Abstract

The method given in Part I for obtaining emission coefficients from emitted spectral intensities is generalized here to include asymmetrical sources as well. In this method the emission coefficient is expanded in terms of a complete set of functions which are invariant in form to a rotation of axes and the integral equation relating the emission coefficient to the emitted spectral intensity is used to determine the unknown expansion coefficients. The method has been checked by means of a hypothetical example corresponding to a source whose emission coefficient is a displaced gaussian. This same hypothetical example is used to test the numerical method which was developed for summing the series representation for the emission coefficient in situations where the emitted spectral intensity is given in the form of experimental data. Finally the numerical method is used to obtain the spatial distribution of the emission coefficient corresponding to an atomic spectral line of argon emitted by a free-burning argon arc which is distorted by an external magnetic field.

© 1966 Optical Society of America

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References

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  1. C. D. Maldonado, A. P. Caron, and H. N. Olsen, J. Opt. Soc. Am. 55, 1247 (1965); this paper is referred to as I, and any of its equations will be identified by the prefix I.
    [Crossref]
  2. S. I. Herlitz, Arkiv Fys. 23, 571 (1963).
  3. C. D. Maldonado, J. Math. Phys. 6, 1935 (1965).
    [Crossref]
  4. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill Book Co., Inc., New York, 1963), Vols. I and II.
  5. M. H. Stone, Ann. Math. (2) 29, 1 (1927); G. Sansone, Orthogonal Functions (Interscience Publishers, Inc., New York, 1959), Chap. IV.
    [Crossref]
  6. H. N. Olsen, C. D. Maldonado, G. D. Duckworth, and A. P. Caron, “Investigation of the Interaction of an External Magnetic Field with an Electric Arc,” Final Report under Contract AF33(615)-1105, ARL 66-0016 (January1966).
  7. H. N. Olsen, J. Quant. Spectry. Radiative Transfer 3, 305 (1963).
    [Crossref]
  8. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill Book Co., Inc., New York, 1963), Vol. I.
  9. W. Gröbner and N. Hofreiter, Integraltafel erster teil Unbestimmte Integrale (Springer-Verlag, Vienna, 1961).

1965 (2)

1963 (2)

S. I. Herlitz, Arkiv Fys. 23, 571 (1963).

H. N. Olsen, J. Quant. Spectry. Radiative Transfer 3, 305 (1963).
[Crossref]

1927 (1)

M. H. Stone, Ann. Math. (2) 29, 1 (1927); G. Sansone, Orthogonal Functions (Interscience Publishers, Inc., New York, 1959), Chap. IV.
[Crossref]

Caron, A. P.

C. D. Maldonado, A. P. Caron, and H. N. Olsen, J. Opt. Soc. Am. 55, 1247 (1965); this paper is referred to as I, and any of its equations will be identified by the prefix I.
[Crossref]

H. N. Olsen, C. D. Maldonado, G. D. Duckworth, and A. P. Caron, “Investigation of the Interaction of an External Magnetic Field with an Electric Arc,” Final Report under Contract AF33(615)-1105, ARL 66-0016 (January1966).

Duckworth, G. D.

H. N. Olsen, C. D. Maldonado, G. D. Duckworth, and A. P. Caron, “Investigation of the Interaction of an External Magnetic Field with an Electric Arc,” Final Report under Contract AF33(615)-1105, ARL 66-0016 (January1966).

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill Book Co., Inc., New York, 1963), Vols. I and II.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill Book Co., Inc., New York, 1963), Vol. I.

Gröbner, W.

W. Gröbner and N. Hofreiter, Integraltafel erster teil Unbestimmte Integrale (Springer-Verlag, Vienna, 1961).

Herlitz, S. I.

S. I. Herlitz, Arkiv Fys. 23, 571 (1963).

Hofreiter, N.

W. Gröbner and N. Hofreiter, Integraltafel erster teil Unbestimmte Integrale (Springer-Verlag, Vienna, 1961).

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill Book Co., Inc., New York, 1963), Vols. I and II.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill Book Co., Inc., New York, 1963), Vol. I.

Maldonado, C. D.

C. D. Maldonado, J. Math. Phys. 6, 1935 (1965).
[Crossref]

C. D. Maldonado, A. P. Caron, and H. N. Olsen, J. Opt. Soc. Am. 55, 1247 (1965); this paper is referred to as I, and any of its equations will be identified by the prefix I.
[Crossref]

H. N. Olsen, C. D. Maldonado, G. D. Duckworth, and A. P. Caron, “Investigation of the Interaction of an External Magnetic Field with an Electric Arc,” Final Report under Contract AF33(615)-1105, ARL 66-0016 (January1966).

