1, 1957
2a Semestre
SUPPL:E-~EbTTO AL V O L U M E VX, S E R I E X
N.
DEL NUOVO CIhfENTO
Mathematical Analysis of the Universal Physical Constants.
E. I~. C0~E~
Atomics International
A Division of North American Aviation, Inc., Canoga Park, California
and California Institute of Technology - Pasadena, California
I n t h e previous p a p e r Professor I)U~C[OND has presented a wealth of experi m e n t a l d a t a on the universal physical constants (primarily those k n o w n as
the a t o m i c constants). This m a t e r i a l is m a d e up of a score or m o r e individua[
e x p e r i m e n t a l results. N o t all of these are of equal e x p e r i m e n t a l accuracy.
S o m e of t h e m m a y be combined a m o n g themselves to yield a simple d a t u m
which represents an average value of several i n d e p e n d e n t evaluations of t h e
same physical quantity. I n w h a t e v e r w a y this m a y be done we still find ourselves with m o r e d a t a t h a n are required to obtain an evaluation of the f u n d a m e n t a l physical constants. This m a y be expressed b y a classical e x a m p l e
(the e x a m p l e is classical in t h a t it represents one of the first instances in which
the existence of the o v e r d e t e r m i n e d n a t u r e of the e x p e r i m e n t a l d a t a b e c a m e
forcefully evident.)
The value of the f a r a d a y constant as m e a s u r e d b y the silver v o l t a m e t e r
is [11 F = 5Ve = 9 6 5 1 . 2 9 ~ 0.19 e m u / m o l e ; t h e a t o m i c weight of the electron
as inferred f r o m the isotopic shift i n the B a l m e r series for h y d r o g e n a n d deut e r i u m is [2] N m = (54.895 ± 0.008).10 -~ amu. F r o m this we can calculate
a value for t h e specific electronic charge
e/m =- (1.75814 !
0.00026).107 e m u / g .
I n comparison with this t h e vatue of elm d e t e r m i n e d f r o m m e a s u r e m e n t ~
of the acceleration and deflection of electrons in electromagnetic fields yielda
a value [3]
elm ~- (1.75959 :j: 0.00038).107 e m u / g .
)fA_TItEM[ATICAL A N A L Y S I S OF T H E
UNIVERSAL PHYSICAL CONSTANTS
]1I'
We could of course have equally well used the measured values of the
atomic weight of the electron and the specific electronic charge to calculate
an (( indirect >) value of the F a r a d a y constant~ or conversely use the specific
electronic charge a n d the F a r a d a y to find an indirect value of the electron'~
atomic weight. These calculations are summarized in Table I. W e are then
TABLE I. -- Compariso~ o] direct and i~tdirect values ]rom a sim ole overdetermined system.
1~ = 2¢e
(cmu/mole)
Direct Values
Indirect Values
Nm
(amu)
e/m
(emu/g)
1.759 59 ±0.000 36). 10v
9651.29±0.19
(54.895±0.008)' I0 -~
(Nm) (e/m)
W(elm)
WNm
9659.32±1.83
54.849 :j::O.O06
1.758 14 :~0.000 26
presented with the embarrassing problem of choosing which consistent set o f
three numbers ought to be used; each choice is certainly poor in itself since
each is contradicted b y the other two. I t should be emphasized t h a t the
numbers presented in this table are illustrative and although based on direct
measurements~ do not represent in any w a y a selection of ~<best >~data or are
t h e y to be considered as r e c o m m e n d e d numbers. They are presented in order
to show the nature of the difficulty (which is actually c o m p o u n d e d many-fold)
when we a t t e m p t to analyze the experimental d a t a on the atomic constants,
considering these d a t a as a single unified complex of experimental results.
The experiments described in the preceding paper measure various combinations of the atomic constants. I n some experiments the same combination
or the same constant is measured in different w a y s ; in others a m e a s u r e m e n t
is made of a q u a n t i t y whose value is also deducible from combining the results
of two or more different and independent experiments. Thus we are confronted
with an inter-related complex which is similar (in an abstract form) to the
situation which is found most often in geodetic triangulation. Let us therefore look more closely at the structure of our present problem .
E a c h of the quantities whose measurement is described in the preceding
paper can be expressed in terms of h, e, m, N , A , c, and a few other auxiliary
quantities whose numerical values are accurately known. The R y d b e r g constant,
however, which is a function of h, e, m, and e, is m u c h more accurately m e a sured t h a n are a n y of its component factors. I t is therefore convenient to
consider the g y d b e r g constant as an exact numerical q u a n t i t y and to use it
to express the electron/mass as a k n o w n function of the other variables. I n
addition, to the accuracy required in the discussion to follow, it is convenient
to treat the velocity of light, c, as an exactly k n o w n quantity. F u r t h e r m o r e ,
112
E.R. COHEN
because of the fact t h a t the fine structure constant, s, is m o r e accurately det e r m i n e d b y direct observation of fine structure in h y d r o g e n t h a n it can be
c o m p u t e d b y combining, say m e a s u r e m e n t s of e, h/e and c, it is also convenient
to use s as a variable in place of h. Thus we shall choose to express all of the
e x p e r i m e n t a l results in t e r m s of the quantities ~, e, £V, a n d A which we m a y
call the p r i m a r y unknowns of our analysis. The result of a n y e x p e r i m e n t
can t h e n be described as measuring (except for quantities which can be considered as accurately calculable correction factors) some p r o d u c t of pbwers
of the p r i m a r y variables of the f o r m
(1)
](s, e, 2V, A) = c~%bN~A ~ = A(1 ± a ) .
T h e second expression is not actually an equation; it is a s h o r t - h a n d m e t h o d
for indicating t h a t we do not k n o w the true value of the r i g h t h a n d side of
t h e equality b u t t h a t the e x p e r i m e n t can be i n t e r p r e t e d Us indicating a value
which is defined only to the e x t e n t of a p r o b a b i l i t y distribution with m e a n
v a l u e A and with relative s t a n d a r d deviation a. This i n t e r p r e t a t i o n is n o t
t h e only one however a n d we shall discuss a different i n t e r p r e t a t i o n later.
F o r convenience in analyzing the d a t a it is useful to linearize our equations.
W e a d o p t origin values s0, eo, No, Ao, which h a v e been chosen sufficiently
close to our expected solution t h a t a n y set of values s, e, .Y, A (any point
in our (( constants-space ))) in which we are likely to be interested will differ
f r o m the origin values b y small amounts. (~Small ~>is d e t e r m i n e d here b y the
m a g n i t u d e s of the second derivatives of the function ]; we assume t h a t we
can e x p a n d f(e, s, 2¢,A) as a m u l t i v a r i a t e T a y l o r series a b o u t the origin point,
and t h a t in such an expansion only the linear t e r m s need be retained. I t is
in practice m o s t convenient to e x p a n d the logarithm of ] in a Taylor series;
we t h e n deal with relative deviations a n d if the function is a simple p r o d u c t
of powers of the variables as is the case here, we obtain, in place b y E q u a tion (1), the linearized equation
(2)
axe, -~ bx~ -~ cx~ ~ dx A = h ~ a,
where x~ ~ ( ~ - - ~ 0 ) / S o ; x~ = (e--eo)/eo, etc., a n d h = ( A - - A o ) / A o in which
A0 is the value of the function / e v a l u a t e d with the origin values So, eo, No, Ao.
1. - A g e o m e t r i c a l
interpretation.
There are various w a y s in which we can i n t e r p r e t the e x p e r i m e n t a l d a t a
geometrically. Professor BIRGE in his p a p e r has discussed various t y p e s of
diagrams [4-7] which he has used to aid him in evaluating (~best )> or (~m o s t
~ I A T I t E M & T I C A L A N A L Y S I S O F TIn~ U N I V E R S A L P H Y S I C A L C O N S T A N T S
113
consistent )) values of the atomic constants whose n u m e r i c a l m a g n i t u d e s he
seeks. These geometrical descriptions are all representations (in t e r m s of
various t y p e s of projections) of the following geometrieM structure.
W e m a y represent eaeh e x p e r i m e n t a l d e t e r m i n a t i o n as defining a single
functional relationship a m o n g the various physical constants. W e m a y consider t h a t an i n d e p e n d e n t set of constants forms an o r t h o g o n a l co-ordinate
s y s t e m in a (( constants-space ~). E a c h functional relationship (each experiment)
t h e n is represented b y a surface in this space. An a r b i t r a r y p o i n t in t h e space
{a given set of values of the constants, say ~, e, N, A) is in a g r e e m e n t with
t h e e x p e r i m e n t a l result if it lies in the surface. A p o i n t ( t h a t is, a set of values
for the constants) is therefore consistent with several e x p e r i m e n t M results
if it lies a t the c o m m o n intersection of the surfaces. I n general if we h a v e
m o r e e x p e r i m e n t s (surfaces or e q u a t i o n s ) t h a n we h a v e constants to be determ i n e d (dimensionality of the space) it is not a priori evident t h a t there will
be a point t h r o u g h which every surface passes. I f the e x p e r i m e n t s are reasona b l y consistent there will be a point, however, which is close to all of the
surfaces. The best choice for the set of physical constants is t h a t point which
is (( closest ~) to all of the su1"faces. The definition of (~distance ~) in this space
(that is, the definition of a metric) we shall discuss below; for the m o m e n t
let us assume t h a t we can define distance. W e should t h e n w a n t to consider
a metric function defined with regard to the point in question a n d the existing
surfaces. The p r o b l e m we t h e n set for ourselves is to find t h e point which
minimizes this function.
This leads us to a second geometrical r e p r e s e n t a t i o n of the problem. A t
each point in our (( constants-space )) we h a v e defined t h e v a l u e of a certain
function. W e now consider a space of one higher dimensionality and consider
a hyper-surface whose distance f r o m an orthogonal subspace (the original
constants-space), is defined at each point b y the value of a certain function
(the exact f o r m of this function we h a v e n o t y e t specified). W e t h e n seek
the point on this surface which is a m i n i m u m . I t is possible to m a k e several
general r e m a r k s a b o u t the f o r m of the function, which we shM1 call Q. I t
m u s t p r i m a r i l y be defined in such a w a y t h a t its value is i n d e p e n d e n t of the
actual variables used to describe it. This m e a n s t h a t Q is a scalar with respect
to a t r a n s f o r m a t i o n or r o t a t i o n in the (( constants-space ~). I n addition it should
also be i n v a r i a n t with respect to non-essential modifications in the f o r m in
which the cxperime~atal results are presented. Again a simple e x a m p l e m a y
b e in order.
L e t us suppose t h a t one of our e x p e r i m e n t a l equations can be w r i t t e n in
the f o r m
(3)
ax~ ÷ bx~ ÷ cx~ = h ± a,
which states t h a t the e x p e r i m e n t a l result h was obtained with a s t a n d a r d
8 - Supplemento
al N u o v o
Cimento.
