Units of Measure in Spectroscopy, Part I: It's the Volume, Folks!

February 14, 2014
Jerome Workman Jr.
Jerome Workman Jr.

Jerome Workman Jr. serves on the Editorial Advisory Board of Spectroscopy and is currently with Unity Scientific LLC. He is also an adjunct professor at Liberty University and U.S. National University. He can be reached at jworkman04@gsb.columbia.edu.

Spectroscopy

Spectroscopy, Spectroscopy-02-01-2014, Volume 0, Issue 0

The data show that different units of measurement have different relationships to the spectral values, for reasons having nothing to do with the spectroscopy. This finding disproves the assumption that different measures of concentration are equivalent except, perhaps, for a constant scaling factor.

The data show that different units of measurement have different relationships to the spectral values, for reasons having nothing to do with the spectroscopy. This finding disproves the unstated, but near-universal, assumption that different measures of concentration are equivalent except, perhaps, for a constant scaling factor.

Over the course of these "Chemometrics in Spectroscopy" columns we have introduced or explained various concepts related to multivariate calibration and assessment using mostly near-infrared (NIR) spectroscopy, but also delineating concepts that are applicable to other molecular spectroscopy techniques. One of the elusive problems associated with quantitative analysis using these techniques is the unexplained error in analysis, and the use of reference techniques to calibrate spectroscopic methods relegates them to being referred to as "secondary techniques." Recent installments of this column have introduced a subject that has mystified analysts over the past several decades (1–11).

For as long as we've been working with NIR (since 1976 for one of us), we've recognized that modern NIR analysis is subject to a near-universal, but widely ignored problem. In the early days everybody knew that. Since then, the application of chemometrics has covered up the problem and enabled calibration models that "worked." Even though these applications are valuable, they represent game-playing, not science. Newcomers accept the paradigms used without much thought given to what is happening "under the hood." Experienced practitioners of NIR think that something is wrong, however, and the methodology is sensed as not being part of the universe of science. Many characteristics of NIR appear to fly in the face of conventional science, constituting a set of symptoms:

  • the apparent need for, and use of, more variables in calibrations than can reasonably be justified

  • algebra dictates that no more equations should be needed than variables

  • difficulty in reproducing calibrations for the same constituents in the same type of samples;

  • inability to reproduce wavelength sets (for multiple linear regression [MLR] models)

  • difficulty or inability to relate wavelengths chosen (for MLR) or prominent bands (for principal components regression [PCR] or partial least squares [PLS]) to spectral features

  • standard error of calibration (SEC) should drop precipitously to the noise level

  • unexpected and unexplained (or unexplainable) "outliers"

  • SEC and standard error of prediction (SEP) should drop precipitously when the number of wavelengths or factors equals the number of variations in the samples (in general that doesn't happen)

  • spectroscopic measurements should be accurate over the entire range of concentrations, not only dilute solutions or a small range of values

  • calibrations should be extrapolatable, with a calculated reduced accuracy at the extremes of the range, and

  • calibration transfer should be as easily and readily performed as comparing two mid-IR spectra.

Taken together, these discrepancies from all previous knowledge of chemistry, spectroscopy, physics, and mathematics constitute a set of "mysteries" that nobody in the affected fields had been able to explain.

These various symptoms of an unknown problem have been attributed to a variety of causes, such as optical scatter effects, reference laboratory error, instrument noise, instrumental nonlinearities, stray light, detector saturation, calibration issues, the "wrong" transforms of spectral data, the "wrong" calibration algorithm, incorrect calibration parameters, an unknown reflectance relation between spectrum and composition, and sample inhomogeneity.

To be sure, all of these effects exist and affect the spectral readings and the nature of the calibration models achieved. However, they failed to satisfy.While attempts to mitigate these effects "worked" to some greater or lesser extent in individual situations, they failed to improve calibration performance as often as they succeeded in that attempt, and a "shotgun" approach to trying different corrections was often needed. Here again, the reasons for that behavior were unknown, and it was impossible to use the knowledge to predict, for a new calibration situation, whether any given "fix" would succeed. Therefore, while the various methods developed to address the symptoms allowed NIR to enjoy the widespread success it had achieved, the lack of understanding of the underlying problem prevented solving that problem in the scientific sense, and the net result was simply to replace one set of mysteries with another. Something was missing.

