International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 4.4, pp. 444-454

Section 4.4.4. Scattering lengths for neutrons

V. F. Searsg

4.4.4. Scattering lengths for neutrons

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The use of neutron diffraction for crystal-structure determinations requires a knowledge of the scattering lengths and the corresponding scattering and absorption cross sections of the elements and, in some cases, of individual isotopes. This information is needed to calculate unit-cell structure factors and to correct for effects such as absorption, self-shielding, extinction, thermal diffuse scattering, and detector efficiency (Bacon, 1975[link]; Sears, 1989[link]). Table 4.4.4.1[link] lists the best values of the neutron scattering lengths and cross sections that are available at the time of writing (January 1995). We begin by summarizing the basic relationships between the scattering lengths and cross sections of the elements and their isotopes that have been used in the compilation of this table. More background information can be found in, for example, the book by Sears (1989[link]).

Table 4.4.4.1| top | pdf |   interactive version
Bound scattering lengths, b, in fm and cross sections, σ, in barns (1 barn = 100 fm2) of the elements and their isotopes

Z: atomic number; A: mass number; I(π): spin (parity) of the nuclear ground state; c: % natural abundance (for radioisotopes, the half-life is given instead in annums); bc: bound coherent scattering length; bi: bound incoherent scattering length; σc: bound coherent scattering cross section; σi: bound incoherent scattering cross section; σs: total bound scattering cross section; σa: absorption cross section for 2200 m s−1 neutrons (E = 25.30 meV, k = 3.494 Å−1, λ = 1.798 Å); i = [{\sqrt-1}].

Element[Z][A][I(\pi)][c][b_c][b_i][\sigma_c][\sigma_i][\sigma_s][\sigma_a]
H 1       −3.7390(11)   1.7568(10) 80.26(6) 82.02(6) 0.3326(7)
    1 1/2(+) 99.985 −3.7406(11) 25.274(9) 1.7583(10) 80.27(6) 82.03(6) 0.3326(7)
    2 1(+) 0.015 6.671(4) 4.04(3) 5.592(7) 2.05(3) 7.64(3) 0.000519(7)
    3 1/2(+) (12.32a) 4.792(27) −1.04(17) 2.89(3) 0.14(4) 3.03(5) 0
                     
He 2       3.26(3)   1.34(2) 0.00 1.34(2) 0.00747(1)
    3 1/2(+) 0.00014 5.74(7) −2.5(6) 4.42(10) 1.6(4) 6.0(4) 5333.(7.)
          −1.483(2)[i] +2.568(3)[i]        
    4 0(+) 99.99986 3.26(3) 0 1.34(2) 0 1.34(2) 0
                     
Li 3       −1.90(2)   0.454(14) 0.92(3) 1.37(3) 70.5(3)
    6 1(+) 7.5 2.00(11) −1.89(5) 0.51(5) 0.46(2) 0.97(7) 940.(4.)
          −0.261(1)[i] 0.257(11)[i]        
    7 3/2(−) 92.5 −2.22(2) −2.49(5) 0.619(11) 0.78(3) 1.40(3) 0.0454(3)
                     
Be 4 9 3/2(−) 100 7.79(1) 0.12(3) 7.63(2) 0.0018(9) 7.63(2) 0.0076(8)
                     
B 5       5.30(4)   3.54(5) 1.70(12) 5.24(11) 767.(8.)
          0.213(2)[i]          
    10 3(+) 20.0 −0.1(3) −4.7(3) 0.144(8) 3.0(4) 3.1(4) 3835.(9.)
          1.066(3)[i] 1.231(3)[i]        
    11 3/2(−) 80.0 6.65(4) −1.3(2) 5.56(7) 0.22(6) 5.78(9) 0.0055(33)
                     
C 6       6.6460(12)   5.550(2) 0.001(4) 5.551(3) 0.00350(7)
    12 0(+) 98.90 6.6511(16) 0 5.559(3) 0 5.559(3) 0.00353(7)
    13 1/2(−) 1.10 6.19(9) −0.52(9) 4.81(14) 0.034(12) 4.84(14) 0.00137(4)
                     
N 7       9.36(2)   11.01(5) 0.50(12) 11.51(11) 1.90(3)
    14 1(+) 99.63 9.37(2) 2.0(2) 11.03(5) 0.5(1) 11.53(11) 1.91(3)
    15 1/2(−) 0.37 6.44(3) −0.02(2) 5.21(5) 0.00005(10) 5.21(5) 0.000024(8)
                     
O 8       5.803(4)   4.232(6) 0.000(8) 4.232(6) 0.00019(2)
    16 0(+) 99.762 5.803(4) 0 4.232(6) 0 4.232(6) 0.00010(2)
    17 5/2(+) 0.038 5.78(12) 0.18(6) 4.20(22) 0.004(3) 4.20(22) 0.236(10)
    18 0(+) 0.200 5.84(7) 0 4.29(10) 0 4.29(10) 0.00016(1)
                     
F 9 19 1/2(+) 100 5.654(10) −0.082(9) 4.017(17) 0.0008(2) 4.018(14) 0.0096(5)
                     
Ne 10       4.566(6)   2.620(7) 0.008(9) 2.628(6) 0.039(4)
    20 0(+) 90.51 4.631(6) 0 2.695(7) 0 2.695(7) 0.036(4)
    21 3/2(+) 0.27 6.66(19) [\pm]0.6(1) 5.6(3) 0.05(2) 5.7(3) 0.67(11)
    22 0(+) 9.22 3.87(1) 0 1.88(1) 0 1.88(1) 0.046(6)
                     
Na 11 23 3/2(+) 100 3.63(2) 3.59(3) 1.66(2) 1.62(3) 3.28(4) 0.530(5)
                     
Mg 12       5.375(4)   3.631(5) 0.08(6) 3.71(4) 0.063(3)
    24 0(+) 78.99 5.66(3) 0 4.03(4) 0 4.03(4) 0.050(5)
    25 5/2(+) 10.00 3.62(14) 1.48(10) 1.65(13) 0.28(4) 1.93(14) 0.19(3)
    26 0(+) 11.01 4.89(15) 0 3.00(18) 0 3.00(18) 0.0382(8)
                     
Al 13 27 5/2(+) 100 3.449(5) 0.256(10) 1.495(4) 0.0082(7) 1.503(4) 0.231(3)
                     
Si 14       4.1491(10)   2.1633(10) 0.004(8) 2.167(8) 0.171(3)
    28 0(+) 92.23 4.107(6) 0 2.120(6) 0 2.120(6) 0.177(3)
    29 1/2(+) 4.67 4.70(10) 0.09(9) 2.78(12) 0.001(2) 2.78(12) 0.101(14)
    30 0(+) 3.10 4.58(8) 0 2.64(9) 0 2.64(9) 0.107(2)
                     
P 15 31 1/2(+) 100 5.13(1) 0.2(2) 3.307(13) 0.005(10) 3.312(16) 0.172(6)
                     
S 16       2.847(1)   1.0186(7) 0.007(5) 1.026(5) 0.53(1)
    32 0(+) 95.02 2.804(2) 0 0.9880(14) 0 0.9880(14) 0.54(4)
    33 3/2(+) 0.75 4.74(19) 1.5(1.5) 2.8(2) 0.3(6) 3.1(6) 0.54(4)
    34 0(+) 4.21 3.48(3) 0 1.52(3) 0 1.52(3) 0.227(5)
    36 0(+) 0.02 3.(1.) E 0 1.1(8) 0 1.1(8) 0.15(3)
                     
