Emergent Monopoles in Spin and Water Ices
Randall S. Frost*
3454 Smoketree Commons, Pleasanton, California 94566, USA
We provide a phenomenological theory for the presence of magnetic
monopoles in spin ice—as well as in water ice and disordered materials
containing oxygen atoms. Noting that the structure of spin ice is similar
to that of frozen water, with spin orientation playing a similar role to that
of the hydrogen position in the latter, we consider the coexistence of
oxygen vacancies, water molecules, and a magnetic monopole on a Flory
lattice consisting of an isotropic system of rigid rods and an emergent
monopole. The model—driven by a most-probable distribution of
interatomic oxygen distances—predicts that a magnetic monopole having
a frequency of 2.012 THz will be separated out of an isotropic system at
an oxygen interatomic distance of ~3.5 Ǻ based solely on the
spaciogeometric constraints of placing water molecules on an oxygen
vacancy lattice.
Introduction.— Pauling noted in 1935 that the structure of water ice exhibits
degrees of freedom that should be expected to remain disordered even at absolute
1
zero.1 He pointed out that even upon cooling to 0 K, water ice should have an
intrinsic randomness, arguing that this is the case because the structure of ice
contains oxygen atoms with four neighboring hydrogens such that for each oxygen
atom two of the neighboring hydrogen atoms are near and two (i.e., the hydrogen
atoms of a neighboring water molecule) are further away.
The structure of spin ice is similar to frozen water where each oxygen atom
has two short and two long bonds to four hydrogen atoms, with spin orientation in
spin ice playing a similar role to that of the hydrogen position in water ice.2
Flipping a single spin in a spin ice state should result in the creation of a
monopole–antimonopole pair in two adjacent tetrahedra.3 This was demonstrated
by Paulsen et al., who succeeded in producing a nonequilibrium population of
magnetic monopoles in a single crystal of dysprosium titanate (Dy2Ti2O7).4
Recently, Sala et al.5 noted that a description of magnetic monopoles in spin ice in
terms of monopole dynamics alone is insufficient at low temperatures. They
proposed that oxygen deficiencies are the leading cause of magnetic impurities in
as-grown samples of the spin-ice material Dy2Ti2O7.
In this paper, motivated by these observations, we consider a magnetic
monopole coexisting as an emergent particle in a system of oxygen vacancies and
water molecules. To do this, we use a Flory lattice to model the coexistence of the
2
vacancies, water molecules, and an embedded magnetic monopole. By taking into
account the size differences of the water molecules and the oxygen vacancies, we
identify the conditions under which the monopole will be separated out.
The Flory lattice.— In 1956, Flory proposed a theory to describe the
thermodynamic properties of athermal binary mixtures consisting of a solvent and
monodisperse rodlike particles.6 In the theory, the system consists of a mixture of
rodlike particles differing in axis ratio x dispersed in a solvent. The rods are
assumed to have identical diameters, with their lengths variable, and the solvent to
be isodiametric with a diameter equal to the mean thickness of the rods. A solute
segment is defined as the portion of the solute segment having the same volume as,
and geometrically equivalent to, a molecule of solvent. Thus, the parameter x is
equivalent to both the axis ratio and the number of segments comprising the solute
molecule.
The solute molecules, n2 in number, are inserted into a space subdivided into
no lattice sites. Their locations are chosen at random, apart from the requirement
that occupation of a site by a segment be exclusive. The orientation of a molecule j
relative to the domain axis is specified by a disorientation index yj such that each
solute molecule j has an axis ratio xj oriented at angle  with respect to the axis of
a domain. Each molecule is thus divided into yj = xj sin submolecules, each
3
aligned parallel to the domain axis. In the theory, the separate yj are replaced by an
equilibrium disorder parameter y that characterizes the average orientations of all
species for which x > y.
Taking one of the axes of the cubic lattice to be parallel to the domain axis,
each submolecule occupies a sequence of lattice sites parallel to this axis. The
system as a whole thus consists of submolecules confined to such rows, and
calculation of the particle orientations amounts to the mixing of submolecules and
solvent molecules (or vacancies) in one dimension.