The recent discovery about the effect of using different units for the reference values, which we've described over the course of a number previous column installments (1–11), seemed to have the right properties to explain these mysteries. This approach basically required a different kind of data transform, which differed in two key respects from previous data transforms used in conjunction with NIR data:

  • It was a transform of the concentration values.

  • It was based on known physical chemistry.

The experimental finding that electromagnetic spectroscopy is sensitive to the volume percent (or, strictly speaking, the volume fraction) of materials in a sample is the resulting conclusion. There are a variety of measurement errors because of variation in sample presentation and instrumentation, as described above, but fundamentally the spectroscopy is relating to volume fraction and not weight percent (despite the overwhelming use of weight percent as the most common units used for expressing analyte "concentration" in a majority of analytical situations, nor volume irrespective of density and mixture or solvation issues). Understanding the ramifications of this volume effect may lead to an improved understanding of quantitative measurement errors and, eventually, to changing multivariate spectroscopic techniques into primary analytical methods.

Progress

This column and the next few installments are intended to constitute a discussion on the ramifications of the new findings. We base our discussions on the previously reported results (1–11).

To review the findings until now, in our previous column (11) we examined the behavior and performance of nearly ideal samples (clear liquid mixtures) using a nearly ideal calibration algorithm (classical least squares [CLS]) and found several characteristics of our data set that indicate that volume fractions are the operative physical variable that spectroscopy is sensitive to. First, we noticed that the values from the CLS analysis of the spectral data from the second laboratory had values that corresponded to the values that represented the target values in the original experimental design: 0%, 25%, 50%, 75%, and 100%. We also noted that in the samples for the second laboratory, those values corresponded to the volume fractions of the various components.

We then applied the scientific method to our microcosm and formulated the hypothesis that when Beer's law holds, the absorbance spectra of clear solutions are in fact sensitive to the volume fractions of the components in the mixture. Continuing the application of the scientific method, we verified the hypothesis by applying it to the data from the first laboratory, which had not previously been analyzed this way, and found that indeed the performance of the CLS method was much improved when the CLS values were compared to the volume fractions, rather than any of the other concentration units that had been used previously.

This was ascertained by comparing individual values of the concentrations as determined by the various methods. A more robust comparison can be obtained by calculating a statistical figure of merit, similarly to the way accuracy measures are obtained for other quantitative calibration algorithms. There are both graphical methods available and numeric methods. It made sense to use both approaches for the comparison, if for no other reason than that is standard procedure when performing calibrations with the other algorithms, and we decided to examine our CLS results as closely as is normally done. We started with the numeric approach, and computed the root mean square differences and correlation coefficients between the concentration values from the CLS method and the concentration values obtained using other units. The calculations were taken over all 15 of the samples, going from 0% to 100% of each component in accordance with the experimental design used (see Figure 1 in reference 5).

Figure 1: Plots of CLS values versus weight percent and volume percent, for toluene: (a) weight percent versus CLS values; (b) volume percent versus CLS values.

As we saw, the data from the second laboratory were somewhat erratic, a characteristic that was explained by David Heaps, the scientist who performed the data collection in that laboratory, as being caused by fact that the sample cell used was not large enough to encompass the entire beam from the spectrometer. This caused small variations in the position of the cell, which ordinarily would be negligible, and had inordinately large effects on the readings. We also previously noted the erratic nature of the data from the second laboratory in reference 9. Therefore, we present only the results from the first laboratory in Table I.

Table I: Comparison of performance statistics for different units of concentration

Table I makes it abundantly clear that for all sample components, the comparison of the CLS values with the volume fraction, overall, is much better than with any of the other units. Indeed, the values of these statistical tests are competitive with what is usually expected for "typical" calibrations using other chemometric calibration algorithms (such as PCR or PLS)

CLS calculations, done the way described in the previous columns, have several interesting and useful characteristics of their own.

