Abstract:
A system and method for rapidly analyzing elemental abundances in rock or soil samples ( 14 ) under field conditions. The system uses a portable neutron source ( 12 ) to allow neutron activation analysis of elements having identifiable radioactive decay characteristics. A radiation detector ( 18 ) detects radiation released by the sample ( 14 ) and provides radiation testing results to an amplifier ( 26 ) for computing the concentration of trace elements in the sample with a high degree of accuracy.

Description:
RELATED APPLICATION 
     The present application is a divisional of U.S. patent application Ser. No. 09/265,723, entitled “Field Analysis of Geological Samples Using Delayed Neutron Activity Analysis”, filed Mar. 9, 1999, which is a continuation-in-part of U.S. patent application Ser. No. 08/890,187, entitled “System and Method for Field Elemental Analysis by Neutron Activation”, filed Jul. 9, 1997 now abandoned. 
    
    
     TECHNICAL FIELD OF THE INVENTION 
     This invention relates in general to the field of mineralogical exploration and mining analysis, including the practice of analyzing field samples to determine the presence of a selected substance, and, more particularly, to a field portable system using neutron activation for detecting a desired element. 
     BACKGROUND OF THE INVENTION 
     The mineral exploration industry dedicates substantial resources to finding mineral deposits with concentrations of desired minerals that are large enough to allow economic extraction of desired minerals. To find such deposits, mining companies typically send individual technicians to field sites to take samples. Field sites are initially selected by indirect indications of deposits which may be obtained from geological indications such as visible host rock alteration or structural features. Technicians take rock and soil samples from each field site, label the samples according to their original location, and transport them to a laboratory for analysis. Laboratories use exact measurement techniques to determine the presence of desired elements in each sample. For instance, techniques such as mass spectrometry can precisely determine the chemical makeup of each sample. Based upon this laboratory analysis, a mining company may determine the feasibility of commercial extraction of the mineral resource. Often, however, additional testing of field sites having promising samples must occur to support the commercial exploitation decision. 
     Recent advances in the efficiency of extraction technologies allow economic extraction of certain precious metals from extremely low concentrations of mineral deposits. For example, the mineral exploration industry can now economically extract gold from subsurface deposits having gold concentrations of only tens of parts per million. These new technologies can also economically extract gold in open pit deposits in concentrations of less than one part per million. Thus, analysis of field samples should provide detection ranges for gold in the low parts per billion detection range. The industry may need similarly precise measurements for other elements. 
     Although laboratory analysis of field soil and rock samples provides accurate detection capability for trace concentrations of precious metals, laboratory analysis imposes a number of disadvantages. For instance, once a sample is taken from a field site, it may need to be transported long distances to an appropriate laboratory for analysis. En route or during transportation, the sample may be mislabeled or misidentified, leading to inaccurate results. From the time of shipping to the time at which results are available from the laboratory, several weeks may pass. Also, laboratory analysis typically includes input by third parties unrelated to the mineral exploration company, which can lead to security leaks when a concentration of a valuable element is discovered. In the time that it takes to obtain laboratory results, the mineral rights to a valuable property may be lost to commercial espionage. 
     SUMMARY OF THE INVENTION 
     In accordance with the present invention, a system and method is provided that substantially eliminates or reduces disadvantages and problems associated with previously developed methods for analyzing field rock and soil samples. The field sample is irradiated ex situ by exposure to a portable neutron source. The activated sample is then analyzed by a radiation detector to determine the spectrum of radiation it releases. The system compares the radiation spectrum against the spectrum that would occur with known concentrations of selected isotopes, allowing a determination of the concentration of the selected isotope in the sample. 
     More specifically, the present invention uses a portable neutron source, such as a moderator assembly or a subcritical reactor. Examples of potential neutron sources include californium 252, americium-beryllium, plutonium-beryllium, radium-beryllium. The moderator assembly is comprised of a moderator, a neutron source, an irradiation chamber. The subcritical reactor is comprised of the same elements with the addition of a fissile element such as uranium. The method includes placing a field sample in the irradiation chamber long enough to allow absorption of an adequate number of neutrons by nuclides of the desired element. The sample is then removed from the irradiation chamber and placed in a separate gamma ray detector, such as a high purity intrinsic germanium semiconductor detector, to allow detection of gamma ray radiation produced by the activated sample. The gamma ray detector is isolated from the neutron source to avoid interference from radiation produced by the neutron source. In one embodiment, the detector is placed in a detector shield to prevent interference from environmental gamma ray radiation. An analyzer, such as a multichannel analyzer and spectroscopy amplifier operating with a personal computer, provides the results of the gamma ray detector to a technician who can determine the presence and concentration of if selected elements in the field sample by determining the concentration of selected isotopes that decay and if produce gamma rays having predictable characteristics. 
