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Exchange force

The exchange force, more precisely termed the exchange interaction, is a quantum mechanical effect that arises from the indistinguishability of identical particles. According to the principles of quantum statistics, the total wave function of a system of identical fermions must be antisymmetric under particle exchange, while for bosons it must be symmetric. This requirement leads to correlation effects in the wave function, manifesting as an effective interaction energy—often attractive or repulsive depending on spin and spatial overlaps—that influences the behavior of multi-particle systems.^[1]^[2] This interaction plays a crucial role in various physical contexts, including the stability of atomic electron shells, covalent bonding in molecules, magnetic ordering in solids, and the binding of nucleons in atomic nuclei. In nuclear physics, it contributes to modeling the strong nuclear force, which has a short range of approximately 1–2 femtometers, through both statistical exchange effects and the exchange of virtual mesons.^[3] The concept originated in the late 1920s with Werner Heisenberg's work on quantum mechanics for multi-electron atoms, introducing exchange effects to satisfy wave function symmetry for identical fermions. In 1932, following James Chadwick's discovery of the neutron, Heisenberg extended these ideas to nuclear structure, proposing that the strong force between protons and neutrons arises from their exchange, similar to electron exchange in molecules, incorporating natural saturation.^[4] In 1933, Ettore Majorana refined the theory by incorporating spatial exchange, aiding explanations of nuclear saturation and scattering. The 1935 breakthrough by Hideki Yukawa introduced meson exchange via a massive particle (mass ~200 electron masses), yielding the Yukawa potential V(r) = -g^2 \frac{e^{-\mu r}}{r} (with \mu tied to meson mass and g the coupling), matching the short-range attraction; pions were discovered in 1947, confirming this.^[4]^[5]^[3] Characteristics include charge independence (equal action on like and unlike nucleon pairs), spin-spatial dependence with tensor components, and a blend of direct and exchange terms (e.g., ~50% in Serber models) to fit scattering data. Modern views frame it as an effective theory linking quantum chromodynamics at quark scales to nuclear phenomena.^[6]^[3]

Foundations in Quantum Mechanics

Identical Particles and Symmetry

In quantum mechanics, identical particles, also known as indistinguishable particles, are those of the same species whose individual identities cannot be distinguished by any physical measurement, as they share identical intrinsic properties such as mass and charge. This indistinguishability arises fundamentally from the principles of quantum theory, where the state of the system is described by a wavefunction that does not permit labeling particles with unique trajectories or identifiers. In contrast, classical mechanics treats particles as distinguishable entities, allowing for unique labeling based on initial conditions or positions, which enables straightforward permutation of their roles without altering the physical description.^[7] The quantum mechanical treatment of such particles requires the total wavefunction to exhibit definite symmetry under particle exchange, dictated by the symmetrization postulate. For bosons, the wavefunction must be totally symmetric upon interchanging any two particles, while for fermions, it must be totally antisymmetric. This classification is governed by the spin-statistics theorem, which establishes that particles with integer spin (0, 1, 2, ...) obey Bose-Einstein statistics and thus symmetric wavefunctions, whereas those with half-integer spin (1/2, 3/2, ...) follow Fermi-Dirac statistics and antisymmetric wavefunctions.^[8]^[9] These symmetry requirements have profound physical implications. Bosons, such as photons with spin 1, can occupy the same quantum state simultaneously, facilitating phenomena like Bose-Einstein condensation where large numbers of particles "bunch" into the ground state. Fermions, like electrons with spin 1/2, cannot share the same state due to the vanishing of the antisymmetric wavefunction when attempting to do so, which enforces spatial separation and underpins the Pauli exclusion principle. The exchange operator P_{12}, which interchanges the coordinates and spins of particles 1 and 2, formalizes this: for a valid two-particle wavefunction \Psi(1,2), it satisfies \Psi(1,2) = \pm P_{12} \Psi(1,2), with the plus sign for bosons and minus for fermions.^[10] A illustrative example is two non-interacting identical particles confined to a one-dimensional infinite potential well (a "box"). The single-particle eigenfunctions are \psi_n(x) = \sqrt{2/L} \sin(n \pi x / L) for quantum numbers n = 1, 2, \dots and box length L. For distinct states n \neq m, the bosonic two-particle wavefunction is the symmetric combination:

\frac{1}{\sqrt{2}} \left[ \psi_n(x_1) \psi_m(x_2) + \psi_m(x_1) \psi_n(x_2) \right],

while the fermionic one is antisymmetric:

\frac{1}{\sqrt{2}} \left[ \psi_n(x_1) \psi_m(x_2) - \psi_m(x_1) \psi_n(x_2) \right].

