Table I
Peak and Integral Ratio Results (Estimated Standard Errors in Parentheses)
| | Flat B4 | Flat D4 |
|---|
| |
|
|
|---|
| (Å) | θ° | f(%) | η(%) | σ(Å) | σ(Å) |
|---|
| Al–K | 1.000 | 60.5(0.8) | 71.2(0.6) | 15.3(0.5) | 10.2(0.9) |
| (8.34) | 1.125 | 40.7(0.5) | 49.2(0.4) | 14.7(0.4) | 9.3(0.9) |
| 1.250 | 14.0(0.2) | 17.4(0.2) | 14.1(0.5) | 10.3(0.7) |
| 1.375 | 6.7(0.1) | 8.5(0.1) | 13.5(0.5) | 12.3(0.6) |
| 1.500 | 3.5(0.1) | 4.8(0.1) | 14.1(0.5) | 10.5(0.6) |
| Mg–K | 1.000 | 68.5(0.9) | 77.2(0.7) | 15.6(0.8) | 10.5(1.2) |
| (9.89) | 1.125 | | | | 9.4(1.2) |
| 1.250 | 53.0(0.7) | 59.8(0.5) | 12.5(0.6) | 10.0(0.9) |
| 1.375 | | | | 10.3(0.8) |
| 1.500 | 13.3(0.2) | 15.8(0.1) | 12.4(0.5) | 10.5(0.9) |
| 1.625 | | | | 11.6(0.8) |
| 1.750 | 3.9(0.1) | 5.0(0.1) | 12.8(0.6) | 10.7(0.7) |
| Cu–L | 1.000 | 61.0(1.0) | 73.3(0.7) | 25.9(1.1) | 10.8(2.9) |
| (13.34) | 1.250 | 56.0(1.0) | 65.3(0.7) | 19.0(1.0) | 18.1(1.2) |
| 1.500 | 43.8(0.7) | 54.3(0.6) | 18.7(0.7) | 17.7(0.9) |
| 1.750 | 28.3(0.6) | 37.1(0.4) | 18.0(0.7) | 13.7(1.0) |
| 2.000 | 11.6(0.3) | 15.2(0.2) | 15.9(0.8) | 15.1(0.9) |
| C–K | 1.000 | 65.2(2.2) | 74.6(0.8) | 74.8(9.2) | 34.9(14.3) |
| (44.7) | 1.250 | 64.4(2.2) | 75.0(0.8) | 63.8(7.1) | 29.4(11.0) |
| 1.500 | 63.2(2.0) | 73.9(0.9) | 53.7(5.6) | 32.2(7.0) |
| 1.750 | 61.8(2.0) | 73.6(0.8) | 48.5(4.6) | 35.8(4.7) |
| 2.000 | 59.4(1.7) | 70.7(0.9) | 47.1(4.2) | 34.0(3.9) |
Table II
Results from Functional Fits to Scattered Background
| | Lorentzian | Gaussian |
|---|
| |
|
|
|---|
| | | | Flat B4 | | | |
|---|
| | | |
| | | |
|---|
| δ1(μm) | χ2 | σ1(Å) | δ1(μm) | χ2 | λ(Å) | θ° | σ1(Å) |
|---|
| 8.34 | 1.000 | 11.3(0.6) | 190(22) | 9.5(8) | 9.4(0.6) | 212(14) | 15.3(8) |
| 1.125 | 9.2(0.7) | 150(22) | 15.6(8) | 7.5(0.6) | 175(17) | 17.8(8) |
| 1.250 | 8.9(1.0) | 124(27) | 15.6(8) | 7.3(0.9) | 154(23) | 14.4(8) |
| 9.89 | 1.000 | 12.3(1.3) | 136(29) | 17.4(6) | 10.0(1.1) | 211(33) | 14.3(6) |
| 1.125 | 11.9(2.9) | 59(28) | 16.6(6) | 8.9(2.3) | 109(52) | 13.4(6) |
| 1.250 | 8.7(1.3) | 166(48) | 14.0(6) | 7.1(0.9) | 219(31) | 9.2(6) |
| 1.375 | 8.3(1.0) | 124(30) | 10.1(6) | 6.8(1.0) | 159(33) | 8.6(6) |
| | | | Flat D4 | | | |
| | | |
| | | |
| 8.34 | 1.250 | 7.8(1.1) | 231(68) | 5.9(8) | 6.3(0.6) | 221(20) | 5.3(8) |
For
σ1 and
δ1 parentheses contain estimated standard errors.
