Abstract

A least squares analysis is applied to the reduction of complete and incomplete Zeeman patterns. The resulting general formulas provide a simple systematic computation of the “best values” of the Landé g factors and a measure of their precision.

© 1956 Optical Society of America

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References

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  1. P. M. Griffin and K. L. Vander Sluis, J. Opt. Soc. Am. 45, 901(A) (1955).
    [Crossref]
  2. M. A. Catalán, J. Research Natl. Bur. Standards 47, 502 (1951).
    [Crossref]
  3. W. J. Youden, Statistical Methods for Chemists (John Wiley and Sons, Inc., New York, 1951), pp. 50 and 119.

1955 (1)

P. M. Griffin and K. L. Vander Sluis, J. Opt. Soc. Am. 45, 901(A) (1955).
[Crossref]

1951 (1)

M. A. Catalán, J. Research Natl. Bur. Standards 47, 502 (1951).
[Crossref]

Catalán, M. A.

M. A. Catalán, J. Research Natl. Bur. Standards 47, 502 (1951).
[Crossref]

Griffin, P. M.

P. M. Griffin and K. L. Vander Sluis, J. Opt. Soc. Am. 45, 901(A) (1955).
[Crossref]

Vander Sluis, K. L.

P. M. Griffin and K. L. Vander Sluis, J. Opt. Soc. Am. 45, 901(A) (1955).
[Crossref]

Youden, W. J.

W. J. Youden, Statistical Methods for Chemists (John Wiley and Sons, Inc., New York, 1951), pp. 50 and 119.

J. Opt. Soc. Am. (1)

P. M. Griffin and K. L. Vander Sluis, J. Opt. Soc. Am. 45, 901(A) (1955).
[Crossref]

J. Research Natl. Bur. Standards (1)

M. A. Catalán, J. Research Natl. Bur. Standards 47, 502 (1951).
[Crossref]

Other (1)

W. J. Youden, Statistical Methods for Chemists (John Wiley and Sons, Inc., New York, 1951), pp. 50 and 119.

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Figures (1)

Fig. 1
Fig. 1

A typical J=5/2 to J=3/2 Zeeman pattern demonstrating the method for forming the set of symmetric differences yi. In this example k=4 and n=6.

Equations (21)

