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A simple method for calculating voigt-profiles by a programmable hand-calculator HP-34C
.[ Quant. Spectrosc. Radiat. Trans[er Vol. 29, No. I, pp. 89-92, 1983
0022..4073lg31010089--04503.00/0
Printed in Great Britain.
Pergamon Press Ltd.
NOTE
A SIMPLE METHOD FOR CALCULATING VOIGT-PROFILES BY A PROGRAMMABLE HAND-CALCULATOR HP-34C KLAUS G. P. SULZMANN~" EnergyCenterand Departmentof AppliedMechanicsand EngineeringSciences,Universityof California,San Diego, La Jolla, CA 92093,U.S.A.
(Received 22 March 1982) Abstraet--A program is given for the calculation of the Voigt-profilenear the line-centerby a hand-held HP-24Ccalculatorand whichis usefulfor spot-checkson moreextensivecalculationsand for data reductions. The programcan be easily adapted to other hand-heldcalculatorswith equivalentstorage and computational capacities. INTRODUCTION Various optimized methods have been developed in the past (see, for example, Refs. 1 and 2 for details) for fast calculations by computers of the Voigt-profile 3
f+~ exp(-y~)dr J-o~ a2+(~- y)2'
~(~, a ) - P d P ' = / a )
(1)
with a = ( b d b D ) ~ / ( l ~ the line-shape parameter, ~ = [(oJ = ¢oo)/dD]'~/-O~) the line-displacement, and P' --- ( S l b t , ) ~ , all in standard notation. However, for many applications it is desirable to have a program for calculations by a hand-held calculator which can be used for spot-checks or data reductions. We use a hand-held HP-34C calculator for 0 < ~ ~ 2 and 0 ~< a ~ 10 near the line center. The program requires 10 storage registers and 127 operational steps on the HP-34C and can easily be adapted for other hand-held calculators with equivalent storage and computational capacities. RATIONALE For the range of parameters of interest, we use the power series expressions given in Eq. (4-45) through (4-47) in Ref. 3, viz.,
a) = [exp(-~2)] ~ 0 (~2"[n !)Qn(a),
(2)
Q,(a) = (a/2n - 1)[2/x/(-~ - 2aQ,_t(a)]
(2a)
¢,(~, where
and
Qo(a) = [exp(a2)l[erfc(a)] oo
= [exp(aE)](2/X/(-~) f [exp(-x2)] dx. Ja For a ~ 3,
Qo(a) is
(3)
[see Eq. (7.1.26) of Ref. 4] 5
Qo(a) = ~
a,.t m
m=l
tThis work was supported by the U.S. Army Research Officeunder Contract No. DAAG29-80-C-0001.
89
(4)
90
Kt.aus G. P. SULZMANN
with t = 1/(1 + aoa), ao = 0.327 591 117,
a, = 0.254 829 592,
a2 = -0.284496 736,
a3 = 1.421 413 741,
a4 = -1.453 152 027,
a5 = 1.061 405 429.
(4a)
(4b)
The error in Qo(a) increases with increasing a but is less than 8 × 10 -~ for a = 3. For a ~>3, Qo(a) may be expressed as [see Eq. (592) of Ref. 5] Qo(a) = [1/C(-~a]{1 + ~__, [(2m)!/m !](-1/4a2) m},
(5)
which is rapidly converging and should be terminated after the last decreasing term, i.e., for m -< a 2. Its accuracy increases with increasing a and, for example, for a = 3 the error in Qo(a) is less than 4 x 10 -4 if only four terms are taken into account. EXECUTION The HP-34C calculator programs for determination of values for Eq. (2) together with those for Eqs. (4) and (5) are given in Appendices 1 through 3. Before execution of the programs, the values for the coefficients ao through a5 of Eq. (4b) are stored permanently in the storage registers STO 0 through STO 5. 0<~£~<2 AND0~
Note added in proof. Largervaluesfor ~:can be used as longas the storageregister STO 9 does not overflow.For a -> 10the Lorentz profile applies in good approximationfor all values of (, REFERENCES 1. J. H. Pierluissi, P. C. Vanderwood,and R. B. Gomex,JQSRT 18, 555 (1977). 2. J. Humli~ek, JQSRT 27, 437 (1982). 3. S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities, Chapter 4. Addison-Wesley, Reading, Massachussetts (1959). 4. M. Abramowitzand J. A. Stegun,(Eds.), Handbook of Mathematical Functions withFormulas, Graphs, and Mathematical Tables, p. 299. NBS Appl. Mathem. Ser. 55, 3rd print., U.S. GovernmentPrint Office,Washington,DC (1965). 5. H. B. Dwight, Tables o.f Integrals and Other Mathematical Data, p. 136.Macmillan,New York (1961).