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill Book Co., Inc., New York, 1963), Vols. I and II.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill Book Co., Inc., New York, 1963), Vol. I.

Olsen, H. N.

C. D. Maldonado, A. P. Caron, and H. N. Olsen, J. Opt. Soc. Am. 55, 1247 (1965); this paper is referred to as I, and any of its equations will be identified by the prefix I.
[Crossref]

H. N. Olsen, J. Quant. Spectry. Radiative Transfer 3, 305 (1963).
[Crossref]

H. N. Olsen, C. D. Maldonado, G. D. Duckworth, and A. P. Caron, “Investigation of the Interaction of an External Magnetic Field with an Electric Arc,” Final Report under Contract AF33(615)-1105, ARL 66-0016 (January1966).

Stone, M. H.

M. H. Stone, Ann. Math. (2) 29, 1 (1927); G. Sansone, Orthogonal Functions (Interscience Publishers, Inc., New York, 1959), Chap. IV.
[Crossref]

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill Book Co., Inc., New York, 1963), Vol. I.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill Book Co., Inc., New York, 1963), Vols. I and II.

Ann. Math. (2) (1)

M. H. Stone, Ann. Math. (2) 29, 1 (1927); G. Sansone, Orthogonal Functions (Interscience Publishers, Inc., New York, 1959), Chap. IV.
[Crossref]

Arkiv Fys. (1)

S. I. Herlitz, Arkiv Fys. 23, 571 (1963).

J. Math. Phys. (1)

C. D. Maldonado, J. Math. Phys. 6, 1935 (1965).
[Crossref]

J. Opt. Soc. Am. (1)

J. Quant. Spectry. Radiative Transfer (1)

H. N. Olsen, J. Quant. Spectry. Radiative Transfer 3, 305 (1963).
[Crossref]

Other (4)

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill Book Co., Inc., New York, 1963), Vol. I.

W. Gröbner and N. Hofreiter, Integraltafel erster teil Unbestimmte Integrale (Springer-Verlag, Vienna, 1961).

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill Book Co., Inc., New York, 1963), Vols. I and II.

H. N. Olsen, C. D. Maldonado, G. D. Duckworth, and A. P. Caron, “Investigation of the Interaction of an External Magnetic Field with an Electric Arc,” Final Report under Contract AF33(615)-1105, ARL 66-0016 (January1966).

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Figures (4)

Fig. 1
Fig. 1

Qualitative sketch of a typical step approximation to the surface of g(ξ,x) at the point (ξ=ξjξj, x=xixi) together with symbols and nomenclature that are pertinent to the numerical procedure for evaluating the double integrals of Eqs. (4.2) and (4.3).

Fig. 2
Fig. 2

Schematic of hypothetical model used to check out the computer program.

Fig. 3
Fig. 3

Isointensity contours of continuous radiation emitted by the 400-A arc when deflected by a 36-G magnetic field.

Fig. 4
Fig. 4

Variation of the emission coefficients along the entire x axis obtained by inversion of integrated intensities measured in planes 3.18 and 3.8 mm from cathode tip in the plasma deflected by 36-G field.

Tables (1)

Tables Icon

Table I Comparison of numerical results of Eq. (4.1) with theoretical results obtained from Eq. (3.10) with α=4.1303 and d=0.1.

Equations (49)

