Quantum thermodynamics at critical points during melting and solidification processes
QQuantum thermodynamics at critical points during melting andsolidification processes
A D Arulsamy ∗ F-02-08 Ketumbar Hill, Jalan Ketumbar,Taman Cheras, 56100 Kuala-Lumpur, Malaysia a r X i v : . [ c ond - m a t . s t r- e l ] J a n bstract: We systematically explore and show the existence of finite-temperature contin-uous quantum phase transition ( C T QPT) at a critical point, namely, during solidification ormelting such that the first-order thermal phase transition is a special case within C T QPT. Infact, C T QPT is related to chemical reaction where quantum fluctuation (due to wavefunctiontransformation) is caused by thermal energy and it can occur maximally for temperaturesmuch higher than zero Kelvin. To extract the quantity related to C T QPT, we use theionization energy theory and the energy-level spacing renormalization group method toderive the energy-level spacing entropy, renormalized Bose-Einstein distribution and thetime-dependent specific heat capacity. This work unambiguously shows that the quantumphase transition applies for any finite temperatures.
Keywords:
Thermal and quantum phase transitions; Energy-level spacing and time-dependent entropy; Ionization energy theory; Energy-level spacing renormalization groupmethod
PACS Nos.: ∗ Corresponding Author, E-mail: [email protected] . Introduction
A critical point here refers to a melting point of ice or solid such that at the criticalpoint state one does not have a well defined phase of matter or phase boundary (both solidand liquid phases coexist with dynamic phase boundary between them). The fluctuation ata critical point during finite-temperature continuous quantum phase transition ( C T QPT) isspecifically called quantum fluctuation (even for finite temperatures) because it is relatedto the properties of electrons at a constant temperature, and therefore, it is still subject tothe Heisenberg uncertainty principle. For example, little did we know that physically andlogically, not all quantum fluctuation ( ω , , ··· ) has become ‘irrelevant’ when T >
0K [1–4].In other words, not all ω → maximum when T → T →
0K is true, then chemical reaction cannot exist.The argument that a chemical reaction is a quantum mechanical effect can be proven if wecan prove the existence of quantum fluctuation (due to wavefunction transformation) thatbecomes large for finite temperatures (or when
T > QPT) can be found elsewhere [1, 2]. The quantum fluctuationresponsible for C T QPT occurring at a finite temperature quantum critical point ( T QCP) isassociated to quantitative changes to the wavefunction, which is also known as the wave-function transformation [5].Here, we systematically explore the notion of thermal phase transition beyond the struc-tural phase transition [6] and prove the existence of C T QPT during melting and solidification(due to chemical reaction) and the standard first-order thermal phase transition (TPT) asa special case within C T QPT. To achieve the above objectives, we first need to introduce andexplain the ionization energy theory (IET), and this is followed by in-depth analysis on thefirst-order thermal phase transition in molten alkali halides, including why and how thistransition is related to the finite temperature quantum phase transition. Subsequently, wedig deep into the details of quantum phase transition and quantum fluctuation with respectto IET, involving wavefunction transformation and electronic phase transition. Along theway, the theory is also discussed and crosschecked whenever necessary against the availableexperimental observations. Our primary results obtained from the proofs of points statedabove are related to new physics beyond the traditional thermal phase transition and thezero-temperature quantum phase transition.3 . Theoretical details
The ionization energy theory can be compactly captured by the IET-Schr¨odinger equa-tion [7, 8], i (cid:126) ∂ Ψ( r , t ) ∂t = (cid:20) − (cid:126) m ∇ + V IET (cid:21) Ψ( r , t )= H IET Ψ( r , t ) = ( E ± ξ )Ψ( r , t ) , (1)and the ionization energy approximation, ξ quantumsystem ∝ ξ constituentatom . (2)Here ξ is the energy-level spacing or the ionization energy, H IET and V IET are the exactHamiltonian and potential term, respectively, (cid:126) is the Planck constant divided by 2 π , m denotes electron mass and E is ground state energy for temperature ( T ) equals zero Kelvin,in absence of external disturbances such that ξ determines the effects of any perturbation ordisturbances and finally, the ± sign refers to electrons and holes, respectively. Add to that,only the true and real (not a guessed) wavefunction denoted by Ψ( r , t ) can properly capturethe properties of electrons. The notation, ξ quantumsystem denotes the real ionization energy of aquantum system, while ξ constituentatom is the real atomic ionization energy. Accurate values for ξ constituentatom can be directly obtained from any atomic spectra database, or routinely calculatedfrom the density functional theory or other quantum chemical methods by using guessedwavefunctions and adjustable parameters.Firstly, Eq. (1) is not specific such that V IET needs to be expanded and specified for agiven system and after finding V IET , we still need to solve the IET-Schr¨odinger equationvariationally for ξ . The potential energy can be specified following Fig. 1(a) and (b) wherewe have assumed the sketched atoms ( X and Y ) to be polarizable, neutral and identical withdiscrete energy levels. Each atom, X and Y has one equally-polarizable electron, namely, e X and e Y , respectively. Here, the negatively charged electrons and positively