Patent Publication Number: US-7587026-B2

Title: Systems and methods for determining a packing fraction of a substance

Description:
BACKGROUND OF THE INVENTION 
     This invention relates generally to systems and methods for determining a type of substance and more particularly to systems and methods for determining a packing fraction of the substance. 
     The events of Sep. 11, 2001 instigated an urgency for more effective and stringent screening of airport baggage. The urgency for security expanded from an inspection of carry-on bags for knives and guns to a complete inspection of checked bags for a range of hazards with particular emphasis upon concealed explosives. X-ray imaging is a widespread technology currently employed for screening. However, existing x-ray baggage scanners, including computed tomography (CT) systems, designed for detection of explosive and illegal substances are unable to discriminate between harmless materials in certain ranges of density and threat materials like plastic explosive. 
     A plurality of identification systems based on a plurality of x-ray diffraction (XRD) techniques provide an improved discrimination of materials compared to that provided by the x-ray baggage scanners. The XRD identification systems measure a plurality of d-spacings between a plurality of lattice planes of micro-crystals in materials. 
     However, the XRD identification systems for explosives detection and baggage scanning are not yet highly developed. Moreover, the diffraction techniques suffer from a false alarm problem for some classes of substances. There are certain types of explosives in which an explosive component cannot be identified by the XRD identification systems and the lack of identification leads to a high false alarm rate. 
     BRIEF DESCRIPTION OF THE INVENTION 
     In one aspect, a method for determining a type of a substance is described. The method includes determining a packing fraction of the substance from a molecular interference function. 
     In another aspect, a processor for determining a type of a substance is described. The processor is configured to determine a packing fraction of the substance from a molecular interference function. 
     In yet another aspect, a system for determining a type of substance is described. The system includes an x-ray source configured to generate x-rays, a detector configured to receive primary and scattered radiation after the x-rays pass through the substance, and a processor configured to determine a packing fraction of the substance from a molecular interference function. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block diagram of a system for determining a packing fraction of a substance. 
         FIG. 2  is a block diagram of an embodiment of the system of  FIG. 1 . 
         FIG. 3  is a flowchart of an embodiment of a method for determining a packing fraction of a substance. 
         FIG. 4  shows a diffraction profile generated by a processor of the system of  FIG. 2 . 
         FIG. 5  shows a dotted line and a solid curve generated by the processor of the system of  FIG. 2 . 
         FIG. 6  is a continuation of the flowchart of  FIG. 3 . 
         FIG. 7  is a continuation of the flowchart of  FIG. 6 . 
         FIG. 8  shows an independent atom model curve generated by applying the method of  FIGS. 3 ,  6  and  7 . 
         FIG. 9  shows an embodiment of a molecular transfer function and an embodiment of an approximation function generated by applying the method of  FIGS. 3 ,  6  and  7 . 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       FIG. 1  is a block diagram of a system  10  for determining a packing fraction of a substance. System  10  includes an x-ray source  12 , a primary collimator  14 , a secondary collimator (Sec collimator)  16 , and a detector  18 . Detector  18  includes a central detector element  20  or a central detector cell for detecting primary radiation. Detector  18  also includes a plurality of detector cells or detector elements  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  for detecting coherent scatter. Detector  18  includes any number, such as, ranging from and including 256 to 1024, of detector elements. A container  38  is placed on a support  40  between x-ray source  12  and detector  18 . Examples of container  38  include a bag, a box, and an air cargo container. Examples of x-ray source  12  include a polychromatic x-ray tube. Container  38  includes a substance  42 . Examples of substance  42  include an organic explosive, an amorphous substance having a crystallinity of less than twenty five percent, a quasi-amorphous substance having a crystallinity at least equal to twenty-five percent and less than fifty percent, and a partially crystalline substance having a crystallinity at least equal to fifty percent and less than one-hundred percent. Examples of the amorphous, quasi-amorphous, and partially crystalline substances include a gel explosive, a slurry explosive, an explosive including ammonium nitrate, and a special nuclear material. Examples of the special nuclear material include plutonium and uranium. Examples of support  40  include a table and a conveyor belt. An example of detector  18  includes a segmented detector fabricated from Germanium. 
