cement. Note the multiple generations of cementation. Image source: http://www.univ-lille1.fr/geosciences/umr_pbds/techniques/cathodo.html#exemples
Cathodoluminescence is explained using the band theory of solids. Additionally, band theory will be used in the discussion of semiconductor X-ray detectors later in the course; thus, it will be covered in some detail here. Solids consist of an essentially infinite number of atoms, and it is incorrect to consider each atom individually when considering the behavior of electrons in them. Band theory considers the structure of the solid as a whole and provides the a way to describe metals and other solids to explain their unique chemical and physical properties.
In basic molecular orbital theory, one assumes that when atoms are brought together, they form bonding, non-bonding and antibonding orbitals of different energies. For n atomic orbitals in a molecule, n molecular orbitals are produced. Thus, a molecule with three atoms (assuming 1 atomic orbital for each) forms 3 molecular orbitals. As the number of molecular orbitals increases, the energy difference between the lowest bonding and the highest antibonding orbitals increases, while the space between each individual orbital decreases. With an essentially infinite number of atoms, the spacing between the lowest bonding and highest antibonding orbital reaches a maximum and there are so many molecular orbitals so close together that they blur into one another forming a band.
Origin of a Band. Schematic diagram of the formation of a band by ever increasing numbers of molecular orbitals. Diagram redrafted from a figure at: http://www.chembio.uoguelph.ca/educmat/chm729/band/concept.htm
Electrons fill the molecular orbitals within a band starting with the lowest energy orbital and follow the Pauli exclusion principle. No more than two electrons may occupy a single orbital and if two do occupy a single orbital, then their spins must be spin paired. Additionally, Hunds rule must not be broken and when more than one orbital has the same energy, electrons must occupy separate orbitals and have parallel spins. If each molecular orbital contains only one electron, the band is only half full (n electrons for n orbitals), leaving empty levels in the band. Conversely, if each atomic orbital contains 2 electrons, the band is full (2n electrons for n molecular orbitals). Whenever a band is half full, the energy of the uppermost filled molecular orbital at 0 K is called the Fermi energy. At T >0 K, the energy rises above the Fermi energy level because the electrons start occupying higher states due to thermal excitation.
The nature of a material can be characterized by the relationship between the valance band (low energy electrons) and the conduction band (higher energy electrons). Most solids are insulators, which means that there is a large band gap between energies of the valence and conduction bands. All electrons are confined to the valence band, and there is no electrical conduction.
Some materials are semiconductors, in which the valence band is separated from the conduction band by a small energy gap, Eg. There are no electrons in the conduction band at 0 K, but with the addition of energy, a finite electrons can be promoted into it and an electrical current can flow. Semiconductors may be either intrinsic or extrinsic. In the later case a material contains small amounts of impurities that have roughly the same atomic size, but more or fewer valence electrons than the host material. Artificial addition of impurities is termed doping. Impurities can transform a non-cathodoluminescent material into one that is luminescent with the nature of the impurity controlling the CL color.
The valence and conduction bands overlap in conductors, such as metals. There is no real difference between the valence and conduction bands and electrons can move through the material easily, yielding electrical conduction. This free-floating electron distribution is also described as metallic bonding.
Types of Materials. Schematic diagram showing the relationship of the conduction and valence bands in a conductor, insulator, and semiconductor. Eg is the energy of the band gap. Image source: http://elchem.kaist.ac.kr/vt/chem-ed/quantum/bands.htm
The electrical properties of all types of materials depends strongly on temperature. The conductivity of an insulator or semiconductor increases with increasing temperature, whereas, conductors become less conductive.
Cathodoluminescence is produced in materials that have semiconductor properties. Beam electrons provide the energy to knock photoelectrons from the valence band into conduction band. This leaves behind a positively charged valence band “hole”. The average energy required to produce an electron-hole pair (eV) is:
where M is a constant (0 < M < 1) and Eg is the size of the band gap. We may thus calculate the number of electron-hole pairs per beam electron as approximately:
where η = backscattered electron coefficient. Backscattered coefficients in silicates are generally 0.05-0.20, so each incident electron potentially can produce a large number of electron-hole pairs. For example, a single electron with E0 = 30 keV could produces 1300 electron-hole pairs in zircon (η = 0.27, Eg = 5.4 eV). Free electrons recombine with the valence band holes, radiating light with ultraviolet to infrared wavelengths (~160-2000 nm). The resulting CL can be polarized by the crystal structure.
