Francisco P. J. Valero, "Analysis of Interferometrically Measured Wavelengths and Their Application as Secondary Standards of Length," J. Opt. Soc. Am. 60, 1675-1680 (1970)

The present paper presents an interpretation of the validity of calculating weighted averages of interferometrically measured wavelengths to be used as secondary standards of length. Weighted averages of all the reported measurements of the wavelengths of thorium are obtained using the Ritz combination principle. A statistical analysis is also performed and the measurements are tested for internal and external consistency as well as for self-consistency. It is shown that there are systematic shifts between the different experiments which, when eliminated, permit calculation of a highly accurate set of Ritz standards that can be used as secondary standards of length. Finally it is concluded that the major part of the observed systematic shifts results from the differences in the standards used to calibrate the interferometers rather than from the light sources. Some of the conclusions of this work are also applicable to other interferometric measurements.

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Comparison between wavelengths belonging to the sets of interferometrically measured Th wavelengths. Columns 2–4 present the average differences between the wavelengths reported by the authors in column 1 and those reported in columns 2–4. For instance in column 2, 1/N Σ_{n}_{−1}^{N} (λ_{X}−λ_{Lw}) represents the average difference between each one of the sets presented in column 1 and the set of wavelengths measured by Littlefield and Wood. Those lines that show very large deviations according to Chauvenet’s criterion were eliminated. The number of lines compared in each case is shown in parentheses. MS stands for Ref. 5; LW for Ref. 2; D et al. for Ref. 3; G et al. for Ref. 4; V for Ref. 6, and GVC for Ref. 8.

MS
$\u3008\begin{array}{c}3263-4576\hspace{0.17em}\AA \\ 4576-7020\hspace{0.17em}\AA \end{array}$

+3.5 (173)

+4.6 (159)

+1.3 (126)

+1.3 (179)

LW

…

+0.6 (104)

+1.4 (139)

D et al.

+5.0 (18)

+5.5 (10)

G et al.$\u3008\begin{array}{c}2687-3263\hspace{0.17em}\AA \\ 3403-4596\hspace{0.17em}\AA \end{array}$

+5.9 (45)

+9.1 (14)

+4.5 (71)

+4.8 (43)

Table II

Average standard deviations for the sets of wavenumber differences used in the statistical analysis. Column 2 presents the average standard deviation for each set as obtained by use of only wavelengths shorter than 4576 Å; column 3 is the same as column 2, but for wavelengths longer than 4576 Å. The numbers in parentheses in columns 2 and 3 are the average wavenumbers of the sets of lines entering in each case in units of cm^{−1}. Column 4 is the average standard deviation for each set of measurements considered as a whole. The average standard deviations were calculated with the formula
$$\frac{1}{I}{\displaystyle \sum _{i=1}^{l}}{{[{\displaystyle \sum _{n=1}^{N}}({w}_{n}{{V}_{n}}^{2}/(N-1){\displaystyle \sum _{n=1}^{N}}{w}_{n}]}_{i}}^{{\scriptstyle \frac{1}{2}}},$$ where w_{n} is given by Eq. (3). I is the total number of energy-level differences entering in the Ritz-combination-principle test for each set of wavelengths. N is the total number of pairs of wavelengths entering in each individual difference between energy levels. V_{n} are the residuals.

Authors

Average SD for wavelengths shorter than 4576 Å (10^{−3} cm^{−1})

Average SD for wavelengths longer than 4576 Å (10^{−3} cm^{−1})

Average SD for all wavelengths included (10^{−3} cm^{−1})

MS

0.88 (25236)

0.53 (17586)

0.85

LW

2.86 (24973)

1.97 (16210)

1.93

GVC+V

0.77 (25598)

0.51 (18413)

0.43

Table III

Comparison between wavelengths belonging to the sets of Davison et al.3 and Giacchetti et al.4 and all the other sets of interferometric measurements of Th. Th i and Th ii lines are included. Those lines that, according to Chauvenet’s criterion, show very large deviations were eliminated.

Relative weights assigned to the sets of interferometrically measured wavelengths according to Eq. (A1).

Authors

Relative weight w_{i}

LW

1.00

MS^{a,b}

5.14

13.15

GVC+V

20.28

All measurements included.
Wavelengths longer than 4576 Å only.

Tables (4)

Table I

Comparison between wavelengths belonging to the sets of interferometrically measured Th wavelengths. Columns 2–4 present the average differences between the wavelengths reported by the authors in column 1 and those reported in columns 2–4. For instance in column 2, 1/N Σ_{n}_{−1}^{N} (λ_{X}−λ_{Lw}) represents the average difference between each one of the sets presented in column 1 and the set of wavelengths measured by Littlefield and Wood. Those lines that show very large deviations according to Chauvenet’s criterion were eliminated. The number of lines compared in each case is shown in parentheses. MS stands for Ref. 5; LW for Ref. 2; D et al. for Ref. 3; G et al. for Ref. 4; V for Ref. 6, and GVC for Ref. 8.

MS
$\u3008\begin{array}{c}3263-4576\hspace{0.17em}\AA \\ 4576-7020\hspace{0.17em}\AA \end{array}$

+3.5 (173)

+4.6 (159)

+1.3 (126)

+1.3 (179)

LW

…

+0.6 (104)

+1.4 (139)

D et al.

+5.0 (18)

+5.5 (10)

G et al.$\u3008\begin{array}{c}2687-3263\hspace{0.17em}\AA \\ 3403-4596\hspace{0.17em}\AA \end{array}$

+5.9 (45)

+9.1 (14)

+4.5 (71)

+4.8 (43)

Table II

Average standard deviations for the sets of wavenumber differences used in the statistical analysis. Column 2 presents the average standard deviation for each set as obtained by use of only wavelengths shorter than 4576 Å; column 3 is the same as column 2, but for wavelengths longer than 4576 Å. The numbers in parentheses in columns 2 and 3 are the average wavenumbers of the sets of lines entering in each case in units of cm^{−1}. Column 4 is the average standard deviation for each set of measurements considered as a whole. The average standard deviations were calculated with the formula
$$\frac{1}{I}{\displaystyle \sum _{i=1}^{l}}{{[{\displaystyle \sum _{n=1}^{N}}({w}_{n}{{V}_{n}}^{2}/(N-1){\displaystyle \sum _{n=1}^{N}}{w}_{n}]}_{i}}^{{\scriptstyle \frac{1}{2}}},$$ where w_{n} is given by Eq. (3). I is the total number of energy-level differences entering in the Ritz-combination-principle test for each set of wavelengths. N is the total number of pairs of wavelengths entering in each individual difference between energy levels. V_{n} are the residuals.

Authors

Average SD for wavelengths shorter than 4576 Å (10^{−3} cm^{−1})

Average SD for wavelengths longer than 4576 Å (10^{−3} cm^{−1})

Average SD for all wavelengths included (10^{−3} cm^{−1})

MS

0.88 (25236)

0.53 (17586)

0.85

LW

2.86 (24973)

1.97 (16210)

1.93

GVC+V

0.77 (25598)

0.51 (18413)

0.43

Table III

Comparison between wavelengths belonging to the sets of Davison et al.3 and Giacchetti et al.4 and all the other sets of interferometric measurements of Th. Th i and Th ii lines are included. Those lines that, according to Chauvenet’s criterion, show very large deviations were eliminated.