Notebook

Engy-4350: Nuclear Reactor Engineering Spring 2019 UMass Lowell; Prof. V. F. de Almeida 03Feb2019

02e. Nuclear Data and Data Processing: Gamma Capture Cross Sections¶

$ \newcommand{\Amtrx}{\boldsymbol{\mathsf{A}}} \newcommand{\Bmtrx}{\boldsymbol{\mathsf{B}}} \newcommand{\Mmtrx}{\boldsymbol{\mathsf{M}}} \newcommand{\Imtrx}{\boldsymbol{\mathsf{I}}} \newcommand{\Pmtrx}{\boldsymbol{\mathsf{P}}} \newcommand{\Lmtrx}{\boldsymbol{\mathsf{L}}} \newcommand{\Umtrx}{\boldsymbol{\mathsf{U}}} \newcommand{\Smtrx}{\boldsymbol{\mathsf{S}}} \newcommand{\xvec}{\boldsymbol{\mathsf{x}}} \newcommand{\uvar}{\boldsymbol{u}} \newcommand{\fvar}{\boldsymbol{f}} \newcommand{\avec}{\boldsymbol{\mathsf{a}}} \newcommand{\bvec}{\boldsymbol{\mathsf{b}}} \newcommand{\cvec}{\boldsymbol{\mathsf{c}}} \newcommand{\rvec}{\boldsymbol{\mathsf{r}}} \newcommand{\mvec}{\boldsymbol{\mathsf{m}}} \newcommand{\gvec}{\boldsymbol{\mathsf{g}}} \newcommand{\zerovec}{\boldsymbol{\mathsf{0}}} \newcommand{\norm}[1]{\bigl\lVert{#1}\bigr\rVert} \newcommand{\transpose}[1]{{#1}^\top} \DeclareMathOperator{\rank}{rank} \newcommand{\Power}{\mathcal{P}} $

Table of Contents¶

Objectives
Introduction
Atom sizes
Nuclei relative sizes
Radiative capture $(n,\gamma)$ cross section example

Objectives ¶

Demonstrate how to obtain traceable nuclear data and have them available through the notebook for analysis and problem solving.
Develop an intuitive sense of spatial scale of the underlying atomic world.
Develop an intuitive sense for the various (interaction) reactive cross sections in nuclear data in relation to the geometric cross section of the nuclei.

Introduction ¶

Refer to Notebook 02.

Atom sizes (van der Walls radius)¶

Collect the van der Walls radius of all atoms of interest. Use data directly from a periodic table package: Mendeleev python package.

In [34]:

'''Function to build a list of atomic properties'''

def get_atoms(z_max=117):
    '''
    Build a list of atomic properties for all atoms of interest up to `z_max`.
    Use the `mendeleev` package.

    Parameters
    ----------
    z_max: int, optional
        Maximum Z number; default = 117.
    
    Returns
    -------
    atoms: list(namedtuple)
        List of namedtuples with names: name, Z, symbol, vdw_radius.

    Examples
    --------
    '''
    
    # Use the Mendeleev python package (periodic table of elements)
    from mendeleev import element

    from collections import namedtuple
    Atom = namedtuple('Atom', ['name','Z','symbol','vdw_radius'])

    atoms = list()
    z_max = 96  # up to Curium

    for i in range(z_max):
        el = element(i+1)
        #print('%20s vdw radius [pm] = %3.1f'%(e.name,round(e.vdw_radius,1)))
        atm = Atom( name=el.name, Z=el.atomic_number, symbol=el.symbol, vdw_radius=el.vdw_radius )
        atoms.append( atm )
    #van_der_waals_radii
    
    return atoms

In [35]:

'''Get a list of atoms Z <= 96 (Cm)'''

atoms = get_atoms(96)

In [36]:

'''Plot the van der Walls radii for all elements of interest'''

from matplotlib import pyplot as plt # import the pyplot function of the matplotlib package
(fig, ax) = plt.subplots(figsize=(18,6))

ax.plot(range(len(atoms)), [atm.vdw_radius for atm in atoms], 
        '-.',color='black', marker='*',markersize=12)

plt.xticks(range(0,len(atoms),2),[atm.symbol for atm in atoms][::2],rotation=0,fontsize=12)
ax.set_ylabel('vdW radius [pm]',fontsize=16)
ax.set_xlabel('Element',fontsize=16)
ax.xaxis.grid(True,linestyle='-',which='major',color='lightgrey',alpha=0.9)

# create a twin y axis to reconfigure the top x axis
ay1 = ax.twiny()
ay1.set_xlim(ax.get_xlim())
#ay1.xaxis.tick_top()
ay1.set_xticks([])
ay1.set_xticks(range(1,len(atoms),2),[atm.symbol for atm in atoms][1::2])
ay1.set_xticklabels([atm.symbol for atm in atoms][1::2],minor=True,fontsize=12)
#ax1.spines["top"].set_position(("axes", 2))

plt.show()

Nuclei relative sizes ¶

Let's compute the sizes of nuclei relative to the atom they are in. Using the IAEA search engine,

expand the More fields of the NUCLIDE Ground State section,
set the Z range to 1, 96
check the nuclear radius R button
click on the Search button
wait for the results window to appear (list of 3001 nuclides)
check the You requested: line to be 1 ≤ Z ≤96 ≤ R ≤
click on the CSV download button to save a table file
rename the file to nuclides-z1-to-z96.csv.

In [ ]:

'''View the raw data'''

!cat data/nuclides-z1-to-z96.csv

In [37]:

'''Function to read the CSV data'''

def read_csv(file_name):
    '''
    Read csv data into a `pandas` data frame (table).
    
    Parameters
    ----------
    file_name: str, required
        File name and its path relative to this notebook.
    
    Returns
    -------
    df: pandas.df
        `Pandas` data frame (table).

    Examples
    --------
    '''
    import pandas as pd

    # read the data into a data frame (or table)
    df = pd.read_csv(file_name)

    #print(df) # uncomment for a screen output of the data

    # plot the data directly from Pandas (quick check)
    ax = df.plot(x='Z', y='radius',legend=False,
             title='Nuclides Radii', fontsize=14, figsize=(18,6))
    ax.set_ylabel('$r$ [fm]',fontsize=16)
    ax.set_xlabel('Z',fontsize=16)
    plt.xticks(fontsize=16)
    plt.yticks(fontsize=16)
    ax.grid()
    
    return df

In [38]:

'''Import the data as a Pandas table'''

df = read_csv('data/nuclides-z1-to-z96.csv')

**NB:** for all nuclides of interest the sizes vary by a maximum factor of 6 when the Z number vary from 1 to 96.

Let's collect the important size data for all nuclides.

In [39]:

'''Function for creating a nuclide container'''

def get_nuclides( df ):
    '''
    Create a list of named tuple nuclides
    Parameters
    ----------
    df: pandas data frame, required
        Table of data for nuclides.
    
    Returns
    -------
    nuclides: list(namedtuple)
        List of namedtuples. Names: name, Z, A, radius, unc.

    Examples
    --------
    '''
    
    nuclides = list()

    # design a container data structure
    from collections import namedtuple
    Nuclide = namedtuple('Nuclide', ['name','Z','A','radius','unc'])

    # fill in the list of containers
    radius_misses = 0 # counter of nuclides without radius data
    a_max = 0 # maximum A number with radius data present
    z_max = 0 # maximum Z number with radius data present

    import pandas as pd
    
    for row in df.itertuples():
        # note row[0] is the index
        if pd.isnull(row[4]):
            radius_misses += 1
            continue
        z = int(row[1])
        n = int(row[2])
        a = z+n
        a_max = max(a,a_max)
        z_max = max(z,z_max)
        symbol = row[3]
        nuc = Nuclide( name=row[3]+'-'+str(z+n), Z=z, A=a, radius=row[4], unc=int(row[5]) )
        
        nuclides.append(nuc)

    print('Number of nuclides with    radius data = ',len(nuclides))
    print('Number of nuclides without radius data = ',radius_misses)
    print('')
    print('Max Z number with radius data = ',z_max)
    print('Max A number with radius data = ',a_max)
    
    return (nuclides, z_max, a_max)

In [40]:

'''Create a list of nuclides'''

(nuclides, z_max, a_max) = get_nuclides( df )

Number of nuclides with    radius data =  908
Number of nuclides without radius data =  2146

Max Z number with radius data =  96
Max A number with radius data =  248

Let's calculate and plot the geometric cross sectional area of the nuclei.

