{'fraction_non_alphanumeric': 0.039661505556330495, 'fraction_numerical': 0.01669223973589994, 'mean_word_length': 4.6839323467230445, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Fluctuating magnetic moments in liquid metals

We re-analyze literature data on neutron scattering by liquid metals to show that non-magnetic liquid metals possess a magnetic moment that fluctuates on a picosecond time scale. This time scale follows the motion of the cage-diffusion process in which an ion rattles around in the cage formed by its neighbors. We find that these fluctuating magnetic moments are present in liquid Hg, Al, Ga and Pb, and possibly also in the alkali metals.

Unlike ordinary simple fluids, liquid metals can support short-wavelength sound waves far outside the hydrodynamic regime; simple fluids only support very strongly damped density fluctuations beyond the hydrodynamic region [34]. In other words, a density disturbance decays much faster in a simple fluid than it does in a liquid metal under comparable thermodynamic conditions. Typically, a short-wavelength sound wave in a simple liquid does not propagate beyond one wavelength [34]. Presumably, this difference can be attributed to the presence of two interacting systems in a liquid metal: the positively charged ionic liquid and the negatively charged conduction sea. This notion has stimulated the study of the decay mechanism of the density fluctuations, by means of neutron and X-ray scattering experiments, as well as by molecular dynamics (MD) computer simulations in a range of liquid metals, such as Hg [1,2,3,26,27], Cs [4,5,28], K [6,7,8], Rb [9,10], Na [11,12,13,14,29,30], Li [15,16,17,31], Pb [18,19,20,21,32], Al [22,33], and Ga [23].
These studies have by and large confirmed the role of the electron sea as a feedback mechanism, serving to reduce the decay rates of disturbances, and also ensuring that density fluctuations can propagate at a higher velocity than the adiabatic sound velocity.
These studies also showed that similar to ordinary liquids, cage diffusion plays an important part in the decay mechanism of density fluctuations [3,34,35,36,37,38]. Cage diffusion occurs when an atom bounces off neighboring atoms, thereby confining the atom to a "cage." This is in contrast to self-diffusion, the process in which the atom moves through the sample and which is characterized by a net displacement from its starting position over a period of time [34]. In MD simulations, where one follows the position of an atom over time, cage-diffusion and self-diffusion show up as two distinct time scales [1]. Cage-diffusion accounts for a small decrease in correlation between the initial and subsequent position of an atom; this initial decrease in correlation occurs within a few picoseconds. The overall demise of correlation is given by the self-diffusion process, which takes place on a much longer time scale [34] and is determined by the coefficient for self-diffusion D s . These two diffusive processes can also be observed by means of quasi-elastic neutron scattering [2].
Neutron scattering is sensitive to the motion of individual atoms because an atom moves during the time it takes the neutron to interact with it [39]. This motion shows itself as a spread in energy of the scattered neutron wave-packet. Rapid movement (cage-diffusion) results in a large spread in energy; slow movement (self-diffusion) results in a spread with small characteristic energy-width. Both these processes have indeed been observed in liquid metals. For instance, in liquid mercury [1,2,3], the scattered neutron intensity originating from a single atom (the so-called incoherent scattering contribution [39]) corresponds to a superposition of two Lorentzian lines. One line is sharp (in energy), corresponding to self-diffusion, and one line is broad, corresponding to cage-diffusion. A Lorentzian line in energy corresponds to an exponential decay in time [39] of the correlation between the initial and subsequent position of an atom.
A comparison between the neutron scattering data and the MD simulations on liquid Hg revealed a serious discrepancy regarding the effectiveness of the cage-diffusion mechanism [1,2,3]. While both studies agreed on the characteristic time scale for the cage-diffusion process, according to the neutron scattering study [2] cage diffusion accounted for up to 20% of the loss in correlation in the position of an atom, compared to only 0.4% as observed in the MD results [3]. In order to explain this discrepancy, Badyal et al. [2] suggested that a mercury ion might have a fluctuating magnetic moment, resulting in an enhanced neutron scattering cross-section. The idea here is straightforward (see Fig. 1): in a liquid, atoms can approach each other very closely. On such a close approach, an electron from a filled inner shell of the metallic ion can be ejected into the Fermi-sea (Fig. 1b), resulting in an unpaired electron, and hence in a magnetic moment (Fig. 1c). Once the ions move away from each other again, the shell can be re-completed (Fig. 1d). One can thus expect a magnetic moment to pop in and out of existence on the same time scale as the rattling motion of an atom inside its cage. This process automatically leads to a pathway for the neutron to scatter from the atom via the electromagnetic force [39], augmenting the interaction via the strong nuclear force and resulting in an enhanced cross section for the cage-diffusion process. From the strength of the magnetic interaction [39], it can then be determined what fraction of the time an ion has an unpaired electron.

