Space group $Im{\overline 3} m$

Fischer and Bauer (Reference Fischer and Bauer2009) considered the SOD framework in space group $Im{\overline 3} m$ to represent the aristotype phase of a sodalite-structured phase, however, this ignores the fact that the oxygen coordinate has a single degree of freedom (x, x, 1/2) that permits non-ideal tetrahedral bond angle geometry. The tetrahedral site in space group $Im{\overline 3} m$ in Wyckoff position 12d exhibits point-group symmetry ${\overline 4} m2$, and as a result of the tetragonal point-group symmetry, the four T-site–anion bond lengths are constrained to have equal magnitudes. This point-group symmetry also permits two independent tetrahedral bond angles of multiplicities two and four, with the unique roto-inversion axis of the point-group bisecting the bond angles of multiplicity two. For the T-site at (1/4, 0, 1/2) the roto-inversion axis is directed along [${\overline 1}$ 0 0] (or equivalently [1 0 0]) as illustrated in Fig. 2a. The bonded anions lie on Wyckoff positions 24h with point-group symmetry mm2 and a single degree of structural freedom (x) as detailed in Table 1. To derive the true aristotype crystal structure it is necessary to find a real value of x that results in the two independent bond angles becoming equal. The point-group symmetry of the T-site is therefore raised from ${\overline 4} m2$ to ${\overline 4} 3m$, and ideal tetrahedral geometry is therefore realised. This can be determined by equating the inner products of the bond vectors r1.r2 with r1.r3 and solving the resultant quadratic equation x ² – x + 1/8 = 0. Of the two possible solutions, only the solution x = 1/2 – 1/√8 (≈0.14645) results in a crystallographically acceptable bond length. The fractional coordinates of the anion with x = 1/2 – 1/√8 defines the true aristotype SOD framework structure in this space group with a T–O bond length of ${{3a} \over 4}\sqrt {1-{{\sqrt 8 } \over 3}}$Å, intra-polyhedral angle (T–O–T) cos^–1(–√8/3) (≈160.529°) and tetrahedral volume of (a ³/24)(3 – √8)^1.5 Å³ (where a is the lattice parameter).

Fig. 2.

(a) The aristotype SOD topology in space group $Im{\overline 3} m$ with undifferentiated T-site (12d) and the direction of the ${\overline 4}$ axis shown as an arrow. (b) The structural topology of the hettotype phase in space group $Pm{\overline 3} n$ with differentiated T-sites (6c and 6d). The directions of the two ${\overline 4}$ axes from the symmetry independent tetrahedra are shown as arrows. (c) The structural topology of the hettotype phase in space group $I{\overline 4} 3m$ with undifferentiated T-site (12d) and the direction of the ${\overline 4}$ axis shown as an arrow. (d) The structural topology of the hettotype phase in space group $P{\overline 4} 3n$ (the sodalite structure) with differentiated T-sites (6c and 6d). The directions of the two ${\overline 4}$ axes from the symmetry independent tetrahedra are shown as arrows. For atom labelling see Table 1. The active irreducible representations relating two crystal structures are shown in Miller-Love notation (Stokes and Hatch, Reference Stokes and Hatch1988) and designated by an arrow.

Table 1.

Fractional coordinates for the T-sites and coordinating anions of the SOD framework in space groups $Im{\overline 3} m$, $I{\overline 4} 3m$, $P{\overline 4} 3n$ and $Pm{\overline 3} n$.

Space group $Pm{\overline 3} n$

The loss of mirror plane lying at x, x, z and the reduction in symmetry associated with the four-fold rotation axis along 0, 0, z becoming a two-fold rotation axis (and symmetry equivalents under ${\overline 3}$ point-group symmetry) reduces the space-group symmetry from $Im{\overline 3} m$ to $Pm{\overline 3} n$. The descent in symmetry results in the single T-site in $Im{\overline 3} m$ in Wyckoff site 12d becoming two tetrahedral sites with Wyckoff positions 6c (1/4, 0, 1/2) and 6d (1/2, 0, 3/4) but retaining the site point-group symmetry ${\overline 4} m2$ (noting that the Wyckoff notation for the two sites is the reverse of that designated for space group $P{\overline 4} 3n$). The unique roto-inversion axis lies parallel to [${\overline 1}$ 0 0] (equivalently [1 0 0]) for the site 6c at (1/4, 0, 1/2), and parallel to [0 0 ${\overline 1}$] (equivalently [0 0 1]) for site 6d at (1/2, 0, 3/4). The coordinating anions lie on a mirror plane with Wyckoff position 24 k (x, y, 1/2) as listed in Table 1, with the difference in x and y permitting different bond lengths to be associated with the two independent T-sites. The identical, tetragonal point-group symmetry for both T-sites ensures that the four bond lengths associated with either site are identical in length and have two sets of bond angles of multiplicities two and four. The unique roto-inversion axis bisects the bond angles of multiplicity two and the local geometry is illustrated in Fig. 2b. Defining the aristotype sodalite structure for this space group as the fractional coordinate of the anion that results in ideal tetrahedral bond angles at both T-sites, a solution is sought where r1.r2/|r1.r2| = r1.r3/|r1.r3|, r5.r6/|r5.r6|, r5.r7/|r5.r7| = –1/3. Equating r1.r2/|r1.r2| = r5.r6/|r5.r6| finds the single acceptable solution x = y, and equating r1.r2/|r1.r2| = r1.r3/|r1.r3| results in the quadratic equation x ² – x + 1/8 = 0 with solution x = 1/2–1/√8. The aristotype structure is therefore identical to that derived for space group $Im{\overline 3} m$ with a single T–O bond length of ${{3a} \over 4}\sqrt {1-{{\sqrt 8 } \over 3}}$Å, intra-polyhedral angle cos^–1(–√8/3) and tetrahedral volume of (a ³/24)(3 – √8)^1.5 Å³.

