ML20065U297
ML20065U297 | |
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Site: | Clinch River |
Issue date: | 07/31/1980 |
From: | Croff A OAK RIDGE NATIONAL LABORATORY |
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ORNL-5621, NUDOCS 8211040132 | |
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{{#Wiki_filter:i {Y; 4 5737 ORNL-5621 I i !1 A 1 _- j i J l -l ~# "' ORIGEN2-A Revised and Updated l Version of the Oak Ridge j eenca Isotope Generation and i Depletion Code A. G. Croff t s -e -A s# m 'e.. x E f 40"n ' a.a \\ 9 j De c, c n($$ %m. 'E'h Yl c: O NY ~ . o' s ',. y (' #g+
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j -) .4 m) l l . ] ORNL-5621 4 Dist. Category UC-70 1 i f Contract No. W-7405-eng-26 m .f, CHEMICAL TECHNOLOGY DIVISION 1 4 -1 { NUCLEAR FUEL AND WASTE PROGRAMS .I,j Waste Management Analysis for Nuclear Fuel Cycles ) (Activity No. AP 05 25 10 0; FTP/A No. ONL-WH01) i 1 I 1 ORIGEN2 - A REVISED AND UPDATED VERSION OF THE OAK RIDGE ISOTOPE GENERATION AND DEPLETION CODE A. G. Croff f Date Published: July 1980 4. i i 3- -i i 0AK RIDGE NATIONAL LABORATORY I tm Oak Ridge, Tennessee 37830 operated by UNION CARBIDE CORPORATION for the DEPARTMENT OF ENERGY i 1 1 i!
.1 1 iii CONTENTS 1 ~ .i Page i
- .j ABSTRACT..
1 l 1. INTRODUCTION 1 .i ll 1.1 Background. 2 I 1.2 Scope of Revisions and Updates. 3 ,l !j 1.2.1 Revision of the ORIGEN computer code 3 j 1.2.2 Update of the cross section and fission product yield library 3 1.2.3 Update of the decay and photon data. 4 1.2.4 Update of the reactor models 5 1.2.5 Update of miscellaneous input information. 5 1.3 Organization of ORIGEN2 and Its Data Libraries. 5 1.4 Availability of CRIGEN2 6 2. OVERVIEW OF THE ORIGEN2 COMPUTER CODE.. -7 2.1 Description of the Capabilities of ORIGEN2.. 7 2.1.1 Function of ORIGEN2.... 7 2.1.2 ORIGEN2 input features 8 2.1.3 ORIGEN2 calculational features 9 2.1.4 ORIGEN2 output features. 12 2.2 Computer-Oriented Description of ORIGEN2...... 16 3. DESCRIPTION OF THE MATHEMATICAL METHOD USED IN ORIGEN2.. 20 3.1 Matrix Exponential Solution . 21 3.1.1 General solution . 21 3.1.2 Computation of the matrix exponential series . 21 1 3.2 Use of Asymptotic Solutions of the Nuclide l Chain Equations for Short-Lived Isotopes. . 24 I 3.2.1 Short-lived nuclide present initially.. . 25 I. 3.2.2 Short-lived daughter of a f' long-lived parent . 27 1 4 3.3 Applications of the Matrix Exponential Method for Nonhomogeneous Systems. . 28 5. 4 h ^ \\ t e
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.m. * *,.. - ~ -
.~. -... ~,.. = :.-nr 4 ? 1 J iv o Pagg i 3.4 Computation of Neutron Flux and Specific Power. 29 3.4.1 Calculation of neutron flux given specific power. . 29 3.4.2 Calculation of specific power given i neutron flux 31 3.4.3 Other considerations. 31 3.5 Construction of the Transition Matrix. 32 I 4. DESCRIPTION AND SOURCE OF MIXCELLANEOUS ORIGEN2 DATA.. 37 4.1 Neutron Yield per Spontaneous Fission. 37 4.2 Neutron Yield per Neutron-Induced Fission. 37 t ] 4.3 Neutron Yields from (a.n) Reactions 38 4.4 Elemental Chemical Toxicities 41 ~ ! 4.5 Nuclide-Dependent Recoverable Energy 43 per Fission 4.6 ORIGEN2 Flux Parameters THERM, RES, and FAST 44 4.6.1 Derivation of ORIGEN2 flux parameters THERM and RES 46 4.6.2 Derivation of the ORIGEN2 flux parameter FAST. 48 4.6.3 Conversion of ORIGEN2 flux parameters to ORIGEN flux parameters 50 4.7 ORIGEN2 Nuclide and Element Identifiers... 51 5. REFERENCES 52 i s s t l' i 4 i . a, l i 1 1. 4 I i' I +
q 1 j 1 l 'l l ORIGEN2 - A REVISED AND UPDATED VERSION OF THE OAK RIDGE ISOTOPE CENERATION AND DEPLETION CODE ] A. G.-Croff 'l ABSTRACT .l b; ORIGEN2 is a versatile point depletion and decay computer l code for u.e in simulating n'uclear fuel cycles and calculating 'l the nuclide compositions of materials contained therein. This l code reptesents a revision and update of the original ORIGEN computer code which has been distributed world-wide beginning in the early 1970s. Included in it are provisions for incor-porating data generated by more sophisticated reactor physics .l codes, free-format input, the ability to simulate a wide variety of fuel cycle flowsheets, and more flexible and controllable output features. Included in this report are: (1) a summary description of the total effort to update ORIGEN and its data bases, (2) a summary description of ORIGEN2, its capabilities, and its relationship to ORIGEN, (3) a description of the mathe-natical methods used in ORIGEN2 to solve the equations describing the generation and depletion of nuclides, (4) a description of the mathematical methods used in ORIGEN2 to calculate the neutron flux and specific power, and (5) a descrip-tion of the sources of specific data associated with the computer code ORIGEN2. Also included are directions for obtaining ORIGEN2, its data bases, and a separate user's manual. 1. INTRODUCTION The purpose of this report is to give a summary description of a revised and updated version of the original ORIGEN computer code, which has been designated ORIGEN2. The remainder of this section is concerned with describing the background, scope, organization, and availability of ORIGEN2 and its data bases. Section 2 gives a more detailed description of the computer code ORIGEN2. Section 3 describes the methods used by ORIGEN2 to solve the nuclear depletion and decay equations. Finally, Section 4 docunents input information necessary to use ORIGEN2 that has not been documented in supporting reports. li l! L_ _._. _. l:
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1.1 Background
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The ORIGEN computer code was written in the late 1960s and early 1970s by the ORNL Chemical Technology Division as a versatile tool for } calculating the buildup and decay of nuclides in nuclear materials. At that time, the required nuclear data libraries (decay, cross section/ Il fission product yield, and photon) and reactor models (PWR-U, PWR-Pu,
- [
LMFBR, HTGR, and MSBR) were also developed, based on the information avail-able at that time. The computer code was principally intended for use in a generating spent fuel and waste characteristics (composition, thermal power, etc.) that would form the basis for the study and design of fuel reprocessing plants, spent fuel shipping casks, waste treatment and dis-posal facilities, and waste shipping casks. Since fuel cycle operations were being examined generically, and thus were expected to accommodate a wide range of fuel characteristics, it was only necessary that the ORIGEN results be representative of this range. A satisfactory result was obtained by using decay and photon data from the Table of Isotopes, tabu-lated thermal cross sections and resonance integrals, and chain fission product yields.5 The resonance integrals of the principal fissile and fertile species were adjusted to obtain agreement with experiment and more sophisticated calculations. Soon after the ORIGEN computer code was documented, it was made avail-able to users outside ORNL through the Radiation Shielding Information Center (at ORNL). The relative simplicity of ORIGEN, coupled with its
- anvenient and detailed output, resulted in its being acquired by many organizations. Some of these organizations began using ORIGEN for appli-cations that required greater precision in the calculated results than those for which it had originally been intended. These applications were ganerally much more specific than the early ORNL generic fuel cycle i
studies, such as environmental impact studies that required relatively precise calculations of minor isotopes such as H, C, U, and 242,244Cm. The initial responses to these requirements were attempts to update specific aspects of ORIGEN and its data bases. However, incon-sistencies and a large number of different data bases soon resulted from these efforts. e , +" e w r==ge e at 3 s
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1.2 Scope of Revisions and Updates '1l8 In an effort to remedy the problems described above, a concerted f. program was initiated in 1975 to update the ORIGEN computer code and its associated data bases and reactor models. More specifically, the follow-4 s'l ing five aspects of ORIGEN were examined and updated: 1. the computer code itself, . i j 2.* cross sections and fission product yields, j 3. decay and photon data, ~1;j 4. reactor models, l 5. miscellaneous input information. a 1.2.1 Revision of the ORIGEN computer code One of the first aspects of the ORIGEN system to undergo modification was the computer code itself, yielding the code ORIGEN2. These modifica-0 tions are the subject of most of this report and of a companion report l, and will not be described in detail here. To summarize, the method for solving the nuclide generation and depletion equations is essentially unchanged. However, the input, output, and control aspect of ORIGEN have 4 undergone substantial changes to improve its flexibility and capability. The computer code ORIGEN2 and its capabilities will be described in more detail in Sect. 2. A general descriptio3 of the methods used to solve the differential equations representing the tuclear buildup and Jepletion processes is given in Sect. 3. 1.2.2 Update of the cross section and fission product yield library The major activity of the effort to update ORIGEN and its data bases was involved in updating the cross sections and fission product yields. Relatively sophisticated reactor physics calculatioas were undertaken for many different reactor / fuel combinations leading to a calculated neutron [ energy spectrum. This multigroup spectrum was then used to weigh multi-group neutron cross sections and fission product yields and to calculate new values for the ORIGEN flux parameters THERM, RES, and FAST. Spectrum-weighed cross sections were calculated for about 230 different nuclides ) - - ~. ,p.-, ..w - w,,..
