Cliff-Lorimer Ratio Method for EDS Measurements
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A spectrum collected by the EDS measurement can be quantified by the Cliff-Lorimer ratio method to obtain the relative concentrations between two elements from the integrated EDS peak intensities. When the sample satisfies the “thin-foil criterion”, meaning that it is thin enough so that any absorption or fluorescence effects are negligible, direct measurement of the individual factors involved in Equation [4646a] can be avoided and the Cliff-Lorimer ratio method for a binary or multi-component system can be used,

           Cliff-Lorimer ratio -------------------------------------------------------- [4643a]

where IA -- The integrated EDS peak intensity of the element A
           IB -- The integrated EDS peak intensity of the element B
           nA -- The atomic density (in percentage) of the element A
          nB -- The atomic density (in percentage) of the element B
          kAB is Cliff-Lorime sensitivity factor between elements A and B

k factors (e.g. kAB) not only varies with microscope conditions such as accelerating voltage but also depends on the materials such as the identity of the two elements A and B and the atomic number correction factor (Z) considering that the effects of absorption and fluorescence are negligible. This factor can either be experimentally calibrated with a standard containing a known ratio of the two elements or theoretically calibrated based on the first principle. By convention, kAB is normally tabulated for a particular instrument with element B being Si. The choice of Si is partially influenced by the large number of different silicates available. Since the results from calculation are generally not very reliable, the theoretical method is normally used for finding quick answers when accuracy is not important.

If there are more than two elements in the materials, the following formula is applicable,

           EDS -------------------------------------------- [4643b]

where i represents the elements A, B, C, ...

Equation [4643b] means the total atomic density in the materials is 1 (100%).

In the case of multiple elements, different Cliff-Lorime sensitivity factors can be calculated by,

           different Cliff-Lorime sensitivity factors ------------------------------------------------ [4643c]

It should be mentioned that the “thin-foil criterion” sometimes cannot be satisfied because characteristic X-rays can be absorbed by thick specimens, which affect the magnitude of k factor and the compositions are no longer simply proportional to the intensities. X-ray absorption is a function of the energy of X-rays. Low energy peaks will be more strongly absorbed than high energies ones. A quantitative analysis without any kind of absorption correction will therefore penalize light elements. The magnitude of the error will depend on the specimen thickness and density as well as the take-off angle. Corrections are usually necessary in such cases, where the zero-thickness k-factor is related to the actual k-factor at thickness t by,

 Corrections of K factors-- [4643d]

where (µρ)Asp and (µρ)Bsp-- The mass absorption coefficients of the characteristic X-ray lines A and B
          kAB -- The zero-thickness k-factor, which can be obtained from thin film measurements
          ρ -- The density of the specimen
          t -- The thickness of the specimen at the electron beam position
          θ -- The X-ray take-off angle

Note that straightforward correction of k-factor requires correct ρ, t, and θ. They actually are also uncertain unless the sample is perfectly flat and horizontal at zero tilt.    

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