Space group $I{\overline 4} 3m$

The tetrahedral site in space group $I{\overline 4} 3m$ in Wyckoff position 12d exhibits point-group symmetry ${\overline 4}$, and as in space group $Im{\overline 3} m$, the four T-site–anion bond lengths are constrained by the tetragonal symmetry to have equal magnitudes. The point-group symmetry permits two independent tetrahedral bond angles of multiplicities two and four with the unique roto-inversion axis bisecting the bond angles of multiplicity two. The orientation of the roto-inversion axis associated with the T-site at (1/4, 0, 1/2) is unchanged from that in space group $Im{\overline 3} m$, as illustrated in Fig. 2c. The bonded anions lie on a mirror plane in Wyckoff positions 24g and two degrees of structural freedom (x, x, z) as detailed in Table 1. To derive an aristotype structure, values of x and z are therefore sought that result in the two independent bond angles becoming equal. The point-group symmetry of the T-site is then raised from ${\overline 4}$ to ${\overline 4} 3m$ and ideal tetrahedral geometry is realised. Equating the inner products of the bond vectors r1.r2 with r1.r3 results in the polynomial equation x ² – x – z ² + z – 1/8 = 0 indicating that an infinite set of solutions are possible. The range of solutions is clearly reduced by the constraint of an acceptable bond length for the T-site and anion under consideration, but despite this constraint, there is not a unique aristotype framework structure in space group $I{\overline 4} 3m$. For chosen z and calculated x, the aristotype T–O bond length is √3a(Δ – 1/4), intra-polyhedral angle cos^–1((5/4 + 3z ² – 3z – 3Δ/2)/(21/16 + 3z ² – 3z – 3Δ/2)) and tetrahedral volume ((8a ³)/(9√3))(21/16 + 3z ² – 3z – 3Δ/2)^3/2 with Δ defined as (3/8 – z + z ²)^½. Analysis of the intra-polyhedral angle function shows it to exhibit a single maximum at z = 0.5 for acceptable values of z. The solution derived for space group $Im{\overline 3} m$, x = 1/2 – 1/√8, z = 1/2, satisfies the polynomial equation above and therefore represents one of the many aristotype framework structures for space group $I{\overline 4} 3m$. A mineralogical example of the SOD framework within this space group is afforded by the phase bicchulite (Ca₈Al₈Si₄O₂₄(OH)₈) (Sahl, Reference Sahl1980).

Space group $P{\overline 4} 3n$

The symmetry lowering in space group $P{\overline 4} 3n$ from space group$I{\overline 4} 3m$ associated with the loss of body-centring translations gives rise to two tetrahedral sites with 6-fold multiplicity and point-group symmetry ${\overline 4}$, as shown in Fig. 2d. The Wyckoff site 6d at (1/4, 0, 1/2) is identical in character to the position (1/4, 0, 1/2) in Wyckoff site 12d associated with space groups $Im{\overline 3} m$ and $I{\overline 4} 3m$, and 6c in space group $Pm{\overline 3} n$. In an identical manner to Wyckoff site 6d in space group $Pm{\overline 3} n$, the symmetry independent T-site at Wyckoff position 6c (1/2, 0, 3/4) is associated with the roto-inversion axis directed along [0 0 ${\overline 1}$] (or equivalently [0 0 1]), and, as in position 6d, bisects the bond angle with two-fold multiplicity. Anion fractional coordinates are in the general equivalent position in Wyckoff site 24i as listed in Table 1. Making the definition that the aristotype in this space group is the framework structure in which both T-sites are ideal tetrahedra with identical bond lengths, there are additional simultaneous constraints that have to be applied to determine any possible solution. Firstly, the bond length equivalence implies r1.r1 = r5.r5, and secondly, the bond angle equivalence requires that r1.r2/|r1.r2| = r1.r3/|r1.r3|, r5.r6/|r5.r6|, r5.r7/|r5.r7|. Expanding and equating the first and second inner product shows the bond length equivalence results in the equality 1/2(y – x) = 0, i.e. x = y, and substituting this solution into the bond angle constraints results in the identical polynomial equation as found for the space group $I{\overline 4} 3m$; x ² – x – z ² + z – 1/8 = 0. Hence, there are an infinite set of solutions (x, x, z), the valid solutions being constrained by the bond length requirement. The expressions for the T–O bond length, T–O–T intra-polyhedral angle and tetrahedral volume that were derived for space group $I{\overline 4} 3m$ are identical for the potential aristotype phases of space group $P{\overline 4} 3n$. The solution derived for space groups $Im{\overline 3} m$, $Pm{\overline 3} n$ and $I{\overline 4} 3m$, x = y = 1/2 – 1/√8, z = 1/2, satisfies the polynomial equation above and therefore represents one of the many potential aristotype framework structures for space group $P{\overline 4} 3n$.

Mode displacements in a crystal structure in space group $P{\overline 4} 3n$ at a fixed thermodynamic state

Without loss of generality, in the subsequent discussion, the aristotype SOD framework in space group $P{\overline 4} 3n$ is chosen to be that defined by the selection of the z coordinate ($x = 1/2-\sqrt {z^2-z-3/8}$). The methodology is equally valid for a given value of x, where the related z coordinate is calculated as $1/2-\sqrt {x^2-x-1/8}$. The choice of which coordinate to use to define the aristotype crystal structure is arbitrary, however, using the z coordinate for the anion O1 at (~1/2–1/√8, ~1/2 – 1/√8, ~1/2), or the average of the x and y coordinates of the same atom to define the aristotype x would be sensible options. For comparison with an actual or hypothetical crystal structure in space group $Im{\overline 3} m$, setting z = 1/2 would be the best choice. In the subsequent analysis, the two tetrahedral sites are taken separately; site 6d is described in detail, site 6c simply follows in an analogous manner.

