New Method for CODATA Adjustment

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A new approach for improving the estimation of

fundamental physical constants (FPC) values

Béla Szakáts,* Aurel Millea**

* AIDRom, Tel. +4021 320 98 70, E-mail: [email protected]

** Romanian Measurement Society, Tel. +4021 666 32 40, Fax +4021 666 47 00

E-mail: [email protected]

Abstract. A new approach to the FPC adjustment is presented, based on the linearisation of the

observational equations by taking the logarithm of the quantities involved. In order to evaluate themerit of the proposed method, called the “weighted logarithmic adjustment” (WLA), a simulation was

performed by applying it to the input data of the 1986 CODATA adjustment. The results obtained show a significant reduction of the uncertainties of the resulting output values.

1. Introduction

The fundamental physical constants (FPC) appear as invariant quantities throughout all of the

formulations of the basic theories of physics and their applications.They are of such importance in

general that their numerical values must be known to an accuracy as high as possible. A particular interest for attaining the highest possible accuracy in the estimation of the FPC values is in metrology,

since the present-day approach in defining the SI units is essentially based on these constants.

Therefore, much effort has been made in order to improve the degree of knowledge of the numerical

values of the most important FPCs. In 1969 a specialized working group was created, the Committee

on Data for Science and Technology (CODATA), which offered in 1973 a first set of accurate values

for a number of physical and chemical constants [1]. It was followed in 1986 by the publication of the

results of a so-called “adjustment” of the FPCs [2], using a least-squares procedure for determining the

values of more than one hundred constants starting from the best existing experimental data,

associated with their estimated uncertainties. The most recent similar collection of values was

produced in 1998, by the same scientific forum [3].

The basic idea behind the determination of the best values of the FPCs by “adjustment” is to start fromcertain “input” quantities and to modify them taking account of the existing functional relations

between them, arriving at an optimal set of “output” quantities. Since the number of these equations is

higher than that of the quantities involved, they can only approximately be met. The problem is to find

the set of output quantities so that the degree of non-conformance to the equations becomes minimal.

The practical way to do this is to divide the data into two groups: auxiliary constants (“exact” values,

by convention) and stochastic input data, which may be expressed in terms of five quantities: K Ω, the

constant relating the SI ohm Ω to the unit of resistance Ω 85BI as maintained by BIPM; K V, the constant

relating the SI volt V to the unit of voltage V 76BI as maintained by BIPM; α -1, the inverse fine-

structure constant;; d 220, the lattice spacing of a perfect cristal of pure silicon at 22,5 °C in vacuum;

μ μ/ μ p, the ratio of the magnetic moment of the muon to that of the proton. Through a least-squares

adjustment the “best” values of these five quantities are obtained as output quantities.



In the three FPC adjustments made by the CODATA group a kind of classical least-squares algorithm

was used, worked out and adapted by the authors. The adjusted values of the five unknown quantities

and hence all the other values that were derived from them – using the auxiliary constants as well –

have finally been based on this least-squares adjustment. In this way, a consistent set of values was

obtained for the FPCs, together with the associated full covariance matrix (since the uncertainties of

many of these constants are correlated, the full covariance matrix is necessary for estimating the

uncertainties of quantities derived from them).

Nevertheless, some criticism has been addressed the 1973 and 1986 adjustments, the most important

points being (a) the assumption of a gaussian distribution and (b) the use of fractional degrees of

freedom.

In this paper a new approach to the FPC adjustment is presented, based on the linearisation of the

observational equations by taking the logarithm of the quantities involved. As a consequence, the

hypothesis of gaussian distribution becomes more plausible. Also, consequently applying the

maximum likelihood principle avoids the use of fractional degrees of freedom. In order to evaluate the

merit of the proposed method, called the “weighted logarithmic adjustment” (WLA), a simulation was

performed by applying it to the input data of the 1986 CODATA adjustment. The results obtainedshow a significant reduction of the uncertainties of the resulting output values, which is another

advantage of the new approach.

