Difference between revisions of "Saxon-Hutner theorem"
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| − | Making use of the one-to-one correspondence between the real localized atomic potentials and | + | {{TEX|auto}} |
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| + | A theorem concerning energy gaps in one-dimensional random alloy models described by the [[Schrödinger equation|Schrödinger equation]] or the [[Dirac equation|Dirac equation]]. It states that forbidden energies that are common to a pure crystal $ A $ | ||
| + | and a pure crystal $ B $( | ||
| + | with the same lattice constant) will always be forbidden energies in any arrangement of the atoms of $ A $ | ||
| + | and $ B $ | ||
| + | in a substitutional solid solution. | ||
| + | |||
| + | Making use of the one-to-one correspondence between the real localized atomic potentials and $ ( 2 \times 2 ) $- | ||
| + | transfer matrices belonging to any of the isomorphic three-dimensional Lie groups $ { \mathop{\rm SU} } ( 1,1 ) $, | ||
| + | $ { \mathop{\rm SL} } ( 2, \mathbf R ) $ | ||
| + | or $ { \mathop{\rm Sp} } ( 2, \mathbf R ) $, | ||
| + | this can be described in mathematical terms as follows. If $ A ^ {r _ {1} } B ^ {s _ {1} } \dots A ^ {r _ {k} } B ^ {s _ {k} } $ | ||
| + | is an arbitrary linear chain consisting of two types of atoms $ A $ | ||
| + | and $ B $, | ||
| + | each having representatives $ r _ {i} , s _ {i} \in \mathbf Z ^ {+} $ | ||
| + | in the $ i $ | ||
| + | th period. Then the group nature of the individual transfer matrices $ M _ {A} $ | ||
| + | and $ M _ {B} $ | ||
| + | makes it possible to express the total transfer matrix $ M $ | ||
| + | of the elementary cell as the product $ M _ {B} ^ {s _ {k} } M _ {A} ^ {r _ {k} } \dots M _ {B} ^ {s _ {1} } M _ {A} ^ {r _ {1} } $, | ||
| + | and the forbidden energies for electrons propagating there are given by the condition $ | { { \mathop{\rm tr} } ( M ) } | > 2 $. | ||
In the transfer-matrix approach, the theorem takes the form of the following question: Given that | In the transfer-matrix approach, the theorem takes the form of the following question: Given that | ||
| − | + | $$ | |
| + | \left | { { \mathop{\rm tr} } ( M _ {A} ) } \right | , \left | { { \mathop{\rm tr} } ( M _ {B} ) } \right | > 2, | ||
| + | $$ | ||
| − | is it true that for any arrangement | + | is it true that for any arrangement $ A ^ {r _ {_ 1 } } B ^ {s _ {_ 1 } } \dots A ^ {r _ {k} } B ^ {s _ {k} } $ |
| + | of atoms of type $ A $ | ||
| + | and $ B $ | ||
| + | one has | ||
| − | + | $$ | |
| + | \left | { { \mathop{\rm tr} } ( M _ {B} ^ {s _ {k} } M _ {A} ^ {r _ {k} } \dots M _ {B} ^ {s _ {_ 1 } } M _ {A} ^ {r _ {_ 1 } } ) } \right | > 2 ? | ||
| + | $$ | ||
Relying on quite different techniques, several non-equivalent sufficient conditions guaranteeing its validity have been derived. | Relying on quite different techniques, several non-equivalent sufficient conditions guaranteeing its validity have been derived. | ||
Latest revision as of 08:12, 6 June 2020
A theorem concerning energy gaps in one-dimensional random alloy models described by the Schrödinger equation or the Dirac equation. It states that forbidden energies that are common to a pure crystal $ A $
and a pure crystal $ B $(
with the same lattice constant) will always be forbidden energies in any arrangement of the atoms of $ A $
and $ B $
in a substitutional solid solution.
Making use of the one-to-one correspondence between the real localized atomic potentials and $ ( 2 \times 2 ) $- transfer matrices belonging to any of the isomorphic three-dimensional Lie groups $ { \mathop{\rm SU} } ( 1,1 ) $, $ { \mathop{\rm SL} } ( 2, \mathbf R ) $ or $ { \mathop{\rm Sp} } ( 2, \mathbf R ) $, this can be described in mathematical terms as follows. If $ A ^ {r _ {1} } B ^ {s _ {1} } \dots A ^ {r _ {k} } B ^ {s _ {k} } $ is an arbitrary linear chain consisting of two types of atoms $ A $ and $ B $, each having representatives $ r _ {i} , s _ {i} \in \mathbf Z ^ {+} $ in the $ i $ th period. Then the group nature of the individual transfer matrices $ M _ {A} $ and $ M _ {B} $ makes it possible to express the total transfer matrix $ M $ of the elementary cell as the product $ M _ {B} ^ {s _ {k} } M _ {A} ^ {r _ {k} } \dots M _ {B} ^ {s _ {1} } M _ {A} ^ {r _ {1} } $, and the forbidden energies for electrons propagating there are given by the condition $ | { { \mathop{\rm tr} } ( M ) } | > 2 $.
In the transfer-matrix approach, the theorem takes the form of the following question: Given that
$$ \left | { { \mathop{\rm tr} } ( M _ {A} ) } \right | , \left | { { \mathop{\rm tr} } ( M _ {B} ) } \right | > 2, $$
is it true that for any arrangement $ A ^ {r _ {_ 1 } } B ^ {s _ {_ 1 } } \dots A ^ {r _ {k} } B ^ {s _ {k} } $ of atoms of type $ A $ and $ B $ one has
$$ \left | { { \mathop{\rm tr} } ( M _ {B} ^ {s _ {k} } M _ {A} ^ {r _ {k} } \dots M _ {B} ^ {s _ {_ 1 } } M _ {A} ^ {r _ {_ 1 } } ) } \right | > 2 ? $$
Relying on quite different techniques, several non-equivalent sufficient conditions guaranteeing its validity have been derived.
References
| [a1] | D. Saxon, R. Hutner, Philips Res. Rep. , 4 (1949) pp. 81 |
| [a2] | H. Matsuda, Progr. Theor. Phys. , 27 (1962) pp. 81 |
| [a3] | J. Hori, Progr. Theor. Phys. , 32 (1964) pp. 371 |
| [a4] | L. Dworin, Phys. Rev. , 138 A (1965) pp. 1121 |
| [a5] | B. Tong, S. Tong, Phys. Rev. , 180 (1969) pp. 739 |
| [a6] | I. Mladenov, C.R. Acad. Sci. (Sofia) , 38 (1985) pp. 993 |
| [a7] | I. Mladenov, Phys. Lett. A , 137 (1989) pp. 313 |
Saxon-Hutner theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Saxon-Hutner_theorem&oldid=23003