Physical Constants

Internationally recommended values of the fundamental constants in physics and chemistry

Physical Constants

Constants for MATLAB provides easy access to the latest internationally recommended values of nearly 150 fundamental constants from physics and chemistry, along with their uncertainties, units, names, and other metadata.

Key Features and Benefits

• Convenient: Simple dot notation with short, symbol-based variable names.
• Always Up-to-Date: No need to manually look up values - you'll always have the latest, most accurate data.
• Accurate: Automatically generated from the official CODATA source, ensuring 100% accuracy.
• Rich Metadata: Access properties like uncertainty, units, names, and historical data back to 1998.
• Symbolic Math Support: Perform arbitrary-precision calculations, unit conversions, and unit consistency checks.
• Future-Proof: Designed to seamlessly accommodate future CODATA adjustments and higher-precision arithmetic in MATLAB.

Basic Usage

Load constants:

const = Constants;

Tip

For easy access, save Constants.m to your userpath or add its parent folder to the MATLAB search path.

Access individual constants:

format long;
const.R   % molar gas constant
const.NA  % Avogadro constant

Constants have SI units unless the variable name ends in inUnit:

const.hbar             % reduced Planck constant in SI units (J s)
const.hbarineVseconds  % reduced Planck constant in eV s

Look up a constant:

const.find('charge')

Get a list of available constants:

const.info

Advanced Usage

Properties and Metadata

Load all properties and metadata:

const = Constants('all');
const.me  % properties and metadata for the electron mass

Property	Description
id	Current name (and other names, if applicable)
year	Dataset year
value	Value
uncty	Uncertainty
unit	Unit
name	Name (and other names, if applicable) in dataset
isExact	true if the constant is exact
isIrr	true if the constant is irrational and exact
symvalue	Symbolic value (*)
symunit	Symbolic unit (*)
sym	Symbolic constant with unit (*)

(*) Requires the Symbolic Math Toolbox.

Access individual properties:

const.me.value   % value of the electron mass
const.me.uncty   % ... its associated uncertainty
const.me.unit    % ... and units

Load individual properties:

uncty = Constants('uncty');
uncty.me  % uncertainty in the value of the electron mass

Note

The default is value. Calling Constants is equivalent to calling Constants('value').

Historic Data

Calling Constants with a second argument provides access to any dataset since 1998:

• 1998
• 2002
• 2006
• 2010
• 2014
• 2018¹

Load values for a specific dataset:

const = Constants('value', '2006');
const.G  % 2006 value of the Newtonian constant of gravitation

Load values for the latest (most recent) dataset:

const = Constants('value', 'latest');
const.Vm  % current value of the molar volume of an ideal gas (273.15 K, 100 kPa)

Note

This is the default. Calling Constants(arg1) without a second argument is equivalent to calling Constants(arg1, 'latest').

Load values for all datasets (from oldest to most recent):

const = Constants('value', 'all');
const.h(1)    % 1998 value of the Planck constant
const.h(4)    % 2010 value of the Planck constant
const.h(end)  % current value of the Planck constant

Load all properties and metadata for all datasets:

const = Constants('all', 'all');
const.alpha   % properties and metadata for the fine-structure constant (1998 to date)

Example 1

const = Constants('all', 'all');
plot(const.NA.year, const.NA.uncty, ':o', 'LineWidth',2);
title(const.NA.id);
xlabel('Year');
ylabel("Uncertainty (" + const.NA.unit(end) + ")");
xticks(const.NA.year);
grid on;

The Avogadro constant was defined as an exact value (uncertainty: 0) when the SI Units were redefined in 2017.

