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Interpolation formulas for Fermi-Dirac functions
SHORT COMMUNICATIONS
INTERPOLATION FORMULAS FOR FERMI-DIRAC FUNCTIONS* and L. V. KUZ’MINA
N. N. KALITKIN
Moscow (Received 29 April 1974) INTERPOLATION
formulas are constructed
which express some Fermi-Dirac
of others. In the whole of the permissible range of application ensure a relative accuracy of up to lo- 6 -lo1. The Fermi-Dirac
they arise in quantum
functions
in terms
of the argument the formulas
*.
functions
are defmed as follows: c0 nR dn II,(X) = s 1 + exp(q-x) ’ 0
mechanical
k>-1,
problems in the study of a system of fermions (for example,
electrons in an atom, metal, or plasma) for averaging various degrees of impulse over the Fermi-Dirac distribution. In these problems the index assumes integral and half-integral values. Using the relations Ik’(.r) =kIk-! (s), it is possible to redefine the Fermi-Dirac functions for all non-integral kc - 1. The properties Fermi-Dirac functions and detailed tables for the functions most used are given in [l-5] .
of the
In the problems of the quantum-statistical model of the atom not only these, but also other functions, various combinations of them and more complex expressions are required:
f3/2(x) x = In
4
(2)
I
l+aly2+azy’+a3y6+a,y8+a,y’0
gTyz
l+b,yz
(3)
I::’ (5) I’;>’(x)
(4)
[I,,:” (5) 1IJl
(9
I
*Zh. vjchisl. Mat. mat. Fiz., 15, 3, 768-771, 1975.
213
214
N. N. Kalitkin and L. V. Kuz’mina
only a few of them have been tabulated. Moreoi; the tables are unsuitable for the numerical solution of problems by computer. Therefore it is desirable to have an economical algorithm for computing the functions. 2. In some cases expressions for the required functions in terms of the function (11, directly connected with the electron density, are convenient:
We have to approximate the functions over the infinite interval - co
akekX,
23 -00,
c k=O m
/jkX_8k/3,
f .w @se
X+.m.
c
k=O
If we take as argument y= [ 3/21~~r (x) 1% the asymptotic form will appear as follows: f-ayA(14aiy2+.
. .),
as
x+--m, Y-4
(lOa>
(lob) We can search for an approximation as a fractional-rational function of the following form:
215
Short communications
Then the investigation summation
of the asymptotic
in the numerator
behaviour imposes additional
and denominator
constraints
on the limits of
and on the ratio of the principal coefficients
of the
sums (that is, either highest order or lowest order coefficients). Such an approximating
function
cannot satisfy both asymptotic behaviours at the same time, m into at least two and to choose its own
that is, it is necessary to subdivide the interval --n
in each. Moreover, since the structures of the approximating
are not completely
identical,
and original functions
the best accuracy requires rather a lot of free parameters.
in [S] to find an approximation
For example,
formula for (2) it was necessary to subdivide the region of definition
into three intervals, and to obtain an accuracy - lo-*
to take up to 10 variable coefficients
in each
interval. Therefore, taking into account this example, we will require that the approximating should have in the interval of approximation
an expansion
outside it should have the correct asymptotic
behaviour.
