Emission yields

The Problem

The emission yield is the term which relates the number of photons coming from the source to the number of atoms entering the plasma. If it is constant then counting the photons is equivalent to counting the number of atoms in the plasma. It is therefore only changes in emission yield which upset this counting process.

Fortunately in a glow discharge, for a particular line from a particular element, the emission yield doesn't change by much, perhaps by a factor of 2-3 over the full range of conditions and samples. But a factor of 2-3 is still too much for quantitative analysis, so we have to reduce the change in emission yield or correct for any changes.

Unfortunately, the emission yield depends on just about everything that is happening in the plasma: collisions, sputtering, etc., and from outside the plasma we cannot measure all these events. So we are forced to use what we can measure, namely the gas pressure and the electrical parameters, such as current, voltage and power.

The Solution?

For mathematical convenience we use here the term: 'inverse relative emission yield', R_i.

Building on the work of many people, but especially of Bengtson,⁽¹⁾ Payling was able to show that if we consider current i_g, voltage U_g and pressure p_g to be independent then⁽²⁾

where a, b, c and U₀ are constants for a particular emission line. Typically a ~ -1, b ~ 0.5, and c is positive, provided the pressure is not too low. Hence the inverse relative emission yield generally decreases with increasing current, and increases with increasing voltage and pressure.

This work was with a DC source but there is no reason to suppose that an RF source would be essentially different.

Given the normal ranges of these variables, the most important effect is from varying current, next is varying voltage and thirdly varying pressure. If we keep current and voltage constant then the variation in R_i because of varying pressure is only about 10-20%.

This work has been interpreted to mean that we should operate a glow discharge with constant current and voltage and variable pressure, to minimise the variation in R_i. In an otherwise excellent paper, Kim Marshall concludes: "... this weakness severely limits the applicability of the PP [constant power-pressure] mode ..."⁽³⁾

But there are many advantages to keeping the pressure constant, and in fact the work showed we can use any source control and then correct for the variations in R_i and obtain the same result.

A detailed discussion of thie topic can be found in a recent (2006) review article by Arne Bengtson and Thomas Nelis (4)

The Solution!

For each possible way of operating the GD source, there will always be at least one free parameter.

In constant current/constant voltage (IV) mode, it is pressure; in constant power/constant pressure (PP) mode, it is the ratio of voltage to current (impedance). This ratio can be expressed either as a function of voltage or of current since, at constant power, when one increases the other decreases.

Hence we can assume any change in inverse relative emission yield will be a function of this free parameter, f(p). Further, if we use the same conditions in calibration and analysis, any variation in the free parameter will be restricted to a limited range. Hence the function will be nearly linear

where r_i is a fitted parameter determined by regression during calibration and p₀ is a reference value chosen somewhere near the middle of the range, eg 700 V for DC bias voltage.

If the calibration function is linear, then inclusion of a linear function for R_i will also be linear, ie

where K_i⁽ⁿ⁾ are constants determined by regression.

As an example, in RF GD-OES it is common to keep power and pressure constant (PP mode). The free parameter p then becomes the DC bias voltage, V_DC. The following graphs show calibrations using RF GD-OES, for a variety of certified reference materials, including Al-Si alloys, Al-Zn alloys, brass, steel, stainless steel, etc., before correction (blue) and after V_DC correction (pink):

for Si 288 nm

Without correction, the calibration appears as two distinct families, corresponding to differences in emission yield, due mainly to Si in steel at the low values and Si in Al-Si at the high values. With correction, the separation of families disappears and one calibration curve results for all samples.

for Al 396 nm (known to show self-absorption)

Without correction, variations in emission yield increase scatter and disguise the effect of self-absorption. With correction, the scatter is greatly reduced and the effect of self-absorption can be clearly seen in the non-linearity of the calibration curve.

