Contents of Chapter.
CONST
Maximum_Channels = 8192;
MaxNumPks = 50; { Maximum peaks in a FIT, or window}
TYPE
Spectrum_counts = ARRAY [1..Maximum_Channels] OF Real;
Experiment Header;
{ The following record, Expt_InfoRec, comprises the
EXPERIMENT header. }
Expt_InfoRec = RECORD
LastSpect : Integer;
FirstSpec : Integer; { first Spectrum# }
Specimen_ID : String[50];;
MCA_Filename : String[25];;
Specimen_Comment_Field : Str255;
Was_PassWord : String[25];
RefFile: Boolean;
Analyst : String[50];
Detector : Detector_Rec;
Azimuth : real; { degrees }
Elevation : real; { degrees }
Detector_Area : real; { cm sq }
Detector_Thickness : real; { mm }
C_Thickness : real; { um }
Diamond : real; { spare }
Mylar : real; { um }
BN_Thickness : real; { um }
SiN_Thickness : real; { um }
Ice_Thickness : real; { um, as pure oxygen }
Au_Thickness : real; { um }
Al_Thickness : real; { um }
Be_Thickness : real; { um }
Si_Thickness : real; { um }
Moxtek : real; { um }
Paralene: real; { um }
WDS_Resolution: real; { eV }
dE : real;
Si_Resolution : real; { eV at Mn K alpha }
Energy_Intercept : real; { y intercept in eV }
Energy_Slope : real; { }
Number_of_Channels : integer;
kV : real;
Detector_Tilt : real; { Beta }
Quantum : real;
Spare2 : Boolean;
Spare3 : integer;
END;
Detector_Rec = RECORD
Spec : (EDS,WDS,EELS);
ID : integer;
END;
{The SPECTRUM header is made up by combining the following records.}
Specrtrum Header;
Element_InfoRec = RECORD
Atomic_number : Integer;
spare1 : real;
Weight_Fraction : real;
spare2 : real;
Valence : real;
END;
Plot_InfoRec = RECORD
Plot_Connected : Integer;
Plot_Symbol : Integer;
Spectrum_Color : RGBColor; {Red, Green, Blue }
END;
Acq_InfoRec = RECORD
Probed_Area : real;
X_Position : real; { um }
Y_Position : real; { um }
Spare1 : real;
FirstChannel : integer;
LastChannel : integer;
Begin_Faraday : real; { nA }
End_Faraday : real; { nA }
Begin_Time : Longint;
FirstValue : real;
EndValue : real;
spare2 : integer;
Real_Time : real; { The time on the wall }
Live_Time : real;
Slow_Channel_Counts : Longint; { Represents total out counts }
Medium_Channel_Counts : Longint; { If UTW. Approx. input counts below 1 keV }
Fast_Channel_Counts : Longint; { Represents total input counts above 1 keV }
RequestedLiveTime : LongInt;
ActualLiveTime : LongInt;
Acquiring : Boolean;
LLD : Integer; { Acquisition setup dialog box }
Offset : Integer; { Acquisition setup dialog box }
PulseProcessorType : Integer;
PulseProcessorSetting : Integer;
END;
Spectrum_InfoRec = RECORD
Spectrum_Type : String[4];
Spectrum_Comment_Field : Str255;
Spectrum_Number : Integer;
Spectrum_Class : String[25];
Theoretically_Generated : Boolean;
This_is_a_Standard : Boolean;
BkgSubtracted : Boolean;
Maximum_Counts : real;
Minimum_Counts : real;
X_Tilt : real; { degrees }
Y_Tilt : real; { degrees }
Take_Off_Angle : real; { degrees }
Spec_Detector_Distance : real; { mm }
Spare : real;
Specimen_Thickness : real; { cm }
Specimen_Density : real; { g/cm sq. }
Number_of_Elements : Integer;
Element_Info : array[1..15] of Element_InfoRec;
extra_space : array[1..157] of real;
WDS_in_eV : Boolean;
bool2 : Boolean;
Average_Z : real;
Spare1 : real;
Spare2 : Boolean;
Spare3 : Integer;
END;
Element_InfoRec = RECORD
Atomic_number : Integer;
Spare1 : Integer;
Weight_Fraction : real;
Spare2 : real;
Valence : real;
END;
{In the next two RECORD definitions, we combine the above records into the final Work_Spectrum RECORD. The "Results" spectrum, as well as the "scratch" spectra 1-8, are copies of the Work_Spectrum RECORD.}
Spectrum_Structure = RECORD
Spectrum_Info : Spectrum_InfoRec;
Acq_Info : Acq_InfoRec;
END;
Work_Spectrum = RECORD
Expt_Info : Expt_InfoRec;
Plot_Info : Plot_InfoRec;
SpectrumStuff : Spectrum_Structure;
S : Spectrum_counts; { ARRAY [1..Maximum_Channels] OF Real }
END;
A DTSA file of spectra will contain one Expt_InfoRec RECORD followed by as many Work_Spectrum RECORDs as there are "spectra".