2. The observational equations

In the 1986 adjustment the following system of 12 “observational” equations was used:

Ω=Ω Ω K BI 85

A K K A V BI

1

85

−

Ω=

V K V V BI =76

Ω

−

∞

= K K F V R

cE M

BI

em

pm

p 22

485 α

12

485

'

'

)(−

Ω∞= K lo

R

E c

BI p B

p

α γ μ

μ

Ω

−

∞= K K hi V R

E c

BI p B

p22

485

'

'

)( α γ μ

μ

(1)

)()( 220220 Sid Sid =

)()( 3

220

2

82

220 Sid K SiV V

R

E cM

m

em

pm

p −

∞

= α μ

11

021

85)(−

Ω

−= K c R BI H α μ

α α =

p p μ

μ

μ

μ μ μ =



pe

B

p

m

mMhfs

qc R μ

μ μ

μ

μ

μ

α ν 2

3316

)1( +=

∞

where q = 1,00095761(14)

(here and throughout this paper the uncertainty expressed as standard deviation is written in

parantheses at the end of the numerical value) and: 85 BI Ω is the ohm maintained by BIPM (in 1985),

Ω is the SI ohm, and similarly A and V are the corresponding quantities for the amper and the volt

(for the volt the year is 1976); F the farad; M p the molar mass of the proton; c the speed of light in

vacuum; E by definition 483594 GHz/VBI76; R∞ the Rydberg constant; m p/me the proton / electron

mass ratio; α fine-structure constant; γ p the proton gyromagnetic ratio; μ p/ μ B the proton magnetic

moment measured in Bohr magneton; V m the silicon molar volume; μ 0 the magnetic constant; ν Mhfs the

muon fine-structure frequency; me/mμ the electron / muon mass ratio.

3. The WLA method

With the auxiliary constants values given in Table 1, the following five quantities are defined:

(2)

Table 1

Symbol Value Unit Uncertainty (ppm)

M p 1,007276470(12) 10-3

kg/mol 0,01191

c 299792458 m/s 0

R∞ 10973731,534(13) m-1 0,00118

m p/me 1836,152701(37) 0,02015

μ p/ μ B 0,001521032202(15) 0,00986

μ p’/ μ B 0,001520993129(17) 0,01118

μ 0 4π x 10-7 NA

-20

1+ me/mμ 1,00483633218(71) 0,00071

E 483594,0 GHz/VBI76 0

q 1,00095761(14) 0,14

A new set of logarithmic variables y1 - y12 is introduced, defined as follows:

y1 = ln (ΩBI85/Ω) y2 = ln ( ABI85/ A) y3 = ln (V BI76/V ) y4 = ln ( F BI85/ I 1) y5 = ln (γ ’ p(lo)BI85/ I 2) y6 = ln (γ ’ p(hi)BI85/ I 2) (3)

y7 = ln (d 220(Si)) y8 = ln (Vm(Si)/ I 3) y9 = ln (( RH)BI85/ I 4

y10 = ln α y11 = ln ( μ μ/ μ p) y12 = ln (ν Mhfs/ I 5)

as well as the logarithmic unknowns:

p1 = ln K Ω, p2 = ln K V, p3 = ln α , p4 = ln (d 220(Si)), p5 = ln ( μ μ/ μ p). (4)

In this way, the system of observational equations (1) becomes a linear system of 12 equations with 5

unknowns. The 12 rows and 5 columns matrix of this over-determined system is:

14

I em

pm

p

R

cE M =

∞24

'

I R

E c B

p

=∞

μ

μ

382

220 I

R

E cM

em

pm

p =∞

μ

53316

)1( I

qc R

m

me

B

p

=+

∞

μ

μ

μ

4021 I c = μ



(5)

where the uncertainties of the elements are zero.