Production of Cathodoluminescence. The incident beam electrons provide energy E1.
It should be emphasized that the processes that yield CL are, except in rare cases, different than those that produce mineral colors. Mineral coloration reflects the absorption of incident white light. This generally occurs by electron transfers in d suborbitals produced by crystal field splitting or by intervalence charge transfer. The band properties of a mineral can produce color by absorption, but this is a relatively uncommon because most minerals are insulators. Generally, the gaps are too large for visible light to promote an electron to the conduction band and no visible light is absorbed, yielding a white or colorless mineral (diamond, C). In minerals that act as semiconductors, light photons with energy > Eg can promote electrons to the conduction band. In the cases where Eg is less than energy of visible light, a mineral will appear black or gray (e.g., galena, PbS). Larger gaps may only absorb short wavelengths of visible light, leaving a red mineral (e.g., cinnabar, HgS). Minerals with overlapping conduction and valence bands absorb the entire range of colors because electrons can be promoted to conduction band by any wavelength; the result is an opaque mineral.
There are two types of semiconductors. A p-type semiconductor is formed when the dopant has fewer valence electrons than the host (e.g., Al3+ into Si4+). The impurity atoms occupy sites in the structure, but contribute fewer electrons to the valence band than the host atoms. This produces "holes" in the lower-energy valence band, which allow electrons to move from one orbital to another within the valence band with a small input of energy (smaller than required for the semiconductor without the doping). Alternately, in an n-type semiconductor, the impurity has more valence electrons than the host (e.g., P5+ into Si4+) and contribute extra electrons to the valence band. However, because the valence band is already filled, the extra electrons must go into the higher-energy (conduction) band. These electrons partially fill the conduction band, can move easily between the orbitals, and thus throughout the solid.
Very few minerals have intrinsic CL; scheelite (CaWO4) is one of them. In contrast, minerals with extrinsic CL are fairly common, among them quartz, feldspar, and carbonates. The impurities that cause luminescence are called activators and include Mn and the rare earth elements (REE). Other elements, called quenchers, can absorb CL before it escapes a material. Iron is perhaps the most important quencher.
Activators and quenchers have been studied extensively in feldspar, carbonates, and zircon. Activators in carbonates include Mn2+, Sm3+, Tb3+, Eu2+ and Eu3+ at concentrations of more than 10-20 ppm. The transition elements, Fe2+, Fe3+, Ni2+ and Co2+, act as quenchers at concentrations of over 30-35 ppm. Much of the documented variation in cathodoluminescence in calcite involves Mn2+ and Fe2+ with the latter being the quencher. Documented activators in feldspar include Ce3+ (bluish-green, peak at 490 nm), Ti3+ (blue, peak at 460 nm), Eu2+ (blue, peak at 420 nm), Mn2+ (greenish-yellow, peak at 540-560 nm), and Fe3+ (red/infrared, peak at 68-780 nm); all must be present in low concentrations. Blue CL emission has also been linked to Al-O–Al defects in the feldspar structure. Zircon, which commonly displays complex CL zoning, incorporates U, Th, heavy REE, Y, P and Hf into its structure. Studies of synthetic crystals have demonstrated a correlation between REE content (especially Dy3+) and CL. However in zircons from the Carpathian Mountains, CL brightness does not correlate with concentrations of REE, Y, or Th; the authors propose U is probably responsible for quenching or suppressing the CL signals.
Cathodoluminescence in calcite, dolomite, and pyroxene. (left) Image in crossed-polarized light XPL); (right) Cathodoluminescent image. Note the distinct color difference in CL between dolomite (dark red) and calcite (orange-yellow). Images source: http://www.lumic.de/gallery/index.html.
Cathodoluminescence in quartz. This image shows total polychromatic CL intensity as gray scale intensity. The quartz shows both sector and concentric zoning in hydrothermal vein quartz, possibly due to Al3+ variation. Image source: http://www.see.leeds.ac.uk/research/igt/people/lloyd/eg_cl.htm.