Virtually all modern energy dispersive x-ray analytical systems include in their quantitative analysis procedures an option referred to as "standardless analysis". The "standardless" procedure offers a seductive lure to the analyst, particularly the novice, in the form of the ultimate simplicity of procedures. Just record the spectrum of the unknown, without concern for the electron dose as long as the deadtime is acceptable, supply the correct beam energy and the elements to be analyzed, push a button, and seconds later an analysis appears. Encouragingly, the analysis totals exactly 100%! As with most things in life, you get what you pay for. For careful application of the formal methods of quantitative analysis by the "ZAF", "f(rhoz)", or the Bence-Albee (Ziebold-Ogilvie) empirical method with unknowns measured against known standards, the reward is analytical performance described by an error histogram with a deviation of just a few percent. The penalty for the apparent ease of the "standardless" approach is the real possibility of very large errors lying far outside these comfortable bounds.
The underlying assumption of the "standardless" analysis approach is the belief that we can calculate the intensity measured from standards with accuracy equal to measuring them directly, based on certain properties of the single spectrum obtained from an unknown. The "standardless" approach makes use of four equations to predict the x-ray intensity that would be obtained from a pure element standard. The first is the equation that describes the absolute generation of x-rays. The second is an expression to correct the loss of x-ray production because of backscattering, the R-factor. The third is a correction for absorption in the putative standard, the absolute absorption factor equation. The fourth is a calculation of the absolute efficiency of the spectrometer. Normalization is then used to force the results to apparent consistency.
It is useful to consider how well the factors incorporated into these equations are known:
[1] Absolute x-ray generation: The critical factors are the ionization cross section, Qi, and the fluorescence yield, w. While the product Qiw is reasonably well known for the K-family of elements with atomic numbers from 20 - 32, the product becomes uncertain for lower atomic numbers. The product is not well known for L- and M- family x-rays. When an analysis must involve x-ray measurements involving more than one family, inter-family calculations are particularly difficult. Some approaches attempt to rectify this problem by tying the calculation to specific experimental measures, e.g., Cu K - Cu L or Au L - Au M, determined at a typical operating condition, e.g., 20 keV.
[2] The absolute value of the R-factor is reasonably well known, probably within 5%.
[3] The absolute value of the absorption factor is also reasonably well known, probably within 5% for lines above 3 keV, but progressively less accurate below 3 keV, and very poorly known below 1 keV.
[4] The greatest problem probably arises when an absolute calculation of spectrometer efficiency is needed over a wide energy range. The efficiency is reasonably smooth and predictable above 3 keV. Below 3 keV, the efficiency changes rapidly due to absorption in all the components. Comparing 0.5 keV with 3 keV, the efficiency typically drops by a factor of 5 to 10. Moreover, this is the region of the spectrometer response where "spectrometer aging" effects, i.e. , changing efficiency with time in service, are most strongly manifested. The inevitable build-up of ice, even in detectors with protective windows, causes substantial changes in the detector efficiency in this region. It is possible with most older (a few years old) detectors and many new ones to experience substantial loss of counts from x-ray peaks due to incomplete charge collection. The magnitudes of loss are at best about one percent for peaks at any energy, to as many as 20 percent for peaks in the range of about 1.8-3 keV. Thus, a "standardless" analysis that involves soft x-ray lines in the 0.5 - 2 keV range and hard x-ray lines in the 5 - 10 keV range are likely to be in substantial error.
When "standardless" analysis is touted, typically the analysis of stainless steel analysis is demonstrated. Rounding up the usual suspects in steel analyses typically involves measuring the transition elements Fe, Cr, Ni, Mn, Co, and V. The K-lines of these elements span the range 5.0 - 7.5 keV where the spectrometer efficiency is virtually constant and close to unity. The product is well behaved, and a reasonably accurate analysis can be obtained, especially if the procedure has been "optimized" for steel analyses.
The summation to 100% is, of course, an artifact of the normalization procedure, and this total means nothing for the analyst's peace of mind, and in fact, should reduce peace of mind.
Even with a "standardless" analysis that appears to work for steels, it is interesting to examine the effect of changing the beam energy. If the procedure is robust, it ought to work over a reasonable range of beam energies for analysis, e.g., 15 - 25 keV. The following two figures show an example of such a "standardless" analysis for specimens measured at different beam energies. The SRM concentrations are very close to the nominal values 20%, 40%, 60%, and 80% so that the distribution of errors from the "standardless" analysis is uncomfortably large.
It is obvious that by choosing the "right" kilovoltage that we can get close to the correct answers. In this case 18 keV appears to minimize the errors. However, if we do not pick the right kilovoltage, we can generate some serious errors. This type of result can be expected from any "standardless" analysis procedure. Caveat emptor.
When multiple x-ray families must be brought to bear, the situation becomes worse. The above figures show the results of the analysis of NIST Standard Reference Material gold-copper alloys, that can only be analyzed by taking combinations of lines that mix families and/or span a large difference in x-ray energy, e.g., Cu K (8.04 keV) vs. Au L (9.71 keV) or Au M (2.12 keV). Here the errors are really unacceptably large.