charged nucleiare arranged in this manner, + − + − along the same axis. In contrast, V e − ionCoulomb is therenormalized screened Coulomb potential between atomic X and Y , which has been addedby hand into Eq. 3 (the first term on the right-hand side). This particular term ( V e − ionCoulomb )imposes the condition that both X and Y are also single electron atoms, but with a little4wist such that atomic X has the least polarizable electron ( e X ), while atomic Y has theeasily polarizable electron ( e Y ). We will also denote atomic X as an anion (namely, Cl),whereas atomic Y represents a cation, namely, Li. The charge carried by each nucleus, X and Y in both sketches, (a) and (b) in Fig. 1 is + e . In particular, for a two-atom systemdepicted in Fig. 1(a) and (b), the relevant potential energy ( V IET ) is the Ramachandraninteraction (stronger than the usual van der Waals type) [9–12] V IET = ˜ V (cid:48) Ramachandran ( ξ ) = ˜ V e − ionCoulomb + ˜ E − (cid:126) (cid:115) ˜ km , (3)where ˜ E − (cid:126) (cid:115) ˜ km = 12 (cid:126) ω (cid:18) √ − (cid:19)(cid:26) exp (cid:20) λξ X (cid:21) + exp (cid:20) λξ Y (cid:21)(cid:27) , (4)and ˜ V e − ionCoulomb = ( − e )(+ e )4 π(cid:15) | R X − r Y | (cid:26) exp (cid:2) − µr X e − λξ X (cid:3)(cid:27) . (5)From [8] ˜ k = k exp [ λξ ] , (6)where ˜ E , ˜ V e − ionCoulomb and ˜ k are the renormalized energy, electron-ion Coulomb potential andinteraction potential constant, respectively, µ is the screening constant of proportionality, λ = (12 π(cid:15) /e ) a B , a B is Bohr radius of atomic hydrogen, R X and r Y are coordinates fornucleus X and electron e Y , respectively. Here, (cid:126) (˜ k/m ) / = (cid:126) ˜ ω , (cid:126) ˜ ω is the renormalizedenergy when V IET = 0, and ˜ V (cid:48) Ramachandran ( ξ ) is negative guaranteed by (1 / √ − − e in Eq. (5). See Ref. [9] for the derivation of Eq. (3). The term on the right-handside of Eq. (4) assumes that both atoms, X and Y are identical, hence their electrons areequally polarizable. In Eq. (5) however, we have a unidirectional electron-ion attractiveinteraction between the easily polarizable e Y and nucleus X , screened by e X via ξ X (see theterm in the curly bracket in Eq. (5)). The imposed asymmetric polarizability is capturedby ˜ V e − ionCoulomb such that the respective atomic X and Y represent an anion (least polarizable)and a cation (easily polarizable), which can be used to understand the transition fromliquid ( V liquidIET ) to solid ( V solidIET ) phase. Here, the quantum phase transition originates from V liquidIET ( ξ liquid ) → V solidIET ( ξ solid ), which requires wavefunction transformation because we needto transform ξ liquid → ξ solid accordingly. 5o avoid calculations on the basis of variational principle that require guessed wavefunc-tions and variationally adjustable parameters, we have devised an alternative first principlesapproach where one just need to use the analytic ionization energy approximation (Eq. (2))to evaluate the physical properties of a particular quantum system such that any changes to ξ can be traced back to a renormalized physical parameter within the energy-level spacingrenormalization group method [8, 13–15]. For example, the quantum fluctuation introducedabove can be written in the form, (cid:126) ω = ξ , and for different compositions ( y , y , · · · ) theenergies of fluctuation read, (cid:126) ω ( y ) < k B T θ , (7) (cid:126) ω ( y ) = k B T θ , (8) (cid:126) ω ( y ) > k B T θ , (9)where (cid:126) ω ( y ) (cid:54) = k B T θ is the doping ( y )-dependent energy that does not cause any fluctuation,and on the other hand, (cid:126) ω ( y ) = k B T θ is the energy at finite-temperature quantum criticalpoint ( T QCP), where ξ fluctuates due to transforming wavefunction [5] at a critical point.Here, y , y and y are related to different chemical compositions, while θ represents acertain physical property under investigation. For example, for melting, T θ = T meltingpoint .In contrast, for the usual QCP, the relevant inequality is simply defined [1, 2] to be (cid:126) ω ( y ) < (cid:126) ω ( y ) < (cid:126) ω ( y ) because T θ = 0K, or generally one writes k B T < (cid:126) ω for smalltemperatures not far from 0K to justify the validity of QCP.To understand how thermal energy initiates quantum fluctuation during melting andsolidification processes, we have to evaluate the relationship between thermally driven first-order TPT and C T QPT. Here, C T QPT requires quantitative changes to the properties of elec-trons (due to changes in their energy levels) that need some qualitative and/or quantitativechanges to their wavefunctions. These changes are driven by thermal energy, not by thetemperature-independent external tuners such as pressure, electric and magnetic fields. Incontrast, the commonly accepted quantum phase transition for T = 0K ( QPT) is drivenby the temperature-independent external tuners, and requires quantitative changes to theproperties of electrons (due to changes in their energy levels) that also need some qualita-tive and/or quantitative changes to their wavefunctions. The changes for T = 0K are drivenby temperature-independent external tuners alone, not by thermal energy. But this doesnot imply that the thermal energy cannot be the cause for C T QPT. Hence, we should not6eserve all ‘quantum phase transitions’ exclusively for transitions driven by temperature-independent external tuners (for T = 0K), and not for transitions driven by thermal energyfor T > T ≥ T ≈ QPT does not imply QPT cannot exist for
T >
0K where the latter implication (QPTcannot exist for
T > T CDW ) is found to be about 200K, but onlythe pressure-dependent CDW transition for T = 3 .