Transformation of the aristotype at site 6d to the final orthonormal basis is achieved in three steps. Firstly, by a translation of the entire TO₄ group by (1/4, 0, 1/2) bringing the site 6d to the origin of the orthonormal crystallographic space. Secondly, an intermediate basis e_i is defined by the bond vectors illustrated in Fig. 2d as ${\bf e}_{\bf 1} = {{{\bf r2}-{\bf r1}} \over {\vert {{\bf r2}-{\bf r1}} \vert }}$, ${\bf e}_{\bf 2} = {{{\bf r4}-{\bf r3}} \over {\vert {{\bf r4}-{\bf r3}} \vert }}$, ${\bf e}_3 = {\bf e}_1 \times {\bf e}_2$ thus bringing the unique ${\overline 4}$ axis parallel to e₃; transformation matrix M1. For convenience and ease of subsequent matrix manipulations, the parameter S is defined as$S = \sqrt {{( {-1 + 2\Delta } ) }^2 + {( {1-2z} ) }^2}$, Δ as defined earlier.

(2)$${\bf M1} = \left({\matrix{ 0 & {\displaystyle{{-1 + 2\Delta } \over S}} & {\displaystyle{{1-2z} \over S}} \cr 0 & {\displaystyle{{1-2z} \over S}} & {\displaystyle{{1-2\Delta } \over S}} \cr {-1} & 0 & 0 \cr } } \right)$$

The final basis (i, j, k) is derived from the intermediate basis with a passive rotation of π/4 around e₃; rotation matrix R.

(3)$${\bf R} = \left({\matrix{ {\displaystyle{1 \over {\sqrt 2 }}} & {\displaystyle{{-1} \over {\sqrt 2 }}} & 0 \cr {\displaystyle{1 \over {\sqrt 2 }}} & {\displaystyle{1 \over {\sqrt 2 }}} & 0 \cr 0 & 0 & 1 \cr } } \right)$$

As the both matrix M1 and the rotation matrix R are orthogonal, their product is therefore also orthogonal, and its inverse is the transpose of the product (Poole, Reference Poole2014). The absolute coordinates of the aristotype (primed) in the i, j, k basis, in terms of its fractional coordinates in the orthonormal crystallographic basis, is therefore given by the product of the matrices (a T1 = a RM1)

(4)$$\left({\matrix{ {x^{\prime}} \cr y \cr {z^{\prime}} \cr }^{\prime}} \right)_{{\bf i}, {\bf j}, {\bf k}} = {\rm a}\left({\matrix{ 0 & {\displaystyle{{\sqrt 2 ( {\Delta + z-1} ) } \over S}} & {\displaystyle{{\sqrt 2 ( {\Delta -z} ) } \over S}} \cr 0 & {\displaystyle{{\sqrt 2 ( {\Delta -z} ) } \over S}} & {\displaystyle{{\sqrt 2 ( {-\Delta -z + 1} ) } \over S}} \cr {-1} & 0 & 0 \cr } } \right)\left({\matrix{ x \cr y \cr z \cr } } \right)_{\mathop{{\bf a}}\limits^\frown, \mathop{{\bf b}}\limits^\frown, \mathop{{\bf c}}\limits^\frown}$$

where a is the lattice parameter of the hettotype. The coordinates of the aristotype in this basis are T (0, 0, 0), O1 (−l, −l, l), O2 (l, l, l), O3 (l, −l, −l) and O4 (−l, l, −l) where l = a(Δ – 1/4).

To determine the displacement vectors from the aristotype to the hettotype, the hettotype fractional coordinates are translated by (1/4, 0, 1/2) and then transformed by a T1 to the new basis. Subtracting the transformed aristotype coordinates from the corresponding transformed hettotype coordinates gives the four displacement vectors ΔOi, i = 1 – 4. If the manipulations have been carried out correctly, the four displacement vectors obey the symmetry operations of point-group ${\overline 4}$, for example, the displacements at ΔO4 are related to those at ΔO1 by the symmetry operation ${\overline 4}$⁺ (0, 0, z).

For the tetrahedral site 6c, the same sequence of operations is carried out, with the translation vector in this case being (1/2, 0, 3/4), and the intermediate vector basis being defined by ${\bf e}_ 1^{\rm{\prime}} = {{{\bf r}6-{\bf r}5} \over {\vert {{\bf r}6-{\bf r}5} \vert }}$, ${\bf e}_ 2^{\rm {\prime}} = {{{\bf r}8-{\bf r}7} \over {\vert {{\bf r}8-{\bf r}7} \vert }}$, ${\bf e}_ 3^{\rm {\prime}} = {\bf e}_ 1^{\rm {\prime}} \times {\bf e}_ 2^{\rm {\prime}}$; transformation matrix M2. The final basis (i′, j′, k′) is derived from the intermediate basis with a passive rotation of π/4 around ${\bf e}_ 3^{\rm {\prime}}$. The absolute coordinates of the aristotype (primed) in the i′, j′, k′ basis, in terms of its coordinates in the orthonormal crystallographic basis, is therefore given by the product of the matrices (a T2 = a RM2)

(5)$$\left({\matrix{ {x^{\prime}} \cr y \cr {z^{\prime}} \cr }^{\prime}} \right)_{{\bf {i}^{\prime}}, {\bf {\,j}^{\prime}}, {\bf {k}^{\prime}}} = {\rm a}\left({\matrix{ {\displaystyle{{\sqrt 2 ( {1-\Delta -z} ) } \over S}} & {\displaystyle{{\sqrt 2 ( {-\Delta + z} ) } \over S}} & 0 \cr {\displaystyle{{\sqrt 2 ( {-\Delta + z} ) } \over S}} & {\displaystyle{{\sqrt 2 ( {-1 + \Delta + z} ) } \over S}} & 0 \cr 0 & 0 & {-1} \cr } } \right)\left({\matrix{ x \cr y \cr z \cr } } \right)_{\mathop{{\bf a}}\limits^\frown, \mathop{{\bf b}}\limits^\frown, \mathop{{\bf c}}\limits^\frown}$$

where a is the lattice parameter of the hettotype. The coordinates of the aristotype in this basis are T (0, 0, 0), O4 (–l, –l, l), O6 (l, l, l), O7 (l, –l, –l) and O8 (–l, l, –l) where l = a(√8 – 1/4). The displacement vectors ΔOi, i = 4, 6–8 follow accordingly.

