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Symmetries in
Superconductors

By: Hugh Heldenbrand

Yoon
-
Mi Kim

January 15, 2001

Computational Chemistry Seminar

Problem Statement


How can the different symmetries of a crystal be
used to describe it?

Introduction


Superconductors.


State University of New York at Binghamton:
Preparation, Structure and Properties of a High
-
temperature Superconductor
(YBa
2
Cu
3
O
7
).


“Model the system using CAChe software . . .”
(Observations and Questions).


MacSpartan vs. CAChe (demo version)

Computational Method


Our program was CAChe, “the happy union on
computational chemistry and highly sophisticated
graphics.” (
CAChe 4.1 A Chemist’s Guide
)


When we went to enter the space group and
fractional coordinates for YBa
2
Cu
3
O
7

we got an
error message that we thought was related to the
fact that we were using demo version software.


Computational Method


This is the point where the computer would
say “path not found for space groups.”

Point Group Symmetries



There are four main types of symmetry
operations:


Rotation (C
n
)
--
the object appears identical
if rotated about an axis by
a

= 360/
n

= 2
p
/
n

degrees.


Improper Rotation (S
n
)

a combination of
a rotation and a mirror plane reflection
perpendicular to the axis of rotation.


Point Group Symmetries


Inversion
(center of symmetry,
i
)
--
each point in
the object is converted to an identical point by
projecting through a common center and
extending an equal distance beyond this center.


Reflection

(mirror plane,

)
--
each point in the
object is converted to an identical point by
projecting through a mirror plane and extending
an equal distance beyond this plane.



Space
-
Group Symmetries


230 space groups


The International Union of Crystallography
publishes them in Volume A of
International
Tables for Crystallography
.


The Herman
-
Mauguin System


The space group of YBa
2
Cu
3
O
7
is:

P mmm, a= 3.820, b=3.886, c=11.683

Herman
-
Mauguin System


The first letter identifies the centering of the lattice


P = Primitive


I = Body centered


F = Face centered


C = C
-
centered


B = B
-
centered


A = A
-
centered


Our crystal is P, so it has a primitive Bravais lattice (there
are no atoms outside the eight that make up the corners).


Herman
-
Mauguin System


The “mmm” in the space group for YBa
2
Cu
3
O
7
means that it has three “mirror plane” transitions.


A mirror plane is simply a plane through which
the crystal can be reflected identically on both
sides.


Herman
-
Mauguin System


The a= 3.820,
b=3.886, c=11.683 in
the space group gives
the dimensions of the
crystal.


Since a, b, and c are
all different numbers,
the crystal is
orthorhombic.


Database of Superconductors


Here is a website with 3
-
D models of
superconductors, which could be produced with a
program like CAChe.


http://barns.ill.fr/dif/xtal
-
3d.super.html

Conclusion


Why does a crystal’s symmetry matter?

Bibliography


http://www.chem.ox.ac.uk/icl/heyes/structure_of_
solids/Lecture4/Lec4.html


http://imr.chem.binghamton.edu/labs/super/superc.
html


http://barns.ill.fr/dif/xtal
-
3d.super.html


Huheey, James et al.
Inorganic Chemistry:
Principle of Structure and Reactivity
. New York:
Harper Collins, 1993.