PATENT DOCUMENT

Abstract:
Apparatuses, methods, software, and systems for analyzing homogeneous samples containing signal emitting entities, such as, but not limited to, radioisotopes, are disclosed. The apparatuses involve sample-container apparatuses that shape samples into different thicknesses. The methods involve characteristic signal acquisition and processing in order to compute sample self-attenuation of signals emitted from within special sample-container apparatuses. An external radiation reference-source having at least one prominent characteristic signal to beam-through the sample without interfering with the radiation signals emitted by the homogeneous sample, wherein the external reference-source is affixed to the reference-source positioning device, which is affixed to the sample-container. The software pairs characteristic signals from samples of varying thicknesses; computes sample self-attenuation, transmittance, signal detection-efficiency calibration of the detection system, identifies, and quantifies signal-emitters. The systems integrate and support the methods, apparatuses, and software.

Full Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is related to U.S. patent application Ser. No. 13/049,903 filed on Mar. 17, 2011, and entitled “APPARATUSES AND METHODS FOR ANALYSIS OF SAMPLES THROUGH MULTIPLE THICKNESSES”, the disclosures of which are incorporated herein by reference, as if fully stated here, for all purposes. 
     
    
     TECHNICAL FIELD 
       [0002]    The present disclosure is generally related to sample analysis, and in particular, it is related to correcting for sample self-attenuation of signals emitted from within the sample. Emitted signals include gamma-rays (“g-rays”), x-rays, beta-rays, and alpha-rays that often follow the decay of radioisotopes but may also include stimulated x-ray emissions from non-radioactive isotopes or any other type of signal that attenuates as it travels through a volume of homogeneous sample. One important application of this disclosure, among others, is reliable non-destructive nuclear forensic identification and quantitation of radioisotopes in a homogeneous sample. 
       BACKGROUND 
       [0003]    In conventional sample analysis methods, ‘external’ signal-emitting reference sources (‘external’ to standard-samples that are used for detection-system calibration and external to unknown-samples to be analyzed) produce beam-lines that are commonly used specifically to determine the linear attenuation of the sample composition. In this disclosure, an ‘external’ signal-emitting reference source that produces at least one beam-line, and preferably produces only one beam-line of moderately high characteristic energy) is used specifically to determine a ratio of composition-independent (‘sample-free’) counting system signal-detection-efficiencies for at least two different thicknesses of the same sample. This ratio of counting-system signal-detection-efficiencies, using at least one characteristic signal energy and when combined with the inventions of the related patent application Ser. No. 13/049,903, improves both the counting-system signal-detection-efficiency ‘calibration’ over a wide energy range and improves quantification of signal-emitters in unknown samples. 
         [0004]    The United States Environmental Protection Agency (“EPA”) reports that over 1,000 U.S. locations are contaminated with radiation. These sites range in size from small spaces in laboratories to massive nuclear weapons facilities. Such contamination is found in air, water, and soil, as well as in equipment and buildings. Radiation levels around such contaminated sites are closely monitored. Clean-up teams use modern technologies to assess the situation and take appropriate actions to limit potential hazards to people, the environment, the economy, and equipment. Besides such sites, general soil, air, and water sampling is required around mines, wells, basement construction, underground parking garages, and lower-level dwellings to ensure that natural radionuclides left over from the formation of the Earth&#39;s crust pose no elevated health risk. It is estimated that approximately one-third of all lung cancers are due in part to inhalation of radioactive radon gas that arises from the natural radioactive decay chains. If the price and reliability of sample analysis can be improved, then wider knowledge of the local hazards posed by natural ambient radioactivity and radon can be economically measured so that mitigating action can be taken when necessary for health and safety. Then there is the entire nuclear fuel cycle, from prospecting to mining, fuel production, operational sampling, and disposition. Nuclear power plant, hospital cyclotron, and radiopharmaceutical wastes also need sampling and measurement. In addition, scientific aging studies of lake, river, and ocean sediment rely on precise and accurate quantitation of radioisotopes in the soils, especially the radioisotope lead-210 (“Pb-210”). The International Atomic Energy Agency (“IAEA”) conducts sampling for compliance. Lunar and planetary rovers conduct sampling at a great distance. However, these measurements can be expensive and complex. Therefore, there is need for simple, reliable, and economical means for analyzing homogeneous samples purported to contain signal emitters. 
       SUMMARY 
       [0005]    The present disclosure provides an apparatus for detecting radiation signals emitted from an unknown homogeneous sample. This apparatus comprises a sample container that includes a plurality of sample container configurations; each sample container configuration enables measurement of the homogeneous sample via at least two different thicknesses; an external radiation reference source having at least one prominent characteristic signal to allow signal beam-through the sample without interfering with the radiation signals emitted by the homogeneous sample, and the external reference source is held tight onto the sample holder by a positioning device; a detector system detects the radiation signals from different sample thicknesses; and a computer processes the detected signals and analyzes the sample composition by comparing radiation signals at different sample thicknesses by means of a sample analysis software program. 
         [0006]    One of the sample container configurations comprises a plurality of sample cups, each sample cup having a different size and shape from other sample cups, so the homogeneous sample assumes a different thickness when placed into each respective sample cup. The sample cups share at least one opening to allow the homogeneous sample to be transferred from one sample cup to the other. 
         [0007]    One exemplary sample container is comprised of two oppositely placed sample cups connected together at one or more of their shared openings in order to allow the homogeneous sample to be transferred from one sample cup to the other when the sample container is flipped 180 degrees. The two oppositely placed sample cups have the ratio of their diameters equal to √{square root over (2)}:1 so that the sample thickness ratio becomes 1:2 when the homogeneous sample is transferred from one sample cup to the other sample cup. 
         [0008]    The present disclosure provides a method for characterizing radiation signals emitted from an unknown homogeneous sample. The method comprises providing a radiation signal detecting system comprising a plurality of detectors, a computer for analyzing the sample composition, and a sample container, wherein the sample container includes a plurality of sample cups, each sample cup has a different size from other sample cups, such that the homogeneous sample forms different thickness when placed in different sample cups; performing background signal detection for each empty sample cup and determining a background signal count rate for each empty sample cup; performing reference signal detection by measuring a reference source emission having at least one prominent characteristic signal to allow signal beam-through the plurality of empty sample containers and the plurality of containers with the sample; performing calibration signal detection by measuring a standard-sample and a reference source signal sequentially in each sample-container apparatus and determining a standard signal count rate for each sample-container apparatus; subtracting the background signal count rate from standard-sample signals for each sample cup; performing the signal detection for the unknown homogeneous sample in each sample cup; subtracting the background signal count rate from the unknown homogeneous sample signals for each sample cup; measuring the characteristic signal count rates for the unknown-sample in each sample cup; verifying the characteristic signal count rates to be qualified data; and calculating the composition of the unknown homogeneous sample by comparing the characteristic signal count rates of the unknown-sample from different sample cups using a software model. 
         [0009]    The present disclosure provides a software product embedded in a computer readable medium for providing analysis in material spectra characterization, the software product comprising: program codes for reading the emitted signals from the homogeneous sample; program codes for subtracting a background signal; program codes for subtracting a reference source signal; program codes for matching signals emitted from a different thickness of the homogeneous sample; program codes for operating on signal count rates of different thicknesses of the homogeneous sample; program codes for calibrating signal detection using a standard sample signal; and program codes for quantization of the material spectra. 
         [0010]    Another exemplary method consistent with the current disclosure applies different sample masses. The method includes: providing a radiation signal detecting system comprising a plurality of detectors, a computer for analyzing the sample composition, and two sample-containers each having the same shape; filling the first sample-container with a first amount of the unknown homogeneous sample; filling the second sample-container with a second amount of the unknown homogeneous sample; performing background signal detection for each sample-container and determining a background signal count rate for each sample; performing reference source signal detection; performing calibration signal detection by measuring a standard sample and the reference source signal detection sequentially in each sample-container apparatus position and determining a standard signal count rate for each sample-container; subtracting the background signal count rate from standard sample signals for each container; performing the signal detection for the first unknown homogeneous sample in the first sample-container and the second unknown homogeneous sample in the second sample-container; subtracting the background signal count rate from the first and second unknown homogeneous sample signals; measuring the characteristic signal count rates for the first and second unknown samples; verifying the characteristic signal count rates to be qualified data; and calculating the composition of the first and second unknown homogeneous samples by comparing the characteristic signal count rates of the first and second unknown samples using a software model. 
         [0011]    An exemplary software model consistent with the current disclosure applies a software product embedded in a computer readable medium for providing analysis in material spectra characterization, the software product comprising: program codes for reading the emitted signals from the homogeneous sample; program codes for subtracting a background signal; program codes for subtracting a reference source emission signal; program codes for matching signals emitted from a different thickness of the homogeneous sample; program codes for operating on signal count rates of different thicknesses of the homogeneous sample; program codes for calibrating a standard sample signals, including one of the three sets of codes: 1) codes for measuring a first reference source emission signal through one empty sample container; codes for measuring a second reference source emission signal through one sample container with the sample at a first thickness; codes for calculating the ratio of the second to the first signals for the first thickness; 2) codes for measuring a second reference source emission signal through one sample container with the sample at a first thickness; codes for measuring a third reference source emission signal through one sample container with the sample at a second thickness; codes for calculating the ratio of the third to the first signals for the second thickness; 3) codes for measuring the reference source emission signal through the sample of the first and second thickness; codes for calculating the ratio of the signals from the thicker thickness to the thinner thickness; and program codes for quantization of the material spectra. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0012]    The present disclosure will be readily understood by reading the detailed description together with the accompanying drawings, wherein like reference numbers designate like structural elements, and in which: 
           [0013]      FIG. 1  shows a signal counting system setup for acquiring ambient background spectra for an empty nd: md sample container apparatus and associated reference-source positioner apparatuses in the thick (md) and thin (nd) counting positions; 
           [0014]      FIG. 2  shows a signal counting system setup for acquiring sample-free reference-source spectra; 
           [0015]      FIG. 3  shows a counting system setup for nd: md beam-assisted detected-fraction calibration; 
           [0016]      FIG. 4  shows a software model for computing the beam-assisted detected-fraction calibration; 
           [0017]      FIG. 5  software module for computing the sample linear attenuation coefficient by one of three techniques; 
           [0018]      FIG. 6  shows a counting system setup for signal-emitter quantitation; 
           [0019]      FIG. 7  shows a software model for computing signal-emitter quantitation; 
           [0020]      FIG. 8  shows a counting system setup for multi-mass nd: md beam-assisted detected-fraction calibration; 
           [0021]      FIG. 9  shows a counting system setup using nd: md wrap-around sample containers and “rabbit-ear” reference-sample positioners for beam-assisted detected-fraction calibration; and 
           [0022]      FIG. 10  shows a counting system setup using distant reference sources to beam through air, gas, water, or other relatively weakly attenuating sample compositions to assist detected-fraction calibration and signal-emitter quantitation. 
       
    
    
     DETAILED DESCRIPTION 
     Important Terms in this Disclosure 
       [0023]    Beam source. The beam source is also referred to as the “reference source”. The beam source is external to the sample, external on the side of the sample opposite to the detector such that emitted reference-source signals (&#39;beam-source signals&#39;) must pass through the sample in order to be counted by the detection system. Although the reference source may not actually emit a ‘beam’, nevertheless it acts like a ‘beam source’ in the sense that only those emitted characteristic signals that are in the solid angle subtended by the detector play a role in the beam-derived, sample-free detected-fraction calibration. The purpose for the beam source is to allow computation of the ratio of ‘beam-derived’ sample-free detected-fraction calibration values 
         [0000]    
       
         
           
             ( 
             
               
                 RDetF 
                 
                   Ei 
                   , 
                   bm 
                   , 
                   
                     nd 
                     : 
                     md 
                   
                 
               
               = 
               
                 
                   DetF 
                   
                     Ei 
                     , 
                     bm 
                     , 
                     stnd 
                     , 
                     nd 
                   
                 
                 
                   DetF 
                   
                     Ei 
                     , 
                     bm 
                     , 
                     stnd 
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                     md 
                   
                 
               
             
             ) 
           
         
       