Symbolic Math

Constants also provides symbolic representations of all values, units, and constants with units to carry out symbolic manipulations and perform calculations with arbitrary accuracy.

symvalue: Symbolic Values

Exact constants, whose values are defined rather than experimentally determined, can be displayed with an arbitrary number of significant digits.

symVal = Constants('symvalue', '2018');
vpa(symVal.R, 150)     % 2018 value of the molar gas constant, rounded to 150 significant digits
vpa(symVal.hbar, 150)  % 2018 value of the reduced Planck constant, rounded to 150 significant digits

Output:

8.3144626181532399999999999999999999999986025942636401108474665474304748280360408924463157254169942689259187318384647369384765625
0.0000000000000000000000000000000001054571817646156391262428003302280744722889961594431207605200865210242417652521623612880705649658442982842269265184440305099090648037855420322457436

The first value, defined as R = N_A k in the 2018 dataset, is rational and has 128 significant digits. (The Avogadro constant N_A and the Boltzmann constant k are defined as exact, rational values in the dataset.)

The second value, defined as h / (2 π) in the 2018 dataset, is exact but irrational. (The Planck constant h has an exact, rational value in this dataset.)

Note

Even the 64-bit (double precision) floating-point values (value) of exact constants are computed from their definitions in Constants. This makes Constants future-proof should MATLAB ever be released with support for 128-bit (quadruple precision) or 256-bit (octuple precision) arithmetic.

symunit: Symbolic Units

Example 2

Verify units for the Rydberg constant, given by m_e e⁴ / (8 ε₀² h³ c) according to the Bohr model of the H atom:

symUnit = Constants('symunit');
simplify(symUnit.me * symUnit.e^4 / (symUnit.epsilonzero^2 * symUnit.h^3 * symUnit.c))

Output:

1/[m]

Caution

Choose your variable names wisely; sym and symunit are the names of commands in the Symbolic Math Toolbox!

sym: Symbolic Constants

symConst = Constants('sym');
u = symunit;
speed = unitConvert(symConst.c, u.km/u.year)  % speed of light in vacuum in km/year
double(separateUnits(speed))                  % extract value and convert to double

(47303652362904/5)*([km]/[year_Julian])
9.460730472580801e+12

Example 3

According to Planck's blackbody radiation law, the proportionality constant between the total power flux and T⁴ is given by σ = 2 π⁵ k⁴ / (15 c² h³) :

sigma = sym('2') * sym(pi)^sym('5') * symConst.k^sym('4') / ...
(sym('15') * symConst.c^sym('2') * symConst.h^sym('3') )
% compare this with the (exact and irrational) Stefan-Boltzmann constant from 'Constants'
symConst.sigma

Output:

0.00000000018529443369510835899732062307541*pi^5*(([Hz]^3*[J]*[s]^2)/([K]^4*[m]^2))
0.00000000018529443369510835899732062307541*pi^5*([W]/([K]^4*[m]^2))

The values are obviously identical. Let's check if their units match as well:

checkUnits(sigma == symConst.sigma)

Output:

Consistent: 0
Compatible: 0

MATLAB thinks they are different, perhaps because sigma has messy units. Let's simplify it and try again:

sigma = simplify(sigma)
checkUnits(sigma == symConst.sigma)

Output:

0.000000056703744191844294539709967318892*(([Hz]*[kg])/([K]^4*[s]^2))
Consistent: 1
Compatible: 1

We have a match!

Background

Constants was built by scientists for students and researchers in the sciences and engineering with a focus on simplicity and completeness.

The internationally recommended values of the basic constants and conversion factors of physics and chemistry are established by the Committee on Data of the International Science Council's (CODATA) Task Group on Fundamental Physical Constants. CODATA recommended values are adjusted in a four-year cycle and available on the National Institute of Standards and Technology (NIST) website.

For each adjustment since 1998, NIST provides an ASCII file of all values. Constants.m is generated automatically by scraping the data from these ASCII files to avoid copy & paste errors and guarantee accuracy.

Frequently Asked Questions

A list of frequently asked questions is maintained separately from this README file.

