# TEV—A Program for the Determination of the Thermal Expansion Tensor from Diffraction Data

^{*}

## Abstract

**:**

_{ij}from diffraction data. Unit cell parameters determined from temperature dependent data collections can be provided as input. An intuitive graphical user interface enables fitting of the evolution of individual lattice parameters to polynomials up to fifth order. Alternatively, polynomial representations obtained from other fitting programs or from the literature can be entered. The polynomials and their derivatives are employed for the calculation of the tensor components of α

_{ij}in the infinitesimal limit. The tensor components, eigenvalues, eigenvectors and their angles with the crystallographic axes can be evaluated for individual temperatures or for temperature ranges. Values of the tensor in directions parallel to either [uvw]’s of the crystal lattice or vectors (hkl) of reciprocal space can be calculated. Finally, the 3-D representation surface for the second rank tensor and pre- or user-defined 2-D sections can be plotted and saved in a bitmap format. TEV is written in JAVA. The distribution contains an EXE-file for Windows users and a system independent JAR-file for running the software under Linux and Mac OS X. The program can be downloaded from the following link: http://www.uibk.ac.at/mineralogie/downloads/TEV.html (Institute of Mineralogy and Petrography, University of Innsbruck, Innsbruck, Austria)

## 1. Introduction

_{ij}can be described by the following relationship:

_{ij}= α

_{ij}ΔT

_{ij}are the thermal expansion coefficients defining a symmetrically second rank tensor and ΔT is the temperature change. Most oxides, for example, have ambient temperature thermal expansion coefficients in the order of 10

^{−6}/K.

_{m}(m >> 6) is described by an appropriate function of temperature. Depending on the symmetry, the maximal six independent tensor components of α

_{ij}can be obtained from solving an over-determined system of m linear equations [2,3]. Alternatively, the changes in the lattice parameters a, b, c, α, β, γ determined at different temperatures can be used directly to relate them to the tensor components α

_{ij}[4,5]. In reference [5], Schlenker et al. presented a mathematical treatment for the general triclinic case in terms of finite changes between two different temperatures T

_{1}and T

_{2}.

_{E}. The resulting expression for the unit cell volume has been transferred to describe the temperature dependence of the individual unit cell edges: r(T) = r

_{0}+ E/(exp(Θ

_{E}/T − 1)), where r

_{0}(value for the particular lattice parameter r at T = 0 K), Θ

_{E}(effective Einstein temperature) and E (Einstein constant) are the fit parameters [7,8,9]. For the description of more complex relationships between r and T “two-term Einstein models” [10,11], “extended Einstein-models” [12,13], Debye-like expressions [14] or combinations between Einstein- and Debye-like functions including anharmonic contributions [15] have been used. Alternatively, a more straightforward description using polynomials has also been successfully employed in many cases to model thermal expansion over large temperature intervals [6,16].

## 2. Results and Discussion

#### 2.1. Mathematical Background

_{ij}) is usually referred to an orthogonalized coordinate system {

**e**,

_{1}**e**,

_{2}**e**}. In general, there is an infinite number of ways in which this reference system could be selected. In TEV, this system was chosen in such a way that it can be derived from the crystallographic basis vectors {

_{3}**a**,

**b**,

**c**} according to the following relations:

**e**is parallel to

_{3}**c**,

**e**is parallel to

_{2}**b***and

**e**=

_{1}**e**×

_{2}**e**(in order to create a right-handed coordinate system). In more detail, these relationships can be expressed as follows:

_{3}^{2}:

_{ij}components can be related to the lattice parameters and their derivatives according to the following mathematical expressions presented by Paufler and Weber [6]:

**q**whose three components are the direction cosines ${q}_{1}$, ${q}_{2}$ and ${q}_{3}$, i.e., the cosines of the angles between the vector

**q**and the three axes of the orthogonalized reference system {

**e**,

_{1}**e**,

_{2}**e**}:

_{3}_{ij}in directions parallel to a crystallographic direction

**t**= u

**a**+ v

**b**+ w

**c**or parallel to a reciprocal lattice vector

**r***= h

**a***+ k

**b***+ l

**c***(perpendicular to a lattice plane with indices (hkl)) are of special importance. Therefore, the direction cosines of these vectors relative to the reference system {

**e**,

_{1}**e**,

_{2}**e**} must be known. In TEV, the necessary transformations are calculated as follows (V and V* are the unit cell volumes of the direct and the reciprocal lattice, respectively):

_{3}**q**$=$(${q}_{1}$, ${q}_{2}$, ${q}_{3}$) one obtains a geometric representation of the tensor in form of a surface in 3-D space. TEV calculates this representation surface and visualizes it as a surface chart. Alternatively, pre- or user-defined 2-D sections can be drawn. The corresponding figures can be exported in bitmap format (PNG).