Later, Flory and this author7 considered the portioning of a so-called most
probable distribution of rodlike particles between isotropic and anisotropic phases.
The most probable distribution8, 9 describes the molecular size distribution that
results when polymers are formed under the conditions of equal reactivity of all
functional groups. In the most probable distribution, the probability that a molecule
consists of exactly x units is (1 – p)p x –1 (p for each of the x – 1 linkages and 1 – p
for the terminal link), where p may be considered to be the expectation of
perpetuation of a sequence of units to include at least one more unit and 1 – p the
expectation of termination of the sequence.
The probability that any molecule selected at random is composed of x units
must equal the mole fraction of x-mers, and the total number of x-mers is given by
4
Nx = N(1 – p)p x–1,
where the total number of molecules of all sizes is
N = N0(1 – p)
and N0 is the total number of units. Therefore,
Nx = N0 (1 – p)2p x–1.
If the added weight of the end groups in the polymerization is neglected, the
molecular weight of each species will be directly proportional to x. The volume
fraction of species x can then be written
vx/v2 = xNx/N0 = x(1 – p)2p x–1.
Here, v2 = –p–1 ln p denotes the mean volume fraction for all species x and vx is the
corresponding volume fraction of solute particles consisting of x units.7 The
number average size or axis ratio xn is
xn = N0/N = 1/(1 – p).
In the case that there is a portioning of rodlike molecules conforming to the
most probable distribution in a system consisting of coexisting isotropic and
anisotropic phases, our theory shows that an incipient anisotropic phase can
emerge from the parent isotropic phase only under very specific conditions. The
5
character of this emergent phase is characterized by three parameters: the number
average size or axis ratio in the anisotropic phase xn', the disorder parameter y, and
the volume fraction of solute in the anisotropic phase v2'. Here, the prime refers to
the anisotropic phase. According to the theory,7 in order for the isotropic and
anisotropic phases to coexist, the following conditions have to arise: xn' = ∞, y = 0,
and v2' =1. Under these conditions, the solute in the isotropic phase is transferred in
its entirety to the anisotropic phase.
The emergent anisotropic phase thus consists of perfectly ordered species of
infinite length. These conditions for its existence apply regardless of the average
axis ratio in the isotropic phase, xn = 1/(1–p), provided that xn > 2.3102.7 The
minimum value of xn corresponds to a complete absence of solvent or vacancies in
the isotropic phase (i.e., v2 = 1). If the anisotropic phase contains a solvent, is
finitely disoriented, or consists of rods of finite length, it cannot coexist in
equilibrium with the parent isotropic phase in which the solute comprises rods
having a most probable distribution of lengths.
Emergent monopole in water ice.—In order to model the coexistence of a
magnetic monopole in a system of water molecules and oxygen vacancies, we
consider a lattice of oxygen vacancies in which a magnetic monopole has been
embedded, which we will then fill with water molecules. Because the water
6
molecules are larger than the oxygen vacancies, they will not fit exactly into the
lattice sites.
To describe the accommodation of the water molecules by the oxygen
vacancy lattice, we imagine the water molecules to undergo the following
"reactions" to form rigid rods made up of oxygen atoms:
H2O  O + 2H
O + H2O  O + O + 2H
or
(O)n–1 + H2O  (O)n + 2H,
(1)
where (O)n is a chain consisting of n oxygen atoms.
One should note that these reactions are not meant prima facie to correspond
to physicochemical reactions. They are at this point simply a theoretical construct
to represent the constraints of placing the water molecules on the oxygen vacancy
lattice. That is to say they model the physical constraints of filling the oxygen
lattice vacancies with water molecules. The extent to which they might correspond
to physical phenomena will be discussed later in the paper.