     
         [0000]    for at least two different standard-sample thicknesses (nd and md) for at least one characteristic (E i ) ‘beam-source signal’. Three different techniques are described that each use the ‘beam-source signals’ to determine RDetF Ei,bm,nd:md , and those three different techniques are taught in the discussion of Equations [2a] through [20b] and  FIGS. 4 and 5 . 
         [0024]    Beam-through-derived terms and their values. Refers to terms and their values that are derived or computed by the use of an external reference source that acts like a beam source. See Beam source and see Standard-sample-derived terms and their values. 
         [0025]    Characteristic signal. Characteristic signals have an emission energy (E i ) that can be used to identify their signal-emitting source. Detectable gamma-ray and x-ray photons often follow nuclear decay, and they are just two types of characteristic signal. 
         [0026]    Counting system. See Detection system. 
         [0027]    Counts, count rates, and lines. Refers to characteristic peaks that make up an emission spectrum. 
         [0028]    Depth (of the sample). Refers to the thickness of a sample in the direction of the detector. 
         [0029]    Detection system. Consists of the sample-container apparatus (if there be one), reference-source positioner apparatus (if there be one), detector, vacuum system (if there be one), pulse-shaping electronics, computer control, software, and any other part or subsystem that helps the detection system collect, shape, remember, or present emitted signals. 
         [0030]    Detected-fraction calibration. In the literature, the sample-specific and sample-free detected-fraction calibration terms are often collectively referred to as the “detection-efficiency calibration”, “counting-efficiency calibration”, “energy-efficiency calibration”, or simply the “efficiency calibration”, among others. 
         [0031]    Multiplet (peaks). A set of overlapping peaks in a spectrum containing multiple characteristic peaks so close in characteristic energy in relation to each other that the resolving fidelity of the detector is unable to resolve them into individual singlet peaks (i.e. a series of non-overlapping characteristic peaks). See also Singlet (peak). 
         [0032]    Operator. An operator is a general title of a person that might operate or implement the apparatuses, methods, software, and systems of this disclosure. An operator, depending on the particular activity described in this disclosure, might also be known as a technician, spectroscopist, spectrometrist, or scientist, among other related and appropriate titles. 
         [0033]    Quantitate. To compute or calculate a quantity. Synonymous with “quantify”. 
         [0034]    Radioisotope. Synonymous with Radionuclide. See also Signal emitter. 
         [0035]    Reference source. See Beam source. 
         [0036]    Sample. Any homogeneous substance that has volume. To be considered as a “sample” in this disclosure, the substance must contain at least one signal emitter. In this disclosure, samples are primarily referred to as “standard-sample”, “unknown-sample”, or simply “sample” when discussing samples in general. Standard-samples have at least some of their contents known and are used to calibrate the characteristic signal-detection efficiency of a detection system (See Detection system). 
         [0037]    Signal emitter. Any entity that emits characteristic signals. Signal emitting entities include radioisotopes; nonradioactive isotopes; and excited elements and molecules, etc. 
         [0038]    Singlet (peak). One non-overlapped, statistically significant characteristic peak. If there are other characteristic peaks in a given spectrum, then they will not overlap a singlet. Overlapping characteristic peaks are called multiplets. See also Multiplet (peaks). 
         [0039]    Specific activity (SpA). The disintegration rate that occurs in a unit of mass of sample, which is commonly defined in units of curies per gram of sample (Ci/g). In this disclosure, specific activity generalizes to include the rate or quantity of total emission of any type of signal emitter. 
         [0040]    Standard-sample-derived terms and their values. Refers to terms and their values that are derived or computed by the use of a standard-sample. See Beam-through-derived terms and their values. 
         [0000]    I. “nd: md” Beam-Assisted Sample Analysis 
         [0041]    In the multiple-sample-thickness analysis methods described in the related U.S. patent application Ser. No. 13/049,903, the sample self-attenuation is determined by setting equal the sample-free detected-fraction calibration terms (DetF Ei,smplFr,nd ≈DetF Ei,smplFr,md ) through at least two different sample thicknesses (nd and md where 0&lt;n&lt;m, and d is a unit of sample thickness). It is acknowledged in the related U.S. patent application Ser. No. 13/049,903 that sample-specific detected-fraction calibration terms computed through different thicknesses of sample are not exactly equal. To help quantify the difference between the sample-specific detected-fraction calibration terms for different sample-thickness orientations (nd and md) relative to the detection system, three new beam-thru techniques that each uses a signal-emission reference-source to beam-thru standard-samples are taught. Any one of the three techniques alone can be used to determine the ratio (RDetF n:m ) of the sample-specific detected-fraction calibration terms (DetF Ei,smplFr,nd  and DetF Ei,smplFr,md ), where: 
         [0000]    
       
         
           
             
               
                 
                   
                     RDetF 
                     
                       n 
                       : 
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                           DetF 
                           
                             Ei 
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                             , 
                             nd 
                           
                         
                         
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                             , 
                             md 
                           
                         
                       
                       ≈ 
                       
                         Constant 
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                         for 
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         [0042]    Normally, in conventional methods, a beam-through reference source provides numerous characteristic peaks to cover the energy range of interest. But in this disclosure, only one single characteristic line is needed, so long as the line is measured through two or more unequal sample thicknesses. The ratio of the characteristic (E i ) beam peak count rates (CR Ei,bm,smplFr,nd , CR Ei,bm,smplFr,md ) through any two thicknesses of standard-sample (e.g. nd and md) allows determining the ratio (RDetF n:m ) of the sample-free detected-fraction calibration terms (DetF Ei,smplFr,nd , DetF Ei,smplFr,md ), which ratio is nearly constant over a wide energy range, as indicated by Equation [1]. Multiple different characteristic beam lines can confirm this assumption, but may induce elemental fluorescence peaks, which add extra spectral noise that may interfere with characteristic peaks from the standard-sample. None of the three beam-thru techniques need be applied to unknown-samples because only standard-samples are used to determine the values of the sample-free detected-fraction calibration terms (DetF Ei,smplFr,nd , DetF Ei,smplFr,md ) for a particular standard-sample shape, position, and orientation relative to a particular detection system. 
         [0043]    All three beam-thru techniques can use the same type of reference signal source. The preferred properties of the signal source include (1) a single high-energy characteristic signal that doesn&#39;t interfere with the characteristic peaks from the standard-sample, but (2) not so high in energy that other negative effects occur; e.g. if the characteristic gamma-ray energy is much higher than 2-MeV, then pair production and matter-antimatter annihilation raise the background noise and degenerate the statistics of other portions of the standard-sample&#39;s characteristic emission spectrum. 
         [0000]    “nd: md” Beam-Assisted System Setups 
         [0044]    The nd: md beam-assisted system setups can be described in five main parts: (1) ambient background emission spectrum acquisition; (2) ambient background acquisition with reference-source auxiliary apparatus; (3) sample-free reference-source spectrum acquisition; (4) beam-assisted detected-fraction calibration; and (5) unknown-sample signal-emitter quantitation. 
         [0000]    Part 1. “nd: md” Ambient Background Emission Spectrum Acquisition 
         [0045]    The Part-1 system setup, i.e. ambient background emission spectrum acquisition, is disclosed in the related U.S. patent application Ser. No. 13/049,903 and will not be described again here. 
         [0000]    Part 2. “nd: md” Ambient Background with Auxiliary Apparatus ( FIG. 1 ) 
         [0046]      FIG. 1  shows a counting system setup  5200  for acquiring thick (md)  5210  and thin (nd)  5260  ambient background spectra  5222  and  5272  for the counting system; nd: md sample container  5212 ; reference-source positioners  5216  and  5266 ; and possibly other auxiliary apparatuses; e.g. reference-source back-scatter shield (not shown in  FIG. 1 ). For this discussion, we presume that the empty nd: md sample container  5212  is first counted in the thick (md) position  5210  relative to the signal detection, processing, preservation, and presentation subsystem  1630  (and which is disclosed in detail in FIG. 16 of the related U.S. patent application Ser. No. 13/049,903. Subsystem  1630  detects, processes, preserves, and presents the nd and md ambient background spectra  5222  and  5272 . The size, shape, and positions of the empty nd: md sample container with respect to the detector, should be the same as those planned for containers holding standard-samples used to calibrate the counting system and for containers holding unknown-samples to be measured and analyzed by the counting system. 
         [0047]    In addition to the nd: md sample container  5212 , there is also a reference-source positioner apparatus  5216  consisting of an edge  5218  for fastening the positioner apparatus to the wide diameter of the nd: md sample container, and consisting of a reference-source holder  5214  to secure the reference source in place (the reference source is not shown, nor is it used here). The base of the reference-source holder  5214  is a thin window of low-z material, so as to minimize the attenuation of the reference-source emission spectrum in the direction of subsystem  1630 . 
         [0048]    After an amount of counting time (t bkgd,md ) the counting is stopped and the empty sample-container  5212  is flipped 180 degrees to the thin (nd) position  5260  relative to subsystem  1630 . The nd: md sample-container  5212  has two different diameter bases. In position  5260 , a smaller-diameter reference-source positioner apparatus  5266  is installed, and it also consists of an edge  5218  for fastening the positioner apparatus to the narrow diameter of the sample container  5212 , and consists of a reference-source holder  5214  to secure the reference source in place. After an amount of counting time (t bkgd,nd ) the counting is stopped. 
         [0000]    Part 3. “nd: md” Sample-Free Ref-Source Emission Spectrum ( FIG. 2 ) 
         [0049]      FIG. 2  illustrates the system setup  5300  for acquiring the sample-free reference-source emission spectra. Just above the empty sample container  5212  is the reference-source positioner apparatus  5216 , into which is placed a signal-emitting reference-source  5314 . A fraction of the characteristic signals  5318  are emitted from this reference-source within the solid angle subtended by the detector (not shown in  FIG. 2 ). The detector is part of subsystem  1630 . Because reference-sources usually have elevated (“hot”) signal-emission activity in order to provide good counting statistics and to facilitate short counting times, signal back-scatter by the surrounding materials may elevate the background noise in the detected spectrum. To minimize this noise, a collimator or back-scatter shield may be added around the reference-source (neither of which is shown in  FIG. 2 ). A reference-source positioner apparatus  5216  fastens the reference-source  5314  to the wide diameter of the sample container  5212  when in the thick (md) position  5310 . When in the thin (nd) position  5360 , a reference-source positioner apparatus  5266  fastens the reference-source  5314  to the narrow diameter of the sample container  5212 . 
         [0050]    The size, shape, and positions of the empty nd: md sample container  5212 , with respect to the detector, should be the same as those planned for sample containers that hold standard-samples used to calibrate the counting system  5400  in  FIG. 3 , and for sample containers holding unknown-samples to be analyzed by the counting system. Subsystem  1630  normalizes and subtracts-out an ambient background spectrum  5222  (in  FIG. 1 ) from the gross sample-free beam spectrum (not shown) to produce a net sample-free beam spectrum  5320 . 
         [0051]    The ambient background spectrum is acquired by the ambient background spectrum subsystem  5210 . The illustrated spectrum  5320  shows a singlet characteristic beam peak  5324  toward the high-energy range, but it is not too high in energy that the annihilation gamma-ray peak (labeled ‘ag’ in  5320 )—which adds noise to the spectrum  5320 —does not result in excessive degradation of the spectrum itself. After a period of counting time (t bm,smplFr,md ), and after a statistically ‘good enough’ beam-peak count rate is acquired (CR Ei,bm,smplFr,md , 5324 ), the counting is stopped. 
         [0052]    The reference source  5314  and its wide-diameter positioner  5216  are removed from the sample container  5212 . The empty sample-container  5212  is flipped 180 degrees to the thin (nd) position  5360  relative to subsystem  1630 . In position  5360 , a smaller-diameter reference-source positioner apparatus  5266  and the reference source  5314  are installed, and then a second counting begins. After a period of counting time (t bm,smplFr,nd ), and after a statistically ‘good-enough’ beam-peak count rate (CR Ei,bm,smplFr,nd ,  5374 ) is acquired, the counting is stopped. Subsystem  1630  detects, processes, preserves, and presents the nd and md sample-free beam spectra  5320  and  5370 . 
         [0053]    If the net reference-source count rates from the two counting positions  5310  and  5360  differ insignificantly, then their associated peak count rates  5324  and  5374  should be equal (CR Ei,bm,smplFr,md =CR Ei,bm,smplFr,nd ), in which case, two options are available to the operator; namely, (1) to sum the two count rates to improve the counting statistics, or (2) to count only one sample-container position (either the thick md or thin nd position, but not both) until the desired counting statistics are achieved. 
         [0000]    Part 4. “nd: md” Beam-Assisted Detected-Fraction Calibration ( FIG. 3 ). 
         [0054]    Before signal detection systems are used to quantify signal-sources in unknown-samples, they usually first require a detection-efficiency calibration of some kind.  FIG. 3  illustrates one such system  5400  for calibrating signal detection-efficiency, where a compositionally well-known standard-sample  5414  is filled to a depth (md) in the same type of sample container  5212 , and placed in the same position relative to the detector, as were the empty sample containers that were used to acquire the ambient background spectra  5222  and  5272  in  FIG. 1  and the sample-free beam spectra  5320  and  5370  in  FIG. 2 . 
         [0055]    Just above the standard-sample container is the same signal-emitting reference-source  5314  as that used to produce the sample-free beam spectra  5320  and  5370  in  FIG. 2 . The reference-source  5314  acts like a beam-source in the sense that only those signals emitted within the solid angle subtended by the detector, which is part of subsystem  1630 , have a chance to pass through the standard-sample  5414  and be detected and registered by subsystem  1630 . 
         [0056]    Although it is possible to carefully align the reference-source  5314  relative to the standard-sample and detector by many methods, one preferred method is to use a positioner  5216  that assures (1) that the sample container doesn&#39;t get contaminated or damaged by the reference-source, and (2) that the reference-source is always positioned in the same spot to achieve reproducible results. 
         [0057]    Characteristic signals emanate (dashed lines  5418 ) from the signal-emitting reference-source  5314  and a fraction of them pass through the sample-container  5212  walls and the standard-sample  5414  contained therein. Those reference-source signals within the solid angle subtended by the detector, act somewhat like a beam penetrating a slab-of-thickness (md) of sample  5414 . Some fraction of the beam passes through the standard-sample (stnd) unattenuated (dashed lines  5418 ), which fraction is called the beam-through-derived (bm), sample-specific beam-transmitted-fraction (BmTrnsF Ei,bm,stnd,md ). 
         [0058]    Subsystem  1630  acquires at least three spectral components as a single composite gross spectrum for each standard-sample counting; among the spectral components are the ambient background, reference-source beam, and the standard-sample emission. (The gross composite spectrum is not shown in  FIG. 3 .) To remove the ambient background component, subsystem  1630  normalizes the characteristic ambient background spectra  5222  and  5272  (in  FIG. 1 ) to the standard-sample counting times, and then subtracts-out those normalized ambient background component spectra from the gross composite spectra to produce the net composite spectra  5426  and  5476 , which are still comprised of at least two spectral components; namely, the reference-source beam peaks  5424  and  5474 , and all the standard-sample spectral peaks (the solid lines in  5426  and  5476 ). 
         [0059]    The nd: md beam-assisted detected-fraction ‘DetF’ calibration software  5500  computes the beam-through-derived, linear attenuation coefficient (μ Ei,bm,stnd ) for the composition of the standard-sample by one or more techniques. Three such techniques are described after this description of the system setup  5400 . Once the beam-through-derived, characteristic linear attenuation coefficient (μ Ei,bm,stnd ) of the standard-sample is determined, software  5500  computes the beam-through-derived, discrete sample-specific escaped-fraction values  5432  and  5482  (identified as the small open circles in  FIG. 3 ), and then computes their associated beam-through-derived, discrete sample-free detected-fraction calibration values  5442  and  5492  (identified as the two small open circles). 
         [0060]    Thus, the ratio of the nd and md beam-through-derived, sample-free detected-fraction calibration values is determined, which ratio allows software  5500  to compute standard-sample-derived (stnd), discrete sample-specific escaped-fraction values  5434  and  5484  in graphs  5430  and  5480 , respectively, and to compute standard-sample-derived, discrete sample-free detected-fraction calibration values  5444  and  5494  in graphs  5440  and  5490 , respectively. The discreet, sample-specific escaped-fraction values are fitted to functions  5436  (dotted line) and  5486  (dotted line) in graphs  5430  and  5480 , respectively, and the discrete sample-free detected-fraction values are fitted to functions  5446  (dotted line) and  5496  (dotted line) in graphs  5440  and  5490 , respectively. The counting system is now ready to be used to analyze homogeneous unknown-samples. 
       nd:md Software Model For Detection-System Calibration (FIG. 4). 
       [0061]      FIG. 4  is a flowchart  5500  of the nd: md software model for detection-system calibration. Software module  2010  reads-in nd and md spectral data, standard-sample data, reference-source data, and signal-emitter signal yield-fraction data (YF Rj,Ei ). 
         [0062]    The data qualification software module  2018  identifies those standard-sample-derived (stnd), characteristic nd and md peak pairs (CR Ei,stnd,nd ,CR Ei,stnd,md ) that are useful for computing associated values of standard-sample-derived, sample-specific beam-transmitted-fraction values (BmTrnsF Ei,stnd,nd , BmTrnsF Ei,stnd,md ). 
         [0063]    For each characteristic beam-peak pair (or n-tuple of characteristic peaks from n-tuple different sample depths, should three or more standard-sample thicknesses be counted), software module  5520  performs one of three techniques to compute the beam-through-derived (bm),sample-specific beam-transmitted-fraction (BmTrnsF Ei,bm,stnd ) and the beam-through-derived, sample-specific linear attenuation (μ Ei,bm,stnd ) terms. These techniques are referred to as Technique- 1  ( 5524 ), Technique- 2  ( 5528 ), and Technique- 3  ( 5532 ) in  FIG. 4 . 
         [0064]    Software module  5540  computes the system-specific ratio of the beam-through-derived, sample-free detected-fraction calibration values (RDetF Ei,nd:md ) using reference-source beam peaks transmitted through different sample thicknesses. 
         [0000]    Technique- 1  for Standard-Sample Lin. Atten. Computation ( FIG. 5 ) 
         [0065]      FIG. 5  shows that Technique- 1  ( 5524 ) independently computes the beam-through-derived (bm), sample-specific beam-transmitted-fraction (BmTrnsF Ei,bm,stnd ) and beam-through-derived linear attenuation coefficient (μ Ei,bm,stnd ) for each of two or more standard-sample thicknesses (e.g. nd and md) and thus, because the linear attenuation coefficient is the same for a given composition of the same density no matter the absolute thickness of the composition, provides independent confirmation of the value for the characteristic linear attenuation coefficient (μ Ei,bm,stnd ) for any given homogeneous standard-sample composition. 
         [0066]      FIG. 5  shows  5600  the characteristic (BO reference-source  5314  beam-peak (bm) count rates (CR Ei,bm,stnd,nd    5474 , CR Ei,bm,stnd,md    5424 ) through each thickness (nd and md) of standard-sample (stnd)  5414  are compared to their corresponding characteristic sample-free (smplFr) reference-source beam-peak count rates, (CR Ei,bm,smplFr,nd    5374 , CR Ei,bm,smplFr,md    5324 ), to determine the beam-through-derived, standard-sample characteristic linear attenuation coefficient (μ Ei,bm,stnd ) for each standard-sample-attenuated and sample-free beam-peak pair. The sample-free md beam-peak ( 5324  in  FIG. 2 ) count rate (CR Ei,bm,smplFr,md ) is paired with the standard-sample-attenuated beam-peak ( 5424  in  FIG. 3 ) count rate (CR Ei,bm,stnd,md ) to make a ‘peak pair  5612 ’. Similarly, the sample-free nd beam-peak ( 5374  in  FIG. 2 ) count rate (CR Ei,bm,smplFr,nd ) is paired with the standard-sample-attenuated beam-peak ( 5474  in  FIG. 3 ) count rate (CR Ei,bm,stnd,nd ) to make another ‘peak pair  5614 ’. Thus, the nd and the md peak pairs are 
         [0000]      {CR Ei,bm,smplFr,nd ,CR Ei,bm,stnd,nd } and {CR Ei,bm,smplFr,md ,CR Ei,bm,stnd,md}   [2a]
 