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g ( ξ , x ) = - + d y f ( x , y ) ,
f ( x , y ) = k = 0 m = 0 m [ C m + 2 k m ( α ) U m + 2 k m ( α x , α y ) + C m + 2 k - m ( α ) U m + 2 k - m ( α x , α y ) ] × exp [ - α 2 ( x 2 + y 2 ) ] ,
L k m [ α 2 ( x 2 + y 2 ) ] = s = 0 k ( - 1 ) s [ ( m + k ) ! / ( k - s ) ! ( m + s ) ! s ! ] × [ α 2 ( x 2 + y 2 ) ] s ,
U m + 2 k ± m ( α x , α y ) = ( - 1 ) k ( α / π 1 2 ) [ k ! / ( m + k ) ! ] 1 2 [ α 2 ( x 2 + y 2 ) ] m / 2 × exp ( ± i m φ ) L k m [ α 2 ( x 2 + y 2 ) ] ,
g ( ξ , x ) = k = 0 m = 0 m [ 1 / k ! ( m + k ) ! 2 2 ( m + 2 k ) ] 1 2 × [ C m + 2 k m ( α ) exp ( i m ξ ) + C m + 2 k - m ( α ) × exp ( - i m ξ ) ] H m + 2 k ( α x ) exp ( - α 2 x 2 ) ,
- π π d ξ exp ( ± i m ξ ) exp ( i ν ξ ) × - + d x H m + 2 k ( α x ) H ν + 2 β ( α x ) exp ( - α 2 x 2 ) = ( 2 π 3 2 / α ) [ ( m + 2 k ) ! ( ν + 2 β ) ! 2 m + 2 k 2 ν + 2 β ] 1 2 × δ m ν δ m + 2 k ν + 2 β ,
C m + 2 k ± m ( α ) = ( α / 2 π 3 2 ) [ ( k ! ( m + k ) ! ) 1 2 / ( m + 2 k ) ! ] × - π π d ξ exp ( i m ξ ) - d x g ( ξ , x ) H m + 2 k ( α x )
f ( x , y ) = ( α / π 3 2 ) k = 0 m = 0 m [ ( k ! ( m + k ) ! ) 1 2 / ( m + 2 k ) ! ] × Re { [ - π π d ξ exp ( - i m ξ ) × - + d x g ( ξ , x ) H m + 2 k ( α x ) ] × U m + 2 k m ( α x , α y ) } exp [ - α 2 ( x 2 + y 2 ) ] ,
f ( x , y ) = B exp { - β 2 [ ( x - d ) 2 + y 2 ] } ,
g ( ξ , x ) = B ( π 1 2 / β ) exp [ - β 2 ( x - d cos ξ ) 2 ] .
f ( x , y ) = ( α / π ) 2 k = 0 m = 0 m ( - 1 ) k [ k ! / ( m + 2 k ) ! ] × [ - π π d ξ cos ( m ξ ) - + d x g ( ξ , x ) H m + 2 k ( α x ) ] × [ α 2 ( x 2 + y 2 ) ] m / 2 cos ( m φ ) × L k m [ α 2 ( x 2 + y 2 ) ] exp [ - α 2 ( x 2 + y 2 ) ] ,
f ( x , y ) = 2 B ( α / β ) 2 m = 0 m ( α d ) m [ α 2 ( x 2 + y 2 ) ] m / 2 × cos ( m φ ) k = 0 [ k ! / ( m + k ) ! ] [ 1 - ( α / β ) 2 ] k × L k m [ α 2 ( x 2 + y 2 ) ] L k m [ ( α d ) 2 / ( 1 - ( α / β ) 2 ) ] × exp [ - α 2 ( x 2 + y 2 ) ] .
k = 0 [ k ! / ( m + k ) ! ] L k m ( x ) L k m ( y ) z k = [ 1 / ( 1 - z ) ( x y z ) m / 2 ] I m [ 2 ( x y z ) 1 2 / ( 1 - z ) ] × exp [ - ( x + y ) z / ( 1 - z ) ] ,
k = 0 [ k ! / ( m + k ) ! ] L k m [ α 2 ( x 2 + y 2 ) ] × L k m [ ( α d ) 2 / ( 1 - ( α / β ) 2 ) ] [ 1 - ( α / β ) 2 ] k = ( β / α ) 2 ( α d ) - m [ α 2 ( x 2 + y 2 ) ] - m / 2 × I m [ 2 β 2 d ( x 2 + y 2 ) 1 2 ] × exp [ ( α 2 - β 2 ) ( x 2 + y 2 ) - ( β d ) 2 ] .
f ( x , y ) = 2 B m = 0 m cos ( m φ ) I m [ 2 β 2 d ( x 2 + y 2 ) 1 2 ] × exp [ - β 2 ( x 2 + y 2 ) - ( β d ) 2 ] .
exp ( z cos φ ) = 2 m = 0 m cos ( m φ ) I m ( z ) ,
