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出典: International Tables for Crystallography, Vol. C. Table 6.1.1.4. Coefficients for analytical approximation to the scattering factors

  • E: exact
  • C: correlated
  • HF: non-relativistic Hartree-Fock by Thakkar & Smith (1992) and Cromer & Mann (1968)
  • RHF: relativistic Hartree-Fock based on the calculations of Doyle & Turner (1968) using the wavefunction of Coulthard (1967)
  • *RHF: relativistic Hartree-Fock based on the calculation of Cromer & Waber (1968) using the wavefunctions of Mann (1968)
  • *DS: modified Dirac-Slater by Cromer & Waber (1968)

X線に対する原子散乱因子 \(f\) は以下の近似式で求めることが出来ます。$$ f(\sin\theta/\lambda) = \sum_{i=1}^4 a_i \exp(-b_i \sin^2\theta/\lambda^2)+c$$

ZSymbolNote\(a_1\) \(b_1\)\(a_2\)\(b_2\)\(a_3\)\(b_3\)\(a_4\)\(b_4\)\(c\)
1HH:HF0.49300210.51090.32291226.12570.1401913.142360.0408157.79970.003038
H1-: HF0.89766153.13680.56561615.1870.415815186.5760.1169733.567090.002389
2HeHe: RHF0.87349.10370.63093.35680.311222.92760.1780.98210.0064
3LiLi: RHF1.12823.95460.75081.05240.617585.39050.4653168.2610.0377
Li1+: RHF0.69684.62370.78881.95570.34140.63160.156310.09530.0167
4BeBe: RHF1.591943.64271.12781.86230.5391103.4830.70290.5420.0385
Be2+: RHF6.26030.00270.88490.83130.79932.27580.16475.1146-6.1092
5BB: RHF2.054523.21851.33261.0211.097960.34980.70680.1403-0.1932
6CC: RHF2.3120.84391.0210.20751.58860.56870.86551.65120.2156
C val: HF2.2606922.69071.561650.6566651.050759.756180.83925955.59490.286977
7NN: RHF12.21260.00573.13229.89332.012528.99751.16630.5826-11.529
8OO: RHF3.048513.27712.28685.70111.54630.32390.86732.90890.2508
O1-: HF4.191612.85731.639694.172361.5267347.0179-20.307-0.0140421.9412
9FF: RHF3.539210.28252.64124.29441.5170.26151.024326.14760.2776
F1-: HF3.63225.277563.5105714.73531.260640.4422580.94070647.34370.653396
10NeNe: RHF3.95538.40423.11253.42621.45460.23061.125121.71840.3515
ZSymbolNote\(a_1\) \(b_1\)\(a_2\)\(b_2\)\(a_3\)\(b_3\)\(a_4\)\(b_4\)\(c\)
11NaNa: RHF4.76263.2853.17368.84221.26740.31361.1128129.4240.676
Na1+: RHF3.25652.66713.93626.11531.39980.20011.003214.0390.404
12MgMg: RHF5.42042.82752.173579.26111.22690.38082.30737.19370.8584
Mg2+: RHF3.49882.16763.83784.75421.32840.1850.849710.14110.4853
13AlAl: RHF6.42023.03871.90020.74261.593631.54721.964685.08861.1151
Al3+: HF4.174481.938163.38764.145531.202960.2287530.5281378.285240.706786
14SiSi: RHF6.29152.43863.035332.33371.98910.67851.54181.69371.1407
Si4+: HF4.439181.641673.203453.437571.194530.21490.416536.653650.746297
Si val: HF5.662692.66523.0716438.66342.624460.9169461.393293.54581.24707
15PP: RHF6.43451.90674.179127.1571.780.5261.490868.16451.1149
16SS: RHF6.90531.46795.203422.21511.43790.25361.586356.1720.8669
17ClCl: RHF11.46040.01047.19641.16626.255618.51941.645547.7784-9.5574