Cathodoluminescence in zircon. This image shows polychromatic CL intensity as gray scale intensity. The zircon shows complex zoning. Image source: http://cda.morris.umn.edu/~jonesjv/mineralogy/zircon.jpg
Cathodoluminescence can be used for determining modal abundances in cases where minerals display different luminescence (as in the calcite-dolomite sample above; however, CL is not always present or characteristic. CL also has application in discriminating synthetic from natural gems. It is particularly valuable in documenting the distribution of trace elements in minerals such as carbonates and zircon, which reflect chemical changes during overgrowth and dissolution. Spectral analysis of CL allows the identification of specific trace elements. For example, the CL zoning in zircon reflects variations in Dy concentration.
All forms of electromagentic radiation (X-rays, light, etc.) are attenuated by absorption from elements when it passes through a material. This results from two processes:
- Compton scattering (σ), in which photons are diverted in directions different from that of the primary beam, and
- Photoelectric absorption (τ), which produces fluorescence or Auger electrons from a material
These two effects can be combined into a single mass absorption coefficient (µ):
Throughout this discussion we will use the notation, µ, for the mass attenuation coefficient. However, more accurately these coefficients are µ/ρ, and are tabulated as such. The mass absorption coefficient depends upon both wavelength (energy) of the photon of interest and the composition of the material. In geological materials photoelectric absorption comprises ~95% of the total absorption and Compton scattering can be largely ignored (it is not significant at λ < 1 Å).
Selected Mass Absorption Coefficients for Kα Radiation.
Lambert's law describes the amount of attenuation (absorption) that radiation undergoes passing through a thickness of material:
where, I0 = initial intensity, I = final intensity, ρ = density (g/cm3), t = thickness (cm), and µ = mass absorption coefficient (cm2/g). For our purposes, intensities will be measured in X-ray counts per second (cps).
In a multi-element material one multiplies the mass absorption coefficients at the wavelength of interest, µi, by the mass fraction each element, wi, and sums the result to yield a bulk mass absorption coefficient:
Absorption in a material may be characterized by a half-thickness, i.e., the thickness required to halve the intensity of the radiation. The expression for half-thickness is derived from Lambert's Law using I = ½ I0:
Taking the natural log of both sides, we get:
The natural log of 0.5 = -0.6931, so the final equation is:
The concept of a half-thickness can be applied to the penetration and any energy deposition by photons (x-ray, γ-ray, bremsstrahlung, etc.) in biological materials, solids, and complex materials as long as mass absorption coefficients are available. The NIST site, http://www.physics.nist.gov/PhysRefData/XrayMassCoef/cover.html, provides a compilation of absorption coefficients from 1 keV to 100 MeV and a detailed discussion of the mass attenuation coefficients.
As an example, consider the absorption of Ni-Kα in forsterite, Mg2SiO4 (ρ = 3.25 g/cm3). The weight fractions of elements in forsterite are: Mg = 0.3455, Si = 0.1996, and O = 0.4549. The values of µ for Ni-Kα radiation for these elements are 48, 74 and 14, respectively, so µNi-Kα = 0.3455 x 48 + 0.1996 x 74 + 0.4549 x 14 = 37.7 cm2/g. The half-thickness is 0.0057 cm or 56 µm. Given that X-ray production in a mineral is only ~3 µm deep, absorption of Ni-Kα X-rays will not be a major problem in forsterite. Using Lambert's Law we see that the observed intensity will be 96.4% of the original assuming a 3 µm path length.
Bulk mass absorption coefficients decrease with increasing X-ray energy; more energetic radiation is absorbed less than less energetic radiation. However, there is strong absorption if the radiation is energetic enough to cause inner shell ionization in an another element (i.e. Ephoton > Ec). This marked increase in absorption produces "teeth" on a curve of absorption vs. photon energy that are called absorption edges.
Absorption coefficient versus energy for Sm. Although the absorption in Sm generally decreases with increasing X-ray energy, there are several absorption edges that correspond to values of Ec (identities and energies indicated) for different Sm X-ray lines.