         [0067]    In principle, the two sample-free beam-peak count rates (CR Ei,bm,smplFr,nd    5374 , CR Ei,bm,smplFr,nd    5324 ) in [2a] should have the same value, i.e. CR Ei,bm,smplFr,nd =CR Ei,bm,smplFr,md , in which case only one sample-free beam-peak counting is needed, and thus, either of the following pairs [2b] or [2c] can be used alone in place of the nd and the md peak pairs described in [2a], so that the use of: 
         [0000]      {CR Ei,bm,smplFr,nd ,CR Ei,bm,stnd,nd } and {CR Ei,bm,smplFr,nd ,CR Ei,bm,stnd,md }  [2b]
 
         [0000]      is synonymous with 
         [0000]      {CR Ei,bm,smplFr,md ,CR Ei,bm,stnd,nd } and {CR Ei,bm,smplFr,md ,CR Ei,bm,stnd,md}   [2c]
 
         [0000]    Nevertheless, the following computations use the peak pairs of [2a] while acknowledging that the peak pairs shown in [2b] or [2c] can also be used. 
         [0068]    Using the thin (nd) sample-free and standard-sample-attenuated beam-peak pair count rates  5614 , i.e. {CR Ei,bm,smplFr,nd , CR Ei,bm,stnd,nd } in Pair [2a], the beam-derived, beam-transmitted-fraction (BmTrnsF Ei,bm,stnd,nd    5622 ) and linear attenuation coefficient (μ Ei,bm,stnd,nd    5624 ) of the standard-sample composition  5414  can be computed as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       BmTrnsF 
                       
                         Ei 
                         , 
                         bm 
                         , 
                         stnd 
                         , 
                         nd 
                       
                     
                     = 
                     
                       
                          
                         
                           
                             - 
                             μ 
                           
                           · 
                           nd 
                         
                       
                       = 
                       
                         
                           CR 
                           
                             Ei 
                             , 
                             bm 
                             , 
                             stnd 
                             , 
                             nd 
                           
                         
                         
                           CR 
                           
                             Ei 
                             , 
                             bm 
                             , 
                             smplFr 
                             , 
                             nd 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     so 
                      
                     
                         
                     
                      
                     that 
                      
                     
                       : 
                     
                   
                 
               
               
                 
                   [ 
                   
                     3 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     μ 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       stnd 
                       , 
                       nd 
                     
                   
                   = 
                   
                     
                       ln 
                        
                       
                         ( 
                         
                           BmTrnsF 
                           
                             Ei 
                             , 
                             bm 
                             , 
                             stnd 
                             , 
                             nd 
                           
                         
                         ) 
                       
                     
                     
                       - 
                       nd 
                     
                   
                 
               
               
                 
                   [ 
                   
                     3 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0069]    Using the thick (md) sample-free and standard-sample-attenuated peak-pair count rates  5612 , i.e. {CR Ei,bm,smplFr,md , CR Ei,bm,stnd,md } in Pair [2a], the beam-derived, beam-transmitted-fraction (BmTrnsF Ei,bm,stnd,md    5626 ) and the linear attenuation coefficient (μ Ei,bm,stnd,nd    5628 ) of the standard-sample composition can be computed as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       BmTrnsF 
                       
                         Ei 
                         , 
                         bm 
                         , 
                         stnd 
                         , 
                         md 
                       
                     
                     = 
                     
                       
                          
                         
                           
                             - 
                             μ 
                           
                           · 
                           md 
                         
                       
                       = 
                       
                         
                           CR 
                           
                             Ei 
                             , 
                             bm 
                             , 
                             stnd 
                             , 
                             md 
                           
                         
                         
                           CR 
                           
                             Ei 
                             , 
                             bm 
                             , 
                             smplFr 
                             , 
                             md 
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     so 
                      
                     
                         
                     
                      
                     that 
                      
                     
                       : 
                     
                   
                 
               
               
                 
                   [ 
                   
                     4 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     μ 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       stnd 
                       , 
                       md 
                     
                   
                   = 
                   
                     
                       ln 
                        
                       
                         ( 
                         
                           BmTrnsF 
                           
                             Ei 
                             , 
                             stnd 
                             , 
                             md 
                           
                         
                         ) 
                       
                     
                     
                       - 
                       md 
                     
                   
                 
               
               
                 
                   [ 
                   
                     4 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0070]    If peak pairs for two or more standard-sample thicknesses are acquired, then an improvement in the statistics  5632  for the beam-derived, beam-transmitted-fraction through 1 cm of standard-sample (BmTrnsF Ei,bm,stnd,1cm    5634 ) and for the linear attenuation coefficient (μ Ei,bm,stnd,1cm    5636 ) can be computed by summing the count rates for each of the individual standard-sample thicknesses, as follows: 
         [0000]      CR Ei,bm,stnd,nd   30  CR Ei,bm,stnd,md =(CR Ei,bm,smplFr,nd )(e −μd )+(CR Ei,bm,smplFr,md )(e −μd ) m   [5a]
 
         [0000]    where d is a unit of sample thickness (e.g. d=1 cm). Putting Equation [6a] into standard form yields: 
         [0000]      (CR Ei,bm,smplFr,nd )(e −μd ) n +(CR Ei,bm,smplFr,md )(e −μd ) m −(CR Ei,bm,stnd,nd +CR Ei,bm,stnd,md )=0  [5b]
 
         [0000]    The count rates of the reference-source sample-free beam-peaks through the thick (md) and thin (nd) positions of the sample-free (empty) sample container should be the same. Thus, Equation [5b] is rewritten using the statistically ‘best’ of the two (or more) sample-free beam-peak count rates, which for the purpose of this discussion is the sample-free beam-peak count rate through the thin (nd) counting position, so that Equation [6b] becomes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                            
                           
                             
                               - 
                               μ 
                             
                              
                             
                                 
                             
                              
                             d 
                           
                         
                         ) 
                       
                       n 
                     
                     + 
                     
                       
                         ( 
                         
                            
                           
                             
                               - 
                               μ 
                             
                              
                             
                                 
                             