... -........ ~... ~ - --- -
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for each reactor / fuel combination. Independent fission product yields were obtained for over 1100 nuclides. This was then reduced to a list of l about 850 nuclides by adding the yields of the very short-lived nuclides to their longer-lived progeny. For those nuclides where multigroup cross-section data were not available, the 2200 m/s cross sections and resonance integrals were updated using the most recent information. The results of this effort and the reports where the results are or will be documented are summarized in Table 1.1. Other reactor models will be added in the y future as the need arises. i Table 1.1. Summary of reactor / fuel combinations for which ? updated cross sections and reactor models were developed Reactor / fuel type Reference Uranium-plutonium cycle PWRs and BWRs 9 Alternative fuel cycle PWRs 10 Uranium-plutonium and alternative-cycle LMFBRs 11 Uranium-cycle CANDUs 12 1.2.3 Update of the decay and photon data i The decay and photon libraries were updated for about 450 of the principal radioactive nuclides using evaluated data in the Evaluated Nuclear Structure Data File (ENSDF) at ORNL. The information in ENSDF is normally published in the Nuclear Data Sheets.15 Information concerning decay half-lives, branching ratios, energies, intensities, photons (gamma and x-ray), and beta particles was abstracted from the file. The beta-particle data was used as input to a computer code that calculated the amount of bremsstrahlung resulting from deceleration of the beta particles. Additional information was also developed concerning photons from spon-taneous fission and (a.n) reactions and fission products that decay via neutron emission (i.e., delayed neutron precursors). All of this 4 information was combined to yield updated decay and photon libraries for ORIGEN. [ ~
i I I e 5 f 1.2.4 Update of the reactor models The updating of the reactor models includes such items as fuel en-richments, fuel specific power during irradiation, fuel burnup, amount and composition of fuel assembly structural materials (e.g., cladding) per unit of fuel, and the impurity concentrations in the fuel itself. This information was updated to include more recent reactor designs and fuel cycle concepts. The results of this update are or will be contained in the references listed in Table 1.1. 1.2.5 Update of miscellaneous input information The category " miscellaneous input information" includes many types of data whose only relationship is that they are not associated with any type of the major ORIGEN data libraries. These data include 1. neutrons per spontaneous fission, 2. neutrons per neutron-induced fission, 3. neutrons per (a,n) reaction, 4. chemical toxicities of the elements, 5. recoverable energy per fission, 6. the ORIGEN flux parameters THERM, RES, AND FAST. This information is discussed in Sect. 4 of this report. 1.3 Organization of ORIGEN2 and Its Data Libraries The ORIGEN2 computer code requires three different computer-readable libraries for complete operation: decay, cross-section/ fission product i yield, and photon. These libraries are maintained at ORNL as master libraries. The primary characteristics of the master libraries are that l (1) the data for each nuclide are listed only once, and (2) the data for all reactor / fuel combinations are listed in the libraries. This mechanism f ensures that only the most recent data are being used in all cases since l superseded data are deleted when the master libraries are updated. How-ever, the large amount of data in these libraries (the cross-section/ fission product yield library has over 25,000 cards) and the fact that they are not l1, I l! I
~- i 6 1 i organized into the requisite ORIGEN groupings (i.e., activation products, actinides and daughters, and fission products) makes these libraries un-suitable for use directly by ORIGEN2. ] The solution to this was to write a series of small computer programs j which access each of the three master libraries, select the data for the 'I desired reactor / fuel combination (cross-section/ fission product yield
- i
-l library only), organize it into the traditional ORIGEN three-group struc-ture, assign each group a unique number, and then output the resulting 1 libraries. These libraries are used by ORIGEN2 and distributed to outside j users. 1 It should be noted that it is possible to have more than one photon i library because the intensity of bremsst: ahlung depends heavily on the 'f medium in which the beta particle decelerates. This necessitates multiple bremsstrahlung libraries which are then combined with the single gamma-ray /x-ray master to yield the ORIGEN2-readable library. The bremsstrahlung library normally used with ORIGEN2 assumes a UO matrix. This will be 2 conservative (i.e., result in the maximum number of photons) for most ap-plications. * 'f 1.4 Availability of ORIGEN2 A computer code package has been deposited with the Radiation Shielding Information Center at ORNL. Inquiries or requests for the code I should be mailed to: i 1 Codes Coordinator Radiation Shielding Information Center Bldg. 6025 l Oak Ridge National Laboratory Oak Ridge, Tennessee 37830 or telephoned to: (615)-574-6176 or FTS 624-6176. f l
] 1 7 'l 1 t 1 i l 2. OVERVIEW OF THE ORIGEN2 COMPUTER CODE 1 The purpose of this section is to give a more detailed description of the ORIGEN2 computer code and its capabilities. Although the details of the mathematical methods used in ORIGEN2 (see Sect. 3) remain substantially unchanged from the original version of ORIGEN, the outward characteristics 1 appear to be substantially different. This detailed description-is divided into.two components: (1) a conceptual description of the capabilities of ORIGEN2, and (2) a description of the computer-related features of ORIGEN2. 4 2.1 Description of the Capabilities of ORIGEN2 2.1.1 Function of ORIGEN2 The general function of the ORIGEN2 computer code is to calculate the nuclides present in various nuclear materials and outpet the results in common engineering units and in a lucid format. The principal calcula-tional task involved in doing this is to determine the buildup and depletion of nuclides in these materials during irradiation and decay. Additional functions necessary to realistically simulate nuclear fuel cycles include the reprocessing (i.e., chemical separation) of nuclear materials and the continuous feed, removal, and accumulation of nuclear materials. In general, these features were present in the original version of ORIGEN although their exact scope has undergone some changes in ORIGEN2. The buildup and depletion of nuclides during irradiation is calcu-lated by ORIGEN2 using zero-dimensional (i.e., point) geometry and quasi-one-group neutron cross sections (see Sect. 2.1.3). This means that ORIGEN2 cannot account for spatial or resonance self-shielding effects or changes in the neutron spectrum other than those encoded initially. Thus ORIGEN2 (or ORIGEN) is not suitable for performing depletion calculations J on materials until the appropriate cross sections have been obtained from I more sophisticated reactor physics codes. The ORIGEN2 Computer code's internal operations use g-atoms as the measure of the amount of a specific nuclide that is present in some mixture. 9 ie -- - m.
__..._-.._e 't I l The variety of common engineering un!.ts available in the ORIGEN2 output is obtained by multiplying the g-atoms of each nuclide by constants and by j nuclide-dependent values from the decay library to convert to more useful
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units, such as grams, curies, photons, in: watts. The nuclide values are 1 then summed to form element values and tested to generate summary tables. g 2.1.2 ORIGEN2 input features a OAIGEN2 has three principd1 input features other than the standard 4 reading of libraries, instructions, and material compositions: free-format ".j Input, listing of input data, and substitute esta library cards. The free-I format input feature means that all of the input to ORIGEN2, including the 1 j data libraries, can appear any place on a card (1r card image) as long as i the data are in the correct order, of the correct type (i.e., real or integer), and separated by a comms and/or at least one space. Real numbers can be in either flonting point or exponential notation. This feature makes input preparation much simpler and reduces the :tumber of bad computer runs due to a value being in the wrong position on the card. It also permits comments to be placed on virtually every card sie e ORIGEN2 ceases to scan the card once the last datum expected on the card is encountered. A listing of the input data to ORIGEh2 is desirable so that the user 4 can determine what input was used to generate the output on hand at a later date. The data libraries (decay, cross-section/ fission product yield, and photon) have always been listed and are not included in this discussion. 1 The input listing now also contains all of the card input to ORIGEN2, which generally includes the controlling instructions and the material 4 i compositions. These data are read, printed on paper, and written to a t ,t scratch file. The scratch file is then rewound and ORIGEN2 reads the data 1 from the scratch file. q The substitute data library cards feature will read a limited number l} of substitute data cards from the card input unit. The data on these cards will then override the data in the primary library, which is most likely on a direct-access device or a tape. This feature eliminated the need to read in an entire library on cards simply to change a few data values. s k t t i j
9 2.1.3 ORIGEN2 calculational features ~ j The changes in the calculational features of ORIGEN that have been I incorporated in ORIGEN2 constitute a much larger and more significant set than either the modifications to the input or output aspects. A list of the more significant features is as follows: 1. The user can now define the flowsheet to be simulated in much more detail. ] 2. The cross sections of the principal fissile and fertile actinides l vary with burnup in ORIGEN2. I 3. The recoverable energy per fission (e.g., 200 MeV per fission) is now nuclide-dependent. 4. The calculated composition of materials in ORIGEN2 can be output and read back in at a later date. 5. A mechanism is included to account for the fission product yield of actinides that do not have explicit fission product yields 0 ). (e.g., U 6. Provisions have been made for including " nonstandard" neutron-induced reactions (e.g. (n He)]. 7. The fractional recoveries (i.e., separation factors) for the ORIGEN2 reprocessing operation ca's be specified by individual i element or by element group. The most visible change in going from ORIGEN to ORIGEN2 is in the method of defining the case to be calculated. The ORIGEN2 case is specified by a series of individual " commands," each of which defines a single operation (such as reading the data libraries), a single irradi-ation step, or a single output. The commands are relatively simple, 4 generally consisting of a three-or four-character keyword and a few l j numbers defining the conditions of the command. By using the commands, the user, in effect, writes a small computer program that defines the case. The program is read by ORIGEN2, " compiled" (i.e., translated and stored), and then executed. Within the bounds of logic, the user can simulate a l diverse assortment of situations using the ORIGEN2 and the presently exist-ing 30 commands. For example, the user.has detailed control over the movement, summation, and placement of material compositions in the columns t som e =, -,e ,a w -m,se,-,e * .e e
m- -_m... __.._.___._____.___z__m m c .m - i i 10 i visible in the ORIGEN2 output and in storage areas, which are similar in appearance. Commands are available which make the simulation of recycle situations straightforward. A different command will blend two material compositions so that the product's infinite multiplication factor (LHF) is equal to a specified value or to the IMF of a third material. Other commands, which control the ORIGEN2 output, will be discussed in Sect. 2.1.4. Two far less visible, but extremely important, features of ORIGEN2 are the variable actinide cross sections and the nuclide-dependent re-coverable energy per fission (REPF). Early in the program to update ORIGEN and its data bases, it was discovered that the use of a constant, average 1 cross section from a sophisticated reactor physics code for the principal 235,238 239-242 actinide nuclides such as U and Pu would not give correct depletion results. The cause of this was determined to be (1) the cross sections of some of these isotopes did not vary linearly with burnup, and (2) the cross section near the end of irradiation is more important in determining the discharge fuel composition than the cross section near the beginning of irradiation. The solution to this problem was to obtain cross sections as a function of fuel burnup for the principal actinides and use these in ORIGEN2. These cross sections are in DATA statements in ORIGEN2 in the form of discrete interpolation points. At the beginning of each irradiation step the burnup is estimated, the variable cross sections are calculated by interpolation, and the values substituted in the matrix of differential equations being solved by ORIGEN2. If a fission cross section of a nuclide having direct fission product yields is altered 3 (e.g., 235 ), then the array containing the product of the fission product U yields and the fission cross section is also adjusted to reflect this ! 4 change. Each different reactor / fuel combination can have either a dif-ferent set of variable cross sections, a different set of actinides with variable cross sections, or both. The variable cross sections for each of the reactor / fuel combinations are listed in the reports discussed in l Sect. 1.2.2. A second aspect of the original ORIGEN had to be changed before the cross sections from the sophisticated reactor physics codes could be used to predict the correct results - the previously constant t i l-L .-x- . l
] 3 1 1 3 11 1 i. REPF value of 200 MeV per fission had to be replaced by a REPF appropriate i for each nuclide. If this was not done, the flux required to sustain a given amount of power was not accurate and, thus, the amounts of neutron capture products in the discharged material was in error. For plutonium-enriched systems, the error it. the flux could be on the order of 5%. A more detailed discussion of the nuclide-dependent REPF is given in Sect. 4.5. The ability of ORIGEN2 to output the calculated compositions of various materials and to read them at a later time has already proven to be a very useful option. The composition is in g-atoms and is written in the same format as that for manually punched compositions. The mass of each nuclide is tested before being written and, if it is less than some -25 cutoff value (10 being typical), the nuclide is not output. This feature's principal use thus far has been in allowing irradiation calcu-lations to be' separated from decay calculations. An irradiation calculation is performed with ORIGEN2, and the charge and discharge compositions are written to some data storage device. A separate ORICEN2 run then reads this information and performs the decay and reprocessing calculations, which are generally the desired product. This procedure eliminates the necessity for repeating the irradiation calculation, which is very time consuming, every time a different decay time or output table is desired. Additionally, the set of equations to be solved in the decay calculations is significantly smaller than that for the irradiation calcu-lations since flux-dependent reactions are not included. A mechanism has been included in ORIGEN2 to account for fission product yields from those actinides that have nonzero fission cross sections but which are not explicitly included in the data base as are the yields for the principal actinides (233,235,238 239,241 232 g p Th). At the be-ginning of each irradiation step ORIGEN2 sums the fission rates of all of the actinides that do not have explicit fission product yields (i.e., unconnected actinides) and also determines the largest contributor in this category. The identity of the largest unconnected actinide contributor is used to find the nearest neighboring actinide that does have explicit fission product yields (i.e., the nearest connected actinide). The fission 9 P 1 r +
.. ~. _ _.. _ __._. -.._. __ __ _ - m 12 product yields of the nearest connected actinide are adjusted upward to ] compensate for the yield of all of tne unconnected actinides. In thermal j reactors, this correction is asually small (<0.5%). However, in a fast reactor the correction factor is on the order of 7%, principally due to 240Pu. I Provisions have been made in ORIGEN2 for the inclusion of nonstandard, i flux-dependent reactions in the calculation. A nonstandard reaction is f j defined as a reaction for which there is no provision in the cross-section/ l fission product yield library. Although an extremely large number of reactions could fall into this class, only a very few are of any signifi-I cance. Examples of this type of reaction are Pu(n,4n) Pu and 6 (n, He) 0 C. s j Finally, the fractional recoveries that are used by ORIGEN2 to calcu-late the products of a reprocessing (chemical separation) operation can be specified by element group as well as by individual element. In the element group method, each element is assigned to a group based on its chemical behavior during separation. For example, all of the noble gases would be assigned to the same group. Then each group has associated with it a fractional recovery which is appropriate for all of the elements in j the group. This approach greatly reduces the number of fractional recoveries that have to be specified or changed since 10 to 15 groups are usually adequate and the element membership of a group is relatively Constant. 2.1.4 ORIGEN2 output features The basic appearance of the ORIGEN2 output is the same as that in the original version of ORIGEN. However, several features have been added to make the output more versatile and easy to use: } 1. Additional output table types have been incorporated. l 2. " Road map" features have been added to enable the user to more i easily find the desired information. 3. Dual output units have been incorporated to minimize the amount of output on paper. I i 1
- 1 y.
