J. Phys. Chem. C 2008, 112, 7757–7760
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Photon Refraction In Dielectric Crystals Using a Modified Gladstone-Dale Relation David K. Teertstra Euclid Geometrics, 509.5 Morningside DriVe, Albuquerque, New Mexico, 87108 ReceiVed: January 22, 2008; ReVised Manuscript ReceiVed: February 28, 2008
The Gladstone-Dale relation1 gives the index of refraction of light through a material as a sum of the optical properties of its oxide components. However, the calculated optical properties of ionic solids are only approximated from the composition, as the refraction also depends on the crystal structure. The Gladstone-Dale relation is now modified so that the specific refractivity of an ion depends on the distance (d) and angle to the surrounding ions (i.e., the structure) and falls off as 1/d2 (in a Coulomb-like manner). Using single values for the refractivity of ions, the calculated and measured indices of refraction agree within n ( 0.0005 (4σ) for the garnet group minerals. Alternatively, with the structural formula from a measured composition, the density can be calculated from the index of refraction or the unit cell volume. 1. Conflicting Theories A great deal of labor is required to use the electromagneticwave theory of light to calculate the optical properties of materials, and the results are not that accurate. It is faster to use the Gladstone-Dale relation1 (explaining the optical properties of a bulk material as the sum of the optical properties of the constituent oxides), and excellent agreement with measurement is common.2 The problem remains that the wave theory of light requires an homogeneous index of refraction of materials and excludes a consideration of local optical interactions with individual molecules or atoms (e.g., the photon as a probe of atomic structure). The Gladstone-Dale relation, as currently written, cannot deal with birefringent materials or polymorphs, but this problem is readily repaired. A recent article in this journal by Rocquefelte et al.3 exemplifies these problems. Rocquefelte and co-workers3 use the Gladstone-Dale relation to describe changes of index of refraction (n) and density (D) as dependent on the interlayer distance of a TiO2 sheet structure. This is a new use of the Gladstone-Dale relation. Gladstone and Dale (1864) varied the weight fraction (w) of mixed liquids and found that n and D were related by a constant (k) such that n/D ) Σ(kw). This equation indicates that each chemical component has characteristic contributions to the optical properties, mass (m), and volume (V) of the bulk material. However, the Gladstone-Dale relation was not designed to deal with isochemical structural changes and only crudely describes the physical properties of phases as dependent on variable structure at constant composition (e.g., variable temperature; the polymorphs of silica). And Rocquefelte and co-workers3 stop short of investigating the very interesting physical consequences of moving the ions Ti4+ and O2- apart. Rocquefelte and co-workers3 also use the dielectric function of the wave theory of light (the Kramers–Kronig relation) to find a linear relation between n and V for the TiO2 sheet structures, even though the wave theory is not designed to deal with the optical properties of materials at an atomic scale. The requirement of wave theory is that the electric waves of each atom bound in a chemical compound add to one another to give a resultant wave and a homogeneous index of refraction for the bulk material. As a result, Rocquefelte and co-workers3 predict that an individual TiO2 sheet is invisible. By contrast, the Gladstone-Dale relation indicates that individual elements or
molecules have characteristic optical properties, either in isolation or if bound as ions in a dielectric solid, and so light must interact with an individual TiO2 sheet. If the ions Ti4+ and O2- differ optically (by virtue of a differing local electric structure), then it is not possible to maintain a uniform wavefront on refraction of light through the material. Light must refract by interaction with individual elements. This sort of local interaction is also required by the theory of atomic absorption, in which light is a local particle emitted or absorbed by individual atoms. The specific problem, mathematically speaking, is that the dielectric function relies on an n2 term, and Rocquefelte and co-workers3 use an n2 term in relation to the distance (d) between the bound layers of a TiO2 structure. As light is electromagnetic in nature and probably undergoes refraction due to the action of the electric force about each ion, it seems more intuitive to use n in direct relation to a 1/d2 term, as the electric force about each ion (described by the Coulomb equation) falls off with 1/d2. The index of refraction is thus a function of the interionic distance and increases with bond strength. That said, it is trivial to modify the Gladstone-Dale relation to give accurate calculated values of n. 2. A Revised Theory The equation of Gladstone and Dale1 [n/D ) Σ(kw)] indicates that each element has three characteristic properties that contribute to the net physical properties of the bulk material. The mass of each element contributes to the formula weight, the volume of each element contributes to the molecular volume (and the ratio Σm/V is the density D), and the specific refractivity of each element contributes to the index of refraction n. Constant values of k for the refractivity of elements can be used to calculate the index of refraction of numerous materials2 but only to a first approximation. This is because solid-solution mechanisms such as Mg-1Fe give characteristic changes of n, D, and interionic distance in a variety of dielectric crystals, but because of differences in structure, the calculated values of n are inexact. The use of constant or average ki values can result in significant differences between calculated and measured values of n. Indices of refraction are commonly measured to an accuracy of n ( 0.001. For crystals, density can be calculated