Cl1-: RHF18.29150.00667.20841.17176.533719.54242.338660.4486-16.378
18ArAr: RHF7.48450.90726.772314.84070.653943.89831.644233.39291.4445
19KK: RHF8.218612.79497.43980.77481.0519213.1870.865941.68411.4228
K1+: RHF7.957812.63317.49170.76746.359-0.0021.191531.9128-4.9978
20CaCa: RHF8.626610.44217.38730.65991.589985.74841.0211178.4371.3751
Ca2+: RHF15.6348-0.00747.95180.60898.437210.31160.853725.9905-14.875
ZSymbolNote\(a_1\) \(b_1\)\(a_2\)\(b_2\)\(a_3\)\(b_3\)\(a_4\)\(b_4\)\(c\)
21ScSc: RHF9.1899.02137.36790.57291.6409136.1081.46851.35311.3329
Sc3+: HF13.40080.298548.02737.96291.65943-0.286041.5793616.0662-6.6667
22TiTi:9.75957.85087.35580.51.699135.63381.9021116.1051.2807
Ti2+: HF9.114237.52437.621740.4575852.279319.53610.08789961.65580.897155
Ti3+: HF17.73440.220618.738167.047165.25691-0.157621.9213415.9768-14.652
Ti4+: HF19.51140.1788478.234736.670182.01341-0.292631.520812.9464-13.28
23VV: RHF10.29716.86577.35110.43852.070326.89382.0571102.4781.2199
V2+: RHF10.1066.88187.35410.44092.288420.30040.0223115.1221.2298
V3+: HF9.431416.395357.74190.3833492.1534315.19080.01686563.9690.656565
V5+: HF15.68870.6790038.142085.401352.030819.97278-9.5760.9404641.7143
24CrCr: RHF10.64066.10387.35370.3923.32420.26261.492298.73991.1832
Cr2+: HF9.540345.660787.75090.3442613.5827413.30750.50910732.42240.616898
Cr3+: HF9.68095.594637.811360.3343932.8760312.82880.11357532.87610.518275
25MnMn: RHF11.28195.34097.35730.34323.019317.86742.244183.75431.0896
Mn2+: RHF10.80615.27967.3620.34353.526814.3430.218441.32351.0874
Mn3+: HF9.845214.917977.871940.2943933.5653110.81710.32361324.12810.393974
Mn4+: HF9.962534.84857.970570.2833032.7606710.48520.05444727.5730.251877
26FeFe: RHF11.76954.76117.35730.30723.522215.35352.304576.88051.0369
Fe2+: RHF11.04244.65387.3740.30534.134612.05460.439931.28091.0097
Fe3+: RHF11.17644.61477.38630.30053.394811.67290.072438.55660.9707
27CoCo: RHF12.28414.27917.34090.27844.003413.53592.348871.16921.0118
Co2+: RHF11.22964.12317.38830.27264.739310.24430.710825.64660.9324
Co3+: HF10.3383.909697.881730.2386684.767958.355830.72559118.34910.286667
28NiNi: RHF12.83763.87857.2920.25654.443812.17632.3866.34211.0341
Ni2+: RHF11.41663.67667.40050.24495.34428.8730.977322.16260.8614
Ni3+: HF10.78063.54777.758680.223145.227467.644680.84711416.96730.386044
29CuCu: RHF13.3383.58287.16760.2475.615811.39661.673564.81261.191
Cu1+: RHF11.94753.36697.35730.22746.24558.66251.557825.84870.89
Cu2+: HF11.81683.374847.111810.2440785.781357.98761.1452319.8971.14431
30ZnZn: RHF14.07433.26557.03180.23335.165210.31632.4158.70971.3041
Zn2+: RHF11.97192.99467.38620.20316.46687.08261.39418.09950.7807
ZSymbolNote\(a_1\) \(b_1\)\(a_2\)\(b_2\)\(a_3\)\(b_3\)\(a_4\)\(b_4\)\(c\)
31GaGa: RHF15.23543.06696.70060.24124.359110.78052.962361.41351.7189