The foundation for achieving accurate quantitative electron probe microanalysis, whether by "ZAF", "f(rhoz)", or the Bence-Albee (Ziebold-Ogilvie) empirical method, rests upon the standardization step, which consists of measuring an element in the unknown against that same element in a standard under identical conditions of beam energy, dose (current x livetime), spectrometer efficiency, and choice of x-ray line. By forming the k intensity ratio as a ratio between the same type of radiation, several important terms in the expression for absolute x-ray intensity, and most importantly, the spectrometer efficiency, cancel in the ratio. It simply is not worth the apparent convenience of the "standardless approach" to concede the accuracy of the robust methods. Additionally, the loss of meaning in the analytical total of "standardless" analysis will inevitably hide those situations where an unexpected constituent appears at the analyzed location, or the specimen differs significantly from ideal bulk material because of voids, surface relief, etc.
We summarize with a few suggestions on how to test (read as "break") a "standardless" procedure, based on the above observations. Use a material that contains light and heavy components, such as Cu-Au. Use X-Ray lines from different families such as K and M. Try the analysis at several voltages such as 15 kV and 25 kV. Try the analysis at low and high count rates, and try taking and processing the data at several x-ray take-off angles. Any of the above should produce the same, correct, answers. "Standardless" analysis has a place in the laboratory. What we are warning the reader to beware of is the temptation to be lured into complacency and to report the data as "quantitative". "Standardless" analysis is simply more convenient than quantitative analysis. It absolutely is a semi-quantitative tool and is not to be mistaken for one that is quantitative.
This release of DTSA provides 1 menu item for interfacing user written quantitation functions into the DTSA framework. This item is an interface to the well-known CITZAF (California Institute of Technology ZAF) program written in BASIC by John Armstrong and converted to Pascal and made into a Macintosh program by Chuck Fiori and John Armstrong. The program will do all standard algorithms including several particle analysis corrections, most known ZAF corrections, most known Phi(RhoZ ) corrections and a Bence Albee correction for geological specimens.
The microprobe assay of a specimen must provide both a mean and the variance about this mean. Since we are concerned with microanalysis, the mean refers to the estimate of the weight or atom concentration at a single analytical point, or some local grouping of points, from a homogeneous region of the specimen. The variance about this mean then represents the uncertainty due to counting statistics plus those aspects of the data reduction procedure that will contribute uncertainty such as peak unraveling and continuum suppression. The accuracy of the estimate is a measure of the closeness of our estimate to the true value of the concentration. The task of predicting the variance about this estimated concentration can range from easy to quite difficult. As noted in a previous section, as the specimen is further examined at many points, any variance greater than that determined above will represent true compositional variation.
As with all measuring devices, the energy dispersive x-ray analysis system has, for a given set of conditions, a sensitivity that translates into a minimum concentration of analyte that can be reported with a certain level of confidence (MDL). The estimation of MDL can also range from easy to quite difficult. We affect the MDL by choices of experimental conditions such as the operating potential, the analytical x-ray lines used for analysis, choice of data reduction algorithm, etc.
This section will discuss a procedure to estimate the MDL accurately for a trace analysis and to estimate the variance accurately about each concentration in an analysis that encompasses a range of concentrations from minor to major element. The procedure will work when either of the above situations does not lend itself to treatment by conventional statistical methods, a frequently occurring situation. We will begin by describing those analytical conditions whereby standard statistical methods are difficult to apply for the determination of MDL and concentration variance.
As with the WDS, a spectrum observed with an energy dispersive spectrometer (EDS) consists of x-rays arising from both the characteristic and the continuum process. The x-ray peaks arising from the characteristic process contain the analytical information we seek. Often the peaks we wish to determine will overlap with the peaks from other elemental constituents of the specimen. Furthermore, the peaks are always superposed onto a smoothly varying spectrum of x-rays arising from the continuous process, and both the characteristic and continuum signals are modulated by the effects of counting statistics.
The MDL and variance about a measured concentration depend on the magnitude of the peak and background intensities, the degree of peak overlap, and the algorithms used to extract the required peak intensity and background intensity values below the peak. In general there is no straightforward way of estimating the quantities required for a standard statistical treatment as discussed in the previous sections. Therefore, some analysts, when faced with the problem of providing good error estimates, will resort to the time consuming but extremely reliable technique of direct measurement. In this method the specimen is sampled n times each at a number of representative locations. For each of the n replicate measurements at each location, one will go through all the spectral processing (removal of background and peak overlap effects) and data reduction (such as ZAF) steps required to arrive at an elemental concentration. From the n results at each location, the analyst can then predict by conventional statistical methods the expected variance for each of the elemental concentrations at the various presumably representative locations. Knowing, then, the expected variances, the analyst can proceed with a strategy of single measurements at each analytical point in the specimen. For specimens with many phases or a wide range of compositions, this procedure can be quite daunting.
