Appendix C — Physical Constants and Conversion Factors
This appendix collects the numerical constants and conversion factors used throughout this textbook. In nuclear physics, we routinely mix SI, Gaussian, and "natural" units, and we measure energies in MeV, lengths in fm, masses in u or MeV/$c^2$, and cross sections in barns. The tables below are organized to make unit conversion as painless as possible.
All fundamental constants are from the 2018 CODATA adjustment (Tiesinga et al., Rev. Mod. Phys. 93, 025010, 2021) unless otherwise noted. Particle properties are from the Particle Data Group (Workman et al., Prog. Theor. Exp. Phys. 2022, 083C01). Astrophysical constants follow IAU 2015 nominal values where applicable.
Table C.1: Fundamental Constants
| Quantity | Symbol | Value | Unit |
|---|---|---|---|
| Speed of light in vacuum | $c$ | $2.99792458 \times 10^{8}$ | m s$^{-1}$ (exact) |
| Planck constant | $h$ | $6.62607015 \times 10^{-34}$ | J s (exact) |
| Reduced Planck constant | $\hbar = h/(2\pi)$ | $1.054571817 \times 10^{-34}$ | J s |
| $6.582119569 \times 10^{-22}$ | MeV s | ||
| Elementary charge | $e$ | $1.602176634 \times 10^{-19}$ | C (exact) |
| Boltzmann constant | $k_B$ | $1.380649 \times 10^{-23}$ | J K$^{-1}$ (exact) |
| $8.617333 \times 10^{-5}$ | eV K$^{-1}$ | ||
| Avogadro constant | $N_A$ | $6.02214076 \times 10^{23}$ | mol$^{-1}$ (exact) |
| Electric constant (vacuum permittivity) | $\varepsilon_0$ | $8.8541878128 \times 10^{-12}$ | F m$^{-1}$ |
| Magnetic constant (vacuum permeability) | $\mu_0$ | $1.25663706212 \times 10^{-6}$ | N A$^{-2}$ |
| Newtonian gravitational constant | $G$ | $6.67430 \times 10^{-11}$ | m$^3$ kg$^{-1}$ s$^{-2}$ |
Table C.2: Particle Masses
| Particle | Symbol | Mass (u) | Mass (MeV/$c^2$) | Mass (kg) |
|---|---|---|---|---|
| Proton | $m_p$ | 1.007276466621 | 938.27208816 | $1.67262192 \times 10^{-27}$ |
| Neutron | $m_n$ | 1.00866491595 | 939.56542052 | $1.67492750 \times 10^{-27}$ |
| Electron | $m_e$ | 0.000548579909065 | 0.51099895000 | $9.10938371 \times 10^{-31}$ |
| Muon | $m_\mu$ | 0.1134289259 | 105.6583755 | $1.88353163 \times 10^{-28}$ |
| Deuteron | $m_d$ | 2.013553212745 | 1875.61294257 | $3.34358375 \times 10^{-27}$ |
| Alpha particle | $m_\alpha$ | 4.001506179127 | 3727.37941 | $6.64465724 \times 10^{-27}$ |
| $^{12}$C atom | — | 12 (exact) | 11177.929 | $1.99264687 \times 10^{-26}$ |
| Atomic mass unit | $u$ | 1 (exact) | 931.49410242 | $1.66053907 \times 10^{-27}$ |
Neutron-proton mass difference: $m_n - m_p = 1.29333236$ MeV/$c^2$ = 0.001388449 u. This mass difference governs beta decay energetics (Chapter 14) and the neutron-to-proton ratio during Big Bang nucleosynthesis (Chapter 24).
Table C.3: Nuclear Physics Specific Constants
These are the composite constants that appear most frequently in nuclear physics calculations. Memorizing the first three will save you significant time.
| Quantity | Symbol / Expression | Value | Unit |
|---|---|---|---|
| $\hbar c$ | $\hbar c$ | 197.3269804 | MeV fm |
| Coulomb constant $\times$ $e^2$ | $e^2/(4\pi\varepsilon_0)$ | 1.4399764 | MeV fm |
| Fine structure constant | $\alpha = e^2/(4\pi\varepsilon_0 \hbar c)$ | $1/137.035999084$ | (dimensionless) |
| Nuclear magneton | $\mu_N = e\hbar/(2m_p)$ | $3.15245125844 \times 10^{-14}$ | MeV T$^{-1}$ |
| $5.0507837461 \times 10^{-27}$ | J T$^{-1}$ | ||
| Bohr magneton | $\mu_B = e\hbar/(2m_e)$ | $5.7883818060 \times 10^{-11}$ | MeV T$^{-1}$ |
| $9.2740100783 \times 10^{-24}$ | J T$^{-1}$ | ||
| Proton magnetic moment | $\mu_p$ | $2.7928473446\;\mu_N$ | |
| Neutron magnetic moment | $\mu_n$ | $-1.9130427\;\mu_N$ | |
| Classical electron radius | $r_e = e^2/(4\pi\varepsilon_0 m_e c^2)$ | $2.8179403262 \times 10^{-15}$ | m (2.818 fm) |
| Electron Compton wavelength | $\lambdabar_e = \hbar/(m_e c)$ | $3.8615926796 \times 10^{-13}$ | m (386.2 fm) |
| Proton Compton wavelength | $\lambdabar_p = \hbar/(m_p c)$ | $2.10308910336 \times 10^{-16}$ | m (0.2103 fm) |
| Bohr radius | $a_0 = \hbar/(m_e c \alpha)$ | $5.29177210903 \times 10^{-11}$ | m |
| Rydberg energy | $R_\infty hc$ | 13.605693122994 | eV |
| Thomson cross section | $\sigma_T = 8\pi r_e^2/3$ | $6.6524587321 \times 10^{-29}$ | m$^2$ (0.6652 b) |
| Fermi coupling constant | $G_F/(\hbar c)^3$ | $1.1663788 \times 10^{-5}$ | GeV$^{-2}$ |
Useful nuclear radius parameter: $r_0 \approx 1.2$–$1.3$ fm (from $R = r_0 A^{1/3}$; the precise value depends on the experimental probe — charge radius, matter radius, or nuclear potential radius). See Chapter 2.