Ga3+: HF12.6922.812626.698830.227896.066926.364411.006614.41221.53545
32GeGe: RHF16.08162.85096.37470.25163.706811.44683.68354.76252.1313
Ge4+: HF12.91722.537186.700030.2058556.067915.479130.85904111.6031.45572
33AsAs: RHF16.67232.63456.07010.26473.431312.94794.277947.79722.531
34SeSe: RHF17.00062.40985.81960.27263.973115.23724.354343.81632.8409
35BrBr: RHF17.17892.17235.235816.57965.63770.26093.985141.43282.9557
Br1-: RHF17.17182.20596.333819.33455.57540.28713.727258.15353.1776
36KrKr: RHF17.35551.93846.728616.56235.54930.22613.537539.39722.825
37RbRb: RHF17.17841.78889.643517.31515.13990.27481.5292164.9343.4873
Rb1+: RHF17.58161.71397.659814.79575.89810.16032.781731.20872.0782
38SrSr: RHF17.56631.55649.818414.09885.4220.16642.6694132.3762.5064
Sr2+: RHF18.08741.49078.137312.69632.565424.5651-34.193-0.013841.4025
39YY: *RHF17.7761.402910.294612.80065.726290.1255993.26588104.3541.91213
Y3+: *DS17.92681.354179.153111.21451.7679522.6599-33.108-0.0131940.2602
40ZrZr: *RHF17.87651.2761810.94811.9165.417320.1176223.6572187.66272.06929
Zr4+: *DS18.16681.214810.056210.14831.0111821.6054-2.6479-0.102769.41454
ZSymbolNote\(a_1\) \(b_1\)\(a_2\)\(b_2\)\(a_3\)\(b_3\)\(a_4\)\(b_4\)\(c\)
41NbNb: *RHF17.61421.1886512.014411.7664.041830.2047853.5334669.79573.75591
Nb3+: *DS19.88120.01917518.06531.1330511.017710.16211.9471528.3389-12.912
Nb5+: *DS17.91631.1244613.34170.02878110.7999.282060.33790525.7228-6.3934
42MoMo: RHF3.70250.277217.23561.095812.887611.0043.742961.65844.3875
Mo3+: *DS21.16640.01473418.20171.0303111.74239.536592.3095126.6307-14.421
Mo5+: *DS21.01490.01434518.09921.0223811.46328.788090.74062523.3452-14.316
Mo6+: *DS17.88711.0364911.1758.480616.578910.058881000.344941
43TcTc: *RHF19.13010.86413211.09488.144874.6490121.57072.7126386.84725.40428
44RuRu: *RHF19.26740.8085212.91828.434674.8633724.79971.5675694.29285.37874
Ru3+: *DS18.56380.84732913.28858.371649.326020.0176623.0096422.887-3.1892
Ru4+: *DS18.50030.84458213.17878.125344.713040.364952.1853520.85041.42357
45RhRh: *RHF19.29570.75153614.35018.217584.7342525.87491.2891898.60625.328
Rh3+: *DS18.87850.76425214.12597.844383.3251521.2487-6.1989-0.0103611.8678
Rh4+: *DS18.85450.76082513.98067.624362.5346419.3317-5.6526-0.010211.2835
46PdPd: *RHF19.33190.69865515.50177.989295.2953725.20520.60584476.89865.26593
Pd2+: *DS19.17010.69621915.20967.555734.3223422.5057005.2916
Pd4+: *DS19.24930.68383914.797.148332.8928917.9144-7.94920.00512713.0174
47AgAg: RHF19.28080.644616.68857.47264.804524.66051.046399.81565.179
Ag1+: *DS19.18120.64617915.97197.191235.2747521.73260.35753466.11475.21572
Ag2+: *DS19.16430.64564316.24567.185444.370921.4072005.21404
48CdCd: RHF19.22140.594617.64446.90894.46124.70081.602987.48255.0694
Cd2+: *DS19.15140.59792217.25356.806394.4712820.2521005.11937