Direct calculation with formulas (2) using the auxiliary constants values of Table 1 gives the following

values for the quantities I i (i = 1,…,5):

I 1 = 1811870597(60) CBI85/mol;

I 2 = 5023594039(62) × 103

CBI85/kg or 1/(sTBI85)

I 3 = 2334122528(78) × 1017

1/mol (6)

I 4 = 188,365156730885… Ω

I 5 = 263294549(40) × 105

s-1

4. Simulation using the 1986 CODATA adjustment data

The values of the input data used in the 1986 CODATA adjustment [2] and those of the

corresponding logarithmic variables are given in table 2.

Table 2

Quantity Date (Unit) Laboratory Value y

1. ohm, Ω BI85 Ω -0,000001534(69)

1.1 1964-85 NML 0,99999854(14) -0,000001460(140)

1.2 1973 NBS 0,99999830(11) -0,000001700(110)

1.3 1977-84 ETL 0,99999840(23) -0,000001600(230)

1.4 1981 NPLI 0,99999850(36) -0,000001500(360)

1.5 1983 NPL 0,99999868(14) -0,000001320(140)

2. amper, ABI85 A -0,000002069(2533)

2.1 1956 NBS 0,9999974(84) -0,000002600(8400)

2.2 1963 NPL 0,9999982(59) -0,000001800(5900)

2.3 1966-69 VNIIM 0.9999986(61) -0,000001400(6100)

2.4 1967 NBS 1,0000027(97) -0,000002700(9700)

2.5 1972 ASMW 1,0000032(79) -0,000003200(7900)

2.6 1976 NPL 1,0000062(41) -0,000006200(4100)

3. volt, V BI76 V -0,000007855(582)

3.1 1982-83 LCIE 0,9999967(24) -0,00000330(2400)

3.2 1985 NML 0,99999186(60) -0,000008140(600)

10200

10000

00100

00101

03120

01000

00221

00201

00221

00010

00011

00001

−−

−

−

−

−

−



4. farad, F CBI85/mol -9,8404724(14)

4.1 1975-84 NBS 96486,00(13) -9,8404724(14)

5. gyrom. ratio (lo) 104

s-1

/TBI85 -9,840486754(209)

5.1 1968 ETL 26751,180(87) -9,840492502(3265)

5.2 1971-76 NPL 26751,177(15) -9,840492614(573)5.3 1977 NIM 26751,399(22) -9,840484315(835)

5.4 1978 NBS 26751,3719(64) -9,840485328(252)

5.5 1980 VNIIM 26751,241(17) -9,840490222(648)

5.6 1981 ASMW 26751,412(57) -9,840483829(2143)

6. gyrom. ratio (hi) 104

CBI85/kg -9,840475269(927)

6.1 1961-64 Kh GIMIP 26751,30(15) -9,840488016(5620)

6.2 1974 NPL 26751,676(27) -9,840473961(1022)

6.3 1981 NIM 26751,564(96) -9,840478148(3601)

6.4 1983 ASMW 26751,466(86) -9,840481811(3227)

7. Si lattice spacing pm -22,373443185(91)

7.1 1973-76 NBS 192,015904(19) -22,373442914(99)

7.2 1981 PTB 192,015560(45) -22,373444706(234)

8. Si molar vol. 10-6

m3/mol -72,0405637(12)

8.1 1973 NBS 12,058808(14) -72,0405637(12)

9. Hall resistance ΩBI85 4,920245154(62)

9.1 1985 PTB 25812,8469(48) 4,920245187(186)

9.2 1983-84 NBS 25812,8495(31) 4,920245288(120)

9.3 1984 ETL 25812,8432(40) 4,920245044(155)

9.4 1984 NPL 25812,8427(34) 4,920245024(132)

9.5 1984 VSL 25812,8397(57) 4,920244908(221)

9.6 1984-85 LCIE 25812,8502(39) 4,920245315(151)

10. 1/α 4,920243627(65)

10.1 1981-84 W/Co. U. 137,0359942(89) 4,920243623(65)10.2 1984 Yale U. 137,036041(82) 4,920243964(598)

11. μ μ/ μ p 1,15793269(29)

11.1 1982 Los Alamos U. 3,1833461(11) 1,15793288(35)

11.2 1982 SIN 3,1833441(17) 1,15793225(53)