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y 1 = a 11 g 1 + a 12 g 2 y i = a i 1 g 1 + a i 2 g 2 y n = a n 1 g 1 + g n 2 g 2 ,
( y 1 y i y n ) = ( a 11 a 12 a i 1 a i 2 a n 1 a n 2 ) ( g 1 g 2 ) ,
Y = A G ,
1 n ( y i obs - y i calc ) 2 = min
B - 1 A T Y = B - 1 A T A G = G .
σ g i 2 = b i i - 1 s 2 ,
s 2 = 1 n ( y i obs - y i calc ) 2 ( n - 2 ) .
g 1 = 1 a c - b 2 ( c z 1 + b z 2 ) , g 2 = 1 a c - b 2 ( b z 1 + a z 2 ) , σ g 1 2 = c a c - b 2 s 2 , σ g 2 2 = a a c - b 2 s 2 .
a = 1 n a i 1 2 , b = - 1 n a i 1 a i 2 , c = 1 n a i 2 2 , z 1 = 1 n a i 1 y i , z 2 = 1 n a i 2 y i .
a 11 y 1 a 12 a i 1 y i a i 2 a n 1 y n a n 2 .
J 1 = J 2 = J g 1 < g 2 .
J 1 > J 2 = J g 1 < g 2 .
J 1 > J 2 = J g 1 > g 2 .
σ { - 2 ( J - 1 ) y 1 + 2 J - 2 ( J - 2 ) y 2 + 2 ( J - 1 ) + 2 J y k - 2 ( J - 1 ) π { - 2 J y ( k + 1 ) + 2 J - 2 ( J - 1 ) y ( k + 2 ) + 2 ( J - 1 ) - 2 ( J - p + 1 ) y n + 2 ( J - p + 1 ) ,
a = c = + 2 J ( 2 J 2 + J + 1 ) , b = + 2 J ( J + 1 ) ( 2 J - 1 ) , z 1 = 1 k - 2 ( J - i ) y i - k + 1 n 2 ( J + k + 1 - i ) y i , z 2 = 1 k 2 ( J + 1 - i ) y i + k + 1 n ( 2 J - i + k + 1 ) y i , g 1 = 1 a 2 - b 2 ( a z 1 + b z 2 ) g 2 = 1 a 2 - b 2 ( b z 1 + a z 2 ) , σ g 1 2 = σ g 2 2 = a a 2 - b 2 s 2 .
g 1 = 1 a 2 - b 2 ( a z 1 + b z 2 ) = 1.2338 g 2 = 1 a 2 - b 2 ( b z 1 + a z 2 ) = 1.5014 s = [ 1 n ( y i obs - y i calc ) 2 n - 2 ] 1 2 = 0.0017 σ g = [ a a 2 - b 2 ] 1 2 s = 0.0004.
σ { - 2 ( J - 1 ) y 1 + 2 J - 2 ( J - 2 ) y 2 + 2 ( J - 1 ) + 2 ( J + 1 ) y k - 2 J π { - 2 J y ( k + 1 ) + 2 J - 2 ( J - 1 ) y ( k + 2 ) + 2 ( J - 1 ) - 2 ( J - p + 1 ) y n + 2 ( J - p + 1 ) a = 2 J ( 2 J 2 + J + 1 ) + 4 ( J + 1 ) 2 b = c = 2 J ( J + 1 ) ( 2 J + 1 ) z 1 = - 1 k 2 ( J - i ) y i - k + 1 n 2 ( J - i + 1 + k ) y i z 2 = 1 k 2 ( J - i + 1 ) y i + k + 1 n 2 ( J - i + 1 + k ) y i g 1 = 1 a - b ( z 1 + z 2 ) g 2 = 1 a - b ( z 1 + a z 2 b ) σ g 1 2 = 1 a - b ( s 2 ) σ g 2 2 = a b ( a - b ) ( s 2 ) .
( 5 / 2 ) g 1 = 1 a - c ( z 1 + z 2 ) = 1.1978 ( 3 / 2 ) g 2 = 1 a - c ( z 1 + ( a / b ) z 2 ) = 1.3140 s = [ 1 n ( y i obs - y i c a l c ) 2 n - 2 ] 1 2 = 0.0019 σ g 1 = ( 1 a - c ) 1 2 s = 0.0005 σ g 2 = ( a ( a - c ) b ) 1 2 s = 0.0006.
r g 1 = t 50 , 4 σ g 1 = 0.741 × σ g 1 = 0.0004 r g 2 = t 50 , 4 σ g 2 = 0.741 × σ g 2 = 0.0004.
σ { + 2 ( J + 1 ) y 1 - 2 J + 2 ( J ) y 2 - 2 ( J - 1 ) - 2 ( J - 1 ) y k + 2 J π { + 2 J y ( k + 1 ) - 2 J + 2 ( J - 1 ) y ( k + 2 ) - 2 ( J - 1 ) + 2 ( J - p + 1 ) y n - 2 ( J - p + 1 ) a = 2 J ( 2 J 2 + J + 1 ) + 4 ( J + 1 ) 2 b = c = 2 J ( J + 1 ) ( 2 J + 1 ) z 1 = 1 k 2 ( J + 2 - i ) y i + k + 1 n 2 ( J + 1 + k - i ) y i z 2 = - 1 k 2 ( J + 1 - i ) y i - k + 1 n 2 ( J + 1 + k - i ) y i g 1 = 1 a - b ( z 1 + z 2 ) g 2 = 1 a - b ( z 1 + a z 2 b ) σ g 1 2 = 1 a - b s 2 σ g 2 2 = a ( a - b ) b s 2 .
( 7 / 2 ) g 1 = 1 a c - b 2 ( c z 1 + b z 2 ) = 1.3755 ( 5 / 2 ) g 2 = 1 a c - b 2 ( b z 1 + a z 2 ) = 1.0902 s = 0.0060 σ g 1 = 0.0022 σ g 2 = 0.0028.