5K is recognized as a QPT, which is anassumption restricted by an ad hoc condition, k B T < (cid:126) ω . We claim here that if the abovepressure-dependent CDW transition, for any constant temperatures (including T = 0K),is accompanied by this continuous transformation, ξ ( P > P
CDW ) → ξ ( P < P
CDW ) suchthat ξ ( P > P
CDW ) (cid:54) = ξ ( P < P
CDW ), then the CDW transition even for T ≈ k B T < (cid:126) ω is no longera restriction to study QPT where the stated ad hoc condition is just an assumption. Weapply our theory to explain the doping- and temperature-dependent melting process in alkalihalides and the freezing phenomenon in water in the presence of hydrogen bonds.7 .2. Thermal and quantum fluctuations Here, we first justify the reason why we have selected salts made up of single-valent cationsand anions from group 17 of the periodic table. Table 1 lists the diatomic bonding ener-gies and melting points of these salts such that the melting points decrease systematicallywith decreasing bonding energies from F to I. Such a decreasing trend can be precisely over-lapped with decreasing ionization energies of anions, namely, ξ F + (1681 kJmol − ) > ξ Cl + (1251kJmol − ) > ξ Br + (1140 kJmol − ) > ξ I + (1008 kJmol − ). The values in these inequalities aregiven in Table 2, follow the values marked with “ * ”. The ionization energy values wereobtained from Ref. [21]. Interestingly however, for a given anion, the decreasing meltingpoints do not agree with the ionization energy values for cations: Li, Na, K and Rb (followtheir values listed in Table 2, marked with “ † ”).For example, the melting points for NaF (996 o C) > KF (858 o C) > LiF (848 o C) > RbF(833 o C) cannot be overlapped with ξ Li + (520 kJmol − ) > ξ Na + (496 kJmol − ) > ξ K + (419kJmol − ) > ξ Rb + (403 kJmol − ). This is not surprising because melting is a process directlyproportional to bonding strengths (see Table 1), which have been discussed earlier [22]within IET where large ionization energy values of cation-like ions ( ξ C < ξ O ) in a givenmolecule (C O − or O − (1) O ) do not necessarily imply stronger bonds. Fortunately, wehave proven that a stronger bond is predictable from IET if one considers the ionizationenergies for anions (say O in H O), because oxygen defines the ability to attract electronsfrom an atomic hydrogen [23]. In this case, smaller ξ O means weaker O − H bond, which isconsistent with the above-stated ionization-energy inequalities for anions (group 17 elements)and the melting points. Here, even though melting points can be captured by changing theinteratomic distance, but this does not imply that the valence electrons at the meltingpoint of molten alkali halide are static because these valence electrons are always dynamicsuch that they constantly rearrange themselves due to chemical reaction (during melting orsolidification) in the presence of mobile ions or atoms.In Table 2, we have deliberately selected systems consisting of single valent cations andanions from group 17 to avoid effects from different electronic interactions due to differentnumber (and type) of constituent atoms in a given molecule. To understand this point, weagain use molecular systems, in which, for a NO molecule, N acts as a cation, while O as ananion, which means that we need to consider the valence state, 4+ since four electrons have8een transfered from N to O − . This electron-transfer is due to ξ N < ξ O (see Table 1).In contrast, N is an anion in NH molecule due to ξ N + > ξ H + where three electrons fromthree hydrogen atoms are transfered to nitrogen, giving N − H +3 . Thus far, the analysis iscorrect. However, the polarizability for molecule NO does not solely depend on these fourelectrons contributed by nitrogen if we compare N O − with N − H +3 because we cannotsimplify the analysis by comparing ξ N with ξ H + (3 electrons contributed by three hydrogenions) only. If we do so, then obviously we have ξ N > ξ H + that falsely allows us to conclude α NO d < α NH d because α d ∝ exp( − ξ ) where α d is the displacement polarizability.In other words, we cannot use the above inequality ( ξ ) to directly compare α d betweenNO and NH molecules because N acts as a cation in a NO molecule, while it is an anionin NH . This implies that we need to take both the cationic- and anionic-effect into accountexplicitly for an accurate logical analysis, which have been correctly invoked in Ref. [23].However, the anionic-effect can be neglected when we compare CO with O because in thesemolecule, oxygen is the anion and therefore α CO d < α O d is valid [22]. On the other hand,the justification required to neglect the cationic-effect for doped-Pnictide superconductorsis given in Ref. [24]. Finally, taking both cationic- and anionic-effect into account meansthat for large (many-electron) cations such as K ( Z = 19) and Rb ( Z = 37), there willalways be some significant amount of polarization (due to large screening) from the outercore electrons, even though effectively, K + and Rb + are single-valent ions. This second-or third-electron polarization is negligible for few-electron atoms (due to small screening),namely, Li ( Z = 3) and interestingly, also for Na ( Z = 11). It should be clear now that inview of Eqs. (1) and (2), one cannot use electron affinities to evaluate melting or solidificationprocess (see following sections).Furthermore, we need smaller and larger ionization energies for cations and anions, re-spectively for stronger ionic bonds. Therefore, incorporating this additional amount ofpolarization (for cations) implies larger ξ cationic or equivalently, smaller ξ anionic that shouldreduce the melting point as has been observed experimentally (see Fig. 2(a)). In Fig. 2(a),melting points of Li(I, Br, Cl and F), Na(I, Br, Cl and F), K(I, Br, Cl and F) and Rb(I, Br,Cl and F) salts are plotted against the anionic (I, Br, Cl and F) ionization energies, while inFig. 2(b), the vertical down-arrow (solidification) denotes the first-order thermal phase tran-sition (from liquid to solid) for LiCl when temperature is reduced from T > o C to T =552 o C. The horizontal arrow pointing left captures the continuous (purely electronic) quan-9um phase transition (from solid to liquid) at constant T = 552 o C by changing the chemicalcomposition systematically, LiCl − a Br a − a I a such that LiCl → LiBr → LiI, y = 1 − a , y = a − a and y = a . Importantly, the above correction does not apply to anionsif a given system consists of mostly ionic bonds because anions are judged solely on theirability to attract electrons from cations. On the contrary, for systems with mostly covalentbonds, we need to consider both cationic- and anionic-effect simultaneously as carried outin Ref. [23]. The existence of this additional polarization is consistent with experimentalresults shown in Fig. 2(a) when one compares the slope, d T θ /d ξ for salts containing Li andNa, for instance (d T θ /d ξ ) Li ≈ (d T θ /d ξ ) Na , while (d T θ /d ξ ) K ≈ (d T θ /d ξ ) Rb (follow the solidlines in Fig. 2(a)). Here, we have shown that thermal energy can cause fluctuation in ξ ,which could give rise to QPT. Later, we will expose why and how the fluctuation in ξ (at acritical point) forces us to impose time-dependence into ξ .