     X-ray source  12  emits x-rays in an energy range, which is dependent on a voltage applied by a power source to x-ray source  12 . Using primary collimator  14 , a primary beam  44 , such as a pencil beam, is formed from the x-rays generated. Primary beam  44  passes through container  38  arranged on support  40  to generate scattered radiation, such as a plurality of scattered rays  46 ,  48  and  50 . Underneath support  40 , there is arranged detector  18 , which measures an intensity of primary beam  44  and photon energy of the scattered radiation. Detector  18  measures the x-rays in an energy-sensitive manner by outputting a plurality of electrical output signals linearly dependent on a plurality of energies of x-ray quanta detected from within primary beam  44  and the scattered radiation. 
     Detector elements  20 ,  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  are geometrically arranged so that a scatter angle or alternatively an incident angle of the scatter radiation detected by each detector element  20 ,  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  is constant. For example, an incident angle  52  at which scattered ray  46  is incident on detector element  30  is equal to an incident angle  54  at which scattered ray  48  is incident on detector element  34  and incident angle  54  is equal to an incident angle  56  at which scattered ray  50  is incident on detector element  36 . As another example, scattered ray  46  is parallel to scattered rays  48  and  50 . Central detector element  20  measures an energy or alternatively an intensity of primary beam  44  after primary beam  44  passes through container  38 . Detector elements  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  separately detect the scattered radiation received from container  38 . 
     Secondary collimator  16  is located between support  40  and detector  18 . Secondary collimator  16  includes a number of collimator elements, such as sheets, slits, or laminations, to ensure that the scatter radiation arriving at detector  18  have constant scatter angles with respect to primary beam  44  and that a position of detector  18  permits a depth in container  38  at which the scatter radiation originated to be determined. The number of collimator elements provided is equal to or alternatively greater than a number of detector elements  20 ,  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  and the collimator elements are arranged such that the scattered radiation between neighboring collimator elements each time is incident on one of the detector elements  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36 . The collimator elements are made of a radiation-absorbing material, such as, a copper alloy or a silver alloy. In one embodiment employing a fan-beam geometry, a plurality of origination points, within container  38 , of the scatter radiation are detected by the detector elements  22 ,  24 ,  26 , and  28 , aligned in a first direction and detector elements  30 ,  32 ,  34 , and  36  aligned in a second direction opposite to and parallel to the first direction. Examples of the constant scatter angle values include values ranging from 0.1 degrees for a high-energy device, such as an x-ray tube radiating x-ray photons having an energy of 1 mega electronvolts (MeV) to four degrees for low-energy systems, such as an x-ray tube radiating x-ray photons having an energy of 150 kilo electronvolts (keV). Detector  18  detects the scattered radiation to generate a plurality of electrical output signals. In an alternative embodiment, system  10  does not include primary and secondary collimators  14  and  16 . 
       FIG. 2  is a block diagram of an embodiment of a system  100  for determining a packing fraction of a substance  42 . System  100  includes central detector element  20 , detector elements  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36 , a plurality of pulse-height shaper amplifiers (PHSA)  102 ,  104 ,  106 ,  108 ,  110 ,  112 ,  114 ,  116 , and  118 , a plurality of analog-to-digital (A-to-D) converters  120 ,  122 ,  124 ,  126 ,  128 ,  130 ,  132 ,  134 , and  136 , a plurality of spectrum memory circuits (SMCs)  138 ,  140 ,  142 ,  144 ,  146 ,  148 ,  150 ,  152 , and  154  allowing pulse height spectra to be acquired, a plurality of correction devices (CDs)  156 ,  158 ,  160 ,  162 ,  164 ,  166 ,  168 , and  170 , a processor  190 , an input device  192 , a display device  194 , and a memory device  195 . As used herein, the term processor is not limited to just those integrated circuits referred to in the art as a processor, but broadly refers to a computer, a microcontroller, a microcomputer, a programmable logic controller, an application specific integrated circuit, and any other programmable circuit. The computer may include a device, such as, a floppy disk drive or CD-ROM drive, for reading data including the methods for determining a packing fraction of a substance from a computer-readable medium, such as a floppy disk, a compact disc—read only memory (CD-ROM), a magneto-optical disk (MOD), or a digital versatile disc (DVD). In another embodiment, processor  190  executes instructions stored in firmware. Examples of display device  194  include a liquid crystal display (LCD) and a cathode ray tube (CRT). Examples of input device  192  include a mouse and a keyboard. Examples of memory device  195  include a random access memory (RAM) and a read-only memory (ROM). An example of each of correction devices  156 ,  158 ,  160 ,  162 ,  164 ,  166 ,  168 , and  170  include a divider circuit. Each of spectrum memory circuits  138 ,  140 ,  142 ,  144 ,  146 ,  148 ,  150 ,  152 , and  154  include an adder and a memory device, such as a RAM or a ROM. 