49InIn: RHF19.16240.547618.55966.37764.294825.84992.039692.80294.9391
In3+: *DS19.10450.55152218.11086.32473.7889717.3595004.99635
50SnSn: RHF19.18895.830319.10050.50314.458526.89092.466383.95714.7821
Sn2+: RHF19.10940.503619.05485.83784.564823.37520.48762.20614.7861
Sn4+: RHF18.93335.76419.71310.46553.418214.00490.0193-0.75833.9182
ZSymbolNote\(a_1\) \(b_1\)\(a_2\)\(b_2\)\(a_3\)\(b_3\)\(a_4\)\(b_4\)\(c\)
51SbSb: RHF19.64185.303419.04550.46075.037127.90742.682775.28254.5909
Sb3+: *DS18.97550.46719618.9335.221265.1078919.59020.28875355.51134.69626
Sb5+: *DS19.86855.4485319.03020.4679732.4125314.1259004.69263
52TeTe: *RHF19.96444.8174219.01380.4208856.1448728.52842.523970.84034.352
53II: RHF20.14724.34718.99490.38147.513827.7662.273566.87764.0712
I1-: RHF20.23324.357918.9970.38157.806929.52592.886884.93044.0714
54XeXe: RHF20.29333.928219.02980.3448.976726.46591.9964.26583.7118
55CsCs: RHF20.38923.56919.10620.310710.66224.38791.4953213.9043.3352
Cs1+: RHF20.35243.55219.12780.308610.282123.71280.961559.45653.2791
56BaBa: RHF20.33613.21619.2970.275610.88820.20732.6959167.2022.7731
Ba2+: *DS20.18073.2136719.11360.2833110.905420.05580.7763451.7463.02902
57LaLa: *RHF20.5782.9481719.5990.24447511.372718.77263.28719133.1242.14678
La3+: *DS20.24892.920719.37630.25069811.632317.82110.33604854.94532.4086
58CeCe: *RHF21.16712.8121919.76950.22683611.851317.60833.33049127.1131.86264
Ce3+: *DS20.80362.7769119.5590.2315411.936916.54080.61237643.16922.09013
Ce4+: *DS20.32352.6594119.81860.2188512.123315.79920.14458362.23551.5918
59PrPr: *RHF22.0442.7739319.66970.22208712.385616.76692.82428143.6442.0583
Pr3+: *DS21.37272.645219.74910.21429912.132915.3230.9751836.40651.77132
Pr4+: *DS20.94132.5446720.05390.20248112.466814.81370.29668945.46431.24285
60NdNd: *RHF22.68452.6624819.68470.21062812.77415.8852.85137137.9031.98486
Nd3+: *DS21.9612.5272219.93390.19923712.1214.17831.5103130.87171.47588
ZSymbolNote\(a_1\) \(b_1\)\(a_2\)\(b_2\)\(a_3\)\(b_3\)\(a_4\)\(b_4\)\(c\)
61PmPm: *RHF23.34052.562719.60950.20208813.123515.10092.87516132.7212.02876
Pm3+: *DS22.55272.417420.11080.18576912.067113.12752.0749227.44911.19499
62SmSm: *RHF24.00422.4727419.42580.19645113.439614.39962.89604128.0072.20963
Sm3+: *DS23.15042.3164120.25990.17408111.920212.15712.7148824.82420.954586
63EuEu: RHF24.62742.387919.08860.194213.760313.75462.9227123.1742.5745
Eu2+: *DS24.00632.2778319.95040.1735311.803411.60963.8724326.51561.36389
Eu3+: *DS23.74972.2225820.37450.1639411.850911.3113.2650322.99660.759344
64GdGd: *RHF25.07092.2534119.07980.18195113.851812.93313.54545101.3982.4196
Gd3+: *DS24.34662.1355320.42080.15552511.870810.57823.714921.70290.645089
65TbTb: *RHF25.89762.2425618.21850.19614314.316712.66482.95354115.3623.58324
Tb3+: *DS24.95592.0560120.32710.14952512.247110.04993.77321.27730.691967