#### 2.2. Using the Program—General Remarks

_{ij}from

- (i)
- the evaluation of a data set containing a sequence of lattice parameters measured as a function of temperature T;
- (ii)
- the evaluation of an already existing polynomial description of the lattice parameters obtained from another fitting program or from the literature.

**(i)**must be stored in plain-text (ASCII) format. The file can be prepared with a standard text editor. People working with Excel or OpenOffice can simply save the data as character-separated values (CSV).

Triclinic | T_{1}; a_{1}; b_{1}; c_{1}; α_{1}; β_{1}; γ_{1} |

T_{2}; a_{2}; b_{2}; c_{2}; α_{2}; β_{2}; γ_{2} | |

⁞ | |

Monoclinic | T_{1}; a_{1}; b_{1}; c_{1}; oblique angle β_{1} or γ_{1} |

T_{2}; a_{2}; b_{2}; c_{2}; oblique angle β_{2} or γ_{2} | |

⁞ | |

Orthorhombic | T_{1}; a_{1}; b_{1}; c_{1} |

T_{2}; a_{2}; b_{2}; c_{2} | |

⁞ | |

Rhombohedral | T_{1}; a_{1}; α_{1} |

T_{2}; a_{2}; α_{2} | |

⁞ | |

Hexagonal/Tetragonal | T_{1}; a_{1}; c_{1} |

T_{2}; a_{2}; c_{2} | |

⁞ | |

Cubic | T_{1}; a_{1} |

T_{2}; a_{2} | |

⁞ |

**(ii)**these values must be separated by a vertical bar “|” (also referred to as the “pipe” symbol), i.e., the coefficients of the function $a\left(T\right)={p}_{0}+{p}_{1}\cdot T+{p}_{2}\cdot {T}^{2}+{p}_{3}\cdot {T}^{3}$ must be provided as ${p}_{0}\left|{p}_{1}\right|{p}_{2}|{p}_{3}$.

**(i)**, three data sets (triclinicESU.crs, monoclinic.crs and hexagonal.crs) are available. For testing of option

**(ii)**, existing polynomials from the literature have been implemented as default values for the triclinic and monoclinic case. The triclinic data refer to the values given in the paper of Paufler and Weber [6] and can be used for direct comparison of the results.

#### 2.3. Examples

_{2}Ca

_{3}(SO

_{4})

_{3}F [17] (point group 6/m). In the range between 25 °C and 600 °C both lattice parameters were fitted with second-order polynomials (see Figure 1).

**Figure 1.**Evolution of the (

**a**) a-; and (

**b**) c-lattice parameters for hexagonal Na

_{2}Ca

_{3}(SO

_{4})

_{3}F.

_{11}and α

_{33}has been calculated in steps 25 °C. It is obvious from Figure 2 that at temperatures below ≈ 190 °C the thermal expansion parallel to [001] is smaller than perpendicular to [001]. Above 190 °C, however, this trend is reversed. This observation is also reflected in the comparison between the shapes of the representation surfaces at ambient temperature and 600 °C, for example, which change from an oblate to a prolate form (see Figure 3). For hexagonal symmetry, these surfaces must be rotationally symmetric along the direction of the sixfold rotation axis of the point group.

_{4}(point group $\overline{1}$). The coefficients of the polynomials up to third order for all six metrical parameters have been directly taken from their publication. The following Figure 4 shows the 3-D representation surface of the thermal expansion tensor for 70 K.

**Figure 2.**Temperature dependency of α

_{11}and α

_{33}. The figure has been produced from the numerical output of Thermal Expansion Visualizing (TEV) using the program Gnuplot [19].

