\(\renewcommand\AA{\text{Å}}\)
Neutron scattering potentials¶
periodictable.nsf
¶
Neutron scattering factors for the elements and isotopes.
For details of neutron scattering factor values, see Neutron
.
The property is set to None if there is no neutron scattering information
for the element. Individual isotopes may have their own scattering
information.
Example¶
Print a table of coherent scattering length densities for isotopes of a particular element:
>>> import periodictable
>>> for iso in periodictable.Ni:
... if iso.neutron.has_sld():
... print("%s %7.4f"%(iso,iso.neutron.sld()[0]))
58Ni 13.1526
60Ni 2.5575
61Ni 6.9417
62Ni 7.9464
64Ni 0.3379
Details¶
There are a number of functions available in periodictable.nsf
neutron_energy()
 Return neutron energy given wavelength.
neutron_wavelength()
 Return wavelength given neutron energy.
neutron_wavelength_from_velocity()
 Return wavelength given neutron velocity.
neutron_scattering()
 Computes scattering length density, cross sections and penetration depth for a compound.
neutron_sld()
 Computes scattering length density for a compound.
neutron_composite_sld()
 Returns a scattering length density for a compound whose composition is variable.
energy_dependent_table()
 Lists isotopes with energy dependence.
sld_table()
 Lists scattering length densitys for all elements in natural abundance.
absorption_comparison_table()
 Compares the imaginary bound coherent scattering length to the absorption cross section.
coherent_comparison_table()
 Compares the bound coherent scattering length to the coherent scattering cross section.
total_comparison_table()
 Compares the total scattering cross section to the sum of the coherent and incoherent scattering cross sections.
For private tables use init()
to set the data.
The neutron scattering information table is reproduced from the Atomic Institute for Austrian Universities[1] (retrieve March 2008):
http://www.ati.ac.at/~neutropt/scattering/table.html
The above site has references to the published values for every entry in the table. We have included these in the documentation directory associated with the periodictable package.
[1]  (1, 2) Rauch, H. and Waschkowski, W. (2003) Neutron Scattering Lengths in ILL Neutron Data Booklet (second edition), A.J. Dianoux, G. Lander, Eds. Old City Publishing, Philidelphia, PA. pp 1.11 to 1.117. (https://www.ill.eu/fileadmin/user_upload/ILL/1_About_ILL/Documentation/NeutronDataBooklet.pdf) 
[2]  Rauch, H. and Waschkowski, W. (2000) Neutron scattering lengths. Schopper, H. (ed.). SpringerMaterials  The LandoltBörnstein Database (http://www.springermaterials.com). doi: 10.1007/10499706_6 
[3]  Koester, L., Rauch, H., Seymann. E. (1991) Atomic Data Nuclear Data Tables 49, 65. doi:10.1016/0092640X(91)90012S 
[4]  Lynn, J.E. and Seeger, P.A. (1990) Resonance effects in neutron scattering lengths of rareearth nuclides. Atomic Data and Nuclear Data Tables 44, 191207. doi:10.1016/0092640X(90)90013A 
[5]  (1, 2) Sears, V. F. (2006) 4.4.4 Scattering lengths for neutrons. In Prince, E. ed. Intl. Tables for Crystallography C. Kluwer Academic Publishers. pp 444454. (https://it.iucr.org/Cb/ch4o4v0001/sec4o4o4/) doi: 10.1107/97809553602060000103 
[6]  Sears, V.F. (1992) Neutron scattering lengths and cross sections. Neutron News 3, No. 3, 2637. 
[7]  (1, 2) May, R.P., Ibel, K. and Haas, J. (1982) The forward scattering of cold neutrons by mixtures of light and heavy water. J. Appl. Cryst. 15, 1519. doi:10.1107/S0021889882011285 
[8]  Mildner, D.F.R., Lamaze, G.P. (1998) Neutron Transmission of SingleCrystal Sapphire. J Appl Crystallogr 31, 835–840. doi:10.1107/S0021889898005846 
[9]  Smith, G.S. and Majkrzak, C.M. (2006) 2.9 Neutron reflectometry. In E. Prince ed. Intl. Tables for Crystallography C. Wiley InterScience. pp 126146. doi: 10.1107/97809553602060000584 
[10]  Glinka, C.J. (2011) Incoherent Neutron Scattering from Multielement Materials. J. Appl. Cryst. 44, 618624. doi: 10.1107/S0021889811008223 