DTSA can generate from first principles an x-ray spectrum that will be more than sufficiently close in all the germane physical and statistical properties to represent an actual spectrum from a real specimen. From this generated spectrum, one can then deduce accurate estimates of the variance about mean compositional values. One may also accurately estimate the MDL of any analyzable stable element in any stable matrix without the need to produce a set of calibration standards. Furthermore, one may adjust the experimental parameters to determine the optimum ones that will produce the lowest MDL. All of this can be done relatively rapidly, using only a small computer, before even presenting a specimen to the electron beam.
One can enter a hypothetical set of concentrations and analytical conditions, and generate a "spectrum" that will have the correct number of counts for the entered conditions. This spectrum will not have counting "noise" on it and represents the true "mean" spectrum. One can add counting noise and then go through all the procedures involved in extracting net peak areas and conversion of these to elemental concentrations. By repeatedly adding new counting noise to the same "mean" spectrum and repeating all the steps to get a concentration for each of the resulting "spectra", one can obtain a set of "measurements" from which it is possible to determine with a high degree of certainty the MDL and variance about a concentration mean for an element in a given chemical matrix. The variance term, is, of course, what is required to derive the statistical quantity "sensitivity".
This procedure of adding new noise to a "mean" spectrum is merely an application of the "Monte Carlo" method in statistics which is becoming very popular with the advent of powerful and inexpensive computers, and is very easy to do in DTSA.
The energy distribution of characteristic x-rays of a single line is well described by the Lorentzian probability distribution:
where Yi is the amplitude in the ith channel, g = FWHM÷2, AC is the amplitude at the center energy of the peak, EC is the center energy, and Ei is the energy of the ith channel away from the center energy.
The peak shape of a characteristic x-ray line, even if it assumed to be a perfect step function (i.e., infinitely narrow), as observed by an energy dispersive system is well described by a Gaussian distribution:
where, again, Yi is the amplitude in the ith channel, g = FWHM÷2, AC is the amplitude at the center energy of the peak, EC is the center energy, and Ei is the energy of the ith channel away from the center energy. The ln(2) term is a scaling constant to permit the use of FWHM÷2 in the equation, rather than the usual "standard deviation", s found in statistical applications. The Gaussian distribution is a good approximation for many instrumental response functions and is often used to describe the peak shapes observed by Si and Ge detectors. In reality there is a small distortion on the low energy side of the peak due to incomplete charge collection, but this is often neglected.
The relative shape of these two distributions, for the same amplitude, energy and width, is shown in the following figure:
Notice the relatively long tails of the Lorentzian distribution. These tails can extend a considerable distance (up to 1 keV) and are the reason that background correction is important in WDS when performing a trace element analysis using an x-ray line in the vicinity of a large line from another element in the specimen. Note also that the Lorentzian distribution is narrower at the top than a Gaussian and is therefore better able to register the presence of the Ka2 peak on the left side of the distribution.
The instrumental broadening of the energy dispersive detector is large. Typically the FWHM at the energy of Mn Ka is 135-165 eV, while the natural width at Mn Ka1,2 is just a few eV. Consequently, the Gaussian shape of the energy dispersive detector dominates the Lorentzian shape of the natural x-ray line and the resulting observed peak appears to be purely Gaussian (and much wider than in the above figure).
For the case of a wavelength dispersive spectrometer, with its considerably better energy resolution, the shape of the natural line width cannot be neglected. Often, the peak shapes as observed by a WDS have strong Lorentzian contributions. Stated mathematically, the observed peak shape is the convolution of the natural line profile and the instrumental response function. In the case of the WDS, the instrumental response curve itself has a Lorentzian contribution due to deliberately induced imperfections (polygonization) in the structure of several types of diffracting crystals. One last complicating factor is the effect of chemical bonding on the natural x-ray line distributions that will distort the Lorentzian profile. This last effect is most pronounced for the Ka x-ray lines from Be-Si.
In general, the instrumental response curve for a wavelength dispersive spectrometer is not predictable. It can change dramatically depending on the choice of diffracting crystal, slit width, etc. The shape can vary from near Gaussian, for the new synthetic multilayer devices, to strongly Lorentzian, for a Lithium Fluoride crystal.
It is possible, however, to convolve a given quantity of Lorentzian with a given quantity of Gaussian, and provide an instrumental response curve that very closely matches the observed distribution from a given wavelength dispersive spectrometer. The relative proportion of these two contributions is called the Hypermet number in the WDS dialog. This dialog is obtained if the WDS radio button is chosen and the Detector Parameters button is pushed.
The SIMPLEX procedure has the option of fitting WDS peak shapes using a user provided Hypermet number. Once this number is determined for a given element on a given spectrometer, it should remain constant for long periods of time. Accurate determination of peak areas from a WDS is only important for a few elements, i.e., the above Be-Si K lines. In many laboratories, it may be necessary to determine a Hypermet number for only oxygen. This matter is discussed in more detail in the SIMPLEX section.
Recently, there has been much interest in analyzing materials using lower beam voltages. The correction procedures use the so-called "overvoltage" term (beam energy/excitation energy). Uncertainties in the beam voltage will generate larger errors at low overvoltage than at high overvoltage. The user may be faced with an instrument that does not have a very precise voltmeter to measure the beam voltage, or the reported voltage may not represent the potential drop from the filament to ground (specimen). The true beam energy may differ from the measured potential drop by several hundred volts.
