Testing and refining the crystalline-field model for the magnetic susceptibility of nitric oxide adsorbed on silica gel: low-temperature behavior, hindered rotation, and site heterogeneity

Nitric oxide (NO) adsorbed on silica gel shows an anomalous magnetic susceptibility: the effective moment stays near 1.83-1.99 Bohr magnetons between 181 and 293 K, instead of dropping toward zero below the spin-orbit splitting as free-molecule theory predicts. Huang and Van Vleck (1969) explained this with a 'frozen-axis' model in which a surface crystalline field of strength $A$ mixes the $^{2}\Pi_{1/2}$ and $^{2}\Pi_{3/2}$ spin-orbit states, but the fit forces $A$ to increase with temperature, and the authors themselves listed what would be needed to test and improve the model: magnetic data over a much wider temperature range, magnetic resonance, a treatment of hindered rotation instead of a completely frozen molecular axis, a distribution of $A$ reflecting heterogeneous adsorption sites, and low-pressure heats of adsorption. Electron spin resonance spectra of NO on silica gel have since been measured and give $g$ -factors and two adsorption-site families with different crystal-field splittings (Iwaizumi, Kubota and Isobe, 1974), while wide-temperature-range susceptibility exists only for a different host (mesoporous MCM-41, where dimerization dominates below about 80 K); no measurement or refined calculation on silica gel itself has ever tested the model's predicted low-temperature Curie-type term or its required temperature-dependent crystal field. The problem of establishing, experimentally and theoretically, the magnetic ground state of NO adsorbed on amorphous silica and whether the temperature-dependent crystalline field is real therefore remains open.

Background

Nitric oxide (NO) is an odd-electron molecule whose electronic ground state is a doubly degenerate orbital ( $\Pi$ ) state, split by spin-orbit coupling into a lower $^{2}\Pi_{1/2}$ level and an upper $^{2}\Pi_{3/2}$ level separated by about 121 cm $^{-1}$ , equal to $k_B T$ at about 173 K. Each spin-orbit level is a Kramers doublet, meaning a pair of states that time-reversal symmetry keeps exactly degenerate for an odd number of electrons. Van Vleck's classic theory for the free molecule predicts that the effective magnetic moment, quoted in Bohr magnetons as the effective magneton number $\mu$ , approaches 2 when $k_B T$ is large compared with the spin-orbit splitting and approaches 0 when $k_B T$ is much smaller, because in the $^{2}\Pi_{1/2}$ level the orbital and spin contributions to the magnetic moment nearly cancel.

Silica gel is a highly porous, amorphous form of $\mathrm{SiO_2}$ with a large internal surface containing a distribution of adsorption sites of different binding strength; the differential heat of adsorption (the heat released per mole as additional molecules adsorb at a given coverage) decreases with coverage on such heterogeneous surfaces. Solbakken and Reyerson (1959) measured the magnetic susceptibility of NO adsorbed on silica gel at only four temperatures and found $\mu = 1.98$ at 181 K, $1.99$ at 193 K, and $1.83$ at 273 K and 293 K - values close to the free-molecule high-temperature limit at temperatures where the free-molecule theory predicts a substantial reduction, and larger at lower temperature. Their tentative explanation (all molecules in the upper spin-orbit state) is physically untenable.