91
A simple method for calculating Voigt-profiles Appendix 1. HP-MC program for the evaluation of Qo(a) from Eqs. (4) through (4b). ol
hLBL
oz
STO 7
O3
RCL
19
A
=a
0
×
21
+
×
22
O5
1
23
+
O8
STO 6
09
RCL
=t
1
ll
RCL gx
13
RCL
26
×
Z7
+ RCL
Z9
6
2
RCL
4
28
X
2
6
hy x
25
h I/x
I0
RCL
24
O7
3
20
04
06
RCL
4
6
5
30
hyx
31
RCL
5
14
X
32
x
15
+
33
+
34
STO 6
35
R/S
16
RCL
17
6 3
=Q0(a)
hyx
18
Appendix 2. HP-34C program for the evaluation of ~P(~, a) from Eqs. (2) and (2a). 36
hLBL
37
STO 8
B
38
0
39
STO 9
= ~
x
63
1
64
40
0
41
STO fI
42
RCL
43
STO 9
44
hLBL
45
2
46
62
=n
6
= An 0
h~
65
+
66
STO 6
67
RCL fI
68
hx!
69
RCL
71
RCL
47
73
48
74
49
RCL
6
50
RCL
7
51
+
70
7Z
= Qn(a); n = l . . . . .
8 fI 2
x hy x
N+I
75
×
76
STO
X
77
R/S
52
2
78
GTO 0
53
X
79
hLBL
80
81
RCL 2 gx
= a
54 55
RCL
56
7
X
82
CHS
57
I
83
ge x
58
RCL fl +
84
RCL
59 6O
STO f I
61
2
= n+l
85
×
86
R/S
+9
=A |
8
9
= n~_l A n n~
n=l,...,N.
N.
92
KLAUSG. P. SULZMANN Appendix 3. HP-34C program for the evaluation of Oo(a) from Eq. (5). 87
hLBL 2
108
88
STO.7 z
109
RCL fl
II0
Z
89
gx
90
4
III
91
×
112
92
CI-tS
113
)<
93
h I/x
114
STO +9
94
STO 8
115
R/S
95
0
116
GTO 4
96
STO 9
117
hLBL
97
0
118
RCL 9
98
STO f I
99 .00
= (.X/4a z) N-1 =7
Bn
hx!
(N)
119
I
hLBL 4
120
+
RCL 8
121
=n-I
RCL
122
102
I
123
hv
103
+
1 24
f27-
STO f I
125
3
7
RCL f l
--H
= Bn, n = 1 , 2 . . . . ,N.
#
I01
.04
= n~=i Bn
÷
105
h7 x
1 26
STO 6
106
RCI~ fI
127
RyS
107
hx:
=Qo(a)
0022..4073lg31010089--04503.00/0
Printed in Great Britain.
Pergamon Press Ltd.
NOTE
A SIMPLE METHOD FOR CALCULATING VOIGT-PROFILES BY A PROGRAMMABLE HAND-CALCULATOR HP-34C KLAUS G. P. SULZMANN~" EnergyCenterand Departmentof AppliedMechanicsand EngineeringSciences,Universityof California,San Diego, La Jolla, CA 92093,U.S.A.
(Received 22 March 1982) Abstraet--A program is given for the calculation of the Voigt-profilenear the line-centerby a hand-held HP-24Ccalculatorand whichis usefulfor spot-checkson moreextensivecalculationsand for data reductions. The programcan be easily adapted to other hand-heldcalculatorswith equivalentstorage and computational capacities. INTRODUCTION Various optimized methods have been developed in the past (see, for example, Refs. 1 and 2 for details) for fast calculations by computers of the Voigt-profile 3
f+~ exp(-y~)dr J-o~ a2+(~- y)2'
~(~, a ) - P d P ' = / a )
(1)
with a = ( b d b D ) ~ / ( l ~ the line-shape parameter, ~ = [(oJ = ¢oo)/dD]'~/-O~) the line-displacement, and P' --- ( S l b t , ) ~ , all in standard notation. However, for many applications it is desirable to have a program for calculations by a hand-held calculator which can be used for spot-checks or data reductions. We use a hand-held HP-34C calculator for 0 < ~ ~ 2 and 0 ~< a ~ 10 near the line center. The program requires 10 storage registers and 127 operational steps on the HP-34C and can easily be adapted for other hand-held calculators with equivalent storage and computational capacities. RATIONALE For the range of parameters of interest, we use the power series expressions given in Eq. (4-45) through (4-47) in Ref. 3, viz.,
a) = [exp(-~2)] ~ 0 (~2"[n !)Qn(a),
(2)
Q,(a) = (a/2n - 1)[2/x/(-~ - 2aQ,_t(a)]
(2a)
¢,(~, where
and
Qo(a) = [exp(a2)l[erfc(a)] oo
= [exp(aE)](2/X/(-~) f [exp(-x2)] dx. Ja For a ~ 3,
Qo(a) is
(3)
[see Eq. (7.1.26) of Ref. 4] 5
Qo(a) = ~
a,.t m
m=l
tThis work was supported by the U.S. Army Research Officeunder Contract No. DAAG29-80-C-0001.
