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The antisymmetry of distortions.

VanLeeuwen BK, Gopalan V - Nat Commun (2015)

Bottom Line: Distortions are ubiquitous in nature.The symmetry of a distortion pathway is then uniquely defined by a distortion group; it has the same form as a magnetic group that involves time reversal.Distortion symmetry has important implications for a range of phenomena such as structural and electronic phase transitions, diffusion, molecular conformational changes, vibrations, reaction pathways and interface dynamics.

View Article: PubMed Central - PubMed

Affiliation: Materials Research Institute and Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, USA.

ABSTRACT
Distortions are ubiquitous in nature. Under perturbations such as stresses, fields or other changes, a physical system reconfigures by following a path from one state to another; this path, often a collection of atomic trajectories, describes a distortion. Here we introduce an antisymmetry operation called distortion reversal that reverses a distortion pathway. The symmetry of a distortion pathway is then uniquely defined by a distortion group; it has the same form as a magnetic group that involves time reversal. Given its isomorphism to magnetic groups, distortion groups could have a commensurate impact in the study of distortions, as the magnetic groups have had in the study of magnetic structures. Distortion symmetry has important implications for a range of phenomena such as structural and electronic phase transitions, diffusion, molecular conformational changes, vibrations, reaction pathways and interface dynamics.

No MeSH data available.


Related in: MedlinePlus

The application of distortion symmetry to a distortion of β-BaB2O4.The mostly rigid rotation of the B3O6 rings leads to two variants of β-BaB2O4 with R3c symmetry group, the λ=−1 variant (inset in orange in a and the λ=+1 variant (inset in cyan in a), transforming through a transition state at λ=0 (inset in magenta in a) with a symmetry of . The symmetry of this path, , requires that the energy profile in a is symmetric. (b) The superimposed images of β-BaB2O4 along the distortion pathway; their colour varies from orange, through magenta to cyan as λ varies from −1 through 0 to +1. (c) The optical second harmonic generation tensor coefficients along this pathway calculated by Cammarata and Rondinelli37 (red, green and blue circles) and a polynomial fit (red, green and blue lines) using only the coefficients that are consistent with  point-group symmetry. Distortion symmetry predicts that these coefficients will be odd functions of the distortion parameter, λ, and zero when λ=0.
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f4: The application of distortion symmetry to a distortion of β-BaB2O4.The mostly rigid rotation of the B3O6 rings leads to two variants of β-BaB2O4 with R3c symmetry group, the λ=−1 variant (inset in orange in a and the λ=+1 variant (inset in cyan in a), transforming through a transition state at λ=0 (inset in magenta in a) with a symmetry of . The symmetry of this path, , requires that the energy profile in a is symmetric. (b) The superimposed images of β-BaB2O4 along the distortion pathway; their colour varies from orange, through magenta to cyan as λ varies from −1 through 0 to +1. (c) The optical second harmonic generation tensor coefficients along this pathway calculated by Cammarata and Rondinelli37 (red, green and blue circles) and a polynomial fit (red, green and blue lines) using only the coefficients that are consistent with point-group symmetry. Distortion symmetry predicts that these coefficients will be odd functions of the distortion parameter, λ, and zero when λ=0.

Mentions: Next we demonstrate the application of distortion groups in predicting allowed energy couplings that are odd powers in the distortion parameter and may appear at first to be disallowed by conventional symmetry groups. We will use beta barium borate, β-BaB2O4, a widely used nonlinear optical crystal, as an example. Using a parent structure (λ=0) with symmetry37, we construct a distortion with R3c variants at λ=−1 and λ=+1. This distortion pathway has symmetry. The calculated energy profile, ΔE(λ), is shown in Fig. 4a and is symmetric with respect to λ. This is a consequence of the starred symmetry operations, just as with the PF5 example. In Fig. 4b, we depict the sequence of intermediate structures along the distortion pathway by superimposing them using a colour scale. From the blurred pattern, we can see that this distortion is mostly the nearly rigid rotation of the B3O6 rings. For β-BaB2O4 and distortion group (no. 4306 in the complete double-antisymmetry space group (DASG) listing3839), the B3O6 rings are on the 12c site. From referring to the listing, this means that there are rings located at {0, 0, z}, {0, 0, −z+½}, {0, 0, −z} and {0, 0, z+½} with rotation vectors of [0, 0, ωz], [0, 0, ωz], [0, 0, −ωz] and [0, 0, −ωz], respectively. This tells us that the symmetry requires that the rotation (ω) of the rings is only along the z axis and alternates every two rings along the column, that is, clockwise, clockwise, counterclockwise, counterclockwise and so on. The distortion symmetry listing also tells us that the displacement of the rings is only along the z axis and all the rings displace in the same direction with the same magnitude. This is just one of the many ways in which the concept of distortion symmetry can be used to make useful predictions about distortions.


The antisymmetry of distortions.