  • It's the nearest thing we have to an "absolute" analytical method, and therefore, the analytical results can be obtained without ever doing a conventional "calibration" of the sort we usually think of.

  • Because results can be obtained without a conventional calibration, you don't have to worry about laboratory error; there's no "lab error" because there are no laboratory values to deal with (except for validation).

  • CLS values are linear with the known concentrations (as long as they are expressed in appropriate units) over the range 0–100% (or, strictly speaking, volume fractions in the range 0–1).

  • Because of this linearity, a conventional calibration based on volume fractions (or one of the scaled variants) should be extrapolatable.

Given that these CLS values are obtained without any reference laboratory values, and using zero PLS or PCR factors (which also means they can't be overfit), the results for the statistical values for performance of volume fractions in Table I are pretty impressive.

We can further confirm the nature of the agreements by plotting the CLS values for each sample versus the concentration expressed in the two different key units of interest. Figures 1–3 show this view of the data for each of the three different components. The plots for the three components versus weight percents (Figures 1a, 2a, and 3a) show varying amounts of scatter. In Figure 1a, the plot for the CLS values for weight percent of toluene does not show any curvature in the plot to the naked eye. Figures 1a and 3a show a considerable amount of scatter, although Figure 2a does not.

Figure 2: Plots of CLS values versus weight percent and volume percent, for dichloromethane: (a) weight percent versus CLS values; (b) volume percent versus CLS values.

The other three plots of CLS values versus volume fractions (Figures 1b, 2b, and 3b), show decreased amounts of scatter (except for the dichloromethane). Figures 1–3 present a graphical comparison that provides a convenient "eyeball" comparison method. Figures 1a, 2a, and 3a show that the reference values are constant, at each level of the analyte, at the value specified by the experimental design. Figures 1b, 2b, and 3b show that all three exhibit linear relations between the CLS and volume fractions over the range 0–100%. Neither do any of the plots of CLS values versus volume fractions show appreciable amounts of scatter; indeed, the use of volume fractions as the unit for the reference values has caused all the data points to fall on a line that is visibly straight, for all three components.

Figure 3: Plots of CLS values versus weight percent and volume percent, for n-heptane: (a) weight percent versus CLS values; (b) volume percent versus CLS values.

But we would also like to be able to apply a more objective, mathematical comparison method. Therefore, we also applied a statistical linearity test (described in reference 12) to the results. This test computes a t-value against the null hypothesis so that there is no nonlinearity in the relationship between paired data points. A statistically significant value for the computed t-value indicates that there is enough nonlinearity to be definitely detected by an objective criterion, while a value that is too small to be statistically significant means that any nonlinearity, if present, is too small to be detected. Table II shows the t-values for the comparisons when comparing the CLS values against other units.

Table II: t-values from linearity test, spectroscopy versus indicated units

The t-values in Table II, for the comparison of the spectroscopic results with the gravimetric values, indicate highly significant amounts of curvature in the relationship between the CLS values and the known weight percents for dichloromethane, in agreements with our visual observation. The corresponding t-values obtained when the spectroscopic results were compared to the volumetric values indicate no detectable nonlinearity for this comparison. These statistics agree with what we infer from Figure 2.

Interestingly, neither toluene nor the n-heptane exhibit statistically significant t-values for either nonlinearity test. Whereas Figure 1a appears to be adequately straight, Figure 3a visually shows what appears to be a curvature of the relationship for the n-heptane; both ends of the relationship appear to bend upward from a line that follows the rest of the data. Table II, however, shows that the t-value for this case is only 3.25, a marginally significant value at best. In the cases of toluene and n-heptane, the large amount of scatter seen in this plot is masking any systematic effect that might actually be present and prevent us from demonstrating that the systematic portion is more than the large amount of scatter could account for, giving us a nonsignificant result.