     Central detector element  20  is coupled to pulse-height shaper amplifier  102 , and detector elements  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  are coupled to pulse-height shaper amplifiers  104 ,  106 ,  108 ,  110 ,  112 ,  114 ,  116 , and  118 , respectively. Central detector element  20  generates an electrical output signal  196  by detecting primary beam  44  and detector elements  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  generate a plurality of electrical output signals  198 ,  200 ,  202 ,  204 ,  206 ,  208 ,  210 , and  212  by detecting the scattered radiation. For example, detector element  22  generates electrical output signal  198  for each scattered x-ray photon incident on detector element  22 . Each pulse-height shaper amplifier amplifies an electrical output signal received from a detector element. For example, pulse-height shaper amplifier  102  amplifies electrical output signal  196  and pulse-height shaper amplifier  104  amplifies electrical output signal  198 . Pulse-height shaper amplifiers  102 ,  104 ,  106 ,  108 ,  110 ,  112 ,  114 ,  116 , and  118  have a gain factor determined by processor  190 . 
     An amplitude of an electrical output signal output from a detector element is proportional to an integrated intensity of an x-ray quantum that is detected by the detector element to generate the electrical output signal. For example, an amplitude of electrical output signal  196  is proportional to an integrated intensity of an x-ray quantum in primary beam  44  detected by detector element  20 . On the other hand, an amplitude of electrical output signal  198  is proportional to an integrated intensity of an x-ray quantum within the scattered radiation that is detected by detector element  22 . 
     A pulse-height shaper amplifier generates an amplified output signal by amplifying an electrical output signal generated from a detector element. For example, pulse-height shaper amplifier  102  generates an amplified output signal  214  by amplifying electrical output signal  196  and pulse-height shaper amplifier  104  generates an amplified output signal  216  by amplifying electrical output signal  198 . Similarly, a plurality of amplified output signals  218 ,  220 ,  222 ,  224 ,  226 ,  228 , and  230  are generated. An analog-to-digital converter converts an amplified output signal from an analog form to a digital form to generate a digital output signal. For example, analog-to-digital converter  120  converts amplified output signal  214  from an analog form to a digital format to generate a digital output signal  232 . Similarly, a plurality of digital output signals  234 ,  236 ,  238 ,  240 ,  242 ,  244 ,  246 , and  248  are generated by analog-to-digital converters  122 ,  124 ,  126 ,  128 ,  130 ,  132 ,  134 , and  136 , respectively. A digital value of a digital output signal generated by an analog-to-digital converter represents an amplitude of energy or alternatively an amplitude of intensity of a pulse of an amplified output signal. Each pulse is generated by an x-ray quantum, such as an x-ray photon. For example, a digital value of digital output signal  234  output by analog-to-digital converter  122  is a value of an amplitude of a pulse of amplified output signal  216 . 
     An adder of a spectrum memory circuit adds a number of pulses in a digital output signal. For example, when analog-to-digital converter  122  converts a pulse of amplified output signal  216  into digital output signal  234  to determine an amplitude of the pulse of amplified output signal  216 , an adder within spectrum memory circuit  140  increments, by one, a value within a memory device of spectrum memory circuit  140 . Accordingly, at an end of an x-ray examination of substance  42 , a memory device within a spectrum memory circuit stores a number of x-ray quanta detected by a detector element. For example, a memory device within spectrum memory circuit  142  stores a number of x-ray photons detected by detector element  24  and each of the x-ray photons has an amplitude of energy or alternatively an amplitude of intensity that is determined by analog-to-digital converter  124 . 
     A correction device receives a number of x-ray quanta that have a range of energies and are stored within a memory device of one of spectrum memory circuits  140 ,  142 ,  144 ,  146 ,  148 ,  150 ,  152 , and  154 , and divides the number by a number of x-ray quanta having the range of energies received from a memory device of spectrum memory circuit  138 . For example, correction device  156  receives a number of x-ray photons having a range of energies from a memory device of spectrum memory circuit  140 , and divides the number by a number of x-ray photons having the range received from a memory device of spectrum memory circuit  138 . Each correction device outputs a correction output signal that represents a range of energies within x-ray quanta received by a detector element. For example, correction device  156  outputs a correction output signal  280  representing an energy spectrum or alternatively an intensity spectrum within x-ray quanta detected by detector element  22 . As another example, correction device  158  outputs correction output signal  282  representing an energy spectrum within x-ray quanta detector element  24 . Similarly, a plurality of correction output signals  284 ,  286 ,  288 ,  290 ,  292 , and  294  are generated by correction devices  160 ,  162 ,  164 ,  166 ,  168 , and  170 , respectively. 
     Processor  190  receives correction output signals  280 ,  282 ,  284 ,  286 ,  288 ,  290 ,  292 , and  294  to generate a momentum transfer x, measured in inverse nanometers (nm −1 ), from an energy spectrum r(E) of energy E of x-ray quanta within the scattered radiation detected by detector  18 . Processor  190  generates the momentum transfer x by applying
 