VanLeeuwen BK, Gopalan V - Nat Commun (2015)

The application of distortion symmetry to a distortion of β-BaB2O4.The mostly rigid rotation of the B3O6 rings leads to two variants of β-BaB2O4 with R3c symmetry group, the λ=−1 variant (inset in orange in a and the λ=+1 variant (inset in cyan in a), transforming through a transition state at λ=0 (inset in magenta in a) with a symmetry of . The symmetry of this path, , requires that the energy profile in a is symmetric. (b) The superimposed images of β-BaB2O4 along the distortion pathway; their colour varies from orange, through magenta to cyan as λ varies from −1 through 0 to +1. (c) The optical second harmonic generation tensor coefficients along this pathway calculated by Cammarata and Rondinelli37 (red, green and blue circles) and a polynomial fit (red, green and blue lines) using only the coefficients that are consistent with  point-group symmetry. Distortion symmetry predicts that these coefficients will be odd functions of the distortion parameter, λ, and zero when λ=0.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4660051&req=5

f4: The application of distortion symmetry to a distortion of β-BaB2O4.The mostly rigid rotation of the B3O6 rings leads to two variants of β-BaB2O4 with R3c symmetry group, the λ=−1 variant (inset in orange in a and the λ=+1 variant (inset in cyan in a), transforming through a transition state at λ=0 (inset in magenta in a) with a symmetry of . The symmetry of this path, , requires that the energy profile in a is symmetric. (b) The superimposed images of β-BaB2O4 along the distortion pathway; their colour varies from orange, through magenta to cyan as λ varies from −1 through 0 to +1. (c) The optical second harmonic generation tensor coefficients along this pathway calculated by Cammarata and Rondinelli37 (red, green and blue circles) and a polynomial fit (red, green and blue lines) using only the coefficients that are consistent with point-group symmetry. Distortion symmetry predicts that these coefficients will be odd functions of the distortion parameter, λ, and zero when λ=0.
Mentions: Next we demonstrate the application of distortion groups in predicting allowed energy couplings that are odd powers in the distortion parameter and may appear at first to be disallowed by conventional symmetry groups. We will use beta barium borate, β-BaB2O4, a widely used nonlinear optical crystal, as an example. Using a parent structure (λ=0) with symmetry37, we construct a distortion with R3c variants at λ=−1 and λ=+1. This distortion pathway has symmetry. The calculated energy profile, ΔE(λ), is shown in Fig. 4a and is symmetric with respect to λ. This is a consequence of the starred symmetry operations, just as with the PF5 example. In Fig. 4b, we depict the sequence of intermediate structures along the distortion pathway by superimposing them using a colour scale. From the blurred pattern, we can see that this distortion is mostly the nearly rigid rotation of the B3O6 rings. For β-BaB2O4 and distortion group (no. 4306 in the complete double-antisymmetry space group (DASG) listing3839), the B3O6 rings are on the 12c site. From referring to the listing, this means that there are rings located at {0, 0, z}, {0, 0, −z+½}, {0, 0, −z} and {0, 0, z+½} with rotation vectors of [0, 0, ωz], [0, 0, ωz], [0, 0, −ωz] and [0, 0, −ωz], respectively. This tells us that the symmetry requires that the rotation (ω) of the rings is only along the z axis and alternates every two rings along the column, that is, clockwise, clockwise, counterclockwise, counterclockwise and so on. The distortion symmetry listing also tells us that the displacement of the rings is only along the z axis and all the rings displace in the same direction with the same magnitude. This is just one of the many ways in which the concept of distortion symmetry can be used to make useful predictions about distortions.

Bottom Line: Distortions are ubiquitous in nature.The symmetry of a distortion pathway is then uniquely defined by a distortion group; it has the same form as a magnetic group that involves time reversal.Distortion symmetry has important implications for a range of phenomena such as structural and electronic phase transitions, diffusion, molecular conformational changes, vibrations, reaction pathways and interface dynamics.

View Article: PubMed Central - PubMed

Affiliation: Materials Research Institute and Department of Materials Science and Engineering, Pennsylvania State University, University Park, Pennsylvania 16802, USA.

ABSTRACT
Distortions are ubiquitous in nature. Under perturbations such as stresses, fields or other changes, a physical system reconfigures by following a path from one state to another; this path, often a collection of atomic trajectories, describes a distortion. Here we introduce an antisymmetry operation called distortion reversal that reverses a distortion pathway. The symmetry of a distortion pathway is then uniquely defined by a distortion group; it has the same form as a magnetic group that involves time reversal. Given its isomorphism to magnetic groups, distortion groups could have a commensurate impact in the study of distortions, as the magnetic groups have had in the study of magnetic structures. Distortion symmetry has important implications for a range of phenomena such as structural and electronic phase transitions, diffusion, molecular conformational changes, vibrations, reaction pathways and interface dynamics.

No MeSH data available.


Related in: MedlinePlus