Consideration of Table 2 shows the displacement vector ΔO1 is related to the amplitudes and phase of the symmetry-adapted basis-vectors di via the matrix DO1 where

(6)$$\left({\matrix{ {\Delta {\rm O}1x} \cr {\Delta {\rm O}1y} \cr {\Delta {\rm O}1z} \cr } } \right) = \left({\matrix{ {-1} & {-1} & 1 \cr {-1} & {-1} & {-1} \cr 1 & {-2} & 0 \cr } } \right)\left({\matrix{ {{\rm d1}} \cr {{\rm d2}} \cr {{\rm d3}} \cr } } \right)$$

and hence

(7)$$\left({\matrix{ {{\rm d1}} \cr {{\rm d2}} \cr {{\rm d3}} \cr } } \right) = \left({\matrix{ {-1/3} & {-1/3} & {1/3} \cr {-1/6} & {-1/6} & {-1/3} \cr {1/2} & {-1/2} & 0 \cr } } \right)\left({\matrix{ {\Delta {\rm O}1x} \cr {\Delta {\rm O}1y} \cr {\Delta {\rm O}1z} \cr } } \right). $$

Writing d1′, d2′ and d3′ for the amplitude and phase of the displacements of the symmetry-adapted basis-vectors associated with the 6c site and noting ΔO4 shown below is given in the (i′, j′, k′) basis, then

(8)$$\left({\matrix{ {{\rm d{1}^{\prime}}} \cr {{\rm d{2}^{\prime}}} \cr {{\rm d{3}^{\prime}}} \cr } } \right) = \left({\matrix{ {-1/3} & {-1/3} & {1/3} \cr {-1/6} & {-1/6} & {-1/3} \cr {1/2} & {-1/2} & 0 \cr } } \right)\left({\matrix{ {\Delta {\rm O4}x} \cr {\Delta {\rm O4}y} \cr {\Delta {\rm O4}z} \cr } } \right). $$

Furthermore, writing the deviation of the fractional coordinates of the hettotype anion O1 (x _h, y _h, z _h) from the aristotype as u = x _h – x, v = y _h – x and w = z _h – z and carrying these calculations through to completion gives di and di′ directly in terms of u, v, w:

(9)$$\eqalign{& {\rm d1} = \displaystyle{{\sqrt 2 a} \over {3S}}\left[{\displaystyle{{-Su} \over {\sqrt 2 }} + ( {1-2\Delta } ) v + ( {2z-1} ) w} \right]\cr & {\rm d2} = \displaystyle{a \over {3\sqrt 2 S}}\left[{\sqrt 2 Su + ( {1-2\Delta } ) v + ( {2z-1} ) w} \right]\cr & {\rm d3} = \displaystyle{a \over {\sqrt 2 S}}[ {( {2z-1} ) v + ( {1-2\Delta } ) w} ] \cr & {\rm d{1}^{\prime}} = \displaystyle{{\sqrt 2 a} \over {3S}}\left[{( {1-2\Delta } ) u-\displaystyle{{Sv} \over {\sqrt 2 }} + ( {2z-1} ) w} \right]\cr & {\rm d{2}^{\prime}} = \displaystyle{a \over {3\sqrt 2 S}}\left[{( {1-2\Delta } ) u + \sqrt 2 Sv + ( {2z-1} ) w} \right]\cr & {\rm d{3}^{\prime}} = \displaystyle{a \over {\sqrt 2 S}}\big[ {( {2z-1} ) u + ( {1-2\Delta } ) w} \big] } $$

The absolute coordinates of the two tetrahedral sites for the i, j, k and i′, j′, k′ bases are listed in Table 3.

Table 3.

Absolute coordinates for the 6c, 6d tetrahedral sites in sodalite (T-sites at the origin).

x = a(–0.25 + (0.375 – z + z ²)^0.5)

Using the normal mode amplitudes, the derived structural parameters are given by:

1. (a) Tetrahedral bond lengths

   (10)$${\rm r}1 = \sqrt {3a^2{\left({\Delta -\displaystyle{1 \over 4}} \right)}^2 + 6a\left({\Delta -\displaystyle{1 \over 4}} \right){\rm d}1 + 3{\rm d}1^2 + 6{\rm d}2^2 + 2{\rm d}3^2} $$
   (11)$$ \hskip-2pc {\rm r5} = \sqrt {3a^2{\left({\Delta -\displaystyle{1 \over 4}} \right)}^2 + 6a\left({\Delta -\displaystyle{1 \over 4}} \right){\rm d}{1}^{\prime} + 3{\rm d}{{1}^{\prime}}^2 + 6{\rm d}{{2}^{\prime}}^2 + 2{\rm d}{{3}^{\prime}}^2} $$

2. (b) Tetrahedral edge lengths of multiplicity two

   (12)$$\sqrt 8 \sqrt {a^2{\left({\Delta -\displaystyle{1 \over 4}} \right)}^2 + {\rm d}1^2 + {\rm d}2^2 + {\rm d}3^2 + 2a\left({\Delta -\displaystyle{1 \over 4}} \right)( {{\rm d}1 + {\rm d}2} ) + 2{\rm d}1{\rm d}2} $$
   (13)$$\sqrt 8 \sqrt {a^2{\left({\Delta -\displaystyle{1 \over 4}} \right)}^2 + {\rm d}{{1}^{\prime}}^2 + {\rm d}{{2}^{\prime}}^2 + {\rm d}{{3}^{\prime}}^2 + 2a\left({\Delta -\displaystyle{1 \over 4}} \right)( {{\rm d}{1}^{\prime} + {\rm d}{2}^{\prime}} ) + 2{\rm d}{1}^{\prime}{\rm d}{2}^{\prime}} $$