3
In this paper we show that the cage-diffusion process in liquid metals is indeed accompanied by a fluctuating magnetic moment. We do this by revisiting published neutron scattering data on Hg, Cs, K, Rb, Na, Li, Pb, Al, and Ga. We observe a small effect in the alkali metals, but find that the ions in Ga and Hg have unpaired electrons for up to 20% of the time. Not only do these magnetic moments provide an additional means for studying cage-diffusion by means of neutron scattering, they provide an additional long-range interaction mechanism for the ions in the liquid.

II. THEORY
In this section we briefly review the various contributions that make up the neutron scattering cross section of a liquid. We use the data by Badyal et al. [2] on mercury to illustrate the various contributions, and to demonstrate under what conditions one can observe the proposed fluctuating magnetic moments.
A neutron interacts with the nucleus of an atom via the strong nuclear force, and with the magnetic moments of electrons present in the system via the electromagnetic force [39].
Thus, the total number of neutrons with initial energy E i that are scattered every second into a solid angle dΩ having final energies between E f and E f + dE is given by the double differential cross section and can be separated into a nuclear and a magnetic term [39]: For mono-atomic systems, such as the ones considered in this paper, the nuclear contribution for single scattering events is given by S coh (q, E) is the dynamic structure factor and represents the collective response of the liquid as a function of momentum q and energy E transferred from the neutron to the liquid, while S inc (q, E) describes the dynamics of a single atom [39]. The cross-sections σ coh and σ inc are element dependent; σ inc arises because the strong interaction depends on the spin state of the nucleus and the number of neutrons in the nucleus. Thus, the nuclear scattering cross-section carries information about the collective behavior of the atoms, such as soundwaves, and information about the motion of indiviual atoms, such as self-diffusion.
The static structure factor S(q) is given by the sum-rule [39] while the incoherent dynamic structure factor satisfies a similar sum-rule In neutron diffraction experiments aimed at measuring S(q), the energy integration in Eq.
3 is carried out by the neutron detector. Because of the term k f /k i in Eq. 2, this procedure leads to small errors in the determination of S(q); however, these errors are small under suitable experimental conditions and can be corrected for using standard methods [39]. A further source of errors is that Eq. 2 is only valid for neutrons that are scattered once by the sample. Again, by choosing sufficiently small samples, the errors introduced by multiple scattering events and events in which a scattered neutron is absorbed by the sample can be corrected for [40]. Therefore, provided these corrections have been carried out, one can check the accuracy of the data reduction procedure by comparing the measured cross-sections σ coh and σ inc to the known values. Even in the case where the absolute values of σ coh and σ inc cannot be inferred from the experiment, their ratio can still be determined using the fact that S(q) oscillates around 1 for large q.
The magnetic contribution to the scattered intensity is only visible in neutron scattering experiments on liquids provided that atoms with unpaired electrons exist [39]. The angular momentum associated with these unpaired electrons, hJ, interacts with the intrinsic magnetic moment of the neutron. The conduction electrons present in liquid metals do not contribute to the scattering at finite q; an electron moves so fast compared to the neutron that the scattered waves only add up coherently at q=0, the forward direction. However, if an electron is localized around an atom, all scattered waves originate from the region of the partially filled orbital, and the scattered waves can be observed for a range of q-values. For this reason the form factor for magnetic scattering F (q), which describes the variation of scattered intensity with q and which is given by the spatial extent of the electron cloud, falls off more rapidly with increasing q than the form factor for nuclear scattering [the so-called Debye-Waller factor W (q)]. The latter reflects the fact that nuclear scattering originates in the much smaller volume of the nucleus.
The number of ions with unpaired electrons at any given moment determines the magnetic cross-section for a liquid with fluctuating magnetic moments. The total number of neutrons that are scattered per second per metallic ion into solid angle dΩ is given by the paramagnetic approximation for the differential cross section [39] dσ is small compared to the incident energy of the neutron [39]. This is the same requirement that allows one to determine S(q) from a liquid without doing an energy analysis of the scattered neutron, and we will therefore assume that this requirement is satisfied for all published datasets discussed in this paper.
Analyzing quasi-elastic neutron scattering experiments on liquid Hg at room temperature, Badyal et al. [2] observed that the scattered signal at small momentum transfers consisted of two contributions (see Fig. 2), attributable to self-diffusion and cage-diffusion, respectively.
However, the relative strength (area under the curves in Fig. 2) of the cage-diffusion contribution compared to the self-diffusion contribution was found to be 22% (corresponding to a differential cross section of ∼ 1.5/4π barn). A relative strength of the order of 0.3% was expected based on MD simulations [3] and on an order of magnitude calculation [2].