10.1021/jp800634c CCC: $40.75 2008 American Chemical Society Published on Web 04/29/2008
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TABLE 1: Physical Properties (n, a, D) of End-Member Silicate Garnets species
formula
n
a (nm)
Dcalc
Pyr – pyrope Alm – almandine Sps – spessartine Grs – grossular And - andradite Gld - goldmanite Uvr - uvarovite
Mg3Al2Si3O12 Fe3Al2Si3O12 Mn3Al2Si3O12 Ca3Al2Si3O12 Ca3Fe3+2Si3O12 Ca3V2Si3O12 Ca3Cr2Si3O12
1.714 1.830 1.800 1.734 1.889 1.834 1.865
1.1459 1.1526 1.1621 1.1851 1.2058 1.2070 1.1996
3.559 4.318 4.190 3.595 3.851 3.765 3.851
with good accuracy because the accuracy of chemical analysis by electron microprobe is commonly within 1-2%, and the unitcell volume is measured by X-ray diffraction to less than 0.01% error. Any values reported in Tables 1–4 that contain additional significant figures do so to give calculated values of n that round up to 0.001. Teertstra4 showed that the disagreeable calculated values of n may be corrected if the ki values depend on structural variations (this article also gives important references to studies of the Gladstone-Dale relation). It is known that the effective refractivity of an ion depends on the local bonding environment in each structure type. So for an exact calculation of n, a modified Gladstone-Dale relation must also account for the effects of structural variation. For ions in a dielectric solid, the Gladstone-Dale relation is rewritten as n/D ) Σ(kiwi). The ki value is a molar quantity with units of cm3/g (for density in g/cm3), but one may factor out atomic weight (AW) to get the effective refractive volume (si′) of a single ion. Calculating si′ ) kiAW/AN in nm3, where AN is Avogadro’s number, the refractive volume is found to be similar to the ionic volume. The composition expressed as a weight fraction of ions (wi) gives a simple relation to the normalized density, as the weight fraction of ions is equivalent to the fractional density (wi ) di, and Σdi ) 1). The fractional density is the mass of each ion divided by the formula weight (di ) mi/FW). The absolute contribution of each ion to the bulk density D is mi/V, and for crystals X-ray diffraction measures the unit cell volume V. So the only remaining concern is relating the refractivity of an ion to the structure, and this is done as follows. To explain the additive properties of the Gladstone-Dale relation at an atomic scale (Figure 1), suppose that the refractivity si of a free O2- anion has a value of 3, and that the refractivity si of a free Mg2+ cation has a value of 7. If the ions are separated at an interionic distance (di) that is greater than some interionic cutoff distance (dc, indicating that the pair is nonbonding), then the refractivity of one ion does not measurably affect the refractivity of the other ion. But if the ions are close enough to bond, then Mg2+ places charge at O2- to give O2- an effective refractivity (si′) of, for example, 4, whereas the effective refractivity (si′) of Mg2+ is just 7.2. Now, if we add a third ion of Mg2+ to this MgO molecule the effective refractivity of Mg2+ will decrease (related to cation–cation repulsion), and the effective refractivity of O2- will increase (related to attraction). The effective refractivity of a cation is thus dominated by its coordination polyhedra of anions and is decreased somewhat by the surrounding next-nearestneighbor cations. In this qualitative description, the refractivity is directly proportional to the electric charge about the ion. Now if the force on a refracting photon varies in proportion to 1/d2, then a function indicating the decreased contribution of refractivity with distance might include a term with the form si(dc - di)2 (Figure 2).
Figure 1. A general model of the relations between the polarization of light, the refractivity and identity of ions, and interionic distances and angles. The model is applicable to aperiodic molecules, liquids and solids, and crystals. The force on a photon near an ion depends on the distance and angle to surrounding ions.
Figure 2. Ion refractivity si vs cutoff distance dc. If a photon refracts at a specific level of energy at a radial distance from the nucleus, the charge beyond that radius of refraction falls off by 1/d2. An ion contributes a fraction of its refractivity si to a neighbor at di, also falling off as 1/d2. Beyond the cutoff distance dc, the ion does not contribute to the refractivity of a neighbor. Increasing the cutoff distance (from solid curve to dashed curve) is equivalent to increasing the radius of refraction; more charge is placed at di then the refractivity of the neighboring ion is increased.