Cl 17       9.5770(8)   11.526(2) 5.3(5) 16.8(5) 33.5(3)
    35 3/2(+) 75.77 11.65(2) 6.1(4) 17.06(6) 4.7(6) 21.8(6) 44.1(4)
    37 3/2(+) 24.23 3.08(6) 0.1(1) 1.19(5) 0.001(3) 1.19(5) 0.433(6)
                     
Ar 18       1.909(6)   0.458(3) 0.22(2) 0.683(4) 0.675(9)
    36 0(+) 0.337 24.90(7) 0 77.9(4) 0 77.9(4) 5.2(5)
    38 0(+) 0.063 3.5(3.5) 0 1.5(3.1) 0 1.5(3.1) 0.8(2)
    40 0(+) 99.600 1.830(6) 0 0.421(3) 0 0.421(3) 0.660(9)
                     
K 19       3.67(2)   1.69(2) 0.27(11) 1.96(11) 2.1(1)
    39 3/2(+) 93.258 3.74(2) 1.4(3) 1.76(2) 0.25(11) 2.01(11) 2.1(1)
    40 4(−) 0.012 3.(1.) E   1.1(8) 0.5(5) 1.6(9) 35.(8.)
    41 3/2(+) 6.730 2.69(8) 1.5(1.5) 0.91(5) 0.3(6) 1.2(6) 1.46(3)
                     
Ca 20       4.70(2)   2.78(2) 0.05(3) 2.83(2) 0.43(2)
    40 0(+) 96.941 4.80(2) 0 2.90(2) 0 2.90(2) 0.41(2)
    42 0(+) 0.647 3.36(10) 0 1.42(8) 0 1.42(8) 0.68(7)
    43 7/2(−) 0.135 −1.56(9) 0.31(4) 0.5(5) E   0.8(5) 6.2(6)
    44 0(+) 2.086 1.42(6) 0 0.25(2) 0 0.25(2) 0.88(5)
    46 0(+) 0.004 3.6(2) 0 1.6(2) 0 1.6(2) 0.74(7)
    48 0(+) 0.187 0.39(9) 0 0.019(9) 0 0.019(9) 1.09(14)
                     
Sc 21 45 7/2(−) 100 12.29(11) −6.0(3) 19.0(3) 4.5(5) 23.5(6) 27.5(2)
                     
Ti 22       −3.370(13)   1.427(11) 2.63(3) 4.06(3) 6.43(6)
    46 0(+) 8.2 4.725(5) 0 2.80(6) 0 2.80(6) 0.59(18)
    47 5/2(−) 7.4 3.53(7) −3.5(2) 1.57(6) 1.5(2) 3.1(2) 1.7(2)
    48 0(+) 73.8 −5.86(2) 0 4.32(3) 0 4.32(3) 8.30(9)
    49 7/2(−) 5.4 0.98(5) 5.1(2) 0.12(1) 3.3(3) 3.4(3) 2.2(3)
    50 0(+) 5.2 5.88(10) 0 4.34(15) 0 4.34(15) 0.179(3)
                     
V 23       −0.3824(12)   0.01838(12) 5.08(6) 5.10(6) 5.08(2)
    50 6(+) 0.250 7.6(6)   7.3(1.1) 0.5(5) E 7.8(1.0) 60.(40.)
    51 7/2(−) 99.750 −0.402(2) 6.435(4) 0.0203(2) 5.07(6) 5.09(6) 4.9(1)
                     
Cr 24       3.635(7)   1.660(6) 1.83(2) 3.49(2) 3.05(8)
    50 0(+) 4.35 −4.50(5) 0 2.54(6) 0 2.54(6) 15.8(2)
    52 0(+) 83.79 4.920(10) 0 3.042(12) 0 3.042(12) 0.76(6)
    53 3/2(−) 9.50 −4.20(3) 6.87(10) 2.22(3) 5.93(17) 8.15(17) 18.1(1.5)
    54 0(+) 2.36 4.55(10) 0 2.60(11) 0 2.60(11) 0.36(4)
                     
Mn 25 55 5/2(−) 100 −3.750(18) 1.79(4) 1.77(2) 0.40(2) 2.17(3) 13.3(2)
                     
Fe 26       9.45(2)   11.22(5) 0.40(11) 11.62(10) 2.56(3)
    54 0(+) 5.8 4.2(1) 0 2.2(1) 0 2.2(1) 2.25(18)
    56 0(+) 91.7 9.94(3) 0 12.42(7) 0 12.42(7) 2.59(14)
    57 1/2(−) 2.2 2.3(1) 0.66(6)   0.3(3) E 1.0(3) 2.48(30)
    58 0(+) 0.3 15.(7.) 0 28.(26.) 0 28.(26.) 1.28(5)
                     
Co 27 59 7/2(−) 100 2.49(2) −6.2(2) 0.779(13) 4.8(3) 5.6(3) 37.18(6)
                     
Ni 28       10.3(1)   13.3(3) 5.2(4) 18.5(3) 4.49(16)
    58 0(+) 68.27 14.4(1) 0 26.1(4) 0 26.1(4) 4.6(3)
    60 0(+) 26.10 2.8(1) 0 0.99(7) 0 0.99(7) 2.9(2)
    61 3/2(−) 1.13 7.60(6) [\pm]3.9(3) 7.26(11) 1.9(3) 9.2(3) 2.5(8)
    62 0(+) 3.59 −8.7(2) 0 9.5(4) 0 9.5(4) 14.5(3)
    64 0(+) 0.91 −0.37(7) 0 0.017(7) 0 0.017(7) 1.52(3)
                     
Cu 29       7.718(4)   7.485(8) 0.55(3) 8.03(3) 3.78(2)
    63 3/2(−) 69.17 6.43(15) 0.22(2) 5.2(2) 0.006(1) 5.2(2) 4.50(2)
    65 3/2(−) 30.83 10.61(19) 1.79(10) 14.1(5) 0.40(4) 14.5(5) 2.17(3)
                     
Zn 30       5.60(5)   4.054(7) 0.077(7) 4.131(10) 1.11(2)
    64 0(+) 48.6 5.22(4) 0 3.42(5) 0 3.42(5) 0.93(9)
    66 0(+) 27.9 5.97(5) 0 4.48(8) 0 4.48(8) 0.62(6)
    67 5/2(−) 4.1 7.56(8) −1.50(7) 7.18(15) 0.28(3) 7.46(15) 6.8(8)
    68 0(+) 18.8 6.03(3) 0 4.57(5) 0 4.57(5) 1.1(1)
    70 0(+) 0.6 6.(1.) E 0 4.5(1.5) 0 4.5(1.5) 0.092(5)
                     
Ga 31       7.288(2)   6.675(4) 0.16(3) 6.83(3) 2.75(3)
    69 3/2(−) 60.1 7.88(2) −0.85(5) 7.80(4) 0.091(11) 7.89(4) 2.18(5)
    71 3/2(−) 39.9 6.40(3) −0.82(4) 5.15(5) 0.084(8) 5.23(5) 3.61(10)
                     
Ge 32       8.185(20)   8.42(4) 0.18(7) 8.60(6) 2.20(4)
    70 0(+) 20.5 10.0(1) 0 12.6(3) 0 12.6(3) 3.0(2)
    72 0(+) 27.4 8.51(10) 0 9.1(2) 0 9.1(2) 0.8(2)
    73 9/2(+) 7.8 5.02(4) 3.4(3) 3.17(5) 1.5(3) 4.7(3) 15.1(4)
    74 0(+) 36.5 7.58(10) 0 7.2(2) 0 7.2(2) 0.4(2)
    76 0(+) 7.8 8.21(1.5) 0 8.(3.) 0 8.(3.) 0.16(2)
                     