If n = m, the fermionic combination vanishes identically, prohibiting both particles from occupying the same state, whereas the bosonic case allows it with the unsymmetrized form \psi_n(x_1) \psi_n(x_2). This demonstrates how symmetry enforces distinct behavioral patterns without invoking interactions.^[11]

Fermions and the Pauli Exclusion Principle

Fermions are particles characterized by half-integer spin values, such as electrons, protons, and neutrons, which must obey specific symmetry requirements in quantum mechanics.^[12] The Pauli exclusion principle, formulated by Wolfgang Pauli in 1925, states that no two identical fermions can occupy the same quantum state simultaneously.^[13] This principle emerges directly from the antisymmetry of the total wavefunction for a system of identical fermions, ensuring that exchanging any two particles changes the sign of the wavefunction.^[12] For a system of two electrons, the principle dictates the form of the wavefunction based on spin configuration. If the spins form a symmetric triplet state (total spin 1), the spatial part of the wavefunction must be antisymmetric to maintain overall antisymmetry. Conversely, for an antisymmetric singlet spin state (total spin 0), the spatial wavefunction is symmetric.^[14] This coupling between spin and spatial degrees of freedom enforces the exclusion by prohibiting configurations where both electrons share identical spatial and spin quantum numbers. A key consequence of this antisymmetry is the formation of an "exchange hole" in the fermion density distribution. The exchange hole reduces the probability of finding two fermions with parallel spins at the same location, effectively creating a repulsive correlation even in the absence of classical forces.^[15] This phenomenon arises purely from the wavefunction's symmetry and underlies many quantum effects in multi-particle systems. In atomic physics, the Pauli exclusion principle governs the filling of electron shells, allowing at most two electrons per orbital with opposite spins to satisfy antisymmetry.^[16] This arrangement explains the periodic table's structure and prevents all electrons from collapsing into the lowest energy state. Furthermore, the principle ensures the stability of matter by counteracting attractive Coulomb forces; without it, electrons could occupy arbitrarily small volumes, leading to gravitational or electromagnetic collapse, as seen in the bounded energy of atomic systems.^[17]

Mathematical Formulation

Wavefunction Antisymmetrization

In quantum mechanics, the wavefunctions of identical fermions must be antisymmetric under the exchange of any two particles to satisfy the Pauli exclusion principle. This requirement ensures that the total wavefunction vanishes if two fermions occupy the same state.^[18] For a system of two identical fermions, the antisymmetric spatial wavefunction is constructed as a linear combination of the individual orbital products that changes sign upon particle interchange:

\Psi_A(1,2) = \frac{1}{\sqrt{2}} \left[ \phi_a(1)\phi_b(2) - \phi_a(2)\phi_b(1) \right],

where \phi_a and \phi_b are single-particle spatial orbitals, and the labels 1 and 2 denote the particles' coordinates. This form guarantees antisymmetry: \Psi_A(2,1) = -\Psi_A(1,2).^[19] The normalization of this wavefunction requires accounting for the overlap between the orbitals. Assuming the individual orbitals are normalized (\int |\phi_a|^2 dV = 1, \int |\phi_b|^2 dV = 1), the exact normalization factor is \frac{1}{\sqrt{2(1 - S^2)}}, where S = \int \phi_a^* \phi_b \, dV is the overlap integral. This ensures \int |\Psi_A|^2 dV_1 dV_2 = 1. If the orbitals are orthogonal (S = 0), the simpler factor \frac{1}{\sqrt{2}} suffices. Such antisymmetric wavefunctions are orthogonal to symmetric ones and to those with identical orbitals, as \Psi_A(1,1) = 0.^[19] For N identical fermions, the antisymmetric wavefunction is generalized using a Slater determinant:

\Psi(1, \dots, N) = \frac{1}{\sqrt{N!}} \det \begin{bmatrix} \phi_1(1) & \phi_2(1) & \cdots & \phi_N(1) \\ \phi_1(2) & \phi_2(2) & \cdots & \phi_N(2) \\ \vdots & \vdots & \ddots & \vdots \\ \phi_1(N) & \phi_2(N) & \cdots & \phi_N(N) \end{bmatrix},

where the \phi_i are orthonormal spin-orbitals (including spatial and spin parts). This determinant automatically enforces antisymmetry under any particle permutation and vanishes if any two spin-orbitals are identical. For the two-particle case, the Slater determinant reduces to the explicit form:

