58
K. D. Dahm and D. J. Dahm
function A(R n ,T n , which was introduced in Eq. 3.25 of Sect. 3.8). The value of this
function is identical regardless of the assumed thickness of the sample.
Table 3.1 also shows the separation of the effects of absorption and scatter through
the “absorbance,” −log 10 (1 − A), as defined in Eq. 3.4. According to Beer’s Law,
absorbance should be linear with sample thickness. Thus, if the data in Table 3.1
were following Beer’s Law, the absorbance on each row would be one-half the
value of the row above it. This is clearly not the case at the top of the table, which
represents the actual sample. However, if one views the results for thicknesses of ~
1/32 cm and below, one sees that absorbance is indeed linear with thickness. Thus,
if we produce layers that are thin enough that the effects of absorption and scatter
have been successfully isolated and separated, Beer’s Law is a good model, even for
scattering samples. Applying the fact that Beer’s Law is linear with thickness, we can
use the absorbance that was determined for a thin layer (e.g., 1/128 cm or 1/256 cm)
to compute an expected absorbance for the actual sample thickness of 1 cm:
Absorbance = (128)(0.0101) = 1.29
(3.26)
Absorbance = (256)(0.00505) = 1.29
(3.27)
Notice the values obtained using these two thin layers are essentially identical to
each other but very different from the measured absorbance from the original sample,
0.658. The value of 1.29 can be termed the “scatter-corrected absorbance.” It can be
interpreted as the absorbance that would be observed from a 1 cm thick sample of
a hypothetical material that had the same absorbing power as the real sample, but
with a complete absence of scatter. It was noted in Sect. 3.1 that the absorbance of
a material is not repeatable from one sample to another because absorbance is not
solely a measure of the absorbing power of the sample material; it is also influenced
by sample size, geometry, etc. The “scatter-corrected absorbance” has the potential
to address these limitations and provide a more genuine metric for the absorbing
power of a material.
Mathematically, one can continue halving the thickness of a layer indefinitely,
until the obtained value of the scatter-corrected absorbance and/or the absorption
coefficient becomes constant, to as many significant figures as desired. Recall that
one of the assumptions underlying Benford’s equations was that the sample was
uniformly distributed; the approach presented throughout this section relies upon
this assumption. Even if this assumption is reasonable for the sample, one must take
care to consider whether the hypothetical “thin layers” are physically meaningful.
Suppose the original sample as illustrated in Table 3.1, which was 1 cm thick, was
made up of particles that were approximately 1 mm in diameter. Table 3.1 includes
a row in which d = 1/1024 cm, or approximately 1 mm. (The calculated A and R
values for this particular row are similar to those as shown in Fig. 3.9). Thus, one
could plausibly consider a layer of thickness d = 1/1024 cm to be approximately
“one particle thick” and representative of the sample as a whole. Mathematically,
K. D. Dahm and D. J. Dahm
function A(R n ,T n , which was introduced in Eq. 3.25 of Sect. 3.8). The value of this
function is identical regardless of the assumed thickness of the sample.
Table 3.1 also shows the separation of the effects of absorption and scatter through
the “absorbance,” −log 10 (1 − A), as defined in Eq. 3.4. According to Beer’s Law,
absorbance should be linear with sample thickness. Thus, if the data in Table 3.1
were following Beer’s Law, the absorbance on each row would be one-half the
value of the row above it. This is clearly not the case at the top of the table, which
represents the actual sample. However, if one views the results for thicknesses of ~
1/32 cm and below, one sees that absorbance is indeed linear with thickness. Thus,
if we produce layers that are thin enough that the effects of absorption and scatter
have been successfully isolated and separated, Beer’s Law is a good model, even for
scattering samples. Applying the fact that Beer’s Law is linear with thickness, we can
use the absorbance that was determined for a thin layer (e.g., 1/128 cm or 1/256 cm)
to compute an expected absorbance for the actual sample thickness of 1 cm:
Absorbance = (128)(0.0101) = 1.29
(3.26)
Absorbance = (256)(0.00505) = 1.29
(3.27)
Notice the values obtained using these two thin layers are essentially identical to
each other but very different from the measured absorbance from the original sample,
0.658. The value of 1.29 can be termed the “scatter-corrected absorbance.” It can be
interpreted as the absorbance that would be observed from a 1 cm thick sample of
a hypothetical material that had the same absorbing power as the real sample, but
with a complete absence of scatter. It was noted in Sect. 3.1 that the absorbance of
a material is not repeatable from one sample to another because absorbance is not
solely a measure of the absorbing power of the sample material; it is also influenced
by sample size, geometry, etc. The “scatter-corrected absorbance” has the potential
to address these limitations and provide a more genuine metric for the absorbing
power of a material.
Mathematically, one can continue halving the thickness of a layer indefinitely,
until the obtained value of the scatter-corrected absorbance and/or the absorption
coefficient becomes constant, to as many significant figures as desired. Recall that
one of the assumptions underlying Benford’s equations was that the sample was
uniformly distributed; the approach presented throughout this section relies upon
this assumption. Even if this assumption is reasonable for the sample, one must take
care to consider whether the hypothetical “thin layers” are physically meaningful.
Suppose the original sample as illustrated in Table 3.1, which was 1 cm thick, was
made up of particles that were approximately 1 mm in diameter. Table 3.1 includes
a row in which d = 1/1024 cm, or approximately 1 mm. (The calculated A and R
values for this particular row are similar to those as shown in Fig. 3.9). Thus, one
could plausibly consider a layer of thickness d = 1/1024 cm to be approximately
“one particle thick” and representative of the sample as a whole. Mathematically,