     The present invention provides important technical advantages for mineral exploration. For instance, the present invention allows timely determinations of the concentration of trace elements in field samples by avoiding delays inherent to laboratory analysis, such as those caused by transport. Detection accuracy at field sites is adequate to allow determinations of the feasibility for commercial exploration of a given mineral resource. The accuracy can be easily adjusted to measure different concentrations of trace elements in accordance with the economic feasibility of mining each trace element. The detection of a number of elements is possible by varying the duration for which the sample is exposed to the neutron source and by measuring for different types of radiation. Secrecy of the results for each sample can be more easily maintained, and the results can be used to focus an investigation of a given field site at promising areas. Thus, mineral exploration companies can form, implement and modify exploration strategies and property purchase decisions in a timely manner as field test results become available. Additionally, the measurements provided by the present invention can enable differentiation among lithologic formations and can support geologic mapping for structural analysis, seismic risk studies, oil exploration and other purposes. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a diagram of the radioactive decay scheme for gold 198. 
     FIG. 2 is a diagram of field procedures used for neutron activation analysis with a portable neutron source in accordance with the invention. 
     FIG. 3 is a side view of one embodiment of the invention, adapted for placement on a truck. 
     FIGS. 4 and 5 illustrate an example of a moderator assembly. 
     FIG. 6 illustrates the neutron flux in the moderator as a function of distance from a neutron source in the radial center of the moderator. 
     FIG. 7 sets out the results of a first set of experiments, performed with SiO 2  samples doped with a gold chloride solution. 
     FIG. 8 sets out the results of a second set of experiments, performed with USGS standard gold-samples. 
     FIGS. 9 and 10 are a top view and side view, respectively, of a subcritical reactor. 
     FIG. 11 is a cross-sectional view of the subcritical reactor of FIGS. 9 and 10. 
     FIG. 12 is a plot of k ∞  as a function of volume fraction of uranium metal where the remainder of the volume is polyethylene. 
     FIG. 13 illustrates various parameters of two subcritical reactors. 
     FIG. 14 illustrates neutrons as a function of time, where k eff  is 0.8 and the neutron source emits one neutron every mean neutron lifetime, Λ. 
     FIGS. 15 and 16 illustrate volume averaged neutron flux as a function of distance from the centers of the smaller and larger moderator assemblies of FIG. 13, respectively. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     System Overview 
     Neutron activation analysis (NAA) determines the presence and concentration of selected elements in a sample by measuring the characteristics of the radioactive decay of certain isotopes of the element. Typically, a linear accelerator is used to bombard a sample with neutrons. Nuclides of elements in the sample absorb the neutrons to become radioactively unstable isotopes. Linear accelerators provide a large supply of high energy neutrons to quickly activate the sample and allow rapid sampling. For instance, in U.S. Pat. No. 5,237,594 by Carroll and in U.S. Pat. No. 4,568,510 by Caldwell, a sample is irradiated in situ, meaning in its undisturbed position, by a linear accelerator. The linear accelerator is pulsed, or turned off and on incrementally, to allow measurement of the sample after it has been activated. Although the linear accelerators of Carroll and Caldwell provide a large flux of neutrons, the total flux over a time period is difficult to predict. Further, in situ sampling does not allow precise measurement of the activated sample due to variations in distance to the sample, the presence of environmental radiation, the necessity of measuring only prompt gamma radiation, and other variable factors. 
     Unstable isotopes in samples activated by neutrons from a linear accelerator are allowed to decay to a stable form and, during decay, release a variety of radiation. Different elements have different isotopes with unique decay characteristics. For instance, referring to FIG. 1, the radioactive decay characteristics of gold 198 are depicted. Naturally-occurring gold is 100% gold 197. When naturally-occurring gold absorbs a neutron to become gold 198, it becomes unstable and beta decays, releasing beta and gamma radiation having unique energy levels. By measuring this released radiation over a time, the concentration of gold in the original inactivated sample can be determined. 