It is difficult and potentially exciting to measure the beam voltage with a calibrated voltmeter. However, the physics of the beam-specimen interaction provides a method that is built into the energy-dispersive x-ray detector. Continuum x-rays are generated up to the incident energy of the electron beam (the Duane-Hunt limit). The beam voltage can then be determined with a calibrated energy dispersive spectrometer (EDS) by examining the high energy portion of the spectrum. While an approximate answer can be obtained by simply looking at the spectrum, the action of the detector response function distorts the apparent Duane-Hunt limit from the true value. DTSA uses a method that will provide consistently unbiased values.
The intensity of the continuum x-rays at a given energy E can be described with an equation derived by Kramers:
1
Where Z is the atomic number, k is a constant, E0 is the beam voltage, dE is the energy interval and I(E)dE is the x-ray intensity per energy interval. A plot of the high energy end of the x-ray continuum of copper is shown in Figure 1.
Figure 1. A segment of a synthetic copper spectrum measured with the beam voltage set at a nominal value of 15 keV. The solid line is the generated x-rays computed with the above equation. The dashed line is the generated x-rays convolved with the Gaussian detector function. The data have all been transformed back to intensity by dividing by E.
The solid line represents the generated x-rays as defined by equation 1, and crosses the x-axis at exactly the energy of the electron beam, E0. When this continuum is convolved with the Gaussian detector function, the dashed line in Figure 1 is produced, that intercepts the x-axis about 100 eV higher in energy than the original expression. This distorts the shape of the curve at the Duane-Hunt limit making it more difficult to fit the data accurately. Because of the 1/E term in equation 1, the plot of intensity versus energy is slightly curved near the Duane-Hunt limit (Figure 2 is a plot of the continuum tails for carbon, copper and gold)
Figure 3. A segment of an experimental spectrum of copper at a nominal beam voltage of 15 keV together with a least squares fit to a second degree polynomial. The data have all been transformed back to intensity by dividing by E. The polynomial fit is strongly affected by the pile-up events.
Figure 4. A segment of an experimental spectrum of copper at a nominal beam voltage of 20 keV together with a least squares fit to a second degree polynomial (dashed line) and a fit using the robust estimation method described here (solid line). The data have all been transformed back to intensity by dividing by E. Note that the polynomial fit never crosses the horizontal axis.
In Figure 3, the pulse pile-up above the Duane-Hunt limit tends to pull the intercept to a higher energy and even though the quadratic fit shown in Figure 4 appears to be a good model of the data, it only produces imaginary roots.
Another difficulty to be reckoned with is the statistical noise associated with the x-ray spectrum. For the most part, this noise can be modeled by a normal distribution however, near the Duane-Hunt limit the normal distribution is not valid. A basis for the method of least squares is that the data be normally distributed. Therefore, attempting to use a method for the modeling of normally distributed data to model a Poissón process would be mathematically naive, since the data go from relatively large numbers, where the normal approximation is valid, to zero.
DTSA uses a "robust estimator" to fit a straight line by minimizing the sum of the absolute deviations (instead of the usual sum of the squares of the deviations, which is inappropriate in this application). The function to be minimized is therefore
2
instead of c2 as required for least squares. Figure 7 is a dramatic example of the effectiveness of the robust estimator, where the presence of the gold peak does not significantly influence the quality of the fit through the continuum.
If we multiply the x-ray intensities in equation [1] by their energies, the result should be a straight line, the x-intercept of which is the Duane-Hunt limit. Experimentally, a slight deviation from a straight line arises because of imperfections in Kramers equation and the Gaussian response shown in Figure 1. Because it is physically impossible to have non-zero intensities beyond the Duane-Hunt limit, the next step is to eliminate all the data beyond the root from the fitting procedure and refit the data. An iterative procedure has been devised to eliminate systematically data points with energies higher than the value of the root of the equation. The procedure always uses 300 channels (a channel is defined as an energy interval and is usually 5, 10 or 20 eV) on the low energy side of the best estimate of the root and, on the high energy side, 150 channels, which on successive iterations, is reduced to 50 channels, then 10, 7, 5, 3 and finally 1 channel. The procedure is not allowed to terminate until after the number of channels on the high energy side is reduced to 1. The resulting expression is used to compute predicted values of the intensities, which are then plotted with the measured values (Figures 5-7).
Figure 5. A segment of an experimental spectrum of carbon at a nominal beam voltage of 15 keV together with a fit using the robust estimation method described here. The data have all been transformed back to intensity by dividing by E.
Figure 6. A segment of an experimental spectrum of copper at a nominal beam voltage of 15 keV together with a fit using the robust estimation method described here. The data have all been transformed back to intensity by dividing by E. Note that the pileup events do not affect the fit.
Figure 7. A segment of an experimental spectrum of gold at a nominal beam voltage of 15 keV together with a fit using the robust estimation method described here. The data have all been transformed back to intensity by dividing by E. Note that the gold peaks have very little effect on "pulling" the background fit higher. This is a good example of the power of a "robust estimator" in this kind of application where conventional least squares is inappropriate.
