To the Question of Validity Grüneisen Solid State Equation

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1 Word Journa of Condensed atter Physics, 0,, Pubished Onine ovember 0 ( o the Question of Vaidity Grüneisen Soid State Euation Vadimir Kh. Kozovsiy Jewish Scientific Society, Berin, Germany. Emai: Kozovsiy.V@gmx.de, os@utranet.mai.ru Received August 7 th, 0; revised September 9 th, 0; accepted September 7 th, 0 ABSRAC he first justified theory of soid state was proposed by Grüneisen in the year 9 and was based on the viria theorem. he forces of interaction between two atoms were assumed as changing with distance between them according to inverse power aws. But ony viria theorem is insufficient to deduce the euation of state, so this author has introduced some reations, which are correct, when the forces ineary depend on dispacement of atoms. But with such aw of interaction the phase transitions cannot tae pace. Debye received Grüneisen euation in another way. He deduced the expression for thermocapacity, using Pan formua for energy of harmonic vibrator. aing into account the dependence of atomic vibration freuency from distance between atoms, when the forces of interaction are anharmonic, he received the euation of state, which in cassica imit turns to Grüneisen euation. he uestion, formuated by Debye is How can we come to phase transitions, when Pan formua for harmonic vibrator was used? Debye soved this uestion not perfecty, because he was born to sma anharmonicity. In the presented wor a chain of atoms is considered, and their movement is anaysed by means of reations, euivaent to viria theorem and theorem of Lucas (disappearing of mean force). Both are the resuts of variation principe of Hamiton. he Grüneisen euation for ow temperature (not very ow, where uantum expression for energy is essentia) was obtained, and a famiy of isotherms and isobars are drown, which show the existence of spinodas, where phase transitions occur. So, Grüneisen euation is an euation of state for ow temperatures. Keywords: Grüneisen Euation; Phase ransitions; Atomic Chain; Atomic Interactions. Introduction he moecuar-dynamica soid state euation, based on viria theorem, was derived by Sutherand [], ie [], Grüneisen [,]. Another way of deriving, based on the Debye theory of heat capacity at ow temperatures [5], which used the Pan expression for the energy of harmonic vibrator, was chosen by Ratnowsy [], Eisenmann [7], Grüneisen [8], Debye [9,0]. In cassica imit the uantum euation of state turns to cassica Grüneisen euation. But here arises a uestion, which Debye formuated in such manner: because an expression of harmonic vibrator energy is used, can the Grüneisen euation be regarded as euation of state that describes phase transitions, which are non inear phenomena? In cassica deduction of euation of state Grüneisen besides the viria theorem (which aone is insufficient to deduce euation of state) used some euaities that are stricty correct for forces that depends ineary from dispacement of atoms. Debye tried to sove the formuated uestion, but, because he was based on reations correct for sma anharmonicity ony, the anaysis cannot be regarded as compete. We empoy another way for euation of state deduction that begins from variation principe of Hamiton, which can be a base for deriving of the viria theorem []. he approximations, used in soving of euations, are not burned to sma anharmonicity. Here is given a brief account of euations deduction, which in detais is pubished esewhere []. he cacuations for atomic chain show that Grüneisen euation is vaid as euation of state for no high temperatures. he famiies of isotherms and isobars iustrate the picture of phase transitions in the chain.. he Basic Euations of Soid State he deduction of viria theorem from Hamiton variation principe is briefy described beow. he variation of mean time Lagrange function L is transformed as foows t t L L d t t t t t t L L t dt t L L L dt t t t t t t () he first term of ast expression is zero as conseuence Copyright 0 SciRes. WJCP