3. (c) Tetrahedral edge lengths of multiplicity four

   (14)$$\sqrt 8 \sqrt {a^2{\left({\Delta -\displaystyle{1 \over 4}} \right)}^2 + {\rm d}1^2 + \displaystyle{5 \over 2}{\rm d}2^2 + \displaystyle{1 \over 2}{\rm d}3^2 + a\left({\Delta -\displaystyle{1 \over 4}} \right)( {{\rm 2d}1-{\rm d}2} ) -{\rm d}1{\rm d}2} $$
   (15)$$\sqrt 8 \sqrt {a^2{\left({\Delta -\displaystyle{1 \over 4}} \right)}^2 + {\rm d}{{1}^{\prime}}^2 + \displaystyle{5 \over 2}{\rm d}{{2}^{\prime}}^2 + \displaystyle{1 \over 2}{\rm d}{{3}^{\prime}}^2 + a\left({\Delta -\displaystyle{1 \over 4}} \right)( {{\rm 2d}{1}^{\prime}-{\rm d}{2}^{\prime}} ) -{\rm d}{1}^{\prime}{\rm d}{2}^{\prime}} $$

4. (d) Bond angles of multiplicity two

   (16)$$\cos ^{{-}1}\left({\displaystyle{{-a^2{\left({\Delta -\displaystyle{1 \over 4}} \right)}^2-2a\left({\Delta -\displaystyle{1 \over 4}} \right)( {{\rm d}1 + 4{\rm d}2} ) -{\rm d}1^2 + 2{\rm d}2^2-2{\rm d}3^2-8{\rm d}1{\rm d}2} \over {{\rm r}1^2}}} \right)$$
   (17)$$\cos ^{{-}1}\left({\displaystyle{{-a^2{\left({\Delta -\displaystyle{1 \over 4}} \right)}^2-2a\left({\Delta -\displaystyle{1 \over 4}} \right)( {{\rm d}{1}^{\prime} + 4{\rm d}{2}^{\prime}} ) -{\rm d}{{1}^{\prime}}^2 + 2{\rm d}{{2}^{\prime}}^2-2{\rm d}{{3}^{\prime}}^2-8{\rm d}{1}^{\prime}{\rm d}{2}^{\prime}} \over {{\rm r} 5^2}}} \right)$$

5. (e) Bond angles of multiplicity four

   (18)$$\cos ^{{-}1}\left({\displaystyle{{-a^2{\left({\Delta -\displaystyle{1 \over 4}} \right)}^2-2a\left({\Delta -\displaystyle{1 \over 4}} \right)( {{\rm d}1-2{\rm d}2} ) -{\rm d}1^2-4{\rm d}2^2{\rm} + {\rm 4d}1{\rm d}2} \over {{\rm r}1^2}}} \right)$$
   (19)$$\cos ^{{-}1}\left({\displaystyle{{-a^2{\left({\Delta -\displaystyle{1 \over 4}} \right)}^2-2a\left({\Delta -\displaystyle{1 \over 4}} \right)( {{\rm d}{1}^{\prime}-2{\rm d}{2}^{\prime}} ) -{\rm d}{{1}^{\prime}}^2 \hbox{-}{\rm 4d}{{2}^{\prime}}^2{\rm} + {\rm 4d}{1}^{\prime}{\rm d}{2}^{\prime}} \over {{\rm r} 5^2}}} \right)$$

6. (f) The intra polyhedral angle by

   (20)$$\eqalignno{& \cos ^{{-}1}\left({\left({\displaystyle{1 \over {{\rm r}1{\rm r}5}}} \right)( {{\rm A} + {\rm B} + {\rm C}} ) } \right)\cr & {\rm A} = \displaystyle{{a^2\left({\Delta -\displaystyle{1 \over 4}} \right)} \over {\sqrt 8 S}}\left[{3\sqrt 8 \Delta S + 2\Delta -2\sqrt 2 S-1} \right]\cr & {\rm B} = a\left[{\rm d}1\left(3\left({\Delta -\displaystyle{1 \over 4}} \right) + \left({\displaystyle{3\sqrt 8 \Delta S + 2\Delta -\sqrt 8 S-1} \over {\sqrt 8 S}} \right) \right) \right. \cr & \quad \left. + {\rm d}2\left({\displaystyle{{2\Delta + \sqrt 2 S-1} \over {\sqrt 8 S}}} \right) + {\rm d}3\left({\displaystyle{{1-2z} \over {\sqrt 8 S}}} \right)} \right]\cr & {\rm C} = 3{\rm d}1^2 + 6{\rm d}2^2 + 2{\rm d}3^2. &(20)} $$

Utilising crystallographic results from sodalite (Hassan and Grundy, Reference Hassan and Grundy1984), a = 8.882 Å, Al (1/4, 0, 1/2), Si (1/2, 0, 3/4) and O (0.1390, 0.1494, 0.4383), then with the aristotype z = 0.4383 (x = 0.14110, Δ = 0.3589), d1 = 0.03806 Å, d2 = 0.00969 Å, d3 = –0.02082 Å, d1′ = –0.03263 Å, d2′ = 0.02053 Å and d3′ = 0.00529 Å. The observed (recalculated) bond lengths and bond angles are compared with the calculations from mode decomposition in Table 4 where the agreement is found to be close to exact.

Table 4.

Comparison of derived structural parameters for sodalite based on crystal structure refinement and mode decomposition.

The aristotype bond length is given by √3a(Δ – 1/4) (1.6753 Å), the calculated quadratic elongation and tetrahedral angle variance (Robinson et al., Reference Robinson, Gibbs and Ribbe1971) of the AlO₄ site (6d) in sodalite are 1.0008 and 1.41°^² respectively. At the SiO₄ site (6c), the corresponding parameters are 1.0025 and 7.49°^². Approximating the ratio of the observed bond lengths to the aristotype bond length as a binomial expansion to first order, the dominant term in the expansion is found to be d1(′)/a(Δ – 1/4). Hence, with the choice of the aristotype z fractional coordinate made in this analysis, the effect of the amplitude and phase of d1 and d1′ is to increase the Al–O bond length from the aristotype value with the converse found for the Si–O bond length as d1 > 0 and d1′ < 0. The bond length changes are in agreement with the observed Al–O and Si–O bond lengths in sodalite, and to a good approximation in this aristotype definition, d1(′) act as pure breathing modes. The polyhedral distortion that transforms as the irreducible representation E(α) is associated with the positive mode amplitudes d2 and d2′, where d2′ > d2, consistent with the larger quadratic elongation and tetrahedral angle variance in the SiO₄ group in sodalite. The AlO₄ tetrahedron is rotated –0.44° away from the aristotype orientation around k, the SiO₄ tetrahedron rotates in the opposite direction by 0.11° ($\tan \varphi {( }^{\prime}) = {{d3{( }^{\prime}) } \over {a( {\Delta -1/4} ) }}$) as d3 < 0 and |d3| > d3′. Note that this interpretation as a rotation about the unique axis is only valid in small displacements of d3 and d3′ i.e. in this case, setting aristotype z to be the z fractional coordinate for the anion at (~1/2 – 1/√8, ~1/2 – 1/√8, ~1/2).