close to the original function,
function and
Such are, for example, the following
functions:
z = a,,yk
2 =
(lla)
a,y’
y
p=o
(1lb)
p=0
TABLE 1
k
a1
-
0.50625059+0
;
1 i
4
1
0.93656,560-l0.74482877-20.21519656-3 0.12159475+0 0.49155758-2 0.21224834+1 0.82099570+0 0.75814482-l 0.10527238+0 0.65348667-2 0.41869012+2 0.35723113+2 0.10230906+2 0.30565347+0 0.18557766-I 0.12266054+0 -0.57855355-l-0.27446412-2
I’ABLE 2
(3)
0.2396@388+0
0.25351970-l 0.16001700-2 0.6'234299-4 0.19780713-5
216
N. N. Kalitkin and L. V. Kuz’mina
TABLE 3 -
-
-I
FUIlCtiOl”
ii
3.75 3.52 3.33 3.60 3.52 6.82 3.02 3.02
Q,bE3KC
-I
-I
EoOL" 2.10-t
aa
k
br
b2
0.33951152-2
0.29736031+0 0138095321-1 0.14333481-2 0.12298369$-O0.15638204-l 0.74910583-l 0.63190618-20.38103053-3 0.41073189+2 0.27484776+-l0.37645845+0 0.25542452+0 0.33350389-l0.16203847-2 0.83750130+0 -0.49086260-l-0.10565505-l 0.25565924+0
1: ‘,;I:
7'10-6
-
0.10730909+0 0.91947157+0 0.17236824$-l 0.50426376+0
3.10-S 3.10-7
Y--
I
-
-
-0.26354443-l
-
TABLE 4 -
-
-I
ai
-
a4
a3
0.21851397+5 0.18917921+4 0.57806497+2 0.58432930+5 0.11424270+4 -0.86438821+1 0.77986304+~ -0.60370205+3 0.11593822-j-4 -0.55280085+1 0.16998461-i-4 0.12572393/4-0.37259280+2 -0.53567595-j-4 0.71234954+7 0.124iSO53+5 -0.14531154+4 0.52161098+3 0.65956384+1 0.72771980+3 -0.117543601-40.32688108+3-0.22333565+1
-
4 -
ii
%aKc
-
TABLE 5 $2
-
-
$3
I
B4
i 0.11348583+4 0.41356686+2 3.75 . 7 0.21793023+5 0.40308111+40.11313165+4-0.53318459+1 3.52 3.33 3.10-e &11914686+5 0.12573112+40.14403693+2 0.18311342+6 0.20011738+50.17382306+4 0.43573326+1 3.60 0.~~44+~ 0.11315848+4-0.32377772-j-2 3.52 f::F"6 6.82 3.10-6 -0.75010872+9 0.14486184+90.74861861+7 0.12453952+5 3.02 6.10-' 0.13113651+4 0.56602247+30.10547260+2 0.33~9903+3 0.10575554+1 3.02 2.10-7
EC6
Especially convenient are Y=‘/~ and p==‘,$.Theprincipal coefficients and the limits of summation are determined by means of the expansion (10). As a rule we find form (1Oa) ao, k and 1, from (lob) we determine o. and (m - n). For example, for (2) we have to put a$)=$
k=2,
1=2,
a0=2/5r
(m-r&)=2.
We make the following interesting remark: if for (2) we put n=O and choose ao and Q2 in a) correspondingly, then the formula will simultaneously yield the principal terms of the expansions (10) and satisfactorily approximate the original function in the entire domain of definition. For ai=0.481,therefore, we arrive at the formula
(11
Short communications
217
which has an error of not more than 2% for any values of the argument. this formula is convenient
Because of its simplicity
for physical estimates.
3. Taking into account the nature of the subsequent
use of the approximation
advisable to assume the following criterion of best approximation:
a function
formulas? it is
of the given class
Z(J, ao, at, . . . . u,) is the best approximation to F(X), if the parameters a~, al, . . . , a, are so chosen that @=maxl z/F-l I is minimal. In other words, our purpose is to obtain formulas with a small relative error. Obviously, the parameters
of formulas (11) are not only the coefficients a, b, (11and 0, but
also the quantity 7. The limits of summation determine
the number of the coefficients
Preliminary
calculations
cannot be regarded as free parameters,
and thereby the class of approximating
have shown that the addition
since they
functions.
of each extra pair of free coefficients
increases the accuracy of the formulas by a factor of 10 approximately
(from the form of the
formula it follows that the number of coefficients can be changed only by an even number). Good accuracy is ensured by approximately six free coefficients. This number was in fact chosen for further calculations. The coefficients
were found by an interpolation
process. We did not succeed in proving its
convergence theoretically, but in practice the process always converged, although fairly slowly. We will not describe this process since it is not of independent mathematical interest. The actual coefficients
of the formulas, and also their accuracy are given in Tables l-5
(Tables 1,3 for (1 la) with v=‘/s, Table 2 for (3), Tables 4,5 for (1 lb) with PL=~/~). Translated by J. Berry REFERENCES 1.
MCDOUGALL, J. and STONER, E. S., The computation of Fermi-Dirac functions. Philos. Trans. Roy. Sot. London,
A237,67-104,1939.
2.
RHODES, P., Fermi-Dirac functions of integral order. Proc. Roy. Sot., A204, No. 1078, 396405, 1950.
3.
BEER, A. C., CHASE, M. N. and CHOQUARD, P. F. Extension of McDougall-Stoner tables of the FermiDirac functions. Helv. phys. acta, 28,529-542, 1955.
4.
LATTER, R., Temperature behaviour of the Thomas-Fermi statistical model for atoms. Phys. Rev., 99, 1854-1870, 1955.
5.