To account for solid solution in a crystal, the function si(dc - di)2 is multiplied by the site occupancy f of an individual ion, giving fsi(dc - di)2. To account for the polarization of light, the effective force on the photon varies by cos θ, where θ is the angle to the photon due to a radial line of charge from a nearby ion (Figure 1) and including this polarization factor gives cos θfsi(dc - di)2. The effective refractivity of an individual ion thus depends on a number of surrounding ions, each with its own cutoff distance, and varies with the structure factor Σ[cos θfsi(dc - di)2]. If a crystal has a known structure and composition with known values of n and V, then the interionic distances can be calculated and only the values dc and si need to be found. The value of cos θi(dc - di)2 is calculated from the structure, then the value of si′ is estimated and converted to a ki value so that the calculated value n ) DΣ(kiwi) can be compared to the measured value of n. This iteration is strongly convergent on distinct values of si and dc for each ion. As an example, consider a strongly ionic and isotropic crystalline solid solution such as garnet. By free iteration, it has not been possible to solve the ki values of ions in the cubic garnet group for the full compositional space of (Mg, Fe, Mn,
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TABLE 2: Values of Refractivity (si), Cut-off Distance (dc), Radius of Refraction (R) and Radius (r) of Free Ions ion 2+
Mg Fe2+ Mn2+ Ca2+ Al3+ Fe3+ V3+ Cr3+ Si4+ O2-
TABLE 3: Site Structure Factors for End-Member Species of Garnet
si (nm3)
dc (nm)
R (nm)
r (nm)
R/r
species
CX
CY
CZ
0.02158 0.03130 0.03200 0.03436 0.00168 0.02680 0.02170 0.02130 0.01400 0.02500
0.38 0.42 0.43 0.43 0.30 0.58 0.60 0.64 0.50 0.50
0.174 0.196 0.197 0.202 0.074 0.186 0.173 0.172 0.150 0.181
0.089 0.091 0.098 0.112 0.053 0.065 0.064 0.062 0.026 0.140
1.9 2.2 2.0 1.8 1.4 2.9 2.7 2.8 5.8 1.3
pyrope almandine spessartine grossular andradite goldmanite uvarovite
0.00792 0.00757 0.00732 0.00677 0.00373 0.00443 0.00296
0.00877 0.00808 0.00769 0.00592 0.00649 0.00654 0.00576
0.01041 0.00938 0.00891 0.00828 0.00535 0.00615 0.00453
Ca)3 (Al, Fe3+, V, Cr)2Si3O12. In the structural formula X3Y2Z3O12, the isotropic species are simply ordered with the divalent cations Mg, Fe, Mn, and Ca in a distorted-cubic X-site, the trivalent cations Al, Fe, Cr, and V in an octahedral-Y site and tetravalent Si in a tetrahedral-Z site. There are linear changes of n and unit-cell edge a between end-members (and V ) a3), so these values can be calculated from the structural formula by n ) nPyr + ∆n/XFe + ∆n/XMn + ∆n/XCa + ∆n/YFe3+ + ∆n/YV + ∆n/YCr and a ) aPyr + ∆a/XFe + ∆a/XMn + ∆a/ XCa + ∆a/YFe3+ + ∆a/YV + ∆a/YCr.4 The values of n and a for end-members are reported in Table 1. For each composition, it is necessary to calculate the interionic distances and angles (from the structure data of ref 5), the weight fraction of each ion (wi), the density (D), and each ldil ) wiD. It is convenient to calculate the angle θ for a photon refracting with a polarization parallel to an a-axis of garnet. To account for the net refractivity contributions of all surrounding ions, the sum of the changes to the si of a free ion now bound at a particular site in the structure is the structure factor C ) Σ[cos θfsi(dc - di)2]. It is convenient to calculate various values of cos θ(dc - di)2 from the structure data and to plot a curve with dc for each ion; depending on the value of dc, one excludes those distant ions with dc > di. For example, Al in the Y-site affects the refractivity of cations in the X-site and the Z-site and is itself affected by Al ions in the surrounding Y-sites. A high value of dc means that a greater number of ions of Al at greater distance must be accounted for. In practice, this is a few tens of ions no more distant than about half a unit cell length (for example, Cr3+ with a range of measurable influence of 0.64 nm). From the general structural formula of cubic garnet X3Y2Z3O12, the structure factors of the four sites are CX ) CXO - CXX - CXY - CXZ, CY ) CYO - CYX - CYY - CYZ, CZ ) CZO - CZX - CZY - CZZ, CO ) COX + COY + COZ - COO. For example, the structure factor applicable to cations in the X-site of spessartine is CX ) Σ[cos θfs(O2-)(dc - di)2] - Σ[cos θfs(Mn2+)(dc - di)2] - Σ[cos θfs(Al3+)(dc - di)2] - Σ[cos θfs(Si4+)(dc - di)2]. If the si of Mn2+ is 0.032 and CX is 0.007, then the effective si′ of Mn2+ in the X-site is 0.039 nm3. The si′ values are found to have linear variation between garnet endmembers. With the s′(Al3+) of pyrope as a reference point, the effective s′(Al3+) in andradite varies with the number of atoms of Fe3+ pfu (per formula unit), and the effective s′(Al3+) in grossular varies with the number of atoms of Ca2+ pfu. 3. Results The site structure factors CX, CY, and CZ are dominated by the values for the refractivity s(O2-) and cutoff distance dc(O2-) of oxygen (Table 2). The magnitudes of the site structure factors of pyrope vary in the order CX < CY < CZ (Table 3), indicating