The analysis presented above is equally valid for the other three space groups discussed. For example, if the crystal structure of bicchulite (Sahl, Reference Sahl1980) in space group $I{\overline 4} 3m$ is analysed with the undifferentiated tetrahedral site (T) at (0.25, 0.50, 0.0) and the oxygen anion (O) at (0.1407, 0.1407, 0.4220), mode decomposition yields d1(′) = 1.873 × 10^–3 Å, d2(′) = 0.0131 Å, and d3(′) = –8.4631.873 × 10^–3 Å, for aristotype z = 0.4220. The calculated and observed bond lengths and angles are T–O = 1.7164 Å calc., 1.716(1) Å obs., O–T–O (multiplicity 2) = 111.615° calc., 111.6(1)° obs. (typographical error in table 2 of Sahl (Reference Sahl1980) reads 111.9°), O–T–O (multiplicity 4) = 108.410° calc., 108.4(1)° obs., T–O–T = 130.712° calc. and obs. See later section for discussion of mode and crystallographic degrees of freedom in this space group.

Mode displacements in a crystal structure in space group $P{\overline 4} 3n$ as a function of thermodynamic variable

The methodology detailed above is particularly advantageous in the study of the SOD framework as a function of thermodynamic variables such as temperature, pressure, or composition in a solid solution. In these cases, the aristotype is defined at a fixed value of the thermodynamic variable, and the evolving crystal structure as a function of this variable is derived in terms of the mode amplitudes. The basis of the analysis is the observation that an aristotype in space group $P{\overline 4} 3n$ may be defined for any set of fractional coordinates (x, x, z) where x and z satisfy the polynomial expression x ² – x – z ² + z – 1/8 = 0. In mode decomposition as a function of thermodynamic variable, the optimal choice of aristotype would be therefore be based on the z coordinate of the anion with fractional coordinates (~1/2 – 1/√8, ~1/2 – 1/√8, ~1/2) at ambient pressure for a study in pressure, the lowest temperature for a study in temperature, or an end-member for a study in composition. The aristotype is transformed to absolute coordinates, as shown in the previous section, with lattice parameter a _a and the hettotype is transformed identically, but in this case using its own lattice parameter a. The displacement vectors from identically labelled anions in the aristotype and hettotype are mode decomposed using the matrices shown earlier. The expressions for the bond lengths and bond angles are unchanged from those listed earlier, only the lattice parameter a requires replacing with a _a. The intra-polyhedral angle, however, is a more complex function than shown earlier, as the calculation involves taking into account the requirement that the site 6c has to be transformed into the i, j, k basis. Writing Δa = a – a _a, the intra-polyhedral angle is given by:

(21)$$\eqalign{& \cos ^{{-}1}\left({\left({\displaystyle{1 \over {{\rm r}1{\rm r}5}}} \right)( {{\rm A} + {\rm B} + {\rm C} + {\rm D}} ) } \right)\cr & {\rm A} = \displaystyle{{a_{\rm a}^2 \left({\Delta -\displaystyle{1 \over 4}} \right)} \over {\sqrt 8 S}}\left[{3\sqrt 8 \Delta S + 2\Delta -2\sqrt 2 S-1} \right]\cr & {\rm B} = a_{\rm a}\left[{\left({\displaystyle{{3\sqrt 8 \Delta S + 2\Delta -\sqrt 8 S-1} \over {\sqrt 8 S}}} \right){\rm d}_1}\right. \cr & \quad + \left.\left({\displaystyle{{2\Delta + \sqrt 2 S-1} \over {\sqrt 8 S}}} \right){\rm d}_2 + \left({\displaystyle{{1-2z} \over {\sqrt 8 S}}} \right){\rm d}_3} \right] \cr &{\rm C} = a_a\left(\Delta -\displaystyle{1 \over 4} \right)\left[{3{\rm d}_1 + \displaystyle{{\Delta a} \over {\sqrt 8 S}}\left({2\Delta -1-\displaystyle{S \over {\sqrt 2 }}} \right)} \right] \cr & {\rm D} = 3{\rm d}_1^2 + 6{\rm d}_2^2 + 2{\rm d}_3^2 -\displaystyle{{\Delta a} \over {\sqrt 8 S}}\left[{\left({-2\Delta + 1 + \displaystyle{S \over {\sqrt 2 }}} \right){\rm d}_1 }\right.} \cr & \quad + \left.\left({-2\Delta + 1-\sqrt 2 S} \right){\rm d}_2 + ( {2z-1} ) {\rm d}_3} \right]} . $$

For the aristotype, where Δa = 0, the expression reduces to that shown previously for mode decomposition at a fixed thermodynamic state.

Taylor (Reference Taylor1972) has defined the polyhedral tilt angles (φ_6c,d) for the two T-sites as the angle between a tetrahedral edge and the closest of the two reciprocal lattice vectors which are co-planar with the edge in question. The change in φ_6c,d is a fundamental structural response to a change in thermodynamic state, and as it is related to the anion fractional coordinates, it can be described as a function of condensed mode amplitudes and phases. For a hettotype phase, the two polyhedral tilt angles generally have different magnitudes, however, in the aristotype phase these angles are identical (φ) and additional mode-related rotational terms are required to be included to correctly evaluate the polyhedral tilt angles at the two T-sites.