     FIG. 2 is a functional illustration of the present invention, used to measure trace element concentrations ex situ. Moderator assembly  10  holds a neutron source  12  adjacent to an irradiation chamber  16 . Sample  14 , which has a known and regular geometry, is placed in irradiation chamber  16  to allow sample  14  to absorb neutrons and become activated, meaning that an adequate number of nuclides of the desired element have absorbed a neutron and become radioactively unstable. The activated sample  14  is then placed proximate to a radiation detector  18  to allow measurement of the radiation resulting from the decay of unstable elements in the activated sample  14 . 
     Neutron source  12  can include any substance that will produce a predictable flux of neutrons. For instance, a radionuclide or combination of radionuclides that decay through neutron emission, such as californium 252, americium-beryllium, plutonium-beryllium, or radium-beryllium, can be placed in a non-critical reactor having a medium for moderating the source neutrons. The moderating medium can be a material with low neutron-absorbing qualities, such as carbon, polyethylene, deuterium, water, liquid hydrocarbons, or solid hydrocarbons. Alternatively, a multiplying medium, such as uranium nuggets inserted in polyethylene, can create a subcritical reactor by operating in combination with a radionuclide neutron source to boost the neutron flux level within moderator assembly  10 . Moderator assembly  10  may include a shielding material to prevent leakage of radiation to the environment. Sample  14  may be activated by exposure to neutron source  12  for a predetermined exposure time based upon the number of neutrons produced by neutron source  12 , the neutron absorption cross section of the element or elements to be detected, the geometry of the moderator assembly and the desired accuracy of the analysis. 
     Referring now to FIG. 3, one embodiment of the present invention adapted for use in a field environment is depicted. Moderator assembly  10  is constructed of steel and/or lead shielding filled with polyethylene or some other hydrocarbon in its interior. A supply of californium 252 acts as neutron source  12  adjacent to irradiation chamber  16 . A convenient opening allows access to irradiation chamber  16  in moderator assembly  10  for insertion of sample  14  for the predetermined exposure time. Once sample  14  becomes activated, it can be removed from moderator assembly  10  and analyzed with radiation detector  18 . Detector shielding  20  fits over detector  18  to ensure that environmental and other radiation does not skew the results of detector  18 . A supply of liquid nitrogen  22  maintains detector  18  at low temperatures for proper detector function. Detector  18  is coupled to a multichannel analyzer and spectroscopy amplifier  26  that includes a processor such as a personal computer. Analyzer  26  includes software that allows a technician to analyze the electrical signals produced by the detection of radiation from the activated sample  14 . A convenient power supply  28  provides power to analyzer  26  for use in remote situations. The entire system can be mounted on truck  30  to allow testing to occur in field conditions. 
     In operation the present invention is adapted to travel to a field site, for instance, by mounting on truck  30 . A neutron source  12  is selected to provide a supply of neutrons for activating samples. Although many radionuclides or combination of radionuclides could provide a neutron source, californium 252 is well adapted for field use. Californium 252 has a half-life of 2.65 years and therefore provides a fairly constant flux of neutrons over a normal field operation cycle without requiring replacement. The constant flux of neutrons allows greater predictability for the number of neutrons that will interact with a given sample over an exposure time. 
     Moderator assembly  10  is built to a size adequate to hold a supply of neutron source  12 . Moderator assembly  10  is of a sturdy construction to prevent any leakage of radioactive material during relocations to field sites. For instance, a sturdy steel or lead casing encloses the neutron source and a medium for holding the neutron source firmly in place, even in the presence of shocks from movement in field conditions, while allowing neutrons to pass from the neutron source to the sample. For instance, polyethylene or composite materials made of hydrocarbons provide mediums that will not allow movement of the neutron source during movement of moderator assembly  10 , will not crack or break, but will allow neutrons to pass through to irradiation chamber  16  with little interference. To enhance the production of neutrons by the neutron source, the medium can include multiplying media, such as uranium nuggets, which will absorb neutrons and then a fission to produce additional neutrons, thereby boosting the neutron flux level. In this way, a multiplying media can make a non-critical reactor containing only a neutron source into a sub-critical reactor with enhanced neutron output. 
     The present invention is transported by truck or other means to a field site. At the field site, a technician takes rock or soil samples from selected locations. Typically, the samples are approximately one-inch plugs having a uniform size. Irradiation chamber  16  can be adapted to accept appropriately-sized samples. When the technician wants to sample soil or rock outcroppings, such samples can be crushed and placed in a vial or other sample holder that matches the size of plugs produced from drilling. In this way, sample  14  has a constant cross section in irradiation chamber  16 , which increases the predictability of the number of neutrons directed towards the sample from neutron source  12 . 