For
χ2 parentheses contain the number of degrees of freedom.
Table III
Results of Fitting All Flat B4 Measured Roughness Values in Table I to the Wavelength and Angle Dependence Expected from Particle Scattering
| Transparent particles |
| Estimated true roughness σ = 14.0(0.1)Å |
| NR4 = 6.3(0.6) × 10−14 cm2 |
| e.g., for N = 106 cm−2, R = 0.16 μm |
| Opaque particles |
| Estimated true roughness σ = 10.7(0.1)Å |
| NR2 = 2.6(0.2) × 10−4 |
| e.g., for N = 106 cm−2, R = 0.16 μm |
Table IV
Effect of Surface Particles on Measured Roughness
| | | Predicted |
|---|
| | |
|
|---|
| λ(Å) | θ° | Measured σm (Å) | Transparent case σm (Å) | Opaque case σm (Å) |
|---|
| Al–K | 1.000 | 15.3(0.5) | 14.4 | 16.3 |
| 8.34 | 1.125 | 14.7(0.4) | 14.3 | 14.9 |
| 1.250 | 14.1(0.5) | 14.2 | 13.9 |
| 1.375 | 13.5(0.5) | 14.2 | 13.2 |
| 1.500 | 14.1(0.5) | 14.2 | 12.7 |
| Mg–K | 1.000 | 15.6(0.8 | 14.8 | 18.1 |
| 9.89 | 1.125 | | | |
| 1.250 | 12.5(0.6) | 14.4 | 15.0 |
| 1.375 | | | |
| 1.500 | 12.4(0.5) | 14.3 | 13.4 |
| 1.625 | | | |
| 1.750 | 12.8(0.6) | 14.2 | 12.6 |
| Cu–L | 1.000 | 25.9(1.1) | 16.5 | 22.4 |
| 13.34 | 1.250 | 19.0(1.0) | 15.3 | 17.8 |
| 1.500 | 18.7(0.7) | 14.8 | 15.3 |
| 1.750 | 18.0(0.7) | 14.5 | 13.9 |
| 2.000 | 15.9(0.8) | 14.4 | 13.0 |
| C–K | 1.000 | 74.8(9.2) | 98.4 | 66.7 |
| 44.7 | 1.250 | 63.8(7.1) | 71.0 | 48.8 |
| 1.500 | 53.7(5.6) | 54.7 | 38.3 |
| 1.750 | 48.5(4.6) | 44.3 | 31.6 |
| 2.000 | 47.1(4.2) | 37.2 | 26.8 |
Clearly neither of the predictions above gives a statistically good fit—the
χ2 deducible for either is an order of magnitude greater than the expected value of 17. This is hardly surprising considering the extreme nature of the two cases, and the possibility of subtler effects also being present. The model does show that particles play an important role in the variation of
σm.
Table V
Useful Formulas
| This table lists x-ray beam intensities and widths. The typical values given are for the present experiment with the following values for the parameters concerned: |
|---|
| w = 46 μm, λ = 8.34 Å, θ = 1.25°, η = 17.4% |
|---|
| σ = 12 6 Å, δ = 150 μm, N = 106 cm−2, R = 0.16 μm |
|---|
| X-ray component | Central intensity at detector | Typical value | FWHM | Typical value (sec of arc) |
|---|
| Optimized direct beam | I0 | I0 | (λ/r)1/2 | 3.5 |
| Reflected beam | I1 = η exp-(4πθσ/λ)2 · I1 | I1 = 0.15 I0 | ~(λ/r)1/2 | 3.5 |
| Surface scattered background (Lorentzian autocorrelation) | 32π2θ3(σ/λ)2(δ/w) · I1 | 2.2 × 10−2I1 | λ/πθδ | 16.7 |
| Surface scattered background (Lorentzian autocorrelation) | ½(π)1/2 × above | 2.0 × 10−2I1 | 2(ln2)1/2 × above | 27.9 |
| Transparent particle background (unit density) | 1.6 × 1023(λ2/wθ)NR5 · I0 | 1.2 × 10−5I0 | [(5 ln2)1/2/π]λ/R | 1075 |
| Opaque particle background | 4(π3/5)1/2(1/wθ)NR3 · I0 | 4.1 × 10−4I0 | ~[(5 ln2)1/2/π]λ/R | 1075 |
The transparent particle intensity formula requires cgs units.
The opaque particle formulas assume the diffracted intensity distribution is similar in form to the scattered distribution from transparent particles.
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