2 0 o the Question of Vaidity Grüneisen Soid State Euation of movements euations, the second for infinite time interva. he product of a coordinate with variabe factor represents a varied coordinate:, so,. hese variations substituted in the eft side of ast Euation () give a reation L L 0 () If the inetic energy W does not depend on coordinates (in crystas rectiinear coordinates of atoms are used), it foows the viria theorem W W he represented variation procedure may be generaised. he variabe coordinates are represented in the form, where, are variabe parameters (further generaization of variation procedure is described in []). he variations of coordinates and veocities are expressed as, and after substitution in () a reation foows () L L L 0 () So the euations are obtained L 0 (5) L L W () where the theorem about uniform distribution of inetic energy is used. Further step in cacuations consists in representation of coordinates as functions of time in the form s u () t (7) where s are the mean vaues of coordinates, u is root mean suare ampitude of vibrations, so the vibration functions satisfy the reations t t (8) 0, If the inetic energy depends ony on veocities then (5) is represented as 0 (9) s and L u s u (0) s u Introducing in () taing into account (7) and (9) we obtain u () u If the system of atoms interacts with externa corps, the positions of which are determined by means of coordinates ai, the mean forces acting on the system are determined by expressions Ai () a By shifting of the corps the forces produce a wor, so the variation of energy by variation of a parameters is given by the expression H W () s u Aiai s u i aing into account (9), () a reation wi be obtained i H Aia i n u () It can be seen that the uantity which ti now is ony the indication for mean inetic energy, an integrant divisor for the uantity of heat is, hence proportiona to absoute temperature. he expression for entropy is the foows S n u (5) We introduce a function of variabes as an anaogy of thermodynamic potentia n u Aa () i i i (items depending on temperature ony are omitted). he condition of minimum Ф by constant temperature eads to Euations (9), (), () and aows to choose stabe vaues of variabes. In a crysta the mean vaues of dispacements and ampitudes are eua for atoms beonging to the same subattice, so the number of unnowns may be not great. Because in crystas the rectiinear coordinates are empoyed, the inetic energy wi not depend on coordinates.. Anharmonic Vibrator with wo Euiibrium Positions Additiona information about dynamica variation procedure is obtained through considering of a simpe an- Copyright 0 SciRes. WJCP

3 o the Question of Vaidity Grüneisen Soid State Euation harmonic vibrator with coordinate x and potentia energy of the form c b x x ; c0, b0 (7) which is represented on Figure It is necessary to determine mean time potentia energy, so it wi be written x s u t (8) (9) x s su t u t x s s u t s u t su t u t (0) x s u ; x s s u u () where is put 0, because t is an accidenta function with zero mean vaue, so with eua probabiity receives positive and negative vaues, and in such manner behaves t. he mean vaue of is /, if the function t is harmonic, but is greater, if on the path is potentia barrier, which reduces the speed. Later we show what vaue of is worthwhie to put in euations. So c b s u s s u u () he euations of state wi be written as foows 0; s c bs bu 0 () s u It foows two soutions u ; cu bs u bu s 0; cu bu () (5) c bs bu 0; cu 9 bu () he first soution corresponds to vibrations above the potentia barrier, the second to vibrations in eft or right potentia pit. If we put =, then the eft sides of second euaities in (5), () differ ony through mutipier, conseuenty intersect the abscissa axis in the same point. For this case the Euations (5) and () are represented graphicay on Figure. Parts of the curves that have physica meaning, correspond to positive temperature. Heating or cooing eads to transitions from one curve to the other, which is manifested in jumps of ampitude. Braunbe [] considered the phenomena of crysta meting, using the euation of subattice motion. he potentia curve was represented by a sinusoida function of coordinate. he integration coud be fufied with the hep of eiptica functions. By ow temperatures the atoms moves in one of minimums of potentia curve, by more high temperature above the maxima. his corresponds to transition from crystaine to non crystaine state. On the Figure is shown the cacuated