     Sample  14  is placed in irradiation chamber  16 , which is proximate to neutron source  12 . In one embodiment, a plurality of irradiation chambers are disposed within moderator assembly  10 , each irradiation chamber adapted to receive a predetermined number of neutrons from neutron source  12 . Several neutron sources  12  can be placed around irradiation chamber  16  along with several rods, or any other convenient configuration. Sample  14  is exposed to neutron source  12  in irradiation chamber  16  for a predetermined exposure time which depends upon the strength of the neutron source, the geometry of moderator assembly  10 , the accuracy of the measurement required, and the neutron absorption cross-section of the element of interest. The exposure time can vary from minutes to hours according to variations in the above factors, and can be calculated and provided to the technician by software loaded on a personal computer in analyzer  26 . For instance, the exposure time of a sample being tested for the presence of gold would be reduced by gold&#39;s relatively high neutron capture cross-section of 98.8 barns, but increased by the low concentrations at which gold is likely to be measured to determine the feasibility of economic extraction. 
     Once the sample  14  has had an adequate exposure time, it can be removed from moderator assembly  10  and placed in radiation detector  18 . Radiation detector  18  will detect radiation, such as gamma radiation or beta radiation, exiting from activated sample  14  as radioactive isotopes in sample  14  decay. For instance, gamma rays having an energy level of 0.411794 MeV would indicate that gold 197 present in the field sample was activated to gold 198 and then decayed into mercury 198, as is depicted in FIG.  1 . Activated sample  14  is placed directly on radiation detector  18  for a testing time which varies according to the degree of activation of sample  14  by exposure to nuclear source  12 , the half-life of the isotope being detected and the desired accuracy of the reading. For instance, gold 198 has a half-life of 2.70 days. Thus, a given gamma ray flux density having energy levels of 0.411794 MeV over a given period of time for a given sized sample with a given exposure time will indicate a sample having a specific quantity of gold. The testing time can be calculated and provided to the technician by software loaded on analyzer  26 . 
     A number of radiation detectors can be used to detect radiation from radioactive decay ex situ to analyze trace element compositions, but a high-purity, intrinsic germanium semi-conductor gamma detector which accepts activated sample  14  directly on its surface will provide a highly accurate count of the gamma ray radiation commonly released during radioactive decay. The function of this radiation detector necessitates cooling, such as can be obtained from liquid nitrogen. The accuracy is further enhanced if the sample  14  is isolated during testing, as can be accomplished with a lead or steel detector shielding that covers radiation detector  18  and separates radiation detector  18  from moderator assembly  10 . In field conditions, a technician can perform each of these steps with little training or supervision. Other embodiments may use a beta ray or alpha ray detector. 
     A technician can use the results of field samples to build a profile for a field site. The timely feedback provided by the present invention allows the technician to concentrate a field search in a promising area during a single visit to the field site. The technician can provide test data to decision makers for prompt decisions regarding property investments without the risk that testing information will be compromised to competitors. The accuracy of the testing can be easily adapted according to the accuracy required for a desired mineral. For instance, gold can be tested for in small concentrations by lengthening the exposure time and testing time as needed. Other elements, such as silver, cobalt, mercury, sodium, potassium, silicon, cadmium, magnesium, or manganese, can be tested for in larger concentrations as needed. The testing process itself leaves no permanent residue, as could result from chemical testing, since the radiation used is of a relatively low concentration level. Further, the testing method requires minimal training since the exposure time, testing time, and concentration calculations can be programmed into a personal computer incorporated in amplifier  26 . In addition, a small personal computer will require very little power and, thus, the present invention can easily be used in field conditions for extended periods of time with a small power supply and relatively low operating costs. 
     Sample Geometry and Moderator Assembly 
     As indicated above, samples taken from the field are first processed to provide a controlled geometry and to enhance the ability to detect smaller amounts of the substance of interest. Solid samples are powdered, and the resulting particles are mixed. The result is a “homogenous” sample, that is, a sample whose constituents are evenly distributed throughout the sample. For irradiation, the sample is placed in a vial  14 , which gives it a regular geometry. For example, a cylindrical vial  14  provides a cylindrical sample geometry. Thus, the cross-section of the sample is constant, in terms of both geometry and make-up. 
     FIGS. 4 and 5 are a top view and cross sectional view, respectively, of a neutron moderator  40 . Moderator  40  is suitable for use as the moderator assembly  10  of FIG.  2 . Referring again to FIG. 2, such a moderator assembly  10  contains the neutron source  12  and has at least one irradiation chamber  16  for containing one or more sample vials  14  during sample activation. 
