The principal axes systems for the elastic properties of monoclinic gallia

We discuss the principal axes systems of monoclinic and triclinic crystals regarding their elastic properties. Explicit formulas are presented for the orientation of these coordinate systems for monoclinic crystals. In this context, theoretical results from literature on the elastic properties of monoclinic (space group C2/m) gallia and alumina are critically discussed.


Definition of the crystal system
The crystal is described with respect to a Cartesian coordinate system x = (1, 0, 0) T , ỹ and z . It must be the same as used for the crystal stress-strain relation (12) given below. A vector in this system is denoted as r.
The lattice vectors of the unit cell are a 1 = (a 11 , a 12 , a 13 ) T , a 2 and a 3 . A vector in the crystal r is related to r via with with a 1 = Tx , a 2 = Tỹ , and a 3 = Tz.
A minimum of six non-zero components is required for the most general case. The standard choice for a triclinic crystal is 19,20 , (1) r = Tr www.nature.com/scientificreports/ with The monoclinic system is obtained by setting α = γ = π/2, The ỹ-direction is perpendicular to the ( x,z)-plane.

Rotation transformation of the coordinates
The spherical angles θ and φ define the rotational transformation of vectors r in the crystal system into vectors r ′ in another Cartesian coordinate system. A rotation of the crystal is generally described by a rotation matrix R, We consider the rotation around the z-axis by the angle φ, and subsequently the rotation around the ỹ-axis by the angle θ, An arbitrary direction can be generated with the combined rotation ( Fig. 1) The angles have a useful range of −π/2 ≤ θ ≤ π/2 and 0 ≤ φ ≤ 2π.

Stress-strain relation in the crystal
The stress-strain relation in the crystal system reads with the stiffness components C ij for the 6-tuples of stress σ and strain e in the Voigt notation, The (symmetrized) strain components are derived from the displacement u via ǫ ij = (∂u i /∂x j + ∂u j /∂x i )/2 . The 6 × 6 matrix C contains the elastic (stiffness) constants and is given with respect to the same (x , ỹ , z) coordinate system as chosen in (7). The matrix C is symmetric, i.e. C ij = C ji . For the triclinic system, all entries are non-zero, yielding 21 components; by special choice of coordinate system, the number can be reduced to 18 independent constants 1 . For the monoclinic system, 13 non-zero components remain; by special choice of coordinate system, x y z θφ Figure 1. Schematic of Cartesian coordinate system x , ỹ , z , with a crystal direction (grey arrow) and the angles θ and φ . After the rotation according to Eq. (11), the grey arrow points along z. www.nature.com/scientificreports/ the number can be reduced to 12 independent constants 1 . Special forms of C are given for all crystals in 1,21 and contain many zeros for suitable choices of coordinate system. For a monoclinic material (mirror plane for y = 0 ) (12) reads

Scientific
The technicalities of the transformation of the matrix C under rotation into C ′ are discussed at length in 8,9 . We define C 5 = C 15 + C 25 + C 35 . For monoclinic (and triclinic) materials, the special PA-R coordination system can be found for which Here, for isotropic dilation, i.e. e 1 = e 2 = e 3 , without shear strains, i.e. e 4 = e 5 = e 6 = 0 , the tangential forces vanish, i.e. σ 4 = σ 5 = σ 6 = 0 and it is evoked only by normal forces. The reciprocal equation, contains the compliances S ij with S = C −1 . For the rotated system, S ′ = C ′ −1 . The coordination system fulfilling equations (20)- (22) is the principal axes system of elastic deformation (PA-D).
Here, for hydrostatic pressure, i.e. isotropic normal forces, σ 1 = σ 2 = σ 3 and σ 4 = σ 5 = σ 6 = 0 , the shear strains vanish, i.e. e 4 = e 5 = e 6 = 0 , meaning that a rectangular box with sides aligned to this coordinate system keeps its right angles For any crystal except monoclinic or triclinic the two PA-D and PA-R coordinate systems coincide. Only for these two low symmetry crystal classes, they have different orientations. We note that a parameter (and criterion) for triclinicity has been given in 22 .