13 4. The user can control the labeling of the ORIGEN2 output columns. l'i 5. A more flexible testing procedure for summary tables has been i incorporated. 1 6. Auxiliary information is printed on separate output units. ORIGEN2 has provisions for writing 22 different types of output l Tables. These table types are listed in Table 2.1. The 22 available 5 table types are much larger than the 7 available in the original version, principally because of the addition of the " fractional" tables which calculate the fraction of a table total constituted by each nuclide and element. Aside from these tables, the new types of tables are as follows: 1. Isotopic composition of each element. Gives the isotopic composition of each element in a table in atom fraction or weight fraction (i.e., each element totals to 1.0, and the table total is equal to the number of elements). 2. Chemical ingestion toxicity. Gives the amount of water required to dilute a material to chemically acceptable levels. Obtained by dividing the grams of each element by the maximum permissible concentration in g/m water; see Sect. 4.4 for further details. 3. Neutron absorption rate. Gives the absorption rate of neutrons in each nuclide and element in neutrons /s. 4. Neutron-induced fission rate. Gives the fission rate of the actinide nuclides and elements in fissions /s. 5. Radioactivity (alpha). Gives the alpha decay rate for each actinide nuclide and element in curies. These table types were incorporated to alleviate the need to perform extensive manual calculations to obtain certain types of information. Provisions for controlling the output tables are also included in ORIGEN2 so only those tables that are desired need to be output. j " Road map" features have been added to ORIGEN2 to aid the user in finding the information of interest. These features.:re the following: 1. Each output page is labeled " Activation Products," " Actinides + daughters," or " Fission Products," according to the contents of )
.._____. m _._. _ _ _ __ _._ ....-m.mu__.__._._.._.m .i d Lj
- i 14 d
G 'l Table 2.1. ORIGEN2 output table description i' Table description Units
- d Isotopic composition of each element atom fraction j
Isotopic camposition of each element weight fraction Composition g-atoms Composition atom fraction Composition g Composition wt fraction Radioactivity (total) Ci .] Radioactivity (total) fractional q ~ Thermal power watts N ~ Thermal power fractional Radioactive inhalation toxicity a air Radioactive inhalation toxicity fractional Radioactive ingestion toxicity m water Radioactive ingestion toxicity fractional Chemical ingestion toxicity m water Chemical ingestion toxicity fractional Neutron absorption rate neutronsIs i Neutron absorption rate fractional Neutron-induced fission rate fissions /s Ncutron-induced fission rate fractional Radioactivity (a) Ci Radioactivity (a) fractional (a.n) neutron production neutrons /s Spontaneous fission neutron production neutrons /s Photon emission rate photons /s, MeV/s, MeV/ watt-s t b I. i i, w....,..-..f 9
~ a 15 the page. 2. Each output page is numbered sequentially. 3. A table of contents is printed during execution that lists each ~ table printed and the page on which it begins. The first of these greatly aids in distinguishing the activation product tables from the fission product tables since both have many of the same "j nuclides. Hopefully, the combination of the latter two features will enable the user to scan the table of contents, identify the table of interest and its page number, and then proceed directly to the correct t page. Dual output units have also been incorporated into ORIGEN2. The principal objective of having dual output units is to allow a reduced 4 amount of output to be printed on the first unit while writing a much larger amount of output to a storage device or on microfiche using the second unit. This procedure is currently in use at ORNL and results in considerable information being available with only the most commonly used information (i.e., mass, radioactivity, thermal power, neutron activity, and photon emission rate) appearing on paper. The user can also control the heading of the ORIGEN2 output columns by using a command that allows a column heading of up to 10 characters to i be specified. This heading will remain associated with the column during 4 1 most of the operations carried out in ORIGEN2. Commonly used alphanumeric ? headings are CHARGE, DISCHARGE, HLW, etc. f A more flexible testing procedure for the output summary tables has been included. The summary tables contain only those nuclides or elements that contribute more than a certain fraction of the table total. In the past, the contribution of each nuclide in a single ORIGEN column was com-pared to a specified cutoff value. If it was greater, the nuclide was included in the summary table; if it was less than the cutoff value, it was excluded. In ORIGEN2, the cutoff value is not a specific number, but rather a fraction of the column total. Thus the contribution of a nuclide '}* or element is tested against the cutoff value (fraction) times the colu=n total. Commonly used cutoff values are 0.1% and 1%. A second change in the summary tables is that the user can specify whether one column is to t t f l i q j
=-. ~. u. .:. : _.. :. _s. a . :...... a.z. =. - . :.=- :. 3 lD 16 1 i j be tested, all columns except one are to be tested, or all columns are to be tested. If multiple columns are tested, a nuclide is printed in the sunmary table if it is larger than the cutoff value times the column total for any of the columns tested. j The final major output change is the printing of various auxiliary information used internally in ORIGEN2. One type of auxiliary information printed concerns the variable cross sections that are being used in each i ORIGEN2 irradiation step and the adjustment of the fission product yields to account for actinides without explicit fission product yields (see Sect. 2.1.3). Another type of auxiliary information output is a one-line message printed at the beginning of execution of each ORIGEN2 command. This serves to inform the user of exactly where in the stream of commands an error occurred. Also included in this output stream is internal infor-mation concerning the calculation of the neutron flux, specific power, and other parameters not directly specified by the user. A final type of auxiliary information, which is printed at the beginning execution of each output command, is the amount of space required thus far in the calcula-tion. This type of information is required to make proper use of the variable dimensioning feature of ORIGEN2 (see Sect. 2.2.1). 2.2 Computer-Oriented Descr.iption of ORIGEN2 The ORIGEN2 computer code is currently comprised of about 7300 source statements. It is written entirely in the FORTRAN computer language. Certain IBM-specific features -(e.g., partial-length words) have been used in the version at ORNL to conserve space during execution. However, these features are specified in such a way that their elimination is not partic-ularly difficult. A CDC-compatible version of ORIGEN2 is also available from the ORNL Radiation Shielding Information Center (see Sect. 1.4). There are currently about 60 subroutines in the ORIGEN2 source deck. The number of subroutines will continue to increase slowly since the addition of new reactor models necessitates the addition of cross sections contained in a new subroutine. Since there are so many subroutines and the ORIGEN2 internal logic,can be somewhat complicated, a list of the ') + 1 -.x
. k* - 1 17 subroutines and their general function (s) is given in Table 2.2. The subroutine scopes have been defined so as to facilitate the use of the OVERLAY function which places only the necessary subroutines in core at f* a given time and thus minimizes the amount of space required during } execution. The subroutines are also set up to facilitate the use of variable dimensions. The dimensions of all major arrays can be varied in the j primary calling routine (MAIN), which is comprised of about 80 cards. I The other primary function of MAIN is to call the subroutines MAIN 1, MAIN 2, MAIN 3, and LISTIT. The dimensions of the arrays in MAIN for a wide variety of cases are given in the ORIGEN2 user's manual along with 8 i an estimate of the amount of space required for a particular case. { Finally, it might be useful for the user to have some feeling for the order in which the ORIGEN2 subroutines are called to facilitate input specification and debugging. MAIN is the primary routine which calls LISTIT, MAIN 1, MAIN 2, and MAIN 3 sequentially. MAIN 2 calls the appropriate XSECan sybroutine. MAIN 3 calls a large number of subroutines with the order depending significantly on the user-specified commands.
- However, in general, the first subroutines called from MAIN 3 are NUDATl (plus DECRED), NUDAT2 (plus SIGRED), NUDAT3, and NUDOC, which read the data libraries, and ANSF, which sets up the spontaneous fission and (a,n) 4 neutron data for later use.
PHOLIB would then be called to read the - j photon library. Next, initial material compositions would be read by MAIN 3. Then the irradiation / decay subroutines FUDGE, FLUXO, DECAY, TERM (plus MATREN.), and EQUIL are called, in the order listed, to calculate 'he nuclida generation and depletion. The final operation is to output ' i the.'sulcs. This involves calls to OUTPUT, OUT1, OUT2, NUTRON, and i GAMHA. "'e other subroutines in Table 2.2 are called in a wide variety of places at various times. ( 6 f I
l, - &m. a zn.> .a. . ~, s. .'~-.+z. A. t s .:1.1 18 N Table 2.2. Functional description of ORIGEN2 subroutines Subroutine or j function subprogram Description i:( ?j MAIN 1 Initializes and/or reads data such as 'l neutrons per spontaneous and neutron-j induced fission, chemical toxicities, j .and fractional reprocessing recoveries '? MAIN 2 Reads and stores the information on the I ORIGEN2 commands; obtains the correct ] set of variable cross sections from a -j XSECan subroutine (see below) if required ijj MAIN 3 Executes the ORIGEN2 commands read by MAIN 2; this is actually a control sub-routine since most of the calculations J are done in the subroutines called by l MAIN 3 OUTPUT Controls output of most tables OUT1 Sums columns to generate totals for use in printing fractional tables and sum-mary tables OUT2 Writes most tables } NUDATI Reads and temporarily stores the decay library i j NUDAT2 Reads and temporarily stores the cross-section/ fission product yield library NUDAT3 Accesses the temporarily stored infor-mation from NUDAT1 and NUDAT2 and rearranges it to form the matrix of equations solved by ORIGEN2 (see .j Sect. 3.5) NUDOC Prints of ORIGEN2 statements pointing to the documentation-DECRED ~ Reads the decay data for a nuclide; called by NUDATl SIGRED Reads the cross-section/ fission product yield data for a nuclide; called by NUDAT2 ANSF Accesses, combines, and stores internal information related to the production of spontaneous fission and (a,n) neutrons for future use t L }
l 19 Table 2.2 (Continued) Subroutine or function subprogram Description PHOLIB Reads, organizes, and atores the photan i library NUTRON Outputs the spontaneous fission and (a.n) neutron production rate tables based on information stored by ANSF GAMMA Outputs the photon tables based on infor-mation stored by PHOLIB HEAD Writes the mixture of ORIGEN2-generated and ur,er specified column headings; called mostly by OUTPUT, OUT2, CAMMA, .} and NUTRON 'f l TOC Prints the table of contents FLUXO Calculates the neutron flux from specific power or vice-versa; nuclide-dependent recoverable energy per fission and the fission product-yield adjustments are also accomplished here FUDGE Calculates and incorporates the correc a variable actinide cross sections i DECAY, TERM, Solve buildup and depletion equations MATREX, EQUIL NOAH Converts six-digit integer nuclide identi-l fiers (see Sect. 4.7) into an alpha-numeric element symbol, three-digit atomic mass, and one character ground / excited state identifier i j RMASS Returns the atomic mass of a nuclide to the calling subroutine i (i RTIME Returns a factor for converting the i specified time units to seconds to the l calling subroutine i IPAGE Returns the current page number to the calling subroutine LISTIT Prints the card input data on paper and writes it to a temporary data set; called from MAIN l t 1 1 I l s.-.
