In statistical physics , the Fermi-Dirac integral (after Enrico Fermi and Paul Dirac ), with index, is defined as
j
{\ displaystyle j}
F.
j
(
x
)
=
1
Γ
(
j
+
1
)
∫
0
∞
t
j
exp
(
t
-
x
)
+
1
d
t
{\ displaystyle F_ {j} (x) = {\ frac {1} {\ Gamma (j + 1)}} \ int _ {0} ^ {\ infty} {\ frac {t ^ {j}} {\ exp (tx) +1}} \, dt}
where is the gamma function . The lower limit of the integral is given as an argument of the function
Γ
(
⋅
)
{\ displaystyle \ Gamma (\ cdot)}
F.
j
(
x
,
b
)
=
1
Γ
(
j
+
1
)
∫
b
∞
t
j
exp
(
t
-
x
)
+
1
d
t
{\ displaystyle F_ {j} (x, b) = {\ frac {1} {\ Gamma (j + 1)}} \ int _ {b} ^ {\ infty} {\ frac {t ^ {j}} {\ exp (tx) +1}} \, dt}
then one speaks of the incomplete Fermi-Dirac integral .
Application for F 1/2
The function occurs, among other things, in solid-state physics in connection with the distribution of electrons in the crystal lattice. Often the integral has to be calculated there (see: density of states ). Substitute for the second equal sign as well so that :
F.
1
/
2
(
x
)
{\ displaystyle F_ {1/2} (x)}
t
: =
E.
-
E.
c
k
T
{\ displaystyle t: = {\ tfrac {E-E_ {c}} {kT}}}
x
: =
μ
-
E.
c
k
T
{\ displaystyle x: = {\ tfrac {\ mu -E_ {c}} {kT}}}
d
E.
=
k
T
d
t
{\ displaystyle \ mathrm {d} E = kT \, \ mathrm {d} t}
n
=
N
∫
E.
c
∞
E.
-
E.
c
exp
(
E.
-
μ
k
T
)
+
1
d
E.
=
N
(
k
T
)
3
2
π
2
2
π
∫
0
∞
t
exp
(
t
-
x
)
+
1
d
t
=
N
(
k
T
)
3
2
π
2
F.
1
/
2
(
x
)
{\ displaystyle n = N \ int _ {E_ {c}} ^ {\ infty} {\ frac {\ sqrt {E-E_ {c}}} {\ exp \ left ({\ frac {E- \ mu} {kT}} \ right) +1}} \, \ mathrm {d} E = N \ left (kT \ right) ^ {\ frac {3} {2}} {\ frac {\ sqrt {\ pi}} {2}} {\ frac {2} {\ sqrt {\ pi}}} \ int _ {0} ^ {\ infty} {\ frac {\ sqrt {t}} {\ exp \ left (tx \ right) +1}} \, \ mathrm {d} t = N \ left (kT \ right) ^ {\ frac {3} {2}} {\ frac {\ sqrt {\ pi}} {2}} F_ {1 / 2} (x)}
Approximation for F 1/2
The integral can be solved approximately for different value ranges of :
F.
1
/
2
(
x
)
{\ displaystyle F_ {1/2} (x)}
x
{\ displaystyle x}
F.
~
1
/
2
(
x
)
=
{
1
e
-
x
+
0.27
if
-
∞
<
x
<
1.3
4th
3
π
(
x
2
+
π
2
6th
)
3
/
4th
if
1.3
≤
x
<
∞
{\ displaystyle {\ tilde {F}} _ {1/2} (x) = {\ begin {cases} {\ frac {1} {e ^ {- x} +0.27}} & {\ text {if} } \ - \ infty <x <1.3 \\ {\ frac {4} {3 {\ sqrt {\ pi}}}} \ left (x ^ {2} + {\ frac {\ pi ^ {2}} { 6}} \ right) ^ {3/4} & {\ text {if}} \ \, 1.3 \ leq x <\ infty \ end {cases}}}
The relative error of this approximate solution is a maximum of 3% (maximum deviation at and at ). For a large distance from the origin, two functions can be used to approximate:
(
F.
~
1
/
2
(
x
)
-
F.
1
/
2
(
x
)
)
/
F.
1
/
2
(
x
)
{\ displaystyle \ left ({\ tilde {F}} _ {1/2} (x) -F_ {1/2} (x) \ right) / F_ {1/2} (x)}
x
=
0
{\ displaystyle x = 0}
x
=
1.3
{\ displaystyle x = 1.3}
F.
1
/
2
(
x
)
{\ displaystyle F_ {1/2} (x)}
F.
1
/
2
(
x
)
≈
e
x
{\ displaystyle F_ {1/2} (x) \ approx e ^ {x}}
For
-
x
≫
1
{\ displaystyle -x \ gg 1}
F.
1
/
2
(
x
)
≈
4th
3
π
x
3
/
2
{\ displaystyle F_ {1/2} (x) \ approx {\ frac {4} {3 {\ sqrt {\ pi}}}} x ^ {3/2}}
For
x
≫
1
{\ displaystyle x \ gg 1}
Representation with polylogarithms
Using the polylogarithm , the Fermi-Dirac integral can be represented as
F.
j
(
x
)
=
-
L.
i
j
+
1
(
-
e
x
)
{\ displaystyle \ mathrm {F} _ {j} (x) = - \ mathrm {Li} _ {j + 1} (- e ^ {x})}
.
Because of
d
d
x
L.
i
n
(
x
)
=
1
x
L.
i
n
-
1
(
x
)
{\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ mathrm {Li} _ {n} (x) = {\ frac {1} {x}} \ mathrm {Li} _ {n-1} (x)}
follows from this
d
d
x
F.
j
(
x
)
=
F.
j
-
1
(
x
)
{\ displaystyle {\ frac {\ mathrm {d}} {\ mathrm {d} x}} \ mathrm {F} _ {j} (x) = \ mathrm {F} _ {j-1} (x)}
.
Web links
literature
JS Blakemore: Approximations for Fermi-Dirac Integrals . Solid-State Electronics, 25 (11): 1067-1076, 1982.
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">