As 33 75 3/2(−) 100 6.58(1) −0.69(5) 5.44(2) 0.060(10) 5.50(2) 4.5(1)
                     
Se 34       7.970(9)   7.98(2) 0.33(6) 8.30(6) 11.7(2)
    74 0(+) 0.9 0.8(3.0) 0 0.1(6) 0 0.1(6) 51.8(1.2)
    76 0(+) 9.0 12.2(1) 0 18.7(3) 0 18.7(3) 85.(7.)
    77 1/2(−) 7.6 8.25(8) [\pm]0.6(1.6) 8.6(2) 0.05(26) 8.65(16) 42.(4.)
    78 0(+) 23.5 8.24(9) 0 8.5(2) 0 8.5(2) 0.43(2)
    80 0(+) 49.6 7.48(3) 0 7.03(6) 0 7.03(6) 0.61(5)
    82 0(+) 9.4 6.34(8) 0 5.05(13) 0 5.05(13) 0.044(3)
                     
Br 35       6.795(15)   5.80(3) 0.10(9) 5.90(9) 6.9(2)
    79 3/2(−) 50.69 6.80(7) −1.1(2) 5.81(12) 0.15(6) 5.96(13) 11.0(7)
    81 3/2(−) 49.31 6.79(7) 0.6(1) 5.79(12) 0.05(2) 5.84(12) 2.7(2)
                     
Kr 36       7.81(2)   7.67(4) 0.01(14) 7.68(13) 25.(1.)
    78 0(+) 0.35   0   0   6.4(9)
    80 0(+) 2.25   0   0   11.8(5)
    82 0(+) 11.6   0   0   29.(20.)
    83 9/2(+) 11.5   185(30.)        
    84 0(+) 57.0   0   0   0.113(15)
    86 0(+) 17.3 8.1(2) 0 8.2(4) 0 8.2(4) 0.003(2)
                     
Rb 37       7.09(2)   6.32(4) 0.5(4) 6.8(4) 0.38(4)
    85 5/2(−) 72.17 7.03(10) 6.2(2) 0.5(5) E 6.7(5) 0.48(1)
    87 3/2(−) 27.83 7.23(12) 6.6(2) 0.5(5) E 7.1(5) 0.12(3)
                     
Sr 38       7.02(2)   6.19(4) 0.06(11) 6.25(10) 1.28(6)
    84 0(+) 0.56 7.(1.) E 0 6.(2.) 0 6.(2.) 0.87(7)
    86 0(+) 9.86 5.67(5) 0 4.04(7) 0 4.04(7) 1.04(7)
    87 9/2(+) 7.00 7.40(7) 6.88(13) 0.5(5) E 7.4(5) 16.(3.)
    88 0(+) 82.58 7.15(6) 0 6.42(11) 0 6.42(11) 0.058(4)
                     
Y 39 89 1/2(−) 100 7.75(2) 1.1(3) 7.55(4) 0.15(8) 7.70(9) 1.28(2)
                     
Zr 40       7.16(3)   6.44(5) 0.02(15) 6.46(14) 0.185(3)
    90 0(+) 51.45 6.4(1) 0 5.1(2) 0 5.1(2) 0.011(5)
    91 5/2(+) 11.32 8.7(1) −1.08(15) 9.5(2) 0.15(4) 9.7(2) 1.17(10)
    92 0(+) 17.19 7.4(2) 0 6.9(4) 0 6.9(4) 0.22(6)
    94 0(+) 17.28 8.2(2) 0 8.4(4) 0 8.4(4) 0.0499(24)
    96 0(+) 2.76 5.5(1) 0 3.8(1) 0 3.8(1) 0.0229(10)
                     
Nb 41 93 9/2(+) 100 7.054(3) −0.139(10) 6.253(5) 0.0024(3) 6.255(5) 1.15(5)
                     
Mo 42       6.715(2)   5.67(3) 0.04(5) 5.71(4) 2.48(4)
    92 0(+) 14.84 6.91(8) 0 6.00(14) 0 6.00(14) 0.019(2)
    94 0(+) 9.25 6.80(7) 0 5.81(12) 0 5.81(12) 0.015(2)
    95 5/2(+) 15.92 6.91(6) 6.00(10) 0.5(5) E 6.5(5) 13.1(3)
    96 0(+) 16.68 6.20(6) 0 4.83(9) 0 4.83(9) 0.5(2)
    97 5/2(+) 9.55 7.24(8) 6.59(15) 0.5(5) E 7.1(5) 2.5(2)
    98 0(+) 24.13 6.58(7) 0 5.44(12) 0 5.44(12) 0.127(6)
    100 0(+) 9.63 6.73(7) 0 5.69(12) 0 5.69(12) 0.4(2)
                     
Tc 43                  
    99 9/2(+) (2.13×105a) 6.8(3) 5.8(5) 0.5(5) E 6.3(7) 20.(1.)
                     
Ru 44       7.03(3)   6.21(5) 0.4(1) 6.6(1) 2.56(13)
    96 0(+) 5.5 0 0 0.28(2)      
    98 0(+) 1.9 0 0 < 8.0      
    99 5/2(+) 12.7 6.9(1.0)          
    100 0(+) 12.6 0 0 4.8(6)      
    101 5/2(+) 17.0 3.3(9)          
    102 0(+) 31.6 0 0 1.17(7)      
    104 0(+) 18.7 0 0 0.31(2)      
                     
Rh 45 103 1/2(−) 100 5.88(4) 4.34(6) 0.3(3) E 4.6(3) 144.8(7)
                     
Pd 46       5.91(6)   4.39(9) 0.093(9) 4.48(9) 6.9(4)
    102 0(+) 1.02 7.7(7) E 0 7.5(1.4) 0 7.5(1.4) 3.4(3)
    104 0(+) 11.14 7.7(7) E 0 7.5(1.4) 0 7.5(1.4) 0.6(3)
    105 5/2(+) 22.33 5.5(3) −2.6(1.6) 3.8(4) 0.8(1.0) 4.6(1.1) 20.(3.)
    106 0(+) 27.33 6.4(4) 0 5.1(6) 0 5.1(6) 0.304(29)
    108 0(+) 26.46 4.1(3) 0 2.1(3) 0 2.1(3) 8.5(5)
    110 0(+) 11.72 7.7(7)E 0 7.5(1.4) 0 7.5(1.4) 0.226(31)
                     
Ag 47       5.922(7)   4.407(10) 0.58(3) 4.99(3) 63.3(4)
    107 1/2(−) 51.839 7.555(11) 1.00(13) 7.17(2) 0.13(3) 7.30(4) 37.6(1.2)
    109 1/2(−) 48.161 4.165(11) −1.60(13) 2.18(1) 0.32(5) 2.50(5) 91.0(1.0)
                     
Cd 48       4.87(5)   3.04(6) 3.46(13) 6.50(12) 2520.(50.)
          −0.70(1)[i]          
    106 0(+) 1.25 5.(2.) E 0 3.1(2.5) 0 3.1(2.5) 1.
    108 0(+) 0.89 5.4(1) 0 3.7(1) 0 3.7(1) 1.1(3)
    110 0(+) 12.51 5.9(1) 0 4.4(1) 0 4.4(1) 11.(1.)
    111 1/2(+) 12.81 6.5(1) 5.3(2) 0.3(3) E 5.6(4) 24(3.)
    112 0(+) 24.13 6.4(1) 0 5.1(2) 0 5.1(2) 2.2(5)
    *113 1/2(+) 12.22 −8.0(2) 12.1(4) 0.3(3) E 12.4(5)   20600(400.)
          −5.73(11)[i]          
    114 0(+) 28.72 7.5(1) 0 7.1(2) 0 7.1(2) 0.34(2)
    116 0(+) 7.47 6.3(1) 0 5.0(2) 0 5.0(2) 0.075(13)
                     