_ - a.m_.. . ____...__.. m m __._ -- -.. _.. -. _. m._ _ _ c ._.m _s w_. t. i d 20 t t . h, ] Table 2.2 (Continued) Subroutine or function subprogram Description ADDMOV Adds and moves ORIGEN2 columns; used extensively internally BLOCK DATA Initializes a large number of variables and labels AREAD, DREAD, QQPACK, Read free-format input; QQPACK is QQREAD, READ, IREAD machine-dependent i q XSEC01 TO XSECnn Contain the variable cross sections j with one reactor / fuel type per sub-routine S J 3. DESCRIPTION OF THE MATHEMATICAL METHOD USED IN ORIGEN2 This section presents a summary description of the mathematical methods used to solve (1) the differential equations describing the build-up and depletion of nuclides, and (2) the equations used to calculate the neutron flux level and the specific power during irradiation. A relatively detailed description of the methods used to generate and store the matrix to be solved is also given. These descriptions are based heavily on the original write-up given in ref. 1 because the mathematical m2thods used in ORIGEN2 are fundamentally the game as those used in ORIGEN. 6 ) A general expression for the formation and disappearance of a nuclide by nuclear transmutation and radioactive decay may be written as follows: 1 N N L 'ik kk ~ i 1} i ij j j + + ~
- )
(1} d j=1 h=1 i where X is the atom density of nuclide i., A is the radioactive dis-integration constant for nuclide i, a is the spectrum-averaged neutron + t y
4 1. 21 are the fractions absorption cross section of nuclide i, and i ) and fik g of radioactive disintegration and neutron absorption by other nuclides I which lead to the formation of species 1. Also in,Eq. (1), i is the
- l i
position-and energy-averaged neutron flux. Rigorously, the system of i equations described by Eq. (1) is nonlinear since the neutron flux and it il crots sections will vary with changes in the composition of the fuel.
- j.
However, the variation with time is slow and, if they are considered to be v.
- )
constant over short time intervals, the system of Eq. (1) is a homogeneous ,j set of simultaneous first-order ordinary differential equations with constant coefficients, which may be written in matrix notation as X=AX. (2) ~ =~ 3.1 Matrix Exponential Solution 3.1.1 General solution Equation (2) has the known solution X(t) = exp (At) X(0), (3)
- g where X(0) is a vector of initial atom densities and A is a transition
= l matrix containing the rate coefficients for radioactive decay and neutron capture. The function exp (At) in Eq. (3) is the matrix exponential 2 function, a matrix of dimension N, which is defined as (At) (At)" = ~ exp (At) = I + At + +... = (4) 2., m., ~ = = -l m=0 1 If one can generate this function accurately from the transition matrix, ? then the solutien of the nuclide chain equations is readily obtained. 3.1.2 Computation of the matrix exponential series l Two principal difficulties are encountered in employing the matrix r exponential technique to solve large systems of equations: (1) a large amount of memory is required to store the transition matrix 4 t j t
e.-- ..._._m__2.._. c._ -...._.-.._=.u.m._.z. u 2 1 '.} 22 l l and the matrix exponential function, and (2) computational problems are .{ encountered in applying the matrix exponential method to systems of l equations with widely separated eigenvalues. The generation and storage j of the transition matrix are explained in Sect. 3.5. The computation and storage of the matrix exponential function have been facilitated by j developing a recursion relation for this function which does not require
- f storage of the entire matrix. Thus it is possible to derive an expression for one nuclide in Eq. (3) which is given by a
x (t) = C", (5) n=0 where C is generated by use of a recursion relation 0C = x (0) (6a) 1 g N C". (6b) C = a j=1 Here, a is an element in the transition matrix that is the first-order rate constant for the formation of species i_ from species 1 This algo-rithm requires storage of only one vector C" in addition to the current value of the solution. In performing the summation indicated by Eq. (5), it is necessary to ensure that precision in the answer will not be lost due to the ad-1 dition and subtraction of nearly equal large numbers. In the past, this objective has been accomplished by scaling the time step by repeatedly dividing by two until the norm of the matrix is less than some acceptable small value, computing the matrix exponential function for the reduced time step, and repeatedly squaring the resulting matrix to obtain the 17"l9 desired time step. Such a procedure would be impractic.able for a computation involving large numbers of nuclides (many of which have short .. - -. ~... .~.
23 l half-lives) corresponding to large norms of the At matrix. However, it is = just for these short-lived isotopes that the conditions of secular and transient equilibrium are known to apply. Thus in the computations per-formed by ORIGEN2, only the compositions of those nuclides whose diagonal 4 4 matrix elements are less than a predetermined value are computed by the 1 j matrix exponential method. The concentrations of the isotopes with large diagonal matrix elements are computed using an analytical expression for q the conditions of secular or transient equilibrium, as described in Sect. 3.2. 's 19 Lapidus and Luus have shown that the accuracy of the computed matrix exponential function can be maintained at any desired value by con-1 trolling the time step such that the norm of the matrix At is less than a J 'i predetermined value which is fixed by the word length of the digital computer used in the calculations. They define a norm of the matrix A, denoted by (A), as the smallar of the maximum-row absolute sum and the maximum-column absolute sum: [A] = min max a , max a (7) wherela ldenotestheabsolutevalueoftheelementa They show that the maximum term in the gummation for any element in the matrix exponential functioncannotexceedh,wherenisthelargestintegernotlargerthan [ Alt. Consideration of the word length of the computer used to perform the calculations will indicate the maximum value of n that can be used while obtaining a desired degree of significance in the results. Using i double precision arithmetic, the IBM 360 operating system can perform i operations retaining 16 significant decimal figures. In the ORIGEN2 code, the norm of the transition matrix is restricted to be less than ~ [A] < -2 in 0.001 = 13.8155, so that the maximum term that will be calcu-l lated will be approximately 49,000. Thus a value as small as -6 exp (-13.8155) = 10 can be computed, while retaining five significant figures. A sufficient number of t'erms must be added to the infinite summation given by Eq. (5) to ensure that the series has converged. The Y l l j !~ x F j '. - - - - - ~. - - _ n A
-..r.
- c. u.
==:. .a. : :.-.u o... a :. z.ea..=.::.=. =. -. u =. : w x ,) 'a .1 24 d! ^ ] mth term in the series for e[A] is equal to [A]m, which, for large values 4 of m, can be approximated by ( )"(2mn) , using Stirling's approxi- ~ f mation. The value of the norm, [A], is calculated by the code; and m_ is -I 7 set equcl to the largest integer in 7 [A] + 5, which has been determined as a " rule of thumb" for the number of terms necesnary to limit the error J i to <0.1%. Thus for [A] equal 13.8155, 53 terms will be required in the 4
- j summation. The absolute value of the last term added to the summation in
~1
- ]
this case will be <6.4 x 10 , which is sufficiently small compared with [ ~0 10 It has been observed that the norm is usually less than its maximum N value, and, in most cases, 30 or fewer terms are required to evaluate the series. It has been mentioned that, in previous applications of the matrix i exponential method, the restriction of the size of the norm of the tran-sition matrix necessary to treat nuclides with large eigenvalues was 1 accomplished by repeatedly dividing the matrix by 2, and the final value of the matrix exponential function was obtained by repeatedly squaring the i resulting intermediate matrix exponent,ial function. In the present appli-cation, the suggestion of Ball and Adams that the transitions involving isotopes with large decay constants be considered " instantaneous" was adopted; that is, if A + B + C and if the decay constant for B is large (i.e., B is short-lived), the matrix is reformulated as if C were formed from A directly, and the concentration of B is obtained by an alternative technique. Similarly, if the time constant for A is very large, the transition matrix is rewritten as if the amount of isotope B initially present were equal to A + B, and only the transition B + C is obtained by the matrix exponential technique. This reduction of the transition matrix and the generation of the solution by the matrix exponential method are performed by the subroutine TERM. 3.2 Use of Asymptotic Solutions of the Nuclide Chain Equations for Sho'rt-Lived Isotopes The numerical techniques described in Sect. 3.1 are applied only to obtain the solutions for isotopes that are sufficiently long-lived'to satisfy the criterion that the norm of the transition matrix be <2 in 1000. + t. t - - ~ - - -
l 25 1 Short-lived isotopes are treated by using linear combinations of the homogeneous and particular solutions of the nuclide chain equations that are computed using alternative procedures. 3.2.1 Short-lived nuclide present initially The quantity of a short-lived nuclide (originally present at the be-ginning of an interval) that remains at the end of the interval is computed in subroutine DECAY using a generalized form of the Bateman 3 l equations which treats an arbitrary forward-branching chain. The gener- .'l alized treatment is achieved by searching through the transition matrix .l and forming a queue of all short-lived precursors of an isotope. The i j Bateman equation solution is then applied to this queue. The queue is terminated when an isotope having no short-lived precursors is en-e countered. The algorithm also has provisions for treating two isotopes with equal eigenvalues and for treating cyclic chains. Bateman's solution for the ith member in a chain at time t_ may be written in the form N(t)=Ny0)e g ~ 1-1 1-1 p exp (-d t) - exp (-d t) 1-1 a +1,n (8) 3 i n + b k(0) (d - d)) " + 1,j n k d - dj j n 1 "d k=1 .j=k - where N (0) is the amount of isotope j initially present and the members of the chain are numbered consecutively for simplicity. This method of solution used the convention that a +1,n is equal to the product n
- k+1,k, *k+2,k+1****i,1-1, and that the empty product (k > i) is equal to 4
unity. The notation a for the first-order rate constant is the same as that described in Sect. 2.1, and d = -a In the present y 1 application, Eq. (8) is recast in the form 1-e k 5 i4
a.:.:
- .u.-
---.a.w.a.:... = = a:- - a..;. .-.~: W o 26 N (t) = N (O)e g g 1-1 1-1 exp (-d t) - exp (-d t) 1-1 d 1-1 a +1,n 3 i n n + f (0) H d H i b d j (di-d) n=k n d - dj n=k n j ,x =1 j=k nej q 1-1 by multiplication and division by H d. The first product in the outer n=k i summation of Eq. (9) has significance because it is the fraction of atoms d of isotope k that follow a particular sequence of decays and captures. -6 If this product becomes <10 , contributions from nuclide k and its pre-cursors to the concentration of nuclide i are neglected. The inner summa-q tion in Eq. (9) is performed in double precision arithmetic to preserve accuracy. This procedure is unnecessary for evaluating the outer summation because all the terms in this sum are known to be positive. The diffi-1 culties described by Vondy in applying the Bateman equations for small values of d t do not occur in the present application since, when this i condition occurs, the matrix exponential solution is employed. The matrix [l exponential method and the Bateman equations complement each other; that is, the former method is quite accurate when the magnitude of the characteristic values of the equations to be solved is small, whereas the Bateman solution encounters numerical problems in this range. For the case where two isotopes have equal removal constants (di = dj), the second } summation in Eq. (9) becomes t i'i 1-1 4 -d t 1-1 d 3 n dte n (10,' j e - dj n=k n j =k ntj An analogous expression is derived tor che case when d =d. These forms ,of the Bateman equations are applied when two isotopes in a chain have the same diagonal element or when a cyclic chain is encountered, in which case a nuclide is censidered to be its own precursor. i j-.. - _..
