Nature of novel criticality in ternary transition-metal oxides

There are the chains of transition-metal cations alternating with the anions of oxygen in ternary transition-metal oxides. When a p-orbital of the oxygen connects the half-filled and empty d-orbitals of adjacent transition-metal cations, double-exchange ferromagnetism takes place. Although double exchange has been well explored, the nature of novel criticality, induced by it, is yet not uncovered. We explored the magnetic-field scaling in the heat capacity of a Sm0.55Sr0.45MnO3 manganite, one of the best ternary transition-metal oxides as it is completely ferromagnetic, and found novel criticality - unordinary critical exponents which are the consequence of coherence of Coulomb lattice distortion and ferromagnetism. The coherence is caused by the trinity of the mass, the charge and the spin of an electron. When the d and p orbitals overlaps, it quickly walks from one site to the another due its lightest mass. And due to its electric charge, it equalizes the valences of the transition-metal cations in the chains and so diminishes the Coulomb lattice distortion. At last, its spin forces magnetic moments of transition-metal cations to ferromagnetically arrange. The disappearance of Coulomb distortions widens the overlap and lowers the elastic lattice energy, so that not only the spin of an electron, but also its electric charge strengthens ferromagnetism. That nonlinear effect strengthens the critical behaviour and critical exponents come off any known universality classes. Thus, the symbiotic coherence of annihilating Coulomb distortions and arising ferromagnetism is a reason of the novel criticality.

could not be explained on the conventional universality classes. Also two critical behaviours take place in an manganese nitride Cu 0.9 NMn 3.1 : ordinary mean field theory below T C , but above T C robust critical fluctuations with (β, γ) = (0.532, 1.63) out of any universal classes (it should be noted that these values lead to unusually large negative α = 2(1 − β) − γ = −0.694 out of any universal classes too), which explained by the short-range antiferromagnetic ordering above T C

9
. Critical exponents β, γ, δ in an organic ferromagnetic tetrakis(dimethylamino)ethylene fullerene[60] differ significantly from the 3D-Heisenberg and, in addition, do not obey the scaling γ = β(δ − 1) and superscaling γ + 2β = dν relations 2 , that are attributed to the presence of disorder in the C 60 molecular orientations near T C . The magnetization of UGe 2 and URhGe shows strong uniaxial magnetic anisotropy, nevertheless, their universality class is no 3D-Ising and cannot be explained via previous approaches to the critical phenomena 8 . It is obvious that the unconventional criticality is also caused by lattice distortions with appearing the strong itinerant character of the 5f electrons in the ferromagnetic superconductive state. Thus, not only the percolation of the 3d electrons but also the delocalization of the 5f electrons can lead to a novel criticality. Pressure 3,4,6,7 , modifying critical behavior by way of sizably reducing the lattice spacing, hints at a key role of lattice distortions in novel criticality. Pressure 6 enhances critical ferromagnetic behaviour in BaRuO 3 , whereas keeps mean-field behavior in SrRuO 3 . The application of pressure 7 to (Sm 0.7 Nd 0.3 ) 0.52 Sr 0.48 MnO 3 single crystal increases the T C and suppresses the hysteresis that makes the transition a second order with (β, γ, δ) = (0.358, 1.297, 4.536) close to the 3D-Heisenberg at 12.1 kbar. The quasi-two dimensional organic insulator κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl shows a pressure-induced critical behaviour inconsistent with any universal classes, hence novel criticality is inherent in quasi-two dimensional systems, in which correlated electrons form a many-body system with anomalous collective behaviour, which cannot be understood from known spin models and, therefore, it needs some new explanation for the unconventional exponents 4 (β, γ, δ) ≈ (1, 1, 2).