Higher Order Calibration

When there is appreciable self-absorption, ie when it is necessary to use third or fourth or even fifth order then inclusion of a linear function for R_i becomes a little cumbersome. Fortunately for the types of corrections envisaged in GD-OES, the product r_i.(p-p₀) is less than 1. For example, for a pressure correction, r_i.(p-p₀) is about 0.1, for a V_DC correction, r_i.(p-p₀) is about 0.4. The inclusion of powers of R_i is then given exactly by

where

References:
 (1) A Bengtson and M Lundholm, J. Anal. Atom. Spectrom. 3, 879 (1988).
 (2) R Payling, Surf. Interface Anal. 23, 12 (1995).
 (3) K A Marshall, J. Anal. Atom. Spectrom. 14, 923 (1999).
(4) A.Bengtson, Th.Nelis; "The concept of constant emission yield in GDOES"; Anal. Bioanal. Chem.; 385; (2006); 568-586; DOI 10.1007/S00216-006-0412-7

First published on the web: 1 June 2000.

Authors: Richard Payling and Thomas Nelis

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Hydrogen effect and Gaseous Contamination

For some years we have known that gaseous contamination of the argon plasma gas can have deleterious, or at least interesting, effects on the plasma.⁽¹⁾ Very recently, excitement has grown over the effect of small amounts of hydrogen contamination.^(2-5) Hydrogen is now known to alter emission yields, making some lines more intense (eg, O I 130.1 nm) and some less intense (eg, Fe I 371.9 nm). It also increases background signals by creating molecular bands (hydrides and molecular hydrogen) particularly in the region 220 nm to 440 nm.⁽³⁾ The problem is that it doesn't matter whether the hydrogen comes from hydrogen deliberately added to the plasma gas (something we could avoid) or from hydrogen sputtered from the sample being analysed.

Hydrogen Correction

The question then arises as to how best to correct for the effect of hydrogen, or any other gas, in the plasma gas. If we assume that the effect is proportional to the measured hydrogen signal, then it is a relatively straightforward matter to include both additive and multiplicative corrections. 

The effect of hydrogen on the background signal can be treated by an additive correction proportional to the hydrogen signal. Such an additive correction can then be included like any other spectral interference, see Quantification.

The effect of hydrogen on emission yields can be treated as a multiplicative correction, ie one effect of hydrogen is to change the slope of the calibration curves. Initially we thought these changes to emission yield could be treated in the same way as other multiplicative parameters, such as pressure or DC bias voltage, see Emission Yield. However, these methods assume the range of values is relatively small (eg 500-1000 Pa for pressure or 500-800 V for DC bias voltage) but this is not the case for hydrogen since the hydrogen content in samples can vary from near 0% in many materials to perhaps 10% (by mass) in some polymeric materials. We therefore need to know more about the real function relating changes in emission yield and hydrogen before we can proceed.

Fortunately Hodoroaba et al. have measured the variation in intensity of a variety of emission lines as a function of hydrogen content in the plasma gas.⁽⁴⁾

Their data was used to construct the following two graphs.⁽⁶⁾ For Si 288.1, the intensities were normalised to the value at zero hydrogen. Similar graphs were obtained for N 149.26, C 156.14, and S 180.73. For Mn 403.4, the intensities were first inverted and then normalised to the value at zero hydrogen. Similar graphs were obtained for Ti 337.27,  Mo 386.41, Al 396.15, Cr 425.43, and Fe 249.3.

These graphs suggest that some line intensities (non-metals?) increase nearly linearly with hydrogen and others (metals?) decrease such that their inverse intensity increases nearly linearly with hydrogen.

These graphs can be represented by either of the following two equations

or

where R_i is the inverse relative emission yield for element i, I_H is the measured hydrogen signal and h_i is a fitted parameter.

Authors: R Payling, Surface Analytical, Australia and M Aeberhard, EMPA, Switzerland

References:
 (1) W Fischer, in R Payling, D G Jones and A Bengtson (Eds), Glow Discharge Optical Emission Spectrometry, John Wiley, Chichester (1997), pp 403-9.
 (2) A Bengtson and S Hägström, Proc. 5th Internal. Conf. on Prog. in Anal. Chem. in Steel and Metals Industries, European Communities, Luxembourg (1999) pp 47-54.
 (3) V-D Hodoroaba, V Hoffmann, E B M Steers, K Wetzig, J. Anal. Atom. Spectrom. 15 (2000) 951.
 (4) V-D Hodoroaba, V Hoffmann, E B M Steers, K Wetzig, J. Anal. Atom. Spectrom. 15 (2000) B0023671 (on web).
 (5) A Bogaerts and R Gijbels, J. Anal. Atom. Spectrom. 15 (2000) on web.
 (6) R Payling, M Aeberhard and D Delfosse, J. Anal. Atom. Spectrosc. 16 (2001) 50. Available on the web at http://www.rsc.org/ej/gd/2000/b007543o.pdf

First published on the web: 1 June 2000 and extended on 5 March 2001.