114
~. n.
COHEN
error ~= a. If we t r a n s f o r m variables f r o m x~, x2, x3 to a new set of variables
Y~, Y2, Y3, we obtain in place of this equation the expression
(3a)
T h e function Q must be defined in terms of the coefficients a, b, c, (and t h e
corresponding coefficients of all of the other experimental data) in such a w a y
t h a t Q(~, fi, y; y l , y.2, ya) should be equal to Q(a, b, c; xl, x2, x~). F u r t h e r m o r e
Q must be invariant with respect t o t r a n f o r m a t i o n of the equations; the most
easily recognized t y p e of function transformation would be the r e p l a c e m e n t
of equation (3) b y the expression!
naxl ÷ nbx2 ÷ ncx3 = nh 4- n(~ .
(3b)
W h e n this is done it is perhaps suggestive t h a t all of the e x p e r i m e n t a l equations ought to be written in a normalized form
a
(4)
-xl
b
÷-x~
c
÷-x3
=-
d
± ].
W h e n the equations are written in this form all of the equations have equal
accuracy and we might t h e n in~er t h a t t h e y ought to be given equal weight
in the analysis. At this point we use the word (( weight ~ in a qualitative sense;
later we shall give a q u a n t i t a t i v e definition which will formalize the intuitive
notion t h a t data with small s t a n d a r d errors should be given more weight or
importance and data with larger errors should be given less weight in a n y
analysis.
2. -
The
criterion
of l e a s t s q u a r e s .
Until now we have purposely made only v e r y general s t a t e m e n t s a b o u t
the problem of finding the best or most consistent set of unknowns from a
set of overdetermined equations. I t m a y perhaps be a p p a r e n t t h a t we would
like to introduce the criterion of least squares in a logical and completely
deductive manner. I t is almost certain t h a t this cannot be done; we can
however carry our deductive development to a certain point, t h e n inductively
introduce the least squares criterion and finally justify the choice (at least
partially) b y considering the conditions which it must satisfy. W e could
specify t h a t the function Q Which has been previously mentioned shall be
defined as the sum of the squares of the deviation of the point from each of
Z[ATHEI~ATICAL A N A L Y S I S OF THE
UNIYERSAL P~IYSICAL CONSTANTS
115
the experimentally defined surfaces, each deviation normalized b y dividing it
b y the s t a n d a r d deviation of the experimental measurement. The eondition~
t h a t Q be a m i n i m u m is therefore the usual condition of (~least squares ~) which
was first f o r m u l a t e d b y GAuss a b o u t 1820 [8].
There h a v e been m a n y who have objected to the use of the m e t h o d of least
squares on the grounds t h a t it assumes the Gaussian curve for the p r o b a b i l i t y
distribution of the error in a n y particular measurement. The objection is
indeed valid t h a t the assumption of the Gaussian distribution is often unw a r r a n t e d in m a n y experimental configurations. The real problem at issue,
however, is one of determining a (( best ~) set of values t h a t can be c o m p u t e d
from an over-determined system of equations, and this is essentially the problem
of determining an analytic basis on which one can define the adjective (,' best. ~
The condition of least squares serves as one such analytic criterion.
This says nothing in itself of w h a t could be calldd the physical interpretation of the criterion. I t is recognized, in general, t h a t the m e t h o d of
least squares corresponds to the ((Axiom of M a x i m u m Likelihood ~), if t h e
distribution functions of all the errors are Gaussian [9]. GAuss himself was,
able to justify the m e t h o d on a m u c h wider base, and in 1821 he published
a t h e o r y which replaces this axiom with an (( A x i o m of Minimum E r r o r ~) or
(( Axiom of M a x i m u m Weight. ~) [10]. The definition of (( best ~ is n o t to b e
m a d e on the basis of t h a t solution which is most likely to be correct, b u t on
t h a t combination of d a t a which yields the most accurate result and to which
can, therefore, be a t t a c h e d the greatest statistical weight. (The statistical
weight of a statistical variable is defined in the present sense as the reciprocal
of the variance, i.e., the reciprocal of the mean square error, of the q u a n t i t y . )
Consider t h e n an overdetermined set of n equations expressing relationships
between q variables. We can find an infinite n u m b e r of solutions for the q
variables depending on how we choose to combine the equations. W e ca~
think of this process as one in which some set of ~ - 1 of the equations are
used to express q - 1 of the variables in terms of one particular variabler
s~y x~. These values are then to be substituted into the remaining n - q + ]
equations to give a set of n - q + l values for the variable x~. A n y of these
numerical values is a possible choice for the variable x~ and, in general we
would want to take some weighted average of these numerical values. The
numerical value of such an average depends both on the weights a t t a c h e d to
the elements t h a t go to m a k e up the average and on the numerical values of
these elements. Ultimately, therefore, the value ascribed to x~ depends on
n - - q independent and urbitrary parameters which specify the m a t h e m a t i c a l
form of the average and n quantities which are either the observational numerics or quantities directly deduced from t h e m (such as relative deviations
from a set of origin values) which determine the numerical value of the average.
There are only n - - q a r b i t r a r y parameters r a t h e r t h a n n - - q + l
because t h e
116
E.R.
COHE~
choice of the n - q ~ - i weights is restricted b y the condition t h a t their s u m
is unity.
T h e n u m e r i c a l value of xl is thus expressed as a linear c o m b i n a t i o n of n
n u m e r i c a l quantities, each of which has associated with it a m e a n square error.
I f t h e numerical quantities are observationMly independent, we can assert
t h a t t h e m e a n square error of xl is the s u m of certain coefficients depending
on the n - - q free p a r a m e t e r s , times the m e a n square errors of the n observ a t i o n M numerics. The A x i o m or Condition of M i n i m u m E r r o r states t h a t
t h e ~ best ~) choice for xl is t h a t one whose error is a m i n i m u m with respect
to the possible v a r i a t i o n of the free p a r a m e t e r s . This condition is equivalent
to the condition of least squares, a l t h o u g h the results cannot, in general, be
identified as corresponding to t h a t set which has m a x i m u m likelihood except
in the case when the distribution function for the errors is specified to be
Gaussian. This is however, a n a d v a n t a g e for one can easily c o n s t r u c t distributions (for example, r e c t a n g u l a r distributions) for which the condition of
m a x i m u m likelihood has no unique solution. F u r t h e r m o r e , the d e v e l o p m e n t
of the condition of m i n i m u m error is quite general in regard to t h e forms of
t h e error distribution functions; all t h a t is specified is the m e a n square error,
so t h a t t h e range of applicability of the t h e o r y of least squares is e x t e n d e d
f r o m Gaussian distributions to t h e m u c h larger class of distributions with
finite second m o m e n t s .
As an e l e m e n t a r y e x a m p l e of t h e m e t h o d a n d in order ot clarify the concepts involved, let us consider a p r o b l e m which is perhaps the simplest
possible example. W e h a v e two m e a s u r e m e n t s of the q u a n t i t y x; these two
m e a s u r e m e n t s are a~ and as, in general, al=/=a~. W e also assume t h a t each
m e a s u r e m e n t represents a single selection f r o m a universe of values. W e let
t h e p r o b a b i l i t y distribution of the first m e a s u r e m e n t be PI(~) such t h a t t h e
p r o b a b i l i t y is P~(~)d~, t h a t the result of a m e a s u r e m e n t of the q u a n t i t y x
b y the specified procedure shall lie b e t w e e n the values ~ a n d ~ - d ~ . Similarly,
t h e second m e a s u r e m e n t of x is t o be characterized b y t h e p r o b a b i l i t y distrib u t i o n P~(~). W e need not m a k e a n y detailed specification of the f o r m of the
distribution functions P1 und P~; it is n o t even necessary t h a t the two functions h a v e similar form. W e impose u p o n t h e m only the restriction t h a t the
distributions huve finite second m o m e n t s .
I f we h a v e two m e a s u r e m e n t s of x t h e n we can t a k e some a v e r a g e value
to use as t h e (~best)~ choice. There are, however, several different choices
for this average and we can write
(5)
xo(~) = ~a~ ÷ (1
--
~)a~,
where ~ is a n y real number, although intuitively we would prefer t h a t 0 < ~ < 1
because this is the condition t h a t xo lies between the values a~ a n d as. Defined in this way, the m e a n of the universe of values of x0 is
MATHEMATICAL
(6)
ANALYSIS
"~o - - f f [ ~
OF
+
THE
UNIVERSAL
PItYSICAL
CONSTANTS
11~
(1 --a)~I]P~(~)P2(~) d~ d ~ .
E a c h p r o b a b i l i t y d i s t r i b u t i o n is a s s u m e d t o b e n o r m a l i z e d a n d if t h e r e a r e
n o s y s t e m a t i c e r r o r s i n e i t h e r m e a s u r e m e n t , t h e e x p e c t a t i o n v a l u e of e a c h
m e a s u r e m e n t is x.
Therefore we find
(7)
5o = ~ x + ( 1 - - ~ ) x
= x,
so t h a t t h e e x p e c t a t i o n v ~ l u e of t h e a v e r a g e is i n d e e d t h e q u a n t i t y w e a r e
t r y i n g t o m e a s u r e , i n d e p e n d e n t of t h e p a r a m e t e r ~ w h i c h d e t e r m i n e s t h e p a r t i c u l a r a v e r a g e . B u t w e n o w a s k t h e q u e s t i o n : (( H o w a c c u r a t e is t h i s a v e r a g e ;
w h a t is i t s s t a n d a r d d e v i a t i o n ? ~ I f w e l e t s 2 b e t h e m e a n s q u a r e d e v i a t i o n
of t h e u n i v e r s e f r o m w h i c h xo is e x t r a c t e d , w e h ~ v e , b y t h e u s u a l d e f i n i t i o n ,
(s)
e2 : f f [ a ~ -}- (1
- - a)~] - -
x]"P~(~)P2(*])d~ d~ =
T h e t w o i n t e g r a l s in t h e l a s t f o r m of E q u a t i o n (8) a r e t h e v a r i a n c e s (the m e a n
s q u a r e e r r o r s ) , r e s p e c t i v e l y , of t h e first a n d of ~he s e c o n d m e a s u r e m e n t s . T h e s e
q u a n t i t i e s a r e d e f i n e d t o b e a l2 a n d a2"
~ T h e e x p r e s s i o n for t h e e r r o r i n t h e
a v e r a g e in t e r m s of t h e e r r o r s of t h e n u m b e r s e n t e r i n g i n t o t h e a v e r a g e is,
therefore,
(9)
2
22
W e f i n d t h a t , a l t h o u g h t h e e x p e c t a t i o n v a l u e of t h e a v e r a g e is i n d e p e n d e n t
of a, t h e e r r o r i n t h e a v e r a g e is a f u n c t i o n of ~. F o r ~ ~ 0 w e h a v e s 2 = a~
a n d for a = 1 w e h a v e s 2 = a l2., w e m a y r e a s o n a b l y a s k w h e t h e r a p r o p e r
choice of a m i g h t n o t r e s u l t i n ~ v a l u e of e 2 w h i c h is s m a l l e r t h a n e i t h e r of
t h e s e a n d i n d e e d w h a t v a l u e of ~ l e a d s t o a m i n i m u m v a l u e of s *. I t is e a s i l y
s h o w n t h a t t h e m i n i m u m v a l u e of s 2 is
(lO1
w h i c h is a c h i e v e d w h e n ~ =
22
2
~o2 = ~1~/(G1+~1
2
2
a~/(a~
+ a~)
a n d t h e c o r r e s p o n d i n g v a l u e of xo i s
118
~. R. COHEN
The (( w e i g h t s , to be a t t a c h e d to m e a s u r e d q u a n t i t i e s in order to c o m p u t e
t h e (~b e s t )) average are therefore p r o p o r t i o n a l to t h e reciprocal of the m e a n
square error of each m e a s u r e m e n t . I f we define the statistical weights
~. =11.~,
w~ =1/o~,
we h a v e two i m p o r t a n t formulas
(w~a~+ w~a~)/(w~+ ,w~) ,
(125)
xo =
(]2b)
wo = 1/e~ = w~ + w~.