Absorption edges can produce dramatic and unexpected effects. Consider the absorption of Ni-Kα and Co-Kα in pyrite, which consists of 0.4655 Fe and 0.5345 S (element weight fractions). The bulk mass absorption coefficients are:
- µNi-Kα = 0.5345 x 108 + 0.4655 x 382 = 236
- µCo-Kα = 0.5345 x 133 + 0.4655 x 56 = 97
Ni-Kα is more energetic than Co-Kα (7.48 keV vs. 6.91 keV), but is absorbed over twice as much. Why is this? The Ni-Kα radiation is energetic enough to excite Fe-Kα radiation (Ec = 7.112 keV) from the pyrite, a process called secondary fluorescence (figure below). Note that the result is enhanced Fe-Kα intensity. Some elements undergo self-absorption, in which the K-line X-rays of an element may be absorbed to produce L-line X-rays in the same element.
Absorption coefficient versus energy for Fe, Co and Ni. The Ec values for the Kα X-rays of each of the elements are indicated.
Significant amounts of heat are produced within a sample because electron excitation of X-rays is not very efficient. Most low energy continuum photons and low-energy inelastically scattered electrons do not escape the sample and their energy is transformed into higher vibrational energies of the bonds (heat).
A crystal lattice consists of bonded atoms, which cannot vibrate independently. Instead, the entire lattice vibrates in a coordinated fashion, with vibrations taking the form of collective modes, which propagate through the material. These harmonic vibrations can be decomposed into elementary vibrations called phonons. The total number of phonons in a vibrating crystal is related to its temperature of the system. At higher temperatures, vibration of an object is stronger and the number of phonons larger. As every phonon carries a quantum of vibrational energy, this means that the internal energy of the object is also larger.
Phonons. (left) Movement of longitudinal phonons in a one-dimensional lattice. Note the large number of potential vibrational modes (only a some of which are shown). (right) Rayleigh phonon on the (100) face of a fcc solid (amplitude exaggerated 10x). Image sources: (left) http://en.wikipedia.org/wiki/Phonon; (right) http://www.fhi-berlin.mpg.de/th/personal/hermann/pictures.html
Castaing (1951) demonstrated that the maximum temperature rise for a material can be expressed as:
where, E0 = accelerating voltage (keV), bi = bean current (µA), Ct = thermal conductivity (W/cm·K), and d0 = beam diameter (µm).
Consider an accelerating voltage of 15 keV, a specimen current of 0.05 µA (= 50 nA), and a beam diameter of 1 µm. A material like copper with a high thermal conductivity (Ct = 3.97 W/cm·K) has ΔT = 0.9°C, whereas, zircon (Ct = 0.042) has ΔT = 86°C. Note that going to a beam diameter of 10 µm would drop this latter number by a factor of 10 to 8.6°C. Actually, epoxy will decompose ("burn") long before attaining ΔT in excess of 150°C. In contrast, most minerals can survive temperatures well in excess of 200°C. Values of ΔT at 15 keV for some materials are given below.
|1 µm diameter spot||5 µm diameter spot|
|10 nA||25 nA||50 nA||10 nA||25 nA||50 nA|
Epoxy decomposes ("burns") long before attaining ΔT > 150 °C. However, most minerals easily survive temperatures >200 °C. Thermal conductivity is a strongly vectorial property, meaning that it varies with crystallographic orientation. For example, quartz ranges from 0.065 along the a axis to 0.12 along c axis. The table below summarizes the sparce data available for for minerals.
|Calcite||0.038 (||a), 0.044 (||c)||Mica||0.042 (||a,b), 0.007 (||c)|
|Diamond||5.4, 20 (synthetic)||Orthoclase||0.042|
|Epoxy||0.002||Quartz||0.065 (||a), 0.12 (||c)|
|Fluorite||0.01||Rutile||0.088 (||a), 013 (||c)|
The strongest region in the electron energy spectrum is due to secondary electrons (SE), which are defined as those emitted with energies less than 50 eV.
Electron Energy Spectrum. Secondary electrons (SE) form a large low-energy peak. Auger electrons (AE) produce relatively small peaks on the backscattered electron (BSE) distribution. Figure after Goldstein et al. 1981.
Secondary electrons are produced when an incident electron excites an electron in the sample and loses some of its energy in the process. The excited electron moves towards the surface of the sample undergoing elastic and inelastic collisions until it reaches the surface. Here it can escape if its energy exceed the surface work function, Ew, which defines the amount of energy needed to remove electrons from the surface of a material. One of the major reasons for coating a non-conductive specimen with a conductive materials is to increase the number of secondary electrons that will be emitted from the sample (decrease Ew).