In 49     4.065(20) 2.08(2) 0.54(11) 2.62(11) 193.8(1.5)    
          −0.0539(4)[i]          
    113 9/2(+) 43 5.39(6) [\pm]0.017(1) 3.65(8) 0.000037(5) 3.65(8) 12.0(1.1)
    115 9/2(+) 957 4.01(2) −2.1(2) 2.02(2) 0.55(11) 2.57(11) 202(2.)
          −0.0562(6)[i]          
                     
Sn 50       6.225(2)   4.870(3) 0.022(5) 4.892(6) 0.626(9)
    112 0(+) 1.0 6.1(1.) E 0 4.5(1.5) 0 4.5(1.5) 1.01(11)
    114 0(+) 0.7 6.2(3) 0 4.8(5) 0 4.8(5) 0.114(30)
    115 1/2(+) 0.4 6.(1.) E 4.5(1.5) 0.3(3) E 4.8(1.5) 30(7.)  
    116 0(+) 14.7 5.93(5) 0 4.42(7) 0 4.42(7) 0.14(3)
    117 1/2(+) 7.7 6.48(5) 5.28(8) 0.3(3) E 5.6(3) 2.3(5)  
    118 0(+) 24.3 6.07(5) 0 4.63(8) 0 4.63(8) 0.22(5)
    119 1/2(+) 8.6 6.12(5) 4.71(8) 0.3(3) E 5.0(3) 2.2(5)  
    120 0(+) 32.4 6.49(5) 0 5.29(8) 0 5.29(8) 0.14(3)
    122 0(+) 4.6 5.74(5) 0 4.14(7) 0 4.14(7) 0.18(2)
    124 0(+) 5.6 5.97(5) 0 4.48(8) 0 4.48(8) 0.133(5)
                     
Sb 51       5.57(3)   3.90(4) 0.00(7) 3.90(6) 4.91(5)
    121 7/2(+) 57.3 5.71(6) −0.05(15) 4.10(9) 0.0003(19) 4.10(9) 5.75(12)
    123 5/2(+) 42.7 5.38(7) −0.10(15) 3.64(9) 0.001(4) 3.64(9) 3.8(2)
                     
Te 52       5.80(3)   4.23(4) 0.09(1) 4.32(4) 4.05(5)
    120 0(+) 0.096 5.3(5) 0 3.5(7) 0 3.4(7) 2.3(3)
    122 0(+) 2.60 3.8(2) 0 1.8(2) 0 1.8(2) 3.4(5)
    123 1/2(+) 0.908 −0.05(25) −2.04(9) 0.002(3) 0.52(5) 0.52(5) 418(30.)
          −0.116(8)[i]          
    124 0(+) 4.816 7.96(10) 0 8.0(2) 0 8.0(2) 6.8(1.3)
    125 1/2(+) 7.14 5.02(8) −0.26(13) 3.17(10) 0.008(8) 3.18(10) 1.55(16)
    126 0(+) 18.95 5.56(7) 0 3.88(10) 0 3.88(10) 1.04(15)
    128 0(+) 31.69 5.89(7) 0 4.36(10) 0 4.36(10) 0.215(8)
    130 0(+) 33.80 6.02(7) 0 4.55(11) 0 4.55(11) 0.29(6)
                     
I 53 127 5/2(+) 100 5.28(2) 1.58(15) 3.50(3) 0.31(6) 3.81(7) 6.15(6)
                     
Xe 54       4.92(3)   3.04(4)     23.9(1.2)
    124 0(+) 0.10   0   0   165.(20.)
    126 0(+) 0.09   0   0   3.5(8)
    128 0(+) 1.91   0   0   < 8.
    129 1/2(+) 26.4           21.(5.)
    130 0(+) 4.1   0   0   < 26.
    131 3/2(+) 21.2           85.(10.)
    132 0(+) 26.9   0   0   0.45(6)
    134 0(+) 10.4   0   0   0.265(20)
    136 0(+) 8.9   0   0   0.26(2)
                     
Cs 55 133 7/2(+) 100 5.42(2) 1.29(15) 3.69(3) 0.21(5) 3.90(6) 29.0(1.5)
                     
Ba 56       5.07(3)   3.23(4) 0.15(11) 3.38(10) 1.1(1)
    130 0(+) 0.11 −3.6(6) 0 1.6(5) 0 1.6(5) 30(5.)
    132 0(+) 0.10 7.8(3) 0 7.6(6) 0 7.6(6) 7.0(8)
    134 0(+) 2.42 5.7(1) 0 4.08(14) 0 4.08(14) 2.0(1.6)
    135 3/2(+) 6.59 4.67(10)   2.74(12) 0.5(5) E 3.2(5) 5.8(9)
    136 0(+) 7.85 4.91(8) 0 3.03(10) 0 3.03(10) 0.68(17)
    137 3/2(+) 11.23 6.83(10)   5.86(17) 0.5(5) E 6.4(5) 3.6(2)
    138 0(+) 71.70 4.84(8) 0 2.94(10) 0 2.94(10) 0.27(14)
                     
La 57       8.24(4)   8.53(8) 1.13(19) 9.66(17) 8.97(5)
    138 5(+) 0.09 8.(2.) E 8.(4.) 0.5(5) E 8.5(4.0) 57.(6.)  
    139 7/2(+) 99.91 8.24(4) 3.0(2) 8.53(8) 1.13(15) 9.66(17) 8.93(4)
                     
Ce 58       4.84(2)   2.94(2) 0.00(10) 2.94(10) 0.63(4)
    136 0(+) 0.19 5.80(9) 0 4.23(13) 0 4.23(13) 7.3(1.5)
    138 0(+) 0.25 6.70(9) 0 5.64(15) 0 5.64(15) 1.1(3)
    140 0(+) 88.48 4.84(9) 0 2.94(11) 0 2.94(11) 0.57(4)
    142 0(+) 11.08 4.75(9) 0 2.84(11) 0 2.84(11) 0.95(5)
                     
Pr 59 141 5/2(+) 100 4.58(5) −0.35(3) 2.64(6) 0.015(3) 2.66(6) 11.5(3)
                     
Nd 60       7.69(5)   7.43(10) 9.2(8) 16.6(8) 50.5(1.2)
    142 0(+) 27.16 7.7(3) 0 7.5(6) 0 7.5(6) 18.7(7)
    143 7/2(−) 12.18 14.2(5) E [\pm]21.1(6) 25.(7.) 55.(7.) 80.(2.) 334.(10.)
    144 0(+) 23.80 2.8(3) 0 1.0(2) 0 1.0(2) 3.6(3)
    145 7/2(−) 8.29 14.2(5) E 25.(7.) 5.(5.) E 30.(9.) 42.(2.)
    146 0(+) 17.19 8.7(2) 0 9.5(4) 0 9.5(4) 1.4(1)
    148 0(+) 5.75 5.7(3) 0 4.1(4) 0 4.1(4) 2.5(2)
    150 0(+) 5.63 5.3(2) 0 3.5(3) 0 3.5(3) 1.2(2)
                     
Pm 61                  
    147 7/2(+) (2.62a) 12.6(4) [\pm]3.2(2.5) 20.0(1.3) 1.3(2.0) 21.3(1.5) 168.4(3.5)
                     
