# Formation of Bragg Band Gaps in Anisotropic Phononic Crystals Analyzed With the Empty Lattice Model

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## Abstract

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## 1. Introduction

## 2. Empty Lattice Model

#### 2.1. Analytical Model

#### 2.2. Empty Lattice Model for Scalar Waves in an Isotropic Medium

#### 2.3. Empty Lattice Model for Vector Waves in an Isotropic Solid

#### 2.4. Empty Lattice Model for Vector Waves in an Anisotropic Solid

## 3. Formation of Bragg Band Gaps in Phononic Crystals

## 4. Conclusions

## 5. Materials and Methods

#### 5.1. Materials

#### 5.2. Methods

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References and Notes

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**Figure 1.**Graphical construction of the empty lattice model for anisotropic phononic crystals. (

**a**) The unit cell of a two-dimensional square-lattice phononic crystal. When the matrix (A) and the inclusion (B) are composed of the same material, the model reduces to the unit cell of the empty lattice. (

**b**) The dispersion relation for Bloch harmonics in reciprocal space is a set of cones originating from reciprocal lattice points. The vertical axis is the angular frequency. The horizontal plane is the reciprocal space, and the black dots are for reciprocal lattice points labeled as ${\Gamma}_{{\alpha}_{1},{\alpha}_{2}}$ with ${\alpha}_{1}$ and ${\alpha}_{2}$ being arbitrary integers. The central square centered at ${\Gamma}_{0,0}$ (black solid line) is the boundary of the first BZ for the square lattice with high symmetry points X and M. The colored lines centered around each reciprocal lattice points represent the isofrequency curves of an arbitrary anisotropic material. The shape of isofrequency curves is a frequency-scaled version of the slowness curve for the considered bulk wave shown in (

**c**). ψ is the direction angle defined with respect to the ${s}_{1}$ axis.

**Figure 2.**Empty lattice model for out-of-plane polarized waves in epoxy. (

**a**) The slowness curve is a circle in this case. (

**b**) The empty-lattice band structure for the out-of-plane polarized waves of epoxy is shown. Numerical and analytical results are shown with solid circles and lines, respectively. Each branch is labeled by an index corresponding to the reciprocal lattice point from which the generating cone originates. Points S${}_{1}$ and S${}_{2}$ are examples of intersections between branches located inside the first Brillouin zone (BZ). (

**c**) Isofrequency curves are plotted for the first band. The white dashed lines mark the first intersection between cones centered at adjacent reciprocal lattice points.

**Figure 3.**Empty lattice model for in-plane polarized waves in epoxy. (

**a**) The slowness curves for pure shear (dashed line) and for pure longitudinal (solid line) waves are both circles. (

**b**) The empty-lattice band structure for in-plane polarized waves is shown. Numerical and analytical results are shown with solid circles and lines, respectively. The color scale represents the polarization in the propagation direction, from 0 (shear) to 1 (longitudinal). Each dispersion branch is labeled by an index corresponding to the reciprocal lattice point from which the generating cone originates. Indices for longitudinal branches are underlined for clarity. Points I${}_{1}$ and I${}_{2}$ are the first intersections between shear and longitudinal branches. (

**c**) Isofrequency curves are plotted for the second band. The thick dashed lines show the first intersection between shear and longitudinal cones, while the thin dashed lines show the first intersection between pure shear cones.

**Figure 4.**Empty lattice model for in-plane polarized waves in Z-cut rutile with the X$-{15}^{\circ}$ orientation. (

**a**) The three slowness curves are for a pure shear wave (long dashed line), a quasi-shear (QS) wave (short dashed line) and a quasi-longitudinal (QL) wave (solid line). The empty-lattice band structures for (

**b**) out-of-plane and (

**c**) in-plane polarized waves are shown. Numerical and analytical results are shown with solid circles and lines, respectively. The color scale in (

**c**) represents the polarization in the propagation direction, from 0 (shear) to 1 (longitudinal). Each dispersion branch is labeled by an index corresponding to the reciprocal lattice point from which the generating cone originates. Indices for longitudinal branches are underlined for clarity. Points I${}_{1}$, I${}_{2}$, and I${}_{2}^{\prime}$ are intersections between QS and QL branches. (

**d**) Isofrequency curves are plotted for the first band of in-plane waves. The thick dashed lines represent the intersections between adjacent QS cones. (

**e**) Isofrequency curves are plotted for the second band of in-plane waves. The thick dashed lines show the first intersection between QS and QL cones and thin dashed lines show intersections between adjacent QS cones.

**Figure 5.**Empty lattice model for in-plane polarized waves in Z-cut rutile. (

**a**) The three slowness curves are for a pure shear wave (long dashed line), a QS wave (short dashed line) and a QL wave (solid line). (

**b**) Isofrequency curves are plotted for the first band of in-plane waves. The thick dashed lines represent the intersections between adjacent QS cones. (

**c**) Isofrequency curves are plotted for the second band of in-plane waves. The thick dashed lines show the first intersection between QS and QL cones.

**Figure 6.**Band structures for (

**a**) the out-of-plane polarized waves and (

**b**) the in-plane polarized waves of epoxy with air holes ($d/a=0.2$). The color scale represents the polarization amount in the propagation direction, from 0 (shear) to 1 (longitudinal). Solid lines correspond to the empty lattice model.

**Figure 7.**Band structures for (

**a**) the out-of-plane polarized waves and (

**b**) the in-plane polarized waves of Z-cut rutile for the X$-{15}^{\circ}$ orientation with air holes ($d/a=0.2$). The color scale represents the polarization amount in the propagation direction, from 0 (shear) to 1 (longitudinal). Solid lines correspond to the empty lattice model.

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**MDPI and ACS Style**

Wang, Y.-F.; Maznev, A.A.; Laude, V.
Formation of Bragg Band Gaps in Anisotropic Phononic Crystals Analyzed With the Empty Lattice Model. *Crystals* **2016**, *6*, 52.
https://doi.org/10.3390/cryst6050052

**AMA Style**

Wang Y-F, Maznev AA, Laude V.
Formation of Bragg Band Gaps in Anisotropic Phononic Crystals Analyzed With the Empty Lattice Model. *Crystals*. 2016; 6(5):52.
https://doi.org/10.3390/cryst6050052

**Chicago/Turabian Style**

Wang, Yan-Feng, Alexei A. Maznev, and Vincent Laude.
2016. "Formation of Bragg Band Gaps in Anisotropic Phononic Crystals Analyzed With the Empty Lattice Model" *Crystals* 6, no. 5: 52.
https://doi.org/10.3390/cryst6050052