Phase diagram of solid hints new fundamental constant and sees atomic orbital


 Cold and pressure transform gas into liquid and then into solid. Van der Waals understood the phase diagram of liquefiable gas with the molecular volume and intermolecular attraction, however, was silent on how solid behaved1. Unfortunately, solid-state phase diagram have remained uncomprehended mystery; only its straight boundary2,3 was explained by struggle of order vs. chaos. Here we show that the volume of orbital overlap has its own energy, with the universal density 8.941 eV/Å3 announced as new fundamental atomic constant that determines the transition temperature TC. Furthermore, we devised solid-state tomography, valid to 5 TPa, - imaging orbital through the baric dependencies of TC. Triangle-shaped pattern of the diagram is explained by the only possible way, just as only one plane passes through triangle: -inflation of the intersection volume during the transition determines hysteresis, but its disappearance does triple point; -approaching ions, whose orbitals overlap, curves the line from zero-field-cooling (ZFC) TC to triple point; -the straight line between zero-field-heating (ZFH) TC and triple point is a consequence of straightening tilting angle. Diamond melting point, calculated from volumes of the tetrahedral covalent bonds, excellently agrees with real; furthermore, the points up to 2 TPa agree with experiment4. Our findings open up way to interpret antiferromagnetism and steric effect in mono, binary, and ternary transition-metal oxides and sulfides5-11, and advance in unravelling unconventional superconductivity12,13, ascertaining the roles of s- and p-hybridizations. Thereby, the importance of the solid-state tomography for organic conductors12,13 being high-compressible and interior of stars can scarcely be exaggerated.

In 1873, Van der Waals showed that the volume of molecules of the gas has the matter and they interact between them with the force named later in his honour 1 , however, was silent on how solid behaves. Van der Waals' studies allowed Kamerlingh Onnes 14 to liquefy helium and discover superconductivity in mercury; so unravelling the mystery of phase chart of gas had advanced not only physics of gas and liquid, but also solid. Solid is the last aggregate state of substance, which only changes its structural, electronic, magnetic and other properties till either its temperature has become infinitely close to absolute zero or its destruction under enormous temperature and pressure in the core of star. Unfortunately, up to now the phase chart of solid capable to advance science remains an uncomprehended mystery. Boundaries between the different properties of solid often behave intriguing manner. The phase charts of organic and cuprate superconductors where four different states meet at the multicritical point 12 is especially interesting. The phase diagram of iron no less attractive puts up, however, knotty issue about whether the number of solid phases should be raised to five 15 . This work reveals the nature of the phase diagram of solid, that is, answers the question been silent by Van der Waals.
We have asked ourselves why the phase boundaries of Sm0.55Sr0.45MnO3 manganite behave straightly with different slopes along the lines BC and DE and curvedly on AC (Fig. 1) Table S1)] means that magnetic field only suppresses the thermal agitation and so the Curie point is linear: Our attempts to interpret the steep energetic pitch 2.37 of BC as easy as DE failed. To reveal the secret of BC was previously necessary to explain AC. We supposed that the transition temperature Ttr is proportional to the volume of the double spheric cap = 2ℎ 2 ( − ℎ/3) formed at the intersection height h of two orbitals with the radius R of their spheric ends (Fig.  2a). So the intersection volume has the energy that equals the phase-transition energy kBTtr. By introducing the energy density , The subscript tr points out that Exp. (2) suited for different-type transitions -Curie, Neel, Mott and melting points. The indirect intersection of half-filled orbital with empty through full-filled (that is, diamagnetic) orbital leads to double exchange, but with half-filled to superexchange which, in turn, forbids the Mott's conductance. The direct intersection of half-filled orbitals creates covalent bond whose energy and number yield the melting point.  Obviously, the intersection height h = a, a is half a distance between Mn 3+(4+) and O 2and  a relative striction. A magnetic cation shrinking solid wastes its magnetostatic energy on enhancing the elastic energy of unit-cell volume V:  • = ε 2 /2, E is Young's modulus. Then ε = √2 • / and so h = a√2 • / , as ZFC Curie point is nonzero, it needs to add the threshold intersection height hA to this. Substitution of ℎ = √2 • / + ℎ into Exp.
Magnetic field supports magnetism along AC same as on DE, therefore, the best physical description of AC is the sum of Exps. (3) and (1) at cos linearly turning down from A to C⪆DE (cos linearly depends both on temperature 16 and on the rare-earth-cation radius 17 , therefore to preset its magnetic-field linearity is quite reasonable). We found the best fit with the root-mean-square error (RMSE) 0.58 K at the initial and fitted quantities mentioned in Table 1.
So the curvilinearity of AC is a consequence of the curvilinear field dependence of the intersection volume at the striction, that is, approaching Mn 3+(4+) and O 2in the paramagnetic state just before appearing the ferromagnetism.