66DyDy: *RHF26.5072.180217.63830.20217214.559612.18992.96577111.8744.29728
Dy3+: *DS25.53951.980420.28610.14338411.98129.349724.5007319.5810.68969
67HoHo: *RHF26.90492.0705117.2940.1979414.558311.44073.6383792.65664.56796
Ho3+: *DS26.12961.9107220.09940.13935811.97888.800184.9367618.59080.852795
68ErEr: *RHF27.65632.0735616.42850.22354514.977911.36042.98233105.7035.92046
Er3+: *DS26.7221.8465919.77480.1372912.15068.362255.1737917.89741.17613
69TmTm: *RHF28.18192.0285915.88510.23884915.154210.99752.98706102.9616.75621
Tm3+: *DS27.30831.7871119.3320.13697412.33397.967785.3834817.29221.63929
70YbYb: *RHF28.66411.988915.43450.25711915.308710.66472.98963100.4177.56672
Yb2+: *DS28.12091.7850317.68170.1599713.33358.183045.1465720.393.70983
Yb3+: *DS27.89171.7327218.76140.1387912.60727.644125.4764716.81532.26001
ZSymbolNote\(a_1\) \(b_1\)\(a_2\)\(b_2\)\(a_3\)\(b_3\)\(a_4\)\(b_4\)\(c\)
71LuLu: *RHF28.94761.9018215.22089.9851915.10.2610333.7160184.32987.97628
Lu3+: *DS28.46281.6821618.1210.14229212.84297.337275.5941516.35352.97573
72HfHf: *RHF29.1441.8326215.17269.599914.75860.2751164.3001372.0298.58154
Hf4+: *DS28.81311.5913618.46010.12890312.72856.762325.5992714.03662.39699
73TaTa: *RHF29.20241.7733315.22939.3704614.51350.2959774.7649263.36449.24354
Ta5+: *DS29.15871.5071118.84070.11674112.82686.315245.3869512.42441.78555
74WW: *RHF29.08181.7202915.439.225914.43270.3217035.1198257.0569.8875
W6+: *DS29.49361.4275519.37630.10462113.05445.936675.0641211.19721.01074
75ReRe: *RHF28.76211.6719115.71899.0922714.55640.35055.4417452.086110.472
76OsOs: *RHF28.18941.6290316.1558.9794814.93050.3826615.6758948.164711.0005
Os4+: *DS30.4191.3711315.26376.8470614.74580.1651915.0679518.0036.49804
77IrIr: *RHF27.30491.5927916.72968.8655315.61150.4179165.8337745.001111.4722
Ir3+: *DS30.41561.3432315.8627.1090913.61450.2046335.8200820.32548.27903
Ir4+: *DS30.70581.3092315.55126.7198314.23260.1672525.5367217.49116.96824
78PtPt: *RHF27.00591.5129317.76398.8117415.71310.4245935.783738.610311.6883
Pt2+: *DS29.84291.3292716.72247.3897913.21530.2632976.3523422.94269.85329
Pt4+: *DS30.96121.2481315.98296.6083413.73480.168645.9203416.93927.39534
79AuAu: RHF16.88190.461118.59138.621625.55821.48265.8636.395612.0658
Au1+: *DS28.01091.3532117.82047.739514.33590.3567526.5807726.404311.2299
Au3+: *DS30.68861.219916.90296.8287212.78010.2128676.5235418.6599.0968
80HgHg: RHF20.68090.54519.04178.448421.65751.57295.967638.324612.6089
Hg1+: *DS25.08531.3950718.49737.6510516.88830.4433786.4821628.226212.0205
Hg2+: *DS29.56411.2115218.067.0563912.83740.2847386.8991220.748210.6268
ZSymbolNote\(a_1\) \(b_1\)\(a_2\)\(b_2\)\(a_3\)\(b_3\)\(a_4\)\(b_4\)\(c\)
81TlTl: *RHF27.54460.6551519.15848.7075115.5381.963475.5259345.814913.1746
Tl1+: *DS21.39851.471120.47230.51739418.74787.434636.8284728.848212.5258