It should be noted that double exchange is qualitatively different from ordinary exchanges -direct exchange, s-d-, s-f-and RKKY-exchanges, and superexchange; as it can alter critical properties and even introduce new universality classes 13 , lattice distortions and "metallization" being co-occurred. Whereas direct exchange, s-d-, s-f-and RKKY-exchanges, and superexchange do not qualitatively change the electrical behavior. A ferromagnetic with direct exchange, s-d-, s-f-and/or RKKY-exchanges is metallic below and above Curie point. And an antiferromagnetic with RKKY-exchange or superexchange is insulating or semiconducting below and above Neel point. In contrast to the ordinary ferromagnetic exchanges antagonistic to superexchange, the double exchange mutually coexists with the superexchange: 'temperature -doping level' phase diagrams of TTMO shows adjacent ferromagnetic and antiferromagnetic areas, where the Neel point gradually transits into the Curie point and reverse. Superexchange, at which a p-electron of an anion stays in the d-orbitals of the adjacent Mn 2+ -cations only about 2% of the time, is virtual, as the electron must come back. That small time is not able to cause lattice distortions as the ions valent states below (Mn 1.98+ -O 1.98− Mn 1.98+ in MnO or Mn 2.98+ -O 1.98− Mn 2.98+ in an undoped manganite) and above (Mn 2+ -O 2− Mn 2+ or Mn 3+ -O 2− Mn 3+ ) Neel point are practically same and charge symmetry relative to an anion remains. Magnetics with s-d, s-f, or RKKY exchange also do not reveal sufficient lattice striction, as their d or f electrons do not walk between atoms but only polarize conduction electrons. Overlapping d-orbitals in ferromagnetic with direct exchange is less than 1% (the overlapping more 1% leads to antiferromagnetism) that makes it impossible colossal spontaneous striction. Double exchange involves actual electron hopping, as opposed to virtual electron hopping in the superexchange scenario: one electron from a Mn 3+ -cation transfers through intervened O 2− anion into the adjacent Mn 4+ -cation and can either go back or go on to the next Mn 4+ -cation. In electric and/or magnetic fields, the onward motion prevails and electric current arises, which can switch magnetic structures and spintronic devices [14][15][16][17][18] . In contrast to all other kinds of exchange practically weakly influencing on lattice constants, double exchange, turning a mixed-valent system (Mn 3+ -O 2− Mn 4+ ) into the fractional valent one (Mn 3.5+ -O 2− Mn 3.5+ ), vanishes the mixed-valent-induced lattice distortions, the colossal spontaneous striction occuring. That striction strengthens the overlapping of the d-and p-orbitals that increases the integral of double exchange. Thus, double exchange not only gives birth to the double-exchange ferromagnetism but also www.nature.com/scientificreports www.nature.com/scientificreports/ governs its exchange integral through Coulomb distortion. That property of double exchange is unique. Then, in a double-exchange magnet, ferromagnetism grows near the Curie point much more quickly than in a canonical ferromagnet, that is, with d-d, s-d, s-f, and/or RKKY exchanges, and that TTMO manifest novel criticality must not be a surprise. The peculiarities of double exchange, mentioned here, allow us to consider it responsible for the novel criticality in a solid. Remarkable, binary transition-metal oxides, being antifferomagnetic, stay within the conventional criticality: ReMnO 3 manganites (Re is a rare earth) belong either to the 3D-Heisenberg or to the 3D-XY in the critical exponents and amplitudes of the magnetization and heat capacity near Neel point 19 ; The electrical conductivity of the antiferromagnetic (V 1−x Cr x ) 2 O 3 Mott insulators reveals the mean field which goes into 3D-Ising at the critical pressure 3 , i.e., even pressure cannot take them out of an ordinary critical behaviour.
Having made a survey of the above-mentioned substances with unordinary criticality, we found their one common pattern -their lattice structures had the adjacent cation-anion elements. When the cation objects, separated by the anion unit, have the different valences, numbers of holes, and/or occupations of orbitals (for example, in manganites, Mn 3+ and Mn 4+ spaced by O 2− ), double exchange creates ferromagnetism and eliminates Coulomb distortions. Together both these processes increase the magnetic interaction range and the magnetic order parameter much quicker than in canonical magnetics, and the novel criticality induced. For a instance, in the k-(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl organic insulator, the cation unit is a BEDT-TTF dimer with one hole (i.e., the band is half-filled) and an anion element an insulating anion layer. The half-filled orbitals indirectly (via the anion complex) hybridizing, the Pauli exclusion principle forbids the double-exchange. That is why, the antiferromagnetic insulator is only allowable in the k-(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl instead of the ferromagnetic conductor.