**Figure 4.**Representation surface of the thermal expansion tensor for α-CuMoO

_{4}at 70 K. Red parts of the surface indicate directions with negative values of thermal expansion. The crystallographic system

**{a**,

**b**,

**c}**, the orthonormal system

**{e**,

_{1}**e**,

_{2}**e**and the coordinate system of the eigenvectors

_{3}}**{EV1, EV2, EV3}**are indicated as well.

**e**,

_{1}-e_{2}**e**,

_{2}-e_{3}**e**planes) and a section that was explicitly defined by the two crystallographic directions [111] and [120], for example. The latter option can be of special interest for the interpretation of thermal expansion data in terms of the atomic arrangements in a given crystal structures, e.g., anisotropy of thermal expansion within a specific plane defined by layer-like building units. Surface plots and sections can be saved in a bitmap format (PNG).

_{3}-e_{1}**Figure 5.**Three default (

**a**–

**c**) and specific (

**d**) sections through the 3-D representation surface for α-CuMoO

_{4}at 70 K.

**EV1, EV2, EV3**} and the crystallographic axes {

**a**,

**b**,

**c**}, for example. Figure 6 shows the temperature dependency of the angle between eigenvector

**EV2**and the three basis vectors {

**a**,

**b**,

**c**}. While

**<**(

**EV2**,

**b**) and

**<**(

**EV2**,

**c**) show only relatively small changes as a function of T, the angle

**<**(

**EV2**,

**a**) exhibits a strong variation and a pronounced non-linear behavior.

_{4}) taken from Knight [7]. For comparison, data were fitted using an Einstein-function with three variable parameters as well as polynomials of orders two, three and four. In this case fitting was performed with the program Gnuplot [19]. It is obvious that a second order polynomial cannot be used to model the data adequately. However, the inclusion of third and fourth order terms resulted in a fit that is almost indistinguishable from the Einstein function. In summary, one can say that polynomial functions offer a great variability of applications concerning the coverage of different temperature regions.

**Figure 6.**Variation of the angle between eigenvector

**EV2**and

**a**(triangles),

**b**(spheres) and

**c**(rhombs) for the T-range from 20 to 300 K.

**Figure 7.**Comparison between an Einstein-type function and polynomials of order two, three and four for the modeling of the low-temperature behavior of the a lattice parameter of crocoite (PbCrO

_{4}).