To test the fitting method, energy dispersive spectra are synthesized with equation 1 so that the "true" Duane-Hunt limit can be defined. For this test case, the proposed fitting method is exact. When specimen absorption, detector efficiency, spectrometer broadening and Poissón noise are added to a synthesized copper spectrum, the fit is still within 0.33% of the "true" value for 10 repeated syntheses with random noise.
An examination of Figures 2, 5, 6 and 7 demonstrates why the results are better with high atomic number targets than with low. Since the continuum intensity increases with atomic number, the higher atomic number targets provide more x-ray counts in the spectrum, and therefore better statistics. Also, the higher number of counts makes the method more robust, as demonstrated by the fit shown in Figure 7. Here, two of the gold L-series x-ray lines occur within the range of the data to be fit. Note that the fitted line is still a good estimation of the data even though the gold peaks are present. In fact, if the peaks are stripped from the spectrum and the fit is repeated, the Duane-Hunt limit is only lowered by 4 channels (40 eV).
As with any procedure of this type, the results depend on the information supplied by the user. The spectrum should be accumulated for a sufficient length of time and at a low enough detector dead-time to ensure good statistics and to minimize any pulse pile-up. It is necessary that the detector be properly calibrated to avoid miscalibration errors propagating through to the final result. Further testing of the method will be carried out to identify any possible systematic errors that might arise from the deviation in linearity from the assumed response given by equation 1.
One last warning: since the procedure currently uses 300 channels of data for fitting, attempting to fit spectra collected at low beam voltages (less than 5 keV) may produce errors. We will examine this area and try to make the procedure work at these lower voltages.
Hardware Interfacing of Spectral Acquisition; to DTSA with a Macintosh CDEF
The following figure is the interface into the acquisition capability. The two rectangular boxes that contain "0" are actually a combination analog and digital count-rate meter. As the count-rates from the fast and slow channels (Input and Output) change the boxes will move up or down (the analog part) and the number in the box will show the digital value.
All hardware functions are controlled from the Acquire Menu. The items in this menu are designed to collect data from more than one detector.
The "Assign Display(s)" item allows the user to select into which of the ten available displays a spectrum will be accumulated and displayed on the screen. It is no longer necessary to accumulate into the "WORK" display; however, you can only save a spectrum from "WORK".
Select the detector from the pop-up menu and the display with the appropriate button.
If the "Hardware Set-up" item is invoked, then the following dialog will appear for each detector:
This dialog permits, via the CDEF capability of the Macintosh, to pass the above information back and forth to an acquisition Nu-Buss card.
Spectra Acquisition can be interfaced to DTSA using the Control Manager and a custom control definition function (CDEF). Source code for the dummy acquisition CDEF implemented in DTSA is included below and the reader should be familiar with control definition functions (see Inside Macintosh volume 1 chapter 10 "Control Manager").
Currently the dummy acquisition control is accessed by a popup menu that is activated by holding down the mouse button while inside the boundary of the control. From this menu, one can select 1 of the 4 following functions: Clear/Start, Stop, Continue and Set-up. One could use these functions to control the spectra acquisition. DTSA accesses the acquisition CDEF each time the main event loop is executed. The control is set to a value of five (AcqUpdate) to force entry into the CDEF code. A "feature" of the Macintosh operating system must be bypassed to guarantee entry into the CDEF. If the value of the control is the same as the value requested, then the Mac OS will NOT call the CDEF. This CDEF gets around this problem by saving a copy of the current value (or acquisition state as it would be) and setting the control value to zero (AcqNull). When the CDEF is called with a value of AcqUpdate, the CDEF should check the state of acquisition and if actively acquiring a spectra it should update the internal DTSA spectra structure "WORK". The DTSA internal structure associated with the "Work" spectra can be accessed with a handle that is stored in the control's ContrlData Handle when the control is created by DTSA. After processing the AcqUpdate message, if the CDEF requires the DTSA program to process the information transferred to the "WORK" structure, then it should set the control value to one (AcqStart) to force an update, otherwise the value should be set to AcqNull. This update will update and redraw the input and output count rate controls as well as the spectra and other information that has been transferred to the "WORK" structure.
Since a CDEF is a resource that is located in the resource fork of the DTSA program, one can use ResEdit or replace the existing dummy CDEF with a real CDEF that is capable of spectra acquisition. Remember that the CDEF needs to be the same number as the existing CDEF or DTSA will not load it in place of the existing CDEF. An example CDEF is available from NIST on request. The example code is in Pascal. This can be compiled and linked to the program using the Apple Macintosh Programmers Workshop (MPW).
If the particular model Macintosh used by you has an extended keyboard with F keys and a numeric keypad, then it is possible to use certain predefined keys to accomplish certain tasks. The assignment of these keys is as follows:
Numeric keypad "8" will do a ScaleUp.
Numeric keypad "2" will do a ScaleDown.