3. Results and discussion
Figure 2 shows direct proportionality between melting points and ionization energies ofthese constituent anions (I, Br, Cl and F). The respective vertical and horizontal arrowsin Fig. 2(b) indicate first-order TPT and C T QPT. In order to expose the existence of thesephase transitions, we need to recall the first and second laws of thermodynamics. These twolaws can be combined to obtaind U = δQ + δW = T d S − P d V, (10)in which, the change in the internal energy, d U of a closed system equals the amount ofheat ( Q ) absorbed and the amount of work ( W ) done by that system. Here, δ is not anexact differential because the changes in Q and W depend on the thermodynamic path (orindependent of a particular system and process), and therefore, Q and W are not statefunctions. Here, P and V denote pressure and volume, respectively and we consider onlyreversible processes. Unlike Q and W , the thermodynamic variables U , P and V are statefunctions that are unique for a given system and process.The second law is given by, δQ = T d S where S is the entropy, another path-independentstate function. All we need now is the relationship connecting S to ξ that can be obtained10rom the derivation of the ionization energy based Fermi-Dirac statistics ( i FDS). Denoting N as the total number of particles with n particles have energy ( E ± ξ ) , n particles withenergy ( E ± ξ ) and so on implies that N = n + n + · · · + n m . As a consequence, thenumber of ways for q unoccupied quantum states to be arranged among n particles isΩ( n , q ) = q ! n !( q − n )! , (11)satisfying the condition that one empty quantum state can accommodate only one particle.Subsequently, the total number of ways for q quantum states ( q = q + · · · + q m ) to bearranged among N particles Ω( N, q ) = (cid:89) i q i ! n i !( q i − n i )! . (12)The most probable configuration for certain T can be obtained by maximizing the numberof ways one can arrange n i particles in q i empty quantum states or, we need to maximizeΩ( N, q ) subject to the restrictive conditions, (cid:88) i n i = N, (cid:88) i dn i = 0 , (13) (cid:88) i ( E ± ξ ) i = E ± ξ = E, (cid:88) i ( E ± ξ ) i dn i = 0 . (14)These conditions impose that the total energy, E and the total number of particles, N for agiven system are always constant. Here, E is the total energy for 0K. Using the standardprocedure, and after some algebraic rearrangements, one gets Nq = 1exp (cid:2) µ + λ B ( E ± ξ ) (cid:3) + 1 , (15)which is the i FDS. Taking exp (cid:2) µ + λ B ( E ± ξ ) (cid:3) (cid:29) µ = − λ B E and dividing all the termsby T will lead us to the energy-level spacing entropy S ξ = − ( E ± ξ ) + E T = 1( λ B , λ ) T ln Nq , (16)in which, λ B = 1 k B T for constant ξ and (17) λ = 12 a B π(cid:15) e for constant T > . (18)11he derivation for λ B is well known in classical thermodynamics and is also given in Ref. [7]within IET formalism, while the proof for λ is available in Ref. [25]. Here, E denotesthe Fermi level for T = 0K, independent of any disturbance. The entropy S ξ given inEq. (16) decreases logarithmically ( S ξ → −∞ ) when N/q → < N/q < −∞ min < S ξ < max . However, physicallyit is meaningless to have −∞ min (unbounded from below) that will also lead to ( − S ξ ) / m where m is a natural number. Consequently, we will convert all our entropy equations tobe positive definite as required in natural sciences. But in any case, within the set of realnegative numbers, including zero ( S ξ ∈ R − ) and Eq. (16), we have the correct correspondencebetween T → S ξ → −∞ (ordered). Apart from that, it is straightforward to identifythe intensive and extensive parameters in the above equations—for example, from Eq. (15),one can readily verify N and q as extensive parameters, while N/q and all the variables onthe right-hand side of Eq. (15) are intensive parameters. This means that Eq. (15) capturesthe changes to the intensive parameters in the presence of external disturbances, namely,temperature and doping.The other relevant entropy is due to atomic-disorder, S D = k B ln D where D ( N D , q D ) = (cid:89) i ( n i + q i − n i !( q i − ≈ (cid:89) i ( n i + q i )! n i ! q i ! , (19)by taking n i (cid:29) q i (cid:29) N D /q D as the ratio of the total number of identicalatoms and empty lattice sites. Here, N D = (cid:80) i n i , q D = (cid:80) i q i , and N D /q D corresponds to theprobability of excited identical particles from their lattice sites. One of the two restrictiveconditions needed to maximize Eq. (19) remains the same (Eq. (13)) because the totalnumber of particles (atoms) is also conserved. While the condition for the total energy hasbeen renormalized, (cid:88) i (cid:2) Ee λξ (cid:3) i n i = (cid:126) (cid:115) ˜ km = ˜ E, (20)to imply that the total energy (for the atoms, not electrons) is also conserved. Moreover, wehave renormalized the total energy of identical atoms ( E ), not of the electrons, to obtain ˜ E so that the electronic polarizability effect is taken into account. This means that the atomsare no longer considered to be the rigid vibrating ions, independent of their surroundingelectrons. Somewhat similar to i FDS derived earlier, we also can derive the renormalized12ose-Einstein statistics ( r BES) N D q D = 1exp (cid:2) λ B Ee λξ (cid:3) − . (21)The constant, α = µ denotes the minimum energy prior to any excitation of atoms fromtheir equilibrium lattice sites, somewhat similar to Fermi level for electrons, and obviously,it is zero here. It is important to note here that q D ≥ N D is required so that the probabilityis normalized to one, even if there can be any number of n particles allowed to occupy asingle