To understand hysteresis AB is to know why the ZFH Curie point is higher than . We think that hA triggers double exchange, which removes the Coulomb distortion 18 (certainly the chemical distortion, defined by the tolerance factor, remains), the orbital overlap inflating till hB. The geometric constructions (Fig. 2b) for the points A and B give hB = (a-R+hA)cosB/cosA-aB+R, aB is half a distance between Mn 3+(4+) and O 2at B. Expression (2) with h = hB and the entries for B (Table 1) Magnetic field supports magnetism on BC same as on AC and DE because the best graph of BC is also a sum of Exp. (2) with h = hBC and Exp. (1). BC excellently traces the data with RMSE 1.37 K only with the initial entries, without fitting (Table 1). So the linearity of BC is a consequence of the linear field dependence of the intersection volume at straightening (B) in ferromagnetic state.  In the case of the pressure diagram, the Hooke law P = E(P) (the Young's modulus E(P) is supposed to depend on pressure as well as temperature 19 , magnetic order 20 and even form 21 ) gives ℎ = / ( ) + ℎ B which turns Exp. ( Expession (6) with the initial entries (Table 1) and only with fitting the prefactor and power in E(P) = 325P 1/3 + E0 excellently fits the TC(P) diagram 18 in (Sm1-xNdx)0.55Sr0.45MnO3 (Fig. 3). Finally, our calculation of the diamond melting-point is the brightest approbation of our discovery. Diamond starts to melt when its atom receives the energy enough to break all its covalent bonds. The diamond sp 3 -orbitals intersection is the carbon p-orbital length minus the diamond covalent radius 22 : h = r pr c , where rp is obtained as the carbon-hydrogen distance in methane ("gaseous diamond") plus the hydrogen covalent radius and minus Bohr radius:  Table S1). To further affirm our theory, proceeding from the warrantable assumption that the diamond phase persists under enormous pressure 23 , we calculated the baric dependence of the melting-point in diamond, Ttr(P), through its (P) taken from Extended Data Table 24 1. The baric dependence of the covalent radius  At last, the hit of our work is an orbital tomographythe reconstruction of the form of the orbital by means of the layer-by-layer scan of its radii R versus the covalent radii rc, which uses the temperature-pressure diagram as microscope. The stress-induced decrement of the covalent radius rc(P) increases the orbital intersection on the disk-shaped layer 2R(rc) 2 rc(P) which induces the increment of the melting energy per half an sp 3 covalent bond kBTtr(P)/8 = 2R(rc) 2 rc(P). Then the measurements of the derivative Ttr(P)/rc(P) allow us to profile the orbital via its radii  (Fig. 4b).
Thus, the announced atomic constant  = 8.941 eV/Å 3 obtained from fitting AC has been also confirmed by drawing BC and TC(P), the calculation of the hysteresis, the melting point and orbital tomography in diamond at ultrahigh pressures. Our discovery qualitatively explains the diagrams of antiferromagnetics 5-10 not clear for decades. Pressure, holding the transition-cation planes away from the anion planes in NiS, V2O3 and RNiO3 (R = Pr, Nd, Nd0.7La0.3), narrows the eg-p orbital overlappings, so that TN lowers and the eg electrons, which cannot migrate because of antiferromagnetism, gradually become current carriers. The BaVS3 diagram 8 become understandable if to take into account that the pressure-induced orbital overlaps in the V-S chains favours to antiferromagnetism which impedes conductance. The linear TN(P) of the Gd, Pr and Tm orthoferrites 9 means the fulfillment of Hooke law at their small rare-earth-cation radii, but the more radius in the La orthoferrite 9 shortens the Fe-O bond and so inflates the 3d and 2p orbital intersection that causes the lift and nonlinearity of its TN(P). Expession (6) with the chemical pressure Pi ~ 1y/yp (yp is a doping index of the paramagnetic composition) instead of P could fit the TN(y) diagram 10 of Ni1-xS1-ySey. The coincidence of the Sm0.55Sr0.45MnO3 Curie point with the MnO Neel point hints at their same intersection volumes, that is, distinct-type transitions can have their near critical points, then certain substances must pose on the near points of the diagram "intersection volume-TC", revealing that their phase transitions caused by the orbital overlapping. Although the wave function is smear, the outline of orbital is sharp (Fig.  4b), that is, electron is like a son who was forbidden by his father to leave the yard, but he sometimes leaves and returns; so that chemists, specifying the orbital overlapping, can tailor necessary substance properties without complicated quantum simulations.  28 . R is about a fourth of the orbital length that is standard for d and p orbitals 29 . The values of a and V are calculated from the neutron-diffraction data 18,30 . aB, B and hB are fitted parameters of the point B. The values of aB, A and B are in good consistency with the neutron-diffraction data 18,30 . And C is in good agreement with DE, that is, C⪆DE. All the calculations were carried out in Supplementary Table S1. As seen, the fitted (■) data of AC are initial (■) for BC, B and TC(P). And the fitted values of B are initial for BC and TC(P). So, BC is drawn only with the initial data, without fitting. TC(P) is traced only with fitting the prefactor and power in the pressure-depended Young's modulus E(P) = 325P 1/3 + E0.