Instrumentation & Measurement Magazine 23-6 - 6

interferometer is used to control the system to achieve a constant velocity when moving up or down. From Faraday's law
we get:
VC  BL  	(2)

	

Combining the two phases we get the Kibble equation:
m g   I  VC	(3)

	

Because the right-hand term, I VC, has the unit watt, this
concept was often called a 'watt balance.' Since Bryan Kibble's
death in 2016, the metrology community has used the term
'Kibble balance' in his honor.
The Kibble balance is indeed a linkage between mechanical and electrical units but relating it to the Planck constant
requires two more things: the Josephson effect [10] and the
quantum Hall effect [11]. By passing the weighing phase current through a resistor, R, the Kibble equation is converted
to the product of two voltages divided by a resistance. If the
voltage developed across the resistor, R, is measured with the
Josephson effect, VR = n1 f1 h/2e, and if R is measured with the
quantum Hall effect, R = n3−1h/e2, and, if the coil voltage of the
moving phase is also related to the Josephson effect, Vc = n f2
h/2e, then the Kibble equation expands to a simple relationship of the Planck constant where ni are Josephson or quantum
Hall integers (quantum or step numbers) and fi are Josephson
microwave frequencies:

	


m g 

VR

 V
R C

 n f h / 2e  n
1 1

n

 1
3

f h / 2e 
  n1n2 n3  f1 f2  h / 4 	(4)

2 2

h/e

2



In this way the test mass is related to the acceleration due
to gravity, the velocity in the moving phase, the two operating frequencies and steps of a Josephson array voltage system,
the operating step of the quantum Hall system and Planck's
constant divided by 4. Note, for simplicity we have omitted any offsets or scaling factors in the voltage and resistance
measurements.
To achieve measurements with relative uncertainties of
< 1×10−7, a large number of design issues must be considered.
Typically, the measurements are made in vacuum to eliminate the buoyancy correction of the test mass and the index
of refraction corrections for the interferometer, and extensive monitoring of temperature, humidity, pressure etc. are
required.
Because most permanent magnets have large temperature
coefficients (∼ -350×10−6 K−1), extreme temperature stability is
necessary.
The local acceleration due to gravity must be determined
with an absolute gravimeter in close proximity to the position
of the test mass in the Kibble balance. Accurate corrections for
the tidal variations, laboratory gradients in the gravity profile,
atmospheric pressure and variations of the center of mass of
the different test masses must be taken into account. A number
6	

of alignments are required [12], [13] including the optical axis
of the interferometer which measures the velocity and position
of the coil must be aligned with respect to the gravity vector
which defines the vertical.
The two phases of the operation of the Kibble balance
are generally performed successively but simultaneous
techniques are possible. Various modes of operation of the
Kibble balance have been proposed as well as reciprocal designs that cancel alignment errors, systematic errors or noise
[14].
A recent comprehensive review of present Kibble balance
techniques used around the world is available [15], as well as
a review of the history of accurate Planck constant determinations [16]. To date there are four projects with published results
< 1×10−7 and nine more with larger uncertainties or under construction [5].

X-ray Crystal Density Method
The x-ray crystal density (XRCD) method essentially determines the number of atoms in a silicon crystal. The ratio of the
mass of an electron to the mass of a silicon atom is known with
high accuracy [7], and the mass of an electron is accuartely
given by the Rydberg and fine structure constants:
2 h  R
m  e   
  	(5)
c   2

	

where c is the velocity of light in vacuum and α is the fine structure constant. The coefficient R∞/(c α 2) is known with a relative
uncertainty below 1 × 10−9 [7].
Physically counting 2 × 1025 Si atoms for 1 kg of mass is impossible so the regularity of these near perfect single crystals
is utilized. The XRCD method measures the distances between
lattice planes in the Si single crystal and calculates the volume
V0 that can be attributed to one atom in the crystal. The sample
volume V the number of atoms is given by:
V
8 V
        3    	(6)
N
V0
a

	

Here 8 is the number of atoms in the unit cell of silicon,
which has the edge length (or lattice parameter) a.
The mass of the crystal, m, is related to the number of unit
cells in the crystal, and in turn is related to the electron mass,
m(e),
	

Ar  Si 
m  a3
m
   m  e  
  	(7)
atom
8 V
Ar  e 

and the relative atomic masses of the silicon atom, Ar(Si), and
of the electron, Ar(e).
Natural silicon has three stable isotopes, 28Si, 29Si and 30Si,
thus Ar(Si) in (7) has to be interpreted as the mean relative
atomic mass in the sample:
	

30

 

Ar  Si   xi Ar i Si   	(8)
i 28

where xi is the amount of substance fraction of the isotope iSi
(Σ xi = 1) [17].

IEEE Instrumentation & Measurement Magazine	

September 2020



Instrumentation & Measurement Magazine 23-6

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