empty quantum state, q i . Similar to S ξ , S D ∈ R − .We are now ready to track the solidification of liquid LiCl from T > o C to T = 552 o Cas shown in Fig. 2(b). In this case, the dominant entropy change is due to the transition fromthe disordered (liquid state) to an ordered (solid) state in which, the liquid state is definedwithin 610 o C < T ≤ T and the solid state is bounded within T ≤ T < o C. The value610 o C ( T θ ) is the critical point, or the melting point of LiCl and S liquid D > S solid D indicates theexistence of TPT, qualitatively. To claim liquid has a larger entropy than solid means wehave invoked an additional condition | S liquid D − S solid D | > | S liquid ξ − S solid ξ | because the changeof entropy due to electronic excitation is always smaller compared to an atomic-disorderinduced entropy (due to broken translational symmetry) within a particular system. Ofcourse, in the absence of this temperature-induced atomic-arrangement asymmetry, electroninduced disorder or S ξ is the dominant entropy.Apparently, the change in entropy during the above-stated transition (solidification) isa first-order TPT because S liquid D : T >T θ > S solid D : T
1, then − N q ln N q > − N q ln N q , (27)which will guarantee S ≥ S ≥ S > S because N/q increases or decreases fasterthan ln (
N/q ). For example, taking N = 1, q = x , f ( x ) = 1 /x and g ( x ) = ln (1 /x ), one canwrite h [ f, g ] = (1 /x ) ln (1 /x ), subsequently it is straightforward to show d m f /d x m > d m g /d x m is always true when x ≥ m ≥ x ∈ R + and m ∈ N ∗ . In addition, S > S isconsistent with increasing entropy if N/q gets larger, and consequently we will always have { S ξ (+) , S + ξ , S D (+) , S + D } ∈ R + where R + and N ∗ are the set of real positive numbers andpositive integers, including zero, respectively. In summary, from the above analyses, we canconstruct the following statement, Statement 1 : Any first-order TPT has got to go through C T QPT at constant T ,which will be shown to be true in the subsequent section.15 .3. Finite-temperature continuous quantum phase transition Thus far, we have exposed the existence of C T QPT during LiCl solidification at T = T θ (see the vertical arrow in Fig. 2)(b), which is commonly accepted as the first-order TPTwithout going into the details of C T QPT. Proving
Statement 1 requires one to track C T QPTor the changes in ξ ( t ) during solidification for constant T = T θ . Alternatively, one may alsoprove Statement 1 by tracking the horizontal arrow pointing left shown in Fig. 2(b) bysystematically changing the chemical composition via substitutional doping of Cl with Brand followed by I such that LiCl → LiBr → LiI at constant T = 552 o C, which is the meltingpoint for LiBr. We will first address C T QPT occurring during solidification (for constant T = 610 o C).The above C T QPT during solidification of LiCl system can be captured by isolating thesystem right at the critical point ( T θ = 610 o C) as shown schematically in Fig. 3. This systemhas been isolated at the critical point when T = T θ = 610 o C such that the solid phase coexistsindefinitely within the liquid phase with a dynamic phase boundary between those twophases. The magnified sketch beneath the main diagram shows the temperature differencesbetween the solid phase ( T sol ) and its surrounding liquid phase ( T liq ) such that T liq > T θ >T sol and T liq − T sol (cid:28) T θ . At the boundary, the system is in extreme nonequilibrium whereLi or Cl from the liquid phase may react to form a rigid ionic bond, and release heat ( − Q )as energy. Conversely, the same amount of heat is absorbed (+ Q ) by Li or Cl in the solidphase so as to break free from the solid phase in order to be part of the liquid phase. Here,the liquid phase Li and Cl are indicated with filled circles, while the same elements in thesolid phase are drawn as circles, but we did not bother to completely fill the solid phasewith circles. These processes are due to chemical reactions between the highly-polarizedLi (small ξ Li ) and the least-polarized Cl (large ξ Cl ) giving rise to the chemical associationbetween them, forming the solid phase, which releases energy as heat ( − Q ) into the liquidphase. This energy transfer increases the kinetic energy of the liquid-phase Li and Cl thatwill eventually collide onto the solid-particle surface with increasing frequency, and thuscould transfer this energy (+ Q ) back into the solid phase to initiate melting of LiCl solid.These two thermal-assisted processes (due to ± Q ) are the causes for this thermal-assisted (orfinite-temperature) C QPT. In a macroscopic point of view, this particular isolated system,containing both solid and liquid phases, is in equilibrium because the average rate of melting16nd solidification is the same, hence the solid-to-liquid and liquid-to-solid transitions ( C T QPT)are in balance. But we need to go deeper to track the physico-chemical processes at thesolid | liquid interface, which can be done with IET. For instance, the energy level spacings inthe liquid and solid phases are ξ LiClliquid and ξ LiClsolid , respectively, and since the valence electronsin the liquid phase are all in the excited states (thermally polarized) then this implies ξ LiClliquid < ξ
LiClsolid . This inequality is strictly valid because the excited energy-level spacings are alwaysnarrower due to weak electron-electron ( e - e ) interaction in the presence of weak electron-nucleus ( e - nuc ) attraction. Conversely, large energy level spacings are inevitable for theenergy levels close to nucleus [7].Substituting ξ LiClliquid < ξ