\Psi_A(1,2) = \frac{1}{\sqrt{2!}} \det \begin{bmatrix} \phi_a(1) & \phi_b(1) \\ \phi_a(2) & \phi_b(2) \end{bmatrix} = \frac{1}{\sqrt{2}} \left[ \phi_a(1)\phi_b(2) - \phi_a(2)\phi_b(1) \right].

This construction was introduced by John C. Slater in 1929 to represent multi-electron wavefunctions in atomic spectra. The total wavefunction for fermions must also incorporate spin degrees of freedom, typically as a product of spatial and spin parts: \Psi_\text{total} = \Psi_\text{spatial} \times \chi_\text{spin}, ensuring overall antisymmetry. For two spin-1/2 particles, the spin part \chi_\text{spin} can be either the antisymmetric singlet state (total spin S=0):

\chi_\text{singlet} = \frac{1}{\sqrt{2}} \left( |\uparrow \downarrow \rangle - |\downarrow \uparrow \rangle \right),

which pairs with a symmetric spatial wavefunction, or one of the three symmetric triplet states (total spin S=1):

\chi_\text{triplet, m=1} = |\uparrow \uparrow \rangle, \quad \chi_\text{triplet, m=0} = \frac{1}{\sqrt{2}} \left( |\uparrow \downarrow \rangle + |\downarrow \uparrow \rangle \right), \quad \chi_\text{triplet, m=-1} = |\downarrow \downarrow \rangle,

which pair with an antisymmetric spatial wavefunction. These spin configurations maintain the required total antisymmetry.^[20]

Exchange Energy Contribution

In the perturbative treatment of two interacting identical fermions, such as electrons, the total Hamiltonian is given by H = H_0 + V, where H_0 = h(1) + h(2) is the sum of single-particle operators and V is the two-body Coulomb interaction V(1,2) = 1/| \mathbf{r}_1 - \mathbf{r}_2 |. Assuming the electrons occupy single-particle orbitals \phi_a and \phi_b that are approximate solutions to the single-particle Schrödinger equation h \phi = \varepsilon \phi with the same eigenvalue \varepsilon for simplicity, the unperturbed energy is E_0 = 2\varepsilon. The first-order correction to the energy is the expectation value of V computed using the properly symmetrized spatial wavefunctions required by the antisymmetry of the total fermionic wavefunction.^[21] The normalized spatial wavefunctions are the symmetric combination for the spin-singlet state (total spin S = 0) and the antisymmetric combination for the spin-triplet state (total spin S = 1):

\Psi_+(1,2) = \frac{\phi_a(1)\phi_b(2) + \phi_a(2)\phi_b(1)}{\sqrt{2(1 + S^2)}}

\Psi_-(1,2) = \frac{\phi_a(1)\phi_b(2) - \phi_a(2)\phi_b(1)}{\sqrt{2(1 - S^2)}}

where S = \int \phi_a^*(\mathbf{r}) \phi_b(\mathbf{r}) \, d\mathbf{r} is the overlap integral between the orbitals. The expectation value \langle \Psi_\pm | V | \Psi_\pm \rangle evaluates to

E_\pm = E_0 + \frac{J \pm K}{1 \pm S^2},

where the Coulomb (or direct) integral J and the exchange integral K are

J = \iint |\phi_a(1)|^2 \, V(1,2) \, |\phi_b(2)|^2 \, dV_1 \, dV_2,

K = \iint \phi_a^*(1) \phi_b(1) \, V(1,2) \, \phi_b^*(2) \phi_a(2) \, dV_1 \, dV_2.