List of Constants

Variable	Constant	Unit
malpha	Alpha particle mass	kg
malphainu	Alpha particle mass in u	u
Malpha	Alpha particle molar mass	kg mol-1
angstromstar	Angstrom star	m
mu	Atomic mass constant	kg
NA	Avogadro constant	mol-1
muB	Bohr magneton	J T-1
muBineVpertesla	Bohr magneton in eV/T	eV T-1
azero	Bohr radius (atomic unit of length)	m
k	Boltzmann constant	J K-1
kineVperkelvin	Boltzmann constant in eV/K	eV K-1
Zzero	Characteristic impedance of vacuum	ohm
re	Classical electron radius	m
lambdaC	Compton wavelength	m
Gzero	Conductance quantum	S
KJninety	Conventional value of Josephson constant	Hz V-1
Aninety	Conventional value of ampere-90	A
Cninety	Conventional value of coulomb-90	C
Fninety	Conventional value of farad-90	F
Hninety	Conventional value of henry-90	H
Omeganinety	Conventional value of ohm-90	ohm
Vninety	Conventional value of volt-90	V
RKninety	Conventional value of von Klitzing constant	ohm
Wninety	Conventional value of watt-90	W
xuCu	Copper x unit	m
gd	Deuteron g factor	(unitless)
mud	Deuteron magnetic moment	J T-1
md	Deuteron mass	kg
mdinu	Deuteron mass in u	u
Md	Deuteron molar mass	kg mol-1
rd	Deuteron rms charge radius	m
ge	Electron g factor	(unitless)
gammae	Electron gyromagnetic ratio	s-1 T-1
gammaeinMHzpertesla	Electron gyromagnetic ratio in MHz/T	MHz T-1
mue	Electron magnetic moment	J T-1
ae	Electron magnetic moment anomaly	(unitless)
me	Electron mass (atomic unit of mass, natural unit of mass)	kg
meinu	Electron mass in u	u
Me	Electron molar mass	kg mol-1
eV	Electron volt	J
eVinhartree	Electron volt-hartree relationship	E_h
e	Elementary charge (atomic unit of charge)	C
F	Faraday constant	C mol-1
alpha	Fine-structure constant	(unitless)
cone	First radiation constant	W m^2
coneL	First radiation constant for spectral radiance	W m^2 sr-1
Eh	Hartree energy (atomic unit of energy)	J
EhineV	Hartree energy in eV	eV
gh	Helion g factor	(unitless)
muh	Helion magnetic moment	J T-1
mh	Helion mass	kg
mhinu	Helion mass in u	u
Mh	Helion molar mass	kg mol-1
sigmah	Helion shielding shift	(unitless)
DeltanuCs	Hyperfine transition frequency of Cs-133	Hz
KJ	Josephson constant	Hz V-1
jouleineV	Joule-electron volt relationship	eV
jouleinhartree	Joule-hartree relationship	E_h
kginu	Kilogram-atomic mass unit relationship	u
a	Lattice parameter of silicon	m
dtwotwozero	Lattice spacing of ideal Si (220)	m
nzero	Loschmidt constant (273.15 K, 100 kPa)	m-3
nzeroSTPold	Loschmidt constant (273.15 K, 101.325 kPa)	m-3
Kcd	Luminous efficacy	lm W-1
Phizero	Magnetic flux quantum	Wb
R	Molar gas constant	J mol-1 K-1
Mu	Molar mass constant	kg mol-1
MtwelveC	Molar mass of carbon-12	kg mol-1
Vm	Molar volume of ideal gas (273.15 K, 100 kPa)	m^3 mol-1
VmSTPold	Molar volume of ideal gas (273.15 K, 101.325 kPa)	m^3 mol-1
VmSi	Molar volume of silicon	m^3 mol-1
xuMo	Molybdenum x unit	m
lambdaCmu	Muon Compton wavelength	m
gmu	Muon g factor	(unitless)