In the Report, having carried out the scaling in the magnetic heat capacity of the Sm 0.55 Sr 0.45 MnO 3 manganite, one of the best TTMO as it is completely ferromagnetic, and having compared our and literature data, we could understand that the nature of the novel criticality in TTMO is caused by the symbiotic coherence of annihilating Coulomb distortions and arising ferromagnetism, which both are triggered by double exchange, the double charge exchange vanishing Coulomb distortions and the double spin one creating ferromagnetism.

Results and Discussion
To prove our concept of a new universality class, the magnetic heat-capacity scaling of Sm 0.55 Sr 0.45 MnO 3 manganite has been carried out. The zero-field and 2.6-T heat capacity in Sm 0.55 Sr 0.45 MnO 3 display a Δ-shaped anomaly and hysteresis reflecting the first-order-like nature of the transition (Fig. 1). The zero-field cooling (ZFC) Curie point, that is, the paramagnetic-to-ferromagnetic transition (p-f), − T (ZFC) The ZFC inflection point at 123.3 K (Fig. 1, the insert), meaning the onset of ferromagnetism, is consistent with the 120-K neitron difraction scan (Fig. 2). Magnetic field, heightening − T C p f faster than − T C f p , narrows their difference, i.e., the hysteresis. Hence, magnetic field suppresses hysteresis and completely vanishes it at 3.5 T and above, the Δ-shaped heat-capacity peak turning into λ-like. Hence, Sm 0.55 Sr 0.45 MnO 3 behaves like an unordinary ferromagnet. That is because the intensification of fluctuations by the field-induced annihilating Coulomb distortion outweighs classic suppressing fluctuations by a magnetic field.
The magnetic heat capacity in Sm 0.55 Sr 0.45 MnO 3 was scaled according to the improved scaling procedure 12 . As Sm 0.55 Sr 0.45 MnO 3 is hysteretic and high-sensitive to magnetic field, we took distinct values of Т C for cooling and warming runs, and for each of magnetic fields and calculated the reduced temperature t = T/T C − 1 and the T onset p f , has been determined: A C p /T representation is very geometrically sensitive to fix − T onset p f , a temperature where heat capacity, because of appearing ferromagnetism, starts to inflect from the approximating straight line. (2019) 9:19328 | https://doi.org/10.1038/s41598-019-55594-w www.nature.com/scientificreports www.nature.com/scientificreports/ magnetic heat capacity C p (t, H) − C p (t, 0) -the difference of heat capacities in a given field and without a field. If at a certain α and ν the scaling exists, dependences C p (t, H) − C p (t, 0) on a scale of H α/2 ν vs. t on a scale of 1/ H 1/2 ν must collapse upon each other 20 ; that is why, this procedure is named so. We have found that the collapse (see Fig. 3) takes place at (α, ν) = (−0.23, 0.7433) out of any known universality classes. Table 1 in its rows 37-48 contains compounds with similar critical exponents, which, altogether with our Sm 0.55 Sr 0.45 MnO 3 , announce a new universality class for alloys and ternary oxides with a 3d-metal cation. As the most of the class is of TTMO, it has been named so. It is intriguing that a high-T c superconductor (the row 39) worms its way into that class.
The collapse is observed practically within the all abscissa range, so that no apparent finite-size effects are near T C . This is no wonder, as, at the mean grain size L ≈ 20000 Å finite-size effects could arise only at |t| < 0.01 when the correlation radius of magnetic order parameter ξ > L|t|ν = 652.27 Å, whereas the real ξ in Sm 0.55 Sr 0.45 MnO 3 calculated from its heat capacity 21 and from its small-angle neutron scattering 22 is much less. One peculiarity in Fig. 3 attracts attention: the 2.6-T graph at heating superimposes better upon the 3.5-T graph than on the 2.6-T graph at cooling. What, at T < T C and 2.6 T, the data at cooling is low than that at heating is thought to be caused by embedding lattice distortions inherent to the paramagnetic state into the ferromagnetic state.  www.nature.com/scientificreports www.nature.com/scientificreports/ Strong magnetic-field effects are emphasized to be distinctive feature of TTMO with the octahedral-centered transition metals: ordering t 2g -orbitals, magnetic field additionally favours to double exchange via the strong spin-orbital interaction of t 2g and e g electrons. Thus, it is natural to suppose that the new TTMO universality class emerges an unknown collective behaviour [1][2][3][4]8 . Heisenberg double Hamiltonian 23 would be useful to take into account lattice distortions during the magnetic transition. In this model, the orbital and spin momenta turn out closely related; and then, varying the spin structure, magnetic field changes the orbital one and so double-exchange interaction. As the result of that, the response of the spin system to magnetic field becomes nonlinear and metamagnetism arises. The interconnection of spins and Jahn-Teller phonons is supposed to account for the inhomogeneities and metamagnetism in the paramagnetic state, resulted in the nonlinear T C -dependence. Metamagnetic features above T C in Sm 0.55 Sr 0.45 MnO 3 are consistent with the colossal magnetostriction as well as the strong magnetostriction in Jahn-Teller crystals is responsible for their unusual metamagnetism. Moreover, the Heisenberg double model 23 predicts a tricritical point on the magnetic field -temperature phase diagram.