7
We assume these reactions to proceed until the lattice is completely filled
(i.e., until all of the lattice vacancies have been filled), so that the volume fraction
of oxygen rods v2 is equal to 1. This corresponds to an isotropic glass state where
the oxygen rods are "frozen" into random positions. If the probability of adding an
oxygen atom to a rod is independent of the number of oxygens that have already
been added, the distribution of oxygen rod lengths will be described by a most
probable distribution. Since the volume fraction of hydrogen atoms produced in the
reactions will be much less than that of the oxygen rods, their contribution will be
neglected when calculating the most probable distribution of rod lengths.9
Based on the theory, as long as the average axial ratio of the (O)n rods
>2.3102 (depending on degree of dilution in the isotropic phase), an incipient
anisotropic phase can coexist with a parent isotropic phase. Under these conditions,
the polymer must ultimately be transferred in its entirety to an anisotropic phase
characterized by xn' = ∞.
We will now limit our discussion to the case where n = 2, corresponding to
diatomic oxygen. The axis ratio xn that appears in the theory then corresponds to
the separation distance between the two oxygen atoms. Taking the separation
distance L between two oxygen atoms at which the repulsive and attractive forces
between the atoms just balance as 3.5 Ǻ from Lennard-Jones potential theory in the
8
harmonic approximation, 10 one has xn = 2.3102 = L/D, where D is the diameter of
the oxygen atom, and we calculate an oxygen atom diameter of 1.515 Ǻ, or a
radius of 0.758 Ǻ.
It follows from our theory, which predicts that species with an average axial
ratio of xn' = ∞ will be separated out with the emergence of the anisotropic phase,
that the coexisting magnetic monopole—which is an infinitely long Dirac
string11,12—that we initially placed on the lattice will be partitioned out from the
undiluted glassy state when the two oxygen atoms in the isotropic phase are
separated by a most-probable distance of about 3.5 Ǻ. We note again that our
theory also predicts that an emergent monopole could coexist with an isotropic
phase at larger values of xn, but only under dilution.
Spin ice.— In spin ice the atoms are arranged in four-sided tetrahedra
stacked together to form a pyrochlore structure. At temperatures near absolute
zero, the electron spins begin to align into their lowest energy state, corresponding
to a configuration where two of the tetrahedron’s four spins point toward the center
of the tetrahedron and two point outward. This “two-in, two-out” configuration is
analogous to the ice rule that describes how hydrogen bonds form in water ice,1
where each oxygen atom is surrounded by two close-in hydrogen atoms and two
farther away.
9
The equivalence of pyrochlore spin ice to water ice follows from identifying
the centers of the pyrochlore tetrahedra with the location of the oxygen atoms. The
spins are then located at the midpoint of the bond between a pair of neighboring
oxygens.13 If some of the tetrahedra end up with three spins pointing in or out and
just one in the opposite direction, a monopole should be produced. This
observation prompted Castelnovo et al.3 to propose a search for confined magnetic
monopoles at temperatures near absolute zero in spin ice materials.
Thermal conductivity.— Zeller and Pohl14 noted a number of years ago that
(1) all noncrystalline solids have thermal conductivities that show a characteristic
plateau around 10 K, (2) their low-temperature specific heat is larger in the glassy
phase than in the crystalline one (suggesting that there are extra modes associated
with the glassy phase), (iii) the specific heat below 1 K in noncrystalline solids
does not follow the Debye T3 law; (iv) all noncrystalline solids have practically
identical thermal conductivities (suggesting that scattering must have a very simple
origin), and (v) the specific heat of Pyrex glass containing iron impurities had been
reported to increase with the concentration of impurity spins.
