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Update C[13] neutron scattering cross sections#115

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pkienzle wants to merge 3 commits intofix-104-type-hintingfrom
update-C-13
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Update C[13] neutron scattering cross sections#115
pkienzle wants to merge 3 commits intofix-104-type-hintingfrom
update-C-13

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@pkienzle pkienzle commented Feb 13, 2026

WARNING This PR is based off of fix-104-type-hinting, not master. Update to master after merging #109.

The paper Fisher (2008) also gave values for b+ and b-, along with equations for computing b_c and b_i from b+ and b- so I created table generators to check whether the values are consistent across all isotopes (see below). The values are in good agreement with the measured coherent and incoherent scattering for many elements. For others the table values are inconsistent and the uncertainties are high.

For incoherence on a mixture of isotopes with natural abundances I used the standard formula (mean(b_i²) + var(b_c)) across the isotopes.

I then verified that the formula for b_i on a single isotope was just the variance in b+ and b- fractionally weighted with a I+1:I ratio for isotope spin I. In an attempt to improve consistency, I changed the mixture of isotopes formula to use variance on b+ and b- instead of the standard formula. This did not change the results significantly.

While running the checks I noticed that some of the spins reported in the tables were suspicious. Odd isotopes should have fractional spin and even isotopes should have integer spin. I corrected those elements using the spins from Tables for Crystallography, Volume C chapter 4. I didn't verify that every spin was correct.

Consistency check output:

In [1]: from periodictable import nsf

In [2]: nsf.bp_bm_bi_comparison_table(tol=0.1)
Comparison of σ_ι and 4 pi/100 var(b_c)
H             80.26    79.90   0.4%
1-H           80.27    79.91   0.4%
3-He           1.60     0.40 304.0%
10-B           3.00     2.72  10.3%
Cl             5.30     5.08   4.2%
35-Cl          4.70     4.46   5.5%
K              0.27     0.39 -31.4%
39-K           0.25     0.39 -35.9%
Ti             2.87     2.51  14.2%
49-Ti          3.30     0.45 639.0%
V              5.08     4.85   4.7%
51-V           5.07     4.84   4.8%
Co             4.80     5.06  -5.1%
59-Co          4.80     5.06  -5.1%
67-Zn          0.28     0.56 -50.4%
Ga             0.16     0.44 -63.9%
69-Ga          0.09     0.52 -82.5%
Ge             0.18     0.32 -43.7%
Zr             0.02     0.12 -83.8%
Rh             0.30     0.05 540.4%
103-Rh         0.30     0.05 540.4%
Pd             0.09     0.40 -76.6%
Cd             3.46     3.00  15.2%
Sn             0.02     0.68 -96.7%
117-Sn         0.30     0.00 6187.6%
119-Sn         0.30     0.00 64861.2%
La             1.13     1.47 -23.2%
139-La         1.13     1.47 -23.3%
Nd             9.20     8.94   2.9%
Sm            39.00    31.41  24.2%
Eu             2.50     2.28   9.8%
Gd           151.00    66.43 127.3%
Dy            54.40    54.63  -0.4%
Er             1.10     1.24 -11.4%
167-Er         0.13     0.87 -85.0%
Yb             4.00     4.30  -6.9%
183-W          0.30     0.01 2498.4%
Ir             0.00     0.17 -100.0%
Hg             6.60     1.13 486.4%
Tl             0.21     0.48 -56.1%
205-Tl         0.01     0.43 -98.4%

In [3]: nsf.bp_bm_bc_comparison_table(tol=0.1)
Comparison of b_c with b_c computed from b+ and b-
14-N           9.37     9.20   1.8%
15-N           6.44     6.63  -2.9%
17-O           5.87     5.37   9.2%
25-Mg          3.62     3.49   3.7%
29-Si          4.70     4.55   3.3%
51-V          -0.40    -0.54 -26.0%
55-Mn         -3.75    -3.48   7.6%
59-Co          2.49    -3.61 -168.9%
69-Ga          8.04     7.88   2.1%
71-Ga          6.17     6.36  -3.0%
93-Nb          7.05     7.19  -1.9%
103-Rh         5.90     7.80 -24.3%
117-Sn         6.59     0.11 6030.2%
119-Sn         6.28     0.11 5881.0%
139-La         8.24     8.38  -1.7%
173-Yb         9.56     7.00  36.6%
183-W          6.59     6.47   1.8%
197-Au         7.90     7.62   3.6%
205-Tl         8.87     6.22  42.6%

@pkienzle pkienzle changed the base branch from master to fix-104-type-hinting February 13, 2026 02:47
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