27 3.2.2 Short-lived daughter of a long-lived parent In the situation where a short-lived nuclide has a long-lived precursor, a second alternative solution is employed. In this instance, j the short-lived daughter is assumed to be in secular equilibrium with its parent at the end of any time interval. The concentration of the parent is obtained from subroutine TERM, and the concentration of the daughter is calculated in a subroutine named EQUIL by setting Eq. (3) equal to zero: .1 1 l N x
0
a x. (11) 1 g j j=1 1 Equation (11), which is a set of linear algebraic equations for the concentrations of the short-lived isotopes, is readily solved by the Gauss-Seidel iterative technique. The coefficients in Eq. (11) have the property that all the diagonal elements of the matrix are negative and all off-diagonal elements are positive. The algorithm involves inverting Eq. (11) and using assumed or previously calculated values for the unknown concentrations to estimate an improved value, that is, N (12) x =- a x j=1 1 j/i The iterative procedure has been found to converge very rapidly since, for these short-lived isotopes, cyclic chains are not usually encoun-tered and the procedure reduces to a direct solution. t I ' l
- l i
{
~ ,........ _.... ~.,.. < -. ~ _. _
- v. u a.:-.
3 l 28 3.3 Application of the Matrix Exponential Method for Nonhomogeneous Systems Certain problems that involve the accumulation of radioactive 'l materials at a constant rate and are of engineering interest require the solution of a nonhomogeneous system of first-order linear, ordinary differential equations. In matrix notation, one writes X=AX+B. (13) ~ ~~ ~ This set of equations has the particular solution '} ~ X(t) = [exp (At) - I] A B, (14) ~ z a a ~ provided that A-1 exists. Substituting the infinite series representa-tion for the matrix exponential function, one obtains At (At)2 X(t) = [I + 3 ;- + + ...] Bt (15a) 2. 3., ~ ~ = / (At)m "( (bl), Bt. (15b) m=0 ~l It should be noted that in many of the cases solved by ORIGEN2, A in Eq. (14) does not exist. However, the series solution [Eqs. (15a) and (15b)] does always exist; when substituted into Eq. (13), the correct solution results. Furthermore, since each term of Eq. (15) is smaller than the corresponding cerm in Eq. (4), Eq. (15)convergesforeachgt for which Eq. (4) converges. The particular solution may also be expressed as the sum of an infinite series x (t) = D", (16) g n=1 l i i
- 1'
.. ~..
1 29 whose terms are generated by use of a recursion relation 1 D =bt (17a) i i N D"g+ D"). (17b) = a y Once again, the algorithm is applied only to long-lived nuclides, and the ,1 concentrations of the short-lived nuclides are obtained by an alternative t' technique. In this situation, Eq. (11) is modified to the form N
- 1 =0=)[]a)xj+b L
i (18) j=1 and is solved by the. Gauss-Seidel method. After the homogeneous and particular solutions have been obtained, they are added to obtain the complete solution of the system of equations. 3.4 Computation of Neutron Flux and Specific Power i To accurately compute changes in fuel composition during irradiation at constant power, it is necessary to take into account changes in the j neutron flux with time as the fuel is depleted. The neutron flux is a l function of the amount of fissile nuclides per unit of fuel, the fission cross section for each fissile nuclide, and the recoverable energy per 4 fission for each nuclide. .4 3.4.1 Calculation of neutron flux given specific power l At the start of the computation, the kncwn parameters are the initial 5 j fuel ccmposition, the constant specific power that the fuel must produce 4 I during a time interval, and the length of the time interval. -l .l The instantaneous neutron flux is related to the constant specific j power at a fixed time by the equation 3 I
.a ...a .a -l 30 -19 $ I x, of R (19) P = 1.602 x 10 g i where P is the specific power, in MW per unit of fuel; x is the amount of f fissile nuclide i present in the fuel in g-atoms per unit of fuel; o is g the microscopic fission cross section for nuclide i; $ is the instanta-neous neutron flux, in neutrons cm~ ~ ; and R is the recoverable energy s g per fission for nuclide i, in MeV/ fission. The constant in Eq. (19) con-verts MeV/sec into megawatts. An approximate expression for the value of the neutron flux as a function of time during the interval is obtained by solving Eq. (19) for the instantaneous neutron flux $ and expansion of the resulting ex-pression in a Taylor series about the start of the interval: 18 0) $(t) = 6.242 x 10 P S(0) -t S(0)2 - S (0) '5 (0) +... 2 S(0) (20a) S(0) ~ ~ (} ( ~ () ( } +... (20b) $(t) = $(0) 1-t S(0) + 2 S(0) In this expression, the parameter S(0) = I x of R at the start of the g time interval. 5(0) and 5(0) are the first and second derivatives of S_ evaluated at the start of the interval. The values of the derivatives of S(0) can be evaluated since X(0) = AX(0) + B(0) and X(0) = AX(0). ~ ~~ ~ ~ .~ The average neutron flux during the interval is obtained by integrating over the interval and dividing by the length of the interval t_: Oh - 0)h(0 $ = $(0). 1-f + +... (21) \\ S(0) / Equation (21) is use.3 in subroutine FLUXO of ORIGEN2 to estimate the average neutron flux during a time interval based on the conditions at the start of the interval. The term involving the second derivative is only employed for the first time interval where, for some nuclides.
31 X is zero but 5(0) is nonzero, 1 4 t 3.4.2 Calculation of specific power given neutron flux l' The average power produced during a time interval for a fuel in a fixed neutron flux is estimated from the initial composition using a lj similarly derived equation: f f P = 1.602 x 10
- 4 S(0) 1 + f S(0) + f - S(0) +...
(22) 4 3.4.3 Other considerations i In order for the above procedures to estimate the average neutron flux or average specific power correctly, the changes in neutron flux during the interval must be relatively small. If the average value of either of these quantities differs from the initial value by more than 20%, a message will be printed out advising the user to employ smaller time increments. It was noted at the beginning of this discussion that the neutron flux is a function of the amount of fissile nuclides present, the nuclide fission cross sections, and the recoverable energy per fission. The variation in the amount of each fissile nuclide is accounted for by the derivatives in the time series described above and will not be considered further here. The nuclide fission cross sections also vary during the time interval but do not ne,cessarily correlate with the variation in nuclide mass. This variation is accommodated by calculating the param-eter S(0) and its derivatives using fission cross sections appropriate for the estimated fuel burnup at the middle of the time interval (see Sect. 2.1.3). Finally, the variation in the recoverable energy per fission is accounted for by using a constant value appropriate for each. nuclide instead of a single constant value for all nuclides. l e '1 i m.. ~. -
- q 1
1 l 32 i -1i 3.5 Construction of the Transition Matrix f An exrensive library of nuclear properties of radioactive isotopes has been compiled for use with the ORIGEN2 code.9-1 The data are in the form of half-lives, fractions of transitions that produce a given nuclear particle, cross sections, and fractions of absorptions that yield certain particles. These data are read from tape, direct access device, or cards and processed into a form for use by the mathematics routines in subroutines NUDAT1, NUDAT2, and NUDAT3 (hereafter called sub- [ routinds NUDATn). It is possible to compute the concentrations of as many as 1700 nuclides using the present code. However, straightforward construction of a generalized transition matrix would require the storage of a 1700 by 1700 array, which would tax the storage capacity of the largest computers available today. On the other hand, the transition matrix is typically very sparse, and storage requirements can be reduced substan-tially by storing only the nonzero elements of the matrix and two relatively small vectors that a::e used to locate the elements,. Sub-routines NUDATn are also used to generate the compacted transition matrix and the two storage vectors. Subroutines NUDATn process the libraries by reading a six-digit identifying numFsr, NUCL(I), for each nuclide, followed by the half-life and the fraction of each decay that occurs by several competing processes from the decay library. Neutron absorption cross sections for (n,Y), (n,a), (n p), (n, n), (n,3n), (n, fission) reactions, and fission product yields are then read from the cross-section library. The six-digit identifying number is equal to Z
- 10,000 + W
- 10 + IS, where Z is the atomic number, W is the atomic weight (in integral atomic mass units),
and IS is either 0 or 1 to indicate a ground state or a metastable state, respectively. This information is processed into a compacted transition matrix, as described below. First, the half-life is used to calculate the radioactive disinte-gration constant, 1. First-order rate constants for various competing decay processes are calculated by multiplying A by the fraction of I
't 1 33 1 transitions to that final state. The product nuclide resulting from i j each nuclear transition is next identified by the addition of a suitable I ~ constant to the six-digit identification number for the parent nuclide. (For example, for a S-decay, 10,000 is added to the parent identifier; for neutron capture, 10 is added; or for isomeric transition the quantity -1 is added.) Two arrays are constructed: the first, NPROD(J,M), contains all the products which can be directly formed from any nuclide i J by the transitions considered in the library; and the second, COEFF(J,M), contains the first-order rate constants for the corresponding transitions. 3;j When these arrays have been construct'ed, a search of the NPROD array is 't1 conducted to identify all the parents of a given nuclide I. [Nuclide J is s ] a parent of I if NPROD(J M) equals NUCL(I) for any reaction of type M.] f f ^ When a parent of nuclide I has been located, the value of the correspond-f ing coefficient a in the transition matrix is equal to COEFF(J,M). q However, direct storage of a in a square array would require an exces-sive amount of storage. Hence this procedure is avoided by incrementing an index, N, each time a coefficicnt is identified. The coefficients are stored sequentially in a one-dimensional array, A(N); the value of J is stored in another one-dimensional array, LOC (N); and the total number of coefficients for production of nuclide I are stored in a third array, NON0(I). When all of the coefficients for every nuclide have been stored, I the NON0(I) array is converted to indicate the cumulative number of matrix coefficients fer all the isotopes up to and including I [i.e., N N0(I + 1) l = N@NO(I) + NON0(I + 1) for all values of I greater than 1). After this procedure has been executed, the NONO array is a monotonically increasing .i list of integers whose final value is the number of nonzero, off-diagonal matrix clements in the transition matrix. This final value is preserved I separacely as the variable NON. For computational convenience, the values of the diagonal matrix elements are stored in a separate vector, D(I). To perform the multiplication of the transition matrix by a vector (e.g., N Ic x ), as is required to execute the algorithm described in I a = 1 g j=1 ',s i Sect. 3.1, the operations described in the flowchart given in Fig. 3.1 are 1 i employed. .i 2 . _ _, _, -. ~. ---..,~._.s.7-L n
~. I 3h 1 ORNL-0WG 72-7938 I=1 N=1 E X(I) = D(I)
- X (I')
F(N4T.N6NO(I) I=I+1
- FALSE
- J= LOC (N)
A (!) = i (I) +A (N)
- X (J) 9 N=N+1 TRUE-F(N.LE. N O N )
i efRLSE* l STOP l Fig. 3.1. Flowchart illustrating computational algorithm executed to perform the matrix calculation k = A X - -... ~. - _... b - -,~._..-- -.
