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Solutions to quick questions to 2.1.1:
Essentials of the Free Electron Gas |
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What happens, if you do not choose
U = U0 = 0 but U =
U1 ? |
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The Energy scale for the total energy
E moves up or down by U1 since
U1 · y can be added to
E · y. |
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What does the
sentence "...a plane wave with
amplitude (1/L)3/2 moving in the direction of the
wave vector k"
mean"? Wave vectors, after all, are defined in reciprocal space with a dimension 1/cm. What,
exactly, is their direction in real
space? |
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The velocity vector of a car in real
space has the dimension cm/s - the dimension 1/cm for wave
vectors thus means nothing. The wave vector comes into being by writing the
components of a plane wave as follows |
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| y(xi, t) |
= |
A · sin |
· |
æ
è |
2p xi
li |
– |
w · t |
ö
ø |
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With vectors we get |
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| y(r, t) |
= |
A · sin |
· |
æ
è |
r · k |
– |
w · t |
ö
ø |
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This defines
k and by definition k is a vector in
real space, pointing in the direction of wave propagation. |
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The better question is:
If we know k in reciprocal space (= Fourier transform of
the real space), how can we conclude on the direction in real space? The answer
is. Reciprocal lattice vectors with components
kh.k.l are perpendicular to the lattice plane
in real space with Miller indices (hkl) - the direction in real space is
thus given |
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Recount what you know bout the
spin of an electron. |
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- Everything contained in this
"basic"
module.
- Everything contained in
this module
describing the relation of spin and magnetic moment.
- The catchword "Spintronics" should
also come up in this context.
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Where does the
(1/L)3/2 in the solution come from? What would one
expect for a crystal woth the dimension Lx,
Ly, Lz? |
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From the normalizing condition. The
factor should change from (1/L)3/2 =
(1/L3)1/2 to
(1/V3)1/2. |
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What kind of information is contained
in the wave vector k? |
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- "Number" of solution or state.
- Wave length l = 2l/|k|
- Momentum p =
 k
- Total energy E via dispersion relation (for free electron gas
E µ k2
- Propagation direction pf plane wave with k
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Consider a system with some given
energy levels (including possibly energy continua). Distribute a number
N of classical particles, of Fermions and of Bosons on these
levels. Describe the basic priciples. |
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Fermions = Fermi-Dirac distribution
Bosons = Bose-Einstein distribution (which we don't know so far)
Classical = Boltzmann distribution and an approximation to the two fundamental
ones
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Do it! Check the
link for details to
the Boltzmann distribution. |
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| Fermi-Dirac distribution |
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1 |
| f(E, T)
= |
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| |
exp |
æ
è |
E – EF
kT |
ö
ø |
+ 1 |
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| Bose- Einstein distribution |
| |
1 |
| f(E, T)
= |
|
| |
exp |
æ
è |
E – EF
kT |
ö
ø |
– 1 |
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| Boltzmann distribution |
| f(E, T) =
exp– |
æ
è |
E
kT |
ö
ø |
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How does on always derive the density of states
D(E)? |
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Volume of "onion skin" in phase
space. |
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Compare the free electron model with and withour
diffraction. |
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Free
elektron gas |
Free elektron gas
with diffraction |
Potential
V(x,y,z) |
V = const = 0 |
Vx =
V0 · cos (2px/a1)
Vy = V0 · cos (2py/a2)
Vz = V0 · cos (2pz/a3)
V0 ® 0 |
Wavefunktion
y(x,y,z) |
| y = |
æ
ç
è |
1
L |
ö
÷
ø |
3/2 |
· eikr |
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| y = |
æ
ç
è |
1
L |
ö
÷
ø |
3/2 |
· eikr |
|
except for wavevectors
kB that are being diffracted. |
Wave vectors
k |
| kx = ± nx · 2p / L |
| ky = ± ny
· 2p / L |
| kz = ± nz
· 2p / L |
| |
| ni = 0, ±1, ±2, ... |
|
| kx = ± nx · 2p / L |
| ky = ± ny
· 2p / L |
| kz = ± nz
· 2p / L |
| |
| ni = 0, ±1, ±2, ... |
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| Energy E |
Total energy = const =
Ekin |
Total energy = const =
Ekin
except for wavevectors
kB that are being diffracted; then some potential
energy comes into play. |
Dispersion function
E(k) |
| E = |
2k2
2m |
|
| E = |
2k2
2m |
except for wavevectors
kB that are being diffracted. |
Density
of states
D(E) |
| D(E) = |
(2me)3/2
23p2 |
E1/2 |
|
| D(E) = |
(2me)3/2
23p2 |
E1/2 |
as a first approximation., Could be rather different, however. |
Probability of
state being
occupied
f(E,T) |
f(E,
T) =
|
1
|
| exp |
æ
è |
Ei –
EF
kT |
ö
ø |
+ 1 |
|
f(E,
T) =
|
1
|
| exp |
æ
è |
Ei –
EF
kT |
ö
ø |
+ 1 |
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| the Fermi distributoin
always obtains! |
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© H. Föll
Zum Index
2.1.1 Das Bindungspotential