 x =( E/hc )sin(θ/2)  (1)
 
     where c is a speed of light, h is Planck&#39;s constant, θ represents constant scatter angles of x-ray quanta of the scattered radiation detected by the detector  18 . Processor  190  relates the energy E to the momentum transfer x by equation (1). Mechanical dimensions of the secondary collimator  16  define the scatter angle θ. The secondary collimator  16  restricts the scatter radiation that does not have the angle θ. Processor  190  receives the scatter angle θ from a user via input device  192 . 
     It is noted that a number of pulse-height shaper amplifiers  102 ,  104 ,  106 ,  108 ,  110 ,  112 ,  114 ,  116 , and  118  changes with a number of detector elements  20 ,  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36 . For example, five pulse-height shaper amplifiers are used for amplifying signals received from five detector elements. As another example, four pulse-height shaper amplifiers are used for amplifying signals received from four detector elements. Similarly, a number of analog-to-digital converters  120 ,  122 ,  124 ,  126 ,  128 ,  130 ,  132 ,  134 , and  136  changes with a number of detector elements  20 ,  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36  and a number of spectrum memory circuits  138 ,  140 ,  142 ,  144 ,  146 ,  148 ,  150 ,  152 , and  154  changes with the number of detector elements  20 ,  22 ,  24 ,  26 ,  28 ,  30 ,  32 ,  34 , and  36 . 
       FIG. 3  is a flowchart of an embodiment of a method for determining a packing fraction of a substance and  FIG. 4  shows a graph  400  or a diffraction profile D(x) generated  401  by processor  190 . Graph  400  is a histogram having a plurality of intensity values at a plurality of momentum transfer values, such as x 1 , x 2 , and x 3 , of the momentum transfer x. As an example, when an operating voltage of x-ray source  12  is 160 kilovolts, processor  190  calculates, by applying equation 1, an energy value E 1  of the energy E to be 160 keV, calculates, by applying equation 1, an energy value E 2  of the energy E to be 140 keV, and calculates, by applying equation 1, an energy value E 3  of the energy value E to be photon energy 120 keV. In the example, the photon energy values E 1 , E 2 , and E 3  correspond, through equation 1, to x 1  of four inverse nanometers, x 2  of 3.5 inverse nanometers, and to X 3  of three inverse nanometers, respectively. Graph  400  represents a histogram of a number of x-ray photons detected by detector  18  versus the momentum transfer x of the x-ray photons. A number of photons detected by detector  18  is plotted along an ordinate  402  and the momentum transfer x is plotted along an abscissa  404 . As an example, abscissa  404  extends from and includes zero inverse nanometers to at most 10 inverse nanometers. An example of a total number of bins of numbers of x-ray photons plotted on ordinate  402  lies between 64 and 1024. An example of a number of x-ray photons detected by detector  18  per examination lies between 1000 and 100,000. 
     The diffraction profile ranging from x≧3 nm −1  is dominated by coherent scatter from free atoms of substance  42 . In a tip region, extending from x 1  to x 3 , of graph  400 , an intensity of the scattered radiation is proportional to a product of density, such as a mean density, of substance  42  and a power, such as ranging between 2.5 and 3.5, of a mean atomic number of a plurality of materials within substance  42 . 
     A cumulative number of photons that are scattered with momentum transfer values between x 1  and x 2  are represented within a band  408  under graph  400 . Processor  190  determines a cumulative number of x-ray photons within band  408  by cumulatively summing a number of photons between momentum transfer values x 1  and x 2  on abscissa  404 . A cumulative number of photons that are scattered with momentum transfer values between x 2  and X 3  are located within a band  410  under graph  400 . Processor  190  determines a cumulative number of x-ray photons within band  410  by cumulatively summing a number of x-ray photons between momentum transfer values x 2  and X 3  on abscissa  404 . 