                              
                             d 
                           
                         
                         ) 
                       
                       m 
                     
                     - 
                     
                       ( 
                       
                         
                           
                             CR 
                             
                               Ei 
                               , 
                               bm 
                               , 
                               stnd 
                               , 
                               nd 
                             
                           
                           + 
                           
                             CR 
                             
                               Ei 
                               , 
                               bm 
                               , 
                               stnd 
                               , 
                               md 
                             
                           
                         
                         
                           CR 
                           
                             Ei 
                             , 
                             bm 
                             , 
                             smplFr 
                             , 
                             nd 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   [ 
                   
                     5 
                      
                     c 
                   
                   ] 
                 
               
             
           
         
       
     
         [0000]    where: 
         [0000]      ( e   −μd )=BmTrnsF Ei,bm,stnd,1cm   [6a]
 
         [0000]    which is the transmitted fraction of a characteristic beam through a unit slab of sample thickness d=1 cm, and which is computed numerically by computer. 
         [0000]      μ Ei,bm,stnd,1cm =−ln(BmTrnsF Ei,bm,stnd,1cm )  [6b]
 
         [0000]    This method of statistical improvement can be extended to any number of measurements through the standard-sample. 
         [0071]    In principle, all determinations of the linear attenuation coefficient for the same homogeneous composition of constant density have the same value, so we define: 
         [0000]      μ Ei,bm,stnd,nd =μ Ei,bm,stnd,md =μ Ei,bm,stnd,1cm →μEi,stnd  [7]
 
         [0000]    Technique- 2  for Standard-Sample Lin. Atten. Computation ( FIG. 5 ) 
         [0072]      FIG. 5  shows that Technique- 2  ( 5328 ) uses only one standard-sample thickness and its corresponding sample-free beam-peak to compute the beam-through-derived (bm), beam-transmitted-fraction (BmTrnsF Ei,bm,stnd ) and linear attenuation coefficient (μ Ei,bm,stnd ) for the standard-sample (stnd). Any one of the thicknesses may be chosen, and in the example setups shown in  FIGS. 2 and 3 , the choices include: 
         [0000]      { CR   Ei,bm,smplFr,nd   ,CR   Ei,bm,stnd,nd }  [8a]
 
         [0000]      or 
         [0000]      { CR   Ei,bm,smplFr,md ), CR   Ei,bm,stnd,md }  [8b]
 
         [0073]    From either set of peak pairs, the sample-specific beam-transmitted-fractions (BmTrnsF Ei,bm,stnd,nd  and BmTrnsF Ei,bm,stnd,md  are computed. As an example, assume that the nd sample-free container position is beamed through (CR Ei,bm,smplEr,nd ) and the nd standard-sample depth is beamed through (CR Ei,bm,stnd,nd    5616 ). In that case, Equation [3a] is used to determine the beam-transmitted-fraction (BmTrnsF Ei,bm,stnd,nd    5642 ) through the nd standard-sample depth, and Equation [3b] is used to determine the linear attenuation coefficient (μ Ei,bm,stnd,nd    5644 ) of the standard-sample composition. 
         [0074]    Should one want to compute the beam-transmitted fraction through any other thickness of the same standard-sample, then one would use Equation [9], as follows, using the thick (md) standard-sample thickness as an example 5646: 
         [0000]    
       
         
           
             
               
                 
                   
                     BmTrnsF 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       smpl 
                       , 
                       md 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         BmTrnsF 
                         
                           Ei 
                           , 
                           bm 
                           , 
                           smpl 
                           , 
                           nd 
                         
                       
                       ) 
                     
                     
                       ( 
                       
                         m 
                         n 
                       
                       ) 
                     
                   
                 
               
               
                 
                   [ 
                   9 
                   ] 
                 
               
             
           
         
       
     
         [0075]    Technique- 2  for computing the standard-sample linear attenuation coefficient cuts down on the number of countings needed and allows for a longer single pair of sample-free and standard sample countings to improve the counting statistics. 
         [0000]    Technique- 3  for Standard-Sample Lin. Atten. Computation ( FIG. 5 ) 
         [0076]      FIG. 5  shows that Technique- 3  ( 5332 ) does not use sample-free beam-peak spectra, per se, but rather acquires at least two reference-source beam spectra through at least two different thicknesses of the standard-sample, and, by processing the differences in the beam peaks, one can determine the beam-derived, beam-transmitted-fraction (BmTrnsF Ei,bm,stnd,Δd ) and linear attenuation coefficient, (μ Ei,bm,stnd ) for the standard-sample (stnd). The beam peak count rate through the thin (nd) sample thickness is taken as the baseline beam peak count rate  5474 , whereas the beam peak count rate  5424  through the thick (md) sample thickness is treated as the attenuated beam peak count rate, so that the beam-derived, beam-transmitted-fraction (BmTrnsF Ei,bm,stnclAd    5652 ) and the linear attenuation (μ Ei,bm,stnd,nd    5654 ) computations are based on the difference (Δd) between the two sample thicknesses, as follows: 
         [0000]      Δ d =( md−nd )  [11]
 
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   
                     BmTrnsF 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       stnd 
                       , 
                       
                         Δ 
                          
                         
                             
                         
                          
                         d 
                       
                     
                   
                   = 
                   
                     
                       
                         CR 
                         
                           Ei 
                           , 
                           bm 
                           , 
                           stnd 
                           , 
                           md 
                         
                       
                       
                         CR 
                         
                           Ei 
                           , 
                           bm 
                           , 
                           stnd 
                           , 
                           nd 
                         
                       
                     
                     = 
                     
                        
                       
                         
                           
                             - 
                             μ 
                           
                           · 
                           Δ 
                         
                          
                         
                             
                         
                          
                         d 
                       
                     
                   
                 
               
               
                 
                   [ 
                   12 
                   ] 
                 
               
             
             
               
                 
                   
                     μ 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       stnd 
                       , 
                       
                         Δ 
                          
                         
                             
                         
                          
                         d 
                       
                     
                   
                   = 
                   
                     
                       ln 
                        
                       
                         ( 
                         
                           BmTrnsF 
                           
                             Ei 
                             , 
                             bm 
                             , 
                             stnd 
                             , 
                             
                               Δ 
                                
                               
                                   
                               
                                
                               d 
                             
                           
                         
                         ) 
                       
                     
                     
                       
                         - 
                         Δ 
                       
                        
                       
                           
                       
                        
                       d 
                     
                   
                 
               
               
                 
                   [ 
                   13 
                   ] 
                 
               
             
           
         
       
     
         [0077]    The beam-derived, beam-transmitted fractions ( 5656  and  5658 ), through the nd and md thicknesses, are then computed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       BmTrnsF 
                       
                         Ei 
                         , 
                         bm 
                         , 
                         smpl 
                         , 
                         nd 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           BmTrnsF 
                           
                             Ei 
                             , 
                             bm 
                             , 
                             smpl 
                             , 
                             
                               Δ 
                                
                               
                                   
                               
                                
                               d 
                             
                           
                         
                         ) 
                       
                       
                         ( 
                         
                           nd 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             d 
                           
                         
                         ) 
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   [ 
                   
                     14 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     BmTrnsF 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       smpl 
                       , 
                       md 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         BmTrnsF 
                         
                           Ei 
                           , 
                           bm 
                           , 
                           smpl 
                           , 
                           
                             Δ 
                              
                             
                                 
                             
                              
                             d 
                           
                         
                       
                       ) 
                     
                     
                       ( 
                       
                         md 
                         
                           Δ 
                            
                           
                               
                           
                            
                           d 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   [ 
                   
                     14 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0078]    The linear attenuation coefficient of the standard-sample composition is always the same through any sample depth, and so we define: 
         [0000]      μ Ei,stnd,Δd =μ Ei,stnd,nd =μ Ei,stnd,md →μ Ei,stnd   [15]
 
       Computing the Ratio of nd and md Detected-Fraction Calibs. (FIG. 4) 
       [0079]    Now that the beam-through-derived (bm), linear-attenuation coefficient (μ Ei,bm,stnd ) for the standard-sample composition at one or more characteristic signal energies is computed by one of the three techniques ( 5524 ,  5528 , and  5532 ) just described, the corresponding characteristic beam-through-derived, sample-specific escaped-fraction (EscF Ei,bm,stnd ) through any thickness of a standard-sample can easily be computed, e.g. 
         [0000]    
       
         
           
             
               
                 
                   
                     EscF 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       stnd 
                       , 
                       nd 
                     
                   
                   = 
                   
                     
                       
                         1 
                         nd 
                       
                        
                       
                         
                           ∫ 
                           0 
                           nd 
                         
                          
                         
                           
                             ( 
                             
                                
                               
                                 
                                   - 
                                   μ 
                                 
                                 · 
                                 x 
                               
                             
                             ) 
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           μ 
                           · 
                           nd 
                         
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                 μ 
                               
                               · 
                               nd 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     16 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     EscF 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       stnd 
                       , 
                       md 
                     
                   
                   = 
                   
                     
                       
                         1 
                         md 
                       
                        
                       
                         
                           ∫ 
                           0 
                           md 
                         
                          
                         
                           
                             ( 
                             
                                
                               
                                 
                                   - 
                                   μ 
                                 
                                 · 
                                 x 
                               
                             
                             ) 
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           μ 
                           · 
                           md 
                         
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                 μ 
                               
                               · 
                               md 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     16 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0080]    Equations [16a] and [16b] can be rewritten in terms of the beam-through-derived, beam-transmitted-fraction (BmTrnsF Ei,bm,stnd ). 
         [0000]    
       
         
           
             
               
                 
                   
                     EscF 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       stnd 
                       , 
                       nd 
                     
                   
                   = 
                   
                     
                       
                         BmTrnsF 
                         
                           Ei 
                           , 
                           bm 
                           , 
                           stnd 
                           , 
                           nd 
                         
                       
                       - 
                       1 
                     
                     
                       ln 
                        
                       
                         ( 
                         
                           BmTrnsF 
                           
                             Ei 
                             , 
                             bm 
                             , 
                             stnd 
                             , 
                             nd 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     17 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     EscF 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       stnd 
                       , 
                       md 
                     
                   
                   = 
                   
                     
                       
                         BmTrnsF 
                         
                           Ei 
                           , 
                           bm 
                           , 
                           stnd 
                           , 
                           md 
                         
                       
                       - 
                       1 
                     
                     
                       ln 
                        
                       
                         ( 
                         
                           BmTrnsF 
                           
                             Ei 
                             , 
                             bm 
                             , 
                             stnd 
                             , 
                             md 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     17 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0081]    The nd and md single characteristic (E i ) beam-through-derived sample-specific escaped-fraction pair (EscF Ei,bm,stnd,nd , ESCF Ei,bm,stnd,md ) are shown as the open circles  5432  and  5482  in  FIG. 3 . Spectra  5426  and  5476  in  FIG. 3  each show only one reference-beam characteristic peak ( 5424  and  5474 ) to which the corresponding nd and sample-free escaped-fraction values are computed for the standard-sample. To solve for the beam-through-derived (bm), sample-free detected-fraction calibration terms (DetF Ei,bm,smplFr,nd ; DetF Ei,bm,smplFr,md ) for the two (or more) spectrally measured thicknesses of the same homogeneous standard-sample, the two count rate balance Equations [18a] and [18b] are defined and then rearranged to solve for the beam-through-derived, sample-free detected-fraction terms, as follows: 
         [0000]      CR Ei,bm,stnd,nd   =M   stnd *SpA Rj,stnd *YF Rj,Ei *EscF Ei,bm,stnd,nd *DetF Ei,smplFr,nd   [18a]
 
         [0000]      CR Ei,bm,stnd,md =M stnd *SpA Rj,stnd *YF Rj,Ei *EscF Ei,bm,stnd,md *DetF Ei,bm,smplFr,md   [18b]
 
         [0000]    where the known mass of the spectrally measured standard-sample is given by (M stnd ); the known signal emitters (R j ) and their specific-activity quantities are given by (SpA Rj,stnd ); the known published signal-emitter characteristic (E i ) emission yield-fractions are given by (YF Rj,Ei ); the two computed beam-through-derived, sample-specific escaped-fraction terms are given by (EscF Ei,bm,stnd,nd , ESCF Ei,bm,stnd,md ); the two computed beam-through-derived, sample-free detected-fraction calibration terms are given by (DetF Ei,bm,smplFr,nd ; DetF Ei,bm,smplFr,md ); and the known measured beam count rate terms are given by (CR Ei,bm,stnd,nd  and CR Ei,bm,stnd,md ). Rearranging the two count rate balance Equations [18a] and [18b] to solve for the beam-through-derived (bm), sample-free detected-fraction calibration terms yield: 
         [0000]    
       
         
           
             
               
                 
                   
                     DetF 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       smplFr 
                       , 
                       nd 
                     
