Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations

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1 Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations W. C. Troy Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.1/46

2 Contents 1. The Models. 2. Einstein Model For Cooling A solid. 3. Fermi-Dirac Two State Paramagnetism Model W. C. T., Quart. Jour. Appl. Math., AMS (2012) W. C. T., Physics Letters A, No. 45 (2012) Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.2/46

3 1. The Models N = N = D e (ǫ µ) kt 1, ǫ > µ (BE) D e (ǫ µ) kt + 1 (FD) N = most probable number of particles with energy ǫ. µ = chemical potential, ǫ = hν = quantum of energy, ν = frequency, h= Planck s constant, T = temperature (K), k = Boltzman s constant. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.3/46

4 Homework Problems N = D e (ǫ µ) kt 1, ǫ > µ (BE) Prove : lim N = 0. (1) T 0 + N = D e (ǫ µ) kt + 1 (FD) Prove : lim N = T 0 + 0, ǫ > µ 1, ǫ < µ (2) Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.4/46

5 Textbooks 1.) D. Halliday and R. Resnick, Fundamentals of Physics 2.) P. Tipler and R. LLewellyn, Modern Physics 3.) D. Schroeder, Thermal Physics 4.) D. Griffiths, Quantum Mechanics Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.5/46

6 2. Einstein Model For A Solid R. Feynman proposed that development of a quantum computer is theoretically possible. 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. D. Powell, Moved By Light. Science News, May 7, 2011 Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.6/46

7 Laser cooling - techniques based on radiation pressure remove energy and reduce vibrations: vibrations in glass doughnuts were reduced by cooling to T=11 mk wobbles in mirrors were reduced by cooling to T=10 mk Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.7/46

8 Further refinements of laser techniques were developed to cool sticks, sails, drums and other multi-atom objects to low temperatures where vibrations are quelled. Papers flowed in as researchers competed to remove every quantum of energy from an object, leaving it in the ground state where quantum effects are expected D. Powell, Moved By Light. Science News, May 7, 2011 Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.8/46

9 quanta left - Park and Wang, Nature Physics quanta left - Schliesser et al, Nature Physics quanta left - Groblacher et al, Nature Physics quanta left - Rocheleau et al, Nature 2010 Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.9/46

10 O Connell et al (Nature, 2010) reduced a quantum drum (one trillion atoms) to its ground state at T 0 20mK - Science magazine 2010 breakthrough of the year. Teufel et al (Nature, 2011) reduced a drum ( kg) to its ground state where it stayed for 100 microseconds, much longer than the 6 nano second result of O Connell et al. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.10/46

11 Theory Our Goal. Answer the question theoretically: can all quanta of thermal energy be drained from a solid? Models. T 0 Lowest Possible Temperature Einstein 1907 Model For A Solid. Stat. Mech. Microcanonical Mds.: T 0 > 0. Goal: show that T 0 > 0. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.11/46

12 Einstein Theory For A Solid Einstein (1907) developed a formula for specific heat C v = 1 n U T in terms of T and the quantum ǫ = hν. 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. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.12/46

13 Petit-Dulong Law of Specific Heat P. L. Dulong ( ) - French physicist and chemist. A. T. Petit ( ) - French physicist Petit-Dulong Law of specific heat: C v = 1 n U T = 5.94 ( ) cal. gm K A. T. Petit and P. L. Dulong, "Recherches sur quelques points importants de la Théorie de la Chaleur," Annales de Chimie et de Physique 10 (1819) Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.13/46

14 Specific Heat: C v = 1 n U T Dulong and Petit (1819): for all solids, C v = 5.94 ( ) cal. gm K Weber (1875), Kopp (1904), Dewar (1904): for many solids at low T. 0 < C v << 5.94 Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.14/46

15 Einstein Formula (1907): C v = 5.94 ( ǫ kt ) 2 exp( ǫ kt ) ( exp( ǫ kt ) 1) 2, 0 < T <. 6 C v Figure 1. C v vs. scaled temperature for diamond. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.15/46

16 θ = temperature where Cv = 1 2 (5.94) 2.97 Lewis and Randall: Thermodynamics And The Free Energy Of Chemical Substances (1923) Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.16/46

17 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. (3) Can T 0 as the number of quanta decreases to zero? To answer this, study the derivation of C v. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.17/46

18 Micro. Canonical Derivation 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)! Simplification: de Moivre - Stirling approximation to n! Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.18/46