dependence of the temperature from the energy of vibrations. he ampitude increases with energy, so it exists a possibiity to compare the graphs, received with both methods of cacuation. he graphs are simiar, and the point of intersection of curves, describing finite and infinite movements (in our mode aso finite, but taing more pace in the space) ies on abscissa axis. So, it is reasonabe to accept. A serious uestion arises by comparing our cacuations with cacuations of Syrin []. On the Figure is shown a form of potentia pit: symmetrica doube minima rectanguar one. Syrin cacuated the dipoe moment in eectric fied for charged atom by means of formua for statistica distribution, integrating over a space of the pit. It turned out that the dipoe moment continuousy decreases with X Figure. Potentia pit with two minima. Figure. Reations between temperature and ampitude for vibrations; Above the barrier; In the region of minimum. Copyright 0 SciRes. WJCP

4 o the Question of Vaidity Grüneisen Soid State Euation W Figure. Reation between temperature and energy according []. Figure. Graph of potentia pit according []. increasing temperature (as poarisabiity for therma poarization according to Debye formua). So, a discrepancy can be mared between dynamica and statistica cacuations, but reay this discrepancy is iusory, because such uestion was considered in the iterature. In the course of Frene [5] is discussed a metastabe state with concusion, that such state corresponds to a region of phase space, where the moecuar system exists ong time and can be observed, then for determination of phase integra the integration must be extended over this region. But Syrin integrates over a space, which incudes both minima and potentia barrier. his means that in no part of the space the atom deays for a ong time, but jumps from one minimum to the other. ode of Syrin corresponds to a hoe in a crysta, where the atom is not bound tight to one minimum. g x on x-axis, where atoms interact with nearest neighbours. he extreme eft atom with number 0 is fixed in the origin and on the extreme right atom a constant force f directed aong the axis is acting (Figure 5). Atom, which has the number, interacts with eft and right atom with numbers and +, which are paced on distances, from the considered. he energy of interaction for the whoe chain is represented by the sum (7) If the mean time distance between neighbouring atoms is identica for a atoms, then the distance in moment t wi be written as u t (8) where the mean suare ampitude is written aso as identica for a atoms. he energy of interaction is further expended in power series with respect to ampitude and the order of derivative is mared by means of Roman numeras I u t u t (9) I IV u t u t For taing the mean time vaue of (9) the reations (8) must be used and the mean time of (9) has the vaue IV u u (0) For the function determined in () and cacuated for one atom we receive, taing into account (0), an expression u () IV u n u f where f is a constant stretching force (hanging oad). he terms depending ony on temperature are omitted. he conditions of minimum have the form u u f I I V IV 0 () u u u 0 () u f. Euations of State of an Atomic Chain K- K K+ x It wi be considered a chain of identica atoms situated Figure 5. he atomic chain. Copyright 0 SciRes. WJCP

5 o the Question of Vaidity Grüneisen Soid State Euation If we negect the terms containing fourth power of ampitude, an euation wi be received I I f () his is Grüneisen type euation. ow we deduce an euation with regard to fourth power of ampitude. We mutipy () with IV and () with V and sub- tract the second from the first, than it turns to be I IV I IV V u IV V f 0 he Euation () wi be soved with respect to IV IV u u (5) () he radica is taen with positive sign, because the ampitude increases with increasing temperature and IV 0, which wi be cear ater. Expanding the radica and retaining terms uadratic in temperature, we receive. u Introducing in (5), we obtain the euation of state IV (7) I V I IV I f (8) Further cacuations concerning the transitions in the chain wi be carried out retaining ony inear in temperature term. 5. he Euation of State for Forces, Changing with Distance According to Inverse Power Law he potentia energy of two atoms, one