Given that the strength of the quasi-elastic coherent contribution for small q-values (given by sum-rules at ∼0.01/4π barn) was negligible [39,41], and given that the characteristic energy width (3 ps −1 ) corresponded to the time scale of the cage diffusion process (1/3 ps), the authors [2] concluded that the broad quasi-elastic line did indeed correspond to 6 cage-diffusion but with a magnetically enhanced cross-section. Using Eq. 5 (S= 1/2, L=2 and J=5/2 and F (q)=e −W (q) = 1 for small q) and noting that crystal electric field effects are absent in a liquid, we find that 19.5% of the Hg-ions have an unpaired d-electron. Should the observed magnetic signal originate from an unpaired s-electron, then the corresponding fraction of magnetic ions would be 82%. We return to this latter possibility in the discussion.  Table I. Unfortunately, the authors did not give the ratio between the narrow and broad component, making it impossible to infer n from their data. In fact, the authors did not attribute this broad mode to cage-diffusion. Instead, it was assumed to be part of the coherent scattering contribution. The latter is inconsistent with their modeling of the rest of the scattered intensity [6], which already completely exhausted the coherent sum-rule (Eq. 3). Given this, and given the very weak dependence of τ on q for q < 1.3Å −1 , we believe that this broad mode represents cage-diffusion. However, whether it is a cage-diffusion process combined with a fluctuating magnetic moment cannot be inferred from this study (as published). with these neutron scattering data (probably due to the large absorbtion cross-section for neutrons). Therefore, we can only give an estimated range for the fraction n of ions with an unpaired electron. Based on the work of Torcini et al. [15], we find the fraction n to be in the range 0 < n < 1%, for S=1/2, L=0, J=1/2 and g(LJS)=2.
In all, the alkali metals do not show unambiguous evidence for the existence of the proposed magnetic cross-section. However, it is interesting to note that small angle x-ray scattering experiments on liquid lithium indicated the presence of an additional cross-section [16], which the authors tentatively attributed to increased correlation between the valence electrons. The mechanism proposed in this paper would offer an explanation for the observed [16] increased correlation. Nonetheless, the evidence for a collision-induced fluctuating moment in the alkali metals is somewhat weak. Much better evidence for its existence comes from scattering experiments on group 3 and 4 metals, which display an enhanced cross-section, similar to the results for liquid mercury.

B. Group 3 and 4 metals
Liquid lead is a good candidate to analyze for the possible presence of a magnetic cross-section since Pb has a negligible incoherent cross-section; therefore, any significant scattering at small momentum transfers (where the coherent cross-section is very small) is indicative of a paramagnetic signal. Reijers et al. [21] measured the static structure factor of liquid lead at 613 K under ambient pressure (see Fig. 3). From Eqs. 2 and 3, we find that the expected neutron scattering intensity at small momentum transfers due to coherent scattering is given by σ coh /4πS(q = 0), with S(q = 0) = 0.009 [42] and σ coh = 11.16 b. The S(q = 0) extrapolated value from the liquid lead experiment is 0.07 (see Fig. 3), implying an additional neutron scattering intensity of 0.7/4π b. Using Eq. 5 with S= 1/2, L=2, J= 5/2 and g(LJS)= 1.2, the fraction n of ions with an unpaired electron is 9%. Assuming the additional cross-section originates from s-electrons (S= 1/2, L=0, J= 1/2 and g(LJS)= 2), we find n= 38% (See Table I).
Liquid aluminum also displays a paramagnetic cross-section. Iqbal et al. [22] performed a study on liquid aluminum at 936K (see Fig. 4). In this study on a liquid with negligible incoherent cross-section, the authors normalized their data to S(q → ∞)= 1; however, the data had not been corrected for multiple scattering effects, which can constitute a major part of the scattering at small q. Based on the dimensions of their cylindrical cell, we have calculated [2,40] the multiple scattering contribution (dashed line in Fig. 4) assuming the energy dependence of S(q, E) to be given by a Lorenzian line shape with half width determined by the coefficient for self-diffusion (D s = 0.4Å 2 /ps [43]). After subtracting the multiple scattering contribution and renormalizing the data accordingly, we find that the neutron scattering data consistently lie above the X-ray data [33] at small q, and that the neutron scattering data do not appear to reach the q → 0 limit S(q=0) = 0.013 [44]. Since a paramagnetic contribution represents a very small correction to X-ray scattering data, we take the difference ∆S = 0.11 between the neutron and X-ray S(q) measurements at q < 1.5Å −1 as the strength of the paramagnetic signal, i.e., dσ magnetic /dΩ= ∆Sσ coh /4π= 0.16/4π b. This corresponds (Eq. 5) to a fraction n= 4% assuming the fluorine electronic configuration for paramagnetic Al-ions; a sodium configuration would correspond to n= 9% (See Table I).