(22)$$\eqalign{& {\rm \varphi }_{6c, d} = {\rm \varphi } \pm {\rm \Delta }{\rm \varphi }_{6c, d} \cr & {\rm \varphi } = {\rm co}{\rm s}^{{-}1}\left({\displaystyle{{1-2{\rm \Delta }} \over {\sqrt {{( {1-2{\rm \Delta }} ) }^2 + {( {1-2z} ) }^2} }}} \right)\cr & {\rm \Delta }{\rm \varphi }_{6c} = {\rm co}{\rm s}^{{-}1}\left({\displaystyle{{a_{\rm a}( {\Delta -1/4} ) + {\rm d}{1}^{\prime} + {\rm d}{2}^{\prime}} \over {\sqrt {a_{\rm a}^2 {( {\Delta -1/4} ) }^2 + 2a_{\rm a}( {\Delta -1/4} ) ( {{\rm d}{1}^{\prime} + {\rm d}{2}^{\prime}} ) + {\rm d}{{1}^{\prime}}^2 + {\rm d}{{2}^{\prime}}^2 + {\rm d}{{\rm {3}^{\prime}}}^2 + 2{\rm d}{1}^{\prime}{\rm d}{2}^{\prime}} }}} \right)\cr & {\rm \Delta }{\rm \varphi }_{6d} = {\rm co}{\rm s}^{{-}1}\left({\displaystyle{{a_{\rm a}( {\Delta -1/4} ) + {\rm d}1 + {\rm d}2} \over {\sqrt {a_{\rm a}^2 {( {\Delta -1/4} ) }^2 + 2a_{\rm a}( {\Delta -1/4} ) ( {{\rm d}1 + {\rm d}2} ) + {\rm d}1^2 + {\rm d}2^2 + {\rm d} 3^2 + 2{\rm d}1{\rm d}2} }}} \right)} $$

The sign for Δφ_6c,d is negative for ${{a_{\rm a}( {\Delta -1/4} ) + {\rm d}1{\rm {( }^{\prime}) } + {\rm d}2{\rm {( }^{\prime}) }-{\rm d}3{\rm {( }^{\prime}) }} \over {a_{\rm a}( {\Delta -1/4} ) + {\rm d}1{\rm {( }^{\prime}) } + {\rm d}2{\rm {( }^{\prime}) } + {\rm d}3{\rm {( }^{\prime}) }}} \ge 1$, otherwise positive.

The temperature dependence of the crystal structure of sodalite

The methodology of mode decomposition with respect to a reference aristotype phase is illustrated using the temperature dependence of the crystal structure of sodalite derived from profile fitting of synchrotron X-ray diffraction data (Hassan et al., Reference Hassan, Antao and Parise2004). It should be noted that the tabulated bond lengths and bond angles are not consistent with the fractional coordinates and lattice parameters given in the publication, differing on average by one estimated standard deviation; the 28°C bond lengths and angles are particularly in error. For the analysis reported here, all bond lengths and angles were recalculated from the listed fractional coordinates and lattice parameters. The aristotype was defined from the 28°C data set, using the z fractional coordinate of 0.43895 and lattice parameter of 8.88696 Å. The crystal structure at each of the 14 temperatures was mode decomposed by the method detailed in the earlier section. As an example, the matrices, absolute coordinates, displacement vectors, mode amplitudes and phases, and bond lengths and angles for the 982°C crystal structure are listed in Table 5; observed (i.e. recalculated) bond lengths and angles shown in italics. The agreement between mode decomposition and calculation (observed) is excellent.

Table 5.

Details of mode decomposition for the crystal structure of sodalite at 982°C (Hassan et al., Reference Hassan, Antao and Parise2004). Bond lengths and angles recalculated from the published lattice parameter and fractional coordinates are shown in italics.

α = 0.917894, β = 0.396825, a _a = 8.88696 Å and z = 0.43895

The temperature dependences of the six modes are illustrated in Figs 4a and b, and, as expected for a phase that is evolving in the absence of a structural phase transition, all are found to vary monotonically with temperature. Percentage changes in the mode amplitudes are –13.2, 63.1, 136.0, 10.9, 39.5, 216.8 for d1, d2, d3, d1′, d2′ and d3′, respectively. The most significant changes are associated with the modes that rotate the two tetrahedral sites relative to the aristotype orientation (d3 and d3′). The percentage variation of the breathing modes (d1 and d1′) are unsurprisingly much smaller and hence show no evidence for disordering of the two tetrahedral sites. The full lines on these figures show quartic polynomial fits to the mode data, noting that extrapolation beyond the range of the fits is inadvisable in such high-order polynomials (Hahn, Reference Hahn1977). The dashed lines on the identical figures show the mode expansivity (${1 \over {{\rm di{( }^{\prime}) }}}{{d{\rm di{( }^{\prime}) }} \over {d{\rm T}}}$), with the exception of d3′, as the magnitude of this mode passes through 0 at a temperature of T = ~82.18°C. In this case, the mode expansivity is approximated by ${1 \over {{\rm d{3}^{\prime}( 500}^\circ {\rm C) }}}{{d{\rm d{3}^{\prime}}} \over {d{\rm T}}}$. Over the entire temperature interval, the mean mode expansion coefficients are –1.43 × 10^–4 °C^–1, 8.21 × 10^–4 °C ^–1, 19.22 × 10^–4 °C^–1, 1.20 × 10^–4 °C^–1, 4.89 × 10^–4 °C^–1 and 28.01 × 10^–4 °C ^–1 for d1, d2, d3, d1′, d2′ and d3′, respectively.

Fig. 4.

The temperature dependence of the mode displacement amplitudes and phases of the SOD framework of sodalite at high temperature derived from the crystallographic results of Hassan et al. (Reference Hassan, Antao and Parise2004). (a) Illustrates the modes in the AlO₄ tetrahedron, (b) the modes in the SiO₄ tetrahedron. In both cases, the calculated mode amplitudes are shown as circles, with the full lines on the figure showing quartic fits to these data. The mode expansion coefficients are shown as the dashed lines on the figure.