LiClsolid into Eq. (26) leads to liq S + ξ > sol S + ξ and liq S + D > sol S + D , whichthen allow one to correctly conclude S liquid > S solid where S liquid = liq S + ξ + liq S + D and S solid = sol S + ξ + sol S + D . However, as we have noted earlier, S solid < S noneqm < S liquid is invalid becausewe need to take the nonequilibrium effect (at the solid | liquid interface) into account. Wenow know that this effect occurs maximally at the critical point (when T = T θ ), and atthe solid | liquid interface where S noneqm = S liquid + S solid + S interface , and therefore S solid < S liquid < S noneqm that guarantees the existence of first-order TPT, while S interface on the other hand,ensures the existence of C T QPT, as well as the coexistence of both solid and liquid phases(see Fig. 3). This means that if S interface = 0, then T (cid:54) = T θ and consequently we have S noneqm → S liquid for T > T θ or S noneqm → S solid for T < T θ . Using Eq. (23), the non-divergent andrenormalized ˜ C v ( T θ , t ) noneqm = C v ( T ) exp (cid:40) − λ (cid:20) (cid:88) j J j ξ solid + (1 − J j ) ξ liquid (cid:21)(cid:41) , (28)in which, we have defined ξ ( t ) = (cid:80) j J j ξ solid + (1 − J j ) ξ liquid where j = { t , t , · · · , t n } , J = X LiClsolid / X LiCltotal and J ∈ [0 , X LiClsolid is the number of Li and Cl atoms in the solidphase only, and X LiCltotal is the total number of Li and Cl atoms in both liquid and solidphases (or in a given system). Moreover, J does not necessarily increase with time, if theheat-exchange fluctuates ( ± Q ), and the total time between t and t n is the time takenfor a complete solidification of liquid LiCl for constant T = T θ . Eq. (28) strictly impliesthat the magnitude of ˜ C v ( t ) noneqm does not change with T only, but also with respect tochanges in the interaction-strength parameter, ξ ( t ). In particular, ˜ C v ( t ) noneqm can change dueto changes in ξ ( t ) from ξ liquid to ξ solid during solidification of LiCl liquid. In this latter case,the heat-exchange as depicted in Fig. 3 is solely used to change the interaction strength via17 solid (cid:10) ξ liquid .If one employs an unrenormalized specific-heat equation, then it is always divergent for T = T θ because it is undefined at this critical point. On the other hand, Eq. (28) is well-defined such that ξ can be exploited at melting points, without any divergence. For example, ξ liquid and ξ solid are constants for T > T θ and T < T θ , respectively, and any heat exchangesthat may exist between a given system and its surrounding only decrease or increase thesystem’s temperature. At the critical point however, the heat exchanges are used only tomodify the system’s physico-chemical properties, hence, the system’s temperature remainsconstant. In other words, the renormalized specific-heat equation (Eq. (28)) reduces to˜ C liquid v ( T ) = C liquid v ( T ) exp (cid:20) − λξ liquid (cid:21) , (29)for liquid LiCl ( T > T θ ), and for solid LiCl ( T < T θ ), one just need to switch the label ‘liquid’with ‘solid’ in the equation above. However, neither of these equations can be applied when T = T θ . We need Eq. (28) for T = T θ . This completes the proof for Statement 1.If we now follow the horizontal arrow given in Fig. 2(b), we can show that both transitions(for vertical and horizontal arrows) are thermal-assisted C T QPT, one occurring during solid-ification (discussed above for the vertical arrow) and the other originates due to changingchemical composition (at a constant temperature). For example, the first C T QPT at the crit-ical point during solidification (constant T = T θ ) is initiated by the heat-exchange betweenliquid and solid LiCl, where d S = δQ/T θ , and from Eq. (16),d | S ξ | = d S = δQT θ = (cid:12)(cid:12)(cid:12)(cid:12) − ( E ± d ξ ) + E T θ (cid:12)(cid:12)(cid:12)(cid:12) , | S ξ | ∈ R + . (30)Equation (30) correctly implies: (a) increasing absorption of heat in a system increasesthe entropy of that system, or vice versa , (b) any amount of change in Q ( δQ ) or ξ (d ξ )corresponds accordingly to a change in entropy, as it should be, however, (c) the entropyof a given system decreases if ξ increases, as strictly required by Eq. (16), (d) systems withlarge ξ need large amount of heat to initiate changes such as melting, for example, fromTable 2, we know [ ξ Br < ξ Cl ] → [ ξ LiBr < ξ
LiCl ] and therefore T LiBr θ < T LiCl θ , and (e) largeamount of heat is required to be removed or added to initiate large changes to ξ , for example ξ LiClliquid (cid:10) ξ LiClsolid . Here, ξ LiClliquid is a constant, while Q is the amount of heat removed to initiatethe change, ξ LiClliquid − Q → ξ LiClsolid . Therefore, (d) refers to entropy-change of a system priorto any phase transition, i.e. , for constant ξ and T < T θ or T > T θ . In contrast, (e) reveals18hanges in the intrinsic physico-chemical properties of a system due to changing ξ when T = T θ . This means that d | S ξ | is only valid at the critical point, or when T = T θ because ξ does not change with T , but it changes significantly when the physico-chemical propertiesof a given system change.In order to understand the existence of thermal-assisted C QPT due to doping (followthe horizontal arrow in Fig. 2(b)), we increase Br content, replacing Cl for constant T toobtain a system defined by LiCl − a Br a . The inequality, ξ Br < ξ Cl implies ξ decreases withincreasing Br content where this doping is carried out for constant T = 552 o C. Now, thecritical point can be obtained for a = 1 and at this point, T = T θ = 552 o C, which is themelting point of LiBr. As a consequence, if ξ LiClsolid → ξ LiBrsolid is achieved through doping at T = T LiBr θ , then