Nuclear Data¶

In this notebook, we will go through the salient features of the openmc.data package in the Python API. This package enables inspection, analysis, and conversion of nuclear data from ACE files. Most importantly, the package provides a mean to generate HDF5 nuclear data libraries that are used by the transport solver.

In [1]:

%matplotlib inline
import os
from pprint import pprint
import shutil
import subprocess
import urllib.request

import h5py
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.cm
from matplotlib.patches import Rectangle

import openmc.data

Physical Data¶

Some very helpful physical data is available as part of openmc.data: atomic masses, natural abundances, and atomic weights.

In [2]:

openmc.data.atomic_mass('Fe54')

Out[2]:

53.939608306

In [3]:

openmc.data.NATURAL_ABUNDANCE['H2']

Out[3]:

0.00015574

In [4]:

openmc.data.atomic_weight('C')

Out[4]:

12.011115164864455

The IncidentNeutron class¶

The most useful class within the openmc.data API is IncidentNeutron, which stores to continuous-energy incident neutron data. This class has factory methods from_ace, from_endf, and from_hdf5 which take a data file on disk and parse it into a hierarchy of classes in memory. To demonstrate this feature, we will download an ACE file (which can be produced with NJOY 2016) and then load it in using the IncidentNeutron.from_ace method.

In [5]:

url = 'https://anl.box.com/shared/static/kxm7s57z3xgfbeq29h54n7q6js8rd11c.ace'
filename, headers = urllib.request.urlretrieve(url, 'gd157.ace')

In [6]:

# Load ACE data into object
gd157 = openmc.data.IncidentNeutron.from_ace('gd157.ace')
gd157

Out[6]:

<IncidentNeutron: Gd157>

Cross sections¶

From Python, it's easy to explore (and modify) the nuclear data. Let's start off by reading the total cross section. Reactions are indexed using their "MT" number -- a unique identifier for each reaction defined by the ENDF-6 format. The MT number for the total cross section is 1.

In [7]:

total = gd157[1]
total

Out[7]:

<Reaction: MT=1 (n,total)>

Cross sections for each reaction can be stored at multiple temperatures. To see what temperatures are available, we can look at the reaction's xs attribute.

In [8]:

total.xs

Out[8]:

{'294K': <openmc.data.function.Tabulated1D at 0x7fd70079e610>}

To find the cross section at a particular energy, 1 eV for example, simply get the cross section at the appropriate temperature and then call it as a function. Note that our nuclear data uses eV as the unit of energy.

In [9]:

total.xs['294K'](1.0)

Out[9]:

142.64747

The xs attribute can also be called on an array of energies.

In [10]:

total.xs['294K']([1.0, 2.0, 3.0])

Out[10]:

array([142.64747   ,  38.6541761 , 175.40019642])

A quick way to plot cross sections is to use the energy attribute of IncidentNeutron. This gives an array of all the energy values used in cross section interpolation for each temperature present.

In [11]:

gd157.energy

Out[11]:

{'294K': array([1.0000e-05, 1.0325e-05, 1.0650e-05, ..., 1.9500e+07, 1.9900e+07,
        2.0000e+07])}

In [12]:

energies = gd157.energy['294K']
total_xs = total.xs['294K'](energies)
plt.loglog(energies, total_xs)
plt.xlabel('Energy (eV)')
plt.ylabel('Cross section (b)')

Out[12]:

Text(0, 0.5, 'Cross section (b)')

Reaction Data¶

Most of the interesting data for an IncidentNeutron instance is contained within the reactions attribute, which is a dictionary mapping MT values to Reaction objects.

In [13]:

pprint(list(gd157.reactions.values())[:10])

[<Reaction: MT=1 (n,total)>,
 <Reaction: MT=101 (n,disappear)>,
 <Reaction: MT=301 heating>,
 <Reaction: MT=2 (n,elastic)>,
 <Reaction: MT=16 (n,2n)>,
 <Reaction: MT=17 (n,3n)>,
 <Reaction: MT=22 (n,na)>,
 <Reaction: MT=24 (n,2na)>,
 <Reaction: MT=28 (n,np)>,
 <Reaction: MT=41 (n,2np)>]

Let's suppose we want to look more closely at the (n,2n) reaction. This reaction has an energy threshold

In [14]:

n2n = gd157[16]
print('Threshold = {} eV'.format(n2n.xs['294K'].x[0]))

Threshold = 6400881.0 eV

The (n,2n) cross section, like all basic cross sections, is represented by the Tabulated1D class. The energy and cross section values in the table can be directly accessed with the x and y attributes. Using the x and y has the nice benefit of automatically acounting for reaction thresholds.