Huang and Van Vleck (1969) analyzed these data with a deliberately primitive model: the molecular axis of each adsorbed NO is regarded as completely frozen in space by a strong axial crystalline potential $V_c$ , whose only important matrix element is $A = \langle \Lambda = +1 | V_c | \Lambda = -1 \rangle$ , where $\Lambda h/2\pi$ is the orbital angular momentum component along the molecular axis. This matrix element ( $\Delta \Lambda = 2$ ) mixes the $^{2}\Pi_{1/2}$ and $^{2}\Pi_{3/2}$ spin-orbit states. They derived closed expressions for the magnetic susceptibilities $\chi_\parallel$ and $\chi_\perp$ parallel and perpendicular to the molecular axis, each carrying a $1/k_B T$ prefactor times a function of $x = (E_1 - E_0)/k_B T$ , where $E_1 - E_0$ is the splitting between the two lowest levels of the adsorbed molecule; at low temperature the model therefore predicts a Curie-type $1/T$ contribution from the ground Kramers doublet whose strength is governed by the admixture of $^{2}\Pi_{3/2}$ character. Fitting the spatially averaged moment to the 1959 data, however, requires the crystalline field parameter to increase with temperature, from $A/k_B$ of about 400 K at 181 K to about 1700 K at 273 K. The authors rationalized this by surface heterogeneity: the differential heat of adsorption of NO on silica gel decreases with coverage, so at higher temperature (lower coverage) molecules sit preferentially in deeper, more strongly binding sites with stronger crystal fields; they estimated the average adsorptive potential $\bar{U}$ as 2.5 kcal/mol at 181 K and 5.2 kcal/mol at 273 K from a two-dimensional van der Waals model, and the ratio of these two values (about 2) is of the same order as the ratio of the required crystalline field parameters. They also checked that dimerization of NO into diamagnetic $\mathrm{N_2O_2}$ at low temperature would push the susceptibility in the wrong direction for the 1959 data.

Electron spin resonance (ESR, equivalently EPR) is a microwave spectroscopy that directly measures the $g$ -factor (the proportionality between magnetic moment and spin angular momentum) of a paramagnetic species and thus the character of its ground state. The 1969 paper's closing discussion states that more refined calculations are not warranted until more extensive experimental data become available: magnetic data over a wider temperature range, possibly magnetic resonance measurements (with the worry that resonance lines on a powdered, heterogeneous adsorbent may be too broad to observe), and heats of adsorption at very low pressures to map the distribution of adsorption potentials. ESR work on other oxides (e.g., Lunsford's 1969 study of NO adsorbed on $\gamma$ -alumina, silica-alumina, and silica-magnesia, where NO serves as a spin probe of surface sites) had shown such spectra to be informative for related surfaces. ESR spectra of NO on silica gel itself, recorded between $-196$ and about $-160$ °C, show the average crystal field acting on the adsorbed molecule decreasing as the temperature is raised, with the lines broadening and fading out above about $-160$ °C (Iwaizumi, Kubota and Isobe, 1974).

Problem Statement

Establish the microscopic magnetic state of nitric oxide adsorbed on silica gel and settle whether the Huang-Van Vleck frozen-axis crystalline-field model, with its required temperature-dependent field parameter, is correct. The central open issue is that the size of the crystalline potential must increase with temperature to explain the experimental data. At low temperature the model's ground Kramers doublet predicts a Curie-type $1/T$ term whose strength is governed by the admixture of $^{2}\Pi_{3/2}$ character into the $^{2}\Pi_{1/2}$ level, and cooperative effects such as dimerization into diamagnetic $\mathrm{N_2O_2}$ could mask or destroy it. It remains open whether the required increase of $A$ with temperature (the fitted $A/k_B$ rising from about 400 K at 181 K to about 1700 K at 273 K) is physically real or a consequence of temperature-dependent site populations, whether the frozen molecular axis with a single fixed crystalline-field parameter $A$ at each temperature is adequate or whether strongly hindered rotation and a distribution of $A$ reflecting the known heterogeneity of adsorption sites are required, and how the distribution of adsorptive potentials, through the average adsorptive potential $\bar{U}$ , relates quantitatively to the crystalline field. The magnetic-resonance sub-question (whether EPR/NMR of adsorbed NO on silica gel is observable and what the $g$ factor says about the ground state) has been answered affirmatively by the 1974 ESR study and is therefore available as input to, rather than a target of, this problem; the unexplained conflict between the ESR temperature trend of the average crystal field and the trend required by the susceptibility fit remains part of the open question.

