In nuclear physics, the semi-empirical mass formula (SEMF; sometimes also called the Weizsäcker formula, Bethe–Weizsäcker formula, or Bethe–Weizsäcker mass formula to distinguish it from the Bethe–Weizsäcker process) is used to approximate the mass of an atomic nucleus from its number of protons and neutrons. As the name suggests, it is based partly on theory and partly on empirical measurements. The formula represents the liquid-drop model proposed by George Gamow, which can account for most of the terms in the formula and gives rough estimates for the values of the coefficients. It was first formulated in 1935 by German physicist Carl Friedrich von Weizsäcker, and although refinements have been made to the coefficients over the years, the structure of the formula remains the same today.

The formula gives a good approximation for atomic masses and thereby other effects. However, it fails to explain the existence of lines of greater binding energy at certain numbers of protons and neutrons. These numbers, known as magic numbers, are the foundation of the nuclear shell model.

The liquid-drop model was first proposed by George Gamow and further developed by Niels Bohr, John Archibald Wheeler and Lise Meitner. It treats the nucleus as a drop of incompressible fluid of very high density, held together by the nuclear force (a residual effect of the strong force): there is a similarity to the structure of a spherical liquid drop. While a crude model, the liquid-drop model accounts for the spherical shape of most nuclei and makes a rough prediction of binding energy.

The corresponding mass formula is defined purely in terms of the numbers of protons and neutrons it contains. The original Weizsäcker formula defines five terms:

The mass of an atomic nucleus, for ${\displaystyle N}$ neutrons, ${\displaystyle Z}$ protons, and therefore ${\displaystyle A=N+Z}$ nucleons, is given by

where ${\displaystyle m_{\text{n}}}$ and ${\displaystyle m_{\text{p}}}$ are the rest mass of a neutron and a proton respectively, and ${\displaystyle E_{\text{B}}}$ is the binding energy of the nucleus. The semi-empirical mass formula states the binding energy is

The ${\displaystyle \delta (N,Z)}$ term is either zero or ${\displaystyle \pm \delta _{0}}$, depending on the parity of ${\displaystyle N}$ and ${\displaystyle Z}$, where ${\displaystyle \delta _{0}={a_{\text{P}}}{A^{k_{\text{P}}}}}$ for some exponent ${\displaystyle k_{\text{P}}}$. Note that as ${\displaystyle A=N+Z}$, the numerator of the ${\displaystyle a_{\text{A}}}$ term can be rewritten as ${\displaystyle (A-2Z)^{2}}$.

Each of the terms in this formula has a theoretical basis. The coefficients ${\displaystyle a_{\text{V}}}$, ${\displaystyle a_{\text{S}}}$, ${\displaystyle a_{\text{C}}}$, ${\displaystyle a_{\text{A}}}$, and ${\displaystyle a_{\text{P}}}$ are determined empirically; while they may be derived from experiment, they are typically derived from least-squares fit to contemporary data. While typically expressed by its basic five terms, further terms exist to explain additional phenomena. Akin to how changing a polynomial fit will change its coefficients, the interplay between these coefficients as new phenomena are introduced is complex; some terms influence each other, whereas the ${\displaystyle a_{\text{P}}}$ term is largely independent.