The mean free path length of secondary electrons in many materials is ~1 nm (10 Å). Thus, although electrons are generated throughout the region excited by the incident beam, only those electrons that originate less than 1 nm deep in the sample escape to be detected as secondary electrons. This volume of production is very small compared with BSE and X-rays. Therefore, the resolution using SE is better than either of these and is effectively the same as the electron beam size. The shallow depth of production of detected secondary electrons makes them very sensitive to topography and they are used for scanning electron microscopy (SEM).
The average number of SE produced per primary electron is called the secondary-electron yield, δ , and is typically in the range 0.1 to 10 (varying between different materials). For a given sample material, δ decreases with increase in incident energy E0 since the probability of inelastic scattering of a primary electron within the escape depth decreases. SE yield depends on the angle of tilt of the specimen relative to the primary-electron beam, φ. The value is lowest for perpendicular incidence (φ = 0) and increases with increasing angle between the primary beam and the surface-normal. This effect may make cracks appear bright in SE images.
Volume of secondary electrons escape. Secondary electrons escape from a depth in the sample, d, measured perpendicular to the surface. Places where the beam -- light gray -- strikes the surface at an angle (right) have larger volumes of electron escape -- dark gray -- than the perpendicular case (left). This SE yield is larger by a factor of 1/cosφ. Image based on figure 7 at: http://laser.phys.ualberta.ca/~egerton/SEM/sem.htm.
The edge effect is modified by the position of the detector relative to the sample. SE detectors are positioned to one side of the sample enhancing topographic effects. Faces oriented towards the detector will be brighter, whereas those in the opposite direction will be dark.
Effect of Topography.
From the above discussion one might conclude that SE production will be independent of the electron beam accelerating voltage. However, only part of the SE signal, the SE1 component, comes from the sample surface. Other components arise from SE produced by backscattered electrons as they exit the specimen (SE2) and when BSE strike the walls of the specimen chamber (SE3).
Generation of SE1 and SE2 electrons. (left) The SE2 and SE3 signals arise from production of secondary electrons from places other than the primary SE region (darker gray). SE2 electrons are formed when backscattered electrons interact with shallow portions of the sample; SE3 electrons are generated outside the specimen when a BSE strikes the objective lens or wall of the sample chamber, for example. (right) Production of SE2 and SE3 defocus the SE signal reducing spatial resolution and making the sample appear more "transparent."
The SE2 component depends on a sample's backscattered coefficient, η, which reflects chemical differences well below the surface. This effect increases with increasing penetration depth, so the sample will appear more "transparent" at higher E0, and less so at low E0. Finer surface structure images can generally be obtained with lower accelerating voltages, At high E0, the SE2 and SE3 signals are larger, reducing image contrast and veiling fine surface structures.
Sample Transparency. The microstructures of the specimen surface are clearly at 5 keV, because penetration of incident electrons is shallow. Image source: A Guide to Scanning Microscope Observation, revised edition, JEOL Corporation. Obtained at http://www.jeolusa.com/tabid/320/DMXModule/692/EntryId/1/Default.aspx.
Backscattered electrons (BSE) are high energy primary electrons that suffer large angle (> 90°) scattering and re-emerge from the entry surface of a specimen. Most BSE have energies slightly lower than that of the primary electron beam, E0, but may have energies as low as ~50 eV (the upper cut-off for secondary electrons). The fraction of beam electrons backscattered from a sample, nb (also symbollized η), depends strongly on the sample's average atomic number, Z, reflecting the increasing charge of the atomic nuclei. Individual scattering events are generally elastic, where a negligible amount of energy is lost by the primary electron in the process. The direction of the electron may be altered, but its energy remains essentially the same. However, an electron that has undergone inelastic scattering (having excited a plasmon, phonon, caused inner shell ionization, or interacted with an electron in the valence band) may subsequently escape the sample surface as a BSE.
A good approximation for the backscattered coefficient (Love & Scott, 1978) is:
where, wi = weight fraction of element i and Zi = atomic number of element i. The energy of a backscattered electron depends upon the number of interactions that it has undergone before escaping the sample surface.
Backscattered Electron Energies. Distribution of BSE energies with an iron target. Note that the energy peak of the BSE for this material occurs ~12 keV. The model was run using the Casino software described at http://www.gel.usherbrooke.ca/casino/What.html.