Theory of Band Structures in Crystals
The ansatz is the special case of electron waves in a periodic crystal lattice using Bloch waves as treated generally in the dynamical theory of diffraction. Every crystal is a periodic structure which can be characterized by a Bravais lattice, and for each Bravais lattice we can determine the reciprocal lattice, which encapsulates the periodicity in a set of three reciprocal lattice vectors (b1,b2,b3). Now, any periodic potential V(r) which shares the same periodicity as the direct lattice can be expanded out as a Fourier series whose only non-vanishing components are those associated with the reciprocal lattice vectors. So the expansion can be written as:
where K = m1b1 + m2b2 + m3b3 for any set of integers (m1,m2,m3).
From this theory, an attempt can be made to predict the band structure of a particular material, however most ab initio methods for electronic structure calculations fail to predict the observed band gap.
Read more about this topic: Electronic Band Structure
Famous quotes containing the words theory of, theory, band, structures and/or crystals:
“[Anarchism] is the philosophy of the sovereignty of the individual. It is the theory of social harmony. It is the great, surging, living truth that is reconstructing the world, and that will usher in the Dawn.”
—Emma Goldman (1869–1940)
“Freud was a hero. He descended to the “Underworld” and met there stark terrors. He carried with him his theory as a Medusa’s head which turned these terrors to stone.”
—R.D. (Ronald David)
“The band waked me with a serenade. How they improve! A fine band and what a life in a regiment! Their music is better than food and clothing to give spirit to the men.”
—Rutherford Birchard Hayes (1822–1893)
“The philosopher believes that the value of his philosophy lies in its totality, in its structure: posterity discovers it in the stones with which he built and with which other structures are subsequently built that are frequently better—and so, in the fact that that structure can be demolished and yet still possess value as material.”
—Friedrich Nietzsche (1844–1900)
“It is clear that everybody interested in science must be interested in world 3 objects. A physical scientist, to start with, may be interested mainly in world 1 objects—say crystals and X-rays. But very soon he must realize how much depends on our interpretation of the facts, that is, on our theories, and so on world 3 objects. Similarly, a historian of science, or a philosopher interested in science must be largely a student of world 3 objects.”
—Karl Popper (1902–1994)