In [15]:

n2n.xs

Out[15]:

{'294K': <openmc.data.function.Tabulated1D at 0x7fd6c4f45940>}

In [16]:

xs = n2n.xs['294K']
plt.plot(xs.x, xs.y)
plt.xlabel('Energy (eV)')
plt.ylabel('Cross section (b)')
plt.xlim((xs.x[0], xs.x[-1]))

Out[16]:

(6400881.0, 20000000.0)

To get information on the energy and angle distribution of the neutrons emitted in the reaction, we need to look at the products attribute.

In [17]:

n2n.products

Out[17]:

[<Product: neutron, emission=prompt, yield=polynomial>,
 <Product: photon, emission=prompt, yield=tabulated>]

In [18]:

neutron = n2n.products[0]
neutron.distribution

Out[18]:

[<openmc.data.correlated.CorrelatedAngleEnergy at 0x7fd6c4f45af0>]

We see that the neutrons emitted have a correlated angle-energy distribution. Let's look at the energy_out attribute to see what the outgoing energy distributions are.

In [19]:

dist = neutron.distribution[0]
dist.energy_out

Out[19]:

[<openmc.stats.univariate.Tabular at 0x7fd6c4f45ac0>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4f45b80>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4f45ca0>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4f45dc0>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4f4a040>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4f4a3a0>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4f4a7c0>,
 <openmc.stats.univariate.Tabular at 0x7fd7007eec10>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4f4f2e0>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4f4f9a0>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4153160>,
 <openmc.stats.univariate.Tabular at 0x7fd6c41539a0>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4157220>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4157b80>,
 <openmc.stats.univariate.Tabular at 0x7fd6c415c580>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4161040>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4161b20>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4165700>,
 <openmc.stats.univariate.Tabular at 0x7fd6c416a340>,
 <openmc.stats.univariate.Tabular at 0x7fd6c416af40>,
 <openmc.stats.univariate.Tabular at 0x7fd6c416fc40>,
 <openmc.stats.univariate.Tabular at 0x7fd6c41739a0>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4179760>,
 <openmc.stats.univariate.Tabular at 0x7fd6c417e580>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4182400>,
 <openmc.stats.univariate.Tabular at 0x7fd6c41882e0>,
 <openmc.stats.univariate.Tabular at 0x7fd6c418d220>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4ed2040>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4ed2f40>,
 <openmc.stats.univariate.Tabular at 0x7fd6c4ed6ee0>]

Here we see we have a tabulated outgoing energy distribution for each incoming energy. Note that the same probability distribution classes that we could use to create a source definition are also used within the openmc.data package. Let's plot every fifth distribution to get an idea of what they look like.

In [20]:

for e_in, e_out_dist in zip(dist.energy[::5], dist.energy_out[::5]):
    plt.semilogy(e_out_dist.x, e_out_dist.p, label='E={:.2f} MeV'.format(e_in/1e6))
plt.ylim(top=1e-6)
plt.legend()
plt.xlabel('Outgoing energy (eV)')
plt.ylabel('Probability/eV')
plt.show()

Unresolved resonance probability tables¶

We can also look at unresolved resonance probability tables which are stored in a ProbabilityTables object. In the following example, we'll create a plot showing what the total cross section probability tables look like as a function of incoming energy.

In [21]:

fig = plt.figure()
ax = fig.add_subplot(111)
cm = matplotlib.cm.Spectral_r

# Determine size of probability tables
urr = gd157.urr['294K']
n_energy = urr.table.shape[0]
n_band = urr.table.shape[2]

for i in range(n_energy):
    # Get bounds on energy
    if i > 0:
        e_left = urr.energy[i] - 0.5*(urr.energy[i] - urr.energy[i-1])
    else:
        e_left = urr.energy[i] - 0.5*(urr.energy[i+1] - urr.energy[i])

    if i < n_energy - 1:
        e_right = urr.energy[i] + 0.5*(urr.energy[i+1] - urr.energy[i])
    else:
        e_right = urr.energy[i] + 0.5*(urr.energy[i] - urr.energy[i-1])
        
    for j in range(n_band):
        # Determine maximum probability for a single band
        max_prob = np.diff(urr.table[i,0,:]).max()
        