Later, this author considered the interactions of phonons with polymeric
glasses and oxygen molecules.15 There, it was shown that when the wavelength of
the phonon is comparable to the dimensions of the disordered (glass) system, solid
10
state theory could be used to describe the interactions of the phonons with the
electrons in the oxygen molecules. Specifically, the case was considered in which a
phonon was created with an electron transferred from state q' to q and the reverse
one in which the phonon was destroyed. It was shown that in this system the
relaxation time tep describing the interactions of phonons with electrons varies
inversely as (N +1)f '(1 – f) – Nf(1 – f '), where N is the Bose-Einstein distribution
function given by
N = [exp(ℏω/kBT) –1]–1,
ℏω is the energy of the phonon created or absorbed in the process, kB is
Boltzmann's constant, and f is the Fermi-Dirac distribution functions given by
f = {exp[(Eq – EF)/kBTs] + 1}–1
with f ' determined analogously for Eq'. Here EF is the Fermi level and Ts is a
paramagnetic spin temperature15 that characterizes the Fermi-Dirac statistics of the
electrons.
When T = Ts
1/tep ~ (N +1)f '(1– f) – Nf(1 – f ') = 0
11
and the phonon-electron relaxation time tep becomes infinite. However, if T ≠ Ts, tep
remains nonzero.
The case in which phonons interact with the electrons in oxygen molecules
was considered in the aforementioned paper.15 Letting Eq' correspond to the energy
of the electron in the singlet state and Eq to that in the paramagnetic state, we now
consider the case where the emergent monopole coexists with paramagnetic and
singlet electrons having nearly the same energy but different electronic states at the
point of phase separation. At the interatomic distance of about 3.5 Ǻ that we
estimated for the onset of phase separation in our oxygen rod system, molecular
orbital calculations do in fact show that the energy of the triplet state electron (with
electron spin ↑ in the first π* orbital and spin ↑ in the second π* orbital) should be
very close that of the singlet state electron (with electron spins ↑ and ↓ in the first
π* orbital and the second π* orbital empty).16 It is at this point that we would
expect the monopole to separate out of the lattice.
We will now evaluate the parameters in our theory. To calculate Ts
corresponding to xn = 2.3102, we note that at absolute zero the oxygen vacancy
lattice will be filled such that xn = 1. Using the thermal expansion coefficient α
from Charles' law of 3.66 x 10–3 K–1, we set α = dxn /dT = (2.3102 –1)/ Ts to
obtain Ts = 358 K.
12
When the probability that the O2 molecule is in the reactive (singlet) state is
just overtaken by the probability that it is in the nonreactive (triplet) state the
oxygen chain polymerization (1) stops. Taking pf '(1– f) as the probability of the
former and (1 – p) f (1 – f ') for the latter, where p as before is the expectation of
chain perpetuation, we have in the limit that Eq' >> EF [15]
p/(1–p) ~ exp [(Eq' – Eq)/kBTs).
For xn = 2.3102 (p = 0.56714), we find Eq' – Eq = 0.27019kBTs = 0.008 eV at Ts =
358 K.
The triplet-singlet electron transition will correspond to the emergence of a
monopole having energy hc/λ and frequency c/λ, where λ is the monopole's
wavelength, h is Planck's constant, and c is the speed of light. Setting hc/λ = 0.008
eV, we solve to obtain λ = 149 µm (about the thickness of a mica sheet) and c/λ =
2.012 THz, which can be taken as signatures of monopole-oxygen interactions.
Discussion.— By considering the placement of water (ice) molecules on a
lattice of oxygen vacancies, we found that when (i) the oxygen vacancy lattice is
completely filled with oxygen atoms and (ii) the most probable distance between
oxygen atoms equals 3.5 Ǻ, a coexisting anisotropic phase appears in which a
monopole having frequency c/λ = 2.012 THz is separated out.17
13
We now assess the reasonableness of our results by considering whether
water (ice) and oxygen molecules (at a most probable oxygen-oxygen distance of
~3.5 Ǻ) would be in resonance with the monopole frequency. In Table I we show
the harmonic frequencies for a standing wave connecting two water molecules
separated by L = 3.5 Ǻ in which the 1st harmonic (natural) frequency is equal to
the monopole frequency of 2.012 THz. The associated wave velocity (frequency
times wavelength) is 1408 m/s. The calculated wave velocity agrees well with the
experimentally determined sound velocity (1402 m/s) in water at 0 oC.18
Table I. Harmonics for two water (ice) molecules whose 1st harmonic frequency is
equal to the monopole frequency. The calculated standing wave velocity is 1408
m/s when the oxygen atoms in the two water ice molecules are separated by 3.5 Ǻ.