An interesting side note here: Both the graphical and numerical results show that volume fraction provides a spectroscopic method that is substantially linear over the range 0–100% (when Beer's law holds, of course) and accommodates the variations in the "matrix" — that is, the rest of the sample. Those properties would not disappear if the concentration values expressed in volume fraction were multiplied by a constant. Convenient constants would be ones like density, moles per unit volume, or similar implicit properties of the analyte. Suitable constants would be constants that are measures of a property per unit volume (as the density and moles per unit volume themselves are). Multiplying one of these constants by the volume fraction (or volume ratio) results in canceling the volume term in the numerator of the volume fraction, and thereby replacing the numerator term in the volume fraction expression by the alternate unit (that is, mass if density is the constant, or moles if moles per unit volume were the constant). The resulting values would then be concentration measures on a volume basis (density, molarity, and so on), which arguably are more familiar to chemists. These other quantities would not necessarily satisfy the more stringent requirements of the CLS algorithm, but would be expected to be linearly related to the volume concentration, and therefore provide a satisfactory basis for expressing the analyte concentration for the more common calibration algorithms that do indeed require the concentration for the calculations.

This is what the data show. The important question now is what it means.

A Short Review

In the first pair of columns in the previous subseries (1,2) we looked at the CLS analysis of water–methanol–acetic acid mixtures and found that the values computed using the CLS algorithm did not agree with the known amounts of the mixture components introduced in the mixtures. In that experiment, we also found that the reconstruction of the mixture spectrum from the spectra of the pure components was rather poor. Based on the poor reconstructions of the mixture spectra, we attributed the inaccurate analytical results obtained to the known changes in hydrogen bonding because of intermolecular interactions between the three components, interactions that distorted their spectra.

Therefore, starting with part III of the previous subseries (3), we redid the experiment using hydrocarbons that would not interact as the materials in the previous experiment did because the materials used for the second experiment did not contain any –OH in their structure. With these mixture components we found that indeed, we were able to reproduce the spectra of the mixtures fairly well.

Lo and behold, despite this ability to reproduce the spectra, we still found large disagreements between the CLS calculations and the accurately known values for the mixture compositions. Considerable sweat was expended trying to reconcile the differences, a process that was recounted over the last few columns.

The starting point for thinking about this is the behavior of the CLS algorithm when applied to data that follow Beer's law. The CLS algorithm is the nearest thing we have in spectroscopy to an absolute method, in that we start with the spectra of the pure components of a mixture and use those to recreate the spectrum of the mixture; we described this above. The coefficients obtained directly represent the contribution of each of those components to the spectrum of the mixture. There is no room for fudging here. A given set of spectra (that is, pure components and the mixture) admits of only one solution, where each coefficient represents the fraction of the corresponding pure component spectrum that contributes to the final mixture spectrum. Thus, the CLS algorithm is concerned only with the relationships between spectra; it says nothing about the relationships between the spectra and the concentrations of the components of the mixture. Therefore, while the CLS algorithm is developed from Beer's law, and requires that Beer's law holds for the materials involved, CLS is not Beer's law.

There is also the question of units. Absorption is dimensionless. A counterargument that has been raised is that in Beer's law, any units for the concentration get taken up in the absorption coefficient. This is true, but beside the point. The point is that the absorption is dimensionless, because it is the logarithm of another dimensionless quantity, the sample transmittance, and it is the absorbance that is used in the CLS algorithm. This is another aspect of the difference between Beer's law and the CLS algorithm. It is immaterial that if an arbitrary unit is used for the concentration, the absorbance coefficients must have units that cancel those concentration units. Because the absorbance is dimensionless, the coefficients calculated for the absorbances by the CLS algorithm are also dimensionless quantities, and those are the fractional amounts of the corresponding pure-component spectra that each contributes to the mixture spectrum.