?> 35 J Two types of data in the nuclear library require a departure from the procedure just described. In the case of neutron-induced reactions, I it is necessary to specify the neutron flux before first-order rate coef-4, ficients can be calculated as products of flux and cross sections. At the ' 1, time the matrix is generated, the neutron flux is unknown. Also, to perform a fuel depletion calculation, the flux must be permitted to vary with time. Thus when the nonzero, off-diagonal matrix elements for isotope I are stored, all those for formation by radioactive decay are grouped first and are followed by those for formation of I by neutron capture. Another vector, KD(I), is also generated and used in a manner analogous to NON0(I). It initially is the number of radioactive parents of isotope I, and the difference NON0(I) - KD(I) represents the number of coefficients for formation of I by neutron capture. The variables A,- LOC, NONO, and KD are all generated in subroutines NUDATn. They are used j to perform calculations in subroutines FLUX 0, DECAY, TERM, and EQUIL. _t The second exception to the standard procedure for constructing the transition matrix involves the coefficients corresponding to the fission f product yields. The nuclear data library contains direct fission yields for the formation of fission product isotopes from several fissionable nuclides. When these yields are multip1'ed by the fission cross section for the fissile nuclide and the neutron flue, the result is a first-order i rate constant for production of fission product isotope I by fission of I nuclide J. Hence for these data, the construction of the arrays NPROD l}lI and COEFF and the subsequent search procedure are not required..The l coefficients are entered directly into the A vector, and the correspond-l ing value of J that identifies the fissioning nucleus is recorded in the LOC array. 't The preceding description summarizes the basic construction of the fj transition matrix in ORIGEN2. This construction process is very nearly identical to that used in ORIGEN.1 However, there are three additional I8l( I l4 features in ORIGEN2 that result in modification of the transition matrix ,.j. as compared to that in ORIGEN. The first is that provisions have been made in ORIGEN2 to incorporate " nonstandard," flux-dependent reactions. These are neutron capture processes that may be important but which are l: i .--n u
i } 4 36 i q la j significant for only one specific nuclide and therefore are not included aj in the cross-section library format (e.g., 16 (n,3He)14 ]. Since the O C 4fl solution methods used in ORIGEN only require that the cross section and ~ the parent and daughter nuclides be specified, these are specified sepa- ] rately and incorporated into the transition matrix. Thereafter, these ) reactions are treated the same as any other flux-dependent reaction. 4 j A second feature of ORIGEN2 that substantially affects the transition ~ matrix is the provision for variable cross sections for certain principa1 ] actinide nuclides (see Sect. 2.1.3). Since these values vary during q i ORIGEN2 irradiation calculations and after the transition aatrix has been .i constructed, it is necessary to modify the appropriate elements of the ] transition matrix each time the cross sections are altered. This is accomplished by simply storing the locations of the variable cross sections in the transition matrix and altering those elements as required. A similar, but more extensive, problem occurs with the fission products since the variation of the actinide fission cross sections also affects the fission product yield. This effect is accounted for in the same t manner as the variation in actinide cross sections: by storing the locations of the fission product yields and adjusting them to compensate for the variation in actinide fission cross sections. The final feature requiring modification of the transition matrix also occurs after the matrix has been generated. ORIGEN2 has provisions to account for the production of fission products from those actinides which do not have explicit yields in the transition matrix (e.g., Np and 240Pu). This is accomplished by calculating the total fission rate in all of the actinides without explicit yields and adjusting the yields for an actinide that has explicit yields (e.g., U) to account for these ad-ditional fissions. This procedure requires that the explicit fission product yields be adjusted at the beginning of every time interval. This adjustment is accomplished in a manner very similar to that use' for accommodating the effect of the variable cross sections on the fission 'I product yields; viz., the locations of the yields in the transition matrix are stored and modified to incorporate the unaccounted-for yields at the beginning of each time interval. i f 6 .-m n.,-.y,.-,..
37 l 4. DESCRIPTION AND SOURCE OF MISCELLANEOUS ORIGEN2 DATA i ? This section describes the nature and sources of a wide variety of I miscellaneous data associated with ORIGEN2. The types of data described i ^ are as follows: I cj 1. neutron yield per spontaneous fission; 2. neutron yield per neutron-induced fission; 3. (a n) neutron yields; l 4. chemical toxicities; n .i 5. the recoverable energy per fission; 6. the methods for calculating values for the ORIGEN2 flux parameters THERM, RES, and FAST; .) j 7. the six-digit ORIGEN2 nuclide and element identifier. i The common basis for describing these data in this report is that (1) they are not associated with a particular reactor or fuel cycle, and (2) they are contained within the ORIGEN2 computer code itself and not in one of the data libraries read by ORIGEN2. l 4.1 Neutron Yield per Spontaneous Fission r, The calculation of the spontaneous fission neutron emission rate l required that the spontaneous fission decay branching ratios and the f neutron yields per spontaneous fission be specified. The values of the i 0 branching ratios are contained in the updated ORIGEN2 decay library and I will not be described here. The spontaneous fission neutron yields are j contained in BLOCK DATA in ORIGEN2. The data being used are given in i -l Table 4.1 and are a combination of measured and calculated values.23'24 4.2 Neutron Yield per Neutron-Induced Fission l 5 j Neutron yields per neutron-induced fission have been incorporated in i the BLOCK DATA subroutine of ORIGEN2 for use in calculating the infinite multiplication factor of the materials output by ORIGEN2. These values ? i i -{' a
f ].'i 38 were generated by weighing energy-dependent ENDF/B neutron yields j with multigroup neutron fluxes calculated as a part of the ORIGEN update effort. Two generic sets of neutron yields have been incorporated into ORIGEN2: (1) one for thermal reactors, based on a PWR-U spectrum; and i (2) one for fast reactors, based on an advanced oxide, plutonium recycle LMFBR. The two sets of neutron yields are listed in Table 4.2. The ORIGEN2 user can control which of the sets of neutron yields are used with a parameter in the MAIN subroutine. However, it is evident by inspection of the values in Table 4.2 that the thermal and fast neutron yields differ I little for the nuclides of significance in most cases. Thus very little error would result if the thermal neutron yields were used. 4.3 Neutron Yields from (a,n) Reactions t The neutrons resulting from (a,n) reactions of light nuclides com-prise a second important source of decay-induced neutrons. These neutrons can result from the interaction of energetic alpha particles with a wide variety of light elements such as beryllium and fluorine. In the co=mer-cial nuclear fuel cycle, these target materials are seldom encountered in concentrated form, and the principal source of (a,n) neutrons is usually 180. Since it is impossible for a single set of (a,n) neutron yields to be valid for all cases of a media-dependent reaction, values appropriate to a heavy-metal oxide matrix have been developed and included in ORIGEN. 1 f These (a,n) neutron yields have been incorporated in two ways. First, the yields of seven principal (a,n) contributors are given explicitly in j the BLOCK DATA subroutine of ORIGEN2. These values were obtained from ll ref. 27 and are given in Table 4.3. For those nuclides which generally contribute to a lesser extent, the (a,n) neutron yield is determined using a semiempirical equation that uses the alpha particle energy as the independent variable. The constants in this equation (Table 4.-3) were I 39 242 based on the measured (a,n) neutron yields for Pu and Cm as given j, in ref. 27. It should be noted that this equation has been significantly altered as compared to the equation in ORIGEN in order to improve the agreement of the equation with measured values. The results predicted t! 1
st 1 1 h 39 Table 4.1. Neutron yields per spontaneous fission used in ORIGEN2 li j. Neutrons per Neutrons per i spontaneous spontaneous j, Nuclide fission Reference Nuclide fission Reference 235 241 U l.695 2 Am 2.383 2 236 242 U l.650 2 Am 2.475 2 237 242m U l.872 2 Am 2.590 2 238 243 U 2.000 2 Am 2.520 2 d 239 244 U 2.048 2 Am 2.657 2 236 244m Np 1.783 2 Am 2.665 2 ,j 236m 241 Np 1.790 2 Cm 2.500 2 242 Np 1.873 2 Cm 2.590 2 i Np 1.963 2 ' Cm 2.687 2 244 'Np 2.053 2 Cm 2.760 2 236 245 Pu 2.220 2 Cm 2.872 2 246 Pu 1.886 2 Cm 3.000 2 248 Pu 2.280 2 Cm 3.320 3 O Pu 2.240 2 Cm 3.560 3 40 249 2 Pu 2.160 2 Bk 3.720 3 j 24bu 2.250 2 Cf 3.440 3 249 242 250 j Pu
- 2. 150 2
Cf 3.360 3 .i 243 252 Pu 2.430 2 Cf 3.725 3 254 Pu 2.300 2 Cf 3.900 3 40 253 Am 2.290 2 Es 3.920 3 4 'l a, e I t a--* ~. e- +g.,- w j 9
.i. _ -.... .... - ~.. ~. ~ ~ I 40 4 s t-j Table 4.2. Spectrum-averaged neutron yields ] per neutron-induced fission (V) ? Nuclide PWR-U LMFBR Nuclide PWR-U LMFBR .j 232 242m y Th 2.418 2.396 Am 3.162 3.311 j 233 243 Pa 2.663 2.631 Am 3.732 3.653 233 242 U 2.499 2.520 Cm 3.746 3.868 i 'U 2.631 2.555 Cm 3.434 3.496 243 235 244 U 2.421 2.468 Cm 3.725 3.743 236 245 U 2.734 2.614 cm 3.832 3.898 l 238 246 U 2.807 2.776 Cm 3.858 3.870 237Np 3.005 2.935 247Cm 3.592 3.680 236 248 Pu 2.870 2.946 Cm 3.796 3.866 238 249 Pu 2.833 3.009 Bk 3.760 3.671 239Pu 2.875 2.946 249Cf 4.062 4.130 240 250 Pu 3.135 3.024 Cf 3.970 3.813 241 251 Pu 2.934 2.978 Cf 4.140 4.227 242 252 Pu 3.280 3.075 Cf 4.126 4.364 'IAm 3.277 3.402 253Cf 4.150 4.151 242Am 3.360 3.361 r p 4 l' 4 i-i' I! ~~ ~ ~~ ~ ' '~ l
- - = .1j 41 Table 4.3. Neutron yields from (a,n) reactions l Neutron yield -l Nuclide (neutrons sec g-1) -1 'k 238 2.00 x 10' Pu 239Pu 45 240Pu 170 j 242P 2.7 ii 241 3 Am 4.00 x 10 242Cm 2.67 x 10 'i 244 5 [j Cm 5.72 x 10 All others 2.152 x 10-18 (E)l4.01, where E = alpha particle energy, MeV by the equation in Table 4.3 agree well with experimental values for alpha particles in the 5.0 to 6.2 MeV range, which is of interest in most commonly encountered situations. For light nuclide targets other 18 ] than 0 or for alpha particle energies significantly outside the 5 to ..1 6 MeV range, a different set of equation constants must be derived and-ij inserted in the ORIGEN2 BLOCK DATA subroutine. .] 4.4 Elemental Chemical Toxicities 'l -i.j Elemental chemical toxicity values have been incorporated into the ORIGEN2 BLOCK DATA subroutine. The values, which were obtained from l ref. 28 and are listed in Table 4.4, are given in units of maximum desired ambient concentration in water (i.e., g/m ). These values are used to calculate a measure of the chemical toxicity of an element l l .1
e.2. n.:...n.- :.:
- 2..
