Measuring Planck’s Constant With Compton Scattering

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Introduction
The System of International Units sets an exact value for Planck's constant based on measurements undertaken by the National Institute of Standards and Technology between 2015 and 2017 using the Kibble balance (Haddad et al., 2017). These measurements reduced uncertainty by more than twofold over previous measurements, achieving a relative standard uncertainty of 1.3 × 10 −10 .
Although the Kibble balance is the preferred method for measuring Planck's constant, it is not the only experimental means for obtaining a high precision measurement. Planck's constant can also be determined from measurements of the electron's Compton wavelength and rest mass, each with a relative standard uncertainty of 3.0 × 10 −10 (NIS, 2018). The relationship between these measurements and the reduced Planck constant is ℏ = C m 0 c = 1.054 571 8176... × 10 −34 kgm 2 /s (1)

Derivation of Results
Equation 1 is obtained by representing Planck's constant with natural units in the constant's unit dimensions-an implicit assumption of Planck's derivation (Humpherys, 2022) Furthermore, it has been shown that an electron's Compton wavelength and rest mass are inversely proportional (Haug, 2022; Humpherys, 2021; Kepner, 2018, making the product of wavelength and mass invariant and equal to the product of Planck length and Planck mass

Planck Scale Metrology
It may be reasonably argued that extensive quantities of Planck length, mass, and time lie beyond the reach of experimental measurements (Adler, 2010). However, certain product and quotient relationships between pairs of Planck units are demonstrably within the reach of modern instrumentation, such as the ratio of Planck length to Planck time. The speed of light has been measured with a relative standard uncertainty of 1.6 × 10 −10 m/s (Jennings et al., 1987), which is an important consideration in the decision to define c in the System of International Units An accurate measurement of the speed of light is possible because the intensive ratio between distance and time can be measured on scales that are much larger than the Planck scale.
The product of Planck length and Planck mass also has a defined value as the ratio of two defined constants: Planck's constant and the speed of light The inversely proportional relationship between wavelength and mass shown in Equation 3 is responsible for the invariance of l P m P and Planck's constant.
The two universal constants give a third defined value in the product of Planck mass and Planck time The significance of these three defined values is much more than academic; the Planck units offer an overlooked pathway for obtaining more accurate values of the gravitational constant and other constants that depend on G. Like Planck's constant, the gravitational constant can be represented in natural units of length, mass, and time The formula indicates that the uncertainty in G lies in the ratio of Planck length to Planck mass, given an exact value of c. Consequently, the gravitational constant has a relative standard uncertainty of 2.2 × 10 −5 which is comparable to the 1.1 × 10 −5 relative standard uncertainty in the CODATA values of Planck length and Planck mass.
Improving the value of G requires a more accurate measurement of at least one of the three undefined values listed in Table 2. This is because the three defined values only provide enough information to constrain the proportions among the Planck units and do not reveal the extensive values themselves.
To see why this is the case, consider the three pairs of Planck units with defined values in Table 2. Note that the set contains either a product relationship or a quotient relationship between a given pair of units, but not both. For example, the ratio l P /t P is defined but l P t P has a large uncertainty by comparison.
If we had precision measurements for both the product and quotient relationships between a pair of units, we could determine a value for the two units with the same level of precision. This is easy to observe in the following manner. Let a and b represent high-precision values of the product and quotient relationships between Planck length and Planck time and From this information we can solve for l P and t P . Restating Eq. 8 and plugging into 9 yields the solution Our misfortune lies in having defined values for three Planck unit pairs without a single set of product and quotient relationships.
The result is that we have greater precision in the proportions among Planck units than in the unit values themselves. This is apparent in the CODATA values of Planck length and Planck time which give a ratio of 299, 792, 423 for the speed of light. Although we know this ratio is inaccurate, we can only improve it with better measurements.  Table 3 gives the formulas for calculating proportionally accurate values of the second and third Planck units when given the value of a first unit.

New Measurement Approaches
The natural units present an opportunity for devising new measurement solutions to improve the accuracy of physical constants while shedding new light on the structure of natural phenomena. Measurements of the universal constants are also measurements of the relationships between natural units, and an improvement in one elevates the other.
One approach to improving the accuracy of the gravitational constant is to continue refining the instruments and methods for measuring l P /m P . However, such measurements depend on accurate measurements of the mass and radius of two bodies http://apr.ccsenet.org Applied Physics Research Vol. 15, No. 1; 2023 and it remains challenging to obtain more precise measurements of the gravitational field between bodies of measurable mass.
An alternate pathway to improve the value of G is to devise new measurement techniques aimed at determining more precise values of m P /t P or l P t P . A more precise measurement of either quantity yields a commensurate increase in the precision of G. This is because a better measurement of any undefined pair in Table 2 improves the values of the Planck units using the formulas in Table 4. In particular, we need better values of l P and m P given the exact value of c 2 . An improvement in these two values will improve the value of G according to Equation 7.
The ratio m P /t P is found in the unit dimensions of force and opens up the possibility of applying more precise measurements of the electromagnetic interaction to these two units. This is perhaps the most promising way to improve the value of G. A greater challenge, however, is measuring l P t P which does not appear in the unit dimensions of common natural phenomena.

Conclusion
The present study is an examination of the structural relationship between universal constants and natural units, while also demonstrating the application of theory by calculating Planck's constant from the electron mass and Compton wavelength. New research may derive further insights by restating universal constants in natural units and examining the formulas in each unit dimension. This approach yields more granular information than setting the constants equal to 1. Such research might also produce a better understanding of physical constants and what they represent.
A better understanding of the natural units may also produce new measurement techniques for improving the values of Planck's constant, the gravitational constant, and other constants which embody natural units. High precision measurements obtained through Compton scattering offer a viable alternative to the Kibble balance for determining Planck's constant because the product of electron mass and Compton wavelength is conserved, and because the ratio of length to time in the speed of light is constant.