The first of these essentially justifies the n o m e n c l a t u r e of (~statistical weight ,~
f o r t h e reciprocal variance, since it is this q u a n t i t y t h a t determines the imp o r t a n c e of the m e a s u r e m e n t in the c o m p u t a t i o n of the average. The second
f o r m u l a shows t h a t when this (( best )~ average is obtained the weight of the
result ( c o m p u t e d as the reciprocal of the variance of the average) is just the
s u m of t h e weights of the c o m p o n e n t s and, furthermore, t h a t this is the
m a x i m u m weight ascribable to a n y average. An other linear c o m b i n a t i o n
of the two observations would h a v e a weight which is less t h a n the sum of
t h e individual weights a n d i s , therefore, an inefficient a v e r a g e t h a t wastes
weight. The statistical weight of the average is a m a x i m u m (equal to the
s u m of the weights of the individual components), w h e n the weighting e m p l o y e d
in c o m p u t i n g the average is d e t e r m i n e d b y the statistical weights of the components.
3. - A g e n e r a l i z e d theory of leav t squares.
The e x a m p l e which we h a v e just considered can be i m m e d i a t e l y generalized
to consider t h e c o m b i n a t i o n of severM m e a s u r e m e n t s instead of only two.
F u r t h e r m o r e it is also r a t h e r s t r a i g h t f o r w a r d (although lengthy) to e x t e n d
t h e concepts introduced a b o v e into the p r o b l e m of finding the best set of
values of several simultaneous variables related b y linear equations. This is
of course the sort of s y s t e m in which we are interested once we h a v e linearized
the equations of observation as previously indicated.
W e write the linearized s y s t e m in the f o r m
allXl
(13)
- ~ a12X2 --~ 5 1 3 2 3 --~ o.. ~ - a l q X q =
C1
a21xl Jr- a2~x2 4:- a~3x3 ~ ... ~ a2~x~ --: c~
.....
• .o..
. . . . . . . . . .
.,
~'VfATHE.¥[ATICAL A N A L Y S I S O F T H E U : N I ~ E R S A L P H Y S I C A L COI~STANTS
119
We shall use R o m a n letters to indicate the indices which range from 1
~o q a n d which refer to the unknowns, and Greek letters for those which range
from 1 to n and refer to the observational equations.
Since the c, are assumed to be independent, the errors ~, in % are urncorrelated and
if v = # ,
(14)
-0
if v :/: #.
We wish to find x~ as a linear combination of the c w i t h appropriate
coefficients, ~ chosen so as to determine the x~ with m a x i m u m accuracy or
m i n i m u m error. Thus we write
(15)
,a
/zj
This equation is to be understood as follows. E a c h of the n individual
equations of observation (identified b y the index # is to be multiplied b y a
number, ~ , there being a set of n such numbers for each variable x~. These
numbers are to be chosen such that, when the n equations are then summed,
the coefficient of each x, other t h a n the specified x~, is zero, while the coefficient of x~ is unity. This implies t h a t there are q conditions on each of the
q sets of n multipliers. This is not sufficient to define the multipliers since
n > q. The additional conditions are obtained b y requiring t h a t we choose
~hat linear combination for each variable x~ which will h a v e the smallest
variance a n d hence the largest possible weight. I t can be shown [11] t h a t
such a condition leads to e x a c t l y the same values of x/ as would be obtained
from the condition t h a t we minimize the quadratic form
(16)
Q = ~(a.~x~ ÷ a~2x~~-... ~-a.¢x~-c~) ~.
tTiz
H e n c e the condition of Minimum E r r o r is just the least squares condition and
is therefore equivalent to the condition of M a x i m u m Likelihood of the more
restricted Gaussian case. I t is, however, m u c h more general t h a n the latter.
No restriction is p u t on the probability distributions of the i n p u t errors, ~ ,
in the % other t h a n t h a t these distributions m u s t have finite second moments,
i.e., the % m u s t exist. Subject to this condition, t h e y m a y h a v e more t h a n
one ~(mode ~ or indeed any arbitrary shape; nor is it necessary t h a t the
distribution functions all have the same form.
120
]~. ~. COHEN
This generality of the significance of the least squares a d j u s t m e n t w h e n
s t a t e d in t e r m s of the s e c o n d m o m e n t s of the error distribution is of g r e a t
i m p o r t a n c e . I t emphasizes also the desirability of a d o p t i n g the r o o t - m e a n square deviation as a m e a s u r e of error in preference to such error m e a s u r e s
as the (~p r o b a b l e error ~), or the (( m e a n absolute error. ~) F o r a gaussian distribution, the three measures, r o o t - m e a n - s q u a r e error, m e a n absolute e r r o r
a n d p r o b a b l e error (that error which divides the distribution c u r v e into equal
areas, so t h a t t h e probabilities of errors of absolute m a g n i t u d e g r e a t e r t h a n
or less t h a n t h e p r o b a b l e error are equal) s t a n d in the ratios1:0.798 0:0.674 5.
I t is h o w e v e r a m i s t a k e to t h i n k of the different ei~or measures as simply
expressing the same error spread on different scales. W h e n we do n o t l i m i t
ourselves to gaussian distribution the r o o t - m e a n - s q u a r e error or s t a n d a r d e r r o r
enjoys a position of far greater statistical significance a n d generality t h a n
do the others. This is because the s t a n d a r d deviation has a simple r e p r o d u c t i o n
p r o p e r t y (for a n y f o r m of p r o b a b i l i t y distribution) which i's not shared b y a n y
other p a r a m e t e r with the same generality.
I t m a y even b e worthwhile to avoid the discussion of p r o b a b i l i t y distributions a n d variances entirely a n d speak only of statistical weights a n d t h e
weight to be assigned to each e x p e r i m e n t a l n u m b e r or to a n y n u m b e r t h a t
results f r o m arithmetic combinations of e x p e r i m e n t a l data.
I t can b e shown~ i n d e p e n d e n t l y of the d e v e l o p m e n t which led to E q n a tion (16) t h a t if the weight to be assigned t o a r a n d o m variable is to be in
accord with certain e l e m e n t a r y axioms regarding the definition of weight a n d
a t the same t i m e be a function only of the s t a n d a r d error of the variable, t h e n
t h a t function is uniquely d e t e r m i n e d to ~be the reciprocal of the variance.
T h e weight of a q u a n t i t y should therefore be defined in such a w a y t h a t i t
is consistent w i t h the identification as the reciprocal of t h e v a r i a n c e in those
cases where a variance can be e x p e r i m e n t a l l y determined. B u t we should
be p r e p a r e d as well to a d m i t to consideration subjective evaluations of t h e
weight of an e x p e r i m e n t b a s e d on an i m p a r t i a l a n d honest appraisal of t h e
i n h e r e n t a c c u r a c y of one e x p e r i m e n t in comparison to a n o t h e r even t h o u g h
no analytic evaluation of the variance exists. I t m a y well be t h a t the introduction of (( weights ~ in place of (~v a r i a n c e s , is purely a semantic s u b t e r f u g e ;
even so, it m a y p r o v e a useful concept.
If, in equation (16), we introduce the notation
(17)
rg = a~lx 1 -]- a ~ x ~ -{- ... ~- a~qxq - - c , ,
we can write Q in a m a t r i x or tensor f o r m [11, 12]
(18)
Q = %z~r~ = R + I I R
Y~cIATHE]VIATICAL A N A L Y S I S OF T h E
UNIVERSAL
:PI{YSICAL C O N S T A N T S
121
in which ~,~ are the elements of a diagonal m a t r i x
%,, = 1/%2
ff = v,
(19)
0
# :/:v,
R is the vector with c o m p o n e n t s r~ a n d R + is the transposed vector. This
f o r m m a y a p p e a r c u m b e r s o m e and unvieldy in ~ctual use, and it is if the
simpler f o r m u l a t i o n is available. However, it represents the point of departure for generalizing the f o r m u l a t i o n and establishing its complete invariance.
I f we p e r f o r m a linear t r a n s f o r m a t i o n on R a n d obtain R ' = T R we must.
write
(20)
Q = R'+T-I+HT-1R'
,
since Q m u s t be a scalar invariant. H e n c e the t r a n s f o r m e d weight m a t r i x H '
is no longer diagonal. I t now has the f o r m
(21)
II'=
T - I + H T -1 = [ T S T + ] -1 ,
where S is the inverse of the diagonal m a t r i x H ; the diagonal elements of S
are the variances % . W e see i m m e d i a t e l y however t h a t the t r a n s f o r m e d
m a t r i x S ' = T S T + is exactly the m a t r i x which now expresses the variances
and covariances of the t r a n s f o r m e d v e c t o r c o m p o n e n t s r:. The elements of S'
are the m e a n error products, /\~,fl~>.
' '
I n the t r a n s f o r m e d s y s t e m the errors ~,.
are not in general i n d e p e n d e n t and hence S' is not diagonal as is the error
m a t r i x in e q u a t i o n (14). Thus, the f o r m given in e q u a t i o n (18) represents
the generalized s t a t e m e n t of the least squares condition; the weight m a t r i x / /
is the inverse of the error m a t r i x , or covariance matrix, of the observational
date. I f the o b s e r v a t i o n a l equations are not independent, the covariance
m a t r i x S a n d the weight m a t r i x / / a r e not diagonal. The off-diagonal elements
of S are directly related to the degree to which the corresponding observational
d a t a are i n t e r d e p e n d e n t .
