When Enrico Fermi analyzed the magnetic interaction between the electron and Na, Cs, and Rb nuclei in 1930, he used the Dirac equation to compute the energy of an electron interacting with a point charge and magnetic dipole (the nucleus) fixed at the origin. After the mathematical dust had settled, he found an interaction that appeared to act only at a single point: the center of the electron wavefunction; it has been called a contact interaction. In 1936, Casimir analyzed the magnetic interaction of proton and neutron spins in the ground state of the deuteron using the Schrödinger equation and classical electrodynamics. Using symmetries appropriate only to s states, and performing an integration by parts, he found that they, too, seemed to interact only at a single common point of the proton and neutron: their center-of-mass. When applied to hydrogen, his result agreed with Fermi's. We present an expanded version of Casimir's important but little-known calculation.

In general, the magnetic hyperfine interaction is the magnetic interaction between the spins of two particles with overlapping wavefunctions. When each of these particles is in an s state, the usual dipole–dipole interaction does not apply because its angular average is zero, yet there is an interaction. Perhaps, the simplest example is a hydrogen atom with a single 1 s electron and a proton.1 

In 1930, Fermi gave the first theory of the magnetic interaction of the spin of a single valence electron with the spin of a nucleus; he specifically considered Na, Cs, and Rb. However, his theory also applies to H (hydrogen). To describe that electron he applied the Dirac equation in the presence of the vector potential of the proton, treated as a classical magnetic moment localized at the origin.2 Only a single coordinate (the electron's) appears in the calculation. In the end, only the value of the electron wavefunction at the origin, ψ ( 0 ), matters. This has been called the contact interaction. Wikipedia lists it as “Fermi contact interaction,”3 but “contact” does not appear in Fermi's paper.2 Although this widely used term suggests a local or singular interaction, we shall see that it arises from mathematical transformations on a non-singular interaction.

In 1936, Casimir gave a related calculation for the magnetic interaction between the proton and neutron in a deuteron, applying the classical current–current interaction to quantum-mechanical (Schrödinger equation) electric current densities due to magnetic moment densities.4 This problem intrinsically involves two coordinates, but the final result, as for Fermi's calculation, involves the value of the wavefunction only where the relative coordinate R is zero. When applied to the hydrogen atom, Casimir's approach gave the same result as Fermi's. As with Fermi's paper, “contact” does not appear in Casimir's paper.

Casimir's brief paper implicitly ignored the center-of-mass coordinate R c m, working with a wavefunction Ψ depending only on the relative coordinate R . Thus, although both the proton and deuteron moment operators appear in the calculation, only their relative coordinate appears explicitly in the wavefunction for the system. Despite the generality and simplicity of Casimir's approach relative to Fermi's, Google Scholar gives his paper a mere six citations, as opposed to 31 for that of Fermi. (The fact that so few papers refer to Fermi may indicate that few have actually read Fermi's article, first published in German, and later in Italian.2)

Casimir's calculation is well worth teaching, given that it is general enough to apply to objects of similar spatial extent. Moreover, the calculation relies only on some vector identities that apply to systems with the simplest symmetry—spherical (the s state)—and the property that 2 r 1 = 4 π δ ( r ). It should be accessible to students during the second semester of an upper division course in electromagnetism. Simpler methods, appropriate only to the interaction between particles of very different mass, were recently presented and reviewed in this journal.5 These methods involve no singularity in the field near either particle.

For hydrogen, with electron and proton magnetic moments m e and m p, and electron wavefunction ψ e ( r e ), the usual hyperfine interaction energy is
E hf = 2 3 μ 0 m e · m p | ψ e ( 0 ) | 2 .
(1)
We will show that the more general result reduces to this.

In Ref. 4, Casimir treats the classical interaction of two magnetic moments, but evaluates the moments using quantum mechanics. The technical details that follow depend on vector identities and on integrations by parts. They also depend on the quantum mechanical wavefunction factorizing into parts depending on the relative coordinate and the center-of-mass coordinate, and that the relative coordinate appears with spherical symmetry (an s state).

Let the magnetization of particle 1 at position r 1 be M 1 ( r 1 ), and similarly for particle 2. From them, we can compute an associated current,
J = × M .
(2)
In the static case, we may employ
B = × A , · A = 0 ,
(3)
which is the Coulomb gauge. With R = r 1 r 2, the vector potential A ( r 1 ) due to J 2 is given by
A ( r 1 ) = μ 0 4 π d r 2 J 2 ( r 2 ) R .
(4)
The magnetic interaction between the two moments is
E m = d r 1 M 1 · B ( r 1 ) = d r 1 M 1 · ( 1 × d r 2 μ 0 4 π J 2 R ) = μ 0 4 π d r 1 M 1 · ( 1 × d r 2 ( 2 × M 2 ) R ) .
(5)
An integration by parts with respect to r 1 combined with Eq. (2) gives E m = ( μ 0 / 4 π ) d r 1 d r 2 J 1 · J 2 / R, which is where Casimir began his calculation.4 
We now transform from r 1 and r 2 to relative and center-of-mass coordinates. We use tildes to distinguish masses ( m ̃) from magnetic moments ( m and mi). The relative and the center-of-mass variables are R = r 1 r 2 and R c m = ( m ̃ 1 + m ̃ 2 ) 1 ( m ̃ 1 r 1 + m ̃ 2 r 2 ). (In changing integration variables, the Jacobian is unity.) After an integration by parts on 2, we use 1 R = R R = 2 R to obtain
E m = μ 0 4 π d r 1 M 1 · [ 1 × d r 2 [ 2 1 R ] × M 2 ] = μ 0 4 π d R d R c m   M 1 · [ R × ( [ R 1 R ] × M 2 ) ] .
(6)
In the above, both gradients operate only on R 1.