The question then becomes, what physical quantities do those spectral fractions correspond to? In this view, being linearly related to the spectral absorbance is a necessary, but insufficient, condition. The quantity that the concentration is expressed in and the numerical values for the concentration the CLS algorithm provides for the spectroscopic contribution of each component spectrum to the mixture spectrum have to be in agreement. From the experiments we did, that physical quantity is the volume fraction. It is true that for purposes of conventional calibration you can scale those values by multiplying them by the density, molecular weight, or some other quantity, but that does not change the nature of the fundamental interaction between the light and the samples, which does seem to be related to the volume fraction.

What we found was that the physical property that agreed with the spectral results is the volume fractions of the components of the mixtures. This leads to some insights into what is happening.

We noted in the previous subseries in part X, Table I (9) (as well as in Table IV in part XI [11]) that there is not a unique conversion between concentration values expressed in different units.

Figure 4: Relationship between weight fractions and volume fractions for toluene in ternary mixtures. The multiple lines represent different relative amounts of the other two components (dichloromethane and n-heptane with compositions at 20% intervals).

There is another aspect to the relationships between different units, which we demonstrate in Figures 4–6. In those figures, we see that the relationship between the different units (weight fraction and volume fraction) is not a linear relationship. Furthermore, those plots also show that the relationship is not unique. For example, a vertical line drawn at any value of volume fraction can intersect any of the lines, and therefore can correspond to different weight fractions of the analyte, depending on the composition of the remainder of the sample (that is, relative amounts of the other components). Similarly, a horizontal line drawn at any value of weight fraction can intersect any of the lines, showing the correspondence to the different volume fractions, again depending on the composition of the rest of the sample. This is the same effect observed in Table I from part IX and Table II from part XI (from the previous columns); the main difference is that the numeric values presented in those tables correspond to the actual samples that were made up for the experiment, while Figures 4–6 indicate the nature and magnitude of the effects for essentially all possible samples.

Figure 5: Relationship between weight fractions and volume fractions for dichloromethane in ternary mixtures The multiple lines represent different relative amounts of the other two components (toluene and n-heptane with compositions at 20% intervals).

Furthermore, there is another, more subtle effect that can be observed in these figures, although it is perhaps easier to see this in Figures 4 and 6 than in Figure 5. Note that the curved lines in each figure are not equally spaced despite the fact that the compositions of the remainder of the samples differ by even 20% intervals. This underscores the fact that not only is the main analyte in each graph exhibiting a nonlinear relationship between the different units, but this nonlinearity exists even between the different solutions of the sample.

We will have more to say about these effects in our next column.

Figure 6: Relationship between weight fractions and volume fractions for n-heptane in ternary mixtures. The multiple lines represent different relative amounts of the other two components (toluene and dichloromethane with compositions at 20% intervals).

The Mathematics

We had just about reached the point of realizing that the data indicated that absorbances of mixtures —when measured in situations where Beer's law applies — are proportional to volume fractions and were preparing a manuscript for publication (13). In preparation for writing up the work, we did a literature search for work dealing with CLS. As reported in that paper, we did not find any reference to the use of volume fractions in conjunction with CLS analysis. However, upon perusing the book about effects of scattering samples written by our good friend Don Dahm (14), down near the bottom of page 5 in his description of a "representative layer," we came across this remarkable statement: ". . . we will require that the volume fraction of each kind of particle in the layer is the same as the volume fraction in the sample."

I (HM) immediately ran to my computer and sent Don an e-mail. After explaining what we had just learned, I asked, "Don, what made you take it into your head to use volume fractions for the description of the concentration in your book?" His response was something equivalent to, "Well, every physicist knows that!"

After a few more back-and-forth exchanges, wherein Don said that it follows from Beer's law, we eventually got to the point where I simply asked him, "Can you prove that? Starting with Beer's law can you derive the fact that components in a mixture absorb in proportion to their volume fractions?" Don then developed a proof. Here is Don's proof exactly as he sent it, except edited only so that the equation numbers conform to our style and numbering:

<Start of derivation>

This mathematical explanation is presented here. We start with Beer's law, which may be stated as

Because absorbance is a dimensionless quantity, the units on the right-hand side must also be dimensionless. For example, if "concentration" is expressed as the mass concentration (g/cm3) and pathlength is measured in centimeters, then the absorptivity must be expressed in cm2/g. "Volume fraction" would be a perfectly reasonable measure to use for "concentration" when applying Beer's law to a mixture of liquids, and because volume fraction is dimensionless, absorptivity would be expressed in cm-1 in such a case.