Author: Richard Payling

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GD Source Impedance

During the operation of a GD source, there is always at least one parameter which is not fixed. In RF operation with constant power and pressure it is the ratio of voltage to current; in DC operation with constant current and voltage it is pressure. What determines the value of the free parameter for a particular sample: impedance.

If we are discussing RMS values in RF operation or DC values in DC operation, we can think of the impedance as the ratio of voltage to current. What determines the impedance for a particular sample (all other things being constant) is secondary electron yield.

The secondary electron yield, g, is the average number of electrons emitted per incisdent ion. It is typically 0.1 for metals, but is estimated to vary from 0.053 for Au up to 0.152 for Ti,⁽¹⁾ i.e. by about a factor of three. For more details about secondary electron yield, click here.

Effect on Emission Yield

Because of the variation in emission yield, for constant voltage and pressure, the current should also vary by about a factor of three. Since emission yield is nearly proportional to current then the emission yield should also change by about a factor of 3.

For constant current and pressure, the voltage (or at least the effective voltage, ie the voltage above threshold) should vary by about a factor of three. Since the emission yield is nearly proportional to the square root of the effective voltage, the emission yield should then vary by about 1.7.

For constant power and pressure, the voltage and current should each change by about 1.7, giving a combined change in emission yield of about 2.3. This is about what we see, for example, for Si in Al-Si compared with Si in Fe-Si.

More information on the source impedance and emission yields can be found in a review article by Arne Bengtson and Thomas Nelis in Analytical and Bioanalytical Chemistry.⁽⁴⁾

References:
 (1) H Hocquaux, in R K Marcus (Ed), Glow Discharge Spectroscopies, Plenum, New York (1993), p 351.
 (2) R A Baragiola, E V Alonso, J Ferron and A Oliva Florio, Surf. Sci. 90 (1979) 915.
 (3) L Ohannessian, PhD Thesis, Université Claude Bernard, Lyon, France (1986).
 (4) A.Bengtson, Th.Nelis; "The concept of constant emission yield in GDOES"; Anal. Bioanal. Chem.; 385; (2006); 568-586; DOI 10.1007/S00216-006-0412-7

First published on the web: 1 June 2000.

Authors: Richard Payling and Thomas Nelis

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Coating mass

Both glow discharge optical emission and massspectrometry determine the sputtered mass rather then the sputtered depth. To determine the sputtered depth a model calculation for the density must be used to convert the sputtered mass and the elemental composition in to the sputtered volume or depth.

1. Elemental Sputtering Rates

To convert a qualitative depth profile of some coating into a quantitative depth profile, we first calculate the elemental sputtering rates, dm_ij, for each element, i, at each point, j, in the depth profile. These elemental sputtering rates tell us how much of each element is being sputtered per second in g/m²/s, ie

where c_ij is the concentration of element i at point j in the depth profile, q_Mj is the sputtering rate per unit area at j, and dt_j the time increment at j.

If we integrate these elemental sputtering rates over time we have the total mass of the element removed, ie we have the coating mass of that element:

The easiest way to do this is to plot c_ijq_Mj vs time and then integrate over the time range of interest. If concentrations are in mass%, sputtering rates in g/m²/s, and time in s, then the result will be in g/m².

Looking at the quantification procedure, we find that GDOES first determines the coating mass and than uses the elemental composition to make an assumption on the density to derive the sputtered depth. In particular, when the density of the analysed layer is not well known, the coating mass will be more reliable than the sputtered depth. Typical bad examples are oxide layers of non-stoechiometric composition.