Noncrystalline water ice
Frequency (THz)
Wavelength (Ǻ)
1st harmonic
2nd harmonic
3rd harmonic
4th harmonic
nth harmonic
2.012
4.015
6.072
8.096
2.012 n
7.0
3.5
2.33
1.75
7.0/n
Calculated standing wave
velocity (m/s)
1408
1408
1408
1408
1408
We next consider the harmonic frequencies for a standing wave connecting
two oxygen atoms in which the 4th harmonic frequency is equal to the monopole
frequency. In this case, the associated wave velocity is 352 m/s. This value
compares favorably with the measured sound velocity (361 m/s) in oxygen at 358
K (the lattice temperature calculated above).19
14
Table II. Harmonics for two oxygen atoms whose 4th harmonic frequency is equal
to the monopole frequency and to the 1st harmonic frequency of two water (ice)
molecules (Table II). The standing wave velocity is calculated to be 352 m/s when
the oxygen atoms are separated by 3.5 Ǻ.
Diatomic oxygen
Frequency (THz)
Wavelength (Ǻ)
1st harmonic
2nd harmonic
3rd harmonic
4th harmonic
nth harmonic
0.503
1.006
1.509
2.012
0.503 n
7.0
3.5
2.33
1.75
7.0/n
Calculated standing wave
velocity (m/s)
352
352
352
352
352
If an oxygen atom is suddenly decoupled from the vibrating diatomic
molecule (corresponding to the formation of an oxygen vacancy), the first
harmonic of the subsequent vibration will then be λ = 4 x 1.75 = 7.0 Ǻ (with a
corresponding frequency of 2.012 / 4 = 0.503 THz. Subsequent harmonic
frequencies will be given by (n+2)(0.503)THz, where n is an odd integer. In other
words, there will no resonance with the monopole frequency, as predicted by our
theory.
Equation (1) can now be interpreted as a representation of the structural
resonance that occurs between water and oxygen molecules at the monopole
frequency.20
Conclusion.—Purcell et al.21 noted over 50 years ago that magnetic
monopoles should experience a repulsion with diamagnetic molecules, causing the
15
monopole to remain free (not bound). In paramagnetic molecules (or diamagnetic
systems containing paramagnetic molecules), however, an isolated paramagnetic
molecule could contain an unpaired electron spin with a monopole some distance
away, so there would be an attractive potential corresponding to the ground state
where the electron spin is polarized favorably along the atom-monopole axis.
As our calculations showed, the monopole frequency is resonant with the
electronic transition in diatomic oxygen from the ground-state triplet to the singlet
electronic state at the same interatomic separation. Thermal vibrations having the
monopole's resonant frequency will selectively couple to this electronic transition,
causing a monopole initially bound by a paramagnetic (triplet-state) oxygen
molecule to be repelled by the resulting diamagnetic (singlet-state) molecule, and
separated out of the isotropic phase. As a result, there can be a very sudden and
rapid appearance of large numbers of monopoles due to thermal interactions with
the oxygen molecules.
Separation out of a magnetic monopole with the transition from the O2
paramagnetic (triplet) state to the O2 diamagnetic (singlet) state is consistent with
the conditions—driven solely by interatomic bond distance—that give rise to an
emergent monopole in our oxygen lattice model. These conditions might
16
reasonably be expected to arise in a wide range of noncrystalline organic and
inorganic materials, as well as in water and spin ices.
*
rfrost808@hotmail.com
References
1
Pauling, L. "The structure and entropy of ice and of other crystals with some
randomness of atomic arrangement." J. Am. Chem. Soc. 57, 2680 (1935).
2
Ramirez, A. P., Hayashi, A., Cava, R, J., Siddharthan, R., and Shastry, B. S.
"Zero-point entropy in 'spin ice'." Nature (London) 399, 333 (1999).