Sm 62       0.80(2)   0.422(9) 39.(3.) 39.(3.) 5922.(56.)
          −1.65(2)[i]          
    144 0(+) 3.1 −3.(4.) E 0 1.(3.) 0 1.(3.) 0.7(3)
    147 7/2(−) 15.1 14(3.) [\pm]11.(7.) 25.(11.) 14.(19.) 39(16.) 57(3.)
    148 0(+) 11.3 −3.(4.) E 0 1.(3.) 0 1.(3.) 2.4(6)
    *149 7/2(−) 13.9 −19.2(1) [\pm]31.4(6) 63.5(6) 137.(5.) 200.(5.) 42080.(400.)
          −11.7(1)[i] −10.3(1)[i]        
    150 0(+) 7.4 14(3.) 0 25(11.) 0 25(11.) 104(4.)
    152 0(+) 26.6 −5.0(6) 0 3.1(8) 0 3.1(8) 206.(6.)
    154 0(+) 22.6 9.3(1.0) 0 11.(2.) 0 11.(2.) 8.4(5)
                     
Eu 63       7.22(2)   6.75(4) 2.5(4) 9.2(4) 4530.(40.)
          −1.26(1)[i]          
    *151 5/2(+) 47.8 6.13(14) [\pm]4.5(4) 5.5(2) 3.1(4) 8.4(4) 9100(100.)
          −2.53(3)[i] −2.14(2)[i]        
    153 5/2(+) 52.2 8.22(12) [\pm]3.2(9) 8.5(2) 1.3(7) 9.8(7) 312.(7.)
                     
Gd 64       6.5(5)   29.3(8) 151.(2.) 180.(2.) 49700.(125.)
          −13.82(3)[i]          
    152 0(+) 0.2 10.(3.) E 0 13.(8.) 0 13.(8.) 735.(20.)
    154 0(+) 2.1 10.(3.) E 0 13.(8.) 0 13.(8.) 85.(12.)
    *155 3/2(−) 14.8 6.0(1) [\pm]5.(5.) E 40.8(4.) 25.(6.) 66.(6.) 61100.(400.)
          −17.0(1)[i] −13.16(9)[i]        
    156 0(+) 20.6 6.3(4) 0 5.0(6) 0 5.0(6) 1.5(1.2)
    *157 3/2(−) 15.7 −1.14(2) [\pm]5.(5.) E 650(4.) 394.(7.) 1044.(8.) 259000.(700.)
          −71.9(2)[i] −55.8(2)[i]        
    158 0(+) 24.8 9.(2.) 0 10.(5.) 0 10.(5.) 2.2(2)
    160 0(+) 21.8 9.15(5) 0 10.52(11) 0 10.52(11) 0.77(2)
                     
Tb 65 159 3/2(+) 100 7.38(3) −0.17(7) 6.84(6) 0.004(3) 6.84(6) 23.4(4)
                     
Dy 66       16.9(2)   35.9(8) 54.4(1.2) 90.3(9) 994.(13.)
          −0.276(4)[i]          
    156 0(+) 0.06 6.1(5) 0 4.7(8) 0 4.7(8) 33.(3.)
    158 0(+) 0.10 6.(4.) E 0 5.(6.) 0 5.(6.) 43.(6.)
    160 0(+) 2.34 6.7(4) 0 5.6(7) 0 5.6(7) 56.(5.)
    161 5/2(+) 19.0 10.3(4) [\pm]4.9(8) 13.3(1.0) 3.(1.) 16.(1.) 600.(25.)
    162 0(+) 25.5 −1.4(5) 0 0.25(18) 0 0.25(18) 194.(10.)
    163 5/2(−) 24.9 5.0(4) 1.3(3) 3.1(5) 0.21(10) 3.3(5) 124.(7.)
    164 0(+) 28.1 49.4(5) 0 307.(3.) 0 307.(3.) 2840.(40.)
          −0.79(1)[i]          
                     
Ho 67 165 7/2(−) 100 8.01(8) −1.70(8) 8.06(16) 0.36(3) 8.42(16) 64.7(1.2)
                     
Er 68       7.79(2)   7.63(4) 1.1(3) 8.7(3) 159.(4.)
    162 0(+) 0.14 8.8(2) 0 9.7(4) 0 9.7(4) 19.(2.)
    164 0(+) 1.56 8.2(2) 0 8.4(4) 0 8.4(4) 13.(2.)
    166 0(+) 33.4 10.6(2) 0 14.1(5) 0 14.1(5) 19.6(1.5)
    167 7/2(+) 22.9 3.0(3) 1.0(3) 1.1(2) 0.13(8) 1.2(2) 659.(16.)
    168 0(+) 27.1 7.4(4) 0 6.8(7) 0 6.9(7) 2.74(8)
    170 0(+) 14.9 9.6(5) 0 11.6(1.2) 0 11.6(1.2) 5.8(3)
                     
Tm 69 169 1/2(+) 100 7.07(3) 0.9(3) 6.28(5) 0.10(7) 6.38(9) 100.(2.)
                     
Yb 70       12.43(3)   19.42(9) 4.0(2) 23.05(18) 34.8(8)
    168 0(+) 0.14 −4.07(2) 0 2.13(2) 0 2.13(2) 2230.(40.)
          −0.62(1)[i]          
    170 0(+) 3.06 6.77(10) 0 5.8(2) 0 5.8(2) 11.4(1.0)
    171 1/2(−) 143 9.66(10) −5.59(17) 11.7(2) 3.9(2) 15.6(3) 48.6(2.5)
    172 0(+) 21.9 9.43(10) 0 11.2(2) 0 11.2(2) 0.8(4)
    173 5/2(−) 16.1 9.56(7) −5.3(2) 11.5(2) 3.5(3) 15.0(4) 17.1(1.3)
    174 0(+) 31.8 19.3(1) 0 46.8(5) 0 46.8(5) 69.4(5.0)
    176 0(+) 12.7 8.72(10) 0 9.6(2) 0 9.6(2) 2.85(5)
                     
Lu 71       7.21(3)   6.53(5) 0.7(4) 7.2(4) 74.(2.)
    175 7/2(+) 97.39 7.24(3) [\pm]2.2(7) 6.59(5) 0.6(4) 7.2(4) 21.(3.)
    *176 7(−) 2.61 6.1(1) [\pm]3.0(4) 4.7(2) 1.2(3) 5.9(4) 2065.(35.)
          −0.57(1)[i] +0.61(1)[i]        
                     
Hf 72       7.77(14)   7.6(3) 2.6(5) 10.2(4) 104.1(0.5)
    174 0(+) 0.2 10.9(1.1) 0 15.(3.) 0 15.(3.) 561.(35.)
    176 0(+) 5.2 6.61(18) 0 5.5(3) 0 5.5(3) 23.5(3.1)
    177 7/2(−) 18.6 0.8(1.0) E [\pm]0.9(1.3) 0.1(2) 0.1(3) 0.2(2) 373.(10.)
    178 0(+) 27.1 5.9(2) 0 4.4(3) 0 4.4(3) 84.(4.)
    179 9/2(+) 13.7 7.46(16) [\pm]1.06(8) 7.0(3) 0.14(2) 7.1(3) 41.(3.)
    180 0(+) 35.2 13.2(3) 0 21.9(1.0) 0 21.9(1.0) 13.04(7)
                     
Ta 73       6.91(7)   6.00(12) 0.01(17) 6.01(12) 20.6(5)
    *180 9(−) 0.012 7.(2.) E 6.2(3.5) 0.5(5) E 7.(4.) 563.(60.)
    181 7/2(+) 99.988 6.91(7) −0.29(3) 6.00(12) 0.011(2) 6.01(12) 20.5(5)
                     