=. :.
- .: 2 :. =;. a:. ar ;s..asuw.;-
= ~ _ b) 1, '2 '} ,,j Table 4.4. Elemental chemical toxicities incorporated in ORIGEN2 .j Toxicity Toxicity Toxicity 3 3 3 i Element (g/m ) Element (g/m ) Element (g/m ) 1 H 3500 Se 0.01 Ho 1.0 1 He 0.2 Br 3.0 Er 0.1 .i .i Li 5.0 Kr 40 Tm 0.2 Be 1.0 Rb 50 Yb 0.1 B 1.0 Sr 10 Lu 0.1 C 400 Y 0.001 Hf 0.05 f N 0.01 Zr 1.0 Ta 1.0 I O 945,500 Nb 0.02 W 100 F 1.0 Mo 0.5 Re 10 Ne 1.0 Tc 100 Os 1.0 Na 1000 Ru 1.0 Ir 0.8 Mg 10 Rh 0.05 Pt 0.3 A1 0.01 Pd 0.05 Au 0.02 Si 5.0 Ag 0.001 Hg 0.002 P 0.01 Cd 0.01 T1 ~0.005 S 50.'0$ In 0.02 Pb 0.01 Cl 0.15 Sn 0.05 Bi 0.1 Ar 10 Sb 0.05 Po 0.2 K 1000 Te 0.2 At 10 Ca 30 I 10 Rn 500 Sc 0.5 Xe 150 Fr 5.0 Ti 0.1 Cs 5.0 Ra 0.001 U 0.1 Ba 0.5 Ac 0.02 4 Cr 0.02 La 1.0 Th 0.0005 Mn 0.01 Ce 2.0 Pa
- 0. 0 5 Fe 0.05 Pr 1.0 U
0.5 Co 0.05 Nd 0.2 Np 0.003 Ni 0.05 Pm 1.0 Pu 0.0008 j Cu 0.01 Sm 0.2 Am 0.04 1 Zn 0.05 Eu 0.2 cm 0.5 i Ga 0.2 Gd 0.2 Lk 0.005 Ge 0.5 Tb 0.5 Cf 0.01 As 0.01 Dy 1.0 Es 0.01 .- g 9 1 4 i
i 43 i mixture by dividing the mass of each element by its chemical toxicity and 4 summing the resulting water volumes. The total water volume thus calcu-i lated is the volume required to dilute the mixture to desired ambient levels and can therefore serve as a measure of the toxicity of the mixture. t e J 4.5 Nuclide-Dependent Recoverable Energy per Fission One of the principal features of ORIGEN2 is that the cross sections j it uses are calculated by more sophisticated reactor physics codes and ] thus are " actual" cross sections. If the neutron flux (or specific power) calculated from these cross sections is to be accurate, then the other parameters used in the calculation must also represent reality. The two .i other parameters involved in the' calculation (see Sect. 3.4) are the i concentrations of the various fissile nuclides and the receverable energy per fission. The fissile nuclide concentration (in g-atoms per unit of fuel) has always been on a "real" basis and need not be considered further. However, in the past the recoverable energy per fission was assumed to be 200 MeV per fission for all fissile nuclides in ORIGEN. This assumption worked well because the fission cross sections being used at that time had been adjusted to account for the difference between the 200 MeV value and the actual value. However, since the cross sections being used in ORIGEN2 are not adjusted, it became necessary to incorporate realistic values of the recoverable energy per fission into ORIGEN2. As.a result of the large number of actinides in ORIGEN2 and the lack of recoverable fission energy data on many of them, it was decided to use a semiempirical equation to calculate the recoverable energy per fission. The form of the selected equation is REPF = C (Z A * ) + C, where Z is y 2 the nuclide's atomic number, A is the nuclide's atomic weight, C and C y 2 are constants, and REPF is the recoverable energy per fission in MeV per fission. Equations of this general form are widely used to correlate fission product kinetic energy,29,30 although most have the exponent of the atomic weight as 0.33 instead of 0.5. However, work by Okolovich et al. l and the results of comparing the general form given above with evaluated data indicate that the 0.5 exponent gives slightly better results. 4 i
. u.. a - . ~ a. -..-..w _.. _.._ w : M 13 1 44 t!:: The constants in the general equation were determined by using the 235 239 REPF values for U and Pu as given in ref. 29. The resulting equation, which has been incorporated in ORIGEN2 subroutine FLUX 0, is 'f REPF(MeV/ fission) = 1.29927 x 10-3 (7 g.5) + 33.12. 20 1 This equation predicts REPF values with a maximum error of 1% for nuclides 3 between Th and ' Pu as compared to the evaluated data given in ref. 29. 4 U f, 4.6 ORIGEN2 Flux Parameters THERM, RES, and FAST f.i The flux parameters THERM, RES, and FAST are needed to allow thermal
- j cross sections, resonance integrals, and fission-spectrum-averaged thresh-old cross sections to be incorporated into ORIGEN2. This feature is highly desirable since multigroup cross sections are only available for about 200 of the 1300 distinct nuclides considered in ORIGEN2 and the only other source of cross-section information is compilations of standard cross sections such as BNL-325.32 Values for these flux parameters were 1
originally developed for ORIGEN based on the Westcott formalisms and generic flux shapes for thermal reactors. However, modern reactor physics codes supply sufficient information to allow these parameters to be derived from the calculated neutron spectrum instead of a generic spectrum. It is the purpose of this section to derive the recipes for calculating these parameters from information available in the output of a code that calculates a static neutron energy spectrum and spectrum-averaged neutron cross sections. l Before launching into the details of the derivation, it is necessary i for the reader to understand the neutron energy groups of interest. These groups are depicted schematically in Fig. 4.1 along with the variables j{ associated with each group. The variables will be defined in detail when 1 they are used in the derivations. The diagram in Fig. 4.1 shows four energy groups which are comprised of two pairs. In addition, there is a single, all-encompassing group characterized by the total flux and the total spectrum-averaged cross section. i l
^ ORNL DWG 79-1368 401 301 l ] p13' #13' RI l 2200 _! NO 1 i l [ j 7 THRESHOLD l I [ l O2' O2 (23' IFS' # 1 FS l THRESHOLD = 1 1 I 1 l I 4,a l 7 i = TOTAL l I i I I I I I I l 1 I l 1 1 E, = 0 E, = 0.5 eV E2 > 2 Mev E3 = 10-20 Mev NEUTRON ENERGY Fig, la.1. Schematic diagram of energy group structure for the calculation of ORIGEN2 flux parameters.
..m m.m.m.. m_ _ _.- =- q .i .] 46 4 4 1 .3 The ORIGEN2 flux parameters are constants applicable to a specific i i reactor-fuel combination. The " stand-J 'f cross sections (e.g., infinite dilution resonance integral), when multiplied by the appropriate flux parameter, yield an average cross section which can be appropriately multiplied by the total flux to yield the correct nuclide reaction rate. This procedure is necessary because the flux-parameter-averaged cross sections are stored in the ORIGEN2 transition matrix in the same manner as the multigroup-spectrum-averaged cross sections and thus must be com- 'j patible with them. } .,i 4.6.1 Derivation of ORIGEN2 flux parameters THERM and RES e The basic approach used in deriving expressions for calculating THERM and RES is to write two reaction rate balances for an unspecified nuclide: one in terms of a thermal cross section and resonance integral and the other in terms of averaged cross sections. For this derivation, the top pair of energy groups and the single-group (bottom) in Fig. 4.1 will be used. The neutron-induced reaction rate of an unspecified nuclide can be written in two ways: &$=C $ 2200 + C2 13
- RI (la)
T 7 01
- 01 + #13 13 3 (1b)
T 01 where -1 $ = Total flux from a reactor physics calculation, neutrons barn T -1 s = Thermal flux (0 to 0.5 ev) from a reactor physics calculation, 01 ~1 -1 neutrons barn s
- 13 = Resonance / fast flux (0.5 eV to maximum) from reactor physics calculation, neutrons barn ~1
-1 s 2200 = Thermal (22 m/s) cross section from source such as ref. 32, barns. r l
i fl 47 RI = Infinite dilution resonance integral (0.5 eV to infinity) from source such as ref. 32, barns. 01 O 1 (i.e., between 0 and l 0 = Effective cross section between E # ], 0.5 eV), from a reactor physics calculation, barns. ?
- f 0
= Effective cross section between E and E3 (i.e., between 0.5 eV 73 y j, and maximum) from reactor physics calculation, barns. I:I U = Effective cross section between E0 "" 3 (i.e., between 0 and
- j the maximum) from reactor physics calculation, barns.
.j y C.C = Constants accounting for effects within energy groups. 1 2 ':t'J Dividing Eqs. (la) and (lb) by the total flux $ yields .] T .c a ~1 C *$
- 02200 + C
- 13 y 0l 2
(2a) a" T T and
- c
- U 01 + @l3 01 13
__a= (2b)
- T
- T The ORIGEN2 flux parameters THERM and RES are now defined as
- D 1 01 THERM E (3a)
$T and 2 @l3 RES E (3b) i T Using these definitions Eqs. (2a) and (2b) can be rewritten as q II = THERM
- 02200 + RES*RI
@a) .j and _a= 4
- a 01 01 + 13 13 (4b) 9T T
i I a s, w, - we s qq _-.