4. -
S t a n d a r d errors a n d c o r r e l a t i o n
coefficients.
The statistical errors to be assigned to the o u t p u t values of a n y least s q u a r e s
a d j u s t m e n t m u s t , in general, be described not only b y stating the s t a n d a r d
deviation for each numerical result b u t also b y specifying the correlations
which exist b e t w e e n each pair of results. The numerical o u t p u t values of
~, e, LV, etc., are of little use unless functions of these quantities can be corn-
122
]~. R. C O H E ~
bined to c o m p u t e other derived values. The s t a n d a r d deviations of such der i v e d values m u s t be c o m p u t e d b y formulae which involve not only the s t a n d a r d
deviations of the values entering into the function b u t also the eovariances v~,
connecting all possible pairs o~ those values. This is because the o u t p u t values
of a least squares a d j u s t m e n t are not in general statistically i n d e p e n d e n t quantities b u t are (( statistically correlated ~).
E a c h q u a n t i t y subject to r a n d o m or accidental error (frequently k n o w n
as a (~random variable )~) m a y conveniently be t h o u g h t of as a sample t a k e n
at r a n d o m f r o m a (( universe ~> of values which group themselves a r o u n d a
m e a n value according to some frequency law. F o r each such r a n d o m variable x~, one is to t h i n k t h e n of the implied universe of values f r o m which t h e
~ a r i a b l e is selected. This universe m a y be described b y giving some of its
p a r a m e t e r s . Thus, if the universe is k n o w n to be Gaussian, for example, t h e n
p r e s c r i p t i o n of its first a n d second m o m e n t s , i.e., its m e a n value, /~, a n d its
variance ~ are sufficient.
Two such r a n d o m quantities are observationally i n d e p e n d e n t if the r a n d o m
selection of a sample value f r o m one universe in no wise affects or biases t h e
free selection of a sample f r o m the other universe. If, for example, two
variables are connected b y a strict functional relationship so t h a t the v a l u e
of either one is uniquely determined b y the other, the variables are completely
correlated a n d the correlation coefficient connecting t h e m has the absolute
value unity. R a n d o m samples can no longer be selected freely a n d indep e n d e n t l y f r o m the two universes because of the functional condition which
~ies the selections rigidly together. H a v i n g selected a sample value f r o m one
universe a t r a n d o m the second selection is now completely specified. On the
other hand, if one of the two r a n d o m variables is a function of the other a n d
also of still other completely independent r a n d o m variables, t h e n these two
will be partially b u t not completely correlated, a n d the correlation coefficient
connecting t h e m will h a v e a value somewhere between - - 1 a n d + 1 . I n
general if we h a v e a set of statistically i n d e p e n d e n t r a n d o m variables, y~,,
t h e n a second set of variables, x~, obtained b y linear t r a n s f o r m a t i o n on the y ,
(22)
Xi =
~ Zi~y ~
,u
will not be statistically independen~ since a given one of the variables y , a n d
hence a given source of error is present in m o r e t h a n one of the variables x~.
This of course is just the situation which exists i n the p r o b l e m of least squares
a d j u s t m e n t . To each outpu~ value, xi, there corresponds a s t a n d a r d deviation, a~, (<~variance ~ v , = a~) and to each pair (x~, x~) there corresponds a
~ c o v a r i a n c e ~, vis = r~jff,aj. The entire set of Variances and covariances f o r m
a s y m m e t r i c m a t r i x which we m a y call the ~ error m a t r i x . ~> The elements
~IAT:[IE~V£ATICA:LANALYSIS OF TIlE UNIVEI~SALPI-IYSICALCONSTANTS
123
'of this m a t r i x are required in order to c o m p u t e the error measures of o t h e r
quantities depending on the x~.
5. - The standard errors of the residues of a least squares a d j u s t m e n t [13].
I n a n y least squares a d j u s t m e n t of d a t a it is obviously i m p o r t a n t to be
able to assign a s t a n d a r d deviation to the difference b e t w e e n the a d j u s t e d
o u t p u t d a t u m a n d the i n p u t d a t u m f r o m which it was obtained. Closely
related to this question is the question of the value which would be obtained
in a least squares analysis if a specific i n p u t d a t u m h a d been omitted. The
input a n d o u t p u t d a t a are certainly correlated a n d it would be incorrect to
calculate the s t a n d a r d deviation of t h e difference w i t h o u t considering this
correlation.
The m e a s u r e d i n p u t d a t a consist of n u m b e r s c, as in equation (13); a f t e r
the least squares solution has been p e r f o r m e d a n d we h a v e o b t a i n e d the adjusted
values for the x's we can insert these values into e q u a t i o n (13) to obtain the
adjusted value of c~ which we shall designate b y c*., The n u m b e r c~* is the
best e s t i m a t e which we can m a k e of the correct value of c, b a s e d u p o n all the
d a t a available to us. The difference between % a n d c~* is thus a measure of
the e x t e n t to which the observed valne c, is consistent with all of the other
data. I n order to e v a l u a t e this consistency we m u s t h a v e an e s t i m a t e of the
m a g n i t u d e of the difference which m i g h t be e x p e c t e d on the basis of statistical
fluctuations. I t can be shown fairly easily t h a t the v a r i a n c e of the difference,
% - c*~ is given b y o,~ - - o~*'~ where o 2, is the v a r i a n c e of t h e i n p u t d a t u m a n d
,2 is the variance of t h e a d j u s t e d value. This is a surprisingly simple result
~nd it justifies the description of t h e ~djusted vMue as being c o m p o u n d e d
of two t e r m s ; one is the direct i n p u t value c, while t h e other is an effective
or indirect value which is d e t e r m i n e d b y the c o m b i n e d action of all of the
other d~ta. This indirect value is the value of c, which would be deduced
f r o m a least squares analysis f r o m which t h e directly observed d a t u m h a d
been omitted.
The indirect value a n d its variance are given b y
(23)
%i - -
2 *
*2
( 7 . C . - - 0". C..
a2__~*2
*2
- - %, +
2 (7. *~
(~$ __ C/k)'
2 *2
(24)
%
~
_#
~.2 •
--
_#
This expression for the variance implies t h a t the statistical weights, which
are proportional to the reciprocal of the variances are related b y the equation
(25)
=
+
p =
ci(72.
124
E.
~. o o H ~
I f we use weights r a t h e r t h a n variances, E q u a t i o n (23) takes on the simple
form
(26)
c*~ - - PI'% -~
~
which is m e r e l y t h e s t a t e m e n t t h a t t h e least squares adjusted v a l u e is t h e
weighted m e a n of the direct i n p u t value a n d the indirecg value.
6. -
Analysis
of d a t a .
W e n o w at last come to t h e p r o b l e m of m a k i n g a specific analysis of a given
set of e x p e r i m e n t a l d a t a in order to d e t e r m i n e best values. W e h a v e form u l a t e d a procedure for doing this; we m u s t now determine the d a t a to which
this p r o c e d u r e is to be applied. One m u s t be especially careful not to a p p l y
t h e m e t h o d of least squares blindly; it is n o t ~ substitute for careful selectior~
of data. No provision exists in the m e t h o d for identifying and isolating system a t i c error; the comparison of g 2 with the theoretical p r o b a b i l i t y t a b l e is
useful in this regard b u t it is not definitive and can only indicate a p r o b a b l e
existence of s y s t e m a t i c error.
I n a least squares fitting each d a t u m is to be assigned a weight which is
inversely p r o p o r t i o n a l to its variance. Such a weighting is n o t a r b i t r a r y ; i t
can be deduced directly f r o m t h e f o r m of t h e q u a d r a t i c e x p r e s s i o n
Q= Z
which we a t t e m p t to minimize. An observation with a large variance therefore carries little weight in determining the value of Q, and hence m a y b e
o m i t t e d w i t h o u t g r e a t l y affecting t h e result. There is, however, a m o r e i m p o r t a n t reason for omitting d a t a ' o f low weight. W h e n an e x p e r i m e n t e r design~
his e x p e r i m e n t he m u s t carefully consider the possible presence of s y s t e m a t i c
error as well as the presence of r a n d o m error. The r a n d o m error of t h e finaI
result can b e reduced b y duplication a n d repetition since these errors, are
different in each repetition; the s y s t e m a t i c errors, on the other h a n d do not
cancel out b u t remain. N o w it is p r o p e r in an e x p e r i m e n t to reduce a n y possible source of s y s t e m a t i c e r r o r to a point where it m a y be of the order of,
say, one t e n t h of the r a n d o m error of a single observation. I n this w a y the
s y s t e m a t i c error will be of the same order as, or smaller t h a n , the r a n d o m error
of the final quoted result. H o w e v e r , it is neither feasible nor practical to do
m u c h b e t t e r t h a n this in the suppression of s y s t e m a t i c error.
Thus an e x p e r i m e n t with a quoted error which is large c o m p a r e d to a n o t h e r
similar b u t m o r e precise m e a s u r e m e n t m a y well be affected with a s y s t e m a t i c
~AT]~IE~C[ATICAL A N A L Y S I S OlV T H E
UNIVERSAL PHYSICAL
CONSTANTS
125
error which is large c o m p a r e d to the accuracy of the second experiment. This
s y s t e m a t i c error would then, a f o r t i o r i , be large c o m p a r e d to the accuracy
which m i g h t be claimed for weighted m e a n of the two results. I t would
~herefore be i n a p p r o p r i a t e to include the less precise observation in a
weighted m e a n .
I t is also necessary to reject d a t a which suffer f r o m serious s y s t e m a t i c error
even if (and perhaps, especially if) the d a t a are precisely measured. Grounds
for the suspicion of such s y s t e m a t i c error m y arise either f r o m e x p e r i m e n t a l
sources (such as a r e e v a l u a t i o n of the conditions u n d e r which the e x p e r i m e n t
was p e r f o r m e d u n d e r such circumstances t h a t corrections for such s y s t e m a t i c
errors can not be m a d e to the existing data) or f r o m theoretical sources (in
which, as in the case of the hyperfine s t r u c t u r e splitting in h y d r o g e n the precision of the e x p e r i m e n t is higher t h a n the esixting s t a t e of the t h e o r y of the
e x p e r i m e n t can handle).
B y the first criterion for rejection m e n t i o n e d a b o v e almost all of the historically i m p o r t a n t early experiments, including m a n y which were considered
i m p o r t a n t as late as 1947 are excluded. Within their estimgted precision
ranges such m e a s u r e m e n t s are not inconsistent with the l a t e r m o r e precise
results b u t t h e y are relatively so m u c h less accurate as to h a v e quite negligible
influence in the present least squares a d j u s t m e n t . As a result of this criterion
v e r y few m e a s u r e m e n t s published prior to 1950 remain.
Professor DU~O~D has surveyed, in the previous paper, the i m p o r t a n t
e x p e r i m e n t s which h a v e a bearing on the values of the atomic constants. We
shall now collect here those results which are to be used in an gnalysis of these
constants.