Q activates the melting process, such that ξ LiBrsolid + Q → ξ LiBrliquid . However, wepoint out that Q = ξ LiBrliquid − ξ LiBrsolid is false. Note here that the above stated doping-induced C T QPT can occur for any constant temperature, and even for T = 0K.In summary, the so-called activation energy in this chemical association should be equalto Q , which is required to complete the liquid-to-solid transition, or vice versa . In otherwords, to complete ξ LiClliquid → ξ LiClsolid or ξ LiBrsolid → ξ LiBrliquid transformation, significant changes to t -dependent many-body wavefunction are necessary, for example, Ψ( t ) LiClliquid → Ψ( t ) LiClsolid . Theexistence of such a transformation in Ψ( t ) has been proven within a new quantum adiabatictheorem developed for chemical reactions [5], which can be used to understand why thewavefunction of unreacted species need to be combined linearly or written in a different formfor the compounds formed after chemical reaction. Interestingly, M¨uller and Goddard [28]have also pointed out such a case must exist during chemical reaction. Thus far, we haveexploited the formalism developed for IET such that the only a priori information onerequired to know is the atomic energy-level spacings listed in Table 2.The technical steps taken to prove the existence of C T QPT required us to first construct
Statement 1 . But before constructing the statement, we have first explained why and howthe first-order thermal phase transition during melting or solidification of alkali halides canbe exactly recaptured with quantum phase transition. For example, in Further Analysis I,we have explained that thermal energy at a critical point (during melting or solidification)can cause fluctuation in ξ , which could give rise to C T QPT. To prove the fluctuation in ξ ,we derived the entropy-change due to electron (see Eq. (15)) and ion (see Eq. (21)) at thecritical point, which are the causes for ξ to fluctuate. Equations (15) and (21) show that (i)19ero-entropy is not required for C T QPT and (ii)
Statement 1 can be constructed becausethe changes to the electronic energy levels (with contributions from electrons and ions) arecaused by thermal energy leading to the wavefunction or ξ solid → ξ liquid transformation.Subsequently, we moved on to show that the inequality, S solid D < S liquid D < S noneqm is re-sponsible to label melting or solidification as the first-order TPT. By digging deeper intothis solidification process, we have proved that ξ continuously switches between ξ liquid and ξ solid due to wavefunction transformation for T = T θ , which unambiguously implies thatthe above process (melting or solidification) satisfies C T QPT such that TPT (independentof wavefunction) is a special case. Next, we derived the renormalized specific heat formulain terms of entropy, which is used to expose that the fluctuation in ξ (at a critical point)forces us to impose time-dependence into the energy level spacing such that ξ → ξ ( t ) (seeEqs. (23) and (28)), and these equations properly prove the correctness of Statement 1 .In addition, we have explained why and how S ξ (see Eq. (30)) can be used to further sup-port that ξ ( t ) liquid → ξ ( t ) solid is equivalent to Ψ( t ) liquid → Ψ( t ) solid that further reinforcesthe correctness of Statement 1 . Finally, our theory and
Statement 1 has been used toexplain the physical processes at the critical point in these systems, namely, alkali halidesand water.
We apply Eq. (28) to water-to-ice thermal phase transition during freezing. Figure 4sketches the specific heat versus temperature curve for constant pressure ( C p ( T )) for water(see the dashed line). The dashed line denotes the usual C p ( T ) curve with a sudden drop( C water p ( T ) > C ice p ( T )) at the freezing point. Whereas, the solid line in Fig. 4 captures thewhole mechanism of phase transition from water to ice, including the freezing curve occurringat the freezing point (273.16K). Obviously, the freezing curve is always hidden at the freezingpoint and appears as a sudden drop in C p ( T ) measurement, when temperature is lowered.Unfortunately, we do not know the explicit microscopic equation for C p ( T ) = d H / d T | P where H is known as the enthalpy. However, the fundamental energy-level spacing renormalizationfactor (the exponential term in Eq. (28)) for ˜ C v ( T θ , t ) noneqm should remain intact for ˜ C p ( T θ , t ) noneqm C p ( T θ , t ) noneqm = C p ( T ) exp (cid:40) − η λ (cid:20) (cid:88) j J j ξ ice + (1 − J j ) ξ water (cid:21)(cid:41) . (31)The only change in the renormalizing factor is the numerical constant, 3 in 3 λ/ η in Eq. (31). This numerical constant, η can only be obtained by solving C p ( T ) =d H / d T | P , but it is irrelevant here. What we actually need are the renormalized specific heatequations for water and ice,˜ C water p ( T ) = C water p ( T ) exp (cid:20) − η λξ water (cid:21) , (32)˜ C ice p ( T ) = C ice p ( T ) exp (cid:20) − η λξ ice (cid:21) . (33)In Eqs. (32) and (33), only C water p ( T ) and C ice p ( T ) are T -dependent variables, and therest are just T -independent variables or constants. This means that Eq. (32) captures thespecific heat for water, whereas, Eq. (33) is for ice (see dashed and solid lines for water andice in Fig. 4). The sketched freezing curve however, follows Eq. (31) and is smooth here,implying there is no fluctuation in Q during freezing, i.e. , there is a continuous extraction ofheat from the water-ice system. Recall here that the freezing curve in Fig. 4 is not observablefrom the C p versus T measurement alone, but exists as a sudden drop in the specific heatfor T = 273 .