Both J and K are positive for like-charged particles and repulsive V, with J > K in typical cases. The term proportional to K constitutes the exchange energy contribution, which originates entirely from the interference between the direct and exchanged configurations in the antisymmetrized wavefunction and has no classical analog.^[21]^[22] When the orbitals are orthogonal (S = 0), the expressions simplify to E_\pm = E_0 + J \pm [K](/page/K). In this case, the exchange term -K in the antisymmetric spatial state (\Psi_-, triplet) reduces the energy relative to the direct Coulomb repulsion J, acting as an effective attraction that favors parallel spins in accordance with Hund's rule. For the symmetric spatial state (\Psi_+, singlet), the +K term increases the repulsion. This exchange contribution is independent of spin and arises purely from the spatial antisymmetry imposed by the Pauli exclusion principle; it explains the energy splitting between singlet and triplet states, such as the $1s2s configuration in the helium atom, where the triplet lies $2K \approx 2.4 eV below the singlet (experimentally \approx 0.8 eV after higher-order corrections). In the general non-orthogonal case (S \neq 0), the denominators $1 \pm S^2 further modulate the exchange effect, with overlap enhancing the magnitude of the contribution in molecular systems like H_2.^[21]^[22]

Historical Development

Heisenberg's Initial Insights

In 1926, Werner Heisenberg applied the newly developed matrix mechanics to the helium atom, addressing longstanding difficulties in treating multi-electron systems within quantum theory. In his paper "Über die Spektra von Atomsystemen mit einem Kern und mehreren Elektronen," he analyzed the interaction between the two equivalent electrons, introducing the concept of quantum-mechanical resonance as a means to account for the observed singlet and triplet spectra of para- and ortho-helium. This resonance arose from the indistinguishability of the electrons, where their wavefunctions overlapped, leading to a linear combination of states that split the energy levels without any classical counterpart, such as orbiting or direct repulsion. Heisenberg described this as a periodic "exchange of positions" (Platzwechsel) between the electrons, a purely quantum effect that lowered the energy of the symmetric state relative to the antisymmetric one, providing the first quantitative insight into electron correlation in atoms.^[23] Building on this, Heisenberg's subsequent work from 1927 to 1929 extended the resonance idea to molecular systems, particularly the hydrogen molecular ion (H₂⁺), demonstrating how exchange could act as a binding mechanism. In "Mehrkörperprobleme und Resonanz in der Quantenmechanik II," he considered the superposition of electronic states where the single electron could be associated with either nucleus, resulting in symmetric and antisymmetric combinations that yielded attractive and repulsive potentials, respectively. The symmetric state produced a lower energy configuration, enabling molecular stability through the overlap of wavefunctions centered on separated nuclei—a breakthrough that resolved issues in perturbation theory for identical particles in the post-matrix mechanics era. This conceptual shift emphasized that the exchange energy stemmed from the symmetry requirements of indistinguishable particles, rather than classical forces, marking the origin of exchange as a quantum resonance phenomenon in atomic physics.^[24] Heisenberg's initial insights were groundbreaking yet limited, as his early formulations neglected electron spin, which was only recently proposed by Uhlenbeck and Goudsmit in 1925 and not fully integrated until later refinements. Without spin, his treatment relied on spatial symmetry alone to enforce the Pauli exclusion principle, leading to approximate calculations that captured the qualitative splitting but required subsequent adjustments for precise spectral predictions. These developments occurred amid the rapid evolution of quantum mechanics, solving key puzzles in identical particle interactions while laying the groundwork for understanding quantum statistics in complex systems.^[23]^[24]

Extensions to Bonding and Nuclei

Following Heisenberg's initial insights into atomic resonance, the concept of exchange interactions was rapidly extended to molecular bonding and nuclear physics in the late 1920s and early 1930s. In 1926, Paul Dirac incorporated exchange effects into quantum mechanics for identical particles.^[25] This laid groundwork for understanding covalent bonds as arising from symmetric or antisymmetric combinations of atomic orbitals. The seminal application came in 1927 with Walter Heitler and Fritz London's valence bond treatment of the H₂ molecule, where the exchange integral in the singlet state (antisymmetric spin, symmetric spatial wavefunction) provides the dominant attractive contribution, explaining the stability of the covalent bond with a binding energy of approximately 4.7 eV and equilibrium distance of 0.74 Å. Fritz London further clarified in 1928 that this exchange mechanism underpins chemical binding in general, distinguishing the bonding singlet from the repulsive triplet state and emphasizing the role of resonance in valence structures. The discovery of the neutron by James Chadwick in 1932 enabled analogous exchange models in nuclear physics, shifting focus from atomic electrons to nucleon interactions. In 1932, Werner Heisenberg proposed exchange forces between protons and neutrons to explain deuteron binding, treating nucleons as identical under charge exchange while incorporating spin-dependent symmetry. Ettore Majorana independently developed a symmetric exchange operator in 1933, capturing proton-neutron equivalence through what later became isospin formalism, which yields a central force saturating at short ranges and binding the deuteron with energy 2.2 MeV.^[4] By the mid-1930s, Eugene Wigner and collaborators integrated these ideas into early nuclear shell models, using exchange to enforce the Pauli principle among nucleons and explain nuclear force saturation—preventing indefinite binding growth—consistent with observed magic numbers like 2, 8, and 20 in stable isotopes. This evolution reframed Heisenberg's atomic "resonance" as "exchange force" in nuclear contexts, explicitly distinguishing it from later quantum field-theoretic virtual particle exchanges, as it relies on wavefunction symmetrization rather than meson mediation.

