# Measurement Modulus of Elasticity Related to the Atomic Density of Planes in Unit Cell of Crystal Lattices

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Experiments

## 3. Methods

#### 3.1. X-Ray Diffraction of Compound X

_{ij}) and elastic compliance values (S

_{ij}) are needed.

#### 3.2. Elastic Stiffness Constant and Elastic Compliance

_{hkl}) can be expressed for cubic and hexagonal crystals respectively as Equations (13) and (14).

#### 3.3. Relationship between Young’s Modulus E_{hkl} and Planar Density of Each Diffracted Plane through X-Ray Diffraction

^{2}l

#### 3.4. Modified W–H (USDM Model)

_{ZnO}related to this material such as Ref [46,47]. In another study, the USDM model was used for yttrium oxide (Y

_{2}O

_{3}) [48] and the reported E value did not correspond to other studies such as Ref [49,50]. In this case, the elastic constant values of sodium chloride (NaCl) were measured via resonant-ultrasound spectroscopy similar to Ref [51,52]. The experimental and theoretical elastic constants and elastic compliance constants in previous research and this study are reported in Table 1 and Table 2, respectively.

^{+}and Cl

^{−}is 0.97 and 1.81$\AA $, respectively. The crystal of NaCl is FCC, and the location of the atom of Cl introduces (000) and Na at ($\frac{1}{2}\frac{1}{2}\frac{1}{2}$) position. According to the X’Pert analysis, the lattice parameter has gained 5.640 Å and it is in good agreement with the values reported in Ref [58]. In addition, crystallographic parameters of NaCl resulting from X’Pert are submitted in Table 3.

_{2}− t

_{1}) in the signals and knowing the length of the specimen, the velocity of ultrasound longitudinal and transverse waves can be calculated through Equation (22).

#### 3.5. Modified Williamson–Hall Method (USDM) for NaCl

## 4. Conclusions

- A new method for measuring the accurate value of the modulus of elasticity of crystalline materials is successfully presented.
- Planar density for the area of total atoms/ions in the plane divided by the plane area is responsible for the modulus of elasticity of that plane.
- Modulus of elasticity of each plane (y axis) is plotted against the planar density of that plane (x axis), by the least squares method, to give the Young’s modulus of the materials at the intercept.
- Case study of NaCl proved the accuracy of the new method in this study, in good agreement with the ultrasonic technique.
- The Williamson–Hall method, especially in the uniform stress deformation model (USDM), can be used in this method to minimize errors in the least squares method and yield a proper modulus of elasticity, much more accurate than the average value.
- The restriction is that XRD data for planar density calculations are applicable in the uniform distribution of atoms in the crystal lattice with a unit cell, so the method cannot be used for amorphous materials.
- This method can be applied for research as well as industrial applications.

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**(

**a**) Longitudinal compression ${\mathrm{C}}_{11}$ and transverse expansion ${\mathrm{C}}_{12}$; (

**b**) shear modulus ${\mathrm{C}}_{44}$ [39].

**Figure 7.**Geometry and the situation of involved atoms in diffracted planes (

**a**) (111), (

**b**) (200) and (

**c**) (220).

**Table 1.**Elastic constant values of NaCl extracted by experimental/theoretical literature and this study.

Elastic Constant (C), (Gpa) | Expt. from Ref (Bartels et al.) [53] | Expt. from Ref (Barsch et al.) [54] | Expt. from Ref (Charles et al.) [55] | Theory. from Ref (Anderson et al.) [56] | This Study |
---|---|---|---|---|---|

C_{11} | 48.99 | 49.00 | 50.00 | 49.50 | 49.11 |

C_{12} | 12.57 | 12.60 | 12.70 | 13.20 | 12.26 |

C_{44} | 12.72 | 12.70 | 14.40 | 12.79 | 13.73 |

**Table 2.**Elastic compliance values of NaCl extracted by experimental/theoretical literature and this study.

Elastic Compliances (S), (Gpa) | Expt. by (Bartels et al.) | Expt. by (Barsch et al.) | Expt. by (Charles et al.) | Theory. by (Anderson et al.) | This Study |
---|---|---|---|---|---|

S_{11} | 0.0228 | 0.0228 | 0.0222 | 0.0227 | 0.0226 |

S_{12} | −0.0046 | −0.0046 | −0.0045 | −0.0047 | −0.0045 |

S_{44} | 0.0786 | 0.0787 | 0.0694 | 0.0781 | 0.0728 |

**Table 3.**Crystallographic parameters of the NaCl (FCC) structure resulting from the X’Pert software.

NaCl | |||||
---|---|---|---|---|---|

Crystal System | a ( $\mathbf{\AA}$) | c ( $\mathbf{\AA})$ | Cell Volume $(\mathbf{\AA}{)}^{3}$ | Crystal Density (g/cm^{3}) | Space Group |