       FIG. 5  shows a dotted line  450  and a solid curve  452  generated by processor  190 . Solid curve  452  represents a theoretical relationship between a ratio of total free atom scatter cross-sections, referred to as total scatter cross-sections or cumulative scatter cross-sections, and an atomic number Z. As an example, processor  190  plots solid curve  452  from an example of the theoretical relationship mentioned in Hubbell, J. H., Veigele, W. J., Briggs, E. A., Brown, R. T., Cromer, D. T., Howerton, R. J., Atomic Form Factors, Incoherent Scattering Functions and Photon Scattering Cross-sections, Journal of Physics and Chemical Reference Data, Volume 4, page 471 (1975), Erratum: Atomic Form Factors, Incoherent Scattering Functions, and Photon Scattering Cross Sections, Journal of Physics and Chemical Reference Data, Volume 6, page 615 (1977). As another example, the theoretical relationship includes an atomic number value of oxygen as eight corresponding to a ratio of 0.68 of total scatter cross-sections calculated for oxygen. As yet another example, the theoretical relationship includes an atomic number value of carbon as six corresponding to a ratio of 0.73 of total scatter cross-sections calculated from carbon. As still another example, processor  190  calculates a ratio of a total scatter cross-section of hydrogen at the momentum transfer value x 3  and a total scatter cross-section of hydrogen at the momentum transfer value x 2 , and plots the ratio on solid curve  452 . As another example, processor  190  calculates a ratio of a total scatter cross-section of flourine at the momentum transfer value x 2  and a total scatter cross-section of flourine at the momentum transfer value x 1 , and plots the ratio on solid curve  452 . As yet another example, processor  190  calculates a ratio of a total scatter cross-section of carbon at the momentum transfer value x 2  and a total scatter cross-section of carbon at the momentum transfer value x 1 , and plots the ratio on solid curve  452 . Processor  190  generates dotted line  450  as a linear fit or linear regression to the theoretical relationship. 
     A plurality of ratios of total scatter cross-sections are plotted along an ordinate  454  and a plurality of atomic numbers Z are measured along an abscissa  456 . For example, a plurality of atomic number values on dotted line  450  extend from an atomic number one of hydrogen to an atomic number nine of flourine and a plurality of ratios of total scatter cross-sections evaluated at momentum transfer values within a first set of regions of bands  408  and  410  and total scatter cross-sections evaluated at momentum transfer values within a second set of regions of bands  408  and  410 . 
     Processor  190  calculates a ratio of cumulative numbers of x-ray photons within bands  408  and  410 . For example, processor  190  determines that R 1  is a ratio of a cumulative number of x-ray photons within band  408  to a cumulative number of x-ray photons within band  410 . Processor  190  determines  458 , by using the solid curve  452 , an effective atomic number Z eff  corresponding to a ratio of a cumulative number of x-ray photons within band  408  and a cumulative number of x-ray photons within band  410 . As an example, processor  190  perpendicularly extends a horizontal line from the ratio R 1  to intersect solid curve  452  at an intersection point  460  and extends a line from intersection point  460  to perpendicularly intersect abscissa  456  at an effective atomic number value Z eff1 . Alternatively, processor  190  determines, by using the dotted line  450 , the effective atomic number Z eff  corresponding to a ratio of a cumulative number of x-ray photons within band  408  and a cumulative number of x-ray photons within band  410 . As an example, processor  190  perpendicularly extends a horizontal line from the ratio R 1  to intersect dotted line  450  at an intersection point and extends a line from the intersection point to perpendicularly intersect abscissa  456  at an effective atomic number value Z eff2 . 
     Processor  190  determines a type or a kind, such as uranium, carbon, oxygen, or plutonium, of substance  42  based on the effective atomic number Z eff , such as Z eff1 , determined from a ratio of cumulative numbers of x-ray photons. For example, processor  190  determines that substance  42  is carbon upon determining that an effective atomic number value 6 corresponds to a ratio of 0.73 of cumulative numbers of x-ray photons detected by detector  18 . Alternatively, processor  190  determines a type or a kind, such as uranium, carbon, oxygen, or plutonium, of substance  42  based on the effective atomic number value Zeff 2  determined from a ratio of cumulative numbers of x-ray photons. 