3
Castelnovo, C., Moessner, R., and Sandhi, S. L. "Magnetic monopoles in spin
ice," Nature (London) 451, 32 (2008).
4
Paulsen, C., et al. "Far-from-equilibrium monopole dynamics in spin ice," Nat.
Phys. 10, 135 (2014).
5
Sala, G., et al. "Vacancy defects and monopole dynamics in oxygen-deficient
pyrochlores," Nat. Mater. 13, 488 (2014).
6
Flory, P. J. "Statistical thermodynamics of semi-flexible chain molecules," Proc.
Phys. Soc. London Sect. A 234, 73 (1956).
17
7
Flory, P. J. and Frost, R. S. "Statistical thermodynamics of mixtures of rodlike
particles. 3. The most probable distribution," Macromolecules 11, 1126 (1978).
8
Examples of the types of reactions that can be modeled by the most probable
distribution include condensation polymerizations that produce molecular chains
having the structure H–ORCO–ORCO–…–ORCO-OH, with water as a byproduct
of the condensation.9
9
See Flory, P. J. Principles of Polymer Chemistry (Cornel, Ithaca, 1953).
10
Oobatake, M., and Ooi, T. "Determination of energy parameters in Lennard-
Jones potentials from second virial coefficients," Prog. Theor. Phys. 48, 2132
(1972). In our model, the equilibrium distance corresponding to the energy
minimum in the Lennard-Jones potential is equal to twice an atom's van der Waals
radius.
11
Dirac, P. A. M. "Quantised singularities in the electromagnetic field," Proc.
Phys. Soc. London Sect. A 133, 60 (1931).
12
Dirac, P. A. M. "The theory of magnetic poles," Phys. Rev. 74, 817 (1948).
13
Moessner, R., and Ramirez, A. P. "Geometrical frustration," Phys. Today 59, No.
2, 24 (2006).
18
14
Zeller, R. C., and Pohl, R. O. "Thermal conductivity and specific heat of
noncrystalline solids," Phys. Rev. B 4, 2029 (1971).
15
Frost, R. S. "Paramagnetic spin-lattice interactions in glass-forming polymers,"
Phys. Rev. B 20, 2163 (1979).
16
Takahashi, Mitsuo, and Fukutome, Hideo, "Unrestricted Hartree-Fock
description of the ground and the lowest singlet states of oxygen molecule," Prog.
Theor. Phys. 59, 1787 (1978).
17
A recent paper by Keen and Goodwin considered the relationship of the structure
of cubic ice to the pyrochlore lattice in which spin-ice monopole behavior has been
observed. In their model, the individual water molecules in cubic ice are randomly
oriented in the cubic lattice, with the oxygen atoms forming the centers of the
water molecules completely aligned. This structure is in accord with our theory,
which predicts such an alignment of oxygen atoms at phase separation. See Keen,
David A., and Goodwin, Andrew L., "The crystallography of correlated disorder,"
Nature (London) 521, 303 (2015). Mostame, S. et al., in "Tunable nonequilibrium
dynamics of field quenches in spin ice," Proc. Natl. Acad. Sci. USA, 111, 640
(2014), give a value of 3.5 Ǻ for the pyrochlore lattice constant.
19
18
Greenspan, Martin, and Tschiegg, Carroll E., "Speed of sound in water by a
direct method," Journal of Research of the National Bureau of Standards 59, 249
(1957).
19
Obtained from the velocity of sound at 273 K (315 m/s) with the sound velocity
varying as the square root of absolute temperature. See Handbook of Chemistry
and Physics, 56th ed., edited by R. C. Weast (CRC Press, Cleveland, Ohio, 1975).
p. F-79.
20
For a discussion of structural resonance, see Pauling, L., The Nature of the
Chemical Bond (Cornell University Press, New York, 1960).
21
Purcell, E. M., Collins, G. B., Fujii, T., Hornbostel, J., and Turkot, F. "Search for
the Dirac monopole with 30-BeV protons." Phys. Rev. 129, 2326 (1963).
20