Some TTMO reveal unconventional criticality and even new universality classes if their magnetic order depends upon a lattice or orbital ordering 24 , hysteresis and tricritical point often observed. Earlier we showed 21 that the droplet of a nascent phase, a fluctuation of a diameter ξ, can appear if the changing of a temperature inside its volume 4 3 πξ 3 is more of the hysteresis width ΔT C = − − − T T C f p C p f . Let us continue the analysis 21 to deduce a relation between ΔT C and ξ. Because of colossal striction, a nascent paramagnetic droplet (such a case takes place just below T C ) must work against the elastic force -the surface tension on the spherical boundary of the area S = πξ 2 with the ferromagnetic. According to Hooke's law, the droplet to grow up from its radius r = ξ/2 to r + dr develops the force F = π(E fm − E pm )ξdξ (E fm and E pm are the Young moduli in the ferromagnetic and the paramagnetic). Then, the energy , consumed to create the droplet, induces the hysteresis

B . This expression at the zero-field
3 K and E fm and E pm from ref. 25 gives ξ = 8.7 Å in a good agreement with ξ yielded from the heat capacity 21 and small-angle neutron scattering 22,26 . Thus, Coulomb distortions accompanied with colossal striction can restrict ξ and so critical behavior.
The tentative TTMO universality class with averaged (β, γ, δ, α, ν) = (0.45, 1.36, 4.14, −0.25, 0.75) also covers the chemically and physically different compounds: monocrystalline Fe-Pt alloys, amorphous Fe-Mn alloys, the high-T C superconductor (Table 1); hence, that class is really universal and its necessity is put forth by us. A peculiarity of the TTMO universality class is that some its members do not obey certain scaling equalities, so that the experimental δ and from the Widom equality δ = 1 + γ/β differ. Also α of the Rushbrook equality α = 2(1 − β) − γ and of the combination of the Widom and Rushbrook equalities α = 2 − β(1 + δ) are different; that is, the scaling equalities with isothermal δ violate. This is because magnetic field directly modifies the lattice behaviour and so changes δ. Such behavior is inherent for so-called "Random fixed point" (Table 1). In some compounds, measured β, γ and δ say about their novel criticality, whereas their α and ν, calculated via scaling relations, belong to ordinary universality classes (Table 1). Hence, α and ν are more preferable in finding the universality class of a solid.
In spite of novel criticality, certain TTMO follow scaling equalities provided their lattice distortions and magnetic order are synchronized by the spin-conservation hops of electrons from one transition-metal cation to the another (Fig. 4a,b). Now, a cause of the coherence of the Coulomb distortion and the double-exchange ferromagnetism pictorially understood, we are able to schematically explain a mechanism of novel criticality in TTMO (see Fig. 5).
The fragmentation of the macroscopic coherent states shown in Fig. 4a,b into mesoscopic clusters with the magnetic dipole-dipole interaction between them instead of double exchange, desynchronizes the lattice and magnetic behaviours, that leads to metamagnetism and so scaling equalities violates (Table 1). That happens when the doping with another transition metal breaks the double-exchange bonds or the doping with another rare earth introduces a disorder in the magnitude of double exchange (see Table 1, rows 4-6). Incidentally, a small drift from the 3d-Heisenberg toward the mean-field model in the inhomogeneous SrFe 0.80 Co 0.20 O 3.0 ferromagnet is supposed to arise from the presence of the dipolar interactions between the Fe 4+ cations 27 .