W 74       4.86(2)   2.97(2) 1.63(6) 4.60(6) 18.3(2)
    180 0(+) 0.1 5.(3.) E 0 3.(4.) 0 3.(4.) 30.(20.)
    182 0(+) 26.3 6.97(14) 0 6.10(7) 0 6.10(7) 20.7(5)
    183 1/2(−) 14.3 6.53(4)   5.36(7) 0.3(3) E 5.7(3) 10.1(3)
    184 0(+) 30.7 7.48(6) 0 7.03(11) 0 7.03(11) 1.7(1)
    186 0(+) 28.6 −0.72(4) 0 0.065(7) 0 0.065(7) 37.9(0.6)
                     
Re 75       9.2(2)   10.6(5) 0.9(6) 11.5(3) 89.7(1.0)
    185 5/2(+) 37.40 9.0(3) [\pm]2.0(1.8) 10.2(7) 0.5(9) 10.7(6) 112.(2.)
    187 5/2(+) 62.60 9.3(3) [\pm]2.8(1.1) 10.9(7) 1.0(8) 11.9(4) 76.4(1.0)
                     
Os 76       10.7(2)   14.4(5) 0.3(8) 14.7(6) 16.0(4)
    184 0(+) 0.02 10.(2.) E 0 13.(5.) 0 13.(5.) 3000.(150.)
    186 0(+) 1.58 11.6(1.7) 0 17.(5.) 0 17.(5.) 80.(13.)
    187 1/2(−) 1.6 10.(2.) E 13.(5.) 0.3(3) E 13.(5.) 320(10.)
    188 0(+) 13.3 7.6(3) 0 7.3(6) 0 7.3(6) 4.7(5)
    189 3/2(−) 16.1 10.7(3)   14.4(8) 0.5(5) E 14.9(9) 25(4.)
    190 0(+) 26.4 11.0(3) 0 15.2(9) 0 15.2(8) 13.1(3)
    192 0(+) 41.0 11.5(4) 0 16.6(1.2) 0 16.6(1.2) 2.0(1)
                     
Ir 77       10.6(3)   14.1(8) 0.(3.) 14.(3.) 425.3(2.4)
    191 3/2(+) 37.3           954.(10.)
    193 3/2(+) 62.7           111.(5.)
                     
Pt 78       9.60(1)   11.58(2) 0.13(11) 11.71(11) 10.3(3)
    190 0(+) 0.01 9.0(1.0) 0 10.(2.) 0 10.(2.) 152.(4.)
    192 0(+) 0.79 9.9(5) 0 12.3(1.2) 0 12.3(1.2) 10.0(2.5)
    194 0(+) 32.9 10.55(8) 0 14.0(2) 0 14.0(2) 1.44(19)
    195 1/2(−) 33.8 8.83(9) −1.00(17) 9.8(2) 0.13(4) 9.9(2) 27.5(1.2)
    196 0(+) 25.3 9.89(8) 0 12.3(2) 0 12.3(2) 0.72(4)
    198 0(+) 7.2 7.8(1) 0 7.7(2) 0 7.6(2) 3.66(19)
                     
Au 79 197 3/2(+) 100 7.63(6) −1.84(10) 7.32(12) 0.43(5) 7.75(13) 98.65(9)
                     
Hg 80       12.692(15)   20.24(5) 6.6(1) 26.8(1) 372.3(4.0)
    196 0(+) 0.2 30.3(1.0) 0 115(8.) 0 115(8.) 3080(180.)
    198 0(+) 10.1   0   0   2.0(3)
    199 1/2(−) 17.0 16.9(4) [\pm]15.5(8) 36.(2.) 30.(3.) 66.(2.) 2150.(48.)
    200 0(+) 23.1   0   0   < 60.
    201 3/2(−) 13.2     7.8(2.0)      
    202 0(+) 29.6   0   0   4.89(5)
    204 0(+) 6.8   0   0   0.43(10)
                     
Tl 81       8.776(5)   9.678(11) 0.21(15) 9.89(15) 3.43(6)
    203 1/2(+) 29.524 6.99(16) 1.06(14) 6.14(28) 0.14(4) 6.28(28) 11.4(2)
    205 1/2(+) 70.476 9.52(7) −0.242(17) 11.39(17) 0.007(1) 11.40(17) 0.104(17)
                     
Pb 82       9.405(3)   11.115(7) 0.0030(7) 11.118(7) 0.171(2)
    204 0(+) 1.4 9.90(10) 0 12.3(2) 0 12.3(2) 0.65(7)
    206 0(+) 24.1 9.22(5) 0 10.68(12) 0 10.68(12) 0.0300(8)
    207 1/2(−) 22.1 9.28(4) 0.14(6) 10.82(9) 0.002(2) 10.82(9) 0.699(10)
    208 0(+) 52.4 9.50(2) 0 11.34(5) 0 11.34(5) 0.00048(3)
                     
Bi 83 209 9/2(−) 100 8.532(2) 0.259(15) 9.148(4) 0.0084(10) 9.156(4) 0.0338(7)
                     
Po 84                  
                     
At 85                  
                     
Rn 86                  
                     
Fr 87                  
                     
Ra 88                  
    226 0(+) (1.60×103a) 10.0(1.0) 0 13.(3.) 0 13.(3.) 12.8(1.5)
                     
Ac 89                  
                     
Th 90 232 0(+) 100 10.31(3) 0 13.36(8) 0 13.36(8) 7.37(6)
                     
Pa 91                  
    231 3/2(−) (3.28×104a) 9.1(3) 1 0.4(7) 0.1(3.3) 10.5(3.2) 200.6(2.3)
                     
U 92       8.417(5)   8.903(11) 0.005(16) 8.908(11) 7.57(2)
    233 5/2(+) (1.59×105a) 10.1(2) [\pm]1.(3.) 12.8(5) 0.1(6) 12.9(3) 574.7(1.0)
    234 0(+) 0.005 12.4(3) 0 19.3(9) 0 19.3(9) 100.1(1.3)
    235 7/2(−) 0.720 10.47(3) [\pm]1.3(6) 13.78(11) 0.2(2) 14.0(2) 680.9(1.1)
    238 0(+) 99.275 8.402(5) 0 8.871(11) 0 8.871(11) 2.68(2)
                     
Np 93                  
    237 5/2(+) (2.14×106a) 10.55(10)   14.0(3) 0.5(5)E 14.5(6) 175.9(2.9)
                     
Pu 94                  
    238 0(+) (87.74a) 14.1(5) 0 25.0(1.8) 0 25.0(1.8) 558.(7.)
    239 1/2(+) (2.41×104a) 7.7(1) [\pm]1.3(1.9) 7.5(2) 0.2(6) 7.7(6) 1017.3(2.1)
    240 0(+) (6.56×103a) 3.5(1) 0 1.54(9) 0 1.54(9) 289.6(1.4)
    242 0(+) (3.76×105a) 8.1(1) 0 8.2(2) 0 8.2(2) 18.5(5)
                     
Am 95                  
    243 5/2(−) (7.37×103a) 8.3(2) [\pm]2.(7.) 8.7(4) 0.3(2.6) 9.0(2.6) 75.3(1.8)
                     
Cm 96                  
    244 0(+) (18.10a) 9.5(3) 0 11.3(7) 0 11.3(7) 16.2(1.2)
    246 0(+) (4.7×103a) 9.3(2) 0 10.9(5) 0 10.9(5) 1.36(17)
    248 0(+) (3.5×105a) 7.7(2) 0 7.5(4) 0 7.5(4) 3.00(26)

4.4.4.1. Scattering lengths

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The scattering of a neutron by a single bound nucleus is described within the Born approximation by the Fermi pseudopotential, [V({\bf r}) =\left({2\pi\hbar^2\over m}\right) b\delta({\bf r}), \eqno (4.4.4.1)]in which r is the position of the neutron relative to the nucleus, m the neutron's mass, and b the bound scattering length. The neutron has spin s and the nucleus spin I so that, if [I\neq0], the Fermi pseudopotential and, hence, the bound scattering length will be spin dependent. Since s = 1/2, the most general rotationally invariant expression for b is [b=b_c+{2b_i\over\sqrt{I(I+1)}}\,{\bf s}\cdot {\bf I}, \eqno (4.4.4.2)]in which the coefficients [b_c] and [b_i] are called the bound coherent and incoherent scattering lengths. If I = 0, then bi = 0 by convention.