that the bulk physical properties are dominated by SiO4 (oxygen in tetrahedral coordination about silicon) with significant contributions from YO6 and moderate contributions from XO8. Each value CX, CY, and CZ is positive, indicating a favorable electrostatic environment for a cation bound in the site. Negative values of CX, CY, or CZ would indicate a net repulsive force at the site, in which case vacancy at the site is favored. Between species, the site structure factors decrease with increasing unit cell volume. As the unit cell volume increases, an ion occupying the X, Y, or Z site experiences a decreased electric field and a decreased contribution of C to its refractivity value si from the increasingly distant surrounding ions, and this is observed. Maintaining spherical ions, iteration of the refractivity (si) and cutoff (dc) values give agreement between calculated and measured values of n to within (0.001 (4σ) for solid solutions of pyrope, almandine, spessartine, and grossular. Each ion has only single values of si and dc, but the system of site structure factors CX, CY, and CZ (Table 3) is highly constrained. For example, high values of s(Fe3+) or dc(Fe3+) place excessive repulsive charge at the X, Y and Z sites and decrease the effective refractivity of all surrounding cations. Similarly, as the X-cations alter the refractivity of the Y-cations, the iteration is strongly convergent on the specific values of si and dc reported in Table 2. Among the s(X2+) values, the poorest fit was for s(Fe2+) in all compositions, but an overly high s(Fe2+) correlates with an unusually short Si-O bond in almandine.5 Correct values of n result if electron density is drawn away from Fe2+ and toward Si4+. Maintaining a constant s(O2-), an aspherical shape of oxygen is produced by decreasing dc(O2-) toward Fe2+ and increasing dc(O2-) toward Si4+. This gives the correct values of n in XFe2+-rich samples, and the accuracy n ( 0.0005 for the common species of garnet allows rounding up of the calculated values to give exact agreement with the measured values. Less accuracy is found for the values of dc, due to a small influence of the most distant ions. Maintaining spherical ions, iteration of the refractivity (si), and cutoff (dc) values gives agreement between calculated and measured values of n to within (0.003 (4σ) for andradite, goldmanite, and uvarovite. Among the s(Y3+) values, the poorest fit was for Fe3+, V3+, and Cr3+ in each of these species but was corrected to n ( 0.0005 by increasing dc(O2-) toward these ions. This anisotropy is related to birefringence in these species. Decreased values of dc(O2-) toward oxygen occur as a result of O-O repulsion, decreasing COO, and enhancing the formation of a distorted tetrahedral orbital. In other words, the valence electrons of the oxygen ion distort from spherical orbitals to provide the closest packing of atoms in the garnet structure and to maximize the density and thermodynamic stability of each species. The refined set of ki values (matching the anisotropic dc(O2-) values) given in Table 4 returns calculated values of n that match the measured values of n in Table 1. For intermediate composi-
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TABLE 4: Values of Molar Refractivity (ki) of ions in end-Member Silicate Garnets Species
Mg2+
Fe2+
Mn2+
Ca2+
Al3+
Fe3+
V3+
Cr3+
Si4+
O2-
Pyr Alm Sps Grs And Gld Uvr
0.7310 0.7223 0.7156 0.6980 0.6205 0.6180 0.5890
0.4223 0.4181 0.4156 0.4106 0.3772 0.3758 0.3634
0.4376 0.4338 0.4310 0.4259 0.3916 0.3905 0.3779
0.6349 0.6298 0.6256 0.6196 0.5726 0.5712 0.5536
0.2332 0.2185 0.2092 0.1687 0.1816 0.1863 0.1653
0.3511 0.3535 0.3541 0.3439 0.3503 0.3529 0.3425
0.3359 0.3386 0.3359 0.3218 0.3282 0.3309 0.3199
0.3220 0.3229 0.3212 0.3091 0.3159 0.3183 0.3073
0.5234 0.5001 0.4904 0.4790 0.4250 0.4231 0.3930
0.4383 0.4529 0.4636 0.4860 0.5494 0.5443 0.5766
