13-04-2013, 04:41 PM
Crystal Structure of Graphite, Graphene and Silicon
ABSTRACT
We analyze graphene and some of the carbon allotropes for which graphene sheets form the
basis. The real-space and reciprocal crystalline structures are analyzed. Theoretical X-ray diffrac-
tion (XRD) patterns are obtained from this analysis and compared with experimental results. We
show that staggered two-dimensional hexagonal lattices of graphite have XRD patterns that differ
significantly from silicon standards.
INTRODUCTION
The wide-variety of carbon allotropes and their associ-
ated physical properties are largely due to the flexibility
of carbon’s valence electrons and resulting dimensionality
of its bonding structures. Amongst carbon-only systems,
two-dimensional hexagonal sheets—graphene—forms of
the basis of other important carbon structures such as
graphite and carbon nanotubes. (:: Say something about
interesting band structure here)
In the following, we will examine the planar lat-
tice structure of graphene and its extension to higher-
dimensional lattice structures, such as hexagonal
graphite. We first analyze the lattice and reciprocal-
space structures of two-dimensional hexagonal lattices of
carbon, and use the resulting structure factors to esti-
mate the x-ray diffraction (XRD) intensities of graphite.
We conclude by comparing its calculated XRD spectra to
experimental spectra of graphene and crystalline silicon.
Atomic form factors
As carbon is the only element present in graphene and
graphite, the atomic form factor is uniform across the
entire crystal, and thus can be factored out when calcu-
lating the structure factor. Thus the atomic form factor
has no effect on the relative intensities of x-ray diffrac-
tion occuring in different planes of graphite. According
to the NIST Physics Laboratory, the atomic form factor
of carbon varies from 6.00 to 6.15 e/atom with incident
radiation ranging from 2 to 433 KeV [2]
ABSTRACT
We analyze graphene and some of the carbon allotropes for which graphene sheets form the
basis. The real-space and reciprocal crystalline structures are analyzed. Theoretical X-ray diffrac-
tion (XRD) patterns are obtained from this analysis and compared with experimental results. We
show that staggered two-dimensional hexagonal lattices of graphite have XRD patterns that differ
significantly from silicon standards.
INTRODUCTION
The wide-variety of carbon allotropes and their associ-
ated physical properties are largely due to the flexibility
of carbon’s valence electrons and resulting dimensionality
of its bonding structures. Amongst carbon-only systems,
two-dimensional hexagonal sheets—graphene—forms of
the basis of other important carbon structures such as
graphite and carbon nanotubes. (:: Say something about
interesting band structure here)
In the following, we will examine the planar lat-
tice structure of graphene and its extension to higher-
dimensional lattice structures, such as hexagonal
graphite. We first analyze the lattice and reciprocal-
space structures of two-dimensional hexagonal lattices of
carbon, and use the resulting structure factors to esti-
mate the x-ray diffraction (XRD) intensities of graphite.
We conclude by comparing its calculated XRD spectra to
experimental spectra of graphene and crystalline silicon.
Atomic form factors
As carbon is the only element present in graphene and
graphite, the atomic form factor is uniform across the
entire crystal, and thus can be factored out when calcu-
lating the structure factor. Thus the atomic form factor
has no effect on the relative intensities of x-ray diffrac-
tion occuring in different planes of graphite. According
to the NIST Physics Laboratory, the atomic form factor
of carbon varies from 6.00 to 6.15 e/atom with incident
radiation ranging from 2 to 433 KeV [2]