(i) The conversion /actor A, f r o m the Siegbahn nominal scale of X - r a y
w~velengths (in X-units) to milliangstroms. We shall use the value r e c o m m e n d e d
b y Sir LAWRENCE :BRAGG since it p r e s u m a b l y supersedes the earlier v a l u e
q u o t e d b y T. T. BII~GE (*):
(27)
A ---- 1.002 020 ~ 0.000 030 •
(ii) The Siegbahn-Avogadro number N's. We use Birge's value c o n v e r t e d
to the physical scale
N~ = N A 3 = (6.061 79 ~: 0.000 23).10 ~3 mole -1 .
(iii) The line structure separation in deuterium, AE~): The frequency
(*) However, very recent information concerning T Y R ~ ' s measurements on X-ray
wavelengths indicates that this work suffers a systematic error of the order of 30 to
50 ppm because of the ommission of the L~mb shift in the calibration of the photographic plates. (See the previous paper by J. W. M. DuMo~D).
126
E. 1~. COHEN
separation between the levels 22P~ and 22P½ measured b y DAY]~o~F, TRIEBWASSE1% and LAMB
(29)
AED =
~2 RDe 1 + g ~2
1 + 7c---- 5.946 ~-~ = (110971.59~-:0.I0) MHz
(iv) The gyromagnetic ratio o] the proton, 7~, obtained at the ~Tational
B u r e a u of Standards b y TttOMAS~DRISCOLL and HIPPLE
(30)
Y, : # ' N e / M , e = (26 725.3 + 0.3) s -1 gauss -1 .
(v) The determination of the ]araday b y electrolysis; although m o r e
accurate values m a y soon be avafluble we are at the m o m e n t confronted with
two somewhut discrepant measurements on silver and on iodine. The iodine
value is
(31)
F = Ne/c = (9 652.15 + 0.13) emu/mole
(physical scale),
a n d the silver is
(32)
F = Ne/e = (9 651.29 =[= 0.19) emu/mole
(physical scale).
(vi) The magnetic moment o / t h e proton in terms of the nuclear m a g n e t o n .
BLocK and JEFF~IES obtain the value (uncorrected for diamagnetism)
(33)
~ ' - - 2.792 36 + 0.000 10
and ttlPP~E, So~I~E]~ and TtIO~IAS obtain for the same q u a n t i t y
(34)
#'---- 2.792 68 -{- 0.000 0 3 .
(vii) The short wavelength limit of the continuous X - m y spectrum is
u n f o r t u n a t e l y not k n o w n experimentally with sufficient accuracy to c a r r y
much weight in a least squares fitting. The difficulties here h a v e been p~eviously discussed [14]. I t appears t h a t the best experimental value is
(35)
hc~feA = (12370.8 + 1.0) e m u .
I n formalating the d a t a on which our input equations of observation are
to be based, care h a d been t a k e n to do so in such a way as to avoid hidden
correlations between the equations of observation. ~ o r example, the measurements resulting from observed values of crystal densities and their X - r a y
grating constants are almost invariably quoted as measurements of Avogadro~s
MATHEMATICAL
ANALYSTS
OF
THE
UNIV]~RSAL PHYSICAl,
CONSTANTS
127
n u m b e r , N. To obtain 2V f r o m these measurements, however, it is necessary
to combine the results whith the cube of the conversion constant, A. Consequently we h a v e e q u a t e d the numeric which was actually m e a s u r e d to t h e
a p p r o p r i a t e function, N A 3, of our p r i m a r y unknown. I n other words we h a v e
been careful to see t h a t no single m e a s u r e d q u a n t i t y shall be involved as a
s u b s t a n t i a l c o n t r i b u t o r to the error measures to two or m o r e of the observ a t i o n a l equations at once.
I t will not be possible here to discuss the numerical details of the a c t u a l
least squares analysis. These m a y b e f o u n d elsewhere in the literature [157 16].
Only the results and their evaluation will be presented. I t has already been
mentioned, p r o b a b l y m o r e t h a n once, t h a t the e x p e r i m e n t a l d a t a are not
entirely self-consister~t. Thus, the two determinations of the F a r a d a y constants
(by iodine and silver v o l t a m e t e r s ) differ b y 89 p a r t s per million although each
m e a s u r e m e n t claims to h a v e an a c c u r a c y of 20 p a r t s per million or better.
The m e a s u r e m e n t of the p r o t o n m a g n e t i c m o m e n t b y BLocH and JEFFR~S
is 115 p a r t s per million higher t h a n the m e a s u r e m e n t b y HIePL]~ SOMMER
and T~OM~S. This is some three times larger t h a n the s t a n d a r d deviation of
the d a t a would imply. The recently published reanalysis of this experiment [17]
has resulted in a reduction of BLOCH and J~PF~IES' result b y 109 p a r t s per
million, thus bringing it in excellent accord with HIP1,LE, SOMMER and THOMAS.
F u r t h e r m o r e , the modification of the inverse cyclotron b y COLLIS~GTOI~,DELLIS~
S ~ D E R S a n d TUlC]~EI~FIELD [18] yields a value which differs f r o m T~mGE~'S
corrected BLOCI~ and JEFFRIES result b y 21 ~ 39 p a r t s per million, and f r o m
NIPPLE, SOMMER and T~OMAS o m e g a t r o n value b y 16 ~= 18 p a r t s per million.
Professor DvMO~D has already fully described the problems associated with
the short w a v e l e n g t h limit.
A p r e l i m i n a r y least squares analysis was carried out in 1952 with the d a t a
t h e n available. This comprised thirteen equations in five variables, ~, e, :V,
A a n d c. The value Z s (which is the m i n i m u m value of the function Q - - the
function whose m i n i m i z a t i o n it is our object to obtain) was 52.1. This is a
disturbingly large figure since there are only eight degrees of freedom and
hence we would expect g ~ to h a v e the m e a n value 8 and to lie between
the limits 2.73 a n d 15.51 with 90~o confidence. The p r o b a b i l i t y is less
t h a n one in a t h o u s a n d t h a t g ~ would be larger t h a n 26.1 entirely as
a result of r a n d o m error. W e m a y say with almost complete c e r t a i n t y t h a t
there is a s y s t e m a t i c error either in the observational d a t a themselves or i n
the assignment of weights to those m e a s u r e m e n t s .
The m o s t convenient m e t h o d for investigating the existence of s y s t e m a t i c
error in a set of d a t a such as the one at h a n d is the comparison of the direct
and indirect values of each i n p u t numeric as has been described earlier. Such
a comparison was m a d e a n d the implications are direct a n d conclusive [13].
The iodine value of the f a r a d a y for e x a m p l e is m u c h m o r e consistent with
128
E.R.
COHEN
the indirect value t h a n is the silver faraday. The difference between the
iodine value and the indirect value is 53 ± 26 parts per million, while the
silver f a r a d a y differs from the indirect value b y 142 =[: 30 parts per million.
The existence of sources of experimental systematic error in this m e a s u r e m e n t
has been pointed out and this physical evidence strengthens the significance
of the statistical implication.
Similarly the indirect value of the p r o t o n magnetic m o m e n t agrees with
HIPPLE, SOMMEI~ and THOMAS' value, thus indicating the possible existence
of an error in the original inverse cyclotron work. This, of course, was later
confirmed b y TRIGGEI~'S reanalysis [17] and the modified experiment carried
out at OxfOrd b y SA~DE~S et al. [18].
The determinations of the short wavelength limit of the continuous X - r a y
spectrum are all in disagreement with the indirect value. Hence we m a y t e n d
to d o u b t the direct data; there are certainly valid experimental reasons w h y
we should, and one is t e m p t e d to reject all of these data.
A word of caution is in order at this point. Of the thirteen items which
were i n c l u d e d in this preliminary analysis we have deduced reasons to reject
five. This represents ~ r a t h e r drastic censoring and should be carried out
with care. As soon as we eliminate one item the values of all of the other
adjusted o u t p u t values change. Hence strictly we should re-evaluate the least
squares solution after each rejection in order to determine a new basis for
the subsequent rejections. F u r t h e r m o r e , the decision to reject a d a t u m ~s
discrepant m a y v e r y well depend upon the order in which other d a t a are
rejected. I t is quite possible, in general, t h a t a particular experimental result
is in disagreement with the indirectly determined value, not because of an
error in the experiment itself, b u t because some other error-ridden experiment
is strongly distorting the indirect value. If the latter experiment were rejected
first, the former one might well be retained. This is particularly liable to occur
if the total degrees of freedom of the system is small so t h a t each experimental
item can contribute significantly to the overall result.
A completely different approach to the problem of determining which
equations of the thirteen are likely to suffer from systematic error is afforded
b y an analysis of variance approach. This analysis involves the calculation
of Z~ for every possible subset of the thirteen equations and the comparison of these values of Z 2 to determine whether the value of 52.1
can be ascribed t0 the effect of one or a few items or whether all of the d a t a
contribute equally to this large value. I n the first instance we would be willing
to say t h a t those items responsible for anomalously large contributions to Z~
were, in fact, suffering from systematic error; in the second instance we would
ascribe an increased standard error to all of the d a t a and admit t h a t if system~tie errors existed t h e y were so wide-spread as to defy unique identification.
5IATIIE)CIATICAL A N A L Y S I S OF T I l E U N I V E R S A L P H Y S I C A L C O N S T A N T S
129
A p a r t i a l s u r v e y , a l t h o u g h n o t complete, is a d e q u a t e to c o n f i r m t h e i m p l i c a t i o n s of T a b l e I I : Silver c o u l o m e t e r v a l u e for t h e f a r a d a y c o n s t a n t a n d
B l o c h a n d Jeffries p r o t o n m a g n e t i c m o m e n t d e t e r m i n a t i o n are i n error a n d
all of t h e s h o r t w a v e l e n g t h l i m i t m e a s u r e m e n t s are s o m e w h a t suspect. A c t u a l l y
s u s p i c i o n does n o t fall s t r o n g l y o n t h e B e a r d e n a n d S c h w a r z v a l u e b u t this
is d u e m o r e to its low w e i g h t t h a n to its close a g r e e m e n t . T h u s a l t h o u g h its
r e s i d u e is i n m o s t c o m p a r i s o n s as large or larger t h a n t h e residues of t h e o t h e r
TABLE II. - Direct a n d indirect evaluation o/ f u n d a m e n t a l constants (1952 A d j u s t m e n t ) .
