3. Chi Integration Overview¶

NOTE THAT THIS FEATURE HAS NOT YET BEEN TESTED Fissionable nuclides can contain one ore more fission reactions in their ACE data. NDPP must parse through each of these reactions (and the associated delayed neutron precursor data), calculate the outgoing energy spectra, $\chi$ for each energy group at each incoming energy point in the ACE tables ( $\chi_g(E')$ ), and then combine these in to a single value of $\chi_g(E')$ (on a single energy grid). To generate $\chi_g(E')$ , NDPP is solving the following equations for the prompt, delayed, and total values of $\chi_g(E')$ :

$\chi_{prompt,g}(E') =\ \sum\limits_{MT}\:\sum\limits_{d} \frac{\sigma_d(E')}{\sigma_f(E')}\int\limits_{E_g}^{E_{g-1}} \chi_{MT,d}\left(E,E'\right)dE'$

$\chi_{delayed,g}(E') =\ \sum\limits_{c}\:\ Y_c(E') \int\limits_{E_g}^{E_{g-1}}\chi_c\left(E,E'\right)dE'$

$\chi_{total,g}(E') =\ \left(1-\beta(E')\right) \chi_{prompt,g}(E') + \beta(E') \chi_{delayed,g}(E')$

In the above equations, $E'$ is the incoming neutron energy, $MT$ is the reaction channel, $d$ is the energy distributions within that channel, $E_g$ and $E_{g-1}$ are the lower and upper energy group boundaries for group $g$ respectively, $\beta(E')$ is the total delayed neutron emission fraction, $\sigma_{d,MT}(E')$ and $\sigma_{f,tot}(E')$ are the microscopic cross-sections of this reaction channel and distribution occuring and the microscopic cross-section for all fission reactions, and $Y_c$ is the yield of precursor group $c$ .

The details of this process are discussed in the following sections.

3.1. Fission Reactions and Initialization¶

To determine if a nuclide is fissionable, NDPP checks for presence of the $MT=18$ reaction channel in the ACE data. This channel is the total fission reaction, and must be present for there to be any possibility of generating values of $\chi_g(E')$ for the nuclide. If the nuclide is fissionable, then the code determines the number of fission reaction channels, outgoing energy distributions, delayed neutron precursor groups, and the number of energy grid points for each.

Then, NDPP progresses through each fission channel (and energy distributions within that channel) and delayed neutron precursor group to obtain values for each of the terms in the $\chi_g(E')$ equation above. The values of $\beta$ , $\sigma_d$ , $\sigma_f$ , and $Y_c$ are calculated by the same means discussed in the OpenMC manual. The calculation of $\int\limits_{E_g}^{E_{g-1}}\chi\left(E,E'\right)dE'$ is described in the next section.

3.2. Outgoing Neutron Energy Distributions¶

Monte Carlo codes must sample single values from probability distribution functions; NDPP, however, must integrate that probability distribution function between upper and lower boundaries. Therefore, the ACE Laws are utilized in different ways than described in the OpenMC manual and will be discussed herein.

All fission reactions are obtained from either ENDF File 5 (Energy Distribution of Secondary Particles) or File 6 (Product Energy-Angle Distributions). The following subsections discuss each of the ACE Laws used to describe fission neutron energy distributions in ENDF/B-7 and how these laws are treated by NDPP.

After the calculation of $\int\limits_{E_g}^{E_{g-1}}\chi\left(E,E'\right)dE'$ for each energy group, the values are normalized to 1.0 to account for any inaccuracies introduced by the interpolation schemes.

3.2.1. ACE Law 4 - Continuous Tabular Distribution¶

This representation is essentially a two-dimensional table which provides points of both the probability distribution function $\chi\left(E,E'\right)$ , a cumulative distribution function (CDF) $\int\limits_{0}^E\chi\left(E,E'\right)dE'$ , and rules for interpolating between each of the E and E’ data sets. To determine $\int\limits_{E_g}^{E_{g-1}}\chi\left(E,E'\right)dE'$ , NDPP must first find the location of the data corresponding to $E'$ data, and then interpolate on the CDF to find teh CDF at $E_{g-1}$ and $E_g$ . The value of $\int\limits_{E_g}^{E_{g-1}}\chi\left(E,E'\right)dE'$ is simply the difference between these two values. The interpolation rules are followed as described in the OpenMC Methods Manual.

3.2.2. ACE Law 7 - Maxwell Fission Spectrum¶

One representation of the secondary energies for neutrons from fission is the so-called Maxwell spectrum. A probability distribution for the Maxwell spectrum can be written in the form

$\chi(E,E') dE' = c E'^{1/2} e^{-E'/T(E)} dE'$

where $E$ is the incoming energy of the neutron and $T$ is the so-called nuclear temperature, which is a function of the incoming energy of the neutron. The ACE format contains a list of nuclear temperatures versus incoming energies. The nuclear temperature is interpolated between neighboring incoming energies using a specified interpolation law. Once the temperature $T$ is determined, we then can analytically determine the value of $\int\limits_{E_g}^{E_{g-1}}\chi\left(E',E\right)dE$ with the following relation:

$\int\limits_{E_g}^{E_{g-1}}\chi\left(E,E'\right)dE' =\ c \left(\frac{1}{2}\sqrt{\pi}\left(T(E)\right)^{\frac{3}{2}} erf\left(\frac{E'}{T(E)}\right)-T(E)\sqrt{E'}\exp{-\frac{E'}{T(E)}}\right)$

This integral is forced to 0 for values of E’ greater than the restriction energy, $U(E)$ .