2. Density

During the calculation we also calculate the density at each point, r_j. So now we can also calculate the coating mass directly from the quantitative depth profile.

First we note the relationship between depth and time:

where dz_j is the change in depth over time increment dt_j.

Substituting equation (3) into equation (1), we get

So if we integrate equation (4) over depth, we have:

The easiest way to do this is to plot c_ij vs depth (ie quantitative depth profile) and then have a special integration over the depth range of interest that first multiplies each c_ij by the density at each j. This special function might appear like magic at first until you realise what it is doing, just solving equation (5).

If concentrations are in mass%, density in g/cm³, and depth in µm, then the result will be in g/m².

First published on the web: 12 September 2000.

Authors: Richard Payling and Thomas Nelis.

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Density

Sometimes it is necessary to estimate the density of a sample or coating from its chemical composition alone. This is especially true of quantitative depth profiling with GD-OES. Currently the most effective method is to calculate the unknown density r from the densities of pure materials, assuming constant atomic volume. That is⁽¹⁾

where c_i is the mass % of element i and r_i the density of pure element i. For a proof of this equation, click here.

How good is this equation? Here is a selection of results:

Generally the approach works well for metal alloys, typically giving results within a few percent of measured values. Non-metals, in particular oxides, do not fair as well, being in error by up to 40%. The assumption of constant volumes is clearly not working for these materials. The main reason lies with the nature of chemical bonds, the stronger the bond the closer the atoms are drawn together and the smaller the atomic volumes; some of the strongest bonds are in oxides. A secondary reason is in the arrangement of atoms, i.e. the crystal structure, since a different structure may have a different number of atoms per unit volume.

So where do we go from here? A more refined model must include additional information about chemical bonds and crystal structure.

Correction for Oxides and Nitrides

Clearly the constant atomic volume equation is not accurate for oxides, it is also not accurate for other compounds with strong covalent bonds, such as nitrides, which reduce the apparent size of the atoms and hence increase the density.

To date, there is no general and easy method to include covalent bonds. But we know from the analysis whether O and N and other covalent bond forming elements are present. So we will assume that if they are present then they will form such bonds, and they will form them preferentially with those elements which give the lowest energy.

We will assume the order of oxide or nitride formation is determined by the electronegativities of the other elements present, beginning with the lowest. Hence, for example, Mg (1.31) will form an oxide before Al (1.61) and Al before Zn (1.65).

Then, as suggested by Analytis,⁽²⁾ we simply look up a table of densities for known oxides and nitrides. To determine the density we then sum the specific densities (inverse densities) of the metals, oxides and nitrides, into the density equation above.

So now we can calculate densities using constant atomic volumes for up to five elements a-e, with oxide and nitride correction.

If, instead, you would like to measure density there is a simple method described by Jukka IsoPahkala of SP Swedish National Testing and Research Institute.

References:
 (1) R Payling, in R Payling, D G Jones and A Bengtson (Eds), Glow Discharge Optical Emission Spectrometry, John Wiley & Sons, Chichester (1997), pp 287-291.
 (2) M Analytis, Spectruma Analytik GmbH, private communication (1998).

Author: Richard Payling

First published on the web: 15 May 2000.

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Prove of density equation:

Let's assume we have a binary alloy of elements a and b. One atom of a has a volume of V_a and one of b has a volume V_b. The volume occupied by one mole of a is V_a x N_A, where N_A is Avogadro's number (the number of atoms in a mole), and for b is V_b x N_A. The mass of one mole of a is w_a x N_A, where w_a is the atomic weight of a, and for b is w_b x N_A. The density r_a of a is therefore

mass/volume = w_a x N_A / V_a x N_A = w_a / V_a,

and for b is w_a / V_a. Now suppose we mix A% of a with B% of b, and assume the atomic volumes do not change. The total volume of one mole is (A x V_b x N_A  + B x V_b x N_A.)/ 100. The mass of one mole is (A x w_a x N_A + B x w_b x N_A) / 100. But instead of the density we will calculate the inverse density (the reason will become apparent)

This process can then be generalised to any number of elements with constant atomic volumes.

First published on the web: 15 February 2000.

Author: Richard Payling

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