19 James Stirling Scottish mathematician Oxford - expelled from Baliol college for correspondence supporting the 1708 Scottish "Gathering of the Brig o Turk" uprising against the Stuarts Venice - feared assassination for discovering trade secrets of Venice glassmkaers. Newton helped him return to England. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.19/46

20 Stirling s Formula de Moivre, "Miscellanes Analytica," N! [constant] N ( N e ) N as N Stirling, "Methodus Differentials," N! 2πN ( N e ) N as N. Statistical Mechanics and Physics textbooks: ln(n!) = N ln(n) N, N >> 1. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.20/46

21 Derivation of C v Apply 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 1) ln(q + N 1) q ln(q) (N 1) ln(n 1) ) Therefore, ( ds dq = k ln 1 + N ) 1. q Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.21/46

22 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 ) 1. q Conclusion I: T 0 = lim q 0 + T = 0. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.22/46

23 Invert Temperature Equation: Specific Heat: C v = 1 n U T. q = N 1 e ǫ kt 1, N = 3nN A. U = qǫ + N ǫ 2 = (N 1)ǫ e ǫ kt 1 + N ǫ 2. 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 + Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.23/46

24 Widely quoted: lim T 0 + C v = 0. (4) Property (4) is mathematically questionable because its derivation is based on ln(q!) = q ln(q) q, which loses accuracy as q 0 +. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.24/46

25 The Error In Stirling s Approximation At Low q: ln(q!) = q ln(q) q When q=10 Relative Error = 13 Percent. When q=2 Relative Error = 188 Percent. When q=2 Stirling s Approximation Gives.69 =.61 When q=1 Stirling s Approximation Gives 0 = 1 Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.25/46

26 New Derivation Replace Stirling s approximation ln(m!) = M ln(m) M with the exact formula ln(m!) = ln(γ(m + 1)), where the Gamma function Γ(z) is defined by Γ(z) = 0 t z 1 e t dt, Re(z) > 0. Basic Property: M! = Γ(M + 1) when M is a positive integer. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.26/46

27 Entropy S = k ln ( ) (q+n 1)! q!(n 1)! becomes S = k ( ln(γ(q + N )) ln(γ(q + 1)) ln(γ(n )) ). U = qǫ + N ǫ 2. 1 T = k ǫ ds 1 T = dq du dq ( Γ (q + N ) ) Γ(q + N ) Γ (q + 1) Γ(q + 1) q 0. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.27/46

28 Temperature: T = ǫ k ( Γ (q + N ) 1 ) Γ(q + N ) Γ (q + 1) > 0 q 0. Γ(q + 1) Specific heat: C V = 1 n du dt = 1 n du dq dt dq = 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) Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.28/46

29 Lowest Temperature: T 0 = lim q 0 + T = νh k ( Γ (N ) 1 ) Γ(N ) + γ > 0. γ = Euler s constant. T 0 is large when ν is 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. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.29/46

30 Derivative Free Method Goal: Derive temperature formula without differentaition. S = k ln ( (q + N ) 1)! q!(n. 1)! U = qνh + N νh 2. 1 T = S U S(q + 1) S(q) = U(q + 1) U(q) = k ( q + N ) νh ln. q + 1 T 0 = lim q 0 T = νh k ln(n ) > T 0 = νh k ( Γ (N ) 1 ) Γ(N ) + γ > 0. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.30/46

31 Comparison With Experiment O Connell et al (Nature, 2010): ν = Hz, one trillion atoms. Ground state at T 0 = 20mK New Temperature Formula: Ground state at q = 0, T 0 = 9.8mK Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.31/46

32 Diamond T 0 = 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 Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.32/46

33 Diamond T 0 = 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 Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.33/46

34 When N! = Γ(N + 1), the new derivation gives lim T = T 0 > 0 and q 0 + lim q 0 + U = N ǫ 2 = Ground State. Does this happen experimentally? Le et al, Science (2011), show how two diamonds can exhibit quantum entanglement at room temperature. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.34/46

35 3. Fermi-Dirac Model N = D e (ǫ µ) kt + 1 (FD) lim T 0 + N = 0, ǫ > µ 1, ǫ < µ (5) Two-state paramagnetism model (Schroeder, Thermal Physics, p. 98): Paramagnetic materials ( e.g. magnesium, molybdenum), are attrated to a magnetic field, B. The material does not retain magnetic properties when B is removed. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.35/46

