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How to Calculate the X-ray Penetration Depth in GIXRD

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If you work with thin films and have access to an X-ray diffractometer with grazing incidence capabilities, choosing the right incidence angle is not just a detail, it determines whether you are actually probing your film or the substrate underneath. In this post, I walk through the theory and calculation of the X-ray penetration depth in Grazing Incidence X-ray Diffraction (GIXRD), using SnO_2 as a worked example.

Why does the incidence angle matter?

In a conventional Bragg-Brentano (\theta–2\theta) XRD measurement, the X-ray beam penetrates several micrometers into the sample. For a thin film of tens or hundreds of nanometers, most of the diffracted signal comes from the substrate, not from your film.

GIXRD solves this by fixing the incident beam at a small grazing angle \alpha_i (typically 0.1°–5°) while scanning the detector in 2\theta. By controlling \alpha_i, you control how deep the X-rays penetrate.

The key question becomes: at what angle does the penetration depth match my film thickness?

The penetration depth formula

For incidence angles well above the critical angle (\alpha_i \gg \alpha_c), the penetration depth \tau defined as the depth at which the beam intensity drops to 1/e of its surface value is given by:

    \[\tau(\alpha_i) = \frac{\sin \alpha_i}{\mu}\]

where:

  • \alpha_i is the grazing incidence angle
  • \mu is the linear absorption coefficient of your film material, in cm^{-1}

This is the practical formula you will use. It tells you that the penetration depth increases linearly with \sin \alpha_i and decreases with stronger absorption (larger \mu).

How to calculate the linear absorption coefficient (µ)

The linear absorption coefficient is not directly tabulated for compounds, you build it from elemental values using Bragg’s mixture rule. The procedure has three steps.

Step 1: Look up the mass absorption coefficients

For each element in your compound, find the tabulated mass absorption coefficient (\mu/\rho)_i at your X-ray wavelength. These values are available in:

Step 2: Calculate the compound mass absorption coefficient

Compute the weight fraction of each element:

    \[w_i = \frac{n_i \, M_i}{M_{\text{compound}}}\]

where n_i is the number of atoms of element i in the formula, M_i is the atomic mass, and M_{\text{compound}} is the molecular weight.

Then apply the mixture rule:

    \[\left(\frac{\mu}{\rho}\right)_{\text{compound}} = \sum_i w_i \left(\frac{\mu}{\rho}\right)_i\]

Step 3: Multiply by the density

    \[\mu = \rho_{\text{compound}} \cdot \left(\frac{\mu}{\rho}\right)_{\text{compound}}\]

This gives you \mu in cm^{-1}, ready to plug into the penetration depth formula.

Worked example: SnO_2 with Cu-K𝛼 radiation

Let’s calculate the penetration depth for a 60 nm SnO_2 thin film measured with Cu-K\alpha (\lambda = 1.5406 Å).

Step 1: Tabulated mass absorption coefficients

From the International Tables for X-Ray Crystallography, Vol. 4, Appendix 8, at Cu-K\alpha:

ElementAtomic mass (g/mol)(\mu/\rho)_i (cm^2/g)
Sn118.71253.3
O15.9911.03

Step 2: Weight fractions and compound mass absorption coefficient

Molecular weight of SnO_2:

    \[M_{\text{SnO}_2} = 118.71 + 2 \times 15.99 = 150.69 \text{ g/mol}\]

Weight fractions:

    \[w_{\text{Sn}} = \frac{118.71}{150.69} = 0.7877\]

    \[w_{\text{O}} = \frac{2 \times 15.99}{150.69} = 0.2122\]

Mass absorption coefficient of the compound:

    \[\left(\frac{\mu}{\rho}\right)_{\text{SnO}_2} = 0.7877 \times 253.3 + 0.2122 \times 11.03\]

    \[= 199.53 + 2.34 = 201.87 \text{ cm}^2/\text{g}\]

Step 3: Linear absorption coefficient

Using the bulk density of SnO_2 at room temperature, \rho = 6.859 g/cm^3:

    \[\mu_{\text{SnO}_2} = 6.859 \times 201.87 = 1382.77 \text{ cm}^{-1}\]

Penetration depth at different grazing angles

Now we can evaluate \tau = \sin\alpha_i / \mu:

\alpha_i (°)\sin(\alpha_i)\tau (cm)\tau (nm)
0.58.727 \times 10^{-3}6.31 \times 10^{-6}63
1.01.745 \times 10^{-2}1.26 \times 10^{-5}126
1.52.618 \times 10^{-2}1.89 \times 10^{-5}189

At \alpha_i = 0.5°, the penetration depth is ~63 nm which matches the 60 nm film thickness. This is the optimal grazing angle: the X-ray beam probes the entire SnO_2 layer with minimal substrate contribution.

At 1.0° and 1.5°, the beam penetrates 126 and 189 nm respectively well into the substrate, contaminating the diffraction pattern with substrate peaks.

Applying this to your own thin film

To adapt this calculation to any material and film thickness:

  1. Identify your compound’s chemical formula and density.
  2. Look up the elemental (\mu/\rho) values for your X-ray source (Cu-K\alpha, Co-K\alpha, Mo-K\alpha, etc.) from the NIST tables or International Tables.
  3. Calculate \mu following Steps 1–3 above.
  4. Set \tau = t (your film thickness) and solve for the incidence angle:

    \[\alpha_i = \arcsin(\mu \cdot t)\]

This gives you the maximum grazing angle at which the X-ray signal is dominated by the film.

References

  1. Cullity, B.D. & Stock, S.R. Elements of X-Ray Diffraction, 3rd ed. Prentice Hall, 2001. — Chapter 1 covers absorption coefficients and the mixture rule.
  2. International Tables for X-Ray Crystallography, Vol. 4, Appendix 8. — Mass absorption coefficients \mu/\rho (cm^2/g) for all elements at common X-ray wavelengths.
  3. Hubbell, J.H. & Seltzer, S.M. “Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients.” NISTIR 5632 (1995). Available online: https://www.nist.gov/pml/x-ray-mass-attenuation-coefficients

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