     As illustrated in FIGS. 4 and 5, moderator  40  is cylindrical in shape. It is primarily comprised of a moderating medium  42  having low neutron absorption and high neutron scattering properties, such as the various materials described above. These materials “moderate” the neutrons so as to increase their interaction with the sample. The materials may be liquid or solid, with a common characteristic being a presence of hydrogen. In the example of FIGS. 4 and 5, the moderating medium  42  is polyethylene. 
     In the case of a liquid medium  42 , moderator  40  has an outer casing (not shown) for containing the liquid. Otherwise, an outer casing is optional, but may be desired to provide radiation shielding. 
     The use of a solid medium  42  permits the medium itself to provide the structural support for neutron source  12  and sample vials  14 . Thus, neutron source  12  may simply be placed inside moderator  40 , at the center. Four irradiation chambers  16  symmetrically spaced around neutron source  12  permit insertion of sample vials  14 . Polyethylene rods (not shown) may be inserted above the vials  14  to provide neutron reflection and to prevent streaming of neutrons from the chambers  16 . 
     Optionally, moderator  40  can be fitted with multiplying media. As explained above, this results in sub-critical enhancement of the neutron flux levels. An example of a subcritical reactor of this type is described below. 
     In the example of FIGS. 4 and 5, moderator  40  has a top portion  40   a  and a bottom portion  40   b . These parts can be separated, to permit insertion of neutron source  12 . As an example of suitable dimensions of moderator  40  (oversized from the standpoint of neutron economy), its outer diameter might be 24 inches and its height might be 20 inches. Neutron source  12  is at the center of moderator  40 , and the sample chambers  16  are spaced a radial distance of 1 inch from the center. Although a two or three inch thickness of medium  42  would be sufficient for neutron reflecting purposes, the additional thickness of medium  42  provides increased shielding. 
     FIG. 6 illustrates the average neutron flux in moderator  40  as a function of the radial distance from the source  12  (at the radial center of the moderator  40 ). The source  12  used for the data of FIG. 6 was a 26 mCi  252 Cf source. This source has a neutron emission rate of 1.1×10 8  neutrons per second. A neutron source of 10 9  neutrons per second was assumed for the calculations. Thermal flux is defined as energy less than 0.05 eV, epithermal plus thermal flux is energy less than 0.5 eV (the cadmium cutoff), and total flux is all neutron energies. The data was generated using a method known as MCNP, which is a Monte Carlo N-Particle Transport method described by J. F. Breismeister, at the Los Alamos National Laboratory. 
     As illustrated in FIG. 6, placement of sample chambers  16  close to source  12  optimizes the thermal, epithermal, and total neutron flux in chambers  16 . In the example of this description, the optimization is for a neutron source that emits 10 9  neutrons per second. The results scale linearly for other source strengths. 
     Although the above description is in terms of a moderator assembly  40  especially designed for DNAA of ex situ geological samples, it should be understood that moderator assembly  40  could be used for any application that calls for an irradiated sample. The moderator&#39;s ability to irradiate a controlled sample with a controlled flux makes it desirable for a wide variety of nuclear activity analysis applications. 
     Sample Analysis 
     In mathematical terms, the goal of neutron activation analysis (NAA) is to determine the atomic number density of the isotope(s) in question, which for purposes of the present invention is  197 Au. Referring again to FIGS. 1 and 2, the sample is irradiated in the neutron flux field of moderator assembly  10  to create a radioactive isotope,  198 Au. During detection, the sample vials  14  are placed proximate to detector  18 . For example, the samples could be placed a distance of 0.9 cm from a detector crystal of approximately 2 inches in diameter. For purposes of example herein, detector  18  is a high-purity intrinsic germanium (HPGe) detector. Detector  18  obtains a radiation signal, which is analyzed by analyzer  26 . This method is a “delayed” NAA method, in the sense that the sample is first irradiated, then the radiation signal is detected. This method is distinguishable from the “prompt” NAA method, in which the radiation is detected as the sample is being irradiated. 
     Thus, delayed NAA can be considered in three phases: irradiation of the sample in a neutron flux field to create a radioactive isotope, measurement of the sample activity level in the radioactive isotope, and determination of the sample concentration in the isotope of interest. 
     For the following description, neutron energy group-averaged quantities are assumed for a single neutron group. It is further assumed that the following quantities are known: 
     φ=volume-averaged neutron flux in the sample chamber 
       197 σ a =averaged neutron absorption cross section for  197 Au 
     It is also assumed that the sample does not strongly affect the neutron field flux, that is, that it contains no strong neutron absorbers in sufficient quantity to significantly perturb the neutron flux field. 
     The rate at which  198 Au atoms are created in the sample per unit volume is: 
     