4.4.4.2. Scattering and absorption cross sections

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When a thermal neutron collides with a nucleus, it may be either scattered or absorbed. By absorption, we mean reactions such as [(n,\gamma)], (n, p), or (n, α), in which there is no neutron in the final state. The effect of absorption can be included by allowing the bound scattering length to be complex, [b=b'-ib''. \eqno (4.4.4.3)]The total bound scattering cross section is then given by [\sigma_s=4\pi\left\langle|b|{}^2\right\rangle, \eqno (4.4.4.4)]in which [\langle\,\rangle] denotes a statistical average over the neutron and nuclear spins and the absorption cross section is given by [\sigma_a={4\pi\over k}\langle b''\rangle, \eqno (4.4.4.5)]where k = 2π/λ is the wavevector of the incident neutron and λ is the wavelength.

If the neutron and/or the nucleus is unpolarized, then the total bound scattering cross section is of the form [\sigma_s=\sigma_c+\sigma_i, \eqno (4.4.4.6)]in which [\sigma_c] and [\sigma_i] are called the bound coherent and incoherent scattering cross sections and are given by [\sigma_c=4\pi|b_c|{}^2, \quad \sigma_i=4\pi|b_i|{}^2. \eqno (4.4.4.7)]Also, [b_c=\langle b\rangle, \eqno (4.4.4.8)]so that the absorption cross section is given by [\sigma_a={4\pi \over k}\, b''_c. \eqno (4.4.4.9)]The absorption cross section is therefore uniquely determined by the imaginary part of the bound coherent scattering length. It is only when the neutron and the nucleus are both polarized that the imaginary part of the bound incoherent scattering length contributes to the value of [\sigma_a].

For most nuclides, the scattering lengths and, hence, the scattering cross sections are constant in the thermal-neutron region, and the absorption cross sections are inversely proportional to k. Since k is proportional to the neutron velocity v, the absorption is said to obey a 1/v law. By convention, absorption cross sections are tabulated for a velocity v = 2200 m s−1, which corresponds to a wavevector k = 3.494 Å−1, a wavelength λ = 1.798 Å, or an energy E = 25.30 meV.

The only major deviations from the 1/v law are for a few heavy nuclides (specifically, 113Cd, 149Sm, 151Eu, 155Gd, 157Gd, 176Lu, and 180Ta), which have an (n, γ) resonance at thermal-neutron energies. For these nuclides (which are indicated by the symbol * in Table 4.4.4.1[link]), the scattering lengths and cross sections are strongly energy dependent. The scattering lengths of the resonant rare-earth nuclides have been tabulated as a function of energy by Lynn & Seeger (1990[link]).

4.4.4.3. Isotope effects

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The coefficients [b_c] and [b_i] in (4.4.4.2)[link] for the bound scattering length depend on the particular isotope under consideration, and this provides an additional source of incoherence in the scattering of neutrons by a mixture of isotopes. If [\langle\,\rangle] is now taken to denote an average over both the spin and isotope distributions, then the expressions (4.4.4.8)[link] for [b_c], (4.4.4.4)[link] for [\sigma_s], and (4.4.4.5)[link] for [\sigma_a] also apply to a mixture of isotopes. Hence, if [c_l] denotes the mole fraction of isotopes of type l, so that [\textstyle\sum\limits_l\,c_l=1, \eqno (4.4.4.10)]then, for an isotopic mixture, [b_c=\textstyle\sum\limits_l\, c_l\,b_{cl}, \eqno (4.4.4.11)] [\sigma_s=\textstyle\sum\limits_l\, c_l\,\sigma_{sl}, \eqno (4.4.4.12)]and [\sigma_a=\textstyle\sum\limits_l\,c_l\,\sigma_{al}. \eqno (4.4.4.13)]The bound coherent scattering cross section of the mixture is given, as before, by [\sigma_c=4\pi|b_c|{}^2, \eqno (4.4.4.14)]while the bound incoherent scattering cross section is defined as [\sigma_i=\sigma_s-\sigma_c. \eqno (4.4.4.15)]Hence, it follows that [\sigma_i\equiv 4\pi|b_i|^2=\sigma_i({\rm spin})+\sigma_i({\rm isotope}), \eqno (4.4.4.16)]in which the contribution from spin incoherence is given by [\sigma_i({\rm spin})=\textstyle\sum\limits_l\, c_l\sigma_{il} = 4\pi\sum\limits_l\, c_l|b_{il}|{}^2, \eqno (4.4.4.17)]and that from isotope incoherence is given by [\sigma_i({\rm isotope})= 4\pi\textstyle\sum\limits_{l\lt l'} \,c_l c_{l'}|b_{cl}-b_{cl'}|{}^2. \eqno (4.4.4.18)]Note that for a mixture of isotopes only the magnitude of [b_i] is defined by (4.4.4.16)[link], and its sign is arbitrary. However, for the individual isotopes, both the magnitude and sign (or complex phase) of [b_i] are defined in (4.4.4.2)[link].

4.4.4.4. Correction for electromagnetic interactions

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The effective bound coherent scattering length that describes the interaction of a neutron with an atom includes additional contributions from electromagnetic interactions (Bacon, 1975[link]; Sears, 1986a[link], 1996[link]). For a neutral atom with atomic number Z, this quantity is of the form [b_c(q)=b_c(0)-b_e[Z-f(q)], \eqno (4.4.4.19)]where q is the wavevector transfer in the collision, [b_c(0)] and [b_e] are constants, and f(q) is the atomic scattering factor (Section 6.1.1[link] ). The latter quantity is the Fourier transform of the electron number density and is normalized such that f(0) = Z.

The main contribution to [b_c(0)] is from the nuclear interaction between the neutron and the nucleus but there is also a small electrostatic contribution ([\le] 0.5%) arising from the neutron electric polarizability. The coefficient [b_e] is called the neutron–electron scattering length and has the value −1.32 (4) × 10−3 fm (Koester, Waschkowski & Meier, 1988[link]). This quantity is due mainly to the Foldy interaction with a small additional contribution (∼10%) from the intrinsic charge distribution of the neutron.

The correction of the bound coherent scattering length for electromagnetic interactions requires a knowledge of the atomic scattering factor f(q). Tables 6.1.1.1[link] and 6.1.1.3[link] provide accurate values of f(q) obtained from relativistic Hartree–Fock calculations for all the atoms and chemically important ions in the Periodic Table. Alternatively, since the correction is small (∼1%), one can often use the approximate analytical expression (Sears, 1986a[link], 1996[link]) [f(q)={Z \over\sqrt{1+3(q/q_0)^2}} \eqno (4.4.4.20)]with [q_0=\gamma Z^{1/3}]. The value γ = 1.90 ± 0.07 Å−1 provides a good fit to the Hartree–Fock results in Table 6.1.1.1[link] for [Z\ge20].

