Nuclear electric dipole moment

The nuclear electric dipole moment (nuclear EDM)${\displaystyle d_{\text{e}}^{N}}$ is an intrinsic property of a nucleus ${\displaystyle N}$ that can be non-zero only if CP violating interactions occur within the atom or the nucleus itself. They have been surmised to exist since at least 1963 ^[1] in both light ^[2] and heavy nuclei ^[3] .

CP-violating properties and interactions that can contribute to a nuclear electric dipole moment include the neutron EDM, the proton EDM as well as CP-violating interactions of nucleons with meson or photons. At a more fundamental level, nuclear EDMs can originate from the electroweak sector, a non-zero ${\displaystyle {\bar {\theta }}_{\text{QCD}}}$ or from physical processes that go beyond the standard model of particle physics.

Experimental Searches

Experimental searches of subatomic EDMs have nearly always been conducted with a powerful external electric field ${\displaystyle \mathbf {E} }$ collinear with an external magnetic field ${\displaystyle \mathbf {B} }$ that exploits the fact that the potential energy depends on the relative orientation of the electromagnetic fields

${\displaystyle U=-(\mathbf {m} _{N}\cdot \mathbf {B} +\mathbf {d} _{\text{e}}^{N}\cdot \mathbf {E} )}$

where ${\displaystyle \mathbf {m} _{N}}$ is the nuclear magnetic moment. The Larmor frequency is proportional to the potential energy which depends on whether the two external fields are parallel or antiparallel to each other. Flipping the direction of ${\displaystyle \mathbf {E} }$ and subtracting the two corresponding Larmor frequencies leads to a result that is proportional to ${\displaystyle d_{\text{e}}^{N}}$. As this technique relies on large electric fields, it is always applied to neutral systems like the neutron or atoms making it difficult to apply to an electrically charged system like a proton or an ionized atom. Furthermore, applying this technique to neutral atoms suppresses the nuclear EDM contribution because the external electric field is nearly zero at the location of the nucleus as must therefore be the contribution of ${\displaystyle d_{\text{e}}^{N}}$ to ${\displaystyle U}$; this suppression is referred to as Schiff screening ^[1] . In fact, for point like non-relativistic nuclei, Schiff screening makes the term proportional to ${\displaystyle d_{\text{e}}^{N}}$ in the potential energy ${\displaystyle U}$ disappear. However, atoms with large atomic numbers have significant relativistic corrections and relatively large nuclear charge radii. Hence, experimental searches of atomic or nuclear EDMs have traditionally used nuclear isotopes with large atomic numbers.

The best upper limit on an atomic EDM was measured on ${\displaystyle ^{199}{\text{Hg}}}$, ${\displaystyle |d_{\text{e}}^{\text{Hg}}|<7.4\times 10^{-30}e\cdot }$cm (95% C.L.) using electric voltages of ${\displaystyle \pm 6~{\text{kV}}}$ and ${\displaystyle \pm 10~{\text{kV}}}$. ^[4]

Nuclear electric dipole moments in metals

Schiff screening in a metal is partially lifted for point like non-relativistic nuclei because some if not all the valence electrons are in the conduction band exposing the nuclear EDM. In fact, the suppression factor of a point like nucleus by the bound electrons is generally ^[5]

${\displaystyle \sigma _{\text{sf}}={\frac {N_{\text{c}}}{Z}}}$

where ${\displaystyle N_{\text{c}}}$ is the number of electrons in the conduction band and ${\displaystyle Z}$ is the atomic number. ${\displaystyle N_{\text{c}}}$ is often equal to the number of valence electrons. The crystal lattice of the bulk material can thus be thought of as having a point like nuclear electric dipole moment at every lattice site.

The total electric field ${\displaystyle \mathbf {E} _{\text{T}}}$ of a point like nuclear electric dipole has two contributions

${\displaystyle \mathbf {E} _{\text{T}}=\mathbf {E_{\lambda }} +\mathbf {E_{\delta }} }$

${\displaystyle \mathbf {E_{\lambda }} \equiv {\frac {1}{4\pi \epsilon _{0}}}{\frac {3(\mathbf {d_{\text{e}}^{N}} \cdot {\hat {\mathbf {r} }}){\hat {\mathbf {r} }}-\mathbf {d_{\text{e}}^{N}} }{r^{3}}}}$

${\displaystyle \mathbf {E_{\delta }} \equiv -\delta ^{3}(\mathbf {r} ){\frac {\mathbf {d_{\text{e}}^{N}} }{3\epsilon _{0}}}}$

where ${\displaystyle \mathbf {d_{\text{e}}^{N}} }$ is the electric dipole moment of nucleus ${\displaystyle N}$ and ${\displaystyle \epsilon _{0}}$ is the permittivity of free space. ${\displaystyle \mathbf {E_{\lambda }} }$ is the long range contribution to the total electric field while ${\displaystyle \mathbf {E_{\delta }} }$ is the contact contribution of the total electric field that only exists at the lattice site and is null otherwise. The mean long range electric field, ${\displaystyle \mathbf {{\bar {E}}_{\lambda }} }$ in a bulk material is called the electrization field. In a closed circuit, the mean, total electric field stemming from the point like nuclear EDMs in the bulk material must be zero with ${\displaystyle \mathbf {{\bar {E}}_{\lambda }=-{\bar {E}}_{\delta }\neq 0} }$.

Electrization field in a superconductor ^[5]

In a superconductor, the Cooper pairs cannot scatter off the lattice sites and their wave functions must be null there implying that they cannot be influenced by the contact electric field ${\displaystyle \mathbf {E_{\delta }} }$. Hence, the Cooper pairs can only be influenced by the electrization field ${\displaystyle \mathbf {{\bar {E}}_{\lambda }} \neq 0}$ if a fraction of the nuclear EDMs are aligned.