Constant
A
Direct value
(experimental)
1.002 020 4,0.000 080
NA a
6 061.79 4,0.23" 102°
(26 752.34,0.6) emu 18-1
[(1)
(9652.154,
4, 0.13) emu/mole
: I(Ag)
(9 651.29 4,
4-0.19) emu/mole
[
f(BJ)
2.79237 4,0.00010
' I (HST/
Indirect value
Least squares
solution
1.002 073 ~0.000 01~ 1.0020 63 4,0.000 013
6 062.904,0.39
26751.94,0.4
9 651.90i0.15
6 062.08 4,0.20
26752.04,0.5
9 652.01 ~=0.10
9652.21±0.11
9 652.01 -t:0.10
2.79270 4, 0.00005
2.79267 4,0.00003
2.79268 4,0.00003
2.79263=}=0.00005
I(FHD) (12 370.02 +0.63) e m u c m 12 372.40~0.17
(12371.034,0.48) e m u c m 12 372.374,0.17
[(BS) (12370.77=[=1.03) emucm 12372.284-0.17
2.79267 :{:0.00003
12 372.23::J:0.16
12 372.23~0.16
12372.234,0.16
R e m a r k s . - B J ~ F. BLOCH and C. D. JEFFRI~S; HST = J. A. HIPPLE,
H. SUM~IER and H. A. THOMAS; F H D = G. F. FELT, J. N. HARRIS and J. W.
M. DuMOND; B J W = J. A. B]~ARDEN, F. T~ JOHNSON and H. M. WATTS;
B S ~ J. n . BEARD:EN and G. SCHWARZ.
X-ray S.W.L. = Short wavelength limit of the continuous X-ray spectrum.
The column marked <<Direct value ~>gives the experimental datum measured
in the experiment which is indicated in the first column. The <~Indirect value >>
is the one which would be deduced from a least squares analysis if the given
experiment were excluded from consideration. If the same experiment is repeated
b y different observers only t,he single experiment in question (and not the others
of the same kind) is excluded. The column marked <,Residue of least squares
solution ~ is the adjusted value of the 1952 analysis. The last column gives,
in pa.rf.s per million, the difference between the 1east squares solution and the
i n p u t datum. The error quoted is the standard error of t h a t difference.
[The fine structure splitting, AE a, and the velocity of light have been omitted
from this table because the direct measurements of these items have so much
more weight t h a n the indirect values that the elimination of numerical accuracy
in the calculation precludes a meaningful evahlation of the latter].
9 - S~tpplemento al Nuovo Cimento.
13o
~. n. com~N
short wavelength limit determinations, its contribution to Z2 is m u c h smaller
because of the m u c h larger standard error assigned to it.
T h e 1955 evaluation of the atomic constant [16] therefore rejected the silver
f a r a d a y and B ~ o c ~ and JEFFI~IES Proton magnetic m o m e n t measurement. I n
addition the short wavelength limit values were replaced b y a single expressiorL
based on the average of all the low voltage d a t a as has been previously mentioned. I t was also considered adequate to consider the velocity of light to
be an auxiliary constant, not subject to least squares analysis. This r a t h e r
drastic censoring of the d a t a reduces the system to seven equations in four
unknowns. The calculated value of Z ~- is t h e n found to be 3.25 which
is deceptively close to the expectation value, since the 90% confidence limits
are 0.35 < g ~ < 7.81. I t can also be argued t h a t the short wavelength limit
measurements should be dropped entirely (and a strict adherence to our selection conditions would require this). I n this case we decrease our s y s t e m
to one with only two degrees of freedom. The value of Z 2 is t h e n
only 0.44. This is at first sight quite low compared with the expectation v a l u e
of 2.00; however the 90% confidence limits are 0.10 and 5.99. I n fact, the
probability t h a t Z2 might be lower t h a n 0.44 is almost 20%, and
this probability is further increased if we recognize t h a t we h a v e rather drast i c a l l y censored all those d a t a which contribute to large values of g 2.
F u r t h e r m o r e , the actual solution is not g r e a t l y changed b y rejecting the short
wavelength limit data entirely; the largest chahge in a n y o u t p u t value is less
t h a n one f o u r t h of the assigned s t a n d a r d deviation.
The system of seven equations in four unknowns which is obtained when
we exclude those data which appear to be discordant with the remainder o f
our data is given below. As has been mentioned above we assume the velocity of light to be an exact constant in this analysis, its value being
(36)
c = 299793.0 k m / s .
(The actual s t a n d a r d deviation to be assigned to this n u m b e r is ~= 0.3 kin/s,
b u t this error has a negligible effect on the numerical results of the leas~
squares analysis.)
The origin values and the linearized variables are t a k e n to be:
(37)
--
~0
xl -- - - " 1 0 5
~o = 0.007297 000 ,
g0
(38)
6--
xo = - - "
60
10 ~
eo = 4.802 200.10 -1° esu ,
~0
N - - No 10 ~ N o = 0.602 500 0"10 ~ mole -~,
No
(39)
x~ --
(40)
- Ao. 105 Ao :
x~ = A -A~
1.002 020 0 .
_YIATIIE~[ATICAL
ANALYSIS
OF
THE
UNIVERSAL
PHYSICAL
CONSTANTS
131
W i t h these definitions the observational equations become:
Experhnental source
Weight
(41)
x~ =
0.0
0.11,
A = 1.002 02 ::t::0.000 03
(~t2)
xa+3x4 =
3.5
0.07,
N A ~ (Birge's average)
:
4.0
4.92,
Fine structure
deuterium
=--
2.3
0.19,
Gyromagnetie ratio of proton
(43)
x1
(44)
3xl--x2
splitting
in
(45)
x~+xa
=
11.1
0.58,
Iodine faraday, electrochemistry
(46)
- - 3x14:-2x2-l-x8
=
1.3.5
0.83,
Magnetic m o m e n t of proton
(omegatron)
(47)
--
0.015,
Short wavelength limit (low
voltage)
x i + x2
--x~ =--
5.6
The n o r m a l equations are formed according to the usual rules; t h e y are
(0¢)
(4s)
14.115xl
- - 5.565x~
- - 2.490x3
+ 0.015x~ = -- 15.162
(e)
- - 5.565xl
@ 4.105x:
-]- 2.240x3
- - 0.015x4
29.201
(N)
- - 2.490xl
~- 2.240xz
-]- 1.480x3
~ 0.210x4 =
17.888
0.015x 1 - - 0.015x~
+ 0.210x3
@ 0.755x4 =
0.819
(A)
It should be noted here t h a t the normal equations are to be formed directly
from the coefficients of the quadratic form, Q, as given in equation (16) and
t h a t no simplification or cancellation should be made in the equations. This
is because the coefficients of the normal equations have a significance which
is not limited to the determination of the values of the x~. A n alteration in
the equations which would n o t alter the solution (such as the cancellation of
a c o m m o n factor in one of the equations, or even the re-ordering of the equations in the set) can destroy the identifications of the coefficients of the normal
equations as the weight m a t r i x of the solution.
The error (or variance) matrix of the solution is the inverse of the m a t r i x
of the coefficients of the normal equations. Hence the error m a t r i x is
(49)
V
0.1989
- - 0.5760
0.5603
0.1633\
c~
0.576 0
3.4478
--4.4319
1.2898~
e
--4.4319
6.7167
--1.9452]
N
0.16o3
1.2898
--1.9452
1.887 9 /
A
c~
e
--0.5603
N
A
132
E . R . co1~]~,-
a n d t h e s o l u t i o n for t h e v a r i n b l e s is
xl =
x~ --~
(5o)
3.92
13.72
x~ =
--2.37
x~ - -
1.94
T h e v n l u e of g ~ for this s o l u t i o n is
(51)
Z ~ = 3.25 .
T h i s is to b e c o m p a r e d to t h e v ~ l u e 52.1 w h i c h was o b t a i n e d i n t h e p r e l i m i n a r y
~ d j u s t m e n t . T h e m a j o r c h a n g e i n t h e v a r i u b l e s c a u s e d b y t h e d e l e t i o n of
t h o s e d ~ t a s u s p e c t e d of s y s t e m a t i c error is ~n i n c r e a s e i n x~ of 2.23 ~ n d ~ n
~ t t e n d a n t decrease i n x4 of 2.35 ; this c o r r e s p o n d s to ~ c h a n g e i n A v o g ~ d r o ' s
n u m b e r ~V of 22.3 p p m u n d a e h u n g e i n t h e c o n v e r s i o n f a c t o r A of - - 23.5 p p m .
T h e v ~ l u e of Z ~ c a n b e e x p r e s s e d e q u i v a l e n t l y i n t e r m s of t h e r a t i o of ext e r n a l to i n t e r n a l c o n s i s t e n c y
re/ri = % / Z 2 / ( ~ - - q) = 1 . 0 4 1 .
T h e 90~o c o n f i d e n c e i n t e r v a l for rJr~ w i t h 3 degrees of f r e e d o m is 0.594
H e n c e t h e r e is n o p a r t i c u l a r c o m p u l s i o n to use i n t e r n a l r ~ t h e r
ro/r~ ~__ 2.795.
t h a n e x t e r n a l c o n s i s t e n c y m e u s u r e s ; o u r d~t~ tell us o n l y t h a t t h e t w o m e a 'sures are c o n s i s t e n t . W e shull, h o w e v e r , q u o t e o u r final error m a t r i x i n t e r m s
of e x t e r n a l c o n s i s t e n c y since this is t h e l~rger m e u s u r e (~lbeit n o t s i g n i f i c a n t l y
so). F u r t h e r m o r e , t h e e r r o r m a t r i x will b e r e - e x p r e s s e d i n t e r m s of r e l a t i v e
parts per million in the p r i m a r y variables.
T h e error m a t r i x u n d c o r r e l a t i o n coefficients ~re g i v e n i n T a b l e s I I I a n d I V .
TABLE I I I . -
Covariance matrix (1955 Adjustme~t).
