Base constants (base-*.ini)ΒΆ
The tables below list the physical constants which are defined in the base-*.ini files. All other constants and units are derived from these [1], and they can be charged to establish various unit systems.
The derived electromagnetic constants also depend on the rational setting. If rational is True, the constants are rationalized (epsilon_0 = 1/(4*pi*k_C)) ala Lorentz-Heaviside; otherwise, epsilon_0 = 1/k_C like the Gaussian units.
The definitions depend on the following items:
- Classes: Quantity
- Mathematical constants: pi
The default is the SI unit system (base-SI.ini), with the following definitions:
| Symbol | Expression | Name & notes |
|---|---|---|
| —— Base physical constants —— | ||
| R_inf | Quantity(10973731.568539*2*pi, 'A/L', 'cyc/m') | Rydberg constant |
| c | Quantity(299792458, 'L/T', 'm/s') | speed of light (aka Planck, Stoney, or natural unit of velocity) |
| k_J | Quantity(483597.870e9*2*pi, 'A*I*T2/(L2*M)', '1/Wb') | Josephson constant |
| R_K | Quantity(25812.8074434/(2*pi), 'L2*M/(A*I2*T3)', 'ohm/cyc') | von Klitzing constant |
| k_F | Quantity(96485.3365, 'I*T/N', 'C/mol') | Faraday constant |
| R | Quantity(8.3144621, 'L2*M/(N*T2*Theta)', 'J/(mol*K)') | gas constant |
| k_Aprime | Quantity(2*pi, 'A', 'rad')*R_K/c | modified Ampere constant (k_A*cyc/alpha) |
| —— Settings —— | ||
| rational | True | True if the unit system is rationalized |
The second argument of the Quantity constructor is comprised of SI dimensions (current (I), length (L), mass (M), amount (N), time (T), temperature (Theta)) and an additional dimension to track angle (A). The value of k_Aprime for SI and most of the other unit systems is chosen so that the radian (rad) has a value of one as defined by [BIPM2006].[2] However, this is not possible in the Planck unit system, so the Josephson constant is (arbitrarily) given a value of one instead.[3]
For the ESU and Gaussian unit systems (base-ESU.ini):
| Symbol | Expression | Name & notes |
|---|---|---|
| —— Base physical constants —— | ||
| R_inf | Quantity(109737.31568539*2*pi, 'A/L', 'cyc/cm') | Rydberg constant |
| c | Quantity(29979245800, 'L/T', 'cm/s') | speed of light (aka Planck, Stoney, or natural unit of velocity) |
| k_J | Quantity(4835978.70*2*pi, 'A*T/(L(3/2)*M(1/2))', 's/(statT*cm3)') | Josephson constant |
| R_K | Quantity(25812.8074434e9/(2*pi), 'L/(A*T)', 'cm/(s*cyc)') | von Klitzing constant |
| k_F | Quantity(9648.53365, 'M(1/2)*L(1/2)/N', 'g(1/2)*cm(1/2)/mol') | Faraday constant |
| R | Quantity(8.3144621e7, 'L2*M/(N*T2*Theta)', 'erg/(mol*K)') | gas constant |
| k_Aprime | Quantity(2*pi, 'A', 'cyc')*R_K/c | modified Ampere constant (k_A*cyc/alpha) |
| —— Settings —— | ||
| rational | False | True if the unit system is rationalized |
The same values are used for both of these unit systems. The difference is in how Maxwell’s equations are expressed, not in the definition of the constants and units. The Lorentz-Heaviside unit system can be established with the same constants but with rational = True.