class
periodictable.nsf.
Neutron
¶ Bases:
object
Neutron scattering factors are attached to each element in the periodic table for which values are available. If no information is available, then the neutron field of the element will be None. Even when neutron information is available, it may not be complete, so individual fields may be None.
The following fields are defined:
 b_c (fm)
 Bounds coherent scattering length.
 total (barn)
 Total scattering cross section \(\sigma_s\). This does not include the absorption cross section. To compute the total collision cross section use \(\sigma_t = \sigma_s + \sigma_a\)
 absorption (barn)
 Absorption cross section \(\sigma_a\) at 1.798 Å. Scale to your beam by dividing by periodictable.nsf.ABSORPTION_WAVELENGTH and multiplying by your wavelength.
Additional fields not used for calculation include:
 b_c_i (fm)
Imaginary bound coherent scattering length. This is related to absorption cross section by \(\sigma_a = 4 \pi b_i/k\) where \(k = 2 \pi/\lambda\) and an additional factor of 1000 for converting between Å·fm and barns. b_c_i is not available for all isotopes for which absorption cross sections have been measured.
 bp, bm (fm)
Spindependent scattering for I+1/2 and I1/2 (not always available). Incoherent scattering arises from the spindependent scattering b+ and b. The Neutron Data Booklet[1] gives formulas for calculating coherent and incoherent scattering from b+ and b alone.
 bp_i, bm_i (fm)
Imaginary portion of bp and bm.
 is_energy_dependent (boolean)
Do not use this data if scattering is energy dependent.
 coherent (barn)
Coherent scattering cross section. This is tabulated but not used. In theory coherent scattering is related to bound coherent scattering by \(\sigma_s = 4 \pi b_c^2/100\). In practice, these values are different, with the following table showing the largest relative difference:
Sc 3% Ti 4% V 34% Mn 1% Cd 4% Te 4% Xe 9% Sm 100% Eu 46% Gd 61% Tb 1% Ho 11% W 4% Au 7% Hg 2%
 incoherent (barn)
Incoherent scattering cross section \(\sigma_i\). This is tabulated but not used. Instead, the incoherent cross section is computed from the total cross section minus the coherent cross section even for single atoms so that results from compounds are consistent with results from single atoms.
For elements, the scattering crosssections are based on the natural abundance of the individual isotopes. Individual isotopes may have the following additional fields
 abundance (%)
 Isotope abundance used to compute the properties of the element in natural abundance.
 nuclear_spin (string)
 Spin on the nucleus: ‘0’, ‘1/2’, ‘3/2’, etc.
Each field
T
above has a correspondingT_units
attribute with the name of the units. For scattering calculations, the scattering length density is the value of interest. This is computed from the number_density of the individual elements, as derived from the element density and atomic mass.Note
1 barn = 100 fm^{2}

has_sld
()¶ Returns True if sld is defined for this element/isotope.

scattering
(wavelength=1.798)¶ Returns neutron scattering information for the element at natural abundance and density.
Parameters: wavelength : float  Å
Returns:  sld : (float, float, float)  10^{6}Å^{2}
(real, imaginary, incoherent) scattering length density
 xs : (float, float, float)  cm^{1}
(coherent, absorption, incoherent) cross sections.
 penetration : float  cm
1/e penetration length.
See
neutron_scattering()
for details.

sld
(wavelength=1.798)¶ Returns scattering length density for the element at natural abundance and density.
Parameters: wavelength : float  Å
Returns:  sld : (float, float, float)  10^{6}Å^{2}
(real, imaginary, incoherent) scattering length density.
See
neutron_scattering()
for details.

absorption
= None¶

absorption_units
= 'barn'¶

abundance
= 0.0¶

abundance_units
= '%'¶

b_c
= None¶

b_c_i
= None¶

b_c_units
= 'fm'¶

bm
= None¶

bm_i
= None¶

bm_units
= 'fm'¶

bp
= None¶

bp_i
= None¶

bp_units
= 'fm'¶

coherent
= None¶

coherent_units
= 'barn'¶

incoherent
= None¶

incoherent_units
= 'barn'¶

is_energy_dependent
= False¶

total
= None¶

total_units
= 'barn'¶

periodictable.nsf.
init
(table, reload=False)¶ Loads the Rauch table from the neutron data book.