F1 is a key which DTSA uses instead of the option key. Because of the way the Macintosh event loop is used in the program it was not possible to use the option key as used in many other programs. The F1 key is used throughout DTSA to speed or slow the effects of certain other keys or arrow icons. See the individual sections for a description.
F2 will rotate work and results. Remember that a rotate will not rotate the headers.
F3 will bring up the "periodic table" for the x-ray database.
F4 will print the spectral display.
F5 will Do a Fit.
F6 will Add A Fit.
F7 will See A Fit.
F8 will clear the work spectrum.
F9 will bring up Spectrum Header Dialog.
F10 will bring up the Experiment Header Dialog.
The radiation emitted from the specimen must pass through several layers of "window" material before it arrives in the "active" part of the detector. The nominal purpose of the first window material is to protect the cooled detector chip from the relatively poor vacuum in the specimen chamber which can be high in water vapor and organic components because of frequent exchanges of the specimen. It is very important that these components do not condense on the cooled detector. Even with a protective window, ice will build up, over a period of time, which will absorb lower energy x-rays. Modern detectors have special heaters to remove this ice buildup, some of which actually originates from the detector assembly and cryostat at the time of manufacture. This problem can be overcome by the use of high vacuum construction practices. Since many modern scanning and transmission electron microscopes themselves are ultra-high vacuum devices, the purpose of the window is actually to protect the microscope vacuum from that of the detector.
Protective windows are made from a variety of different materials each having advantages and disadvantages. Historically, beryllium, about 7.6 mm thick (0.3 mils), has been used. This thickness has the advantage that it can withstand a full atmosphere pressure differential thereby avoiding possible shock associated with specimen changes. Furthermore, the detector can be removed from the microscope without the need for isolation valves. It is also opaque to optical photons, which can easily saturate the detector during operation. About 1985, there was a shortage of 7.6 mm beryllium foil, due to occupational health and safety regulations, and 12 mm was used for a year or so; many of these detectors still exist. Recently, protective window materials with considerably less mass-thickness, still able to withstand a full atmosphere of pressure differential, have gained wide popularity. They are constructed of thin films on a support grid having approximately 85% open space. Low energy x-rays, such as oxygen or carbon, will only pass through the un-backed portions of the film, while more energetic x-rays (>8 keV) can pass both through the film and the support grid. The grid is usually pure silicon or a metal such as nickel and the window materials are either boron nitride, silicon nitride, diamond, or are polymeric. Examples of transmission curves for various window materials currently in use are given in Figure 1. Since thin films of these materials are transparent and the detector is also sensitive to visible light, it is either necessary to eliminate possible light sources or to apply an optically opaque layer to the protective window. Light sources include microscope illumination systems, viewing ports, and light leaks; however, specimen cathodoluminescence is rarely intense enough to cause a serious problem.
The second window layer is an electrical contact (about 10-20 nm and usually gold). Gold has been observed to form "islands" during the evaporation process and hence is not uniform in thickness. The third window layer consists of inactive p-type silicon extending 100-200 nm or less into the detector. This layer is also not uniform and is often called the "dead layer" or "silicon dead zone". The combined effect of these Au and Si layers is often overlooked. In reality, their effect is as great, or greater, than that of any of the new "thin" window materials. It is clear from Figure 1a that there really is no such thing as a "Windowless" detector. In this context, the ability to see pure beryllium is misleading. This metal has very low continuum radiation (Z=4) and hence the beryllium K line seems to be large. To demonstrate this, generate and compare pure beryllium and beryllium in Å90% copper matrix. It will be seen that the practical measurement of Be is difficult at best.
The chief utility of the detection of beryllium is that it demonstrates that the EDS system is performing in a state-of-the-art manner.
Figure 1a. Transmission curve for a "Windowless" detector with a 0.01 µm Au contact and a 0.1 µm silicon dead layer. The energies of the light element K lines are shown along the energy axis. It should be noted that the mass absorption coefficients for energies less than 200 eV are poorly known.
Figure 1b. Transmission curve for several commercially available window materials. Thickness' and compositions were obtained from advertising literature or published values. The energies of the light element K lines are shown along the energy axis. It should be noted that the mass absorption coefficients for energies less than 200 eV are poorly known. The actual transmission of x-rays into the active part of the detector is the combined effect of parts "a" and "b" of this figure. This demonstrates that a consideration of window materials alone results in an overly optimistic expectation of performance.
During passage of x-rays through all of the window layers, absorption occurs. It is important to recall that photoelectric absorption refers to a process in which x-rays are diminished in number, but not in energy, thus the energies of the observed spectral lines are not altered by the effects. In the case of 0.3 mil beryllium protective windows, nearly all x-rays below about 600 eV are eliminated due to absorption effects. Above 2 keV, virtually all x-rays are transmitted. Between these limits, the absorption increases with decreasing energy such that at 1.5 keV about 70% of the x-rays are transmitted, while for an energy of 1 keV, the transmission is 45%. Absorption by the gold and silicon layers is much less significant because of the small mass thickness of these layers. However, a noticeable change in the x-ray continuum is observed at the absorption edge of silicon and to a lesser degree at gold, Figure 1. Just above the energy of the absorption edge, the mass absorption coefficient increases abruptly, resulting in a decrease in the measured continuum x-radiation. The height of the resulting step is an indication of the thickness of the layer. Note that the action of the broadening effect of the detection process causes the absorption edge, which in reality is a sharp change of absorption over a range of about l eV, to be smeared over a much broader range, typically 100 eV for the silicon absorption edge.
