In 1935,
Linus Pauling noted that the hydrogen atoms in water ice would be expected to remain disordered even at
absolute zero. That is, even upon cooling to zero
temperature,
water ice is expected to have
residual entropy,
i.e., intrinsic randomness. This is due to the fact that the hexagonal
crystalline structure of common water ice contains
oxygen atoms with four neighboring
hydrogen atoms. In ice, for each oxygen atom, two of the neighboring hydrogen atoms are near (forming the traditional H2O
molecule), and two are further away (being the hydrogen atoms of two neighboring water molecules). Pauling noted that the number of configurations conforming to this "two-near, two-far"
ice rule grows
exponentially with the system size, and, therefore, that the zero-temperature
entropy of ice was expected to be
extensive. Pauling's findings were confirmed by
specific heat measurements, though pure crystals of water ice are particularly hard to create. Spin ices are materials that consist of regular corner-linked
tetrahedra of magnetic
ions, each of which has a non-zero
magnetic moment, often abridged to "
spin", which must satisfy in their low-energy state a "two-in, two-out" rule on each tetrahedron making the crystalline structure (see figure 2). This is highly analogous to the two-near, two far rule in water ice (see figure 1). Just as Pauling showed that the ice rule leads to an extensive entropy in water ice, so does the two-in, two-out rule in the spin ice systems – these exhibit the
same residual entropy properties as water ice. Be that as it may, depending on the specific spin ice material, it is generally much easier to create large single crystals of spin ice materials than water ice crystals. Additionally, the ease to induce interaction of the magnetic moments with an external magnetic field in a spin ice system makes the spin ices more suitable than water ice for exploring how the residual entropy can be affected by external influences. While
Philip Anderson had already noted in 1956 the connection between the problem of the
frustrated Ising antiferromagnet on a (
pyrochlore) lattice of corner-shared tetrahedra and Pauling's water ice problem, real spin ice materials were only discovered forty years later. The first materials identified as spin ices were the
pyrochlores Dy2Ti2O7 (
dysprosium titanate), Ho2Ti2O7 (holmium titanate). In addition, compelling evidence has been reported that Dy2Sn2O7 (
dysprosium stannate) and Ho2Sn2O7 (
holmium stannate) are spin ices. These four compounds belong to the family of rare-earth pyrochlore oxides. CdEr2Se4, a
spinel in which the magnetic Er3+ ions sit on corner-linked tetrahedra, also displays spin ice behavior. Spin ice materials are characterized by a random disorder in the orientation of the moment of the magnetic
ions, even when the material is at
very low temperatures. Alternating current (AC)
magnetic susceptibility measurements find evidence for a dynamic freezing of the magnetic moments as the temperature is lowered somewhat below the temperature at which the
specific heat displays a maximum. The broad maximum in the
heat capacity does not correspond to a phase transition. Rather, the temperature at which the maximum occurs, about 1K in Dy2Ti2O7, signals a rapid change in the number of tetrahedra where the two-in, two-out rule is violated. Tetrahedra where the rule is violated are sites where the aforementioned monopoles reside. Mathematically, spin ice configurations can be described by closed
Eulerian paths.{{Cite journal|last1=Caravelli|first1=F.|last2=Saccone|first2=M.|last3=Nisoli|first3=C.|title=On the Degeneracy of Spin Ice Graphs, and its Estimate via the Bethe Permanent == Magnetic monopoles ==