The intent of this derivation is to demonstrate that volume fraction is a fundamental property in determining the absorbance of mixtures, regardless of the specific measurement used for concentration. We will do this by examining application of Beer's law to a mixture, expressing "concentration" as mass concentration.

The following symbols and definitions will be used in the derivation:

V = the volume of a solution which is a mixture of pure compounds

A = the absorbance of a solution in a cell of unit pathlength

Ai pure = the absorbance of a pure sample of compound i

Ai = the contribution of compound i to the absorbance of the solution (A)

αi = the absorptivity of the ith component

νi = the volume of the ith pure component

mi = the mass of the ith pure component

ρi = density of the ith pure component (mi/νi)

i} = mass concentration of component i in solution (mi/V)

l = the pathlength.

Consider a sample that is formed by mixing pure materials having known volumes (ν1, ν2, and so on). The final volume of the mixture is given by V = ν0 + Σνi, with the term ν0 representing the volume change upon mixing. Here, we will consider the "ideal mixture" in which ν0 = 0.

Beer's law is, again

Applying Beer's law to a pure component (i), and using the nomenclature established above, gives

While applying Beer's law to a single component (i) within the mixture gives

By definition mi = ρi × νi and γi = mi/V. Combining these definitions gives

Or equivalently:

Note that νi/V is the volume fraction of component i in the mixture. Substituting the definition of mass concentration given by equation 5 into equation 3 gives this expression for Beer's law for a component in a mixture:

Thus, at constant pathlength, the contribution of a component to absorbance is directly proportional to its volume fraction.

Furthermore, note that we can group (αi× ρi × l) together, and apply equation 2, giving

For transmission through a homogeneous solution, from Beer's law we know that the total absorbance of a mixture is the sum of the contribution of each of the absorbers, and we express that mathematically with the following expression:

Combining equations 7 and 8 gives

Thus, according to Beer's law, the absorbance of the mixture is dependent upon the absorbances of the individual pure components, the volume fraction of each, and nothing else.

<End of derivation>

Thus, we were able to confirm our experimental findings with a mathematical derivation based on the physics of Beer's law.

While we will continue our discussion about the effects of curvature of the relationship between different units for concentration in the next column, for now we will finish off with one other point of view. As described above, only the volume fraction agrees with the spectroscopically determined fractions. This is an empirically determined fact, backed by theory, and is a critically important one. The matter is not discussed in the spectroscopic literature, but there seems to have been a tacit assumption made by spectroscopists, and even analytical chemists in general, that different measures of what we consider "concentration" are equivalent to each other, differing at most by perhaps a scaling factor.

As we see in Figures 4–6, however, different units of measure have different relationships to each other, usually relationships that are not linear with respect to each other. As pointed out earlier, if a given unit of measure is linear with respect to the spectroscopy, then other units of measure, which are not linear with respect to that unit of measure, will ipso facto also not be linear with respect to the spectroscopy. For this reason, knowing which unit of measure agrees with the spectroscopy is a key point, and the empirical finding that the volume fraction is the one that agrees with the spectroscopy is crucial.

There is a difficulty in dealing with volume fractions, however, in that it is rarely used as the "natural" unit for expressing concentrations in chemical analysis. On the other hand, the physics of the interactions between light and matter doesn't depend on the mathematical or chemometric calculations applied to spectral data to perform the chemical analysis with. Therefore, while the use of CLS was necessary to determine what physical sample properties conform to the spectroscopy, all other methods of chemometric analysis are subject to the same effects.