16K (freezing point) because C water p ( T ) > C ice p ( T ) or ξ water < ξ ice . This is thereason why temperature is not a proper variable to monitor at the freezing point of anythermally-driven finite-temperature continuous quantum phase transition.In summary, we have proven that ξ ( t ) is the proper quantum variable to capture what isreally happening at the freezing point. For example, at the freezing point, the heat that isbeing extracted from the water-ice system, activates the energy-level spacing transformation, ξ water → ξ ice or the wavefunction transformation Ψ water → Ψ ice due to the formation ofpermanent hydrogen bonds. This transformation results in C water p ( T ) > C ice p ( T ) that implies ξ water < ξ ice . The latter inequality has its origin in hydrogen bonds. For example, both ξ water and ξ ice refer to energy-level spacings due to hydrogen bonds and not due to covalentbonds between O and H in a H O molecule. Obviously, the energy-level spacings due tohydrogen bonds are larger for ice compared to water because the hydrogen bonds in ice21s permanently bonded (static) and therefore, its strength is larger in ice than in water.The hydrogen bonds in water phase is dynamic such that the bonds are broken and formedrandomly. As such, indeed ξ water < ξ ice . Now for water vapor, all H O molecules are isolated,and they do not form any hydrogen bond. As a result, ξ vapor refers to energy-level spacingsdue to covalent bonds (between O and H) within a H O molecule. Since covalent bonds aremuch stronger than hydrogen bonds, one has ξ water < ξ ice < ξ vapor that correctly corroborateswith experimental observations [29, 30], C water p (4.187 kJkg − K − ) > C ice p (2.108 kJkg − K − ) > C vapor p (1.996 kJkg − K − ). This concludes our analytical proofs and technical analyses onH O system, again supported by experimental results.
4. Conclusions
The finite-temperature continuous quantum phase transition ( C T QPT) has been formallyproven to exist and to be responsible for the thermal phase transition during melting orsolidification, as well as during substitutional doping for constant temperature. Along theway, we also have proven that first-order phase transition obtained from constant-pressurespecific heat data between water and ice unambiguously satisfies the notion of C T QPT devel-oped here. In view of the analysis for alkali halide salts and water, we found that standardfirst-order thermal phase transition is a special case within C T QPT. The equations developedherein allow one to obtain the precise physico-chemical mechanisms (without any diver-gences) right at the melting point of a given solid, beyond the usual physics of discontinuous(sudden change) thermal phase transition, specifically, in the first-derivative of specific heatcapacity. Therefore, we can also claim that chemical reaction is a quantum critical pointphenomenon that can be associated to the finite temperature continuous quantum phasetransition.
Acknowledgments
I thank Madam Sebastiammal Savarimuthu, Mr Arulsamy Innasimuthu, Madam AmeliaDas Anthony, Mr Malcolm Anandraj and Mr Kingston Kisshenraj for their financial supportand kind hospitality between August 2011 and August 2013. I also thank Dr Naresh KumarMani for his kind hospitality during my short stay in Cachan, France (March/April 2011)22here part of this work was completed. [1] S Sachdev
Quantum Phase Transitions (New York : Cambridge University Press) Ch 1, p 3(1999)[2] T Vojta
Ann. der Phys.
403 (2000)[3] M Vojta
Phil. Mag. Phil. Mag. B Prog. Theor. Phys.
577 (2011)[6] F Isik, M A Sabaner, A T Akan and A Bayri
Indian J. Phys.
241 (2013)[7] A D Arulsamy
Pramana J. Phys.
615 (2010)[8] A D Arulsamy
Ann. Phys.
541 (2011)[9] A D Arulsamy
J. Chem. Sci. to be published (2014)[10] G N Ramachandran, V Sasisekharan and C Ramakrishnan
J. Mol. Biol.
95 (1963)[11] G N Ramachandran and V Sasisekharan
Adv. Prot. Chem.
283 (1968)[12] C Ramakrishnan and G N Ramachandran
Biophys. J.
909 (1965)[13] R Shankar
Physica A
530 (1991)[14] R Shankar
Rev. Mod. Phys.
129 (1994)[15] R Shankar
Phil. Trans. R. Soc. A
Phys. Rev. Lett. Indian J. Phys.
395 (1928)[18] C V Raman
Indian J. Phys.
387 (1928)[19] C V Raman and K S Krishnan
Nature
501 (1928)[20] C V Raman and K S Krishnan
Indian J. Phys.
399 (1928)[21] M J Winter (cid:104) (cid:105) (2011)[22] A D Arulsamy, K Elerˇsiˇc, M Modic, U Cvelbar and M Mozetiˇc
Chem. Phys. Chem. Phys. Chem. Chem. Phys. J. Supercond. Nov. Magn.
785 (2009)
25] A D Arulsamy
Phys. Lett. A
413 (2005)[26] S C Gairola
Indian J. Phys.
967 (2012)[27] L Cemiˇc
Thermodynamics in Mineral Sciences (Berlin : Springer) Ch 6, p 231 (2005)[28] R P M¨uller and W A Goddard
Valence Bond Theory (New York : Academic Press) Reprintedfrom the Encyclopedia of Physical Science and Technology (2002)[29] V Tchijov
J. Phys. Chem. Solids
851 (2004)[30] S V Lishchuk, N P Malomuzh and P V Makhlaichuk
Phys. Lett. A able 1: Experimental values for the melting points and diatomic bonding energies of saltsobtained from Ref. [21]. The systematic decrease of melting points and diatomic bondingenergies with respect to anions (from F to I with increasing Z ) satisfy the decreasingionization energies for the same anions, from F to I. See text and Table 2 (follow the valuesmarked with “ * ”) for details. Table 2:
Averaged atomic ionization energies ( ξ ) for individual ions and their re-spective valence states ordered with increasing atomic number Z . All experimentalionization energy values were obtained from Ref. [21]. Figure 1: (a): The sketched atoms ( X and Y ) are equally polarizable, neutral andidentical with discrete energy levels. (b): Atomic X has the least polarizable electron ( e X ),while e Y from atomic Y is easily polarizable. Figure 2: (a): Melting points of salts are plotted against the ionization energies ofanions (I, Br, Cl and F). (b): Melting points of Li(I, Br, Cl and F) versus anionsionization energies. The vertical down-arrow (solidification) denotes the first-orderthermal phase transition (TPT: from liquid to solid) for LiCl, while the horizontal left-arrow denotes quantum phase transition (QPT: from solid to liquid) at constant T = 552 o C. Figure 3:
Both atomic Li and Cl in liquid phase are denoted with filled circles,while circles represent ionic Li and Cl in solid phase.
Figure 4: