H, H and H Nuclear Magnetic Dipole Moment Effects on the Electron

Pramana – J. Phys. (2021) 95:104 © Indian Academy of Sciences https://doi.org/10.1007/s12043-021-02142-9

1H, 2Hand3H nuclear magnetic dipole moment effects on the electron energy states

M F ALSHUDIFAT

Department of Physics, Al al-Bayt University, Mafraq 25113, Jordan E-mail: [email protected]

MS received 26 October 2020; accepted 12 March 2021

Abstract. The effect of nuclear magnetic dipole moment of hydrogen, deuterium and tritium isotopes on the electron atomic energy has been studied and analysed in this work. The electromagnetic Hamiltonian was rewritten in terms of the orbital angular momentum Le of the electron, the gyromagnetic ratio γN of the nucleus and its total angular momentum JN . The ﬁrst-order energy correction has been derived, and this correction splits each state into 2JN +1 states. The energy corrections and the differences between the split states of the ﬁrst two lowest energy states for each isotope were calculated and it is found that these two energy states are in the radiowave range.

Keywords. Nuclear magnetic dipole moment; gyromagnetic ratio; vector potential; hydrogen isotopes; radiowaves.

PACS Nos 31.30.Gs; 31.15.aj

1. Introduction atomic hydrogen, ψnm(0) = 0 unless = 0, and the JN ·se interaction in eq. (1) reduces to sN ·se Hydrogen is the most abundant element in the Uni- spinÐspin interaction and the hyperfine splitting affects verse, and it represents the simplest atomic system [1]. essentially the atomic S-states. Hydrogen is studied widely because of its importance. In this work, the magnetic moment of the nucleus Different physical interactions between the electron and is rewritten in terms of the magnetic vector potential, the atomic nucleus are exploited to describe the physical and the interaction between this vector potential and structure of this atom. The hydrogen atom was solved the electron’s angular momentum in the 1H, 2Hand3H quantum mechanically using the predominant Coulomb hydrogen isotopes was considered as a perturbation. The potential between the electrically charged nucleus and first-order energy corrections due to this interaction have the electron, Other interaction potentials have a small been derived, and the energy corrections and differences effect on the atomic energy and are considered as per- between split states for the lowest energy states have turbations. Sommerfeld introduced the fine-structure been calculated in terms of the optical wavelength. As correction [2]. He combined the relativistic correction far as I know, this work has not been done before. with the spinÐorbit coupling corrections. The fine structure kills the degeneracy in azimuthal quantum number . The hyperfine splitting represents the interaction between the magnetic moment of the nucleus and the electron spin in general [3,4]. The first-order energy correction due to the hyperfine splitting can be written as 2. Energy correction analysis [3] The atomic nuclei in this work are characterised by 2 2 the dominant magnetic dipole moment μ = γ J . Ehf = μ0γeγN JN ·se |ψnm(0)| , (1) N N 3 This magnetic dipole moment creates a magnetic vector where γe(γN ) is the electron(nucleus) gyromagnetic potential AN that interacts with the atomic electrons. ratio, J is the total angular momentum of the nucleus The magnetic vector potential is presented in terms of N γ and se is the electron spin. The wave unction of the nuclear gyromagnetic ratio N and the total angular

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3 momentum JN as The expectation value of 1/r was calculated using Kramer’s relation [5]as μ × 0 JN r AN = γN . (2) 1 1 4π r 3 = . (10) r 3 ( + 1/2)( + 1) n3 a3 The Hamiltonian operator in the presence of magnetic e e e vector potential is The first-order energy correction due to the nuclear magnetic vector potential becomes 1 H = (p + eA)2 − eV(r). (3) 2m e μ (hc¯ )2 e H = 0 2m c2 4π a3 The magnetic vector potential of the nucleus A has a e N ( + ) − ( + ) − ( + ) small effect on the atomic energy states and it can be × i i 1 jN jN 1 e e 1 γ , 3 N considered as a perturbation, where e(e + 1/2)(e + 1) n 2 (11) = pe − ( ) H0 eV r where 2me 2 e μ (hc¯ ) − and the perturbed Hamiltonian is 0 = 4.118 × 10 15 eV T s. 2 π 3 2 2mec 4 a e e 2 H = (pe · AN + AN ·pe) + A . (4) Accordingtoeq.(11), the first-order energy correc- 2m 2m N e e tion depends on the nuclear gyromagnetic ratio, energy Substitute AN into eq. (4), and the perturbed Hamilto- state’s principle number n, the nuclear total angular nian becomes momentum, the electron’s orbital angular momentum 2 and its summation. This correction splits each atomic e μ0 1 e H = γ p · (J ×r) + A2 . (5) energy state into 2JN + 1 states. π N 3 e N 2 N me 4 r 4me The vector product p ·(J ×r) rewritten as J ·(r× e N N 3. Energy correction results pe), is simply the scalar product between the angular momentum of the atomic electron and the total angular The interaction between the nuclear magnetic dipole momentum (Le · JN ) of the nucleus. The interaction Hamiltonian becomes moments of the hydrogen, deuterium and tritium isotopes and the orbital angular momentum of the atomic μ e 0 1 electron was derived in this work and the first-order H = γN Le · JN m 4π r 3 e energy correction is presented by eq. (11). The first- 2 order energy correction equation for S-states ( = 0and e2 μ 2 J ×r + 0 γ N . i = j) is reduced to N 3 (6) 2me 4π r −15 4.118 × 10 eVTs Since the second part of eq. (6) is very small compared H =−2 γN . (12) n3 to the first one, it was ignored, and the first-order energy 1 correction E is rewritten as The first-order energy corrections for the first two lowest e μ 1 energy states have been calculated and listed in table 1. H = 0 γ L · J . N 3 e N (7) The energy differences between split energy states ( me 4π r = 1) in terms of eV and the optical wavelengths are This interaction depends on the expectation values of listed in table 1. As the energy correction for higher 3 Le · JN and 1/r . Assume that the sum of Le and energy states (n > 2) becomes less significant, it was JN is given by ignored. = + . The energy correction due to the interaction between I Le JN (8) the nuclear magnetic dipole moment and the orbital Then the expected value of Le · JN according to eq. angular momentum of atomic electron is found to (8)isgivenby increase the S-state atomic potential depth for hydrogen and deuterium by 2.20×10−6 and 3.38×10−7 eV for 2 − − h¯ n = 1 and by 2.75×10 7 and 4.23×10 8 eV for n = 2 Le · JN = (i(i + 1) − e(e + 1 2 respectively. Since the gyromagnetic ratio of tritium is − jN ( jN + 1)). (9) negative [6], the energy correction decreases the S-state Pramana – J. Phys. (2021) 95:104 Page 3 of 3 104

Table 1. The atomic first-order energy corrections due to the nuclear magnetic dipole moment of 1H, 2H and 3H in terms of eV and the optical wavelength units for the ground state and first excited state. −1 −1 1 1 Isotope γN (s T ) jN n iE(eV) E (eV) λ (m) 1H2.68×108 [6]1/2101/2 −2.20 × 10−6 ÐÐ 201/2 −2.75 × 10−7 ÐÐ 211/2 −9.18 × 10−8 1.38×10−7 9.0 213/24.59×10−8 2H4.11×107 [6]1101−3.38 × 10−7 ÐÐ 201 −4.23 × 10−8 ÐÐ 210 −2.82 × 10−8 1.41×10−8 87.9 211 −1.41 × 10−8 2.82×10−8 43.9 222 1.41×10−8 4.23×10−8 29.3 3H −2.04 × 108 [6]1/2101/21.68×10−6 ÐÐ 201/22.10×10−7 ÐÐ 211/26.99×10−8 1.04×10−7 11.8 213/2 −3.50 × 10−8 potential depth by 1.68×10−6 and 2.10×10−7 eV for References n = 1 and 2 respectively. The non-zero state splits into 2 jN + 1 states under this correction, which kills [1] N Frischauf, in Compendium of hydrogen energy, Wood- the degeneracy in . The energy differences between head publishing series in energy edited by M Ball, A the new split states were calculated in terms of the Basile and T N Veziroùglu (Woodhead Publishing, Oxford, optical wavelength and they are found to be in the 2016) pp. 87Ð107 radiowave range. Radiowave lines such as 9.0 m can [2] A Sommerfeld, Naturwissenschaften 28, 417 (1940) be emitted by the hydrogen atom, 29.3, 43.9, 87.9 m [3] D J Griffiths, Am.J.Phys.50, 698 (1982) [4] J E Nafe, E B Nelson and I I Rabi, Phys. Rev. 71, 914 by deuterium, and 11.8 m by tritium. To the best (1947) of the author’s knowledge, no recorded radiowave [5] H Beker, Am.J.Phys.65, 1118 (1997) lines with these wavelengths were found in the litera- [6] R Lässer, Tritium and helium-3 in metals (Springer Sci- ture. ence & Business Media, 2013) Vol. 9