.__,m, ,1 d 48 ] 1 Lj The final step in the derivation is to equate the first terms of Eqs. 3 (4a) and (4b) with each other and the second terms with each other and i solve for the flux parameters. In offect, this procedure equates the two expressions giving the effective cross section in the thermal range (E
- 1) with each other and similarly for the resonance / fast neutron 0
,} group. The result of this procedure is d .I &ot*0ni i THERM = 9,q (Sa) 2200 1
- a l3 y3 RES =
(5b) $ *RI .i i All of the parameters on the right-hand sides of Eqs. (Sa) and (5b) are ) obtai,nable from either a reactor physics (i.e., static neutron spectrum) calculation with the output in two groups with a 0.5 eV cutoff or from standard literature references such as ref. 32. The only difficulty that arises is in choosing which nuclide to use in the calculation since the varying shapes and resonance structures of different nuclides will alter the calculated values for the flux param-eters. The solution to this problem was to use an artificial cross section that varies inversely with neutron velocity (i.e., inversely with the square root of neutron energy), which is the theoretical variation for all nuclides. This nuclide's thermal cross section is defined as 1.0 barn and the resonance integral is 0.45 barn. By spectrum-averaging the multigroup representation of this cross section so that the resulting ) cross sections and fluxes are in two groups, values of THERM and RES can readily be calculated. This is the procedure that has been followed in previously issued reports. O describing updated ORIGEN2 reactor models. 4.6.2 Derivation of the ORIGEN2 flux parameter FAST The derivation of an expression for calculating the ORIGEN2 flux parameter FAST is similar to that for THERM and RES. The energy groups used in this derivation are the middle pair and the single group (bottom) . I
l. 't 49 t ,Jl in Fig. 4.1. However, it is possible to simplify the derivation by noting ] that the parameter FAST is only applicable to nuclear reactions with a l{ threshold (i.e., the reaction cross section is effectively zero below a few MeV, such as (n,2n) and (n,3n)]. Thus the term for the low-energy 4 reaction rate is always zero because the cross section is zero. l. The neutron-induced reaction rate of an unspecified nuclide can be 4 written in either of two ways: l $U=C $ (6a) T 3 23 FS i or (6b) " @FS FS T where = fast neutron flux (from about 100 kev to maximum) from a reactor 23 ~1 ~1 physics calculation, neutrons barn s O = threshold cross section averaged over a fission neutron spectrum, FS ~1 neutrons s ; C = constant accounting for effects within the fast energy group; 3 $pg = fission spectrum neutron flux (from about 100 kev to maximum) ~ based on a reactor physics calculation, neutrons barn s and the other variables are as previously defined. An expression for calculating the flux parameter FAST is derived by (1) dividing such equation by $, (2) defining FAST C *$23 *T, and (3) equating the right-T 3 hand sides of these two equations. The result is: FAST = $p3/$ (8) T which is a relatively simple expression and is free of the nuclide-dependence characterizing the expressions for THERM and RES. There is, however, one difficulty with this equation: the flux $pg is not directly obtainable from a reactor physics calculation over the desired range _g (i.e., about 100 kev to 10 MeV) because the portion of the fission neutron flux below the peak value (about 2 MeV) is distorted due to high-energy f 0 i .f
.c... .-.-...-.:-.- -... _ - ~. ~. -.. - - - - ~. -.... -.. -a 4' [ 50 j
- 4 j
scattering processes. The solution used to avoid this problem is to obtain a partial fission neutron flux from the region above the fission neutron flux peak,(1.e., the undistorted region) and divide it by a constant factor to convert it into the full fission neutron flux. If is defined as the undistorted fission neutron flux and f,is defined F as the constant & / FS.( btainable from sources such as ref. 34, the FS resulting expression for calculating FAST is then: FAST =$pg/(f*Q) (9) T For example, if $'3 is deffued as the fission neutron flux above 2.5 MeV, this value would.'correspoLd to 0.2927 of the total number of fission ~. Thus, the appropriate value of f is 0.2927. It should be noted neutrons. that the exact energy used in this calculation is not important as long as the fission neutron spectrum is undistorted and the corresponding value of f,is used. 4.6.3 Conversion of ORIGEN2 flux carameters to ORIGEN flux parameters 'As is evident from the derivations in Sects. 4.6.1 and 4.6.2, the ORIGEN2 flux parameters are on a total flux (i.e., summed from 0 to maxi-mum energy) basis. However, previous versions of ORIGEN have generally been on a thermal flux basis for thermal reactors (e.g., PWRs). Thus, it may be desirable to convert the total-flux-bases parameters so older versions of ORIGEN can take advantage of more recent data. The most straightforward method for making this change is to multiply ( the total-flux-based parameters by the ratio of the total flux to the l' thermal flux; i.e., by $ / 01 [see under Eqs. (la) and (lb) for more T precise definitions]. However, in many instances the requisite reactor physics informati6S for generating this conversion factor are not readily obtainable. In this case, the thermal-flux-based value of THERM can be 33 i calculated using the expression derived from the Westcott formalism: i: THERM (thermal)=Y o i q 1 1 ,. ~....
1 J 51 .i ~ ), l. where T is the e'ffective moderator temperature in K. The conversion 'l r j factor for the other flux parameters is then simply the ratio of THERM a on s' thermal-flux-basis to THERM calculated on a total-flux basis. Values -l for the ORIGEN flux parameters on a thermal-flux basis will be give in all 9 of the reports describing updated models ORIGEN2 reactor models. l, ..s .J 4.7 ORIGEN2 Nuclide and Element Identifiers Most of the input and internal operations done in ORIGEN2 are based ton the use of a six-digit integer nuclide or element identifier. The l nuclide identifier is defined as NUCLID = 10,000*Z + 10*A + M i where s NUCLID = six-digit nuclide identifier, Z = atomic number of nuclide (1 to 99), A = atomic mass of nuclide (integer), M = state indicator; O = ground state; 1 = excited state. It should be'noted that only one excited state is allowed in ORIGEN2, as N with the previous versions of ORIGEN. The principal functions of the 1 NUCLID are to identify the data associated with it in the input libraries { { and internally in ORICEN2 and to permit construction of the transition matrix using integer operations (see Sect. 3.5). It is also used to l supply 'the atomic weight for converting g-atoms to grams. ORIGEN2 con-tains a single psuedonuclide, 2503g, which is used to collect the fission product mass resulting from spontaneous fission. The six-digit identifier for en element,follows the pattern set by the nuclide identifier NELID = 10,000*Z, ll where NELID is the element identifier and Z is as de, scribed previously. y l' ~ \\ + i -i . t 9 ps-
^a-a ' =-.=. z-.. e 1 A 52 I 5. REFERENCES ,j 1. M. J. Bell, ORIGEN - The ORNL Isotope Generation and Depletion Code, ORNL-4628 (May 1973). 2. C. M. Lederer, J. M. Hollander, and S. Perlman, Table of Isotopes, 6th ed., Wiley, New York, 1967. J. R. Stehn, M. D. Goldberg, B. A. Magurna, and R. Weiner-Chasman, 3. Neutron Cross Sections, 2nd ed., Suppl. 2, BNL-325 (1964). 4. M.' K. Drake, A " Compilation of Resonance Integrals," Nucleonics, 24(8), 108-111 (August 1966). 5. M. E. Meek and B. F. Rider, Summary of Fission Product Yields for 1 235U, 238U, 239Pu, and Pu at Thermal, Fission Spectrum, and 14 MeV 241 1 Neutron Energies, APED-5398-A (Rev.) (1968). 6. C. W. Kee, A Revised Light Element Library for the ORIGEN Code, ORNL/TM-4896 (May 1975). 7. C. W. Kee, C. R. Weisbin, and R. E. Schenter, Processing and Testing of ENDF/B-IV Fission Product and Transmutation Data, Trans. Am. Nucl. Soc. 19, 398-99 (1974). 8. A. G. Croff, A User's Manual for the ORIGEN2 Computer Code, ORNL/TM-7175 (in preparation). 9. A. G. Croff, M. A. Bj erke, G. W. Morrison, and L. M. Petrie, Revised Uranium-Plutonium Cycle PWR and BWR Models for the ORIGEN Computer Code, ORNL/TM-6051 (September 1978). 10. A. G. Croff and M. A. Bjerke, Alternative Fuel Cycle PWR Models for i the ORIGEN Computer Code, ORNL/TM-7005 (February 1980). 11. A. G. Croff and J. W. McAdoo, fMFBR Models for the ORIGEN2 Computer Code, ORNL/TM-7176 (in preparation). 12. A. G. Croff and M. A. Bjerke, CANDU Models for the ORIGEN Computer Code, ORNL/TM-7177 (in preparation). 13. W. B. Ewbank, M. R. Schmorak, F. E. Bertrand, M. Feliciano, and D. J. Horen, Nuclear Structure Data File: A Manual for Preparation of Data Sets, ORNL-5054 (June 1975). i 14. W. B. Ewbank, " Evaluated Nuclear Structure Data File (ENSDF) for Basic and Applied Research," paper presented at the Fifth Interna-tional CODATA Conference, Boulder, Colo., June 1976. >?, i i
.I 53 1 15. Nuclear Data Sheets, published by Academic Press, Inc. Subscription information available on request to Academis Press, Inc., 111 Fifth t Ave., New York, N.Y. 10013. 16. A. G. Croff, R. L. Haese, and N. B. Gove, Updated Decay and Photon Libraries for the ORIGEN Code, ORNL/TM-6055 (February 1979). j 17. B. H. Duane, in Physics Research Quarterly Report, Oct.-Dec. 1963, HW-80020 (1964). 18. H. E. Krug, J. E. Olhoeft, and J. Alsina, Chap. 8 in Suppletentary ] Report on Evaluation of Mass Spectrometric and Radiochemical Analyses I of Yankee Cor I Spent Fuel, Including Isotopes of Elements Thorium through Curium, WCAP-6086 (August 1969). 19. L. Lapidus and R. luus, Optimal Control of Engineering Processes, pp. 45-49, Bleisdell Publishing Co., Waltham, Mass.,1967. 20. S. J. Ball and R. K. Adams, MATEXP, a General Purpose Digital Computer Program for Solving Ordinary Differential Equations by the Matrix Exponential Method, ORNL/TM-1933 (August 1967). 21. D. R. Vondy, Development of a General Method of Explicit Solution to the Nuclide Chain Equations for Digital Machine Calculations, ORNL/TM-361 (Oct. 17, 1962). 22. L. Lapidus, Digital Computation for Chemical Engineers, pp. 259-62, McGraw-Hill, New York, 1962. 23. S. Raman, " General Survey of Applications Which Require Actinide Nuclear Data," presented at the IAEA Advisory Group Meeting on r Transactinium Nuclear Data, Karlsruhe, FRG (November 1975). 24. L. J. King, J. E. Bigelow, and E. D. Collins, Transuranium Processing Plant Semiannual Report of Production Status, and Plans for the } Period Ending June 30, 1973 (March 1974). 25. ENDF/B-IV Library Tapes 401-411 and 414-419, available from the National Neutron Cross Section Center, Brookhaven National Laboratory (December 1974). 1 26. ENDF/B-V Library Tapes 521 and 522, available from the National 3; Neutron Cross Section Center, Brookhaven National Laboratory (July 1979). 9 i i-l
__m; m_ - ~- i 54 27. D. L. Johnson, "Suberitical Reactivity Monitoring: Neutron Yields from (a,n) Reactions in FTR Fuel," section in Core Engineering Technical Progress Report April, May, June 1976, HEDL-TME 76-36 7 (March 1977). 28. G. W. Dawson, The Chemical Toxicity of Elements, BNWL-1815 (June 1974). 29. J. P. Unik and J. E. Gindler, A Critical Review of the Energy Released in Nuclear Fission, ANL-7748 (March 1971). 30. V. E. Viola, Jr., " Correlation of Fission Fragment Kinetic Energy Data," Nuclear Data 1(5), pp. 391-410 (July 1966). 31. V. N. Okolovich, V. I. Bo'l'shov, L. D. Gordeeva, and G. N. Smirenkin, " Dependence of the Mean Kinetic Energy of Fragments on the Fission-able Nucleus Mass," Soviet Atomic Energy M(5), pp.1177-78 (November 1963). 32. S. F. Mughabghab and D. I. Garber, Neutron Cross Sections, Volume 1, Resonance Parameters, BNL-325, 3d ed. (June 1973). 33. C. H. Westcott, Effective Cross Section Values for Well-Moderated Thermal Reactor Spectra, CRRP-960, 3d ed. corrected (November 1960). 34. H. Etherington, Nuclear Engineering Handbook, 1st ed., pp. 7-91, McGraw-Hill, New York (1958).
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