FCC | 5.640 | 5.640 | 181.511 | 2.141 | Fm-3m |

Study | Young Modulus (E), (Gpa) in This Method (Intercept Value) |
---|---|

Expt. by (Bartels et al.) | 33.57 |

Expt. by (Barsch et al.) | 33.53 |

Expt. by (Charles et al.) | 37.24 |

Theory. by (Anderson et al.) | 33.85 |

This study | 35.68 |

NaCl | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

2θ (Degree) | Β = FWHM (Degree) | θ (Degree) | cosθ (Degree) | 1/cosθ (Degree) | Ln(1/cosθ) (Degree) | Β = FWHM (Radian) | Ln β (Radian) | 4 sinθ (Degree) | β(Radian).cosθ (Degree) | hkl |

27.01 | 0.70 | 13.50 | 0.97 | 1.03093 | 0.03046 | 0.01218 | −4.40796 | 0.92 | 0.01181 | 111 |

30.91 | 0.90 | 15.45 | 0.96 | 1.04167 | 0.04082 | 0.01566 | −4.15665 | 1.04 | 0.01503 | 200 |

45.08 | 0.89 | 22.54 | 0.92 | 1.08696 | 0.08338 | 0.01549 | −4.16782 | 1.52 | 0.01425 | 220 |

53.70 | 0.91 | 26.85 | 0.89 | 1.1236 | 0.11653 | 0.01583 | −4.1456 | 1.80 | 0.01409 | 311 |

56.79 | 0.90 | 28.39 | 0.87 | 1.14943 | 0.13926 | 0.01566 | −4.15665 | 1.88 | 0.01362 | 222 |

66.98 | 0.80 | 33.49 | 0.83 | 1.20482 | 0.18633 | 0.01392 | −4.27443 | 2.20 | 0.01155 | 400 |

72.99 | 0.10 | 36.49 | 0.80 | 1.25 | 0.22314 | 0.00174 | −6.35387 | 2.36 | 0.00139 | 331 |

76.05 | 0.70 | 38.02 | 0.78 | 1.28205 | 0.24846 | 0.01218 | −4.40796 | 2.44 | 0.0095 | 420 |

83.93 | 0.81 | 41.96 | 0.74 | 1.35135 | 0.30111 | 0.01409 | −4.26201 | 2.64 | 0.01043 | 422 |

92.60 | 0.10 | 46.30 | 0.69 | 1.44928 | 0.37106 | 0.00174 | −6.35387 | 2.88 | 0.0012 | 511 |

Mechanical Properties | |||||
---|---|---|---|---|---|

Study | $\mathbf{\sigma}$(GPa) | $\mathbf{\epsilon}$ | µ ^{a} (GPa) | $\mathbf{\upsilon}$^{b} | B ^{c} (GPa) |

Expt. by (Bartels et al.) | −0.1757 | −0.00523 | 14.91 | 0.24 | 24.71 |

Expt. by (Barsch et al.) | −0.1754 | −0.00523 | 14.90 | 0.24 | 24.73 |

Expt. by (Charles et al.) | −0.1949 | −0.00523 | 16.10 | 0.23 | 25.13 |

Theory. by (Anderson et al.) | −0.1771 | −0.00523 | 14.93 | 0.25 | 25.30 |

This study | −0.1867 | −0.00523 | 15.60 | 0.23 | 24.54 |

_{44}+ C

_{12}− C

_{11}[65]; (a) Shear modulus: $\mu ={\mathrm{C}}_{44}-\frac{1}{5}\mathrm{H}$ [66]; (b) Poisson’s ratio: $\mathsf{\upsilon}$ = $\frac{{\mathrm{C}}_{12-\frac{\mathrm{H}}{5}}}{2\left({\mathrm{C}}_{12}+{\mathrm{C}}_{44}-2\frac{\mathrm{H}}{5}\right)}$ [65]; (c) Bulk modulus: B = $\frac{{\mathrm{C}}_{11}+2{\mathrm{C}}_{12}}{3}$ [67].

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**MDPI and ACS Style**

Rabiei, M.; Palevicius, A.; Dashti, A.; Nasiri, S.; Monshi, A.; Vilkauskas, A.; Janusas, G.
Measurement Modulus of Elasticity Related to the Atomic Density of Planes in Unit Cell of Crystal Lattices. *Materials* **2020**, *13*, 4380.
https://doi.org/10.3390/ma13194380

**AMA Style**

Rabiei M, Palevicius A, Dashti A, Nasiri S, Monshi A, Vilkauskas A, Janusas G.
Measurement Modulus of Elasticity Related to the Atomic Density of Planes in Unit Cell of Crystal Lattices. *Materials*. 2020; 13(19):4380.
https://doi.org/10.3390/ma13194380

**Chicago/Turabian Style**

Rabiei, Marzieh, Arvydas Palevicius, Amir Dashti, Sohrab Nasiri, Ahmad Monshi, Andrius Vilkauskas, and Giedrius Janusas.
2020. "Measurement Modulus of Elasticity Related to the Atomic Density of Planes in Unit Cell of Crystal Lattices" *Materials* 13, no. 19: 4380.
https://doi.org/10.3390/ma13194380