In this method of background reduction, the continuum component of an x-ray spectrum is viewed as an undesirable signal whose effect can be removed by mathematical filtering or modification of the frequency distribution of the spectrum. Knowledge of the physics of x-ray production, emission, and detection is not required.
Filtering techniques take advantage of the fact that the continuum component of an x-ray spectrum is smooth and slowly varying, as a function of energy, relative to the characteristic x-ray peaks, except at absorption edges. If we mathematically transform an x-ray spectrum from energy space into frequency space, the result would be as shown in Figure 1. The horizontal axis gives the frequency of an equivalent sine wave component. For example, a sine wave with a full period of 0-10keV would be plotted at channel 1 while a sine wave with a full period in 10 eV would be plotted at channel 1000. The vertical axis |F| gives the population of each sine wave. In this representation, the continuum background that has a long period variation is found at the low-frequency end. The characteristic peaks vary more rapidly than the background and are found at higher frequency. The noise is found at all frequencies.
Figure 1. Frequency space (Fourier transform) representation of electron excited x-ray spectrum. (See text for explanation.)
We can now mathematically suppress the frequency components responsible for the slowly varying continuum and additionally suppress a portion of the frequency components responsible for the statistical noise. By then performing a reverse Fourier transform, we go back to energy space with a spectrum that should now be comprised only of characteristic peaks devoid of continuum and that is, furthermore, "smoothed" for statistical variation. Unfortunately, close examination of Figure 1 reveals that the three main components of the transform overlap. Consequently it is not possible to suppress all the undesirable components without sacrificing part of the peak components. Similarly, it is not possible to keep all the peak components without including part of the undesirable components. The result in either case will be a spectrum with undesirable distortions.
A filtering technique that is now widely used is the "top hat" digital filter. The top hat filter is a simple and elegant algorithm-a fact that can easily be obscured by the mathematical formalism required to describe it. Simply stated, the top hat filter is a special way of averaging a group of adjacent channels of a spectrum, assigning the "average" to the center "channel" of the filter, and placing this value in a channel in a new spectrum that we will call the filtered spectrum. The filter is then moved one channel and a new "average" is obtained. The process is repeated until the entire spectrum has been stepped through. The filter in no way modifies the original spectrum; data are only taken from the original to create a new spectrum. The "averaging" is done in the following manner. The filter, see Figure 2, is divided into three sections: a central section, or positive lobe, and two side sections, or negative lobes. The central lobe is a group of adjacent channels in the original spectrum from which the contents are summed together and the sum divided by the number of channels in the central lobe. The side lobes, similarly, are two groups of adjacent channels from which the contents are summed together and the sum divided by the total number of channels in both lobes. The "average" of the side lobes is then subtracted from the "average" of the upper lobe. This quantity is then placed in a new spectrum into a channel that corresponds to the center channel of the filter.
Figure 2. Effect of top hat digital filter on spectrum comprised of a Gaussian peak plus a sloped linear background. The filtered spectrum is plotted immediately below the actual spectrum. The channel correspondence for one calculation of the top hat filter is shown.
The effect of this particular averaging procedure is as follows. If the original spectrum is straight, across the width of the filter, then the "average" will be zero. If the original spectrum is curved concave upward, across the width of the filter, the "average" will be negative. Similarly, if the spectrum is curved convex upward the "average" will be positive. The greater the curvature, the greater will be the "average." The above effects can be observed, for a Gaussian superposed on a linear background, in Figure 2. In order for the filter to respond with the greatest measure to the curvature found in spectral peaks, and with the least measure to the curvature found in the spectral background, the width of the filter must be carefully chosen. In general, the width of the filter for any given spectrometer system is chosen to be twice the full width at half the maximum amplitude (FWHM) of the MnKa peak, with the number of channels in the upper lobe equal to or slightly more than the combined number of channels in the side lobes.
Because the top hat filter "averages" a number of adjacent channels, the effects of counting statistics in any one channel are strongly suppressed. Consequently, in addition to suppressing the background under spectral peaks, the digital filter also "smoothes" a spectrum. Note that in Figure 2, the top hat filter converts the sloped background into a flat background.
In conclusion, the effects of passing a top hat digital filter through an x-ray spectrum as recorded by a Si(Li) spectrometer system are to (1) strongly suppress the background and statistical scatter and (2) significantly alter the shape of the spectral peaks. The result strongly resembles the smoothed second derivative; however, this distortion has no adverse statistical or mathematical effects of any consequence. Clear advantages of the method are simplicity and that an explicit model of the continuum is not required. However, since the continuum has been suppressed, the information it carried (i.e., average atomic number, mass-thickness, etc.) is no longer available.