exp [ 2 β 2 d ( x 2 + y 2 ) 1 2 cos φ ] = 2 m = 0 m cos ( m φ ) I m [ 2 β 2 d ( x 2 + y 2 ) 1 2 ] .
f ( x , y ) = B exp { - β 2 [ x 2 + d 2 - 2 d ( x 2 + y 2 ) 1 2 × cos φ + y 2 ] } ,
f ( x , y ) ( α / π ) 2 k = 0 K m = 0 M m ( - 1 ) k [ k ! / ( m + 2 k ) ! ] × [ B m + 2 k m ( α ) cos ( m φ ) + D m + 2 k m ( α ) sin ( m φ ) ] × [ α 2 ( x 2 + y 2 ) ] m / 2 L k m [ α 2 ( x 2 + y 2 ) ] × exp [ - α 2 ( x 2 + y 2 ) ] ,
B m + 2 k m ( α ) = - π π d ξ cos ( m ξ ) - + d x g ( ξ , x ) H m + 2 k ( α x )
D m + 2 k m ( α ) = - π π d ξ sin ( m ξ ) - + d x g ( ξ , x ) H m + 2 k ( α x )
B m + 2 k m ( α ) i = 0 I - 1 j = 0 J - 1 g ( ξ j + Δ ξ j , x i + Δ x i ) × ξ j ξ j + 1 d ξ cos ( m ξ ) x i x i + 1 d x H m + 2 k ( α x )
D m + 2 k m ( α ) i = 0 I - 1 j = 0 J - 1 g ( ξ j + Δ ξ j , x i + Δ x i ) × ξ j ξ j + 1 d ξ sin ( m ξ ) x i x i + 1 d x H m + 2 k ( α x ) ,
B m + 2 k m ( α ) [ 1 / 2 α m ( m + 2 k + 1 ) ] × i = 0 I - 1 j = 0 J - 1 g ( ξ j + Δ ξ j , x i + Δ x i ) × [ sin ( m ξ j + 1 ) - sin ( m ξ j ) ] × [ H m + 2 k + 1 ( α x i + 1 ) - H m + 2 k + 1 ( α x i ) ]
D m + 2 k m ( α ) - [ 1 / 2 α m ( m + 2 k + 1 ) ] × i = 0 I - 1 j = 0 J - 1 g ( ξ j + Δ ξ j , x i + Δ x i ) × [ cos ( m ξ j + 1 ) - cos ( m ξ j ) ] × [ H m + 2 k + 1 ( α x i + 1 ) - H m + 2 k + 1 ( α x i ) ] .
I m + 2 k ± m ( ξ , α x ) = - + d y U m + 2 k ± m ( α x , α y ) × exp ( - α 2 y 2 )
U m + 2 k ± m ( α x , α y ) = ( - 1 ) k ( α / π 1 2 ) [ k ! / ( m + k ) ! ] 1 2 × L k m [ α 2 ( x 2 + y 2 ) ] s = 0 m ( ± i ) s × [ m ! / s ! ( m - s ) ! ] ( α x ) m - s ( α y ) s ,
I m + 2 k ± m ( ξ , α x ) = ( - 1 ) k ( α / π 1 2 ) × exp ( ± i m ξ ) [ k ! / ( m + k ) ! ] 1 2 × s = 0 m ( ± i ) s [ m ! / s ! ( m - s ) ! ] ( α x ) m - s × - + d y ( α y ) s L k m [ α 2 ( x 2 + y 2 ) ] × exp ( - α 2 y 2 ) .
L k m [ α 2 ( x 2 + y 2 ) ] = r = 0 k L k - r p ( α 2 x 2 ) L r q ( α 2 y 2 ) ,
I m + 2 k ± m ( ξ , α x ) = ( - 1 ) k ( α / π 1 2 ) × exp ( ± i m ξ ) [ k ! / ( m + k ) ! ] 1 2 × n = 0 [ m / 2 ] ( - 1 ) n [ m ! / ( 2 n ) ! ( m - 2 n ) ! ] × ( α x ) m - 2 n r = 0 k L k - r p ( α 2 x 2 ) × - + d y ( α y ) 2 n L r q ( α 2 y 2 ) × exp ( - α 2 y 2 ) ,
- + d y ( α y ) 2 n L r q ( α 2 y 2 ) exp ( - α 2 y 2 ) = ( 1 / α ) 0 d t t n - 1 2 L r q ( t ) exp ( - t )
0 d t t n - 1 2 L r q ( t ) exp ( - t ) = π 1 2 ( 2 n ) ! Γ ( q + r - n + 1 2 ) / n ! 2 2 n Γ ( r + 1 ) Γ ( q - n + 1 2 ) .
L k m - n - 1 2 ( α 2 x 2 ) = r = 0 k [ Γ ( q + r - n + 1 2 ) / Γ ( r + 1 ) Γ ( q - n + 1 2 ) ] L k - r p ( α 2 x 2 )