Ratio $\mu_B/\mu_N$: $\mu_B/\mu_N = m_p/m_e = 1836.15267343$. This is why nuclear magnetic moments are roughly 2000 times smaller than electronic ones, making NMR frequencies much lower than ESR frequencies.
Table C.4: Unit Conversion Factors
Energy
| From | To | Factor |
|---|---|---|
| 1 MeV | J | $1.602176634 \times 10^{-13}$ |
| 1 eV | J | $1.602176634 \times 10^{-19}$ |
| 1 MeV | erg | $1.602176634 \times 10^{-6}$ |
| 1 MeV | K (via $E = k_B T$) | $1.16045 \times 10^{10}$ |
| 1 u $c^2$ | MeV | 931.49410242 |
| 1 J | MeV | $6.24150907 \times 10^{6}$ |
| 1 keV | MeV | $10^{-3}$ |
| 1 GeV | MeV | $10^{3}$ |
Length
| From | To | Factor |
|---|---|---|
| 1 fm | m | $10^{-15}$ |
| 1 fm | cm | $10^{-13}$ |
| 1 pm (picometer) | fm | $10^{3}$ |
| 1 $\text{\AA}$ (angstrom) | fm | $10^{5}$ |
Cross Section
| From | To | Factor |
|---|---|---|
| 1 b (barn) | m$^2$ | $10^{-28}$ |
| 1 b | cm$^2$ | $10^{-24}$ |
| 1 b | fm$^2$ | $10^{2}$ |
| 1 mb (millibarn) | b | $10^{-3}$ |
| 1 $\mu$b (microbarn) | b | $10^{-6}$ |
| 1 nb (nanobarn) | b | $10^{-9}$ |
Note: 1 barn $= 100$ fm$^2$. The name was coined during the Manhattan Project because uranium nuclei presented a cross-sectional area that was "as big as a barn" to neutrons.
Radioactivity
| From | To | Factor |
|---|---|---|
| 1 Ci (Curie) | Bq (Becquerel) | $3.7 \times 10^{10}$ |
| 1 Bq | disintegrations s$^{-1}$ | 1 (exact) |
| 1 mCi | Bq | $3.7 \times 10^{7}$ |
| 1 $\mu$Ci | Bq | $3.7 \times 10^{4}$ |
Radiation Dose
| From | To | Factor |
|---|---|---|
| 1 Gy (Gray) | J kg$^{-1}$ | 1 (exact) |
| 1 Gy | rad | 100 |
| 1 rad | Gy | 0.01 |
| 1 Sv (Sievert) | rem | 100 |
| 1 rem | Sv | 0.01 |
| 1 Sv | J kg$^{-1}$ (for $w_R = 1$) | 1 |
The Sievert is the SI unit of equivalent (biological) dose: $H = D \times w_R$, where $D$ is absorbed dose in Gray and $w_R$ is the radiation weighting factor ($w_R = 1$ for photons and electrons, $w_R = 20$ for alpha particles, $w_R \approx 2$–$20$ for neutrons depending on energy). See Chapters 16, 27, and 29.