12. ν Mhfs kHz -8,68255450(29)

12.1 1982 L.A./Y.U. 4463302,88(62) -8,68255450(29)

By applying the maximum likelihood principle to the over-determined system (5) the following five-

equation five-unknown system is obtained:

; j= 1,…,5 (7)

where xk (i)

(i = 1,…,12, k = 1,…,5) represents the element on row i and column k in matrix (5),

yi = 1,…,12 are the logarithmic input quantities in Table 2 column 5, σ i = 1,…,12 are their uncertain-

ties and pk (0)

(k = 1,…,5) are the unknowns of the adjustment.

Solving the system (7) yields the following results:

p1(0)

= - 0,000001435(49)

p2(0)

= - 0,000006776(285)

p3(0)

= - 4,920243734(44) (8)

p4(0) = -22,373443250(89)

p5

(0)

= 1,157932829(210)

∑ ∑∑ = == =

12

1

12

12

)(

2

)()(5

1

)0(

i i i

i

ji

i

i

j

i

k

k k

x y x x

p σ σ



with the covariance matrix:

2,390 × 10-15

1,330 × 10-16

-8,382 × 10-16

1,811 × 10-17

0,838 × 10-15

1,330 × 10-16

8,104 × 10-14

10,018 × 10-16

2,642 × 10-15

-1,002 × 10-15

-8,382 × 10-16 10,018 × 10-16 1,922 × 10-15 1,345 × 10-18 -1,922 × 10-15

1,811 × 10-17

2,642 × 10-15

1,345 × 10-18

7,960 × 10-15

-1,345 × 10-18

0,838 × 10-15 -1,002 × 10-15 -1,922 × 10-15 -1,345 × 10-18 4,397 × 10-14

which corresponds to the values

K Ω = 0,999998565(49)

K V = 0,999993224(285)

α -1 = 137,0360094(60) (9)

d 220 = 192,015839(17)

μ μ/ μ p = 3,18334595(67)

The comparative presentation of the input quantities, of the 1986 CODATA adjustment results and of

the results of the simulation by the WLA method described here is given in the following table:

Table 3

Input 1986 CODATA adjustment Simulation by the WLA method

K Ω 0,999998466(69) 0,999998476(92) 0,999998565(49)

K V 0,999992145(582) 0,999992760(540) 0,999993224(285)

α -1 137,0359947(88) 137,0359960(110) 137,0360094(60)

d 220 192,015852(18) pm 192,015553(74) pm 192,015839(17) pm

μ μ/ μ p 3,18334551(92) 3,18334571(87) 3,18334595(67)

5. Conclusion

A new method, the “weighted logarithmic adjustment” (WLA), for the FPC adjustment is proposed,

based on the linearisation of the observational equations by taking the logarithm of the quantities

involved. In order to evaluate the merit of the method, a simulation was performed by applying it to

the input data of the 1986 CODATA adjustment, allowing a direct comparison to be made between the

resulting output quantities by the CODATA method and the WLA method. Table 3 shows that the

associated uncertainties are lower for all of the five parameters, the most spectacular improvement

affecting d 220 (uncertainty more than 4 times lower), while for K Ω, K V and 1/α - the improvement isroughly 2 times. At the same time, all the five values are slightly higher than those of the CODATA

adjustment, the significance of which has still to be clarified, being beyond the scope of this paper.

References

[1] CODATA: Recommended Consistent Values of the Fundamental Physical Constants: 1973, CODATABulletin nr. 11, 1-7, 1973

[2] Cohen, E.R and Taylor, B.N: The 1986 Adjustment of the Fundamental Physical Constants, CODATABulletin Nr. 63, Nov. 1986; Reviews of Modern Physics, Vol. 59, No. 4, 1121-1148, 1987

[3] Mohr, P.J. and Taylor, B.N: CODATA Recommended Values of the Fundamental Physical Constants: 1998,Reviews of Modern Physics, Vol. 72, No, 2, 351-495, 2000

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New Method for CODATA Adjustment