I m + 2 k ± m ( ξ , α x ) = ( - 1 ) k exp ( ± i m ξ ) [ k ! / ( m + k ) ! ] 1 2 × n = 0 [ m / 2 ] ( - 1 ) n [ m ! / n ! ( m - 2 n ) ! 2 2 n ] × ( α x ) m - 2 n L k m - n - 1 2 ( α 2 x 2 ) .
L k m - n - 1 2 ( α 2 x 2 ) = ( - 1 ) k [ ( α x ) - ( m - 2 n ) / k ! 2 2 k ] × exp ( α 2 x 2 ) d 2 k d ( α x ) 2 k [ ( α x ) m - 2 n exp ( - α 2 x 2 ) ]
I m + 2 k ± m ( ξ , α x ) = exp ( ± i m ξ ) × [ k ! ( m + k ) ! ] - 1 2 2 - ( m + 2 k ) H m + 2 k ( α x )
H m + 2 k ( α x ) = exp ( α 2 x 2 ) [ d 2 k d ( α x ) 2 k ] × [ exp ( - α 2 x 2 ) H m ( α x ) ] .
J m + 2 k m ( α ) = - π π d ξ cos ( m ξ ) × - + d x g ( ξ , x ) H m + 2 k ( α x ) .
H m + 2 k [ α ( z + d cos ξ ) ] = [ 1 / 2 ( m + 2 k ) / 2 ] s = 0 m + 2 k [ ( m + 2 k ) ! / s ! ( m + 2 k - s ) ! ] × H s ( 2 1 2 α z ) H m + 2 k - s ( 2 1 2 α d cos ξ ) ,
J m + 2 k m ( α ) = B ( π 1 2 / β 2 ( m + 2 k ) / 2 ) × s = 0 m + 2 k [ ( m + 2 k ) ! / s ! ( m + 2 k - s ) ! ] × - π π d ξ cos ( m ξ ) H m + 2 k - s ( 2 1 2 α d cos ξ ) × - + d z exp ( - β 2 z 2 ) H s ( 2 1 2 α z ) .
- + d z H 2 t ( 2 1 2 α z ) exp ( - β 2 z 2 ) = ( π 1 2 / β ) [ ( 2 t ) ! / t ! ] [ 2 ( α / β ) 2 - 1 ] t .
J m + 2 k m ( α ) = B ( π / β 2 2 ( m + 2 k ) / 2 ) × t = 0 ( m + 2 k ) / 2 [ ( m + 2 k ) ! / t ! ( m + 2 k - 2 t ) ! ] [ 2 ( α / β ) 2 - 1 ] t × - π π d ξ cos ( m ξ ) H m + 2 k - 2 t ( 2 1 2 α d cos ξ )
J m + 2 k m ( α ) = B ( π / β 2 2 ( m + 2 k ) / 2 ) × t = 0 ( m + 2 k ) / 2 [ ( m + 2 k ) ! / t ! ] [ 2 ( α / β ) 2 - 1 ] t × p = 0 [ ( m + 2 k - 2 t ) / 2 ] ( - 1 ) p [ 1 / p ! ( m + 2 k - 2 t - 2 p ) ! ] × [ 2 3 2 α d ] m + 2 k - 2 t - 2 p × - π π d ξ cos ( m ξ ) cos ( ξ ) m + 2 k - 2 t - 2 p
H m + 2 k - 2 t ( 2 1 2 α d cos ξ ) = p = 0 [ ( m + 2 k - 2 t ) / 2 ] ( - 1 ) p [ ( m + 2 k - 2 t ) ! / p ! ( m + 2 k - 2 t - 2 p ) ! ] [ 2 3 2 α d cos ξ ] m + 2 k - 2 t - 2 p ,
- π π d ξ cos ( m ξ ) cos ( ξ ) m + 2 k - 2 t - 2 p = 2 π [ ( m + 2 k - 2 t - 2 p ) ! / ( m + k - t - p ) ! ( k - t - p ) ! 2 m + 2 k - 2 t - 2 p ] ,
J m + 2 k m ( α ) = 2 B ( π / β ) 2 ( - 1 / 2 ) k ( α d ) m t = 0 k ( - 1 ) t [ ( m + 2 k ) ! / t ! ( m + k - t ) ! ] × [ 2 ( α β ) 2 - 1 ] t L k - t m ( 2 α 2 d 2 ) .
L k m ( λ x ) = t = 0 k [ ( m + k ) ! / t ! ( m + k - t ) ! ] × λ k - t ( 1 - λ ) t L k - t m ( x ) ,
t = 0 k ( - 1 ) t [ ( m + 2 k ) ! / t ! ( m + k - t ) ! ] × [ 2 ( α / β ) 2 - 1 ] t L k - t m ( 2 α 2 d 2 ) = [ ( m + 2 k ) ! 2 k / ( m + k ) ! ] [ 1 - ( α / β ) 2 ] k × L k m { α 2 d 2 / [ 1 - ( α / β ) 2 ] } .
J m + 2 k m ( α ) = 2 B ( π / β ) 2 [ ( m + 2 k ) ! / ( m + k ) ! ] ( α d ) m × [ ( α / β ) 2 - 1 ] k L k m [ α 2 d 2 / ( 1 - ( α / β ) 2 ) ] .