References

N. L. Huang and J. H. Van Vleck, 'Magnetic Susceptibility of Nitric Oxide Molecules Adsorbed on Silica Gel', The Journal of Chemical Physics 50(7), 2932-2935 (1969). DOI: 10.1063/1.1671486, https://doi.org/10.1063/1.1671486 (source paper; full text paywalled).
Aage Solbakken and Lloyd H. Reyerson, 'Sorption and Magnetic Susceptibility Studies on Nitric Oxide-Silica Gel systems at a Number of Temperature', The Journal of Physical Chemistry 63(10), 1622-1625 (1959). DOI: 10.1021/j150580a014, https://doi.org/10.1021/j150580a014 (experimental data interpreted by the source paper).
Masamoto Iwaizumi, Shozo Kubota, and Taro Isobe, 'An ESR Study of Nitric Oxide Adsorbed on Silica Gel', Bulletin of the Chemical Society of Japan 47(3), 597-599 (1974). DOI: 10.1246/bcsj.47.597, https://doi.org/10.1246/bcsj.47.597.
C. E. Brown and P. G. Hall, 'Physical adsorption of nitric oxide on graphite and silica and adsorption of gases on nitric oxide preadsorbed on carbon', Journal of Colloid and Interface Science 42(2), 334-341 (1973). DOI: 10.1016/0021-9797(73)90297-X, https://doi.org/10.1016/0021-9797(73)90297-X.
Masaki Mito, Tsuyoshi Tatano, Yuki Komorida, Takayuki Tajiri, Hiroyuki Deguchi, Seishi Takagi, Shigemi Kohiki, Masaaki Ohba, Ryotaro Matsuda, and Susumu Kitagawa, 'Magnetic properties of nitric oxide molecules physisorbed into nano-sized pores of MCM-41', Microporous and Mesoporous Materials 132(3), 464-469 (2010). DOI: 10.1016/j.micromeso.2010.03.028, https://doi.org/10.1016/j.micromeso.2010.03.028 (closed access).
J. Lunsford, 'A study of surface interactions on gamma-alumina, silica-alumina, and silica-magnesia using the EPR spectra of adsorbed nitric oxide', Journal of Catalysis 14(4), 379-385 (1969). DOI: 10.1016/0021-9517(69)90328-5, https://doi.org/10.1016/0021-9517(69)90328-5.
Ranjani V. Siriwardane and Jason M. Cook, 'Interactions of NO and SO2 with iron deposited on silica', Journal of Colloid and Interface Science 104(1), 250-257 (1985). DOI: 10.1016/0021-9797(85)90029-3, https://doi.org/10.1016/0021-9797(85)90029-3.
Arafat H. Khan, D. Michel, and M. Bertmer, 'Temperature dependent 15N NMR study of nitric oxide', Solid State Nuclear Magnetic Resonance 101, 38-43 (2019). DOI: 10.1016/j.ssnmr.2019.05.002, https://doi.org/10.1016/j.ssnmr.2019.05.002.
Y. Horii, H. Suzuki, Y. Miyazaki, M. Nakano, S. Hasegawa, Y. Hashikawa, and Y. Murata, 'Dynamics and magnetic properties of NO molecules encapsulated in open-cage fullerene derivatives evidenced by low temperature heat capacity', Physical Chemistry Chemical Physics 23(17), 10251-10256 (2021). DOI: 10.1039/d1cp00482d, https://doi.org/10.1039/d1cp00482d.
Benjamin Barth, Matthias Mendt, Andreas Pöppl, and Martin Hartmann, 'Adsorption of nitric oxide in metal-organic frameworks: Low temperature IR and EPR spectroscopic evaluation of the role of open metal sites', Microporous and Mesoporous Materials 216, 97-110 (2015). DOI: 10.1016/j.micromeso.2015.02.020, https://doi.org/10.1016/j.micromeso.2015.02.020.