Modeling the attenuated total reflectance infrared (ATR-FTIR) spectrum of apatite

Abstract

Attenuated total reflectance (ATR) infrared spectra were measured on a synthetic and a natural fluorapatite sample. A modeling approach based on the computation of the Fresnel reflection coefficient between the ATR crystal and the powder sample was used to analyze the line shape of the spectra. The dielectric properties of the samples were related to those of pure fluorapatite using an effective medium approach, based on Maxwell–Garnett and Bruggeman models. The Bruggeman effective medium model leads to a very good agreement with the experimental data recorded on the synthetic fluorapatite sample. The poorer agreement observed on the natural sample suggests a more significant heterogeneity of the sample at a characteristic length scale larger than the mid-infrared characteristic wavelength, i.e., about 10 micrometers. The results demonstrate the prominent role of macroscopic electrostatic effects over fine details of the microscopic structure in determining the line shape of strong ATR bands.

Appendix

Complex solutions of the uniaxial Bruggeman model

The Bruggeman relation is (Eq. 2):

$$f\left\lfloor {2\frac{{(1 - g_{//} )(\varepsilon_{Br} - \varepsilon_{ \bot } )}}{{(1 + g_{//} )\varepsilon_{Br} + (1 - g_{//} )\varepsilon_{ \bot } }} + \frac{{g_{//} (\varepsilon_{Br} - \varepsilon_{//} )}}{{(1 - g_{//} )\varepsilon_{Br} + g_{//} \varepsilon_{//} }}} \right\rfloor + 3(1 - f)\frac{{\varepsilon_{Br} - \varepsilon_{h} }}{{2\varepsilon_{Br} + \varepsilon_{h} }} = 0$$

The solutions of the equation are the roots of a cubic polynomial with complex coefficients:

$$ax^{3} + bx^{2} + cx + d = 0$$

with

$$a = - 3 - f + 6fg_{//} + 3g_{//}^{2} - 9fg_{//}^{2}$$

$$\begin{aligned} b & = 3(\varepsilon_{h} - \varepsilon_{ \bot } ) + (7\varepsilon_{ \bot } - 5\varepsilon_{h} )f + (6\varepsilon_{ \bot } - 3\varepsilon_{//} )g_{//} + fg_{//} (3\varepsilon_{h} + \varepsilon_{//} - 16\varepsilon_{ \bot } ) \\ & \quad - 3g_{//}^{2} (\varepsilon_{//} + \varepsilon_{ \bot } + \varepsilon_{h} ) + 9(\varepsilon_{//} + \varepsilon_{ \bot } )fg_{//}^{2} \\ \end{aligned}$$

$$\begin{aligned} c & = \varepsilon_{h} \varepsilon_{ \bot } (3 - f) + (3\varepsilon_{h} \varepsilon_{//} - 6\varepsilon_{h} \varepsilon_{ \bot } - 3\varepsilon_{//} \varepsilon_{ \bot } )g_{//} + (9\varepsilon_{//} \varepsilon_{ \bot } + \varepsilon_{h} \varepsilon_{ \bot } - 4\varepsilon_{h} \varepsilon_{//} )fg_{//} \\ & \quad + 3g_{//}^{2} (\varepsilon_{h} \varepsilon_{//} + \varepsilon_{h} \varepsilon_{ \bot } + \varepsilon_{//} \varepsilon_{ \bot } ) - 9\varepsilon_{//} \varepsilon_{ \bot } fg_{//}^{2} \\ \end{aligned}$$

$$d = 3\varepsilon_{h} \varepsilon_{//} \varepsilon_{ \bot } \left( {g_{//} - g_{//}^{2} } \right)$$

The three roots are:

$$\begin{aligned} x_{1} & = - \frac{b}{3a} - \frac{{2^{1/3} ( - b^{2} + 3ac)}}{{3a( - 2b^{3} + 9abc - 27a^{2} d + \sqrt { - 4(b^{2} - 3ac)^{3} + (2b^{3} - 9abc + 27a^{2} d)^{2} } )^{1/3} }} \\ & \quad + \frac{{( - 2b^{3} + 9abc - 27a^{2} d + \sqrt { - 4(b^{2} - 3ac)^{3} + (2b^{3} - 9abc + 27a^{2} d)^{2} } )^{1/3} }}{{3a2^{1/3} }} \\ \end{aligned}$$

$$\begin{aligned} x_{2} & = - \frac{b}{3a} + \frac{{(1 + i\sqrt 3 )( - b^{2} + 3ac)}}{{3a2^{2/3} ( - 2b^{3} + 9abc - 27a^{2} d + \sqrt { - 4(b^{2} - 3ac)^{3} + (2b^{3} - 9abc + 27a^{2} d)^{2} } )^{1/3} }} \\ & \quad - \frac{{(1 - i\sqrt 3 )( - 2b^{3} + 9abc - 27a^{2} d + \sqrt { - 4(b^{2} - 3ac)^{3} + (2b^{3} - 9abc + 27a^{2} d)^{2} } )^{1/3} }}{{6a2^{1/3} }} \\ \end{aligned}$$

$$\begin{aligned} x_{3} & = - \frac{b}{3a} + \frac{{(1 - i\sqrt 3 )( - b^{2} + 3ac)}}{{3a2^{2/3} ( - 2b^{3} + 9abc - 27a^{2} d + \sqrt { - 4(b^{2} - 3ac)^{3} + (2b^{3} - 9abc + 27a^{2} d)^{2} } )^{1/3} }} \\ & \quad - \frac{{(1 + i\sqrt 3 )( - 2b^{3} + 9abc - 27a^{2} d + \sqrt { - 4(b^{2} - 3ac)^{3} + (2b^{3} - 9abc + 27a^{2} d)^{2} } )^{1/3} }}{{6a2^{1/3} }} \\ \end{aligned}$$

Among these three roots, only that with a positive imaginary part has to be selected.