Tl3+: *DS30.86951.100818.34816.5385211.93280.2190747.0057417.21149.8027
82PbPb: RHF31.06170.690213.06372.357618.4428.6185.969647.257913.4118
Pb2+: *DS21.78861.336619.56820.48838319.14066.77277.0110723.813212.4734
Pb4+: *DS32.12441.0056618.80036.1092612.01750.1470416.9688614.7148.08428
83BiBi: RHF33.36890.70412.9512.923816.58778.79376.469248.009313.5782
Bi3+: *DS21.80531.235619.50266.2414919.10530.4699997.1029520.318512.4711
Bi5+: *DS33.53640.9165425.09460.3904219.24975.714146.9155512.8285-6.7994
84PoPo: *RHF34.67260.70099915.47333.5507813.11389.556427.0258847.004513.677
85AtAt: *RHF35.31630.6858719.02113.974589.4988711.38247.4251845.471513.7108
86RnRn: RHF35.56310.663121.28164.06918.003714.04227.443344.247313.6905
87FrFr: *RHF35.92990.64645323.05474.1761912.143923.10522.11253150.64513.7247
88RaRa: *RHF35.7630.61634122.90643.8713512.473919.98873.21097142.32513.6211
Ra2+: *DS35.2150.60490921.673.57677.9134212.6017.6507829.843613.5431
89AcAc: *RHF35.65970.58909223.10323.6515512.597718.5994.08655117.0213.5266
Ac3+: *DS35.17360.57968922.11123.414378.1921612.91877.0554525.944313.4637
90ThTh: *RHF35.56450.56335923.42193.4620412.747317.83094.8070399.172213.4314
Th4+: *DS35.10070.55505422.44183.244989.7855413.46615.2944423.953313.376
ZSymbolNote\(a_1\) \(b_1\)\(a_2\)\(b_2\)\(a_3\)\(b_3\)\(a_4\)\(b_4\)\(c\)
91PaPa: *RHF35.88470.54775123.29483.4151914.189116.92354.17287105.25113.4287
92UU: RHF36.02280.529323.41283.325314.949116.09274.188100.61313.3966
U3+: *DS35.57470.5204822.52593.1229312.216512.71485.3707326.339413.3092
U4+: *DS35.37150.51659822.53263.0505312.029112.57234.798423.458213.2671
U6+: *DS34.85090.50707922.75842.890314.009913.17671.2145725.201713.1665
93NpNp: *RHF36.18740.51192923.59643.2539615.640215.36224.185597.490813.3573
Np3+: *DS35.70740.50232222.6133.0380712.989812.14495.4322725.492813.2544
Np4+: *DS35.51030.49862622.57872.9662712.776611.94844.9215922.750213.2116
Np6+: *DS35.01360.4898122.72862.8109914.388412.331.7566922.658113.113
94PuPu: *RHF36.52540.49938423.80833.2637116.770714.94553.47947105.9813.3812
Pu3+: *DS35.840.48493822.71692.9611813.580711.53315.6601624.399213.1991
Pu4+: *DS35.64930.48142222.6462.890213.359511.3165.1883121.830113.1555
Pu6+: *DS35.17360.47320422.71812.7384814.763511.5532.2867820.930313.0582
95AmAm: *RHF36.67060.48362924.09923.2064717.341514.31363.49331102.27313.3592
96CmCm: *RHF36.64880.46515424.40963.0899717.39913.43464.2166588.483413.2887
97BkBk: *RHF36.78810.45101824.77363.0461917.891912.89464.2328486.00313.2754
98CfCf: *RHF36.91850.43753325.19953.0077518.331712.40444.2439183.788113.2674
ZSymbolNote\(a_1\) \(b_1\)\(a_2\)\(b_2\)\(a_3\)\(b_3\)\(a_4\)\(b_4\)\(c\)

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Stupidity has a certain charm; ignorance does not. Frank Zappa