3.2.3. ACE Law 9 - Evaporation Spectrum¶

Evaporation spectra are primarily used in compound nucleus processes where a secondary particle can “evaporate” from the compound nucleus if it has sufficient energy. The probability distribution for an evaporation spectrum can be written in the form

$\chi(E,E') dE' = c E' e^{-E'/T(E)} dE'$

where $E$ is the incoming energy of the neutron and $T$ is the nuclear temperature, which is a function of the incoming energy of the neutron. The ACE format contains a list of nuclear temperatures versus incoming energies. The nuclear temperature is interpolated between neighboring incoming energies using a specified interpolation law. Once the temperature $T$ is determined, we then analytically determine the value of $\int\limits_{E_g}^{E_{g-1}}\chi\left(E,E'\right)dE'$ with the following relation:

$\int\limits_{E_g}^{E_{g-1}}\chi\left(E,E'\right)dE' =\ -T(E) c \exp{-\frac{E'}{T(E)}}\left(T(E)+E'\right)$

This integral is forced to 0 for values of E’ greater than the restriction energy, $U(E)$ .

3.2.4. ACE Law 11 - Energy-Dependent Watt Spectrum¶

The probability distribution for a Watt fission spectrum can be written in the form

$\chi(E,E') dE' = c e^{-E'/a(E)} \sinh \sqrt{b(E) \, E'} dE'$

where $a$ and $b$ are parameters for the distribution and are given as tabulated functions of the incoming energy of the neutron. These two parameters are interpolated on the incoming energy grid using a specified interpolation law. Once the parameters have been determined, we then analytically determine the value of $\int\limits_{E_g}^{E_{g-1}}\chi\left(E,E'\right)dE'$ with the following relation:

This integral is forced to 0 for values of E’ greater than the restriction energy, $U(E)$ .

3.2.5. ACE Law 61 - Correlated Energy and Angle Distribution¶

This law is very similar to ACE Law 4, except there is another dimension in the table to represent the angular probability distribution function. Since the $\chi$ portion of NDPP is not concerned with the outgoing angle, and therefore this extra dimension can be ignored. Therefore the methods used to calculate $\int\limits_{E_g}^{E_{g-1}}\chi\left(E,E'\right)dE'$ , are the same as is discussed in the Law 4 section.

3.3. Creation of Union Energy Grids for $\chi_g(E)$ ¶

At this stage, NDPP has a tabular representation of $\int\limits_{E_g}^{E_{g-1}}\chi\left(E,E'\right)dE'$ for each incoming energy, $E$ , and outgoing energy group, $g$ for every fission reaction channel and energy distribution as well as for each of the delayed neutron precursor groups. Each of these tables has values on a completely different set of incoming energies (since the ACE data are on separate energy grids as well) and must be combined on to the same energy grid for the prompt, delayed, and total values of $\chi_g(E)$ . This unionized energy grid is made by using all of the energy points in the relevant $\chi_g(E)$ distributions and linearly interpolating between values for points without a data set on the grid. Due to the additional interpolation step, these values are also re-normalized to 1.0. A unioninzed grid exists for each of the prompt, delayed, and total values of $\chi_g(E)$ .

3.4. Thinning of Union Energy Grids¶

Since the unionized grids must be searched by the Monte Carlo code during runtime, it is desirable to have the size of the grid be as small as possible. To this end, NDPP provides the user with an option to thin the energy grid such that $E$ points which provide an increase accuracy of less than the user-specified tolerance when linear interpolation with neighboring points is used instead of the explicit value are discared from the data.

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3. Chi Integration Overview¶

3.1. Fission Reactions and Initialization¶

3.2. Outgoing Neutron Energy Distributions¶

3.2.1. ACE Law 4 - Continuous Tabular Distribution¶

3.2.2. ACE Law 7 - Maxwell Fission Spectrum¶

3.2.3. ACE Law 9 - Evaporation Spectrum¶

3.2.4. ACE Law 11 - Energy-Dependent Watt Spectrum¶

3.2.5. ACE Law 61 - Correlated Energy and Angle Distribution¶

3.3. Creation of Union Energy Grids for $\chi_g(E)$ ¶

3.4. Thinning of Union Energy Grids¶

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3. Chi Integration Overview¶

3.1. Fission Reactions and Initialization¶

3.2. Outgoing Neutron Energy Distributions¶

3.2.1. ACE Law 4 - Continuous Tabular Distribution¶

3.2.2. ACE Law 7 - Maxwell Fission Spectrum¶

3.2.3. ACE Law 9 - Evaporation Spectrum¶

3.2.4. ACE Law 11 - Energy-Dependent Watt Spectrum¶

3.2.5. ACE Law 61 - Correlated Energy and Angle Distribution¶

3.3. Creation of Union Energy Grids for ¶

3.4. Thinning of Union Energy Grids¶

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3.3. Creation of Union Energy Grids for $\chi_g(E)$ ¶