For the electromagnetic unit (EMU) system (base-EMU.ini):
| Symbol | Expression | Name & notes |
|---|---|---|
| —— Base physical constants —— | ||
| R_inf | Quantity(109737.31568539*2*pi, 'A/L', 'cyc/cm') | Rydberg constant |
| c | Quantity(29979245800, 'L/T', 'cm/s') | speed of light (aka Planck, Stoney, or natural unit of velocity) |
| k_J | Quantity(4835978.70*2*pi, 'A*T/(L(3/2)*M(1/2))', '1/Mx') | Josephson constant |
| R_K | Quantity(25812.8074434e9/(2*pi), 'L/(A*T)', 'cm/(s*cyc)') | von Klitzing constant |
| k_F | Quantity(9648.53365, 'L(1/2)*M(1/2)/N', 'abC*s/(cm*mol)')*c | Faraday constant |
| R | Quantity(8.3144621e7, 'L2*M/(N*T2*Theta)', 'erg/(mol*K)') | gas constant |
| k_Aprime | Quantity(2*pi, 'A', 'cyc')*R_K/c | modified Ampere constant (k_A*cyc/alpha) |
| —— Settings —— | ||
| rational | False | True if the unit system is rationalized |
The next two unit systems are fully natural, so all of the quantities are dimensionless. In the definitions, floating point numbers are used instead of the Quantity class.
For the Hartree unit system (base-Hartree.ini):
| Symbol | Expression | Name & notes |
|---|---|---|
| —— Base physical constants —— | ||
| R_inf | 299792458e-7*pi/25812.8074434 | Rydberg constant |
| c | 1/(2*R_inf) | speed of light (aka Planck, Stoney, or natural unit of velocity) |
| k_J | 2 | Josephson constant |
| R_K | 2 | von Klitzing constant |
| k_F | 1 | Faraday constant |
| R | k_F | gas constant |
| k_Aprime | 2*pi*R_K/c | modified Ampere constant (k_A*cyc/alpha) |
| —— Settings —— | ||
| rational | False | True if the unit system is rationalized |
For the Planck unit system (base-Planck.ini):
| Symbol | Expression | Name & notes |
|---|---|---|
| —— Base physical constants —— | ||
| G | 1 | gravitational constant |
| c | 1 | speed of light (aka Planck, Stoney, or natural unit of velocity) |
| k_J | 1 | Josephson constant |
| R_K | sqrt(25812.8074434/(2*299792458*1e-7))/(pi*k_J) | von Klitzing constant |
| k_F | 1 | Faraday constant |
| R | k_F*k_J*R_K*sqrt(pi) | gas constant |
| k_Aprime | 2*(pi*k_J*R_K)**2/c | modified Ampere constant (k_A*cyc/alpha) |
| —— Empirical —— | ||
| R_inf | 10973731.568539*k_J*c**2*sqrt(k_Aprime*6.67384e-11/(G*R_K*25812.8074434*299792458**3))/483597.870e9 | Rydberg constant |
| —— Derived —— | ||
| l_P | sqrt(k_Aprime*G/2)/(c*k_J*R_K*pi) | Planck length |
| M_P | l_P*c**2/G | Planck mass |
| t_P | l_P/c | Planck time |
| E_P | M_P*c**2 | Planck energy |
| T_P | E_P*k_F*k_J*R_K*sqrt(pi)/R | Planck temperature |
| —— Settings —— | ||
| rational | True | True if the unit system is rationalized |
Note that the gravitational constant is included as a base constant. The Rydberg constant is no longer a base constant but is empirically related to the base constants.
References
| [BIPM2006] | (1, 2) International Bureau of Weights and Measures (BIPM), “The International System of Units (SI),” 8th ed., 2006. |
Footnotes
| [1] | ... except for the candela (cd), which is not directly related due to the luminosity function. |
| [2] | However, note that there is a contradiction in the SI unit system. Since rad = 1, it should follow that a cycle or revolution is 2*pi, yet [BIPM2006] defines the hertz (generally accepted as cycles per second) as 1/s. |
| [3] | When considering angle as a dimension, the Planck unit system only places a constraint on the product of the Josephson constant and the von Klitzing constant, not on either constant individually. |