of which is paced in the origin, other on the x-axis on the distance wi be represented by means of a usua function n m (9) n m which may be written in more convenient form n m n m (0) his expression is mathematicay euivaent to given in []. he minimum energy is dispaced on distance, determined through euation n m () n n m m he force constants may be expressed through, n m n, () n m It foows the expression for the energy in the minimum n m () nm Further we find n m I n m m n n I n n m m m () (5) () As can be seen, in the vicinity of the point it must be 0, because n m, and the same is true for IV. hese vaues for derivatives of interaction energy substituted in () give the Grüneisen euation for accepted aw of interaction. We undertae now a mathematica investigation of this euation for the purpose to receive euations for isotherms and isobars and anayse their behaviour in order to discover transitions. As was noticed [7] the cacuations are most simpe for vaues n =, m = that we introduce in above expressions and use the notation (7) he expressions for energy and its derivatives are the foows (8) I (9) 7 (50) I (5) Copyright 0 SciRes. WJCP

6 o the Question of Vaidity Grüneisen Soid State Euation he Euation () now turns to the form 7 f 7 (5) As the mutipier in front of the bracet varies sowy with the ength, we put approximate euation of state and accept foowing 7 f 7 (5) From this euation, as is usua in physica chemistry, a famiy of isotherms and isobars wi be deduced.. Isotherms of the Atomic Chain he expression in bracets, incuding the sign in front of it, consists of two items he first is represented as a paraboa, that goes to negative infinity intersecting abscissa axis in points = 0 and =, the top of which arises above abscissa axis on 0.5. he second item is a fraction where numerator and denominator are inear functions of and it aspires to constant negative vaue when goes to infinity, by diminishing of aso diminishes ti the vaue = 7/, where the fraction turns to negative infinity. So the function (5), when the temperature does not exceed definite vaue, is represented ti this point by a curve with roots, ying between vaues of abscissa 0 and, and with top paced above the abscissa axis. By rising of the temperature the roots and the top coincide on the abscissa axis, and farther the chain can exist ony under the action of negative (compressing) force. As it was noted in [7], the soid corps must resist to the action of stretching force, otherwise it is not a soid corps. he famiy of isotherms is depicted on the Figure. he physica meaning has parts of isotherms to the right side of the top, where condition of stabiity f 0 is fufied [7]. In the top of the curves the transition in other phase taes pace, but the structure of new phase is not described by the present considerations. It may be noticed in passing, that the branch of the curve to the eft side of the top approaches the abscissa axis more abrupty, then that to the right side because the third derivative in the top is positive (when it is so, then in the vicinity of the top f f top top 0 Expanding in series with respect to, we arrive at the formuated condition). Each curve on Figure is characterised by the position and height of maximum and position of intersection points with abscissa axis. he position of maximum m is determined by the euation f 7 Figure. Famiy of isotherms of chain euation of state. 7 m m 7 9 g (5) If this expression of temperature wi be substituted in Euation (5), then the euation of the curve, connecting maxima of the isotherms (spinoda) wi be obtained f mm (55) m m m 7 9 he right side decreases with increasing of the abscissa from the vaue , where the eft side has the vaue 9 0.9, ti the intersection point with abscissa axis, where the top of the isotherm reaches it. As numerica cacuation shows, here m m 0.75; 0.00 (5) he diagram (f, ) is represented in parametric form by expressions (5), (55). ow we determine the positions of the points, where the isotherm branches intersect the abscissa axis. According to (5) one must put f = 0, then sove the euation 7 7 (57) We sha not sove this cubic euation to express the abscissa as a function of temperature, but sha confine us with consideration of soving procedure. he right side is a curve, intersecting the abscissa axis in the points = 7/ and =, between them has a maximum. he eft side is a straight ine parae to abscissa axis, that at no high temperatures intersect the curve in two points, corresponding to roots of Euation (57). By rising of the temperature the roots converge ti the straight ine touches the top of the curve, then the roots and point of maximum coincide. he corresponding vaues of abscissa and temperature are (5). Copyright 0 SciRes. WJCP