tions, the values of ki can be determined by linear interpolation. For example, ki (Mg2+) varies from pyrope to almandine: the change is (0.7223-0.7310)/3 per atom of XFe2+; there is an additional change from pyrope to uvarovite of (0.589-0.7310)/2 per atom of YCr3+. From the formula, by reference to endmember pyrope, ki (Mg2+) ) 0.73100 - 0.00290 Fe2+ 0.005133 Mn - 0.01100 Ca – 0.05520 Fe3+ - 0.05650 V 0.07100 Cr. To estimate the composition of garnet from measurements of index of refraction and unit cell edge or density, a likely composition is proposed and the formula is varied to minimize differences across the equation n/Σ(kidi) ) Σ(aiAW)/VAN, where ai is the number of ions of type i in the unit cell volume. The term n/Σ(kidi) is the density calculated from the index of refraction. The term Σ(aiAW)/VAN is the density calculated from the unit cell parameters. For a single composition, identical values of D (within error) must result if the structure is analyzed by refracted visible-light photons (giving n) or by reflected X-ray photons (giving a). Once the ki values are known, the unit cell volume can be calculated even if V is not determined from X-ray diffraction. The s(X2+) values are characteristic of each ion and increase with ionic radius in the order Mg < Fe < Mn < Ca, leaving little doubt that the refractivity volumes of ions are characteristic atomic properties resulting from the atomic-electric structure specific to each ion. The increased s(X2+) with increased cation radius indicates an increased displacement of light about each ion as 2+ units of charge is shielded to varying degree by the spherical surface of electron orbitals of noble-gas configuration. If the photon path is approximated by displacement about a semicircle, the 2.15 eV yellow-light photon has an effective refracting radius of r ) [0.75si′/π]1/3 nm. This semicircular radius of refraction of each X2+ cation (Table 2) is approximately twice the mean cation radius determined by X-ray crystallography (relative to a 0.140 nm radius of O2-).6 Each cation presents a spherical distribution of charge to an incoming photon, but the force on the photon increases gradually and the path of the photon is more like a bell curve than a semicircle. Once the specific shape of the photon path is solved, the 0.54 eV level of energy (for refracted yellow light) of bonds in the
structure are mapped by linking the refracted paths of light about each ion in a specific direction through the unit cell. The best-fit dc and si values of refractivity of free ions in Table 2 are independent of state and are applicable to the structures of liquid, glass, or crystalline materials. The dc and si values for garnet group minerals have sufficient precision to fit the index of refraction of all samples in the (Mg, Fe, Mn, Ca)3(Al, Fe, V, Cr)2Si3O12 solid solution to n ( 0.0005 but remain mostly dependent on the dc and si values of Si4+ and O2- because solid solution at the Z-site was not considered. The accuracy of this set of dc and si values may be assessed or improved by considering solid solution at the Z-site of garnet or by study of another group of crystals. The modified Gladstone-Dale relation remains at odds with the electromagnetic-wave theory of light. From wave theory in general, one cannot gather atomic-scale information, as the wavelength of visible light is far greater than the diameter of atoms. However, such information gathering relates only to image formation, not to data from nonimage-forming refraction experiments. By using a particle theory of light that requires the interaction of individual photons with individual atoms, it is almost trivial to calculate exactly the degree of refraction of light through a material. By accounting for variation in composition and crystal structure, single values for the refractivity of ions (Table 2) give accurate calculated values for the refraction of light. As the values of n relate directly to the electric force between ions, one can consider that a beam of light consists of local particles that refract by displacement around with individual ions. References and Notes (1) Gladstone, J. H.; Dale, T. P. Phil. Trans. R. Soc. London 1864, 153, 317–343. (2) Mandarino, J. A. Can. Mineral. 2007, 45, 1307–1324. (3) Rocquefelte, X.; Jobic, S.; Whangbo, M.-H. J. Phys. Chem. 2006, 110, 2511–2514. (4) Teertstra, D. K. Can. Mineral. 2005, 43, 543–552. (5) Novak, G. A.; Gibbs, G. V. Am. Mineral. 1971, 56, 791–825. (6) Shannon, R. D.; Prewitt, C. T. Acta Crystallogr. 1969, B25, 925–946.
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