       
           r =φ· 197 σ a · 197   n   (1) 
       
     
     , where  197 n is the atomic concentration of  197 Au in the sample. 
     From Equation (1), a first order differential equation for the time rate of change of the  198 Au atomic concentration in the sample during irradiation is:                       (             198        n          (   τ   )       )            τ       =         -   λ                             198        n          (   τ   )         +   r             (   2   )                                
     , where  198 n(τ) is the time-dependent atomic concentration of  198 Au in the sample, λ is the radioactive decay constant for  198 Au, and τ is the irradiation time. 
     Equation (2) can be solved to determine  198 n(τ): 
     
       
           198   n (τ)= r /λ( 1 − e   −λτ )  (3) 
       
     
     The activity of the sample as a function of irradiation time is: 
     
       
           A (τ)= r ( 1 − e   −λτ ) V   (4) 
       
     
     , where V is the sample volume. For a given value of τ, Equation (4) provides the activity level of the sample. 
     Next, the activity level is related to measurable quantities. Referring again to FIG. 1, the decay of  198 Au produces a 412 keV gamma ray in 95.5 percent of the disintegrations. This is a signature radiation that may be used to detect gold. 
     Analyzer  26  counts gamma rays from the decay of  198 Au in the 412 keV photopeak over a counting interval. The count, I, may be a result of subtracting background counts from total counts. This measured quantity, I, may be expressed as:                    I   =                ∫         A        (   τ   )       ·   ɛ   ·            -   λ                   τ                 t                     =                      A        (   τ   )       ·   ɛ     λ          (              -   λ                   t1       -            -   λ                   th         )                     (   5   )                                
     , where the integration is from t l , the time after the end of irradiation at which the counting interval begins, to t h , the time after the end of irradiation at which the counting interval ends. The total counting efficiency ε is the product of the following efficiencies: the efficiency for decay of  198 Au to produce a 412 keV gamma ray (0.955), the geometric efficiency of the sample/detector arrangement, the intrinsic efficiency for the detector at 412 keV, and the efficiency for the 412 keV gamma ray to escape the sample without interaction. 
     The counting interval (t l  to t h ) need not begin immediately after the end of irradiation; in some cases, it may be desirable to allow the sample to decay for a time to permit isotopes with short half-lives (isotopes other than  198 Au) to disintegrate. These isotopes may result in compton effects in detector  18 , which increases background at 412 keV and makes the measurement less certain. Examples of such isotopes are sodium and manganese,  24 Na and  56 Mn. 
     By combining Equations (1), (4), and (5), a relationship between  197 n and the measured number of counts, I, can be established:                      197        n     =         I                 λ       ϕ                   σ   a               197        ɛ                 V              (     1   -            -   λ                   τ         )       -   1              (              -   λ                   t1       -            -   λ                   th         )       -   1                 (   6   )                                
     Many of the parameters in Equation (6) are either one-group parameters, such as φ and  197 σ a , or are specific to the detector/sample arrangement, such as ε. In either case, these quantities are difficult to quantify based on first principles. As an alternative to using detailed calculations to quantify these parameters, Equation (6) may be rearranged so that they are lumped into a single constant, which may be experimentally determined. The rearranged Equation (6) is: 
     
       
           197   n=k·I ( 1 − e   −λτ ) −1 ( e   −λtl   −e   −λth ) −l   (7) 
       
     
     The following section discusses determination of a value for the experimentally determined constant, k. 
     Sample Analysis; Value of k; Experimental Results 
     The following discussion is of two types of experiments, using the delayed neutron activity analysis (DNAA) techniques that are the subject of the invention. As explained below, the experiments were used to determine a value of k for Equation (7). 
     For these experiments, the moderator and detector configurations are those described above in connection with FIGS. 2-6. Thus, the moderator was the same as moderator  40  with a polyethylene medium. The detector was an HPGe detector. The neutron source was Cf 252  with no multiplying media. The use of a multiplying media would improve the results. 
     FIG. 7 illustrates the results of a first set of experiments. These experiments measured gold concentrations in SiO 2  samples doped with a known amount of dilute gold chloride solution. This type of sample represents optimum conditions for system performance—an absence of strongly interfering species such as sodium or manganese. 
     The standard deviation, σ I , on the measured number of counts, I, can be assumed to have two elements: (1) errors due to fluctuations in the number of radioactive emissions during the counting period, σ st , and (2) errors due to estimating the number of background counts under the photopeak, σ bg . The quantity σ st  can be estimated as the square root of I; σ bg  can be estimated from instrument specifications. Because I is estimated by subtracting background counts from total counts, the standard deviation of I is estimated by adding the two error values in quadrature as follows: 
     
       
         σ I =(σ st   2 +σ bg   2 ) ½   
       
     
     The standard deviation of k, σ k , is calculated by equating its fraction error with that of I, as follows: 
     
