Physics 4340: Introduction to
Solid State Physics
Lecture 32:
Electrons in a periodic potential
Electric toothbrush
USE IT. (and floss too.)
Think about the summation:
e
iK R
R
For K R  2p m the sum is still equal to the
number of unit cells. What are the special K?
A)
B)
C)
D)
Vectors of the direct lattice
Vectors of the reciprocal lattice
K = 2p/R
Something else.
Only 14 distinct
styles of direct
lattice in 3-d.
(Bravis and
Frankenheim)
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/bravais.html
Applies equally to
the Recip. Lattice,
which are the F.T.
of direct lattice.
Special wave
vectors of
periodic direct
lattice are also a
lattice, in k-space
http://hyperphysics.phy-astr.gsu.edu/hbase/solids/bravais.html
For the scattering geometry shown,
which is the direction of:
kincident  kscatter
A)
B)
C)
D)
E)
To the right
To the left
Down the board
Up the board
Out of the board
Electrons in a simple 1-d periodic potential
http://ph.qmul.ac.uk/~anthony/spfm/18.html
Think about the summation:
e
iK R
R
For some arbitrary K, and for a large number
of unit cells (going towards infinitely large)
this sum is likely to yield:
A)
B)
C)
D)
E)
Infinity (infinite number of terms)
Zero (equal pos. and neg. contribs)
Always goes to 1
Number atoms in unit cell
Something else
Think about the summation:
e
iK R
R
For some special K, this sum has a simple
limiting result. What are some special K?
A)
B)
C)
D)
I like K = 0
I like K = 2p
I like K = 2p/R
I like something else for special K.
What is the sum in that limit?
Only 5 distinct
styles of lattice
in 2-d.
http://staff.chess.cornell.edu/~smilgies/xrd/IntroductionSurfaces.html
http://www.atpm.com/8.08/periodic-table.shtml
Wigner-Seitz unit cell examples
BCC lattice
FCC lattice
http://what-when-how.com/electronic-properties-of-materials/
energy-bands-in-crystals-fundamentals-of-electron-theory-part-2/