                   
                   = 
                   
                     
                       CR 
                       
                         Ei 
                         , 
                         bm 
                         , 
                         stnd 
                         , 
                         nd 
                       
                     
                     
                       
                         M 
                         stnd 
                       
                       * 
                       
                         SpA 
                         
                           Rj 
                           , 
                           stnd 
                         
                       
                       * 
                       
                         YF 
                         
                           Rj 
                           , 
                           Ei 
                         
                       
                       * 
                       
                         EscF 
                         
                           Ei 
                           , 
                           bm 
                           , 
                           stnd 
                           , 
                           nd 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     19 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     DetF 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       smplFr 
                       , 
                       md 
                     
                   
                   = 
                   
                     
                       CR 
                       
                         Ei 
                         , 
                         bm 
                         , 
                         stnd 
                         , 
                         md 
                       
                     
                     
                       
                         M 
                         stnd 
                       
                       * 
                       
                         SpA 
                         
                           Rj 
                           , 
                           stnd 
                         
                       
                       * 
                       
                         YF 
                         
                           Rj 
                           , 
                           Ei 
                         
                       
                       * 
                       
                         EscF 
                         
                           Ei 
                           , 
                           bm 
                           , 
                           stnd 
                           , 
                           md 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     19 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0000]    The ratio of the sample-free detected-fraction calibration terms at this particular characteristic beam energy (RDetF Ei,bm,nd:md ) is then computed from: 
         [0000]    
       
         
           
             
               
                 
                   
                     RDetF 
                     
                       Ei 
                       , 
                       bm 
                       , 
                       
                         nd 
                         : 
                         md 
                       
                     
                   
                   = 
                   
                     
                       DetF 
                       
                         Ei 
                         , 
                         bm 
                         , 
                         smplFr 
                         , 
                         nd 
                       
                     
                     
                       DetF 
                       
                         Ei 
                         , 
                         bm 
                         , 
                         smplFr 
                         , 
                         md 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     20 
                      
                     a 
                   
                   ] 
                 
               
             
           
         
       
     
         [0082]    The sample-free detected-fraction calibration terms are comprised of two terms, the geometry-fraction (GF) and the capture-fraction (CapF Ei ). These two terms are disclosed and discussed in detail in the related U.S. patent application Ser. No. 13/049,903. Both terms are independent of the standard-sample composition. Over the energy-range of interest, GF remains constant for a given standard-sample placement orientation with respect to the detection system. Over the energy-range of interest, Cap F Ei  will vary in proportion to signal attenuation through the detection system materials as a function of characteristic signal energy (E i ). In many sample-detector setups, CapF Ei  varies with energy (E i ) in approximately the same proportion for both the nd and md standard-sample orientations. Consequently, the ratio of the nd and md sample-free detected-fraction calibration terms, computed at any given characteristic energy (E i ), should remain relatively constant over the entire energy range of the spectrum. This can be confirmed by using a multi-energy beam source. 
         [0000]      RDetF Ei,bm,nd:md →RDetF nd:md ≈Constant for  E   i ε{low,high}  [20b]
 
         [0083]    So far, the beam-through-derived, sample-specific escaped-fraction terms (EsCF Ei,bm,stnd,nd , ESCF Ei,bm,stnd,md ) and the beam-through-derived, sample-free detected-fraction calibration terms (DetF Ei,bm,smplFr,nd , DetF Ei,bm,smplFr,md ) have only been determined for a single beam energy (E i ) which is shown as the tall peak  5424  in the spectrum  5426 , and the tall peak  5474  in the spectrum  5476 , of  FIG. 3 . 
         [0084]    Software module  5550  uses all of the ‘useful’ standard-sample-derived peak pairs (or n-tuple of characteristic peaks from n-tuple different sample thicknesses should three or more thicknesses of the standard-samples be counted), where ‘useful’ peak pairs are determined by software model  2018  in  FIG. 4 , to compute the standard-sample-derived (stnd), characteristic linear attenuation coefficients (μ Ei,stnd ) and the standard-sample-derived, sample-specific beam-transmitted-fraction (BmTrnsF Ei,stnd,nd ; BmTrnsF Ei,stnd,md) of the standard-sample.    
         [0085]    This is a multi-step process accomplished by first taking the ratio of the two count-rate balance Equations [21a] and [21b] that correspond to each characteristic standard-sample-derived peak pair, (CR Ei,stnd,nd ; CR Ei,stnd,md ); as follows: 
         [0000]      CR Ei,stnd,nd =M stnd *SpA Rj,stnd *YF Rj,Ei *ESCF Ei,stnd,nd *DetF Ei,smplFr,nd   [21a]
 
         [0000]      CR Ei,stnd,md =M stnd *SpA Rj,stnd *YF Rj,Ei *ESCF Ei,stnd,md *DetF Ei,smplFr,nd   [21b]
 
         [0000]    and taking their ratio yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       CR 
                       
                         Ei 
                         , 
                         stnd 
                         , 
                         nd 
                       
                     
                     
                       CR 
                       
                         Ei 
                         , 
                         stnd 
                         , 
                         md 
                       
                     
                   
                   = 
                   
                     
                       
                         M 
                         stnd 
                       
                       * 
                       
                         SpA 
                         
                           Rj 
                           , 
                           stnd 
                         
                       
                       * 
                       
                         YF 
                         
                           Rj 
                           , 
                           Ei 
                         
                       
                       * 
                       
                         EscF 
                         
                           Ei 
                           , 
                           stnd 
                           , 
                           nd 
                         
                       
                       * 
                       
                         DetF 
                         
                           Ei 
                           , 
                           smplFr 
                           , 
                           nd 
                         
                       
                     
                     
                       
                         M 
                         stnd 
                       
                       * 
                       
                         SpA 
                         
                           Rj 
                           , 
                           stnd 
                         
                       
                       * 
                       
                         YF 
                         
                           Rj 
                           , 
                           Ei 
                         
                       
                       * 
                       
                         EscF 
                         
                           Ei 
                           , 
                           stnd 
                           , 
                           md 
                         
                       
                       * 
                       
                         DetF 
                         
                           Ei 
                           , 
                           smplFr 
                           , 
                           md 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     22 
                      
                     a 
                   
                   ] 
                 
               
             
           
         
       
     
         [0086]    Equation [22a] is one equation in four unknowns. The four unknowns are the two standard-sample-derived, sample-specific escaped-fraction terms (EscF Ei,stnd,nd  and EscF Ei,stnd,md ); and the two standard-sample-derived, sample-free detected-fraction calibration terms (DetF Ei,smplFr,nd  and DetF Ei,smplFr,md ). The known terms are the measured nd and standard-sample-derived count rates (CR Ei,stnd,nd  and CR Ei,stnd,md ); the measured standard-sample mass (M stnd ); the reported signal emitters (R j ) and their specific-activity quantity (SpA Rj,stnd ); and the widely published signal-emitter characteristic (E i ) emission yield-fractions (YF Rj,Ei ). Then, cancelling the equal terms and substituting for known ratios of like terms, reduces the number of unknowns in Equation [22a] to only two unknowns in Equation [22b]; i.e. the two sample-specific escaped-fraction terms (EscF Ei,stnd,nd  and ESCF Ei,stnd,md ); 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       CR 
                       
                         Ei 
                         , 
                         stnd 
                         , 
                         nd 
                       
                     
                     
                       CR 
                       
                         Ei 
                         , 
                         stnd 
                         , 
                         md 
                       
                     
                   
                   = 
                   
                     
                       
                         EscF 
                         
                           Ei 
                           , 
                           stnd 
                           , 
                           nd 
                         
                       
                       
                         EscF 
                         
                           Ei 
                           , 
                           stnd 
                           , 
                           md 
                         
                       
                     
                     * 
                     
                       RDetF 
                       
                         nd 
                         : 
                         md 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     22 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0000]    where the ratio of the sample-free detected-fraction calibration terms (DetF Ei,smplFr,nd ; DetF Ei,smplFr,md ) are replaced by the ratio factor RDetF nd:md . 
         [0087]    In many cases, the ratio of the two sample-free detected-fraction calibration terms (RDetF nd:md ) is very close to unity for the detection-system/sample-volume-shape and position setup shown in  FIG. 3 . 
         [0088]    To solve Equation [22b], the two sample-specific escaped-fraction terms are defined in terms of the fraction of a characteristic beam that transmits through a standard-sample whose “per unit” thickness is one centimeter (d=1 cm). In a preliminary step, the two unknown terms in Equation [22b] (ESCF Ei,stnd,nd , EscF Ei,stnd,md ) are redefined using a single new common term, i.e. the characteristic sample-specific linear attenuation coefficient (μ Ei,stnd ). 
         [0089]    The thick (md) and thin (nd) standard-sample orientations yield characteristic sample-specific escaped-fraction terms according to: 
         [0000]    
       
         
           
             
               
                 
                   
                     EscF 
                     
                       Ei 
                       , 
                       stnd 
                       , 
                       nd 
                     
                   
                   = 
                   
                     
                       
                         1 
                         nd 
                       
                        
                       
                         
                           ∫ 
                           0 
                           nd 
                         
                          
                         
                           
                              
                             
                               
                                 - 
                                 μ 
                               
                               · 
                               x 
                             
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           μ 
                           · 
                           nd 
                         
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                 μ 
                               
                               · 
                               nd 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     23 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     EscF 
                     
                       Ei 
                       , 
                       stnd 
                       , 
                       md 
                     
                   
                   = 
                   
                     
                       
                         1 
                         md 
                       
                        
                       
                         
                           ∫ 
                           0 
                           md 
                         
                          
                         
                           
                              
                             
                               
                                 - 
                                 μ 
                               
                               · 
                               x 
                             
                           
                            
                           
                              
                             x 
                           
                         
                       
                     
                     = 
                     
                       
                         1 
                         
                           μ 
                           · 
                           md 
                         
                       
                        
                       
                         ( 
                         
                           1 
                           - 
                           
                              
                             
                               
                                 - 
                                 μ 
                               
                               · 
                               md 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     23 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0090]    Knowing that a beam passing through each of the sample depths (nd and md) attenuates according to: 
         [0000]      BmTrnsF Ei,stnd,nd   =e   −μ·nd =( e   −μd ) n =(BmTrnsF Ei,stnd,1cm ) n   [24a]
 
         [0000]      BmTrnsF Ei,stnd,md   =e   −μ·md =( e   −μd ) m =(BmTrnsF Ei,stnd,1cm ) m   [24b]
 
         [0000]    and where d=1 cm; the linear attenuation per centimeter of thickness (cm −1 ) of the standard-sample composition is: 
         [0000]      μ Ei,stnd =−ln(BmTrnsF Ei,stnd,1cm )  [25]
 
         [0091]    Then, rewriting Equations [23a] and [23b] in terms of a beam passing through 1 cm of standard sample composition, BmTrnsF EscF Ei,stnd,1cm  yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     EscF 
                     
                       Ei 
                       , 
                       stnd 
                       , 
                       nd 
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             BmTrnsF 
                             
                               Ei 
                               , 
                               stnd 
                               , 
                               
                                 1 
                                  
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 m 
                               
                             
                           
                           ) 
                         
                         n 
                       
                       - 
                       1 
                     
                     
                       n 
                       * 
                       
                         ln 
                          
                         
                           ( 
                           
                             BmTrnsF 
                             
                               Ei 
                               , 
                               stnd 
                               , 
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 m 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     26 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     EscF 
                     
                       Ei 
                       , 
                       stnd 
                       , 
                       md 
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             BmTrnsF 
                             
                               Ei 
                               , 
                               stnd 
                               , 
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 m 
                               
                             
                           
                           ) 
                         
                         m 
                       
                       - 
                       1 
                     
                     
                       m 
                       * 
                       
                         ln 
                          
                         
                           ( 
                           
                             BmTrnsF 
                             
                               Ei 
                               , 
                               stnd 
                               , 
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 m 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     26 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0092]    Substituting the expressions of Equations [26a] and [26b] into Equation [22b], and reducing terms, yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         CR 
                         
                           Ei 
                           , 
                           stnd 
                           , 
                           nd 
                         
                       
                       
                         CR 
                         
                           Ei 
                           , 
                           stnd 
                           , 
                           md 
                         
                       
                     
                     = 
                     
                       
                         [ 
                         
                           
                             m 
                             n 
                           
                           * 
                           
                             
                               
                                 
                                   ( 
                                   
                                     BmTrnsF 
                                     
                                       Ei 
                                       , 
                                       stnd 
                                       , 
                                       
                                         1 
                                          
                                         
                                             
                                         
                                          
                                         c 
                                          
                                         
                                             
                                         
                                          
                                         m 
                                       
                                     
                                   
                                   ) 
                                 
                                 n 
                               
                               - 
                               1 
                             
                             
                               
                                 
                                   ( 
                                   
                                     BmTrnsF 
                                     
                                       Ei 
                                       , 
                                       stnd 
                                       , 
                                       