periodictable.nsf.
neutron_energy
(wavelength)¶ Convert neutron wavelength to energy.
Parameters: wavelength : float or vector  Å Returns: energy : float or vector  meV Wavelength is converted to energy using
\[E = 1/2 m_n v^2 = h^2 / (2 m_n \lambda^2)\]where:
\(h\) = planck’s constant in J·s
\(m_n\) = neutron mass in kg

periodictable.nsf.
neutron_wavelength
(energy)¶ Convert neutron energy to wavelength.
Parameters: energy : float or vector  meV Returns: wavelength : float or vector  Å Energy is converted to wavelength using
\[E = 1/2 m_n v^2 = h^2 / (2 m_n \lambda^2) \Rightarrow \lambda = \sqrt{h^2 / (2 m_n E)}\]where
\(h\) = planck’s constant in J·s
\(m_n\) = neutron mass in kg

periodictable.nsf.
neutron_wavelength_from_velocity
(velocity)¶ Convert neutron velocity to wavelength.
Parameters: velocity : float or vector  m/s Returns: wavelength : float or vector  Å Velocity is converted to wavelength using
\[\lambda = h/p = h/(m_n v)\]where
\(h\) = planck’s constant in J·s
\(m_n\) = neutron mass in kg

periodictable.nsf.
neutron_scattering
(compound, density=None, wavelength=1.798, energy=None, natural_density=None)¶ Computes neutron scattering cross sections for molecules.
Parameters:  compound : Formula initializer
Chemical formula
 density : float  g·cm^{3}
Mass density
 natural_density : float  g·cm^{3}
Mass density of formula with naturally occuring abundances
 wavelength 1.798 : float  Å
Neutron wavelength.
 energy : float  meV
Neutron energy. If energy is specified then wavelength is ignored.
Returns:  sld : (float, float, float)  10^{6}Å^{2}
(real, imaginary, incoherent) scattering length density.
 xs : (float, float, float)  cm^{1}
(coherent, absorption, incoherent) cross sections.
 penetration : float  cm
1/e penetration depth of the beam
Raises: AssertionError : density is missing.
The coherent and incoherent cross sections are calculated from the bound scattering lengths for nuclei. The actual cross sections depend on the incoming neutron energy and sample temperature, especially for light elements. For low energy neutrons (cold neutrons), the tabulated cross sections are generally a lower limit. The measured incoherent scattering from hydrogen, for example, can be considerably larger (by more than 20%) than its bound value. For example, the incoherent scattering cross section of H2O is 5.621/cm as computed from these tables compared to ~7.0/cm as measured with 5 meV neutrons at 290K. [7]
The scattering factor tables are not self consistent. The following functions show discrepencies between the various measurements of the scattering potential:
To compute the neutron cross sections we first need to average quantities for the unit cell of the molecule.
Molar mass m (g/mol) is the sum of the masses of each component:
\[m = \sum{n_i m_i}\ {\rm for\ each\ atom}\ i=1,2,\ldots\]Cell volume \(V\) (Å^{3}/molecule) is molar mass \(m\) over density \(rho\), with a correction based on Avogadro’s number \(N_A\) (atoms/mol) and the length conversion \(10^8\) Å/cm:
\[V = m/\rho \cdot 1/N_A \cdot (10^8)^3\]Number density \(N\) is the number of scatterers per unit volume:
\[N = \left.\sum{n_i} \right/ V\]Coherent scattering cross section \(\sigma_c\) of the molecule is computed from the average scattering length of its constituent atoms, weighted by their frequency.
\[b_c = \left.\sum n_i b_c \right/ \sum n_i\]This is converted to a scattering cross section and scaled by 1 barn = 100 fm^{2}:
\[\sigma_c = \left. 4 \pi b_c^2 \right/ 100\]Similarly, the absorption cross section \(\sigma_a\), the incoherent cross section \(\sigma_i\), and the total cross section \(\sigma_s\) can be computed from the corresponding cross sections of the constituent elements,[5] already expressed in barns:
\[\begin{split}\sigma_a &= \left. \sum n_j \sigma_{aj} \right/ \sum n_j \\ \sigma_i &= \left. \sum n_j \sigma_{ij} \right/ \sum n_j \\ \sigma_s &= \left. \sum n_j \sigma_{sj} \right/ \sum n_j\end{split}\]The neutron cross sections are tabulated at wavelength 1.798 Å. In the thermal neutron energy range most absorption cross sections scale linearly with wavelength,[4] and can be adjusted with a simple multiplication:
\[\sigma_a = \sigma_a \lambda / \lambda_o = \sigma_a \lambda / 1.798\]If isotope.neutron.is_energy_dependent() is true for any part of the material, then this relation may not hold, and the returned values are only valid for 1.798 Å.
From the scattering cross sections, the scattering length for a material \(b = b'  i b''\) can be computed using the following relations:[5]
\[\begin{split}\sigma_c &= 4 \pi b_c^2 \\ \sigma_a &= \left. 4 \pi \left< b'' \right> \right/k \ {\rm for} \ k=2\pi / \lambda \\ \sigma_i &= 4 \pi b_i^2 \\ \sigma_s &= 4 \pi \left< b^2 \right>\end{split}\]Transforming we get:
\[\begin{split}b'' &= \left. \sigma_a \right/ (2 \lambda) \\ b_i &= \sqrt{ \sigma_i / (4 \pi) }\end{split}\]The incoherent scattering length \(b_i\) can be treated primarily as an absorption length in large scale structure calculations, with the complex scattering length approximated by \(b = b_c  i (b'' + b_i)\).
The scattering potential is often expressed as a scattering length density (SLD). This is just the number density of the scatterers times their scattering lengths, with a correction for units.
\[\begin{split}\rho_{\rm re} &= 10 N b_c \\ \rho_{\rm im} &= N b'' / 100 \\ \rho_{\rm inc} &= 10 N b_i\end{split}\]with the factors of 10 chosen to give SLD in units of \(\AA^{2}\). The resulting \(\rho = \rho_{\rm re} + i \rho_{\rm im}\) can be used in the scattering equations. Treatment of the incoherent scattering \(\rho_{\rm inc}\) will depend on the equation. For example, it can be treated as an absorption in specular reflectivity calculations since the incoherently scattered neutrons are removed from the multilayer recurrence calculation.
Similarly, scattering cross section includes number density:
\[\begin{split}\Sigma_{\rm coh} &= N \sigma_c \\ \Sigma_{\rm inc} &= N \sigma_i \\ \Sigma_{\rm abs} &= N \sigma_a \\ \Sigma_{\rm s} &= N \sigma_s\end{split}\]The 1/e penetration depth \(t_u\) represents the the depth into the sample at which the unscattered intensity is reduced by a factor of \(e\):
\[t_u = \left. 1 \right/ (\Sigma_{\rm s} + \Sigma_{\rm abs})\]Note that the calculated penetration depth includes the effects of both absorption and incoherent scattering (which spreads the beam in the full \(4\pi\) spherical surface, and so it looks like absorption with respect to the beam), as well as the coherent scattering from the sample. If you instead want to calculate the effective shielding of the sample, you should recalculate penetration depth without the coherent scattering.
Transmission rate can be computed from \(e^{d/t_u}\) for penetration depth \(t_u\) and sample thickness \(d\). This does not include many real world effects, such as single phonon scattering[8] and forward scattering[7], which result in measured transmission significantly different from the values predicted from nuclear properties alone.
In general, the total scattering cross section is not the sum of the coherent and incoherent cross sections, \(\Sigma_{\rm s} \ne \Sigma_{\rm coh}+\Sigma_{\rm inc}\).[10] Instead, we compute \(\Sigma_{\rm inc} = \Sigma_{\rm s}  \Sigma_{\rm coh}\) in accordance with Sect. 4.4.4 of the Internation Tables for Crystallography Volume C.
Including unit conversion with \(\mu=10^{6}\) the full scattering equations are:
\[\begin{split}\rho_{\rm re}\,(\mu/\AA^2) &= (N/\AA^3) \, (b_c\,{\rm fm}) \, (10^{5} \AA/{\rm\,fm}) \, (10^6\,\mu) \\ \rho_{\rm im}\,(\mu/\AA^2) &= (N/\AA^3) \, (\sigma_a\,{\rm barn}) \, (10^{8}\,\AA^2/{\rm barn}) / (2 \lambda\, \AA) \, (10^6\,\mu) \\ \rho_{\rm inc}\,(\mu/\AA^2) &= (N/\AA^3) \, \sqrt{(\sigma_i\, {\rm barn})/(4 \pi) \, (100\, {\rm fm}^2/{\rm barn})} \, (10^{5}\, \AA/{\rm fm}) \, (10^6\, \mu) \\ \Sigma_{\rm coh}\,(1/{\rm cm}) &= (N/\AA^3) \, (\sigma_c\, {\rm barn}) \, (10^{8}\, \AA^2/{\rm barn}) \, (10^8\, \AA/{\rm cm}) \\ \Sigma_{\rm inc}\,(1/{\rm cm}) &= (N/\AA^3) \,(\sigma_i\, {\rm barn}) \, (10^{8}\, \AA^2/{\rm barn}) \, (10^8\, \AA/{\rm cm}) \\ \Sigma_{\rm abs}\,(1/{\rm cm}) &= (N/\AA^3) \,(\sigma_a\,{\rm barn}) \, (10^{8}\, \AA^2/{\rm barn}) \, (10^8\, \AA/{\rm cm}) \\ \Sigma_{\rm s}\,(1/{\rm cm}) &= (N/\AA^3) \,(\sigma_s\,{\rm barn}) \, (10^{8}\, \AA^2/{\rm barn}) \, (10^8\, \AA/{\rm cm}) \\ t_u\,({\rm cm}) &= 1/(\Sigma_{\rm s}\, 1/{\rm cm} \,+\, \Sigma_{\rm abs}\, 1/{\rm cm})\end{split}\]