       FIGS. 6 and 7  are a flowchart of an embodiment of a method for determining a packing fraction of a substance,  FIG. 8  shows an embodiment of an independent atom model (IAM) curve  500  generated by processor  190 , and  FIG. 9  shows a plurality of embodiments of a plurality of graphs s(x) and I(x) generated by processor  190 . The graph s(x) represents a molecular interference function and the graph I(x) represents an approximation function. Processor  190  removes 502 a plurality of crystalline interference peaks from graph  400  by applying a peak removal algorithm. An example of the peak removal algorithm is provided in a software, such as an “OptiFit” computer software, described in Rabiej M, Determination of the Degree of Crystallinity of Semicrystalline Polymers by Means of the “OptiFit” Computer Software, POLIMERY 6, pages 423-427 (2002). In an alternative embodiment, processor  190  removes all crystalline interference peaks that represents a crystallinity of substance  42  and that are located within the diffraction profile D(x) by applying the peak removal algorithm. For example, in case of quasi-amorphous or alternatively partially crystalline substances, a plurality of crystalline interference peaks may be included within graph  400  and processor  190  removes the crystalline interference peaks by applying the peak removal algorithm. The peak removal algorithm is applied to generate a peak-removed graph, such as graph  400 . 
     Processor  190  determines  506  a total scatter cross-section of IAM curve  500  from the effective atomic number Z eff  that is illustrated in  FIG. 5  and that is determined from the scattered radiation. For example, upon determining by processor  190  that the effective atomic number value Z eff1  is a rational number, such as 6.3, processor  190  generates a weighted average of a plurality of IAM functions corresponding to neighboring atomic numbers six and seven. In the example, processor  190  generates the weighted average, such as ⅓ [IAM( 6 )]+⅔[IAM(7)], where IAM(6) is a total scatter cross-section for carbon and IAM(7) is a total scatter cross-section for nitrogen. An example of the IAM functions corresponding to neighboring atomic numbers are available in Hubbell, J. H., Veigele, W. J., Briggs, E. A., Brown, R. T., Cromer, D. T., Howerton, R. J., Atomic Form Factors, Incoherent Scattering Functions and Photon Scattering Cross-sections, Journal of Physics and Chemical Reference Data, Volume 4, page 471 (1975), Erratum: Atomic Form Factors, Incoherent Scattering Functions, and Photon Scattering Cross Sections, Journal of Physics and Chemical Reference Data, Volume 6, page 615 (1977). The weighted average is an example of a total scatter cross-section, determined in  506 , of LAM curve  500 . 
     Alternatively, instead of generating the weighted average, upon determining by processor  190  that the effective atomic number value Z eff1  is the rational number, processor  190  generates a closest total scatter cross-section of an IAM curve corresponding to an atomic number value, which is an integer closest to the rational number and plots, with respect to y-axis  402 , the closest total scatter cross-section. In yet another alternative embodiment, instead of generating the weighted average, upon determining by processor  190  that the effective atomic number value Z eff1  is the rational number, processor  190  generates a universal total scatter cross-section of an IAM curve by scaling the momentum transfer x of IAM curve  500  in  FIG. 8 . As an example, abscissa  404  in  FIG. 8  is scaled by multiplying the momentum transfer x of IAM curve  500  with 0.02Z eff1 +0.12 to generate the universal total scatter cross-section. 
     Processor  190  multiplies  507  a total scatter cross-section, determined in  506 , by an initial amplitude or an initial height to generate a first iteration cycle free atom curve. For example, processor  190  multiplies each value of a total scatter cross-section, determined in  506 , with the initial height to generate the first iteration cycle free atom curve. Processor  190  receives the initial height from the user via input device  192 . Processor  190  calculates 508 the molecular interference function s(x) by dividing a number of x-ray photons represented by graph  400  by the first iteration cycle free atom curve. As an example, processor  190  generates a molecular interference value s 1 (x) of the molecular interference function s(x) by dividing a number of x-ray photons having the momentum transfer value x 1  that lies on graph  400  by a number of x-ray photons having the momentum transfer value x 1  that lies on the first iteration cycle free atom curve. As another example, processor  190  generates a molecular interference value s 2 (x) of the molecular interference function s(x) by dividing a number of x-ray photons having the momentum transfer value x 2  that lies on graph  400  by a number of x-ray photons having the momentum transfer value x 2  that lies on the first iteration cycle free atom curve. 
     Processor  190  calculates 512 the approximation function I(x) as
 
