1
Fermi Function
The argument of the Fermi function is
T  T0 
X 
W0
(1.1)
Note that
X
1
(1.2)

T0
W0
And
X
X
(1.3)

W0
W0
In terms of X, the Fermi function is
1
FX  
1  exp  X 
(1.4)
Figure 1 Fermi function with T0 = 460, W0 = 30
Derivative of Fermi Function
F F X

T X T
X
1

T W0
F  X 
X

(2.1)
(2.2)
 exp  X 
1  exp  X  
2
(2.3)
2
F 2 exp  X 
F

T
W0
bli\DFermi1.zip in bli.zip
(2.4)
Figure 2 dF/dT with T0 = 460 and W0 = 30
The peak is both upside down and symmetrical.
Generalized Fermi Functioni
p


1
p
FG  X   
  F X 
1  exp  X  
bli\GFermi1.zip in bli.zip
(3.1)
3
Figure 3 With p = 1.7 the Fermi and Generalized Fermi are slighlty different.
Partials of Generalized Fermi Function
The fitting function used by polystd.wpj with X defined in (1.1) and F defined in (1.4) is
f A  x   h F  X  (4.1)
The adjustable parameters are h, p, T0, W0, where the last two enter through X.
The constants in this function are
h
Cons(1)
W0
Cons(2)
T0
Cons(3)
p
Cons(4)
f A  x 
 Fp
1)
(4.2)
h
p
f A  x 
F X
 p  h  F p 1
X W0
2) W0
 ph F
p 1
exp  X   X / W0
f A  x 
F X
 p  h  F p 1
T0
X T0
3)
 ph F
p 1
(4.3)
exp  X  / W0
(4.4)
4
f A  x 
 exp  p ln F 
 h
p
p
4)
(4.5)
 h exp  p ln F   ln F
In the work in this folder, data was fitted to the derivative of the Fermi function. This data was then used to
find the Fermi function and its standard deviation in polystd.wpj.
Derivative of Generalized Fermi Function


dFG  X 
1
 p

dT
1  exp  X  
dF
 pF p 1
dT
p 1
d
1
dT 1  exp  X  (5.1)
dF
F2
  exp  X 
(5.2)
dT
W0
So that
dFG  X 
F p 1 exp  X 
(5.3)
 p
dT
W0
bli\Dgfermi1.zip in bli.zip
Figure 4 The blue curve is the derivative of the generalized Fermi function with p = 1.7. The new curve is sharper and less
symmetric than the old.
5
Partials of Derivative of Generalized Fermi Function
f A T ; c   bkg  h
dFG  X 
F p 1 exp  X 
 Bkg  h  p
dT
W0
The constants in this function are
h
Cons(1)
W0
Cons(2)
T0
Cons(3)
P
Cons(4)
Bkg
Cons(5)
dfpoly.for
f A  x 
  p  F p 1 exp( X ) / W0
1)
h
2)
3)
(6.1)
(6.2)
f A T ; c 
F p exp  X  F X h  pF p 1 exp  X  X
 h  p  p  1

W0
W0
X W0
W0
W0
h  pF p 1 exp  X 

W02
So that
f A T ; c 
F p 1 exp  X 
 h p
   p  1 F exp  X  X  X  1 (6.4)
W0
W02
f A  x 
exp  X  F X
exp  X  X
 h  p  p  1 F p
 h  pF p 1
T0
W0 X T0
W0 T0
F
X exp  X 


    p  1
 F  ph F p
X
T0 W0


So that
f A  x 
exp  X 
 hp   p  1 F exp  X   1 F p 1
T0
W02
(6.3)
(6.5)
(6.6)
f A  x 
exp  X 
exp  X   exp   p  1 ln F 
  hF p 1
 ph
p
W0
W0
p
4)
 h
 h
5)
exp  X  p 1
F  p exp   p  1 ln F  ln F
W0


exp  X  p 1
F 1  p ln F 
W0
f A  x 
1
Bkg
(6.8)
(6.7)
6
In the work in this folder, data was fitted to the derivative of the Fermi function using nlfit.wpj. This data was
then used to find the Fermi function and its standard deviation in polystd.wpj.
i
It is a pleasure to acknowledge A. E. S. Green as the inspiration for this section. He introduced me to the generalized Fermi function
and its derivative. He has used these for data ranging from nuclear potentials to the salaries of physicist as a function of age. He
provided an original set of data to fit with the Generalized Fermi function and introduced me to the ivorf concept for varying and not
varying parameters.