Elements of the matrix are in units of (relative parts per million)~.
e
o:
A
N
.F
e
m
h
g
A
N
374
560
685
62
140
- - 480
107
560
940
1057
60
226
- - 778
--218
685
1 057
1 246
103
262
- - 899
216
62
60
103
22
18
--61
2
140
226
262
18
204
--211
--71
- - 480
- - 778
- - 899
--61
211
726
246
-
-
-
-
-
-
F
107
--218
--216
2
--71
246
141
-
-
MATII]~SiATICAL ANALYSIS OF TIlE UNIV:ERSAL P]:[YSICAL CONSTANTS
133
T~BL~: IV. - Correlation coef]icients (1955 Adjustme~tt).
e
Ht
h
A
5'
F
e
m
h
1.00
0.95
0.99
0.70
0.51
0.92
0.47
0.95
0.99
0.97
1.00
0.63
0.52
0.94
0.51
1.00
0.97
0.43
0.52
0.94
0.60
~
A
-0.70
0.43
0.63
N
F
1.00
0.51
0.52
0.52
0.27
0.27
- - 0.48
0.03
1.00
- - 0.92
- - 0.94
- - 0.94
- - 0.48
- - 0.55
- - 0.55
- - 0.42
1.00
0.47
- - 0.60
- - 0.51
0.03
- - 0.42
0.77
0.77
1.00
T h e e r r o r m a t r i x of T a b l e I I I h a s b e e n e x t e n d e d t o i n c l u d e t h e v a r i a b l e s m, h
and F. These variables were eliminated from the least squares solution by the
u s e of a u x i l i a r y e q u a t i o n s . I t is, h o w e v e r , i n c o n v e n i e n t t o b e f o r c e d t o u s e
t h e s e e q u a t i o n s in o r d e r t o e x p r e s s m, h or F i n t e r m s of o u r p r i m a r y v a r i a b l e s
e a c h t i m e one n e e d s t o c o m p u t e c o r r e l a t i o n coefficients b e t w e e n t h e s e v a r i a b l e s a n d o t h e r v a r i a b l e s . T h e v a r i a n c e s a n d c o r r e l a t i o n coefficients for t h e s e
s e c o n d a r y v a r i a b l e s c a n b e c a l c u l a t e d in t e r m s of t h e v a r i a n c e m a t r i x of t h e
p r i m a r y v a r i a b l e s . T h u s , a l t h o u g h w e h a v e p r o d u c e d a m a t r i x of s e v e n t h o r d e r
i t is s t i l l o n l y of t h e f o u r t h r a n k since t h r e e of t h e r o w s (or c o l u m n s ) c a n b e
e x p r e s s e d as l i n e a r c o m b i n a t i o n s of t h e o t h e r r o w s (or c o l u m n s ) . T h e n e a r
e q u a l i t y of m a n d h c o l u m n s of t h e m a t r i x of c o r r e l a t i o n coefficients ( T a b l e I V )
is a r e f l e c t i o n of t h e s t r o n g c o u p l i n g b e t w e e n t h e a d j u s t e d v a l u e s of t h e s e t w o
variables.
This is a r e s u l t of t h e r e l a t i o n s h i p
2R~h
m
--
C~2
'
a n d t h e f a c t t h a t t h e r e l a t i v e s t a n d a r d e r r o r of a is m u c h s m a l l e r t h a n t h e
e r r o r i n e i t h e r m or h.
7.
-
Output
values.
Recommended
(1955)
least
squares
adjustment.
T h e 1955 a d j u s t m e n t is b a s e d , as h a s b e e n e x p l a i n e d e a r l i e r , o n t h e d e t e r r u i n a t i o n of f o u r p r i m a r y u n k n o w n s o:, e, N a n d A , b y l e a s t - s q u a r e s a d j u s t m e n t
f r o m s e v e n i n d e p e n d e n t sources of e x p e r i m e n t a l i n p u t d a t a c o m b i n e d w i t h
a n u m b e r of a u x i l i a r y c o n s t a n t s w h o s e v a l u e s a r e k n o w n so m u c h m o r e accurately than the aforementioned input data that they are negligible error
contributors.
T h e p h y s i c a l scale of a t o m i c w e i g h t s is u s e d a l m o s t e x c l u s i v e l y in t h e s e
tables, the Sackur-Tetrode constant (given on both physical and chemical
scales) b e i n g t h e sole e x c e p t i o n . T h e c o n v e r s i o n f a c t o r b e t w e e n t h e s e scales,
r ~ 1.000 272 ~ 0.000 005 c a l c u l a t e d b y I t . T. BIRGE on t h e b a s i s of t h e a b u n -
134
E.n.
co~E~,
~
dance ratio for the oxygen isotopes, ~60: ~sO: ~70 = (506 ± 10) : 1 : (0.204 ± 0.008)
is a d o p t e d here as a definition for the chemical scale. These abundance ratios,
and the value or r implied b y them, are subject to variation depending on
the source of the oxygen. The value of the ~O/~sO ratio can v a r y from app r o x i m a t e l y 495 for oxygen f r o m air or carbonates to 515 for o x y g e n from
water and rocks. Corresponding to this variation is a variation in the value
of r from 1.002 78 to 1.002 68. The I n t e r n a t i o n a l Commission on A t o m i c Weights
is at present (1956) considering the a r b i t r a r y redefinition of the chemical scale
of atomic weights in terms of the physical scale and the value r = 1.002 75.
The a c c u r a c y ~scribed to our ~dopted value is such t h a t these two numbers
do not differ significantly (approximately one half the s t a n d a r d error).
The new Kelvin scale of t e m p e r a t u r e a d o p t e d October 1954 i n Paris at
the T e n t h General Conference on Weights and Measures is here used. On
this scale the triple point of water is assigned the t e m p e r a t u r e 273.16 °K
exactly. This changes the numerical value of the gas constant, Ro, slightly
from t h a t used in earlier evaluations and gives the value of the ice point as
(273.150 0 :k 0.000 2 ) ° K . Absolute electrical units are used exclusively, the
(~international ~> electrical units h a v i n g been abolished in 1948.
TABLn
V.
-
Auxiliary
constant.is.
Rydberg wave number for infinite mass
R¢~ = (109737.309_4_0.012) e r a - 1
Rydberg wave numbers for the light nuclei
RE = (109 677.576±0.012) c m - 1
R D = (109 707.419 :k0.012) cm -1
(109 717.345 ~:0.012) em 1
(109 722.267Z0.012) cm 1
~ayi e
Velocity of light
C ~ (299 793.0 J:0.3) km s-1
Atomic mass of neutron
1.008982 ~0.000003 (physical scale)
Atomic mass of hydrogen
H = 1.008142~0.000003 (physical ~¢ale)
Atomic. mass ratio of hydrogen to proton
H/}/, = 1.00054461
(computed using atomic mass of electron
2gin = 0.00054875) (physical scale)
Atomic mass of the proton
M s ~ 1.007593-~0.000003 (physical scale)
Atomic mass of deuterium
D = 2.014735±0.000006 (physical scale)
3IATHESIATICAL ANALYSIS OF T H E U N I V E R S A L PHYSICAL CONSTANTS
'TABLE ~:
COl~'~'tte(l.
A t o m i c m a s s r a t i o of d e u t e r i u m to d e u t e r o n
D / M a = 1 . 0 0 0 2 7 2 4 4 ( c o m p u t e d u s i n g a t o m i c m a s s of e l e c t r o n ,
N m = 0.000 548 75) (physical scale)).
R a t i o of e l e c t r o n m a g n e t i c m o m e n t to p r o t o n m a g n e t i c m o m e n t w i t h o u t d i a m a gnetic correction
[ M p / ( N m y ) ] ( 1 ~- ~/2~ - - 2.973~2/~ ~) = 658.228 8 @ 0.000 4
A n o m a l o u s m a g n e t i c m o m e n t of e l e c t r o n
t~¢/tt0 = (1 -}- e / Y z c - 2.973c~2/~r~) = 1,001 145 36
( c o m p u t e d u s i n g t h e v a l u e 1/e = 137.037)
Gas c o n s t a n t p e r mole
irgo = (8.316 96=k0.000 34). 107 erg mole -1 deg -1 (physicalseale)
S t a n d a r d v o l u m e of
a
p e r f e c t gas
Vo = (22 420.7-,L0.6) e m a a r m mole -~ (physical scale)
TABL~ V I . - L e a s t squares a d j u s t e d o u t p u t values, 1965.
T h e q u a n t i t y following e a c h :k sign is t h e - s t a n d a r d error. A t t e n t i o n is called
t o t h e f a c t t h a t t h e q u a n t i t i e s in t h i s T a b l e are o b s e r v a t i o n a l l y c o r r e l a t e d so t h a t
i n t h e c o m p u t a t i o n of t h e e r r o r m e a s u r e s of d e r i v e d v a l u e s d e p e n d e n t o n t w o
or m o r e of t h e v a l u e s in t h i s T a b l e t h e e r r o r m a t r i x of T a b l e I I I m u s t b e used.
Avogadro's constant
2g :
(6.024 86 ± 0 . 0 0 0 16). 1023 mole -1 (physical scale)
Loschmidt's constant
L o = N / V o = (2.687 19 ± 0 . 0 0 0 10). 10 ~9 c m -a (physical scale)
Electronic charge
e = (4.802 86~:0.000 09). 10 -1° esu
e ' = e/c = (1.602 0 6 ~ : 0 . 0 0 0 03)" 10 -2° e m u
Electron rest mass
m :
(9.108 3 : c 0 . 0 0 0 3 ) . 10 -ys g
Proton rest mass
, m =: M~/IY = (1.672 39£=0.000 04). 102a g
Neutron rest mass
m n = Mn/-~r :
(1.674 70£_0.000 04)" 10:24 g
Planck's constant
h = (6.625 17~=0.00023). 10 -27 erg s
l~/2~r = (1.054 43 ~=0,000 04)- 10 -37 e r g s
C o n v e r s i o n f a c t o r f r o m S i e g b a h n X - u n i t s to m i l l i a n g s t r o m s
A =
,~g/,~s -- 1.002 0 3 9 ~ 0 . 0 0 0 0 1 4
135
136
E.
I~. C O H E N
TABLE V I : contir~ued.
Faraday constant
= N e = (2.893 66 ~0.0O0 03). 10 ~ esu m o l e -~
(physical scale)
F ' = N e / c = (9 652.19 ~ 0 . 1 1 ) e m u m o l e -~
C h a r g e - t o - m a s s r a t i o of t h e e l e c t r o n
e/m = (5.273 05~=0.000 07). 10 ~7 esu g - t
(1.758 90~=0.000 02)- 107 e m u g-~
e/mc
R a t i o h/e
h/e = (1.379 4 2 ± 0 . 0 0 0 02). 10 -17 erg s csu -1
Fine structure constant
(7.297 29 :L0.000 03)" 10 -a
137.037 3 ~ 0 . 0 0 0 6
1/~.
