What Does “Infinite Thickness” Mean in Reference to XRF?

Infinite thickness is a term used to describe the minimum thickness a sample must be in order to absorb all the x-rays of the primary x-ray beam emitted from an XRF instrument and emit the characteristic signal from the sample.

  • Penetration Depth – how deep the primary X-ray radiation into the sample?
  • Escape Depth (Analysis Layer)  – how deep the fluorescence (secondary) radiation can be detected from?

Usually we can assume that once we exceed the penetration Depth for our sample that we also have exceeded the escape depth. The Escape depth controls the analyzed layer.

 

Often, people who encounter XRF methodology and data analysis for the first time are confused by this term, because it makes no sense for something to be “infinitely thick” when taken literally. To get a better understanding of infinite thickness, think of it as a sample that is thicker than the distance that the primary x-ray emission (penetrating) and returning x-ray energies (escaping)  can travel through, or at least as thick as the x-rays from your instrument can “penetrate or see.” In other words, a sample is considered infinitely thick for a specific fluorescence energy when it is thick enough to absorb all fluorescence of atoms deep inside the sample:

 

Infinite thickness will be different for every material, and it can be calculated. The more dense the sample, the less thickness it takes to achieve the x-ray fluorescence definition of infinite thickness. For example, infinite thickness in metal may be only a few microns, while infinite thickness in a polymer may be multiple millimeters. In order to obtain accurate quantitative measurements from any given sample, infinite thickness—among other requirements—must be met.

 

Penetration depth for infinite thickness is based on the kV (Tube excitation) settings of your instrument as well as the Tube material (Generally for S1 TITAN and TRACER = Rhodium).

 

Infinite thickness can be calculated both for individual elements and for compounds. For calculating the infinite thickness of a sample, one needs to know the MAC (mass attenuation coefficient) and density of the sample. For pure element samples, obtaining these two data points is a matter of looking at specialized tables of element density and elemental MACs. For compounds, the situation is significantly more complicated.

How to Calculate Infinite Thickness for XRF Analysis

To calculate the MAC of a compound, one needs to know the elemental MAC of the elements of which the compound consists, the specific energy in question (KeV), and the mass fraction of the various elements of the compound. To calculate the MAC of a compound knowing this information, the formula is:

    μs,E= μi,e* Ci+ μj,e* Cj

Where:
μs,E =MAC of a compound or a mixture at a specific KeV
μi,e = MAC of element ‘i’ at a specific KeV
μj,e = MAC of element ‘j’ at a specific KeV
Ci = Concentration of element ‘i’ at a specific energy expressed as weight fraction
Cj = Concentration of element ‘j’ at a specific energy expressed as weight fraction

Now, knowing the MAC of the compound in question, one can apply the formula for calculation of infinite thickness. The formula for infinite thickness (t) for any energy in any sample is:

    loge⁡[1-lt/l]=μs* * ρ * t

Where μs* = Total Effective MAC,  ρ= density, and t = infinite thickness

 

At infinite thickness,  lt/l = 0.99, thus yielding a result:
    μs* * ρ * t0.99=6.91
or
    t0.99=6.91/(μs* *  ρ)

Note, there are some interval calculations that need to be done in order to apply these formulas, including sample density ρ and the weight fractions of the various elements in the compound in questions. The necessary information and formulas for such calculations are not specific to XRF and can be found in a variety of textbooks and online sources.

What if my sample does not meet the infinite thickness requirement?

When your sample does not meet the infinite thickness requirement, it throws off quantitative XRF calibrations (which causes inaccuracies in data obtained) because the calibration is expecting feedback from “a larger” or “more” sample which would attenuate all the primary x-rays and thereby return maximum information by way of fluorescent x-rays. The reading now is not longer just proportional to the concentration but also to the thickness.
 
Since part of the sample is "missing," the calibration interprets the information incorrectly, as if an infinitely thick sample is being presented to the instrument and actually sending that signal. However, when some of your fluorescence are not captured because there is not “enough” sample or your sample “lacks the proper density” to excite at the maximum possible depth (as your calibration is anticipating), the accuracy of your data will be compromised. E.g. you will under report.

Still have questions? Contact a Bruker in house expert for all of your handheld XRF analysis questions!

In short, in order to produce accurate data from XRF analysis, make certain you present the instrument with “enough sample.” “Enough sample” is dependent on the density of the material as well as the MAC of the compounds in the sample. Below are some quick and dirty techniques for getting the most out of your data when analyzing different types of samples in the absence of higher math skills (note: these guidelines should be adhered to at all times regardless of calculating sample densities etc. or not):

Metal Samples:

  • Make certain to cover the entire window of the instrument with your sample
  • For smaller metal samples, use a small spot collimator/weld adapter, or present multiple samples of the same material in order to cover the entire window of the shutter of the analyzer

Soil Samples:

  • Make certain to cover the entire window of the instrument with your sample
  • Make certain your sample is at least an inch thick worth of sample even when compacted
  • Be sure to take elongated measurements paying close attention to your precision numbers
  • Make certain your samples contain less than 20% moisture
  • Avoid small rocks or chunks in your sample; try to make the particle size as homogeneous as possible

Consumer Product Samples:

  • Make certain to cover the entire window of the instrument with your sample
  • Make certain your sample is at least an inch thick worth of sample; stack up multiple layers until thickness is achieved
  • Pay attention to the 2 standard deviation precision numbers during analysis
  • Make certain the sample is flush against the window of the instrument and there is no air between the sample and the shutter; air causes error

Adhering to these guidelines and techniques, you can help you maintain certainty that you are achieving the most from your XRF analysis. As far as accuracy is concerned, if you put insufficient data into the instrument by presenting it with a sample that is not adequate, you can be sure that your data will be lacking ; garbage in, garbage out.

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