        # Determine bottom of band
        if j > 0:
            xs_bottom = urr.table[i,1,j] - 0.5*(urr.table[i,1,j] - urr.table[i,1,j-1])
            value = (urr.table[i,0,j] - urr.table[i,0,j-1])/max_prob
        else:
            xs_bottom = urr.table[i,1,j] - 0.5*(urr.table[i,1,j+1] - urr.table[i,1,j])
            value = urr.table[i,0,j]/max_prob

        # Determine top of band
        if j < n_band - 1:
            xs_top = urr.table[i,1,j] + 0.5*(urr.table[i,1,j+1] - urr.table[i,1,j])
        else:
            xs_top = urr.table[i,1,j] + 0.5*(urr.table[i,1,j] - urr.table[i,1,j-1])
    
        # Draw rectangle with appropriate color
        ax.add_patch(Rectangle((e_left, xs_bottom), e_right - e_left, xs_top - xs_bottom,
                     color=cm(value)))

# Overlay total cross section
ax.plot(gd157.energy['294K'], total.xs['294K'](gd157.energy['294K']), 'k')

# Make plot pretty and labeled
ax.set_xlim(1.0, 1.0e5)
ax.set_ylim(1e-1, 1e4)
ax.set_xscale('log')
ax.set_yscale('log')
ax.set_xlabel('Energy (eV)')
ax.set_ylabel('Cross section(b)')

Out[21]:

Text(0, 0.5, 'Cross section(b)')

Exporting HDF5 data¶

If you have an instance IncidentNeutron that was created from ACE or HDF5 data, you can easily write it to disk using the export_to_hdf5() method. This can be used to convert ACE to HDF5 or to take an existing data set and actually modify cross sections.

In [22]:

gd157.export_to_hdf5('gd157.h5', 'w')

With few exceptions, the HDF5 file encodes the same data as the ACE file.

In [23]:

gd157_reconstructed = openmc.data.IncidentNeutron.from_hdf5('gd157.h5')
np.all(gd157[16].xs['294K'].y == gd157_reconstructed[16].xs['294K'].y)

Out[23]:

True

And one of the best parts of using HDF5 is that it is a widely used format with lots of third-party support. You can use h5py, for example, to inspect the data.

In [24]:

h5file = h5py.File('gd157.h5', 'r')
main_group = h5file['Gd157/reactions']
for name, obj in sorted(list(main_group.items()))[:10]:
    if 'reaction_' in name:
        print('{}, {}'.format(name, obj.attrs['label'].decode()))

reaction_002, (n,elastic)
reaction_016, (n,2n)
reaction_017, (n,3n)
reaction_022, (n,na)
reaction_024, (n,2na)
reaction_028, (n,np)
reaction_041, (n,2np)
reaction_051, (n,n1)
reaction_052, (n,n2)
reaction_053, (n,n3)

In [25]:

n2n_group = main_group['reaction_016']
pprint(list(n2n_group.values()))

[<HDF5 group "/Gd157/reactions/reaction_016/294K" (1 members)>,
 <HDF5 group "/Gd157/reactions/reaction_016/product_0" (2 members)>,
 <HDF5 group "/Gd157/reactions/reaction_016/product_1" (2 members)>]

So we see that the hierarchy of data within the HDF5 mirrors the hierarchy of Python objects that we manipulated before.

In [26]:

n2n_group['294K/xs'][()]

Out[26]:

array([0.000000e+00, 3.026796e-13, 1.291101e-02, 6.511110e-02,
       3.926270e-01, 5.752268e-01, 6.969600e-01, 7.399378e-01,
       9.635450e-01, 1.142130e+00, 1.308020e+00, 1.463500e+00,
       1.557600e+00, 1.640550e+00, 1.688960e+00, 1.711400e+00,
       1.739450e+00, 1.782070e+00, 1.816650e+00, 1.845280e+00,
       1.865409e+00, 1.867240e+00, 1.881558e+00, 1.881560e+00,
       1.881800e+00, 1.894470e+00, 1.869570e+00, 1.821200e+00,
       1.716000e+00, 1.600540e+00, 1.431620e+00, 1.283460e+00,
       1.101660e+00, 1.065300e+00, 9.307300e-01, 8.029800e-01,
       7.777400e-01])

