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-# elastic-stability-conditions
-Necessary and sufficient elastic stability conditions in various crystal systems
+# escpy: Necessary and sufficient elastic stability conditions in various crystal systems
+
+A crystalline structure is stable, in the absence of external load and in the harmonic approximation, if and only if
+
+1. all its phonon modes have positive frequencies for all wave vectors (dynamical stability);
+2. the elastic energy is always positive. This condition is called the *elastic stability criterion*. As first noted by Born (See Ref. 2), it is mathematically equivalent to the following necessary and **sufficient stability conditions**:
+   1. The matrix $\mathrm{ C }$ is definite positive;
+   2. all eigenvalues of $\mathrm{ C }$ are positive;
+   3. all the leading principal minors of $\mathrm{ C }$ (determinants of its upper-left $k \times k$ submatrix, where $1 \le k \le 6$) are positive, a property known as *Sylvester’s criterion*;
+   4. an arbitrary set of minors of $\mathrm{ C }$ are all positive. It can be useful to choose, for example, the trailing minors, or any other set. 
+
+These are 4 possible formulations of the generic Born elastic stability conditions for an **unstressed** crystal. They are valid regardless of the symmetry of the crystal studied, and are not linear. 
+
+Please check Ref. 1 for more information.
+
+## References
+
+1. Mouhat, F. & Coudert, F.-X. Necessary and sufficient elastic stability conditions in various crystal systems. *Physical Review B* **90,** 224104 (2014).
+2. Born, M. On the stability of crystal lattices. I. *Mathematical Proceedings of the Cambridge Philosophical Society* **36,** 160–172 (1940).
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