In [41]:

'''Plot the van der Walls radii for all elements of interest'''
import math

from matplotlib import pyplot as plt # import the pyplot function of the matplotlib package
fig, ax = plt.subplots(figsize=(18,6))

# plot the cross sectional area of the nuclei; 1 barn = 100 fm^2
ax.plot([nc.A for nc in nuclides], [math.pi*nc.radius**2/100 for nc in nuclides], 
        ' ',color='green', marker='.',markersize=12)

ax.set_ylabel('Geometric Cross Sectional Area [b]',fontsize=16)
ax.set_xlabel('Nuclide A Number',fontsize=16)

fig.suptitle(r'%i Nuclides; $1 \leq Z \leq %i$; $1 \leq A \leq %i$ '%(len(nuclides),z_max,a_max),fontsize=20)
ax.grid()
plt.show()

**NB:** for all nuclides of interest the geometric cross sectional area vary to a **maximum of ~1.1 b** when the number of nucleons vary from 1 to 248. Also, the area does not vary significantly for a given value of A as a function of isotopes.

If an atom is the size of a classroom, say 10 m in diameter, what would the corresponding values of the nuclei be?

In [42]:

'''Plot the classroom size nuclei diameter'''
import math

from matplotlib import pyplot as plt # import the pyplot function of the matplotlib package
fig, ax = plt.subplots(figsize=(18,6))

# plot the classroom-sized atom nuclide diameter for a classroom of 10-m diameter
ax.plot([nc.A for nc in nuclides], [2*nc.radius*1e-15 * 5.0/atoms[nc.Z-1].vdw_radius*1e+12 * 1e+6 for nc in nuclides], 
        ' ',color='blue', marker='.',markersize=12)

ax.set_ylabel(' Classroom-Sized Nuclide Diameter [$\mu~m$]',fontsize=16)
ax.set_xlabel('Nuclide A Number',fontsize=16)

fig.suptitle(r'%i Nuclides; $1 \leq Z \leq %i$; $1 \leq A \leq %i$ '%(len(nuclides),z_max,a_max),fontsize=20)
ax.grid()
plt.show()

**NB:** for all nuclides of interest the classroom sizes vary between 100 and 300 micrometers.

In [43]:

'''Plot the classroom size nuclei diameter'''
import math

from matplotlib import pyplot as plt # import the pyplot function of the matplotlib package
fig, ax = plt.subplots(figsize=(18,6))

ax.plot([nc.Z for nc in nuclides], [2*nc.radius*1e-15 * 5.0/atoms[nc.Z-1].vdw_radius*1e+12 * 1e+6 for nc in nuclides], 
        ' ',color='blue', marker='.',markersize=12)

#plt.xticks(range(0,len([nc.Z for nc in nuclides]),2),[nc.Z for nc in nuclides][::2],rotation=0,fontsize=12)
ax.set_ylabel(' Classroom-Sized Nuclide Diameter [$\mu~m$]',fontsize=16)
ax.set_xlabel('Nuclide Z Number',fontsize=16)

# create a twin y axis to reconfigure the top x axis
ay1 = ax.twiny()
ay1.set_xlim(ax.get_xlim())
#ay1.xaxis.tick_top()
ay1.set_xticks([])
ay1.set_xticks(range(1,len(atoms),2),[atm.symbol for atm in atoms][::2])
ay1.set_xticklabels([atm.symbol for atm in atoms][::2],minor=True,fontsize=12)
#ax1.spines["top"].set_position(("axes", 2))

fig.suptitle(r'%i Nuclides $1 \leq Z \leq 96$'%(len(nuclides)),fontsize=20)
ax.grid()
plt.show()