Working with ENDF files¶

In addition to being able to load ACE and HDF5 data, we can also load ENDF data directly into an IncidentNeutron instance using the from_endf() factory method. Let's download the ENDF/B-VII.1 evaluation for $^{157}$Gd and load it in:

In [27]:

# Download ENDF file
url = 'https://t2.lanl.gov/nis/data/data/ENDFB-VII.1-neutron/Gd/157'
filename, headers = urllib.request.urlretrieve(url, 'gd157.endf')

# Load into memory
gd157_endf = openmc.data.IncidentNeutron.from_endf(filename)
gd157_endf

Out[27]:

<IncidentNeutron: Gd157>

Just as before, we can get a reaction by indexing the object directly:

In [28]:

elastic = gd157_endf[2]

However, if we look at the cross section now, we see that it isn't represented as tabulated data anymore.

In [29]:

elastic.xs

Out[29]:

{'0K': <openmc.data.function.ResonancesWithBackground at 0x7fd6c15174f0>}

If you had Cython installed when you built/installed OpenMC, you should be able to evaluate resonant cross sections from ENDF data directly, i.e., OpenMC will reconstruct resonances behind the scenes for you.

In [30]:

elastic.xs['0K'](0.0253)

Out[30]:

998.7871174521487

When data is loaded from an ENDF file, there is also a special resonances attribute that contains resolved and unresolved resonance region data (from MF=2 in an ENDF file).

In [31]:

gd157_endf.resonances.ranges

Out[31]:

[<openmc.data.resonance.ReichMoore at 0x7fd6c20ad6a0>,
 <openmc.data.resonance.Unresolved at 0x7fd6c20add60>]

We see that $^{157}$Gd has a resolved resonance region represented in the Reich-Moore format as well as an unresolved resonance region. We can look at the min/max energy of each region by doing the following:

In [32]:

[(r.energy_min, r.energy_max) for r in gd157_endf.resonances.ranges]

Out[32]:

[(1e-05, 306.6), (306.6, 54881.1)]

With knowledge of the energy bounds, let's create an array of energies over the entire resolved resonance range and plot the elastic scattering cross section.

In [33]:

# Create log-spaced array of energies
resolved = gd157_endf.resonances.resolved
energies = np.logspace(np.log10(resolved.energy_min),
                       np.log10(resolved.energy_max), 1000)

# Evaluate elastic scattering xs at energies
xs = elastic.xs['0K'](energies)

# Plot cross section vs energies
plt.loglog(energies, xs)
plt.xlabel('Energy (eV)')
plt.ylabel('Cross section (b)')

Out[33]:

Text(0, 0.5, 'Cross section (b)')

Resonance ranges also have a useful parameters attribute that shows the energies and widths for resonances.

In [34]:

resolved.parameters.head(10)

Out[34]:

	energy	J	neutronWidth	captureWidth
0	0.0314	2.0	0.000474	0.1072
1	2.8250	2.0	0.000345	0.0970
2	16.2400	1.0	0.000400	0.0910
3	16.7700	2.0	0.012800	0.0805
4	20.5600	2.0	0.011360	0.0880
5	21.6500	2.0	0.000376	0.1140
6	23.3300	1.0	0.000813	0.1210
7	25.4000	2.0	0.001840	0.0850
8	40.1700	1.0	0.001307	0.1100
9	44.2200	2.0	0.008960	0.0960

Heavy-nuclide resonance scattering¶

OpenMC has two methods for accounting for resonance upscattering in heavy nuclides, DBRC and RVS. These methods rely on 0 K elastic scattering data being present. If you have an existing ACE/HDF5 dataset and you need to add 0 K elastic scattering data to it, this can be done using the IncidentNeutron.add_elastic_0K_from_endf() method. Let's do this with our original gd157 object that we instantiated from an ACE file.

In [35]:

gd157.add_elastic_0K_from_endf('gd157.endf')

Let's check to make sure that we have both the room temperature elastic scattering cross section as well as a 0K cross section.