Different critical exponents or even distinct universality classes 10 below and above T C are a consequence of hysteresis induced by the Coulomb distortions, which are present above and annihilate below T C (Fig. 4a,b). When hysteresis comes to nought at a tricritical point, critical exponents everywhere equal in full accordance with the scaling theory. The influence of the Coulomb lattice distortion on the exchange integral can lead not only to novel criticality but also to metamagnetism which violates the scaling equalities between isothermal and isomagnetic exponents (the short-range antiferromagneic correlations in Cu 1−x NMn 3+x is a classic feature of metamagnetism 9 ).
The lattice, Mott, magnetic and ferroelectric transitions inherent TTMO give diversity of their interdependence: the critical temperatures nonlinear and very-sensitive to magnetic and electric fields and pressure; hysteresis and so tricritical points; and, finally, novel criticality which is the culmination of the all. So, discovering the nature of the novel criticality is very important for understanding of amazing properties of TTMO in tailoring principal-new spintronic devices on the base of the film heterostructures 27-30 of the alternated layers of a TTMO with the ferroelectric, insulator, semiconductor, semimetal, or superconductor.

conclusions
In summary, we studied the magnetic heat-capacity scaling of Sm 0.55 Sr 0.45 MnO 3 manganite using heat-capacity measurements in magnetic fields up to 4 T. We found that the critical exponent of the heat capacity α and the correlation radius ν were out of any known universality classes, which was attributable to the existence of Coulomb distortions, as demonstrated by the neutron-diffraction result and as supported with the electrical resistance and the electric conductance. The disappearance of Coulomb distortions near T C leads to the colossal striction and the elastic energy of the lattice lowers, so that not only the spin of an electron, but also its electric charge makes ferromagnetism favourable. That nonlinear effect strengthens the critical behaviour and critical exponents come off any known universality classes. The long-range coherence of the Coulomb distortions and ferromagnetism (2019) 9:19328 | https://doi.org/10.1038/s41598-019-55594-w www.nature.com/scientificreports www.nature.com/scientificreports/ remains scaling equalities, however, its violation breaks off the scaling equalities between isothermal and isomagnetic exponents. The present study announces the new universality class for TTMO and other system with the similar critical exponents (see Table 1). As the similar critical phenomena take place in the diversity of compounds thoroughly reviewed in Table 1, the proposed mechanism of novel criticality would be fruitful in interpreting their unconventional critical properties. Hence, the novelty of the Report is in presenting the successful heat-capacity scaling the manganite; the explanation of the origin of the novel criticality in TTMO; and the new universality class including compounds (Table 1,

Methods
The Sm 0.55 Sr 0.45 MnO 3 manganite was prepared by the chemical homogenizing method 33,34 from aqueous solutions of Sm, Sr, and Mn nitrates with a total concentration of 1 mol/l. The residue left after the burning of these filters was calcined at 973 K, pressed into pellets, and sintered at 1473 K for 12 h. The phase composition of the ceramics  Figure 7. The heat-capacity measuring apparatus has a base 1, a cover 2 and two glass fibres 3 on which there is a sample 4 onto which a thermocouple 5 measuring its temperature oscillations T f is glued. A carbon resistance thermometer 6 is placed in a hole in the base. A heater 7 is wound on the outer surface of the cover. A calorimeter thermocouple 9 measuring the average temperature of the thermal bath T bath is buried on one side of the top cover and on the other side of the cover there is a calorimeter holder tube 10 which also used to pass the thermocouple lead wires and pumping down the thermal bath and for admission the gaseous 4 He. Above the window 8 (glued to the inside of the cover with Araldite to provide a hermetic seal down to the lowest temperatures), there is a light guide 11 which runs to the calorimeter along the tube 12, used also for pumping down and to pass the lead wires of the T bath -thermocouple, the carbon resistance thermometer and the heater. The light guide and wires is sealed with the half-and-half mix of beeswax and colophony 13. and the lattice parameters were characterized by X-ray diffraction (Fig. 2a) with a Siemens D5000 diffractometer. The obtained ceramics were found to be a single-phase perovskite with orthorhombic structure (P nma group) and the lattice parameters: a = 5.424(1) Å, b = 7.678(2) Å, and c = 5.434(2) Å. The value of the orthorhombicity parameter, 0.2%, suggests closeness to the cubic structure. The ratio a < b 2 < c is characteristic of orthorhombic manganites with a tolerance factor of ≈0.92. The sample density is 5.16 g/cm 3 . 