Applications

Atomic and Molecular Bonding

In valence bond theory, covalent bonds form through the overlap of atomic orbitals from adjacent atoms, leading to an exchange attraction that stabilizes the molecule when the electrons occupy a symmetric spatial wavefunction. This exchange interaction arises from the quantum mechanical indistinguishability of electrons, resulting in a lowering of the system's energy due to the symmetric combination of atomic orbitals.^[26] A classic example is the hydrogen molecule (H₂) in its ground state, where the singlet spin state allows the two 1s orbitals to overlap symmetrically, producing an exchange energy that binds the atoms with a dissociation energy of approximately 104 kcal/mol, significantly below the energy of separated atoms. In contrast, the triplet state features an antisymmetric spatial wavefunction, leading to repulsion and no stable bond, as the exchange term becomes positive and destabilizing. In polyatomic molecules, valence bond theory extends this concept through hybridization, where atomic orbitals mix to form hybrid orbitals optimized for maximum overlap with neighboring atoms, enhancing the exchange attraction in directional bonds, as seen in the tetrahedral sp³ hybrids of methane (CH₄). Resonance structures further incorporate exchange by allowing delocalization of electron pairs across multiple equivalent bonding configurations, such as in benzene, where the cyclic overlap contributes to the molecule's stability.^[27] Unlike ionic bonding, which relies on classical electrostatic attractions between charged species, the exchange force in covalent bonding is a purely quantum statistical effect stemming from wavefunction symmetry, without net charge transfer between atoms. The strength of this exchange-driven bond scales with the orbital overlap integral S, where greater overlap correlates with stronger binding, though the exact energy also depends on exchange integrals like J.^[26]

Magnetism in Solids

In solids, the exchange interaction governs magnetic ordering by favoring specific alignments of electron spins on lattice sites, leading to collective phenomena such as ferromagnetism and antiferromagnetism. This arises from the quantum mechanical exchange of electrons between neighboring atoms, which lowers the system's energy when spins align in a manner dictated by the Pauli exclusion principle. The effective description of these interactions in lattice systems is captured by the Heisenberg spin Hamiltonian, given by

H = -\sum_{\langle i,j \rangle} 2J_{ij} \mathbf{S}_i \cdot \mathbf{S}_j,

where the sum is over nearest-neighbor pairs \langle i,j \rangle, \mathbf{S}_i and \mathbf{S}_j are the spin operators at sites i and j, and J_{ij} is the exchange constant. When J_{ij} > 0, parallel spin alignments are energetically favored, resulting in ferromagnetism; conversely, J_{ij} < 0 promotes antiparallel alignments, yielding antiferromagnetism.^[28] Direct exchange is the primary mechanism in many metallic and insulating solids, originating from the short-range spatial overlap of electron orbitals on neighboring atoms, which allows direct electron exchange and Coulomb repulsion effects. This interaction is particularly strong in transition metals like iron, where partially filled d-orbitals enable significant overlap, stabilizing ferromagnetic order. In insulators, direct exchange can still occur but is often weaker due to larger interatomic distances.^[29] For longer-range interactions, superexchange dominates in Mott insulators, such as transition metal oxides, where direct overlap is minimal but virtual electron hopping through intervening non-magnetic ions (e.g., oxygen) mediates the coupling. In these systems, an electron from one magnetic ion virtually transfers to the anion and then to a neighboring magnetic ion, effectively coupling the spins via second-order perturbation theory; this typically favors antiferromagnetic alignment to minimize kinetic energy costs. A classic example is manganese oxide (MnO), where oxygen-mediated superexchange between Mn^{2+} ions (with S=5/2) establishes antiferromagnetic order below the Néel temperature of 116 K.^[30]^[31] In ferromagnetic iron, direct exchange yields a positive J of approximately 0.1 eV, contributing to a high Curie temperature of 1043 K, above which thermal disorder disrupts the parallel spin alignment. This contrasts with MnO's antiferromagnetic superexchange (J ≈ -4 meV), which enforces alternating spins in a type-II structure. Fundamentally, these preferences stem from the Pauli exclusion principle: the antisymmetric wavefunction for fermions requires spin alignment (parallel for ferromagnetism) to allow symmetric spatial parts, thereby reducing kinetic energy through delocalization while avoiding double occupancy.^[32]^[33]

