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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100501.png" /> and a pure crystal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100502.png" /> (with the same lattice constant) will always be forbidden energies in any arrangement of the atoms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100504.png" /> in a substitutional solid solution.
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Making use of the one-to-one correspondence between the real localized atomic potentials and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100505.png" />-transfer matrices belonging to any of the isomorphic three-dimensional Lie groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100507.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100508.png" />, this can be described in mathematical terms as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100509.png" /> is an arbitrary linear chain consisting of two types of atoms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005011.png" />, each having representatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005012.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005013.png" />th period. Then the group nature of the individual transfer matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005015.png" /> makes it possible to express the total transfer matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005016.png" /> of the elementary cell as the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005017.png" />, and the forbidden energies for electrons propagating there are given by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005018.png" />.
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005019.png" /></td> </tr></table>
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$$
 +
\left | { { \mathop{\rm tr} } ( M _ {A} ) } \right | , \left | { { \mathop{\rm tr} } ( M _ {B} ) } \right | > 2,
 +
$$
  
is it true that for any arrangement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005020.png" /> of atoms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005022.png" /> one has
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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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005023.png" /></td> </tr></table>
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$$
 +
\left | { { \mathop{\rm tr} } ( M _ {B} ^ {s _ {k} } M _ {A} ^ {r _ {k} } \dots M _ {B} ^ {s _ {_ 1 }  } M _ {A} ^ {r _ {_ 1 }  } ) } \right | > 2  ?
 +
$$
  
 
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
How to Cite This Entry:
Saxon-Hutner theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Saxon-Hutner_theorem&oldid=23003
This article was adapted from an original article by I.M. Mladenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article