Some mathematical properties of the Bruggeman model

The solutions of Eq. 5 are the roots of a quadratic polynomial equation with complex coefficients:

$$x^{2} + 2bx + c = 0$$

with $x = \frac{{\varepsilon_{Br} }}{{\varepsilon_{h} }}$, $b = - \frac{{(3f - 1)\frac{{\varepsilon_{S} }}{{\varepsilon_{h} }} + (2 - 3f)}}{4}$ and $c = - \frac{{\varepsilon_{S} }}{2{\varepsilon_{h}}}$. ε _h is the dielectric constant of the host, and ε _S  = ε ^′ _S  + iε ^″ _S is the complex dielectric function of the substance.

The corresponding solutions are: $x_{ \pm } = - b \pm \sqrt \Delta$ with Δ = b ² − c. We will now show that the imaginary part of the solution x ₊ is positive.

The imaginary part of $\sqrt \varDelta$ is equal to $\sqrt {\left( {\left| \Delta \right| - \Delta_{r} } \right)/2}$ where Δ = Δ _r  + iΔ _i . The imaginary part of x ₊ is thus:

$$\text{Im} (x_{ + } ) = - b_{i} + \sqrt {\frac{{\left( {\left| \Delta \right| - \Delta_{r} } \right)}}{2}}$$

(6)

where b = b _r  + ib _i

We now show that the absolute value |b _i | is smaller than $\sqrt {\frac{{\left( {\left| \Delta \right| - \Delta_{r} } \right)}}{2}}$. This is equivalent to determining that |Δ|² − (2b ² _i  + Δ _r )² is positive. Replacing the coefficients Δ and b_i by their expressions as a function of the dielectric function of the substance and matrix, we obtain:

$$\left| \Delta \right|^{2} - (2b_{i}^{2} + \Delta_{r} )^{2} = \frac{9}{8}\left( {\frac{{\varepsilon^{\prime\prime}_{S} }}{{\varepsilon_{h} }}} \right)^{2} f(1 - f)$$

which is indeed always positive for 0 ≤ f ≤ 1.

As a consequence, $\text{Im} (x_{ + } )$ is also positive. This property is even stronger than the Herglotz property because ε ^″ _S is not assumed to be positive. In particular for f = 0, Im(x₊) = (ε ^″ _S  + |ε ^″ _S |)/2 ≥ 0 even if ε ^″ _S is negative. The condition that ɛ _h is real and positive thus implies that Im(ε _Br ) is positive, for any function ε _S .

As $- b_{i} = (3f - 1)\frac{{\varepsilon^{\prime\prime}_{S} }}{{4\varepsilon_{h} }}$ and ε ^″ _S  > 0, the first term in Eq. 6 is positive when f > 1/3. This contribution is thus proportional to the dielectric function of the pure bulk substance and always present. For f < 1/3, this contribution becomes negative, but since Im(x ₊) ≥ 0, this negative contribution is compensated for by the second term in Eq. 6. The ε ^″ _S peak then disappears completely from the spectrum. Equation 6 has another interesting consequence. In a region where |Δ_i| is smaller than |Δ_r|, there are two regimes: If Δ_r ≥ 0, then |Δ| − Δ_r ≈ Δ ²_i /(2Δ_r) is small and the first term of Eq. 6 is dominant for f > 0.3; if Δ_r ≤ 0, then |Δ| − Δ_r ≈ 2|Δ_r| and the deformation of the spectrum due to the second term of Eq. 6 becomes large. In other words, the deformation of the spectrum of apatite for f > 0.3 occurs when Δ has a large negative real part.

We now examine the properties of the imaginary part of the solution for f = 0.5 and ε _h  = 1, i.e., the parameters used to model the spectrum of synthetic fluorapatite (Fig. 6). In this case, b ² − c = ɛ _S (ɛ _S  + 34) + 1. For a vanishingly small damping parameter and out of the TO frequency, ε _S is real: ε _S  ≈ ε ^′ _S . Therefore, only the second term contributes to the imaginary part of the solution when b ² − c is negative. This condition is met for ε ^′ _S approximately ranging between −34 and zero, i.e., on the high energy side of the TO frequency and up to about the LO frequency. For ε ^′ _S below −34, the imaginary part of ε _S becomes dominant and the spectrum is proportional to the dielectric function of the pure bulk substance.

Fig. 6

Real part of the model dielectric function (bold line, right axis) and imaginary part of the corresponding effective dielectric function (dotted line, left axis). The small damping parameter (1 cm⁻¹) makes it possible to discriminate the two superposed contributions for a volume fraction of 0.5. The thin contribution at TO frequency displays an almost Lorentzian shape proportional to the imaginary part of the model dielectric function. In contrast, the broad contributions at higher frequency are observed when the real part of the model dielectric function takes values between ~−34 and ~0 (LO frequency), indicated by the horizontal dotted lines