Nuclear Structure

In nuclear physics, the exchange force arises from the indistinguishability of protons and neutrons under the SU(2) isospin symmetry, treating them as two states of a nucleon isodoublet. This symmetry implies that the total wavefunction of nucleons must be antisymmetric under particle exchange to obey the Pauli exclusion principle. In the deuteron, the lightest bound nucleus consisting of one proton and one neutron, the ground state has total isospin T=0 (antisymmetric in isospin) and spin S=1 (symmetric in spin), with a dominant S-wave spatial component that is symmetric. This configuration allows an attractive exchange interaction, contributing to the observed binding energy of approximately 2.225 MeV.^[34]^[35] Nuclear potential models incorporate exchange effects to describe nucleon interactions, often using a Yukawa form modified by spin and isospin dependencies. A common phenomenological representation is V(r) = V_{\text{central}}(r) + V_{\text{exchange}}(r) P_{\sigma} P_{\tau}, where P_{\sigma} and P_{\tau} are the spin-exchange and isospin-exchange operators, respectively, which capture the statistical effects of fermion antisymmetrization. These terms arise naturally in meson-exchange theories, such as one-pion exchange, where the pseudoscalar nature of the pion leads to spin- and isospin-dependent attractions or repulsions depending on the state symmetry. For instance, in the deuteron, the P_{\tau} operator favors the T=0 channel, enhancing binding in the spin-triplet state while the T=1 singlet remains unbound.^[36]^[37] The nuclear shell model relies on antisymmetrized multi-nucleon wavefunctions to enforce the Pauli principle, restricting nucleons to distinct single-particle orbitals and explaining the stability of nuclei with "magic" numbers of protons or neutrons (2, 8, 20, 28, 50, 82, 126). These numbers correspond to completed shells in a mean-field potential with strong spin-orbit coupling, as proposed by Maria Goeppert Mayer and J. Hans D. Jensen, where exchange interactions ensure only Pauli-allowed configurations contribute to the ground state energy. This antisymmetrization prevents overlap of identical fermion states, leading to closed-shell nuclei with enhanced binding and low excitation energies, such as ^{4}\text{He}, ^{16}\text{O}, and ^{208}\text{Pb}.^[38]^[39] Exchange repulsion plays a crucial role in nuclear saturation, balancing the attractive central force to prevent collapse under increasing density. Werner Heisenberg's early model of nuclear exchange forces demonstrated that the antisymmetric nature of the wavefunction introduces a short-range repulsion, limiting the number of nucleons that can bind closely and yielding a near-constant binding energy per nucleon (~8 MeV) across heavy nuclei. This mechanism ensures nuclear matter saturates at a density of about 0.17 nucleons per fm³, as the exchange term grows with the number of pairs while attraction does not scale indefinitely.^[40] A representative example is the alpha particle (^{4}\text{He}), the ground state of which features a fully symmetric spatial wavefunction in the lowest S-shell, paired with a totally antisymmetric spin-isospin component (S=0, T=0) to maintain overall antisymmetry for the four fermions. This configuration, akin to a spin-isospin singlet under SU(4) symmetry, maximizes attraction while respecting Pauli exclusion, resulting in exceptional stability with a binding energy of 28.3 MeV.^[41]

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[38]
[PDF] PoS(EMC2006)002 - SISSA
Oct 6, 2006 · Majorana and Heisenberg must have discussed the exchange theory of nuclear forces during those first days. Heisenberg's third paper [4] was not.
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[PDF] Lecture 7 1.1. If we add two vectors of lengths r and r the sum can ...
The α particle is a spin zero and isospin zero state; it can be thought of as a bound state of four nucleons. The αparticle is the nu- cleus of the abundant ...