4.4.4.5. Measurement of scattering lengths

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The development of modern neutron-optical techniques during the past 25 years has produced a dramatic increase in the accuracy with which scattering lengths can be measured (Koester, 1977[link]; Klein & Werner, 1983[link]; Werner & Klein, 1986[link]; Sears, 1989[link]; Koester, Rauch & Seymann, 1991[link]). The measurements employ a number of effects – mirror reflection, prism refraction, gravity refractometry, Christiansen filter, and interferometry – all of which are based on the fact that the neutron index of refraction, n, is uniquely determined by [b_c(0)] through the relation [n^2 =1-{4\pi\over k^2}\,\rho\,b_c(0), \eqno (4.4.4.21)]in which ρ is the number of atoms per unit volume. Apart from a small (≤0.01%) local-field correction (Sears, 1985[link], 1989[link]), this expression is exact.

In methods based on diffraction, such as Bragg reflection by powders or dynamical diffraction by perfect crystals, the measured quantity is the unit-cell structure factor [|F_{hkl}|]. This quantity depends on [b_c(q)] in which q is equal to the magnitude of the reciprocal-lattice vector corresponding to the relevant Bragg planes, i.e. [q=2k\sin\theta_{hkl}, \eqno (4.4.4.22)]where [\theta_{hkl}] is the Bragg angle. In dynamical diffraction measurements, it is usual for the authors to correct their results for electromagnetic interactions so that the published quantity is again [b_c(0)]. In the past, this correction has not usually been made for the scattering lengths obtained from Bragg reflection by powders. However, these latter measurements are accurate only to ±2 or 3% so that the correction is then relatively unimportant.

The essential point is that all the bound coherent scattering lengths in Table 4.4.4.1[link] with the experimental uncertainties less than 1% represent [b_c(0)] and should therefore be corrected for electromagnetic interactions before being used in the interpretation of neutron diffraction experiments. Failure to make this correction will introduce systematic errors of 0.5 to 2% in the unit-cell structure factors at large q, and corresponding errors of 1 to 4% in the calculated intensities.

Expression (4.4.4.21)[link] assumed that the neutrons and/or the nuclei are unpolarized. If the neutrons and the nuclei are both polarized then [b_c(0)] is replaced by [\langle b(0)\rangle], which depends on both the coherent and incoherent scattering lengths. If the coherent scattering length is known, neutron-optical experiments with polarized neutrons and nuclei can then be used to determine the incoherent scattering length (Glättli & Goldman, 1987[link]).

4.4.4.6. Compilation of scattering lengths and cross sections

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The bound scattering lengths and cross sections of almost all the elements in the Periodic Table, as well as those of the individual isotopes, are listed in Table 4.4.4.1[link]. As in earlier versions of this table (Sears, 1984[link], 1986b[link], 1992a[link],b[link]), our primary aim, has been to take the best current values of the bound coherent and incoherent neutron scattering lengths and to compute from them a consistent set of bound scattering cross sections. In the present version, we have used the values of the coherent and incoherent scattering lengths recommended by Koester, Rauch & Seymann (1991[link]), supplemented with a few more recently measured values, and have computed from them the corresponding scattering cross sections. The trailing digits in parentheses give the standard errors calculated from the errors in the input data using the statistical theory of error propagation (Young, 1962[link]). The imaginary parts of the scattering lengths, which are appreciable only for strongly absorbing nuclides, were calculated from the measured absorption cross sections (Mughabghab, Divadeenam & Holden, 1981[link]; Mughabghab, 1984[link]) and are listed beneath the real parts of Table 4.4.4.1[link].

In a few cases, where the scattering lengths have not yet been measured directly, the available scattering cross-section data (Mughabghab, Divadeenam & Holden, 1981[link]; Mughabghab, 1984[link]) were used to obtain the scattering lengths. Equations (4.4.4.11)[link], (4.4.4.12)[link], and (4.4.4.13)[link] were used, where necessary, to fill gaps in Table 4.4.4.1[link]. For some elements, these relations indicated inconsistencies in the data. In such cases, appropriate adjustments in the values of some of the quantities were made. In almost all cases, such adjustments were comparable with the stated errors. Finally, for some elements, it was necessary to estimate arbitrarily the scattering lengths of one or two isotopes in order to be able to complete the table. Such estimates are indicated by the letter `E' and were usually made only for isotopes of low natural abundance where the estimated values have only a marginal effect on the final results. Apart from the inclusion of new data for Ti and Mn, the values listed in Table 4.4.4.1[link] are the same as in Sears (1992b[link]).

References

Bacon, G. E. (1975). Neutron diffraction, 3rd ed. Oxford: Clarendon Press.
Glättli, H. & Goldman, M. (1987). Nuclear magnetism. Methods of experimental physics, Vol. 23, Neutron scattering Part C, edited by K. Sköld & D. L. Price, pp. 241–286. New York: Academic Press.
Klein, A. G. & Werner, S. A. (1983). Neutron optics. Rep. Prog. Phys. 46, 259–335.
Koester, L. (1977). Neutron scattering lengths and fundamental neutron interactions. Springer Tracts in Modern Physics, Vol. 80, pp. 1–55. Berlin: Springer Verlag.
Koester, L., Rauch, H. & Seymann, E. (1991). Neutron scattering lengths: a survey of experimental data and methods. At. Data Nucl. Data Tables, 49, 65–120.
Koester, L., Waschkowski, W. & Meier, J. (1988). Experimental study on the electric polarizability of the neutron. Z. Phys. A329, 229–234.
Lynn, J. E. & Seeger, P. A. (1990). Resonance effects in neutron scattering lengths of rare-earth nuclides. At. Data Nucl. Data Tables, 44, 191–207.
Mughabghab, S. F. (1984). Neutron cross sections, Vol. 1, Part B: Z = 61–100. New York: Academic Press.
Mughabghab, S. F., Divadeenam, M. & Holden, N. E. (1981). Neutron cross sections, Vol. 1, Part A: Z = 1–60. New York: Academic Press.
Sears, V. F. (1984). Thermal-neutron scattering lengths and cross sections for condensed-matter research. Report AECL-8490. Atomic Energy of Canada Limited, Chalk River, Ontario, Canada.
Sears, V. F. (1985). Local-field refinement of neutron scattering lengths. Z. Phys. A321, 443–449.
Sears, V. F. (1986a). Electromagnetic neutron–atom interactions. Phys. Rep. 141, 281–317.
Sears, V. F. (1986b). Neutron scattering lengths and cross sections. Methods of experimental physics, Vol. 23, Neutron scattering, Part A, edited by K. Sköld & D. L. Price, pp. 521–550. New York: Academic Press.
Sears, V. F. (1989). Neutron optics. Oxford University Press.
Sears, V. F. (1992a). Scattering lengths for neutrons. International tables for crystallography, Vol. C, edited by A. J. C. Wilson, pp. 383–391. Dordrecht: Kluwer Academic Publishers.
Sears, V. F. (1992b). Neutron scattering lengths and cross sections. Neutron News, 3, 26–37.
Sears, V. F. (1996). Correction of neutron scattering lengths for electromagnetic interactions. J. Neutron Res. 3, 53–62.
Werner, S. A. & Klein, A. G. (1986). Neutron optics. Methods of experimental physics, Vol. 23, Neutron scattering, Part A, edited by K. Sköld and D. L. Price, pp. 259–337. New York: Academic Press.
Young, H. D. (1962). Statistical treatment of experimental data. New York: McGraw–Hill.








































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