Another liquid metal for which we can verify the presence of an additional component to the cross-section is liquid gallium. Bellissent-Funel et al. [23] found in their experiments on liquid Ga at 326 K and 959 K that the observed scattered intensities were not consistent with the known values for σ inc and σ coh . Since both uncorrected and corrected data were published in this study [23], and since every step of the data reduction procedure was clearly described, we can infer a very accurate estimate of the paramagnetic cross-section for Ga. Using the dimensions of the sample cell used in the experiments [23], we have calculated [2,40] the multiple scattering contribution (see Fig. 5). Taking into account the S(q = 0) values and the fact that the magnetic contribution will be absent at very large q, we find an additional differential scattering cross-section of 0.88/4π b at 326 K and 0.78/4π b at 959 K. Assuming this scattering to originate from an unpaired electron with quantum numbers S= 1/2, L=2, J= 5/2 and g(LJS)= 1.2, we find n = 11.5% at T= 326 K and n= 10.1% at T=959 K. If we assume the scattering to originate from a s-electron (S=1/2, L=0, J= 1/2 and g(LJS)= 2), we find n= 48% and n= 42%, respectively (see Table I). Thus, gallium displays a large magnetic cross-section, but its magnitude appears to be only weakly temperature dependent.

IV. DISCUSSION
The available neutron scattering data point overwhelmingly to the existence of shortlived magnetic moments in non-magnetic liquid metals. These moments come in and out of existence on the same time scale as the cage-diffusion motion, as observed in the quasi-elastic neutron scattering experiments on liquid Hg [1,2,3]. The alkali metals show only a weak effect, but the effect is much more pronounced in mercury and in group 3 and 4 metals (see Table I).
The actual percentage of ions with unpaired electrons is more difficult to assess than establishing that such ions with unpaired electrons exist. For instance, it is feasible that the unpaired electron in liquid mercury is either an s-electron or a d-electron. The 6s shell in mercury has been drawn in closer to the nucleus because of the relativistic contraction of the underlying shells, so it is definitely conceivable in a liquid that the 6s shell can be completely filled (for some of the time at least). In other words, the observed paramagnetic intensity could originate from a Hg 1+ or from a Hg 3+ -ion. (In liquid lead, it is in fact more likely that the paramagnetic contribution stems from Pb 3+ than from Pb 5+ -ions, given the prevalence of lead to form Pb 2+ in solids.) Should this indeed be the case, then the electrical resistance in liquid mercury does not come solely from electrons being scattered by ions, but also from electrons actually being captured by Hg-ions; far from being unchanging, the Fermi-sea constantly changes in size while interchanging electrons with the ions.
The phenomenon of the additional magnetic cross-section seems to have been mostly overlooked. However, its implications on the interaction mechanisms in a liquid metal cannot be overlooked given the long range of the magnetic dipole interaction and the ability of localized moments to polarize the surrounding conduction electrons. In particular, it would be interesting to see how incorporation of paramagnetic ions and their polarization capability into the interatomic potential used in MD simulations would alter the characteristics of short-wavelength sound propagation.
Finally, this paramagnetic cross-section provides a means of studying the cage-diffusion mechanism at small momentum transfers even in systems that do not exhibit an incoherent 12 cross-section, such as lead and aluminum. We are currently carrying out polarized neutron scattering experiments on liquid gallium in order to verify that the observed additional cross-section is indeed magnetic in origin and to study its temperature dependence close to the solidification transition.   The static structure factor of liquid aluminum just above the melting point as measured by neutron scattering [22] (solid line) and X-ray scattering [33] (stars). The difference between the two data sets is considerably larger than the calculated multiple scattering contribution to the neutron scattering data (dashed-dotted curve). After correcting for these multiple scattering effects, we find that the remaining difference between the two data sets (solid circles and horizontal line) is only weakly dependent on q, indicative of an incompletely filled electronic shell with small radius.
The data point at q=0 (open diamond) is the compressibility limit taken from thermodynamic data [44].
FIG. 5: The unnormalized static structure factor of liquid gallium at two temperatures (solid line with stars) as measured by neutron scattering [23]. The calculated incoherent contribution is given by the dashed dotted lines, the sum of the incoherent and multiple scattering contribution (see text) is denoted by the solid lines. The difference (at small q-values) between the experimental data points and the solid line is ascribed to paramagnetic scattering.