       
         σ k =σ I   /I k   
       
     
     The inverse variance-weighted average value of k and the standard deviation of k can be estimated using known statistical analysis methods. The values from the last column of FIG. 7 indicate that the method yields results that have relative errors of less than about 60 percent down to the 10 ppb level, with no strongly interfering species present. 
     The level of detection in this case is limited by the background count rate at 412 keV, which was measured at 22±1 counts per hour, as well as by the count rate at 412 keV due to compton effects from activated trace contamination decays, which was measured at 29±1 counts per hour for these samples (with no cooling time). 
     FIG. 8 illustrates the results of a second set of experiments, which measured gold concentrations in United States Geological Survey (USGS) standard samples. These samples represent realistic conditions for system performance—in the presence of interfering species such as sodium or manganese. In FIG. 8, each of three samples is shown with its USGS sample designation and certified gold concentration. 
     For purposes of the experiments of FIG. 8, the gold concentration was treated as an unknown and the value of k was assumed to be the value determined in the above-described experiments that used doped SiO 2  samples. The standard deviation of I was calculated by adding the standard deviations from background subtraction and statistical deviation of decays during the counting interval in quadrature. The standard deviation on the final estimate of gold concentration was calculated by adding the relative standard deviations for I and k in quadrature. 
     As indicated by FIG. 8, there is good agreement between reference and measured values of gold concentration in the USGS samples. Because of this agreement and because the measured concentrations were not consistently smaller than known concentrations, it follows that the USGS samples did not significantly affect the neutron flux field, as compared to the doped SiO 2  samples. Thus, interfering species did not significantly affect the neutron flux fields, and the value of k calculated from doped SiO 2  samples is appropriate for use with actual geologic samples. 
     Subcritical Reactor 
     As stated above, the purpose of a subcritical reactor is to provide a source of neutrons that is more powerful than a neutron emitter alone. A moderator assembly, such as moderator assemblies  10  and  40 , may be modified to provide a subcritical reactor. This is achieved by embedding a multiplying material within the moderating material. Like moderator assembly  40 , the subcritical reactor described below is suitable for activating samples for applications other than geological analysis. 
     Polyethylene is a suitable moderating material for the following reasons: (i) it has a large hydrogen concentration, (ii) it is a solid at room temperature, (iii) it has low cost, (iv) it is easy to fabricate, and (v) it minimizes the size of the subcritical reactor. Since polyethylene has a large hydrogen concentration, it is able to slow down neutrons from their birth energies in a short distance, limiting the size of the subcritical reactor which is an advantage for portability. A solid-state moderator is also advantageous for portability. 
     Natural uranium metal is a suitable multiplying material. It may be more suitable than enriched uranium for the following reasons: (i) its relatively greater availability and low expense, and (ii) its inability to be formed into a critical arrangement with hydrogenated materials as the moderator. 
     The following description is in terms of computer-aided models of a subcritical reactor. The polyethylene was modeled as having a density of 0.955 g/cm 3 . The uranium metal was modeled as having a density of 19.0 g/cm 3 . The relative isotopic composition of the uranium was assumed to be 99.2745 percent  238 U, 0.7200 percent  235 U, and 0.0055 percent  234 U. 
     FIGS. 9 and 10 are a top view and side view, respectively, of a subcritical reactor  90 . A right-cylinder with height equal to diameter (hereafter referred to as a right-square cylinder) is suitable for several reasons: (i) it is a stable shape for transportation while having a reasonably small surface area to volume ratio, and (ii) it is easy to machine. An outer polyethylene “blanket”  91  surrounds the perimeter of the moderator assembly  90 . A top piece  92  and bottom piece  93  are at the top and bottom of the assembly  90 . The blanket thicknesses are the minimum thicknesses necessary for obtaining maximum reflection of neutrons back into the subcritical reactor. 
     Uranium fuel rods  94  are arranged in a rectangular grid, embedded in the core area  95  of the polyethylene moderating material. The center rod is omitted to allow for insertion of the neutron source  96 . One or more of the fuel rod locations is used as a sample chamber  97 . 
     FIG. 11 is a cross sectional view of a portion of the subcritical reactor  90 . In the example of FIG. 11, each grid element is 2.54 cm×2.54 cm. Each fuel rod  94  is 1.72 cm in diameter. An analysis of k ∞  may be used to determine the optimal grid spacing and fuel rod size. 
     FIG. 12 is a plot of k ∞  as a function of volume fraction of uranium metal where the remainder of the volume is polyethylene. Four different grid spacings (i.e., the center to center distance of consecutive fuel rods) were examined: 1.27, 2.54, 3.81, and 5.08 cm (0.5, 1.0, 1.5, and 2.0 inches). The volume fraction of uranium was changed by varying the fuel rod diameter. As can be seen from FIG. 12, k ∞  can be maximized with a grid spacing of 2.54 cm. 
     A volume fraction of 0.360 was chosen for the subcritical reactor  90 . This volume fraction is not exactly at the peak of the k ∞  curve for a 2.54 cm grid, however, this location on the curve has a corresponding k ∞  that is within one percent of the maximum and represents a ten percent weight savings over the peak location. This volume fraction corresponds to a rod diameter of 1.72 cm for a 2.54 cm grid spacing. 
     The neutron flux in sample chamber  97  is roughly proportional to 1/(1−k eff ) Thus, the largest value of k eff  that can be achieved is desired, given the restriction that the moderator  90  is to be sufficiently small so as to be field portable. Maximization of k eff  minimizes the amount of uranium in the subcritical reactor  90 , and hence its hence size and weight. 
     FIG. 13 illustrates various parameters of two subcritical reactors  90 . A smaller and a larger reactor have a k eff  of approximately 0.8 and 0.9, respectively. Because the factor by which the flux in sample chamber  97  is amplified by the multiplying material is roughly proportional to 1/(1−k eff ), the higher value of k eff  is preferred. However, the weight of the larger reactor  90  (3,059 kg versus 692 kg for the smaller assembly) may limit its field portability. 
     Analytic methods may be used to determine an optimal location for the sample chamber  97  relative to the neutron source  96 . An optimal location is one that is sufficiently close to the neutron source  96  so as to be within a strong neutron field, but sufficiently distant to permit neutrons to be moderated and hence absorbed. 
     A sample chamber  97  having a volume of about ten cubic centimeters is adequate for activating a sufficient number of gold atoms for detection to the limits of economic importance. This volume is large enough for the sample to contain a sufficient number of gold atoms yet not so large as to significantly perturb the neutron flux level in the assembly. The sample chamber  97  modeled herein is a right-cylinder with a diameter of 2.0 cm and a height of 4.0 cm, with the volume averaged neutron flux inside that volume being the optimized quantity. 
     The neutron source  96  may be modeled as a maxwellian fission energy spectrum having the following probability density function: 
     
       
           p ( E )= CE   ½  exp(− E/a ) 
       