#### 2.4. Final Remarks

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Jessen, S.M.; Küppers, H. The precision of thermal-expansion tensors of triclinic and monoclinic crystals. J. Appl. Cryst.
**1991**, 24, 239–242. [Google Scholar] [CrossRef] - Lonappan, M.A. Thermal expansion of boric acid. Proc. Indian Acad. Sci. A
**1955**, 42, 10–21. [Google Scholar] - Haussühl, S. Physical Properties of Crystals—An Introduction; Wiley-VCH: Weinheim, Germany, 2007; pp. 159–164. [Google Scholar]
- Schlenker, J.L.; Gibbs, G.V.; Boisen, M.B. Thermal expansion coefficients for monoclinic crystals: A phenomenological approach. Am. Mineral.
**1975**, 60, 828–833. [Google Scholar] - Schlenker, J.L.; Gibbs, G.V.; Boisen, M.B. Strain-tensor components expressed in terms of lattice parameters. Acta Cryst. A
**1978**, 34, 52–54. [Google Scholar] [CrossRef] - Paufler, P.; Weber, Z. On the determination of linear thermal expansion coefficients of triclinic crystals using X-ray diffraction. Eur. J. Mineral.
**1999**, 11, 721–730. [Google Scholar] [CrossRef] - Knight, K.S. A neutron powder diffraction determination of the thermal expansion tensor of crocoite (PbCrO
_{4}) between 60 K and 290 K. Mineral. Mag.**1996**, 60, 963–972. [Google Scholar] [CrossRef] - Knight, K.S.; Stretton, I.C.; Schofield, P.F. Temperature evolution between 50 K and 230 K of the thermal expansion tensor of gypsum derived from neutron powder diffraction data. Phys. Chem. Minerals
**1999**, 26, 477–483. [Google Scholar] [CrossRef] - Ballirano, P.; Melis, E. Thermal behaviour and kinetics of dehydration of gypsum in air from in situ real-time laboratory parallel-beam X-ray powder diffraction. Phys. Chem. Minerals
**2009**, 7, 391–402. [Google Scholar] [CrossRef] - Knight, K.S. Low temperature thermoelastic and structural properties of LaGaO
_{3}perovskite in the Pbnm phase. J. Solid State Chem.**2012**, 194, 286–296. [Google Scholar] [CrossRef] - Knight, K.S. A high-resolution powder neutron diffraction study of the crystal structure of neighborite (NaMgF
_{3}) between 9 K and 440 K. Am. Mineral.**2014**, 99, 824–838. [Google Scholar] [CrossRef] - Fortes, A.D.; Wood, I.G.; Knight, K.S. The crystal structure and thermal expansion tensor of MgSO
_{4}–11D_{2}O (meridianite) determined by neutron powder diffraction. Phys. Chem. Minerals**2008**, 35, 207–221. [Google Scholar] [CrossRef] - Fortes, A.D.; Suard, E.; Knight, K.S. Negative linear compressibility and massive anisotropic thermal expansion in methanol monohydrate. Science
**2011**, 331, 742–746. [Google Scholar] [CrossRef] [PubMed] - David, W.I.F.; Evans, J.S.O. Parametric Powder Diffraction. In Uniting Electron Crystallography and Powder Diffraction; NATO Science for Peace and Security Series B. Physics and Biophysics; Kolb, U., Shankland, K., Meshi, L., Avilov, A., David, W., Eds.; Springer Science + Business Media: Dordrecht, The Netherlands, 2012; pp. 149–163. [Google Scholar]
- Senyshyn, A.; Boysen, H.; Niewa, R.; Banys, J.; Kinka, M.; Burak, Ya.; Adamiv, V.; Izumi, F.; Chumak, I.; Fuess, H. High-temperature properties of lithium tetraborate Li
_{2}B_{4}O_{7}. J. Phys. D Appl. Phys.**2012**, 45, 1–15. [Google Scholar] [CrossRef] - Fey, Y. Thermal expansion. In Mineral Physics & Crystallography: A Handbook of Physical Constants; Ahrens, T.J., Ed.; American Geophysical Union: Washington, DC, USA, 1995; pp. 29–44. [Google Scholar]
- Botta, C.; Kahlenberg, V.; Hejny, C.; Többens, D.M.; Bykov, M.; van Smaalen, S. Structural investigations, high temperature behaviour and phase transition of Na
_{6}Ca_{4}(SO_{4})_{6}F_{2}. Mineral. Petrol.**2014**, 108, 487–501. [Google Scholar] - Weber, Th.; Harz, M.; Wehner, B.; Zahn, G.; Paufler, P. Thermal expansion of CuMoO
_{4}below room temperature. Z. Kristallogr.**1998**, 213, 210–216. [Google Scholar] [CrossRef] - Williams, T.; Kelley, C.; et al. Gnuplot 4.6.6: An Interactive Plotting Program. Available online: http://gnuplot.info (accessed on 8 October 2014).
- Download Free Java Software. Available online: http://www.java.com/en/download/ (accessed on 9 February 2015).
- Langreiter, T.; Kahlenberg, V. Thermal Expansion Visualizing—A program for the determination of the thermal expansion tensor from diffraction data. Available online: http://www.uibk.ac.at/mineralogie/downloads/TEV.html (accessed on 12 December 2014).

© 2015 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Langreiter, T.; Kahlenberg, V.
TEV—A Program for the Determination of the Thermal Expansion Tensor from Diffraction Data. *Crystals* **2015**, *5*, 143-153.
https://doi.org/10.3390/cryst5010143

**AMA Style**

Langreiter T, Kahlenberg V.
TEV—A Program for the Determination of the Thermal Expansion Tensor from Diffraction Data. *Crystals*. 2015; 5(1):143-153.
https://doi.org/10.3390/cryst5010143

**Chicago/Turabian Style**

Langreiter, Thomas, and Volker Kahlenberg.
2015. "TEV—A Program for the Determination of the Thermal Expansion Tensor from Diffraction Data" *Crystals* 5, no. 1: 143-153.
https://doi.org/10.3390/cryst5010143