Time
| From | To | Factor |
|---|---|---|
| 1 yr | s | $3.1557 \times 10^{7}$ |
| 1 d | s | $8.6400 \times 10^{4}$ |
| 1 h | s | $3600$ |
Table C.5: Astrophysical Constants
These constants appear in the nuclear astrophysics chapters (Part V) and the neutron star discussion (Chapter 25).
| Quantity | Symbol | Value | Unit |
|---|---|---|---|
| Solar mass | $M_\odot$ | $1.98892 \times 10^{30}$ | kg |
| $1.116 \times 10^{57}$ | $m_p$ | ||
| Solar luminosity | $L_\odot$ | $3.828 \times 10^{26}$ | W |
| Solar radius | $R_\odot$ | $6.957 \times 10^{8}$ | m |
| Solar core temperature | $T_{\text{core}}$ | $\sim 1.57 \times 10^{7}$ | K ($\sim 1.35$ keV) |
| Solar core density | $\rho_{\text{core}}$ | $\sim 1.5 \times 10^{5}$ | kg m$^{-3}$ |
| Chandrasekhar mass | $M_{\text{Ch}}$ | $\approx 1.4\; M_\odot$ | |
| Neutron star mass (typical) | $M_{\text{NS}}$ | $\sim 1.4$–$2.1\; M_\odot$ | |
| Neutron star radius (typical) | $R_{\text{NS}}$ | $\sim 10$–$13$ | km |
| Neutron star central density | $\rho_c$ | $\sim 5$–$10\; \rho_0$ | |
| Nuclear saturation density | $\rho_0$ | $\approx 0.16$ | fm$^{-3}$ ($\approx 2.7 \times 10^{17}$ kg m$^{-3}$) |
| Nuclear saturation energy | $E/A(\rho_0)$ | $\approx -16$ | MeV |
| Nuclear incompressibility | $K_0$ | $\approx 230 \pm 20$ | MeV |
| Symmetry energy at saturation | $J$ | $\approx 32 \pm 2$ | MeV |
| Symmetry energy slope | $L$ | $\approx 50 \pm 15$ | MeV |
| Age of the Universe | $t_0$ | $13.797 \pm 0.023 \times 10^{9}$ | yr |
| Age of the Solar System | $t_\odot$ | $4.567 \times 10^{9}$ | yr |
| Age of the Earth | $t_\oplus$ | $4.54 \times 10^{9}$ | yr |
| Hubble constant | $H_0$ | $67.4 \pm 0.5$ | km s$^{-1}$ Mpc$^{-1}$ |
| Baryon-to-photon ratio | $\eta$ | $(6.12 \pm 0.04) \times 10^{-10}$ | (dimensionless) |
Notes: - The nuclear saturation density $\rho_0 \approx 0.16$ fm$^{-3}$ is the density at which nuclear matter is in equilibrium. It corresponds to about $2.7 \times 10^{17}$ kg m$^{-3}$ — roughly the density inside all heavy nuclei and the crust of a neutron star. - The nuclear equation of state parameters ($K_0$, $J$, $L$) constrain the mass-radius relationship of neutron stars (Chapter 25). The symmetry energy slope parameter $L$ is particularly important: it governs the pressure of pure neutron matter near saturation density, which determines the neutron star crust thickness and radius. - The baryon-to-photon ratio $\eta$ is the single most important cosmological parameter for Big Bang nucleosynthesis (Chapter 24). It determines the primordial abundances of D, $^3$He, $^4$He, and $^7$Li.
Useful Combinations for Quick Estimates
The following composite quantities appear so often in nuclear physics that they are worth committing to memory:
| Quantity | Value | Typical use |
|---|---|---|
| $\hbar c$ | 197.3 MeV fm | Converting between momentum (MeV/$c$) and wavelength (fm) |
| $e^2/(4\pi\varepsilon_0)$ | 1.440 MeV fm | Coulomb energy at 1 fm separation |
| $\hbar^2/(2m_p)$ | 20.74 MeV fm$^2$ | Kinetic energy scale for protons |
| $\hbar^2/(2m_n)$ | 20.72 MeV fm$^2$ | Kinetic energy scale for neutrons |
| $m_p c^2$ | 938.3 MeV | Proton rest energy |
| $m_n c^2$ | 939.6 MeV | Neutron rest energy |
| $(m_n - m_p)c^2$ | 1.293 MeV | Beta decay Q-value threshold |
| $m_e c^2$ | 0.511 MeV | Electron rest energy (PET annihilation photons) |
| $2m_e c^2$ | 1.022 MeV | Pair production threshold |
| $m_\alpha c^2$ | 3727.4 MeV | Alpha particle rest energy |
| 1 u $c^2$ | 931.5 MeV | Mass-energy conversion |
| $k_B \times 10^{10}$ K | 0.862 MeV | Stellar core temperature scale |
Example: The Coulomb barrier height for two protons separated by $R = 1.2 \times 2^{1/3} \approx 1.51$ fm (surface contact of two protons) is:
$$V_C = \frac{e^2}{4\pi\varepsilon_0 R} = \frac{1.440\;\text{MeV fm}}{1.51\;\text{fm}} \approx 0.95\;\text{MeV}$$
This corresponds to a temperature $T = V_C/k_B \approx 1.1 \times 10^{10}$ K — far above the solar core temperature of $1.57 \times 10^7$ K, which is why quantum tunneling (the Gamow factor) is essential for stellar fusion (Chapter 21).
Constants marked "(exact)" are defined values in the 2019 SI redefinition. All others carry experimental uncertainties; the digits shown are significant to the precision relevant for nuclear physics calculations. For the most precise values, consult the NIST CODATA database at physics.nist.gov/cuu/Constants.