     
     , where E is the neutron birth energy, C is a proportionality constant, and a is a shape parameter. A value of 1.2895 MeV for the shape parameter is appropriate for modeling the neutron energy spectrum from  252 Cf. 
     Computer-aided neutron analysis tools, such as tools using MCNP estimation methods, may be used to estimate the volume averaged neutron fluence per unit source particle plus its progeny. When there is no multiplying media, the volume averaged neutron flux for a particular application is then determined by multiplying that estimation by a source strength (the number of neutrons per second emitted from the source). The result of this multiplication is the rate at which fluence is delivered to the sample chamber  97 , which is one definition of flux. When multiplying media is present, the determination of the volume averaged neutron flux from the MCNP estimation is not so straightforward; contributions from one source particle&#39;s progeny can coincide (as opposed to being consecutive) with future particles&#39; contributions to the volume averaged neutron fluence. 
     An enhanced means for estimating volume averaged neutron flux can be used when multiplying material is present, provided that the source rate is larger than the inverse of the mean neutron lifetime. For purposes of this description, mean neutron lifetimes are assumed to be about 0.2 milliseconds, and typical neutron source rates are assumed to be on the order of 10 8  to 10 9  neutrons per second. 
     FIG. 14 illustrates neutrons as a function of time, where k eff  is 0.8 and the neutron source emits one neutron every mean neutron lifetime, A. The fluence from the source neutron plus its progeny is the fluence in the shaded area of FIG.  14 . 
     The quantity of interest is the fluence per unit time (i.e., the flux) in sample chamber  97 . Once a steady state condition has been reached, there is a total of one less than 1/(1−k eff ) progeny contributing simultaneously to the flux for every source neutron. The effect of the simultaneous deposition of fluence is that a source neutron is amplified by a factor of 1/(1−k eff ). Thus, to estimate the neutron flux in the sample chamber  97 , the fluence-per-source-particle-plus-progeny is multiplied by the source rate and by 1/(1−k eff ). 
     Computer-aided neutron analysis tools may be used to estimate flux in terms of discrete energy bins, with the estimated flux being the volume averaged neutron fluence per source particle plus progeny per unit energy integrated over the energy bin. The neutron flux, averaged per unit volume and energy bin, in cm −2  MeV −1 , is given by the following equation:        Φ   =         10   9       1   -     k   eff                ∫     E   =     E   1         E   =     E   u                ∫   v            ∫     t   =          t   =   ∞              ϕ        (     E   ,     r   _     ,   t     )               t               V     V             E                                      
     , where E u  is the upper energy level of the bin, E l  is the lower energy level of the bin, φ(E, r, t) is the neutron flux per unit energy per unit time per source particle plus progeny in cm −2  s −1  MeV −1 , t is time in s, E is energy in MeV, and V is volume in cm 3 . 
     FIGS. 15 and 16 are plots of the thermal (i.e., neutron energy less than 0.05 eV), thermal plus epithermal (i.e., as defined by the cadmium cutoff where neutron energy is less than 0.5 eV), and total flux as a function of distance from the neutron source  96 . The reactors that are the basis for these plots are the smaller and larger reactors  90  described in FIGS. 9-13, with FIG. 15 representing the smaller reactor and FIG. 16 representing the larger reactor of FIG.  13 . The neutron source  96  is at the center of the reactor  90 , and is assumed to provide 10 9  neutrons per second. The sample chamber  97  is assumed to be filled with air and on the same horizontal plane as the neutron source  96 . FIGS. 15 and 16 indicate that placing the sample chamber  97  at the center of the reactor  90  maximizes the total, epithermal, and thermal fluxes. 
     Using one of the fuel rod locations for the sample chamber  97 , the chosen location for the sample chamber  97  was at 2.54 cm radially outward from the neutron source  96 . At this location and assuming a neutron source rate of 10 9  neutrons per second, for the smaller reactor  90 , the volume averaged neutron flux in the sample chamber  97  is 1.72×10 8  cm −2  s −1  total flux, 4.47×10 7  cm −2  s −1  thermal flux (i.e., energy less than 0.05 eV). For the larger reactor  90 , the volume averaged neutron flux in the sample chamber  97  is 3.43×10 8  cm −2  s −1  total flux, 9.09×10 7  cm −2  s −1  thermal plus epithermal flux, and 4.37×10 7  cm −2  s −1  thermal flux. 
     Other Embodiments 
     Although the present invention has been described in detail, it should be understood that various changes, substitutions and alterations can be made hereto without departing from the spirit and scope of the invention as defined by the appended claims.