Using the polynomial parameterisation for the mode amplitudes and phases, and a quadratic fit to the lattice parameter variation with temperature, the derived structural parameters were calculated using the expressions shown earlier for the bond lengths/angles. The mode predicted variations are illustrated in Figs 5a, b and c as the full lines where they are compared with the observed results re-evaluated from the published fractional coordinates and lattice parameters. The agreement is excellent with the mode derived values <1 estimated standard deviation of the experimental values. The temperature invariance of the two T-site–O bond lengths is reproduced by mode decomposition; the apparent oscillatory behaviour of Al–O is within the scatter of the experimental results. Deducing a mechanism for the constant T-sites–O bond lengths with temperature, however, is not easy to determine as ${1 \over {\rm r}}{{d{\rm r}} \over {d{\rm T}}} = {{\bf r} \over {{\rm r}^2}}\cdot {{d{\bf r}} \over {d{\rm T}}}$, and this algebraic expansion is unfortunately too unwieldy to derive the dominant terms for either case.

Fig. 5.

Comparison of the observed, derived structural parameters of sodalite at high temperature (Hassan et al., Reference Hassan, Antao and Parise2004) (shown as circles) with those calculated by mode decomposition (as the full lines). In all cases the bond lengths and angles were recalculated from the published lattice parameters and fractional coordinates as the tabulated data were inconsistent with these results. (a) Shows the results from the AlO₄ tetrahedron, (b) the results from the SiO₄ tetrahedron, whilst (c) shows the intra-polyhedral rotation angle. The agreement between the observed data and the temperature variation calculated by mode parameterisation is excellent for all temperatures indicating the ability for mode decomposition to accurately interpolate the structural parameters at intermediate temperatures.

With the exception of the temperature variations of the bond lengths and angles, structural distortions of the two tetrahedral sites in sodalite were not considered in detail by Hassan et al. (Reference Hassan, Antao and Parise2004). Re-analysis of their published data shows that the volumes of the tetrahedral sites reduce in a linear manner with increasing temperature as illustrated in Fig. 6 where the volume of the AlO₄ tetrahedron varies as (2.7105(1) – 9.6(2) × 10^–6 T) Å³ (T °C) (r ² = 0.99375), and the SiO₄ tetrahedron as (2.1738(3) – 1.29(5) × 10^–5 T) Å³ (T °C) (r ² = 0.97843). The polyhedral distortion parameters of quadratic elongation (QE) and tetrahedral angle variance (TAV) (Robinson et al., Reference Robinson, Gibbs and Ribbe1971) are found to vary linearly with temperature for both sites (QE Al–O = 1.00057 + 2.413 × 10^–6 T (T °C), r ² = 0.99336; QE Si–O = 1.00249 + 4.031 × 10^–6 T, r ² = 0.98836; TAV/degrees² AlO₄ = 2.2345 + 9.23 × 10^–3 T, r ² = 0.99327; TAV/degrees² SiO₄ = 9.6045 + 1.51 × 10^–2 T, r ² = 0.98788) but also vary monotonically with the corresponding value of the mode amplitudes that transform as the irreducible representation E, thus providing further evidence for this mode being related to the tetrahedral distortion. These variations with the E mode amplitude are shown in Fig. 7 where the full and dashed lines are guides to the eye based on quadratic fits to the calculated distortion parameters. It should be noted that whilst the quadratic elongation and tetrahedral angle variance of the two T-sites are well separated as a function of temperature, the variation of the individual distortion parameter for both sites is almost continuous in the E mode amplitude. The lack of high-temperature crystallographic results for other sodalite-structured phases precludes making any conclusion on whether this is typical or atypical behaviour.

Fig. 6.

The linear temperature dependences of the tetrahedral volumes of the AlO₄ and SiO₄ tetrahedra in sodalite.

Fig. 7.

The correlation of the structural distortion parameters of quadratic elongation and tetrahedral angle variance (Robinson et al., Reference Robinson, Gibbs and Ribbe1971) with the amplitudes of the E(α) normal modes in sodalite at high temperatures (Hassan et al., Reference Hassan, Antao and Parise2004). Despite the magnitudes of the quadratic elongation and tetrahedral angle variance of the two tetrahedra being well separated in temperature, they follow identical trends in the E(α) normal mode amplitudes. Further work is necessary to determine whether this is a common structural response of sodalite-structured phases.

Hassan et al. (Reference Hassan, Antao and Parise2004) considered the structural basis for the thermal expansion of sodalite to be related to the temperature-induced weakening of the Na–Cl bonds. The migration of the Na ion towards the six-membered ring causes the tetrahedral sites to rotate relative to one another. Calculated bond valence sums (Brown, Reference Brown2002) for the 7 coordinated NaO₆Cl site show it to be slightly under-bonded at room temperature, and significantly under-bonded at the highest temperatures measured. Manual adjustment of the R ₀ values by allowing them to have a temperature dependence (R ₀(T) = R ₀(25 °C) + cT) (T °C) c ~ 9 × 10^–3 pm °C^–1 for Na – O, and c ~ 6 × 10^–3 pm °C^–1 for Na – Cl) gives a slightly over-bonded, but constant bond valence sum for all temperatures. However, RUM analysis (Hammonds, Reference Hammonds, Dove, Giddy, Heine and Winkler1996; Dove et al., Reference Dove, Pryde, Heine and Hammonds2007) has found the SOD framework to be intrinsically flexible with zero-frequency modes throughout much of the Brillouin zone. It therefore seems more plausible that the thermal expansion of sodalite is governed by the behaviour of these low frequency rigid unit modes which permit the tetrahedra to rotate relative to one another with the Na moving towards the six-membered ring to maintain bond valency. Low frequency external modes (<300 cm^–1) have been observed in the Raman spectrum of sodalite, with a single band measured as low as 59 cm^–1, compared to the ‘free TO₄’ modes that lie in the range 350–1150 cm^–1 (Arai and Smith, Reference Arai and Smith1981).