152 Sm 0.55 Sr 0.45 MnO 3 neutron-diffraction patterns [35][36][37] , measured at different temperatures and a neutron wavelength 2.343 Å on a G4.2 high-resolution neutron powder diffractometer at the neutron guide room of an ORPHEUS reactor (LLB, Saclay, France), show no additional peaks attributed to antiferromagnetism below − T (ZFC) C p f and no phase separation (Fig. 2b). Sm 0.55 Sr 0.45 MnO 3 suffers a phase transition into a homogeneous ferromagnetic metallic-like state 21,22 with the magnetic moment M = 3.36(5) μ B per a Mn atom upon saturation at T = 4 K that corresponds to a complete ferromagnet order without indications of antiferromagnetism. Thus, Sm 0.55 Sr 0.45 MnO 3 possesses the maximum ferromagnetic fraction that allows us to choose it to carry out the renewed scaling procedure 12 . The neutron-diffraction data near − T (ZFC) indicate an abrupt decrease in the cell volume (ΔV C /V C ≈ 0.1%) upon transition into the ferromagnetic state, associated with a displacive Jahn-Teller transition, that is, the annihilation of Coulomb distortions, the space group, P nma , remaining over the all temperatures. The lattice parameters changed in a specific manner: the rhombic base of the unit cell contracted sharply (the temperature dependences of the parameters a and c exhibited jumps), whereas the parameter b changed only slightly (Fig. 6a). The orthorhombic distortion δ = (a − c)/(a + c) ≈ 0.15% which is comparatively large. The Curie point is attended by jump-wise decreasing the Coulomb distortions of the MnO 6 -octahedrons. The neutron diffraction patterns have given detailed data on the Coulomb distortions of the MnO 6 -octahedrons (Fig. 6b). The angle Mn- O 21 -Mn, an angle between the adjacent octahedrons, at room temperature ≈159° but ≈161° near .004 Å at 100 K. So, the changings of the interatomic distances interior the octahedron and the angle between adjacent octahedron indicate noticeable Coulomb distortions above T C and their significant reduction below T C , which obviously affect double exchange and, hence, the critical behavior.
The heat capacity C p was measured using an apparatus (Fig. 7) by an ac-calorimetry 38 . One advantage of the method is the smallest temperature gradient across a sample (<10 mK), which is especially important in studying the critical phenomena near T C . The gradients permit measurements with sensitivity less than 0.01 K or 10 −5 of the reduced temperature |T/T C − 1|. According to the ac-calorimetry, absorbing a light energy per a second dQ from an tungsten lamp, modulated with a frequency f, leads to the sample temperature oscillations T f with the same frequency and increases an average sample temperature of the T with respect to a thermal-bath temperature T bath on T dc , i.e. T = T bath + T dc . By measuring T dc and T f with known the temperature dependence of heat conduction, g(T), of gaseous 4 He filling the thermal bath, and a distance between the sample and the thermal bath, d, the absolute value of C p = dg(T)T dc /T f (T dc used not to exceed 1.5 K). To reach the quickest response to a temperature of the sample, the 25-μm chromel-constantan thermocouple, spot-welded from the flattened ends of the wires, was glued to the backside of the sample with the BF-2 phenol polyvinyl-acetal glue (the analog of the GE-7031 varnish). The sample is a 0.27-mm thickness platelet with the area of 3*3 mm 2 and weight <30 mg. The "cold junction" of the thermocouple is mounted on the thermal bath to simultaneously detect T f and T dc . The front side of the sample was periodically lit by a tungsten lamp with a mechanical chopper. The used frequency f = 2 Hz adopts the ac-calorimetry criteria 38 and allows us to set the 10-s time constant on a lock-in amplifier to maximally avoid the electromagnetic noise. The 0.1-mm copper-constantan thermocouple with one junction glued to the thermal bath and another placed in the ice bath detected T bath . The absolute error 0.8% was estimated as the ratio of the heat capacities of the thermocouple and the adjacent sample volume bounded by the thermal diffusivity length η π = l f / , a distance on which the absorbed heat propagates for a period (η is the thermal diffusivity).