                                         1 
                                          
                                         
                                             
                                         
                                          
                                         c 
                                          
                                         
                                             
                                         
                                          
                                         m 
                                       
                                     
                                   
                                   ) 
                                 
                                 m 
                               
                               - 
                               1 
                             
                           
                         
                         ] 
                       
                       * 
                       
                         RDetF 
                         
                           nd 
                           : 
                           md 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     which 
                      
                     
                         
                     
                      
                     rearrages 
                      
                     
                         
                     
                      
                     to 
                      
                     
                       : 
                     
                   
                 
               
               
                 
                   [ 
                   
                     27 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     
                       
                         [ 
                         
                           
                             CR 
                             
                               Ei 
                               , 
                               stnd 
                               , 
                               nd 
                             
                           
                           
                             CR 
                             
                               Ei 
                               , 
                               stnd 
                               , 
                               md 
                             
                           
                         
                         ] 
                       
                        
                       
                         
                           ( 
                           
                             BmTrnsF 
                             
                               Ei 
                               , 
                               stnd 
                               , 
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 m 
                               
                             
                           
                           ) 
                         
                         m 
                       
                     
                     - 
                     
                       
                         [ 
                         
                           m 
                           n 
                         
                         ] 
                       
                        
                       
                         ( 
                         
                           RDetF 
                           
                             nd 
                             : 
                             md 
                           
                         
                         ) 
                       
                        
                       
                         
                           ( 
                           
                             BmTrnsF 
                             
                               Ei 
                               , 
                               stnd 
                               , 
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 m 
                               
                             
                           
                           ) 
                         
                         n 
                       
                     
                     + 
                     
                       [ 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     m 
                                     n 
                                   
                                   * 
                                   
                                     RDetF 
                                     
                                       nd 
                                       : 
                                       md 
                                     
                                   
                                 
                                 ) 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 
                                   CR 
                                   
                                     Ei 
                                     , 
                                     stnd 
                                     , 
                                     nd 
                                   
                                 
                                 
                                   CR 
                                   
                                     Ei 
                                     , 
                                     stnd 
                                     , 
                                     md 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   [ 
                   
                     27 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0093]    The only unknown in Equation [27b] is the beam-transmitted-fraction through 1 cm of standard-sample depth (BmTrnsF Ei,stnd,1cm ), which is easily solved numerically by computer. 
         [0094]    Although not required to be known to quantify the radioisotopes of interest, nevertheless, there may be an interest to knowing the standard-sample energy-specific linear attenuation (μEi,stnd). Once the value of BmTrnsF Ei,stnd,1cm  is known from Equation [27b], Equation [25] is used to determine the value of the linear attenuation coefficient (μ Ei,stnd ) for the standard-sample composition. 
         [0095]    Software module  5560  calls the count rates of the characteristic peak pairs and the corresponding characteristic values of BmTrnsF Ei,stnd,1cm , and computes the associated sample-specific escaped-fraction values (EscF Ei,stnd,nd , EsCF Ei,stnd,md ) using Equations [26a] and [26b], as well as provides for computing sample-specific escaped-fraction functions or interpolated curves to cover the entire energy range, i.e. 
         [0000]      ESCF Ei,stnd,nd →EscF( E   i ) stnd,nd   [28a]
 
         [0000]      ESCF Ei,stnd,md →EscF( E   i ) stnd,md   [28b]
 
         [0000]    and computes their associated uncertainties. 
         [0096]    Software module  2040  in  FIG. 4  evaluates and processes the ‘useful’ discrete, sample-specific escaped-fraction values and uncertainties for computing sample-specific escaped-fraction functions or interpolated curves. 
         [0097]    Software module  2058  computes the sample-free detected-fraction calibration terms (DetF Ei,smplFr,nd  and DetF Ei,smplFr,md ) using Equations [29a] and [29b], as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     DetF 
                     
                       Ei 
                       , 
                       smplFr 
                       , 
                       nd 
                     
                   
                   = 
                   
                     
                       CR 
                       
                         Ei 
                         , 
                         stnd 
                         , 
                         nd 
                       
                     
                     
                       
                         M 
                         stnd 
                       
                       * 
                       
                         SpA 
                         
                           Rj 
                           , 
                           stnd 
                         
                       
                       * 
                       
                         YR 
                         
                           Rj 
                           , 
                           Ei 
                         
                       
                       * 
                       
                         
                           EscF 
                            
                           
                             ( 
                             
                               E 
                               i 
                             
                             ) 
                           
                         
                         
                           stnd 
                           , 
                           nd 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     29 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     DetF 
                     
                       Ei 
                       , 
                       smplFr 
                       , 
                       md 
                     
                   
                   = 
                   
                     
                       CR 
                       
                         Ei 
                         , 
                         stnd 
                         , 
                         md 
                       
                     
                     
                       
                         M 
                         stnd 
                       
                       * 
                       
                         SpA 
                         
                           Rj 
                           , 
                           stnd 
                         
                       
                       * 
                       
                         YF 
                         
                           Rj 
                           , 
                           Ei 
                         
                       
                       * 
                       
                         
                           EscF 
                            
                           
                             ( 
                             
                               E 
                               i 
                             
                             ) 
                           
                         
                         
                           stnd 
                           , 
                           md 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     29 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0000]    Software module  2058  also computes the associated uncertainties, indicates which of Equations [29a] and [29b] provides the better statistics for the sample-free detected-fraction calibration and provides for computing sample-free detected-fraction calibration functions or interpolated curves to cover the entire energy range, i.e. 
         [0000]      DetF Ei,smplFr,nd →DetF( E   i ) smplFr,nd   [30a]
 
         [0000]      DetF Ei,smplFr,md →DetF( E   i ) smplFr,md   [30b]
 
         [0000]    and formats all the data from each part of the software flow, as just described, for presentation by spreadsheet, comma separated value (CSV) listings, and computer screen presentation. 
         [0098]    The mathematical laws of error analysis are computed as appropriate alongside the mathematical operations on the values of the terms comprising the count rate balance equations. Thus, the terms will have the form ν±Δν, where ν represents the numerical value of a particular term and ±Δν is the uncertainty in ν. 
         [0099]    Once calibrated, the counting system is ready to identify, measure, and compute quantities of signal emitters in unknown-samples. 
         [0000]    Part 5. “nd: md” Signal-Emitter Quantitation ( FIG. 6 ) 
         [0100]    To quantify the individual signal emitter of each characteristic signal (E i ) in unknown-samples (unkn) of homogeneous composition, the values of the sample-specific escaped-fraction (EscF Ei,unkn ) are computed and combined with the computed values for the sample-free detected-fraction calibration (DetF Ei,smplFr ), y quantities which then results in signal emitter (R j ) specific-activity (SpA Rj,unkn ).  FIG. 6  illustrates one such system  5700  for analyzing homogeneous samples of unknown signal-emitter specific-activity quantities (SpA Rj,unkn ). 
         [0101]    It is presumed that the sample-free detected-fraction calibration (DetF Ei,smplFr ) has been performed, after which the unknown-sample is placed in the same type of sample-container apparatus, and in the same position and orientation relative to the detection system, as the sample-container apparatuses (disclosed in detail in FIG. 18 of the related U.S. patent application Ser. No. 13/049,903) that were used to acquire the ambient background emission spectra and the standard-sample emission spectra (the solid spectral lines in graphs  5426  and  5476  of  FIG. 3 ). 
         [0102]    An unknown-sample mass (M unkn ), may be computed where the mass of an empty sample-container apparatus (M cntr ) is subtracted from the combined mass of the sample-container apparatus and the unknown-sample (M cntr+unkn ), as follows: 
         [0000]        M   unkn   =M   cntr+unkn   −M   cntr   [31]
 
         [0103]    It is presumed for this particular discussion that the unknown-sample  5714  is first placed into the md cup  5720  of the cylindrical nd: md sample-container apparatus  5212 . (Note: in the alternative, the operator could have placed the unknown-sample into the nd cup  5470  instead and proceeded accordingly.) Then the sample-container apparatus is sealed securely and placed into the thick (md) position  5710  for counting, and counting then commences. Characteristic signals (E i ) that escape the thick (md) unknown-sample  5714  and register, along with the ambient background signals, in the signal detection, processing, preservation, and presentation subsystem  1630 , produce a “thick” gross composite spectrum (not shown). 
         [0104]    Once the “thick” gross composite spectrum is obtained, then the sample-container apparatus  5212  is flipped 180 degrees  5752  into the thin (nd) position  5760  which ‘reshapes’ the unknown-sample  5714  to fill the wider-diameter cylinder of thinner depth (nd)  5470 . Characteristic signals (E i ) that escape the thin (nd) unknown-sample and register, along with the ambient background signals, in subsystem  1630 , produce a “thin” gross composite spectrum (not shown). 
         [0105]    Count rates can be compared directly. The ambient background signal count rates  1810  and  1860  are subtracted-out from their corresponding “thick” and “thin” unknown-sample gross spectra (not shown) leaving the corresponding net unknown-sample spectra  5726  and  5776 . This can be summarized, as follows: 
         [0000]        CR   Ei,unkn,nd   =GCR   Et,unkn,nd   −BCR   Ei,nd   [32a]
 
         [0000]        CR   Ei,unkn,md   =GCR   Ei,unkn,md   −BCR   Ei,md   [32b]
 
         [0106]    By comparing the nd and md net unknown-sample spectra  5726  and  5776 , there is a noticeable, significant difference between the peak heights at the low-energy portion of both the thick (md) and thin (nd) net unknown-sample spectra  5718  and  5768 . The highly attenuated peaks of the thick (md) unknown-sample counting  5710  are anticipated because, at low energy, more of the characteristic signals (E i ) are attenuated within the thick ‘shape’ of the unknown-sample, relative to its thin ‘shape’. 
         [0107]      FIG. 5  also shows the nd:md Signal-emitter Quantitation Software  5800  that reads input of the ‘thick’ and ‘thin’ spectral peak data and computes, for each useful thick and thin characteristic spectral-peak pair, the discrete characteristic sample-specific escaped-fraction values (EscF Ei,unkn,nd  and EscF Ei,unkn,md , of which only one set of discrete values is shown 5784); interpolated or fitted sample-specific escaped-fraction functions [EscF(E i ) unkn,nd  and EscF(E i ) unkn,md , of which only one function is shown as the dotted line  5786 ]; and signal-emitter identities (R j ) and corresponding specific-activity quantities (SpA Rj,unkn )  5792 . 
         [0108]    All of the discrete characteristic sample-specific escaped-fraction values  5784 , taken together, resemble the outline of a curve that spans the energy range of interest (represented by the dotted line  5786 ). Commonly, a function is fitted to these discrete values  5784  to cover the entire usable energy-detection range of the detection system. The discrete values  5784  and all of the possible fitted values  5786  of the sample-specific escaped fraction are illustrated together in graph  5782 . 
         [0109]    The specific-activity quantities (SpA Rj,unkn ) within the unknown-sample  5714  are computed by software  5800 , where the software  5500  (in  FIG. 4 ) computes the fitted sample-free detected-fraction calibration functions [DetF(E i ) smplFr,nd , DetF(E i ) smplFr,md ]. 
         [0110]    Subsystem  5792  aggregates all of the processed data into user-selected or default formats, e.g. comma-separated-value (CSV) list, spreadsheet, computer screen, or other suitable output format. 
       nd:md Software Model for Signal-Emitter Quantitation (FIG. 7) 
       [0111]      FIG. 7  shows the flowchart  5800  of the new nd:md sample-analysis software. Software module  2210  reads-in (a) the primary unknown-sample nd and md spectral data, which includes the nd and md characteristic-peak net count rates (CR Ei,unkn,nd , CR Ei,unkn,md ); (b) unknown-sample data, such as the unknown-sample mass (M unkn ); (c) the interpolated or fitted sample-free detected-fraction calibration functions [DetF(E i ) smplFr,nd , DetF(E i ) smplFr,md ]; and (d) signal-emitter (R j ) and yield-fraction (YF Rj,Ei ) databases. The Data Qualification software module  2018  in  FIG. 7  identifies those characteristic nd and md peak pairs that are ‘useful’ for computing the associated sample-specific beam-transmitted-fraction values (BmTrnsF Ei,unkn,nd , BmTrnsF Ei,unkn,md ). 
         [0112]    For each characteristic-peak pair, software module  5850  calls the values of the peak pairs and computes the sample-specific beam-transmitted-fraction values (BmTrnsF Ei,unkn,nd ; BmTrnsF Ei,unkn,md ) and the sample-specific linear attenuation coefficient (μ Ei,unkn ). The count rates for each characteristic-peak pair (or n-tuple of characteristic peaks from n-tuple different sample thicknesses, should three or more be counted) and other related terms, are shown in the count-rate balance equations as follows: 
         [0000]      CR Ei,unkn,nd   =M   unkn *SpA Rj,unkn *YF Rj,Ei *EscF Ei,unkn,nd *DetF(E i ) smplFr,nd   [33a]
 
