# Schottky Defects

- Page ID
- 664

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Lattice structures are not perfect; in fact most of the time they experience defects. Lattice structures (or crystals) are prone to defects especially when their temperature is greater than 0 K [1]. One of these defects is known as the Schottky defect, which occurs when oppositely charged ions vacant their sites [1].

## Introduction

Like the human body, lattice structures (most commonly known as crystals) are far from perfection. Our body works hard to keep things proportional but occasionally our right foot is bigger than our left; similarly, crystals may try to arrange it's ions under a strict layout, but occasionally an ion slips to another spot or simply goes missing. Realistically speaking, it should be expected that crystals will depart itself from order (not surprising considering defects occurs at temperature greater than 0 K). There are many ways a crystal can depart itself from order (thus experiences defects); these defects can be grouped in different categories such as Point Defects, Line Defects, Planar Defects, or Volume or Bulk Defects [2]. We will focus on Point Defects, specifically the defect that occurs in ionic crystal structures (i.e. NaCl) called the Schottky Defect.

## Point Defects

Lattice structures (or crystals) undergoing point defects experience one of two types:

- atoms or ions leaving their spot (thus creating vacancies).
- atoms or ions slipping into the little gaps in between other atoms or ions; those little gaps are known as interstitials--since atoms or ions in the crystals are occupying interstitials, they inherently become (create) interstitials.

By the simplest definition, the Schottky defect is defined by type one, while type two defects are known as the Frenkel defect. The Schottky defect is often visually demonstrated using the following layout of anions and cations:

+ | - | + | - | + | - | + | - | + | - | + |

- | + | - | + | - | + | - | + | (vacant) | + | - |

+ | - | + | - | + | - | + | - | + | - | + |

- | + | - | (vacant) | - | + | - | + | - | + | - |

+ | - | + | - | + | - | + | - | + | - | + |

- | + | - | + | - | + | - | + | - | + | - |

**Figure \(\PageIndex{1}\)***: The positive symbols represents cations (i.e. Na ^{+}) and the negative symbol represents anions (i.e. Cl^{-}). *

In addition, this layout is applicable only for ionic crystal compounds of the formula MX--layout for ionic crystals with formula MX_{2} and M_{2}X_{3} will be discussed later--where M is metal and X is nonmetal. Notice the figure has exactly one cation and one anion vacating their sites; that is what defines a (one) Schottky Defect for a crystal of MX formula--for every cation that vacant its site, the same number of anion will follow suit; essentially the vacant sites come in pairs. This also means the crystal will neither be too positive or too negative because the crystal will always be in equilibrium in respect to the number of anions and cations.

It is possible to approximate the number of Schottky defects (n_{s}) in a MX ionic crystal compound by using the equation:

\[N= \exp^{-\dfrac{\Delta H}{2RT}} \label{3} \]

where

- \(\Delta{H}\) is the enthalpy of defect formation,
- \(R\) is the gas constant,
- \(T\) is the absolute temperature (in K), and

N can be calculated by:

\[N = \dfrac{\text{density of the ionic crystal compound} \times N_A}{\text{molar mass of the ionic crystal compound}} \label{4} \]

From Equation \(\ref{3}\), it is also possible to calculate the fraction of vacant sites by using the equation:

\[\dfrac{n_s}{N} = \exp^{-\dfrac{\Delta H}{2RT}} \label{5} \]

## Schottky defects for \(MX_2\) and \(M_2X_3\)

As mentioned earlier, a Schottky defect will always result a crystal structure in equilibrium--where no crystal is going to be too positive or too negative; thus in the case of:

- MX
_{2}: one Schottky defect equals one cation and two anion vacancy. - M
_{2}X_{3}: one Schottky defect equals two cation and three anion vacancy.

## References

- Housecroft, Catherine. Inorganic Chemistry. Pearson Prentice Hall 2008.
- Sólyom, J.(Jenö). Fundamentals of the Physics of Solids. Translated by Attila Piróth. New York : Springer, c2007.
- Tilley, Richard. Understanding Solids. John Wiley & Sons, LTD. 2004.

## Problems

- How does an ionic crystal structure maintain electrical neutrality despite undergoing a Schottky defect?
- How is a Schottky defect defined for a compound with a MX formula? MX
_{2}? M_{2}X_{3}? - Given that the enthalpy of defect formation for LiCl is 3.39 x 10
^{-19}J and the density of LiCl is 2.068 g/cm^{-3}. Calculate the number of Schottky defect at 873 K. - Using the number of Schottky defect solved for question 3, determine the fraction of vacant site for LiCl.
- If a anion and a cation vacant its site and occupies a space between other anions and cations, is it still a Schottky defect?

## Answers

- For a MX compound: one anion and one cation vacant their sites, so the overall charge will remain balanced. This is the same for MX
_{2}and M_{2}X_{3}because appropriate numbers of anions and cations vacant their site thus leaving the overall charge neutral. - MX compound: one Schottky defect is when one anion and one cation leave their sites. MX
_{2}compound: one Schottky defect is when one anion and two cations leave their sites. M_{2}X_{3}is when two anions and three cations leave their sites.