periodictable.nsf.
neutron_sld
(*args, **kw)¶ Computes neutron scattering length densities for molecules.
Parameters:  compound : Formula initializer
Chemical formula
 density : float  g·cm^{3}
Mass density
 natural_density : float  g·cm^{3}
Mass density of formula with naturally occuring abundances
 wavelength : float  Å
Neutron wavelength.
 energy : float  meV
Neutron energy. If energy is specified then wavelength is ignored.
Returns:  sld : (float, float, float)  10^{6}Å^{2}
(real, imaginary, incoherent) scattering length density.
Raises: AssertionError : density is missing.
Returns the scattering length density of the compound. See
neutron_scattering()
for details.

periodictable.nsf.
neutron_composite_sld
(materials, wavelength=1.798)¶ Create a composite SLD calculator.
Parameters:  materials : [Formula]
List of materials
 wavelength = 1.798: float OR [float]  Å
Probe wavelength(s).
Returns: calculator : f(w, density=1) > (real, imaginary, incoherent)
The composite calculator takes a vector of weights and returns the scattering length density of the composite. This is useful for operations on large molecules, such as calculating a set of contrasts or fitting a material composition.
Table lookups and partial sums and constants are precomputed so that the calculation consists of a few simple array operations regardless of the size of the material fragments.

periodictable.nsf.
sld_plot
(table=None)¶ Plots SLD as a function of element number.
Parameters:  table : PeriodicTable
The default periodictable unless a specific table has been requested.
Returns: None

periodictable.nsf.
absorption_comparison_table
(table=None, tol=None)¶ Prints a table comparing absorption to the imaginary bound coherent scattering length b_c_i. This is used to checking the integrity of the data and formula.
The relationship between absorption and b_c_i is:
\[\sigma_a = 2 \lambda b_i \cdot 1000\]The wavelength \(\lambda = 1.798 \AA\) is the neutron wavelength at which the absorption is tallied. The factor of 1000 transforms from Å·fm to barn.
Parameters:  table : PeriodicTable
The default periodictable unless a specific table has been requested.
 tol = 0.01 : float  barn
Show differences greater than this amount.
Returns: None
Example
>>> absorption_comparison_table (tol=0.5) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE Comparison of absorption and (2000 lambda b_c_i) 3He 5333.00 5322.08 0.2% Li 70.50  6Li 940.00 934.96 0.5% B 767.00 755.16 1.6% 10B 3835.00  N 1.90  ...