         [0000]      CR Ei,unkn,md =M unkn *SpA Rj,unkn *YF Rj,Ei *EscF Ei,unkn,md *DetF(E i ) smplFr,md   [33b]
 
         [0113]    Equations [33a] and [33b] are two equations in four unknowns. The four unknowns are the two sample-specific escaped-fraction terms (EscF Ei,unkn,nd , ESCF Ei,unkn,md ); signal-emitter identities (R j ) and their specific-activity quantities (SpA Rj,unkn ); and the associated signal-emitter characteristic (E i ) emission yield-fractions (YF Rj,Ei ). The known terms are the measured nd and md count rates (CR Ei,unkn,nd , CR Ei,unkn,md ); the measured unknown-sample mass (M unkn ); and the two sample free detected-fraction calibration terms [DetF(E i ) smplFr,nd , DetF(E i ) smplFr,md ]. To reduce the number of unknowns, the approach is to take the ratio of the nd and md count-rate balance Equations [33a] and [33b], as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       CR 
                       
                         Ei 
                         , 
                         unkn 
                         , 
                         nd 
                       
                     
                     
                       CR 
                       
                         Ei 
                         , 
                         unkn 
                         , 
                         md 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             
                               M 
                               stnd 
                             
                             * 
                             
                               SpA 
                               
                                 Rj 
                                 , 
                                 unkn 
                               
                             
                             * 
                             
                               YF 
                               
                                 Rj 
                                 , 
                                 Ei 
                               
                             
                             * 
                           
                         
                       
                       
                         
                           
                             
                               EscF 
                               
                                 Ei 
                                 , 
                                 unkn 
                                 , 
                                 nd 
                               
                             
                             * 
                             
                               
                                 DetF 
                                  
                                 
                                   ( 
                                   
                                     E 
                                     i 
                                   
                                   ) 
                                 
                               
                               
                                 smplFr 
                                 , 
                                 nd 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               M 
                               stnd 
                             
                             * 
                             
                               SpA 
                               
                                 Rj 
                                 , 
                                 unkn 
                               
                             
                             * 
                             
                               YF 
                               
                                 Rj 
                                 , 
                                 Ei 
                               
                             
                             * 
                           
                         
                       
                       
                         
                           
                             
                               EscF 
                               
                                 Ei 
                                 , 
                                 unkn 
                                 , 
                                 md 
                               
                             
                             * 
                             
                               
                                 DetF 
                                  
                                 
                                   ( 
                                   
                                     E 
                                     i 
                                   
                                   ) 
                                 
                               
                               
                                 smplFr 
                                 , 
                                 md 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     34 
                      
                     a 
                   
                   ] 
                 
               
             
           
         
       
     
         [0114]    Cancelling the equal terms in Equation [34a] and substituting for the known ratio of the two sample-free detected-fraction calibration terms, leaves one Equation [34b] in two unknowns, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       CR 
                       
                         Ei 
                         , 
                         unkn 
                         , 
                         nd 
                       
                     
                     
                       CR 
                       
                         Ei 
                         , 
                         unkn 
                         , 
                         md 
                       
                     
                   
                   = 
                   
                     
                       
                         EscF 
                         
                           Ei 
                           , 
                           unkn 
                           , 
                           nd 
                         
                       
                       
                         EscF 
                         
                           Ei 
                           , 
                           unkn 
                           , 
                           md 
                         
                       
                     
                     * 
                     
                       RDetF 
                       
                         nd 
                         ; 
                         md 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     34 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0000]    where the ratio of the two terms, [DetF (E i ) smplFr,nd , DetF(E i ) smplFr,md ] is replaced by a single term, RDetF nd:md , whose value was computed in the detection-system calibration Equations [20a] and [20b]. In many cases, such value of this term (RDetF nd:md ), is very close to unity for the system setup shown as in  FIG. 3 . 
         [0115]    Rewriting the sample-specific escaped-fraction terms, (EscF Ei,unkn,nd , EscF Ei,unkn,md ) in terms of a beam through one centimeter of unknown-sample composition (BmTrnsF Ei,unkn,1cm ) yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     EscF 
                     
                       Ei 
                       , 
                       unkn 
                       , 
                       nd 
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             BmTrnsF 
                             
                               Ei 
                               , 
                               unkn 
                               , 
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 m 
                               
                             
                           
                           ) 
                         
                         n 
                       
                       - 
                       1 
                     
                     
                       n 
                       · 
                       
                         ln 
                          
                         
                           ( 
                           
                             BmTrnsF 
                             
                               Ei 
                               , 
                               unkn 
                               , 
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 m 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     35 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     EscF 
                     
                       Ei 
                       , 
                       unkn 
                       , 
                       md 
                     
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             BmTrnsF 
                             
                               Ei 
                               , 
                               unkn 
                               , 
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 m 
                               
                             
                           
                           ) 
                         
                         m 
                       
                       - 
                       1 
                     
                     
                       m 
                       · 
                       
                         ln 
                          
                         
                           ( 
                           
                             BmTrnsF 
                             
                               Ei 
                               , 
                               unkn 
                               , 
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 m 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     35 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0116]    Replacing these sample-specific escaped-fraction terms of Equation [34b] with their equivalent expressions from Equations [35a] and [35b] yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       CR 
                       
                         Ei 
                         , 
                         unkn 
                         , 
                         nd 
                       
                     
                     
                       CR 
                       
                         Ei 
                         , 
                         unkn 
                         , 
                         md 
                       
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           m 
                           n 
                         
                         * 
                         
                           
                             
                               
                                 ( 
                                 
                                   BmTrnsF 
                                   
                                     Ei 
                                     , 
                                     unkn 
                                     , 
                                     
                                       1 
                                        
                                       
                                           
                                       
                                        
                                       c 
                                        
                                       
                                           
                                       
                                        
                                       m 
                                     
                                   
                                 
                                 ) 
                               
                               n 
                             
                             - 
                             1 
                           
                           
                             
                               
                                 ( 
                                 
                                   BmTrnsF 
                                   
                                     Ei 
                                     , 
                                     unkn 
                                     , 
                                     
                                       1 
                                        
                                       
                                           
                                       
                                        
                                       c 
                                        
                                       
                                           
                                       
                                        
                                       m 
                                     
                                   
                                 
                                 ) 
                               
                               m 
                             
                             - 
                             1 
                           
                         
                       
                       ] 
                     
                     * 
                     
                       RDetF 
                       
                         nd 
                         : 
                         md 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     36 
                      
                     a 
                   
                   ] 
                 
               
             
           
         
       
     
         [0000]    which rearranges to: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         [ 
                         
                           
                             CR 
                             
                               Ei 
                               , 
                               unkn 
                               , 
                               nd 
                             
                           
                           
                             CR 
                             
                               Ei 
                               , 
                               unkn 
                               , 
                               md 
                             
                           
                         
                         ] 
                       
                        
                       
                         
                           ( 
                           
                             BmTrnsF 
                             
                               Ei 
                               , 
                               unkn 
                               , 
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 c 
                                  
                                 
                                     
                                 
                                  
                                 m 
                               
                             
                           
                           ) 
                         
                         m 
                       
                     
                     - 
                     
                       
                         [ 
                         
                           m 
                           n 
                         
                         ] 
                       
                        
                       
                         ( 
                         
                           RDetF 
                           
                             nd 
                             : 
                             md 
                           
                         
                         ) 
                       
                        
                       
                         
                           ( 
                           
                             BmTrnsF 
                             
                               
                                 Ei 
                                 , 
                                 unkn 
                                 , 
                                 
                                   1 
                                    
                                   c 
                                    
                                   
                                       
                                   
                                    
                                   m 
                                 
                               
                                
                               
                                   
                               
                             
                           
                           ) 
                         
                         n 
                       
                     
                     + 
                     
                       [ 
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     m 
                                     n 
                                   
                                   * 
                                   
                                     RDetF 
                                     
                                       nd 
                                       : 
                                       md 
                                     
                                   
                                 
                                 ) 
                               
                               - 
                             
                           
                         
                         
                           
                             
                               ( 
                               
                                 
                                   CR 
                                   
                                     Ei 
                                     , 
                                     unkn 
                                     , 
                                     nd 
                                   
                                 
                                 
                                   CR 
                                   
                                     Ei 
                                     , 
                                     unkn 
                                     , 
                                     md 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   [ 
                   
                     36 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0117]    The only unknown in Equation [36b] is the beam-transmitted-fraction through a unit (1-cm) of unknown-sample thickness (BmTrnsF Ei,unkn,1cm ), which is easily solved numerically by a computer. Although not required to be known to quantify the signal-emitters of interest, nevertheless there may be an interest to know the unknown-sample energy-specific linear attenuation coefficient (μ Ei,unkn ). Once the value of BmTrnsF Ei,unkn,1cm  is known from Equation [36b], then Equation [36c] is used to compute the value of the linear attenuation coefficient (μ Ei,unkn ) for the unknown-sample. The characteristic linear attenuation per centimeter of thickness (1 cm) of the unknown-sample is: 
         [0000]      μ Ei,unkn =−ln(BmTrnsF Ei,unkn,1cm )  [36c]
 
         [0118]    Software module  2838  in  FIG. 7  performs a number of functions. It calls the count rates of the characteristic-peak pairs and the corresponding characteristic values of BmTrnsF Ei,unkn,1cm ; computes the associated sample-specific escaped-fraction values (EscF Ei,unkn,nd , ESCF Ei,unkn,md ) using Equations [35a] and [35b]; and then provides for computing sample-specific escaped-fraction functions or interpolated curves to cover the entire energy region of interest: 
         [0000]      ESCF Ei,unkn,nd →EscF( E   i ) unkn,nd   [37a]
 
         [0000]      ESCF Ei,unkn,md →EscF( E   i ) unkn,md   [37b]
 
         [0000]    and computes their associated statistics. 
         [0119]    Software module  2040  in  FIG. 7  evaluates and processes the ‘useful’ discrete sample-specific escaped-fraction values  5784  in  FIG. 6  and its associated statistics to compute a sample-specific escaped-fraction function or interpolated curve  5786 . Only one set of values for discrete sample-specific escaped-fraction values  5782  is illustrated in  FIG. 6  (either EscF Ei,unkn,nd  or ESCF Ei,unkn,md ), because only one of Equations [38a] and [38b] is necessary to determine the signal-emitter quantities (SpA Rj,unkn ). The logical choice is the one that provides the best statistics for SpA Rj,unkn . 
         [0000]    
       
         
           
             
               
                 
                   
                     SpA 
                     
                       Rj 
                       , 
                       unkn 
                     
                   
                   = 
                   
                     
                       CR 
                       
                         Ei 
                         , 
                         unkn 
                         , 
                         nd 
                       
                     
                     
                       
                         M 
                         unkn 
                       
                       * 
                       
                         YF 
                         
                           Rj 
                           , 
                           Ei 
                         
                       
                       * 
                       
                         
                           EscF 
                            
                           
                             ( 
                             
                               E 
                               i 
                             
                             ) 
                           
                         
                         
                           unkn 
                           , 
                           nd 
                         
                       
                       * 
                       
                         
                           DetF 
                            
                           
                             ( 
                             
                               E 
                               i 
                             
                             ) 
                           
                         
                         
                           smplFr 
                           , 
                           nd 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     38 
                      
                     a 
                   
                   ] 
                 
               
             
             
               
                 
                   
                     SpA 
                     
                       Rj 
                       , 
                       unkn 
                     
                   
                   = 
                   
                     
                       CR 
                       
                         Ei 
                         , 
                         unkn 
                         , 
                         md 
                       
                     
                     
                       
                         M 
                         unkn 
                       
                       * 
                       
                         YF 
                         
                           Rj 
                           , 
                           Ei 
                         
                       
                       * 
                       
                         
                           EscF 
                            
                           
                             ( 
                             
                               E 
                               i 
                             
                             ) 
                           
                         
                         
                           unkn 
                           , 
                           md 
                         
                       
                       * 
                       
                         
                           DetF 
                            
                           
                             ( 
                             
                               E 
                               i 
                             
                             ) 
                           
                         
                         
                           smplFr 
                           , 
                           md 
                         
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     38 
                      
                     b 
                   
                   ] 
                 
               
             
           
         
       
     