Orientation of the principal axes system of elastic resistance (PA-R)
We look now for the angles of rotation of the PA-R system relative to the crystal system (x,ỹ,z) . In the monoclinic system for symmetry reasons, the angle φ must be zero and the rotation must lie around the ỹ-axis. Also, if θ 0 is a solution, θ 0 + n π/2 , n ∈ Z 0 must a solution as well. This will come out explicitly.
Therefore, the solutions can be finally written as We chose as solution the angle with the smallest absolute value, i.e. a value in the range −π/4 ≤ θ C ≤ π/4 . The principal axis system is then given by the directions θ C and θ C + π/2 in the ( x,z)-plane and the ỹ direction.
(30) cos 2θ S + ζ sin 2θ S = 0.   www.nature.com/scientificreports/  (a, b) 10 , (c, d) 13 , and (e, f) 18 . Also, the sums according to 11,12 are depicted as black dashed (dash-dotted) lines in (a, b). The vertical dashed lines indicate the zeros of the black solid line sums. www.nature.com/scientificreports/ It should be mentioned that this formula does not depend on C 44 , C 46 and C 66 . The solutions of (30) are given by, n ∈ Z 0 . Again we chose −π/4 ≤ θ S ≤ π/4 . The principal axis system is then given by the directions θ S and θ S + π/2 in the ( x,z)-plane and the ỹ direction.
For these sets we have calculated the angles θ C of the PA-R and θ S for the PA-D system as depicted in Fig. 2. Foremost, all calculations arrive at θ S = θ C , as expected for monoclinic material. The difference θ S − θ C is within about one degree approximately 7 • for all calculations (except FFS), showing that the effect is present but not drastic. For β-Ga 2 O 3 , several independent DFT calculations agree within a few degrees 10-12 that θ C is close to zero. The absolute angles derived from 13 ( 14 ) (010)-plane ( φ=0) as a function of the rotation angle θ for three selected data sets from 10 (blue), 13 (black) and 18 (experimental elastic constants, red). www.nature.com/scientificreports/ within the experimental error), for C ′ 5 = 0 , also C ′ 25 = 0 , i.e. C ′ 15 = −C ′ 35 . This is in contrast to all available DFT calculations where for C ′ 5 = 0 , clearly none of the C ′ i5 components ( i = 1, 2, 3 ) is zero. The experimental data for β-Ga 2 O 3 from 23 yield an angular difference between the PA-D and PA-R systems of about 7.6 • , in agreement with most theories; the absolute angles are closest to the results of 13 .

Young's module
The monoclinic angle β = π/2 also leads to a characteristic distortion of the angular dependence of the Young's module Y ′ = 1/S ′ 11 in the ( x,z)-plane, i.e. the (010) crystallographic plane, away from mirror symmetries that are present for an orthorhombic system. We note that a three-dimensional view of the data from 13 can be found in Ref. 28 . The remaining symmetry is that Y ′ (θ) = Y ′ (θ + π) . The angular dependence in the ( x,z)-plane is visualized in Fig. 3 for three data sets with linear angular scale and as polar plot. The angular positions θ Y,max and θ Y,min of the maximum and minimum values of the Young's module, respectively, in the ( x,z)-plane, are listed in Table 1. There seems to be significant disagreement between different theories. The two experimental data sets yield rather similar values which agree more or less with theories in 13,14 . Notably, the theory of 23 is the only one yielding θ Y,max > π/2.

Summary
We have presented analytical formulas for the orientations of the two symmetry-adapted Cartesian coordinate systems of monoclinic crystals, namely the compression and resistance ellipsoids. Various theoretical and experimental data sets for monoclinic gallia and alumina have been analyzed and significant differences between theories and theories and experiment have been found, making further investigations necessary to correctly capture the anisotropic elastic properties of these technologically important materials.
The data that support the findings of this study are available from the corresponding author upon reasonable request.