Calibration algorithms other than CLS don't require knowledge of all the components in a given sample; most only require knowledge of the concentration of the one component of interest (that is, the analyte). But now we see that the units that the analyte is expressed in makes a difference. So the question arises: Are there any other units to express the analyte concentration in that would also be linear with respect to the spectroscopic measurements, preferably while also being more familiar to the chemist? These other quantities would not necessarily satisfy the more stringent requirements of the CLS algorithm, but would be expected to be linearly related to the volume percent concentration, and therefore provide a satisfactory basis for expressing the analyte concentration for the more common calibration algorithms, that do indeed require the concentration for the calculations.

The answer is, yes, there are other units. Consider, for example, using the unit of measure of weight per unit volume (wt/vol). Expressing volume fraction as vol/voltot we can then perform the calculation of multiplying the numerator of the volume fraction by the weight. This clearly makes the wt/vol a scaling factor applied to the volume fraction, as expressed in equation 10:

Canceling the two volume terms in the numerator of equation 10 results in

Thus, weight per unit volume is seen to be a unit for concentration measure that will be linear with the spectroscopic data because it is one way to scale volume fractions.

Other scaling factors can also be used. For example, molarity is a chemical measure of concentration whose units are moles per volume. This can be similarly introduced into a volume fraction expression to form a different concentration measure that is a scaled variation of volume fraction:

By induction, it becomes clear that any measure of concentration that is expressed as "quantity per unit volume" can be used as a measure of analyte concentration that will be linear with respect to the spectroscopic properties of that analyte. (Note: We extend our thanks to Jim Brown for pointing this relationship out to us.)

When these units of measure are used to describe the concentration of the analyte, then indeed, any of the conventional calibration algorithms will implicitly incorporate whatever scaling factors are needed in the calibration coefficients, to perform the conversions between the spectroscopic quantities measured, and the units in which the analyte concentration is expressed. If other units are used, however, then a single "scaling factor" will not suffice to accommodate the nonlinear relations between the different units of measure.

References

(1) H. Mark and J. Workman, Spectroscopy 25(5), 16–21 (2010).

(2) H. Mark and J. Workman, Spectroscopy 25(6), 20–25 (2010).

(3) H. Mark and J. Workman, Spectroscopy 25(10), 22–31 (2010).

(4) H. Mark and J. Workman, Spectroscopy 26(2), 26–33 (2011).

(5) H. Mark and J. Workman, Spectroscopy 26(5), 12–22 (2011).

(6) H. Mark and J. Workman, Spectroscopy 26(6), 22–28 (2011).

(7) H. Mark and J. Workman, Spectroscopy 26(10), 24–31 (2011).

(8) H. Mark and J. Workman, Spectroscopy 27(2), 22–34 (2012).

(9) H. Mark and J. Workman, Spectroscopy 27(5), 14–19 (2012).

(10) H. Mark and J. Workman, Spectroscopy 27(6), 28–35 (2012).

(11) H. Mark and J. Workman, Spectroscopy 27(10), 12–17 (2012).

(12) H. Mark, J. Pharm. Biomed. Anal. 33, 7–20 (2003).

(13) H. Mark, R. Rubinovitz, D. Heaps, P. Gemperline, D. Dahm, and K. Dahm, Appl. Spect.64(9), 995–1006 (2010).

(14) D.J. Dahm and K.D. Dahm, Interpreting Diffuse Reflectance and Diffuse Transmittance: A Theoretical Introduction to Absorption Spectroscopy of Scattering Materials, 1 Edition (IM Publications, West Sussex, UK, 2007).

Jerome Workman, Jr. serves on the Editorial Advisory Board of Spectroscopy and is the Executive Vice President of Engineering at Unity Scientific, LLC, (Brookfield, Connecticut). He is also an adjunct professor at U.S. National University (La Jolla, California), and Liberty University (Lynchburg, Virginia). His e-mail address is JWorkman04@gsb.columbia.edu

Jerome Workman, Jr.

Howard Mark serves on the Editorial Advisory Board of Spectroscopy and runs a consulting service, Mark Electronics (Suffern, New York). He can be reached via e-mail: hlmark@nearinfrared.com

Howard Mark