**NB:** for all nuclides of interest the classroom sizes vary between 100 and 300 micrometers.

Radiative capture $(n,\gamma)$ cross section example ¶

This is a significant capture reaction emitting a prompt capture gamma ray (photon)

\begin{equation*} ^1_0\text{n} \ + \ ^{235}_{92}\text{U} \longrightarrow \ ^{236}_{92}\text{U}^* \overset{\approx 10^{-14}\,\text{s}}{\longrightarrow} \ ^{236}_{92}\text{U} \ + \ \gamma , \end{equation*}

Note:

a) over much of the neutron energy range in a nuclear reactor, this is the only possible parasitic reaction that influences neutron balance; neutrons of any energy can be captured with gamma emission.
b) often this reaction is the dominat source of radiation heating and bulk shielding in the reactor vessel.
c) the product nuclide is often unstable, decaying via beta emission and delayed gamma rays; the latter continues for years after reactor shutdown.

The microscopic cross section of this neutron interaction with the nuclide $^{235}_{92}\text{U}$ represents the probability of interaction measured in units of area; typical units are: cm$^2\times 10^{-24}$ or barn. Therefore it makes sense to compare against the geometric cross sectional area of the nuclide.

Acquiring cross section data and saving it to a file:

Using the cross section data site click on U in the periodic table and choose $^{235}_{92}$U from the isotope list.
Select the neutron capture reaction cross section $^{235}_{92}$U(n,$\gamma$) and plot the data.
Choose view evaluated data and click on the Text link to view the data in a two-column arrangement (energy versus cross section).
Using the browser, save data to a text file (u-235-sigma-c.dat). Relocate the file in a directory data/ relative to this notebook.

In [ ]:

'''View the raw data'''
!cat data/u-235-sigma-g.dat

Importing and processing tabular data is readily done using the Pandas Python Package:

Pandas documentation.

In [44]:

'''Visualize the data'''

# Pandas: python package for tabular data analysis
import pandas as pd

# read the data into a data frame (or table)
df = pd.read_csv('data/u-235-sigma-g.dat', 
                  names=['energy [eV]','sigma_g [b]'], 
                  skiprows=3)

#print(df) # uncomment for a screen output of the data

# plot the data directly from Pandas
ax = df.plot(loglog=True, x='energy [eV]', y='sigma_g [b]',legend=False,
             title='Capture $\sigma_\gamma$ for $^{235}_{92}$U(n,$\gamma$)', figsize=(16,5))
ax.set(ylabel='$\sigma_\gamma$ [b]')
ax.grid()

To obtain a sense of how significant this reaction is, calculate the cross sectional area of the nucleus and compare with the radiative cross section in the plot.

In [45]:

for nc in nuclides:
    if nc.name == 'U-235':
        r_u235 = nc.radius
        r_u235_unc = nc.unc
print('U-235 r [fm] = ',r_u235,r_u235_unc)

U-235 r [fm] =  5.8337 41

In [46]:

import math
area = math.pi * r_u235**2
print('nucleus geometric cross section area [b] = %5.5f'%(area/100)) # 1 barn = 100 fm^2

nucleus geometric cross section area [b] = 1.06915

From the radiative capture cross section data (plot): $\sigma_\gamma = 101.153$ b at $E=2.4\, 10^{-2}$ eV. Therefore the radiative cross section is

In [47]:

round(101.153/1.06915,1)

Out[47]:

94.6

times greater than the nucleus geometric cross section. Note that $\sigma_\gamma\approx 1$ at $E\approx 10$ keV and this reaction is significant for a large extent of neutron energies.

In [ ]:

02e. Nuclear Data and Data Processing: Gamma Capture Cross Sections¶

Table of Contents¶

Objectives¶

Introduction¶

Atom sizes (van der Walls radius)¶

Nuclei relative sizes¶

Radiative capture $(n,\gamma)$ cross section example¶

Objectives ¶

Introduction ¶

Nuclei relative sizes ¶

Radiative capture $(n,\gamma)$ cross section example ¶