In solid-state physics , the Einstein model (according to Albert Einstein ) describes a method to calculate the contribution of the lattice vibrations ( phonons ) to the heat capacity of a crystalline solid . Since the Einstein model can only be applied to optical phonons , it is not as successful as the Debye model , which describes acoustic phonons .
Basics of the model
The lattice vibrations of the crystal are quantized , i.e. H. the solid can only absorb vibrational energy in discrete quanta . These quanta are also called phonons . The solid is then described as consisting of N quantum harmonic oscillators , each of which can oscillate independently in three directions. The occupation probability of such an oscillation mode (a phonon) depends on the temperature T and follows (since phonons are bosons ) the Bose-Einstein distribution :
ℏ
⋅
ω
E.
{\ displaystyle \ hbar \ cdot \ omega _ {\ mathrm {E}}}
⟨
n
⟩
{\ displaystyle \ langle n \ rangle}
⟨
n
⟩
=
1
exp
(
ℏ
⋅
ω
E.
k
B.
⋅
T
)
-
1
{\ displaystyle \ langle n \ rangle = {\ frac {1} {\ exp \ left ({\ frac {\ hbar \ cdot \ omega _ {\ mathrm {E}}} {k _ {\ mathrm {B}} \ cdot T}} \ right) -1}}}
With
This results in the internal energy U in the solid body (the quantization condition of the harmonic oscillator was used):
U
=
3
N
⋅
ℏ
⋅
ω
E.
⋅
(
⟨
n
⟩
+
1
2
)
=
3
N
⋅
ℏ
⋅
ω
E.
⋅
[
1
exp
(
ℏ
⋅
ω
E.
k
B.
⋅
T
)
-
1
+
1
2
]
{\ displaystyle {\ begin {aligned} U & = 3N \ cdot \ hbar \ cdot \ omega _ {\ mathrm {E}} \ cdot \ left (\ langle n \ rangle + {\ frac {1} {2}} \ right) \\ & = 3N \ cdot \ hbar \ cdot \ omega _ {\ mathrm {E}} \ cdot \ left [{\ frac {1} {\ exp \ left ({\ frac {\ hbar \ cdot \ omega _ {\ mathrm {E}}} {k _ {\ mathrm {B}} \ cdot T}} \ right) -1}} + {\ frac {1} {2}} \ right] \ end {aligned}} }
With
The contribution indicates the zero point energy .
ℏ
⋅
ω
E.
2
{\ displaystyle {\ frac {\ hbar \ cdot \ omega _ {\ mathrm {E}}} {2}}}
The contribution of the phonons to the heat capacity is then:
C.
V
=
(
∂
U
∂
T
)
V
=
c
O
n
s
t
=
3
N
⋅
(
ℏ
⋅
ω
E.
)
2
k
B.
⋅
T
2
⋅
exp
(
ℏ
⋅
ω
E.
k
B.
⋅
T
)
[
exp
(
ℏ
⋅
ω
E.
k
B.
⋅
T
)
-
1
]
2
{\ displaystyle C_ {V} = \ left ({\ frac {\ partial U} {\ partial T}} \ right) _ {V = {\ rm {const}}} = 3N \ cdot {\ frac {(\ hbar \ cdot \ omega _ {\ mathrm {E}}) ^ {2}} {k _ {\ mathrm {B}} \ cdot T ^ {2}}} \ cdot {\ frac {\ exp \ left ({\ frac {\ hbar \ cdot \ omega _ {\ mathrm {E}}} {k _ {\ mathrm {B}} \ cdot T}} \ right)} {\ left [\ exp \ left ({\ frac {\ hbar \ cdot \ omega _ {\ mathrm {E}}} {k _ {\ mathrm {B}} \ cdot T}} \ right) -1 \ right] ^ {2}}}}
With
V
{\ displaystyle V}
: Volume .
The Einstein temperature results in a simpler notation:
Θ
E.
=
ℏ
⋅
ω
E.
k
B.
{\ displaystyle \ Theta _ {\ mathrm {E}} = {\ frac {\ hbar \ cdot \ omega _ {\ mathrm {E}}} {k _ {\ mathrm {B}}}}}
C.
V
(
T
)
=
3
N
⋅
k
B.
⋅
(
Θ
E.
T
)
2
⋅
exp
(
Θ
E.
T
)
[
exp
(
Θ
E.
T
)
-
1
]
2
{\ displaystyle C_ {V} \ left (T \ right) = 3N \ cdot k _ {\ mathrm {B}} \ cdot \ left ({\ frac {\ Theta _ {\ mathrm {E}}} {T}} \ right) ^ {2} \ cdot {\ frac {\ exp \ left ({\ frac {\ Theta _ {\ mathrm {E}}} {T}} \ right)} {\ left [\ exp \ left ( {\ frac {\ Theta _ {\ mathrm {E}}} {T}} \ right) -1 \ right] ^ {2}}}}
Failure at low temperatures
Like the Debye model, the Einstein model provides the correct high temperature limit according to the Dulong-Petit law :
T
→
∞
:
C.
V
→
3
⋅
N
⋅
k
B.
{\ displaystyle T \ rightarrow \ infty: \ \ C_ {V} \ rightarrow 3 \ cdot N \ cdot k _ {\ mathrm {B}}}
In the Limes of lower temperatures, the following results:
T
→
0
:
C.
V
∝
e
-
Θ
E.
/
T
→
0
{\ displaystyle T \ rightarrow 0: \ \ C_ {V} \ propto e ^ {- \ Theta _ {\ mathrm {E}} / T} \ rightarrow 0}
This course of C V (T) for low temperatures deviates considerably from measurements. This is related to the assumption that all harmonic oscillators in the solid state would vibrate with a uniform frequency. However, the conditions in the real solid are much more complicated.
literature
"The Planck theory of radiation and the theory of specific heat", A. Einstein, Annalen der Physik, volume 22, pp. 180–190, 1907. Online
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">