89
(4)
90
Kt.aus G. P. SULZMANN
with t = 1/(1 + aoa), ao = 0.327 591 117,
a, = 0.254 829 592,
a2 = -0.284496 736,
a3 = 1.421 413 741,
a4 = -1.453 152 027,
a5 = 1.061 405 429.
(4a)
(4b)
The error in Qo(a) increases with increasing a but is less than 8 × 10 -~ for a = 3. For a ~>3, Qo(a) may be expressed as [see Eq. (592) of Ref. 5] Qo(a) = [1/C(-~a]{1 + ~__, [(2m)!/m !](-1/4a2) m},
(5)
which is rapidly converging and should be terminated after the last decreasing term, i.e., for m -< a 2. Its accuracy increases with increasing a and, for example, for a = 3 the error in Qo(a) is less than 4 x 10 -4 if only four terms are taken into account. EXECUTION The HP-34C calculator programs for determination of values for Eq. (2) together with those for Eqs. (4) and (5) are given in Appendices 1 through 3. Before execution of the programs, the values for the coefficients ao through a5 of Eq. (4b) are stored permanently in the storage registers STO 0 through STO 5. 0<~£~<2 AND0~
Note added in proof. Largervaluesfor ~:can be used as longas the storageregister STO 9 does not overflow.For a -> 10the Lorentz profile applies in good approximationfor all values of (, REFERENCES 1. J. H. Pierluissi, P. C. Vanderwood,and R. B. Gomex,JQSRT 18, 555 (1977). 2. J. Humli~ek, JQSRT 27, 437 (1982). 3. S. S. Penner, Quantitative Molecular Spectroscopy and Gas Emissivities, Chapter 4. Addison-Wesley, Reading, Massachussetts (1959). 4. M. Abramowitzand J. A. Stegun,(Eds.), Handbook of Mathematical Functions withFormulas, Graphs, and Mathematical Tables, p. 299. NBS Appl. Mathem. Ser. 55, 3rd print., U.S. GovernmentPrint Office,Washington,DC (1965). 5. H. B. Dwight, Tables o.f Integrals and Other Mathematical Data, p. 136.Macmillan,New York (1961).
91
A simple method for calculating Voigt-profiles Appendix 1. HP-MC program for the evaluation of Qo(a) from Eqs. (4) through (4b). ol
hLBL
oz
STO 7
O3
RCL
19
A
=a
0
×
21
+
×
22
O5
1
23
+
O8
STO 6
09
RCL
=t
1
ll
RCL gx
13
RCL
26
×
Z7
+ RCL
Z9
6
2
RCL
4
28
X
2
6
hy x
25
h I/x
I0
RCL
24
O7
3
20
04
06
RCL
4
6
5
30
hyx
31
RCL
5
14
X
32
x
15
+
33
+
34
STO 6
35
R/S
16
RCL
17
6 3
=Q0(a)
hyx
18
Appendix 2. HP-34C program for the evaluation of ~P(~, a) from Eqs. (2) and (2a). 36
hLBL
37
STO 8
B
38
0
39
STO 9
= ~
x
63
1
64
40
0
41
STO fI
42
RCL
43
STO 9
44
hLBL
45
2
46
62
=n
6
= An 0
h~
65
+
66
STO 6
67
RCL fI
68
hx!
69
RCL
71
RCL
47
73
48
74
49
RCL
6
50
RCL
7
51
+
70
7Z
= Qn(a); n = l . . . . .
8 fI 2
x hy x
N+I
75
×
76
STO
X
77
R/S
52
2
78
GTO 0
53
X
79
hLBL
80
81
RCL 2 gx
= a
54 55
RCL
56
7
X
82
CHS
57
I
83
ge x
58
RCL fl +
84
RCL
59 6O
STO f I
61
2
= n+l
85
×
86
R/S
+9
=A |
8
9
= n~_l A n n~
n=l,...,N.
N.
92
KLAUSG. P. SULZMANN Appendix 3. HP-34C program for the evaluation of Oo(a) from Eq. (5). 87
hLBL 2
108
88
STO.7 z
109
RCL fl
II0
Z
89
gx
90
4
III
91
×
112
92
CI-tS
113
)<
93
h I/x
114
STO +9
94
STO 8
115
R/S
95
0
116
GTO 4
96
STO 9
117
hLBL
97
0
118
RCL 9
98
STO f I
99 .00
= (.X/4a z) N-1 =7
Bn
hx!
(N)
119
I
hLBL 4
120
+
RCL 8
121
=n-I
RCL
122
102
I
123
hv
103
+
1 24
f27-
STO f I
125
3
7
RCL f l
--H
= Bn, n = 1 , 2 . . . . ,N.
#
I01
.04
= n~=i Bn
÷
105
h7 x
1 26
STO 6
106
RCI~ fI
127
RyS
107
hx:
=Qo(a)