The Lowest Temperature Of An Einstein Solid Is Positive

Transcription

1 The Lowest Temperature Of An Einstein Solid Is Positive W. C. Troy The Lowest Temperature Of An Einstein Solid Is Positive p.1/20

2 The Question. Can a solid be cooled to a temperature T 0 > 0 where all quanta of thermal energy are drained off, leaving the object in the ground state? If yes, quantum effects are expected (superposition of states). This result may lead to quantum computing devices. Experimental results. Recent studies claim Yes! (1) Nature, Nature Physics , 2009, 2010 (2) Science Magazine Breakthrough of the Year Our Goal. Answer the question theoretically. First step: Study Einstein s specific heat model. The Lowest Temperature Of An Einstein Solid Is Positive p.2/20

3 Specific Heat: C v = 1 n U T 1. Dulong and Petit (1819): for all solids, C v = 5.94 ( ) cal. gm K 2. Weber (1875), Kopp (1904), Dewar (1904): for many solids at low T. 0 < C v << 5.94 The Lowest Temperature Of An Einstein Solid Is Positive p.3/20

4 θ = temperature where Cv = 1 2 (5.94) 2.97 Lewis and Randall: Thermodynamics And The Free Energy Of Chemical Substances (1923) The Lowest Temperature Of An Einstein Solid Is Positive p.4/20

5 (1907) Einstein Formula for C v. A1. Each atom is a 3D quantum oscillator, which is attached to a preferred position by a spring. A2. q > 0 quanta of energy have been added to the solid. A3. Each quantum has energy ǫ = νh. Einstein Formula: C v = 5.94 ( ǫ kt ) 2 exp( ǫ kt ) ( exp( ǫ kt ) 1) 2, 0 < T <. The Lowest Temperature Of An Einstein Solid Is Positive p.5/20

6 Einstein, A. DIE PLANCKESCHE THEORIE DER STRAHLUNG UND DIE THEORIE DER SPEZIFISCHEN WARME. Annalen Der Physik (1907) 6 C v Figure 2. C v vs. scaled temperature for diamond. The Lowest Temperature Of An Einstein Solid Is Positive p.6/20

7 Behavior of C v as T 0 +. C v = 5.94 ( ǫ kt It is easily verified that ) 2 exp( ǫ kt ) ( exp( ǫ kt ) 1) 2, 0 < T <. lim T 0 + C v = 0. (1) But, can T 0 + for an Einstein solid? The Lowest Temperature Of An Einstein Solid Is Positive p.7/20

8 Derivation of C v. N = no. of atoms. N = 3N = no. of degrees of freedom. W = no. of ways to distribute q quanta of energy over N degrees of freedom: W = (q + N 1)! q!(n. 1)! Entropy: S = k ln(w) = k ln ( (q + N ) 1)! q!(n. 1)! The Lowest Temperature Of An Einstein Solid Is Positive p.8/20

9 Stirling Approximation. ln(m!) = M ln(m) M when M >> 1. S = k ( ln((q + N 1)!) ln(q!) ln((n 1)!) ) becomes S = k ( (q + N ) ln(q + N ) q ln(q) (N 1) ln(n 1) ) Therefore, ds (1 dq = k ln + N ). q The Lowest Temperature Of An Einstein Solid Is Positive p.9/20

10 Internal Energy: U = qǫ + N ǫ 2 and du dq = ǫ. Temperature: 1 T = S U = ds dq du dq = k ǫ ln (1 + N ) q. Conclusion I: lim T = 0. q 0 + The Lowest Temperature Of An Einstein Solid Is Positive p.10/20

11 Invert Temperature Equation: q = N e ǫ kt 1, N = 3nN A. U = qǫ + N ǫ 2 = N ǫ e ǫ kt 1 + N ǫ 2. Specific Heat: C v = 1 n U T. C v = 5.94 ( ǫ kt ) 2 exp( ǫ kt ) ( exp( ǫ kt ) 1) 2, 0 < T <. Conclusion II: C v exists for all T > 0 and lim C v = 0. T 0 + The Lowest Temperature Of An Einstein Solid Is Positive p.11/20

12 Widely quoted: lim q 0 + T = 0 and lim T 0 + C v = 0. (2) (1) R. Prathia, Statistical Mechanics, p. 175 (2) D. Schroeder, Thermal Physics, p. 307 Property (2) is mathematically questionable. Why? Because the derivation of property (2) is based on ln(q!) = q ln(q) q, which loses accuracy as q 0 +. The Lowest Temperature Of An Einstein Solid Is Positive p.12/20

13 New Derivation of C v. Replace Stirling s approximation ln(m!) = M ln(m) M with the exact formula ln(m!) = ln(γ(m + 1)). Entropy becomes S = k ( ln(γ(q + N )) ln(γ(q)) ln(γ(n )) ). The Lowest Temperature Of An Einstein Solid Is Positive p.13/20

14 Temperature: T = ǫ k ( Γ (q + N ) 1 ) Γ(q + N ) Γ (q + 1) > 0 q 0. Γ(q + 1) Specific heat: C V = k n ( Γ (q + N ) 2 [ ) Γ(q + N ) Γ (q + 1) d Γ(q + 1) dq > 0 q 0. ( Γ (q + N )] 1 ) Γ(q + N ) Γ (q + 1) Γ(q + 1) The Lowest Temperature Of An Einstein Solid Is Positive p.14/20

15 Lowest Temperature: lim q 0 + T = T 0 = ǫ k ( Γ (N ) 1 ) Γ(N ) + γ > 0. γ is Euler s constant, ǫ = νh. T 0 can be large! Lowest Specific heat: lim q 0 + C v = k n ( Γ (N ) [ 2 ) Γ(N ) Γ (1) d Γ(1) dq ( Γ (q + N ) Γ(q + N ) Γ (q + 1) Γ(q + 1) q=0 )] 1 > 0. The Lowest Temperature Of An Einstein Solid Is Positive p.15/20

16 Diamond lim T = (K) and lim q 0 + C v = q 0 + C v Diamond New C v C v 0.9 x Blowup New C v C T(Kelvin) Einstein C v T 0 T The Lowest Temperature Of An Einstein Solid Is Positive p.16/20

17 Diamond lim T = (K) and lim q 0 + C v = q 0 + C v Diamond New C v C v 0.9 x Blowup New C v C T(Kelvin) Einstein C v T 0 T The Lowest Temperature Of An Einstein Solid Is Positive p.17/20

18 Answer to the original question is Yes! Question. Can a solid be cooled to a temperature T 0 > 0 where all quanta, in excess of ground state energy, are drained off, leaving the object in the ground state? For the Einstein solid we have proved that lim T = T 0 > 0 and q 0 + lim q 0 + U = N ǫ 2 = Ground State. Therefore, as the excess quanta are drained off, the solid cools to temperature T 0 > 0 where its internal energy reaches the ground state. The Lowest Temperature Of An Einstein Solid Is Positive p.18/20

19 The Debye Formula Einstein Specific Heat (1907): C v = 5.94 ( ǫ kt ) 2 exp( ǫ kt ) ( exp( ǫ kt ) 1) 2, 0 < T <. Debye Specific Heat (1913): C v = 9Nk ( ) 3 T T D T T D 0 x 4 e x (e x 1) 2dx, 0 < T <. The Lowest Temperature Of An Einstein Solid Is Positive p.19/20

20 The Lowest Temperature Of An Einstein Solid Is Positive p.20/20