periodictable.nsf.
coherent_comparison_table
(table=None, tol=None)¶ Prints a table of \(4 \pi b_c^2/100\) and coherent for each isotope. This is useful for checking the integrity of the data and formula.
The table only prints where b_c exists.
Parameters:  table : PeriodicTable
The default periodictable unless a specific table has been requested.
 tol = 0.01 : float  barn
Amount of difference to show
Returns: None
Example
>>> coherent_comparison_table (tol=0.5) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE Comparison of (4 pi b_c^2/100) and coherent n 172.03 43.01 300.0% 1n 172.03 43.01 300.0% Sc 18.40 19.00 3.2% 45Sc 18.40 19.00 3.2% 65Cu 13.08 14.10 7.2% 70Zn 5.98 4.50 33.0% 84Sr 3.14 6.00 47.6% ...

periodictable.nsf.
incoherent_comparison_table
(table=None, tol=None)¶ Prints a table of incoherent computed from total and b_c with incoherent.
Parameters:  table : PeriodicTable
The default periodictable unless a specific table has been requested.
 tol = 0.01 : float  barn
Amount of difference to show
Returns: None
Example
>>> incoherent_comparison_table (tol=0.5) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE Comparison of incoherent and (total  4 pi b_c^2/100) Sc 4.50 5.10 11.8% 45Sc 4.50 5.10 11.8% 65Cu 0.40 1.42 71.7% 70Zn 0.00 1.48 100.0% 84Sr 0.00 2.86 100.0% 113Cd 0.30 4.36 93.1% ...

periodictable.nsf.
total_comparison_table
(table=None, tol=None)¶ Prints a table of neutron.total and sum coh,inc for each isotope where these exist. This is used to checking the integrity of the data and formula.
Parameters:  table : PeriodicTable
The default periodictable unless a specific table has been requested.
 tol = 0.01 : float  barn
Amount of difference to show
Returns: None
Example
>>> total_comparison_table (tol=0.1) Comparison of total cross section to (coherent + incoherent) n 43.01  1n 43.01  84Kr 6.60  149Sm 200.00 200.50 0.2% Eu 9.20 9.07 1.4% Gd 180.00 180.30 0.2% 155Gd 66.00 65.80 0.3% 161Dy 16.00 16.30 1.8% 180Ta 7.00 6.70 4.5% 187Os 13.00 13.30 2.3%

periodictable.nsf.
energy_dependent_table
(table=None)¶ Prints a table of energy dependent isotopes.
Parameters:  table : PeriodicTable
If table is not specified, use the common periodic table.
Returns: None
Example
>>> energy_dependent_table() Elements and isotopes with energy dependent absorption: He3 Cd Cd113 Sm Sm149 Eu Eu151 Gd Gd155 Gd157 Yb168 Hg196 Hg199

periodictable.nsf.
sld_table
(wavelength=1, table=None, isotopes=True)¶ Scattering length density table for wavelength 4.75 Å.
Parameters:  table : PeriodicTable
If table is not specified, use the common periodic table.
 isotopes = True : boolean
Whether to consider isotopes or not.
Returns: None
Example
>>> sld_table(wavelength=4.75) # doctest: +ELLIPSIS, +NORMALIZE_WHITESPACE Neutron scattering length density table atom mass density sld imag incoh H 1.008 0.071 1.582 0.000 10.691 1H 1.008 0.071 1.583 0.000 10.691 D 2.014 0.141 2.823 0.000 1.705 T 3.016 0.212 2.027 0.000 0.453 He 4.003 0.122 0.598 0.000 0.035 3He 3.016 0.092 1.054 0.272 0.706 * 4He 4.003 0.122 0.598 0.000 0.035 ... 248Cm 248.072 13.569 2.536 0.000 0.207 * Energy dependent cross sections

periodictable.nsf.
neutron_sld_from_atoms
(*args, **kw)¶ Deprecated since version 0.91:
neutron_sld()
now accepts dictionaries of {atom: count} directly.