7 o the Question of Vaidity Grüneisen Soid State Euation 5 7. he Isobars of Atomic Chain From Euation (5) foows the expression for temperature 7 7 f (58) he right side is a product of a fraction with negative sign, whose numerator and denominator are inear functions of abscissa, and a uadratic function. he graphs of mutipiers are represented on Figure 7 (the etter f on vertica axis means function). he temperature is positive, so physica meaning has the interva on abscissa axis, where the ordinates of both curves have the same (negative) sign that means interva between points, where the fraction and paraboa intersect the abscissa axis. he first point is fixed; the second depends on the vaue of the force and may disappear (if paraboa is paced above the abscissa axis). So, the famiy of isobars has the form, shown in Figure 8. Ony the part to the right side of the top, where 0, have physica meaning, because here the ength of the chain increases with increasing temperature, what is designated as therma diatation (notation 7). he position of the point Q, where paraboa intersects the abscissa axis is given by the expression f Q (59) With increasing force this point moves toward the point with abscissa 7/ and coincides with it by the vaue of the force f f 7 Q Q Q Figure 8. Famiy of isobars of chain euation of state. f 9 By this vaue the isobar disappears, and the system of atoms does not exist as reguar structure. For determination of maximum the expression (58) wi be differentiated, and the resut is f mm (0) m 7 m m 9 his euation has the form of Euation (55). he vaue of temperature maximum on the curves, shown in Figure 8, we receive, if substitute the vaue of force (0) in (58), where the vaue of abscissa is m. he temperature maximum is 7 m m 7 () 9 and has the same mathematica form as (5). It is the euation of spinodae that joints the points of maxima on Figure 8. Z 7 Q 8. he Isobars emperature Ampitude he initia euations are (7), (50), (58) and the probem consists in excuding the uantity. his is made as foows: Excuding the temperature eads to the euation for (here ) 9w ) 8w p 0 () Figure 7. Graphs of mutipiers in Euation (58). where w u f. p he soution has the Copyright 0 SciRes. WJCP

8 o the Question of Vaidity Grüneisen Soid State Euation form where 8w R 9w 8 9 R w w p () Introducing () in (7), eads to the expression for temperature w 8wR 90w R () 9 w he first mutipier turns to zero, when w = 0, the ast mutipier when p = /9 turns to zero aso. When p = 0, the ast mutipier is zero at w = /9. It can be proved directy that for p aying between these extreme vaues the ast mutipier is zero at 9 w p 9 (5) So, the temperature has two roots with a maximum between them. he dependence of the temperature from the ampitude is shown in Figure 9. Physica meaning has increasing part of the curve from zero to maximum. In the vicinity of the maximum the temperature is neary constant, so the suare of vibration freuency, which is proportiona to the fraction, nearing to the maximum. Such concusion is formuated in Reference [8]. Givarry states that root mean suare ampitude divided through mean distance between atoms at transition point has the same vaue for a substances with the same structure [9] and the same assertion is found in the artice [0]. Such concusion foows aso from presented cacuations. From (7) and (50) foows u w, diminishes with 7 () With using the expression (7) for this euaity turns to Figure 9. Famiy of isobars temperature ampitude. P W u 7 (7) By means of euation of state (58), where is put f = 0, this euation transforms to u (8) 7 o receive the vaue of in the transition point, one must differentiate (58) and the derivative euate to zero. nder the condition f = 0 definite vaue of and conseuenty definite vaue of fraction u wi be obtained. 