In [36]:

gd157[2].xs

Out[36]:

{'294K': <openmc.data.function.Tabulated1D at 0x7fd6c4fb35e0>,
 '0K': <openmc.data.function.Tabulated1D at 0x7fd6c12713a0>}

Generating data from NJOY¶

To run OpenMC in continuous-energy mode, you generally need to have ACE files already available that can be converted to OpenMC's native HDF5 format. If you don't already have suitable ACE files or need to generate new data, both the IncidentNeutron and ThermalScattering classes include from_njoy() methods that will run NJOY to generate ACE files and then read those files to create OpenMC class instances. The from_njoy() methods take as input the name of an ENDF file on disk. By default, it is assumed that you have an executable named njoy available on your path. This can be configured with the optional njoy_exec argument. Additionally, if you want to show the progress of NJOY as it is running, you can pass stdout=True.

Let's use IncidentNeutron.from_njoy() to run NJOY to create data for $^2$H using an ENDF file. We'll specify that we want data specifically at 300, 400, and 500 K.

In [37]:

# Download ENDF file
url = 'https://t2.lanl.gov/nis/data/data/ENDFB-VII.1-neutron/H/2'
filename, headers = urllib.request.urlretrieve(url, 'h2.endf')

# Run NJOY to create deuterium data
h2 = openmc.data.IncidentNeutron.from_njoy('h2.endf', temperatures=[300., 400., 500.], stdout=True)

 njoy 2016.67  xxXxx22                                       10/04/22 13:31:26
 *****************************************************************************

 reconr...                                                                0.0s

 broadr...                                                                0.0s
      300.0 deg                                                           0.0s
      400.0 deg                                                           0.1s
      500.0 deg                                                           0.2s

 heatr...                                                                 0.3s

 heatr...                                                                 0.4s

 gaspr...                                                                 0.5s

 purr...                                                                  0.6s

 mat =  128                                                               0.6s

 ---message from purr---mat  128 has no resonance parameters
                          copy as is to nout

 acer...                                                                  0.6s

 acer...                                                                  0.7s

 acer...                                                                  0.9s
                                                                          1.0s
 *****************************************************************************

Now we can use our h2 object just as we did before.

In [38]:

h2[2].xs

Out[38]:

{'300K': <openmc.data.function.Tabulated1D at 0x7fd6c11d88b0>,
 '400K': <openmc.data.function.Tabulated1D at 0x7fd6c1227cd0>,
 '500K': <openmc.data.function.Tabulated1D at 0x7fd6c091ef10>,
 '0K': <openmc.data.function.Tabulated1D at 0x7fd6c1190280>}

Note that 0 K elastic scattering data is automatically added when using from_njoy() so that resonance elastic scattering treatments can be used.

Windowed multipole¶

OpenMC can also be used with an experimental format called windowed multipole. Windowed multipole allows for analytic on-the-fly Doppler broadening of the resolved resonance range. Windowed multipole data can be downloaded with the openmc-get-multipole-data script. This data can be used in the transport solver, but it can also be used directly in the Python API.

In [39]:

url = 'https://github.com/mit-crpg/WMP_Library/releases/download/v1.1/092238.h5'
filename, headers = urllib.request.urlretrieve(url, '092238.h5')

In [40]:

u238_multipole = openmc.data.WindowedMultipole.from_hdf5('092238.h5')

The WindowedMultipole object can be called with energy and temperature values. Calling the object gives a tuple of 3 cross sections: elastic scattering, radiative capture, and fission.

In [41]:

u238_multipole(1.0, 294)

Out[41]:

(array(9.13284265), array(0.50530278), array(2.9316765e-06))

An array can be passed for the energy argument.

In [42]:

E = np.linspace(5, 25, 1000)
plt.semilogy(E, u238_multipole(E, 293.606)[1])

Out[42]:

[<matplotlib.lines.Line2D at 0x7fd6c11a0e20>]

The real advantage to multipole is that it can be used to generate cross sections at any temperature. For example, this plot shows the Doppler broadening of the 6.67 eV resonance between 0 K and 900 K.

In [43]:

E = np.linspace(6.1, 7.1, 1000)
plt.semilogy(E, u238_multipole(E, 0)[1])
plt.semilogy(E, u238_multipole(E, 900)[1])

Out[43]:

[<matplotlib.lines.Line2D at 0x7fd6c0956100>]