To evaluate (6), it is essential to observe that the overall state of the system is a product of functions of R and R c m, not a product of functions of R 1 and R 2, despite the fact that M 1 and M 2 appear. Moreover, because the system has the spherical symmetry of a relative s state, the function of R depends only on | R |. Thus, under d R above, when vector identities appear in the form / R i ( / R j ), they may be replaced by 1 3 δ i j R 2.

With ( M 1 ) i M 1 i and ( M 2 ) n M 2 n, we then have
M 1 i ϵ ijk j ϵ kln l R 1 M 2 n = M 1 i M 2 n j l R 1 ϵ ijk ϵ kln = M 1 i M 2 n j l R 1 ( δ i l δ j n δ i n δ j l ) = M 1 i M 2 j j i R 1 M 1 i M 2 i j j R 1 M 1 i M 2 j 1 3 δ i j 2 R 1 M 1 i M 2 i 2 R 1 = 2 3 M 1 · M 2 2 R 1 .
(7)

We now give the magnetizations more explicitly in terms of the joint particle density n ( r 1 , r 2 ), which depends on the overall wavefunction Ψ ( r 1 , r 2 ). This wavefunction factorizes into a relative and a center-of-mass part as ψ ( R ) ϕ ( R c m ). Let n R ( R ) = | ψ ( R ) | 2 and n c m ( R c m ) = | ϕ ( R c m ) | 2. Then, the joint density is n ( r 1 , r 2 ) = | ψ ( R ) ϕ ( R c m ) | 2 = n R ( R ) n c m ( R c m ).

Further, for s-state symmetry, ψ ( R ) = ψ ( R ). Thus, n ( r 1 , r 2 ) = n R ( R ) n c m ( R c m ), so in terms of magnetic moments
M 1 i M 2 j = m 1 i m 2 j n ( r 1 , r 2 ) = m 1 i m 2 j n R ( R ) n c m ( R c m ) .
(8)
Then, using R 2 R 1 = 4 π δ ( R ) and the normalization of the center-of-mass wavefunction
1 = d R c m n c m ( R c m ) ,
Equation (6) becomes
E m = m 1 · m 2 2 3 μ 0 d R d R c m n R ( R ) n c m ( R c m ) δ ( R ) = 2 3 μ 0 m 1 · m 2 n R ( 0 ) .
(9)
This is equivalent to Eq. (1).

We can now see that the origin of the apparent “contact interaction” is the integration by parts in Eq. (6) and the vector identities in Eq. (7) that lead to 2 acting on R 1. Thus, it is a mathematical identity that gives the apparent contact, rather than a truly singular magnetic field at the site of one particle or the other.

Moreover, we also can see that the reason Fermi did not employ the proton coordinate is that he implicitly ignored the motion of the proton, which is more or less the position of the center of mass, and can be taken to be fixed. Fermi's approach does not apply to the deuteron or to positronium, but Casimir's approach does. We believe that Casimir's calculation, as given above rather explicitly, should not be too difficult for an advanced undergraduate to follow.

I have tried, without success, to identify precisely where in the literature the term “contact interaction” first appears. That term does not appear in Bethe and Salpeter's review article of 1957, where the corresponding equation does appear.6 A sampling of the Russian literature does not show it either.7,8 I would appreciate further information from any reader who knows more about the origin of this term.

The author would like to thank one of the referees of Ref. 5 for an essential suggestion.

The author has no conflict of interest.

1.
If the nuclear magnetic moment is due to a microscopic current loop, then in the vicinity of the origin, its magnetic field points along the nuclear moment; if it is due to microscopic magnetic poles, then the moment points from the negative to the positive part of the pole distribution, and therefore, in the vicinity of the origin, its magnetic field points opposite the nuclear moment. As a consequence, the sign of the s-state magnetic dipole interaction reveals information about the internal composition of matter.
2.
(a)
E.
Fermi
, “
Über die magnetischen Momente der Atomkerne
,”
Z. Phys.
60
(
5–6
),
320
333
(
1930
).
(b)
E.
Fermi
, “
Sui momenti magnetici dei nuclei atomici
,”
Mem. Accad. d'Italia 1, (Fis.)
P69
,
139
148
(1930). For completeness, (b) includes an extract from paper (a). Both titles translate to English as “On the magnetic moments of atomic nuclei.”
3.
See https://en.wikipedia.orgon for “
Wikipedia, the Free Encyclopedia
2023
.
4.
H. B. G.
Casimir
, “
On the magnetic interaction in the neutron
,”
Physica
3
(
9
),
936
938
(
1936
).
5.
W. M.
Saslow
, “
Magnetic hyperfine interaction made easier
,”
Am. J. Phys.
92
(
5
),
367
370
(
2024
).
6.
H. A.
Bethe
and
E. E.
Salpeter
,
Quantum Mechanics of One and Two Electron Atoms
(
Springer Verlag
,
Goettingen
,
1957
), Sec. 12. A search for “contact” within the text revealed that it occurs only in the phrase “contact transformation,” which appears five times on p. 83 and once on p. 177.
7.
L. D.
Landau
and
E. M.
Lifshitz
, “
Quantum mechanics
,”
Course of Theoretical Physics
(
Pergamon Press
,
Oxford
,
1978
), Vol.
3
; specifically, the magnetic hyperfine interaction energy is computed using the Biot–Savart field. See Secs. 115 and 121. The very last problem in this book is on the hyperfine interaction.
8.
M. A.
Andreichikov
,
B. O.
Kerbikov
, and
Yu. A
Simonov
, “
Magnetic field focusing of hyperfine interaction in hydrogen
,”
JETP Lett.
99
(
5
),
246
249
(
2014
).