 I ( x )=[ s ( x )−1]2  (2)
 
     Processor  190  determines  513  a next iteration cycle amplitude I min  or a next iteration cycle height of IAM curve  500  by minimizing an integral of I(x) represented as 
     
       
         
           
             
               
                 
                   
                     ∫ 
                     0 
                     
                       x 
                       max 
                     
                   
                   ⁢ 
                   
                     
                       I 
                       ⁡ 
                       
                         ( 
                         x 
                         ) 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ⅆ 
                       x 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where x max  is the largest value of x on abscissa  404  of graph  400  and IAM curve  500 . For example, processor  190  determines the next iteration cycle height I min  by selecting a minimum from a first and a second calculated value. Processor  190  determines the first calculated value by applying  507 ,  508 ,  512 , and equation (3) to the initial height. Processor  190  determines the second calculated value by applying  507 ,  508 ,  512 , and equation (3) to a changed height instead of the initial height. For example, processor  190  multiplies a total scatter cross-section, determined in  506 , by the changed height to generate a second iteration cycle free atom curve, calculates the molecular interference function s(x) by dividing a number of x-ray photons represented by graph  400  by the second iteration cycle free atom curve, calculates the approximation function I(x) from equation (2), and generates the second calculated value by applying equation (3). Processor  190  generates the changed height by modifying, such as incrementing or decrementing, the initial height. As another example, processor  190  determines the next iteration cycle height I min  by selecting a minimum from a plurality, such as three, of calculated values, such as the first calculated value, the second calculated value, and a third calculated value. Processor  190  generates the third calculated value in a similar manner in which first and second calculated values are generated. For example, processor  190  generates the third calculated value after incrementing or alternatively decrementing the changed height. 
     Processor  190  determines  514  a second moment X2S of I(x) by applying 
     
       
         
           
             
               
                 
                   
                     X 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                     ⁢ 
                     S 
                   
                   = 
                   
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                       ⁢ 
                       
                         
                           x 
                           2 
                         
                         ⁢ 
                         
                           
                             I 
                             min 
                           
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           x 
                         
                       
                     
                     
                       
                         ∫ 
                         0 
                         ∞ 
                       
                       ⁢ 
                       
                         
                           
                             I 
                             min 
                           
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ⅆ 
                           x 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Processor  190  determines  516  a packing fraction η of substance  42  as being linearly proportional, such as equal, to the second moment X2S. The packing fraction η is linearly proportional to the second moment X2S when substance  42  includes a plurality of identical hard spheres. An example of the linearly proportional relationship includes
 
η= a ( X 2 S )  (5)
 
     where a is a coefficient received by processor  190  via input device  192  from the user, a ranges from and including 0.1 to 0.2. 
     Technical effects of the herein described systems and methods include determining a packing fraction of substance  42  to identify a kind of substance  42 . Other technical effects include determining whether substance  42  is a special nuclear material. 
     While the invention has been described in terms of various specific embodiments, those skilled in the art will recognize that the invention can be practiced with modification within the spirit and scope of the claims.