The electric and magnetic dipole moments of a nucleus must be collinear ^[6] because of the Wigner-Eckart theorem. That implies that if a fraction of nuclear magnetic moments in the penetration depth are aligned by an external magnetic field, the precise same fraction of nuclear EDMs will also be aligned with that external magnetic field thus generating an electrization field that can accelerate the Cooper pairs. The energy for the work done by the electrization field on the Cooper pairs ultimately comes from the source of the external magnetic field. If the closed superconducting loop has a self-inductance ${\displaystyle L}$ and a magnetization ${\displaystyle M}$ is induced in the penetration depth, the rate of the change of the supercurrent ${\displaystyle I_{\text{s}}}$ will be

${\displaystyle L{\frac {{\text{d}}I_{\text{s}}}{{\text{d}}t}}=d_{\text{e}}^{N}{\frac {\sigma _{\text{sf}}}{3\epsilon _{0}m_{N}}}M}$

Knowing ${\displaystyle L}$, ${\displaystyle m_{N}}$, the magnetization ${\displaystyle M}$ and measuring the rate of change of the supercurrent allows for an experimental determination of ${\displaystyle d_{\text{e}}^{N}}$.

Significance ^[5]

The extreme sensitivity of this method creates a number of experimental possibilities, for example:

• measure a large number of nuclear EDMs in a way that would over-constrain a smaller number of CP-violating parameters, including the neutron EDM, and thus determine them, in the same way as the large dataset of nuclear magnetic moments permits the determination of parameters that contributes to these moments ^{[7] [8]} ;
• measure the electron EDM in p-wave superconductors ^{[9] [10]} ;
• detect the induced nuclear EDM of the cosmic axion background by searching for an oscillating supercurrent whose frequency is proportional to the axion mass and whose amplitude would be proportional to the local density of the dark matter axions at the location of the laboratory ^[11] .

See also

• Neutron electric dipole moment
• Electron electric dipole moment
• Electron magnetic moment
• Anomalous electric dipole moment
• Anomalous magnetic dipole moment
• Electric dipole spin resonance
• Parity (physics) § Parity violation
• CP violation
• Charge conjugation
• T-symmetry

References

1. Schiff, L. I. (1963). "Measurability of Nuclear Electric Dipole Moments". Physical Review. 132 (5) 5: 2194–2200. Bibcode:1963PhRv..132.2194S. doi:10.1103/physrev.132.2194.
2. Dekens, W.; de Vries, J.; Bsaisou, J.; Hanhart, C.; Bernreuther, W.; Meißner, Ulf-G.; Nogga, A.; Wirzba, A. (2014). "Unraveling models of CP violation through electric dipole moments of light nuclei". JHEP. 07 (7) 69. arXiv: 1404.6082 . Bibcode:2014JHEP...07..069D. doi:10.1007/JHEP07(2014)069.
3. Auerbach, N.; Flambaum, V. V.; Spevak, N. (1996). "Collective T-odd and P-odd Electromagnetic Moments in Nuclei with Octupole Deformations". Physical Review Letters. 76 (23) 23: 4316–4319. arXiv: nucl-th/9601046 . Bibcode:1996PhRvL..76.4316A. doi:10.1103/physrevlett.76.4316. PMID   10061259.
4. Graner, B.; Chen, Y.; Lindahl, E. G.; Heckel, B. R. (2016). "Reduced Limit on the Permanent Electric Dipole Moment of ${\displaystyle ^{199}}$Hg". Physical Review Letters. 116 (16) 16. arXiv: 1601.04339 . doi:10.1103/physrevlett.116.161601. PMID   27152789.
5. Gary Prézeau (2025). "First non-zero measurement of a nuclear electric dipole moment". arXiv: 2510.21768 [nucl-ex].
6. Donoghue, J. F.; Golowich, E.; Holstein, B. R. (1992). Dynamics of the Standard Model. Cambridge University Press. p. 250.
7. Chambers-Wall, G.; Gnech, A.; King, G. B.; Pastore, S.; Piarulli, M.; Schiavilla, R.; Wiringa, R. B. (2024). "Magnetic structure of ${\displaystyle A\leq 10}$ nuclei using the Norfolk nuclear models with quantum Monte Carlo methods". Physical Review C. 110: 054316. doi:10.1103/physrevc.110.054316. OSTI   2478508.
8. Li, Jian; Meng, J. (2018). "Nuclear magnetic moments in covariant density functional theory". Frontiers of Physics. 13 (6) 132109. doi:10.1007/s11467-018-0842-7.
9. Maeno, Y.; Hashimoto, H.; Yoshida, K.; Nishizaki, S.; Fujita, T.; Bednorz, J. G.; Lichtenberg, F. (1994). "Superconductivity in a layered perovskite without copper". Nature. 372 (6506): 532–534. Bibcode:1994Natur.372..532M. doi:10.1038/372532a0.
10. Talantsev, E. F.; Iida, K.; Matsumoto, T.; Crump, W. P.; Strickland, N. M.; Wimbush, S. C.; Ikuta, H. (2019). "p-wave superconductivity in iron-based superconductors". Scientific Reports. 9 (1) 14245. arXiv: 1905.00183 . Bibcode:2019NatSR...914245T. doi:10.1038/s41598-019-50687-y. PMC   6775168 . PMID   31578391.
11. Abel, C. et al (2017). "Search for Axionlike Dark Matter through Nuclear Spin Precession in Electric and Magnetic Fields". Physical Review X. 7 (4) 041034. doi:10.1103/physrevx.7.041034.