9. Concuding Remars On the base of dynamic euations, that foow from variation principe of Hamiton, a therma behavior of inear chain of atoms, interacting with nearest neighbors with forces, obeys inverse power aw, and stretched with externa force was considered. he mean potentia energy is deveoped in series with respect to the ampitude of atomic vibrations and ony second powers were retained. his restricts the temperature from above, and the Grüneisen type euation of state was obtained. he famiies of isotherms and isobars were drowing, and spinodas are mared. So, Grüneisen euation is an euation of state, which describes phase transitions. his is the answer to the uestion, formuated by Debye: Is the Grüneisen euation an euation of state? he origina cacuations contain assertions, which are stricty correct for inear dependence of interacting forces from dispacements of atoms, so no transitions can occur. It is a correct reasoning, and if we put in Euations (7), (8) the numerica vaues (5), we come to resut that inear and uadratic in temperature items are neary eua, so at such temperature uadratic terms cannot be ignored, and Grüneisen euation is an approximation usefu for ower temperature. In a discussion about Grüneisen wor ernst expressed his opinion []: You a have received an impression, that we are so far, that can consider aso the soid state from moecuartheoretica point of view, and one may hope that we soon receive a simiar perfecty true theory aso for soid state, that we for ong time possess for gases. he Grüneisen euation in essentia features turns into ie euation, when in the fraction the terms, arises from attraction interaction, wi be stroed out. So, the foundation for soid state euation deveopment is the Grüneisen euation. he euation that taes into account terms with second potent of temperature wi be essentiay more compicated, than Grüneisen euation. REFERECES [] W. Sutherand, A Kinetic heory of Soids, with an Ex- Copyright 0 SciRes. WJCP

9 o the Question of Vaidity Grüneisen Soid State Euation 7 perimenta Introduction, Phiosophica agazine Series V, Vo., o. 99, 89, pp doi:0.080/ [] G. ie, o the Kinetics heory of One-Atomic Corps, Annas of Physics, Vo., o. 8, 90, pp [] E. Grüneisen, o the heory of One-Atomic Soid Corps, Physica agazine, Vo., o. -, 9, pp [] E. Grüneisen, heory of Soid State One-Atomic Eements, Annas of Physics, Vo. 9, o., 9, pp [5] P. Debye, o the heory of Heat Capacities, Annas of Physics, Vo. 9, 9, pp [] S. Ratnowsy, One-Atomic Soid Corps Euation of State and the Quantum heory, Annas of Physics, Vo. 8, 9, pp [7] K. Eisenmann, One-atomic Soid Corps Euation of State, according to the Quantum heory, Annas of Physics, Vo. 9, o., 9, pp [8] E. Grüneisen, oecuar heory of Soid Corps, he Structure of the atter, Conference of Physics, Bruxees, October 9. [9] P. Debye, Euation of State and Quantum Hypothesis, Physica agazine, Vo., 9, pp [0] P. Debye, he Euation of State and Quantum Hypothesis with Suppement of Heat Conduction, Lectures on the Kinetic heory of atter and Eectricity, Leipzig and Berin, B. G. eubner, 9, pp. 9-. [] V. Foc and G. Krutow, Remars to the Viria heorem of Cassica echanic, Physica agazine of SSR, Vo., o., 9, pp [] V. Ch. Kozovsiy, Dynamica Euations for ime- Averaged Coordinates at herma Euiibrium, Soviet Physics Journa, Vo., o., 99, pp [] W. Braunbe, Gratings Dynamics heory of eting Phenomenon, Physica agazine, Vo. 8, o. -7, 9, pp [] L.. Syrin, o the Question of emperature Dependence of herma Ionic Poarization, Journa of echnica Physics, Vo., o., 95, pp. -5. [5] Ja. I. Frene, Statistica Physics, oscow, Academy of Sciences, Leningrad, 98. [] Yu.. Goryachev, L. L. oiseeno and E. I. Shvartsman, Parameters of Eastodynamic State of Dodecaborides of Rare Earth etas, Soviet Physics Journa, Vo., o. 5, 99, pp [7] D. S. Lemon and C.. Lund, hermodynamics of High emperature ie-grüneisen Soids, American Journa of Physics, Vo. 7, o., 999, pp doi:0.9/.909 [8] F. Lindemann, About the Cacuation of oecuar Own Freuencies, Physica agazine, Vo., o., 90, pp [9] J. J. Givarry, he Lindemann and Grüneisen Laws, Physica Review, Vo. 0, o., 95, pp doi:0.0/physrev.0.08 [0] E. Grüneisen, o the heory of One-atomic Soid Corps, Conference of German Physica Society, Vo., 9, pp Copyright 0 SciRes. WJCP