CODY, W. J. and THACHER, H. C. Jr., Rational Chebyshev approximation orders -%, % and 312. Math. Comput.. 21,97,30-40, 1967.
for Fermi-Dirac integrals of
INTERPOLATION FORMULAS FOR FERMI-DIRAC FUNCTIONS* and L. V. KUZ’MINA
N. N. KALITKIN
Moscow (Received 29 April 1974) INTERPOLATION
formulas are constructed
which express some Fermi-Dirac
of others. In the whole of the permissible range of application ensure a relative accuracy of up to lo- 6 -lo1. The Fermi-Dirac
they arise in quantum
functions
in terms
of the argument the formulas
*.
functions
are defmed as follows: c0 nR dn II,(X) = s 1 + exp(q-x) ’ 0
mechanical
k>-1,
problems in the study of a system of fermions (for example,
electrons in an atom, metal, or plasma) for averaging various degrees of impulse over the Fermi-Dirac distribution. In these problems the index assumes integral and half-integral values. Using the relations Ik’(.r) =kIk-! (s), it is possible to redefine the Fermi-Dirac functions for all non-integral kc - 1. The properties Fermi-Dirac functions and detailed tables for the functions most used are given in [l-5] .
of the
In the problems of the quantum-statistical model of the atom not only these, but also other functions, various combinations of them and more complex expressions are required:
f3/2(x) x = In
4
(2)
I
l+aly2+azy’+a3y6+a,y8+a,y’0
gTyz
l+b,yz
(3)
I::’ (5) I’;>’(x)
(4)
[I,,:” (5) 1IJl
(9
I
*Zh. vjchisl. Mat. mat. Fiz., 15, 3, 768-771, 1975.
213
214
N. N. Kalitkin and L. V. Kuz’mina
only a few of them have been tabulated. Moreoi; the tables are unsuitable for the numerical solution of problems by computer. Therefore it is desirable to have an economical algorithm for computing the functions. 2. In some cases expressions for the required functions in terms of the function (11, directly connected with the electron density, are convenient:
We have to approximate the functions over the infinite interval - co
akekX,
23 -00,
c k=O m
/jkX_8k/3,
f .w @se
X+.m.
c
k=O
If we take as argument y= [ 3/21~~r (x) 1% the asymptotic form will appear as follows: f-ayA(14aiy2+.
. .),
as
x+--m, Y-4
(lOa>
(lob) We can search for an approximation as a fractional-rational function of the following form:
215
Short communications
Then the investigation summation
of the asymptotic
in the numerator
behaviour imposes additional
and denominator
constraints
on the limits of
and on the ratio of the principal coefficients
of the
sums (that is, either highest order or lowest order coefficients). Such an approximating
function
cannot satisfy both asymptotic behaviours at the same time, m into at least two and to choose its own
that is, it is necessary to subdivide the interval --n
in each. Moreover, since the structures of the approximating
are not completely
identical,
and original functions
the best accuracy requires rather a lot of free parameters.
in [S] to find an approximation
For example,
formula for (2) it was necessary to subdivide the region of definition
into three intervals, and to obtain an accuracy - lo-*
to take up to 10 variable coefficients
in each
interval. Therefore, taking into account this example, we will require that the approximating should have in the interval of approximation
an expansion
outside it should have the correct asymptotic
behaviour.
close to the original function,
function and
Such are, for example, the following
functions:
z = a,,yk
2 =
(lla)
a,y’
y
p=o
(1lb)
p=0
TABLE 1
k
a1
-
0.50625059+0
;
1 i
4
1
0.93656,560-l0.74482877-20.21519656-3 0.12159475+0 0.49155758-2 0.21224834+1 0.82099570+0 0.75814482-l 0.10527238+0 0.65348667-2 0.41869012+2 0.35723113+2 0.10230906+2 0.30565347+0 0.18557766-I 0.12266054+0 -0.57855355-l-0.27446412-2
I’ABLE 2
(3)
0.2396@388+0
0.25351970-l 0.16001700-2 0.6'234299-4 0.19780713-5
216
N. N. Kalitkin and L. V. Kuz’mina
TABLE 3 -
-
-I
FUIlCtiOl”
ii
3.75 3.52 3.33 3.60 3.52 6.82 3.02 3.02
Q,bE3KC
-I
-I
EoOL" 2.10-t
aa
k
br
b2
0.33951152-2
0.29736031+0 0138095321-1 0.14333481-2 0.12298369$-O0.15638204-l 0.74910583-l 0.63190618-20.38103053-3 0.41073189+2 0.27484776+-l0.37645845+0 0.25542452+0 0.33350389-l0.16203847-2 0.83750130+0 -0.49086260-l-0.10565505-l 0.25565924+0