~/2~ = (1.161 3 9 8 ± 0 . 0 0 0 005)" 10 -a
=
~2=_ (5.325 04 ~:0.000 05)" 10 ~5
1 - - ( 1 - - ~)+ = (0.266 252 ~:0.000 002)" 10 -~
A t o m i c m a s s of t h e e l e c t r o n
~Vm = (5.487 63~:0.000 06). 10 -a (physical scale)
R a t i o of m a s s of h y d r o g e n t o m a s s of p r o t o n (*)
= 1.000 544 6 1 3 ~ 0 . 0 0 0 000 006
A.tomic m a s s of p r o t o n
M~, = t I - - N m = 1.007593~0.000003
(physical scale)
R a t i o p r o t o n m a s s to e l e c t r o n m a s s
~J(Nm)
= 1836.12±0.02
R e d u c e d m a s s of e l e c t r o n in h y d r o g e n a t o m
# = mM~/H
= (9.103 4~:0.000 3). 10 - ~ g
S c h r S d i n g e r c o n s t a n t for a fixed nucleus
2 m / h ~ = (1.638 36 ~ 0 . 0 0 0 07). 1027 erg - t c m -~
S e h r 6 d i n g e r c o n s t a n t for t h e h y d r o g e n a t o m
2#/h 2 = (1.637 4 8 ~ 0 . 0 0 0 07)- 1027 erg -1 c m -2
First Bohr radius
ao = h2/(me 2) = ~/(4~R¢~) =
:
(5.291 7 2 ~ 0 . 0 0 0 02).10 -9 c m
R a d i u s of e l e c t r o n o r b i t in n o r m a l ~H, r e f e r r e d t o c e n t e r of m a s s
ao
ao(1 _ ~2)½ = (5.291 58 J=0.000 02). 10 -9 c m
S e p a r a t i o n o~ p r o t o n a n d e l e c t r o n in n o r m a l ~H
ao
aoR~/R~=
(5.294 46 =~0.000 02) • 10-~ c m
(*) The binding energy of the electron in the hydrogen atom has been iacluded in the
quantity. The mass oi the electron whea found in the hydrogen atom is not m, but
more correctly m(1 -- l]2a ~+ ...).
MATHEMATICAL ANALYSIS OF THE UNIVERSAL I)IIYSICAL CONSTANTS
137I
TABLE V I : c o n t i ~ u e d .
Compton
w a v e l e n g t h of t h e e l e c t r o n
~e.,o ~ h / ( m c ) ~ o : ~ / 2 R ~ = (24.262 65=0.000 2)- 10 - n c m
~-c.,¢ = ~c,,J2~ : (3.861 51 5=0.000 04). 10 -~1 c m
C o m p t o n w a v e l e n g t h of t h e p r o t o n
~c.,p ~ h / m p c ~ (13.214 15=0.000 2). 10 -1~ c m
kc.,, =: 2c,,p/2~ = ~2.103 085=0.000 03). l 0 -1~ c m
C o m p t o n w a v e l e n g t h of t h e n e u t r o n
~ c . . , - - h / m , c = (13.195 9 5=0.000 2). 10 -1~ c m
kc..~ ~ ~c,,~/2z = (2.100 1 9 ~ 0 . 0 0 0 03). 10 -v~ c m
Classical e l e c t r o n r a d i u s
(2.817 8 5 5 = 0 . 0 0 0 0 4 ) . 1 0 ~3 c m
r o = (7.940 3 0 i 0 . 0 0 0 2 1 ) . 1 0 -26 c m
Thomson
cross
section
(8/3)zr~ = (6.652 05 5=0.00018). 10 -25 c m 2
Fine structure doublet separation in hydrogen
A E ~ = (1/16)R~e2[1 + ~/~ + ( 5 / 8 - 5.946/~2)~, 2] =
= ( 0 . 3 6 5 8 7 1 ± 0 . 0 0 0 0 0 3 ) a m -1
= ( 1 0 9 6 8 . 5 6 5 = 0 . 1 0 ) H z -1
F i n e s t r u c t u r e s e p a r a t i o n in d e u t e r i u m
AED
=
= (0.3659705=0.000003) cm 1
= ( 1 0 9 7 1 . 5 4 5 = 0 . 1 0 ) H z -1
AE~RD/.RH
Zeeman displacement per gauss
e / 4 : ~ m c 2 == (4.668855=t=0.00006). 10 -~ c m -1 G--1
Boltzmann's
constant
= ( 1 . 3 8 0 4 4 5 = 0 . 0 0 0 0 7 ) . 10 -16 e r g dog -1
-~ ( 8 . 6 1 6 7 5 = 0 . 0 0 0 4 ) , 10 -5 e V d e g -1
1/]c : (11605.45=0.5) d e g eV -1
k = Iio/N
First radiation constant
cl~8zhc
~
(4.9918=t=0.0002). 10 -15 e r g c m
Second radiation constant
c2 ~
hc/k, ~
( 1 . 4 3 8 8 0 5 = 0 . 0 0 0 0 7 ) c m de(A"
A t o m i c specific h e a t c o n s t a n t
c2/c=h/]~
=
(4.799 31 5=0.00023). 10 -11 s d e g
W i e n d i s p l a c e m e n t l a w c o n s t a n t (*)
2ma~T = c2/4.96511423 = (0.289782 ± 0 . 0 0 0 0 1 3 ) c m d e g
Stefan-Boltzmann
=
constant
(~/60)(I~4/h%
2) =
(0.56687 5=0.00010). 10 -~ c r g c m -~ deg -~ s -1
(*) The numerical constant 1.96511423 is the root of the transcendental equation
x = 5(1 -- exp [ - x]).
138
E.
~.
COHEN
TABL]~ V I : continued.
Sackur-Tetrode constant
So/Ro = ~ -b In [(2ZRo)~h-32V -4] =
= - - 5.573244-0.00007
So =
- - (46.35244-0.0020). 10 erg m o l e - ~ d e g -~
(physical scale)
Saekur-Tetrode constant
So/Roc h = - - 5.57256 4-0.00007
S O =
- - ( 4 6 . 3 4 6 7 ± 0 . 0 0 2 0 ) . 107 erg mole -1 deg -1
(chemical scale)
Bohr magneton
=
( 0 . 9 2 7 3 1 ~ 0 . 0 0 0 0 2 ) - 1 0 -2° erg G -1
Anomalous electron m o m e n t correction
1
-~ a / 2 a - - 2.973~2/a 2 = .u,/.uo = 1.0011453584-0.000000005
(computed using adjusted value
l / ~ = 137.0373~=0.6000)
M a g n e t i c m o m e n t of t h e e l e c t r o n
#o = (0.928374-0.00002). 10 -~° erg G -~
Nuclear magneton
.u, ~ he/(4um~c) = .Uo2Vm/H+
( 0 . 5 0 5 0 3 8 ± 0 . 0 0 0 0 1 8 ) . 10 -23 erg G - I
Proton moment
.up == (2.792 75 4-0.00003) n u c l e a r m a g n e t o n s
= (1.41044~_0.00004). 10 -2a erg G -1
G y r o m a g n e t i c r a t i o of t h e p r o t o n in h y d r o g e n ( u n c o r r e c t e d for d i a m a g n e t i s m )
y r = (2.675234-0.00004)- 10 ~ r a d s -1 G -1
G y r o m a g n e t i c r a t i o of t h e p r o t o n (corrected)
y = (2.675304-0.00004). 104 t a d s -1 G -1
M u l t i p l i e r of (Curie constant)½ x½, t o give m a g n e t i c m o m e n t p e r m o l e c u l e
(3tc/N(x)½ = 2.6178 4-0.00010). 10 -2° (erg mole deg-1)½
Mass-energy conversion factors
lg
1 electron mass
1 atomic mass unit
1 proton mass
1 neutron mass
=
=
=
=
=
( 5 . 6 1 0 0 0 ± 0 . 0 0 0 1 1 ) . 102¢ MeV
(0.5109764-0.000007) h l e V
( 9 3 1 . 1 4 1 ± 0 . 0 1 0 ) MeV
(938.2114-0.010) MeV
(939.5054-0.010) MeV
Quantum energy, E, conversion factors
1 eV = (1.60206=[-0.00003)'10 -12 erg
F / Y = hc = (1.98618-j=0.00007). 10 -1~ erg c m
E).g = ( 1 2 3 9 7 . 6 7 ± 0 . 2 2 ) - 1 0 - s eV c m
== ( 1 2 3 7 2 . 4 4 ± 0 . 1 6 ) k V X - u n i t s
E / v = (6.62517 4-0.00023). 10 -~7 erg s
= (4.13541 4-0.00007). 10 -is eV s
3~ATIIEMATICAL
ANALYSIS
OF T H E
VNIVa~RSAL PHYSICAL
CONSTANTS
139
TAELI~; V I : contit~,ued.
"~/E = ( 5 . 0 3 4 7 9 ± 0 . 0 0 0 1 7 ) " 1015 c m ~ e r g -1
= ( 8 0 6 6 . 0 3 - t - 0 . 1 4 ) e m -1 e V -1
:~/E = ( 1 . 5 0 9 4 0 ± 0 . 0 0 0 0 5 ) .
1026 s -~ e r g -~
= (2.41814z0.00004)-101~ s 1 eV-1
de B r o g l i e w a v e l e n g t h s
2B.., of e l e m e n t a r y
p a r t i c l e s (*)
Elect, t o n s
(7.273 77 ± 0 . 0 0 0 0 6 ) c m -° s 1/V
( 1 . 5 5 2 2 5 7 ± 0 . 0 0 0 0 1 6 ) . 10 -13 c m (erg)~/(E)~
(1.226 378 ± 0 . 0 0 0 0 1 0 ) . 10 -~ c m (eV)½/(E)~
Protons
( 3 . 9 6 1 4 9 ± 0 . 0 0 0 0 5 ) . 10 -3 c m 2 s - 1 / V
(3.622 53 ± 0 . 0 0 0 0 8 ) . 10 -~5 c m (erg)½/(E)½
(2.86202±0.00004)-10
~ c m (eV)½/(E)½
Neutrons
~B.,n
( 3 . 9 5 6 0 3 = t - 0 . 0 0 0 0 5 ) . 10 3 c m 2 s - 1 / V
( 3 . 6 0 2 0 4 ± 0 . 0 0 0 0 8 ) . 10 -~5 c m (erg-~/(E) ½
( 2 . 8 6 0 0 5 ± 0 . 0 0 0 0 4 ) - 10 -9 c m (eV)½/(E)½
l~]ner~y of 2 200 m / s n e u t r o n
~2200 -- (0.025 297 3 ± 0 . 0 0 0 0 0 0 3) e V
( 2 9 3 . 5 8 5 =t-0.012) ° K
T2200
( 2 0 . 4 3 5 ± 0 . 0 1 2 ) oC
The Rydberg
~nd related
derived constants
Roe = (109 7 3 7 . 3 0 9 ± 0 . 0 1 2 )
e m -~
R¢oc = (3.289 848 ± 0 . 0 0 0 0 0 3 ) - 101~ s -1
l~odtc ~ ( 2 . 1 7 9 5 8 : L 0 . 0 0 0 0 7 ) . 10 n e r g
R h d e i. 1 0 - s
(13.60488±0.00022) eV
Hydrogen
ionization potential
10 = RH(hd/e)(1 + ¼~2 + ...). 1 0 - s =
= (13.597 65 ± 0 . 0 0 0 22) e V
(*) These formulae apply only to nonrelativistic velocities. If the velocity of the ])article
~s not negligible compared to the velocity of light, c, or the energy n o t negligible compared to the
rest mass energy, we m u s t use 2B..D= ~c[e(e +2)] -½ where ,~eis the appropriateCompton wavelenght
of the particle in question and e is the kinetic energy measured in units of the particle rest-mass.
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w.
n.
R.
J.
c.