         [0120]    Software module  2260  in  FIG. 7  searches databases for known signal emitters (R j ) and their characteristic (E i ) yield-fractions (YF Rj,Ei ) that match the spectral peaks arising from unknown-samples. Then, spectral analysis is performed, and the signal-emitters (R j ) and their associated yield-fractions (YF Rj,Ei ) are identified. 
         [0121]    Software module  2272  computes the specific-activity quantities (SpA Rj,unkn ) of the identified signal-emitters and their associated statistics. The interpolated or fitted sample-specific escaped-fraction functions, [EscF(E i ) unkn,nd , EscF(E i ) unkn,md ], are used in place of the discrete-value terms (ESCF Ei,unkn,nd , ESCF Ei,unkn,nd ), except that, in some cases, the operator may use the discrete sample-specific escaped-fraction values. Software module  2272  also identifies which of Equations [38a] or [38b] provides the better statistics for the computed specific-activity values. 
         [0122]    Software module  2276  formats all of the data from each part of the process flow, as just described, for presentation by e.g. spreadsheets, comma-separated-value (CSV) lists, and computer screen presentations. 
         [0123]    The mathematical laws of error analysis are used to compute the statistics for the values associated with each of the terms comprising the count rate balance Equations [33a] and [33b]. Thus, the terms will have the form ν±Δν, where ν represents the numerical value of a particular term in Equations [33a] and [33b], and +Δν is the uncertainty in ν. 
         [0124]    The statistics of the specific-activity quantitation (SpA Rj,unkn ) can be improved by applying the ‘sum and difference’ method (described in the related U.S. patent application Ser. No. 13/049,903) to two or more different-thickness count rates from the unknown-sample (e.g. CR Ei,unkn,nd  and CR Ei,unkn,md ), and by incorporating the new term for the ratio of the reference-beam-derived sample-free detected-fraction calibration (RDetF nd:md ) into the ‘sum-and-difference’ formulas. 
         [0000]    II. “nd: md” Beam-Assisted Multiple-Mass Sample Analysis 
         [0125]    In prior disclosures (described in the related U.S. patent application Ser. No. 13/049,903), a homogeneous sample is ‘shaped’ into at least two different thicknesses within a single sample-container, with respect to one or more detectors. If only one detector is present, the detector (or the sample-container) is repositioned to change the effective thickness of the sample in the direction of the detector. This section discloses new apparatuses and methods that allow for the use of the sample container and detector to be maintained in their same position during a series of multiple countings, but that, for at least two countings, the mass (and hence, the ‘thickness’ of the sample in the direction of the detector) is different, i.e. by adding into, or subtracting from, the already-counted sample mass in the sample-container, in order to perform an additional counting with the new quantity of sample mass. Alternatively, two or more samples of differing mass but the same homogeneous composition can be counted using two or more sample-containers of the same shape, size, and type. Each of such containers is placed into the same position with respect to the detector during each of such individual countings. 
         [0126]    Because the sample-containers that are used to count multiple masses of the same standard-sample composition, are the same in every meaningful respect, then only two ambient background countings—one of an empty sample-container with, and one without, an empty reference-beam positioner—need to be performed in order to improve the statistics of the detection-system signal-detection efficiency calibration. For the same reason, only one sample-free reference-beam counting is needed. All three of these ‘parts’ are performed as taught previously. The teaching in this disclosure begins with a fourth part (Part 4). 
         [0000]    Part 1. “nd: md” Ambient Background Emission Spectrum Acquisition 
         [0127]    The Part-1 system setup, i.e. ambient background emission spectrum acquisition, is disclosed in the related disclosure (Appl. No. 13049903) and will not be taught again here. 
         [0000]    Part 2. “nd: md” Ambient Background With Auxiliary Apparatus ( FIG. 1 ) 
         [0128]    The Part-2 system setup, i.e. “nd: md” Ambient Background with Auxiliary Apparatus ( FIG. 1 ) is taught earlier in this disclosure and will not be taught again here. 
         [0000]    Part 3. “nd: md” Sample-Free Ref-Source Emission Spectrum ( FIG. 2 ) 
         [0129]    The Part-2 system setup, i.e. “nd: md” Sample-Free Ref-Source Emission Spectrum ( FIG. 2 ) acquisition is taught earlier in this disclosure and will not be taught again here. 
         [0000]    Part 4. “nd: md” Beam-Assisted Multiple-Mass Calibration ( FIG. 8 ) 
         [0130]    Before signal detection systems can be used to quantify signal-sources in unknown-samples, they usually first require a signal detection-efficiency calibration of some kind.  FIG. 8  illustrates one such system  5900  for calibrating signal detection-efficiency, where a compositionally well known standard-sample  5916  is filled to a ‘thin’ (nd)  5918  thickness (or depth) relative to the thickness of other standard-samples of the same composition in the same type of sample-container; and placed in the same position relative to the detector, as were the empty sample-containers that were used to acquire the ambient background spectra and sample-free beam spectrum. 
         [0131]    Just above the sample-container that holds the standard-sample is the same signal-emitting reference-source  5914  as that used to produce the sample-free beam spectrum (not shown). The reference-source acts like a beam-source in that only those signals  5924  emitted within the solid angle subtended by the detector are countable. 
         [0132]    Although it is possible to carefully align the reference-source with respect to the standard-sample and detector by many methods, one preferred method is to use a reference-source positioner  5816  that ensures (1) that the sample-containers  5912  and  5962  do not become contaminated or damaged by the reference-source  5914 , and (2) that the reference source is always positioned in the same location with respect to the detector, in order to achieve repeatable results. 
         [0133]    Characteristic signals emanate (dashed lines  5924  and  5974 ) from the signal-emitting reference-source  5914  and a fraction of them pass through the sample-container walls  5908  and  5958 ; sample-container cap  5912  and  5962 , and the standard-sample  5920  and  5970  contained therein. Those reference-source signals within the solid angle subtended by the detector act like a ‘beam’ passing through a ‘slab’ of thickness (nd)  5918  of standard-sample  5920  or a ‘slab’ of thickness (md)  5968  of standard-sample  5970 . Some fraction of the beam passes through the respective standard-sample unattenuated (dashed lines  5924  and  5974 ), and such fraction is called the nd or md sample-specific beam-transmitted-fraction (BmTrnsF Ei,stnd,nd  or BmTrnsF Ei,stnd,md ), respectively. 
         [0134]    The signal detection, processing, preservation, and presentation subsystem  1630  acquires at least three spectral components as a single composite ‘gross’ spectrum (not shown) for each standard-sample counting; among the spectral components are the ambient background (not shown), reference-source beam  5924  and  5974 , and the standard-sample emissions  5926  and  5976 . To remove the ambient background component, subsystem  1630  normalizes the characteristic ambient background spectra to their associated standard-sample counting times, and then subtracts-out such normalized ambient background component spectra from their associated composite ‘gross’ spectra to produce their associated ‘net’ composite spectra (not shown), which are still comprised of at least two spectral components; i.e. the reference-source beam peaks and the standard-sample spectral peaks. 
         [0135]    Once the ‘thin’ (nd) 5918 standard-sample counting is complete, a ‘thick’ (md) 5968 standard-sample can be prepared. There are two cases: In the first case, the cap  5912  to the sample container holding the already counted ‘thin’ (nd) 5918 standard-sample, is opened and additional standard-sample of the same composition is added to the same sample-container, filling the sample-container to create a relatively ‘thick’ (md) 5968 standard-sample, after which the cap to the sample-container is replaced and sealed tightly; and the ‘thick’ (md) 5968 standard-sample counting can begin. 
         [0136]    In the second case, when dealing with standard-samples that may be used in recurring calibrations, two sample-containers of the same shape, size, and type are filled with the same standard-sample composition, but to different thicknesses (which are called, nd  5918  and md  5968 ). In the second case, after the first standard-sample is counted (and it could be either the ‘thin’ or ‘thick’ standard-sample), then the other standard-sample replaces the first in the detection system, and then it is also counted. 
         [0137]    As with the first standard-sample counting, the second and subsequent standard-sample countings (should there be more than two standard-sample thicknesses) have their respective, normalized ambient background spectra subtracted-out from their associated composite ‘gross’ spectra to produce their associated ‘net’ composite spectra (not shown), which are still comprised of at least two spectral components; i.e. the reference-source beam peaks and the standard-sample spectral peaks. 
         [0138]    To determine the sample-free detected-fraction calibration functions for each sample thickness [DetF (E i ) smplFr,nd , DetF (E i ) smplFr,md ], a similar method is carried out as described in the discussion surrounding Equations [2a] to [30b], except the formulas carry an additional mass-ratio factor (RM stnd,nd:md ) that takes into account the ratio of the different masses of counted standard-samples, where: 
         [0000]    
       
         
           
             
               
                 
                   
                     sample 
                      
                     
                         
                     
                      
                     ratio 
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                      
                     
                         
                     
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                       RM 
                       
                         stnd 
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                         , 
                         nd 
                       
                     
                     
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                         stnd 
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                   [ 
                   39 
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                      
                     
                       : 
                     
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                       σ 
                       
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         [0000]    Part 5. “nd: md” Beam-Assisted Multiple-Mass Sample Quantitation 
         [0139]    It is presumed that the sample-free detected-fraction calibrations for each thickness of standard-sample have been determined (e.g., DetF Ei,smplFr,nd  and DetF Ei,smplFr,md ), after which analysis of one or more unknown-samples can begin. 
         [0140]    An unknown-sample composition is prepared in one or more sample-containers to different depths and counted in a manner similar to that just outlined for the standard-sample, but without any external reference sources (i.e. only the unknown-sample itself is deliberately counted). 
         [0141]    The processing of unknown-sample spectra is similar to that outlined in the first section of this disclosure for quantitating characteristic signal emitters in unknown-samples, except that in this case, to compute the specific-activity quantities (SpA Rj,unkn ) of the signal emitters (R i ) in the unknown-sample, the formulas carry an additional mass-ratio factor (RM unkn,nd:md ) that takes into account the ratio of the different masses of counted unknown-sample, where: 
         [0000]    
       
         
           
             
               
                 
                   
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       Reference-Source Positioners for Wrap-Around Type Containers (FIG. 9) 
       [0142]      FIG. 9  illustrates  6000  apparatuses  6004 ,  6012 , and  6066 . Double-sided ‘nd: md’ wrap-around sample-container apparatus  6004  is described in detail in the related U.S. patent application Ser. No. 13/049,903. In brief, the wrap-around sample-container  6004  consists of one side that forms the sample  6032  into a thin (nd,  6060 ) thickness with respect to the other side that forms the sample  6032  into a thicker (md,  6010 ) thickness when the wrap-around sample-container apparatus  6004  is flipped 180 degrees. Both sides of the wrap-around sample-container apparatus  6004  have an inside diameter that wraps around the detector when in one of the two (nd or md) sample counting positions. When in the thin (nd,  6060 ) counting position, some portion of the emitted characteristic signals  6026  from the sample are detected by the detector. When in the thick (md,  6010 ) counting position, some portion of the emitted characteristic signals  6076  from the sample are detected by the detector. 
         [0143]    The narrow-diameter reference-source positioner  6012  consists of a base  6016 ; an elevated ridge  6022  that positions and secures the reference source  6014  in the center; a hollowed out section in the center consisting of a thin light-element (“low-Z”) ‘window’  6024  to improve the transparency to the characteristic signals  6028  emanated by the reference-source  6014  in the direction of the standard-sample  6032  and detector; and raised tabs  6018  (&#39;rabbit ears&#39;) that allow an operator to physically insert and remove the reference-source positioner from the recessed narrow- or wide-diameter portions of the wrap-around sample-container  6004 . The rabbit ears  6018  may consist of any number of shapes to facilitate inserting and removing the reference-source positioner. 
         [0144]    When the container is in the thin (nd,  6060 ) counting position, the narrow-diameter reference-source positioner  6012  can be sized to snugly fit into the small-diameter recess of the wrap-around sample-container  6004 . When the wrap-around sample-container  6004  is in thick (md,  6010 ) counting position, then a reference-source positioner ring (“positioner ring”  6066 ) of the proper inner and outer diameters positions the narrow-diameter reference-source positioner  6012  in the center of the wide-diameter recess of the wrap-around sample-container. Alternatively, an additional wide-diameter reference-source positioner (not shown in  FIG. 9 ) can be used to snugly fit into the wide-diameter recess of the wrap-around sample-container  6004  without the need for a positioner ring. 
         [0145]    The reference-source  6014 , reference-source positioner  6012 , and positioner ring  6066  work combine in part or in whole to ensure repeatable reference-source beam lines that pass through the sample and allow computation of the linear attenuation coefficient of the sample composition and of the ratio of nd and md beam-derived sample-free detected-fraction calibration values (RDetF Eti,bm,nd:md ) as disclosed by Equations [20a] and [20b]. 
       Reference-Source Setup for Distant Signal Measurement (FIG. 10) 
       [0146]      FIG. 10  illustrates  6100  a plurality of reference sources  6112 ,  6114 , and  6116  at different distances from a plurality of detectors. The reference sources permit computing the ratio of nd, md, and ad beam-derived sample-free detected-fraction calibration values in the process disclosed by Equations [20a] and [20b]. The reference sources may be recessed into chambers (not shown) with electronically operated shutters or doors (not shown) that block the signals emitted from the reference sources in the direction of the plurality of detectors.