1: ‘,;I:
7'10-6
-
0.10730909+0 0.91947157+0 0.17236824$-l 0.50426376+0
3.10-S 3.10-7
Y--
I
-
-
-0.26354443-l
-
TABLE 4 -
-
-I
ai
-
a4
a3
0.21851397+5 0.18917921+4 0.57806497+2 0.58432930+5 0.11424270+4 -0.86438821+1 0.77986304+~ -0.60370205+3 0.11593822-j-4 -0.55280085+1 0.16998461-i-4 0.12572393/4-0.37259280+2 -0.53567595-j-4 0.71234954+7 0.124iSO53+5 -0.14531154+4 0.52161098+3 0.65956384+1 0.72771980+3 -0.117543601-40.32688108+3-0.22333565+1
-
4 -
ii
%aKc
-
TABLE 5 $2
-
-
$3
I
B4
i 0.11348583+4 0.41356686+2 3.75 . 7 0.21793023+5 0.40308111+40.11313165+4-0.53318459+1 3.52 3.33 3.10-e &11914686+5 0.12573112+40.14403693+2 0.18311342+6 0.20011738+50.17382306+4 0.43573326+1 3.60 0.~~44+~ 0.11315848+4-0.32377772-j-2 3.52 f::F"6 6.82 3.10-6 -0.75010872+9 0.14486184+90.74861861+7 0.12453952+5 3.02 6.10-' 0.13113651+4 0.56602247+30.10547260+2 0.33~9903+3 0.10575554+1 3.02 2.10-7
EC6
Especially convenient are Y=‘/~ and p==‘,$.Theprincipal coefficients and the limits of summation are determined by means of the expansion (10). As a rule we find form (1Oa) ao, k and 1, from (lob) we determine o. and (m - n). For example, for (2) we have to put a$)=$
k=2,
1=2,
a0=2/5r
(m-r&)=2.
We make the following interesting remark: if for (2) we put n=O and choose ao and Q2 in a) correspondingly, then the formula will simultaneously yield the principal terms of the expansions (10) and satisfactorily approximate the original function in the entire domain of definition. For ai=0.481,therefore, we arrive at the formula
(11
Short communications
217
which has an error of not more than 2% for any values of the argument. this formula is convenient
Because of its simplicity
for physical estimates.
3. Taking into account the nature of the subsequent
use of the approximation
advisable to assume the following criterion of best approximation:
a function
formulas? it is
of the given class
Z(J, ao, at, . . . . u,) is the best approximation to F(X), if the parameters a~, al, . . . , a, are so chosen that @=maxl z/F-l I is minimal. In other words, our purpose is to obtain formulas with a small relative error. Obviously, the parameters
of formulas (11) are not only the coefficients a, b, (11and 0, but
also the quantity 7. The limits of summation determine
the number of the coefficients
Preliminary
calculations
cannot be regarded as free parameters,
and thereby the class of approximating
have shown that the addition
since they
functions.
of each extra pair of free coefficients
increases the accuracy of the formulas by a factor of 10 approximately
(from the form of the
formula it follows that the number of coefficients can be changed only by an even number). Good accuracy is ensured by approximately six free coefficients. This number was in fact chosen for further calculations. The coefficients
were found by an interpolation
process. We did not succeed in proving its
convergence theoretically, but in practice the process always converged, although fairly slowly. We will not describe this process since it is not of independent mathematical interest. The actual coefficients
of the formulas, and also their accuracy are given in Tables l-5
(Tables 1,3 for (1 la) with v=‘/s, Table 2 for (3), Tables 4,5 for (1 lb) with PL=~/~). Translated by J. Berry REFERENCES 1.
MCDOUGALL, J. and STONER, E. S., The computation of Fermi-Dirac functions. Philos. Trans. Roy. Sot. London,
A237,67-104,1939.
2.
RHODES, P., Fermi-Dirac functions of integral order. Proc. Roy. Sot., A204, No. 1078, 396405, 1950.
3.
BEER, A. C., CHASE, M. N. and CHOQUARD, P. F. Extension of McDougall-Stoner tables of the FermiDirac functions. Helv. phys. acta, 28,529-542, 1955.
4.
LATTER, R., Temperature behaviour of the Thomas-Fermi statistical model for atoms. Phys. Rev., 99, 1854-1870, 1955.
5.
CODY, W. J. and THACHER, H. C. Jr., Rational Chebyshev approximation orders -%, % and 312. Math. Comput.. 21,97,30-40, 1967.
for Fermi-Dirac integrals of