36 Two State Model Assumptions: a paramagnetic system has N >> 1 electrons attracted to external field B = B 0ˆk, B0 > 0. N 1 >> 1 electrons are in the up-state, N 2 >> 1 electrons are in the down-state N = N 1 + N 2 Energies: U 1 = µ B B 0, U 2 = µ B B 0 µ B = eh 4πm = Joules Tesla Total Energy: U = µ B B 0 (N 2 N 1 ) = µ B B 0 (N 2N 1 ) Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.36/46

37 ( ) Entropy: S = k ln N! N 1!N 2! = k ln ( N! N 1!(N N 1 )! ) N >> 1,N 1 >> 1,N N 1 >> 1 S k (N ln(n) N 1 ln (N 1 ) (N N 1 ) ln (N N 1 )) Total Energy: U = µ B B 0 (N 2N 1 ) 1 T = S U = ds/dn 1 du/dn 1 N Solve for N 1 : N 1 = exp 2µ B B 0 kt +1 Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.37/46

38 N 1 = exp N ( ) 2µ BB 0 kt + 1 lim N 1 = N T 0 + Solve for T: T = 2µ BB 0 k ln N 1 N 1 lim T = 0 N 1 N These limits are both invalid because the Stirling approximation requires N N 1 >> 1 Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.38/46

39 ( Entropy: S = k ln N! N 1!(N N 1 )! ) = k ln ( Γ(N +1) Γ(N 1 +1)Γ(N N 1 +1) ) Total Energy: U = µ B B 0 (N 2N 1 ) = heb 0 4πm (N 2N 1) 1 T = S U = ds/dn 1 du/dn 1 T = 2πmk heb ( 0 ) Γ (N 1 +1) Γ(N 1 +1) Γ (N N 1 +1) Γ(N N 1 +1) ( Γ (N+1) Γ(N+1) Γ (1) Γ(1) lim T = heb 0 N 1 N 2πmk ) > 0 Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.39/46

40 Application BPPH (2,2-diphenyl-1-picrylhydrazyl) - paramagnetic powder (1) Grobet et al, J. Chem Phys. (1978), p (2) Schroeder, Thermal Physics, pp N = ,B 0 = 2.06,k = ,µ B = lim T = T 1 = N 1 N 2πmk heb ( 0 ) =.05K > 0 Γ (N+1) Γ(N+1) Γ (1) Experiment: > 99 percent magnetization at T 2K Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.40/46

41 Acknowledgement Stefanos Folias S. J. Anderson Richard Field Toby Chapman Stuart Hastings Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.41/46

42 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 <. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.42/46

43 Teufel et al (Nature, 2011) single atom analysis relies on predictions of the Bose-Einstein equation < N > = 1 exp( ǫ kt ) 1, 0 < T <. P 1 : Thermal energy is present at every positive T > 0. P 2 : T 0 as < N > 0. i.e. T 0 = 0. P 2 : Quantum effects become important when < N > < 1. < N > < 1 when T < ǫ k ln(2). Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.43/46

44 One Atom Model Current theory (e.g. Teufel et al, Nature, 2011) - the Bose-Einstein equation < N >= 1 exp( ǫ kt ) 1, 0 < T <, for a single atom represents the entire solid: < N >= ave. no. of quanta in 1 particle state with energy ǫ. ǫ = hν = quantum of energy, ν = frequency, h= Planck s constant, T = temperature (K), k = Boltzman s constant. Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.44/46

45 Teufel et al (Nature, 2011) single atom analysis relies on predictions of the Bose-Einstein equation < N > = 1 exp( ǫ kt ) 1, 0 < T <. P 1 : Thermal energy is present at every positive T > 0. P 2 : T 0 as < N > 0. i.e. T 0 = 0. P 2 : Quantum effects become important when < N > < 1. < N > < 1 when T < ǫ k ln(2). Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.45/46

46 Comparison With Experiment O Connell et al experiment: ground state achieved when < N > =.07, ν = 6GHz and T = 20mK The one atom model predicts < N > =< N/D >= 1 exp( ǫ kt ) 1 Ground State At T = 106mK Low Temperature Properties of the Bose-Einstein and Fermi-Dirac Equations p.46/46

Cooling A Solid To Its Ground State

Cooling A Solid To Its Ground State W. C. Troy Cooling A Solid To Its Ground State p.1/41 Acknowledgement Stefanos Folias Chris Horvat Sashi Marella Brent Doiron S. J. Anderson Richard Field Stuart Hastings