mumu	Muon magnetic moment	J T-1
amu	Muon magnetic moment anomaly	(unitless)
mmu	Muon mass	kg
mmuinu	Muon mass in u	u
Mmu	Muon molar mass	kg mol-1
lambdaCn	Neutron Compton wavelength	m
gn	Neutron g factor	(unitless)
gamman	Neutron gyromagnetic ratio	s-1 T-1
gammaninMHzpertesla	Neutron gyromagnetic ratio in MHz/T	MHz T-1
mun	Neutron magnetic moment	J T-1
mn	Neutron mass	kg
mninu	Neutron mass in u	u
Mn	Neutron molar mass	kg mol-1
G	Newtonian constant of gravitation	m^3 kg-1 s-2
muN	Nuclear magneton	J T-1
muNineVpertesla	Nuclear magneton in eV/T	eV T-1
h	Planck constant	J Hz-1
hineVseconds	Planck constant in eV/Hz	eV Hz-1
lP	Planck length	m
mP	Planck mass	kg
TP	Planck temperature	K
tP	Planck time	s
lambdaCp	Proton Compton wavelength	m
gp	Proton g factor	(unitless)
gammap	Proton gyromagnetic ratio	s-1 T-1
gammapinMHzpertesla	Proton gyromagnetic ratio in MHz/T	MHz T-1
mup	Proton magnetic moment	J T-1
sigmapprime	Proton magnetic shielding correction	(unitless)
mp	Proton mass	kg
mpinu	Proton mass in u	u
Mp	Proton molar mass	kg mol-1
rp	Proton rms charge radius	m
lambdabarC	Reduced Compton wavelength (natural unit of length)	m
hbar	Reduced Planck constant (atomic unit of action, natural unit of action)	J s
hbarineVseconds	Reduced Planck constant in eV s (natural unit of action in eV s)	eV s
lambdabarCmu	Reduced muon Compton wavelength	m
lambdabarCn	Reduced neutron Compton wavelength	m
lambdabarCp	Reduced proton Compton wavelength	m
lambdabarCtau	Reduced tau Compton wavelength	m
Rinfinity	Rydberg constant	m-1
ctwo	Second radiation constant	m K
gammahprime	Shielded helion gyromagnetic ratio	s-1 T-1
gammahprimeinMHzpertesla	Shielded helion gyromagnetic ratio in MHz/T	MHz T-1
muhprime	Shielded helion magnetic moment	J T-1
gammapprime	Shielded proton gyromagnetic ratio	s-1 T-1
gammapprimeinMHzpertesla	Shielded proton gyromagnetic ratio in MHz/T	MHz T-1
mupprime	Shielded proton magnetic moment	J T-1
sigmadp	Shielding difference of d and p in HD	(unitless)
sigmatp	Shielding difference of t and p in HT	(unitless)
c	Speed of light in vacuum (natural unit of velocity)	m s-1
gzero	Standard acceleration of gravity	m s-2
atm	Standard atmosphere	Pa
ssp	Standard-state pressure	Pa
sigma	Stefan-Boltzmann constant	W m-2 K-4
lambdaCtau	Tau Compton wavelength	m
mtau	Tau mass	kg
mtauinu	Tau mass in u	u
Mtau	Tau molar mass	kg mol-1
sigmae	Thomson cross section	m^2
gt	Triton g factor	(unitless)
mut	Triton magnetic moment	J T-1
mt	Triton mass	kg
mtinu	Triton mass in u	u
Mt	Triton molar mass	kg mol-1
u	Unified atomic mass unit	kg
epsilonzero	Vacuum electric permittivity	F m-1
muzero	Vacuum magnetic permeability	N A-2
RK	Von Klitzing constant	ohm
sinsquaredthetaW	Weak mixing angle	(unitless)
bprime	Wien frequency displacement law constant	Hz K-1
b	Wien wavelength displacement law constant	m K

Footnotes

1. At the time of writing, the 2022 dataset has not yet been released. ↩