Introduction to Radioactivity and Nuclear Physics

Eyal Buks Introduction to Thermodynamics and Statistical Physics (114016) - Lecture Notes April 13, 2011 Technion Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Preface to be written Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Con tents 1. The Principle of Larg est Uncertain ty 1 1.1 EntropyinInformationTheory 1 1.1.1 Example- TwoStatesSystem 1 1.1.2 SmallestandLargest Entropy 2 1.1.3 Thecompositionproperty 5 1.1.4 Alternative Definitionof Entropy 8 1.2 LargestUncertaintyEstimator 9 1.2.1 Useful Relations 11 1.2.2 TheFreeEntropy 13 1.3 The Principle of Largest Uncertainty in Statistical Mechanics 14 1.3.1 Microcanonical Distribution 14 1.3.2 CanonicalDistribution 15 1.3.3 Grandcanonical Distribution 16 1.3.4 TemperatureandChemicalPotential 17 1.4 Time Evolutionof EntropyofanIsolatedSystem 18 1.5 Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5.1 ExternallyAppliedPotential Energy 20 1.6 FreeEntropyandFreeEnergies 21 1.7 ProblemsSet1 21 1.8 SolutionsSet1 29 2. Ideal Gas 45 2.1 AParticleinaBox 45 2.2 GibbsParadox 48 2.3 FermionsandBosons 50 2.3.1 Fermi-DiracDistribution 51 2.3.2 Bose-EinsteinDistribution 52 2.3.3 ClassicalLimit 52 2.4 IdealGasintheClassicalLimit 53 2.4.1 Pressure 55 2.4.2 Useful Relations 56 2.4.3 HeatCapacity 57 2.4.4 InternalDegrees of Freedom 57 2.5 ProcessesinIdealGas 60 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Contents 2.5.1 IsothermalProcess 62 2.5.2 IsobaricProcess 62 2.5.3 IsochoricProcess 63 2.5.4 Isentropic Process 63 2.6 CarnotHeatEngine 64 2.7 Limits Imposed Upon the Efficiency 66 2.8 ProblemsSet2 71 2.9 SolutionsSet2 79 3. Bosonic and Fermionic Systems 97 3.1 ElectromagneticRadiation 97 3.1.1 ElectromagneticCavity 97 3.1.2 PartitionFunction 100 3.1.3 CubeCavity 100 3.1.4 AverageEnergy 102 3.1.5 Stefan-Boltzmann Radiation Law 103 3.2 PhononsinSolids 105 3.2.1 OneDimensionalExample 105 3.2.2 The3DCase 107 3.3 FermiGas 110 3.3.1 OrbitalPartitionFunction 110 3.3.2 PartitionFunctionof theGas 110 3.3.3 EnergyandNumberofParticles 112 3.3.4 Example: ElectronsinMetal 112 3.4 Semiconductor Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.5 ProblemsSet3 115 3.6 SolutionsSet3 117 4. Classical Limit of Statistical M echanics 127 4.1 ClassicalHamiltonian 127 4.1.1 Hamilton-Jacobi Equations 128 4.1.2 Example 128 4.1.3 Example 129 4.2 Density Function 130 4.2.1 EquipartitionTheorem 130 4.2.2 Example 131 4.3 NyquistNoise 132 4.4 ProblemsSet4 136 4.5 SolutionsSet4 138 5. Exam Wint er 2010 A 147 5.1 Problems 147 5.2 Solutions 148 Eyal Buks Thermodynamics and Statistical Physics 6 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Con tents 6. Exam Wint er 2010 B 155 6.1 Problems 155 6.2 Solutions 156 References 163 Index 165 Eyal Buks Thermodynamics and Statistical Physics 7 Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com 1. The Principle of Largest Uncertainty In this chapter w e discuss relations between information theory and statistical mechanics. We show that the canonical and grand canonical distributions can be obtained from Shannon’s principle of maximum uncertainty [1, 2, 3]. Moreover, the tim e evolution of the entropy of an isolated system and the H theorem are discussed. 1.1 Entropy in Information Theory The possible states of a given system are denoted as e m ,wherem =1, 2, 3, , and the Introduction to Radioactivity and Nuclear Physics Introduction to Radioactivity and Nuclear Physics Bởi: OpenStaxCollege The synchrotron source produces electromagnetic radiation, as evident from the visible glow (credit: United States Department of Energy, via Wikimedia Commons) There is an ongoing quest to find substructures of matter At one time, it was thought that atoms would be the ultimate substructure, but just when the first direct evidence of atoms was obtained, it became clear that they have a substructure and a tiny nucleus The nucleus itself has spectacular characteristics For example, certain nuclei are unstable, 1/2 Introduction to Radioactivity and Nuclear Physics and their decay emits radiations with energies millions of times greater than atomic energies Some of the mysteries of nature, such as why the core of the earth remains molten and how the sun produces its energy, are explained by nuclear phenomena The exploration of radioactivity and the nucleus revealed fundamental and previously unknown particles, forces, and conservation laws That exploration has evolved into a search for further underlying structures, such as quarks In this chapter, the fundamentals of nuclear radioactivity and the nucleus are explored The following two chapters explore the more important applications of nuclear physics in the field of medicine We will also explore the basics of what we know about quarks and other substructures smaller than nuclei 2/2 Eyal Buks Introduction to Thermodynamics and Statistical Physics (114016) - Lecture Notes April 13, 2011 Technion Preface to be written Con tents 1. The Principle of Larg est Uncertain ty 1 1.1 EntropyinInformationTheory 1 1.1.1 Example- TwoStatesSystem 1 1.1.2 SmallestandLargest Entropy 2 1.1.3 Thecompositionproperty 5 1.1.4 Alternative Definitionof Entropy 8 1.2 LargestUncertaintyEstimator 9 1.2.1 Useful Relations 11 1.2.2 TheFreeEntropy 13 1.3 The Principle of Largest Uncertainty in Statistical Mechanics 14 1.3.1 Microcanonical Distribution 14 1.3.2 CanonicalDistribution 15 1.3.3 Grandcanonical Distribution 16 1.3.4 TemperatureandChemicalPotential 17 1.4 Time Evolutionof EntropyofanIsolatedSystem 18 1.5 Thermal Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5.1 ExternallyAppliedPotential Energy 20 1.6 FreeEntropyandFreeEnergies 21 1.7 ProblemsSet1 21 1.8 SolutionsSet1 29 2. Ideal Gas 45 2.1 AParticleinaBox 45 2.2 GibbsParadox 48 2.3 FermionsandBosons 50 2.3.1 Fermi-DiracDistribution 51 2.3.2 Bose-EinsteinDistribution 52 2.3.3 ClassicalLimit 52 2.4 IdealGasintheClassicalLimit 53 2.4.1 Pressure 55 2.4.2 Useful Relations 56 2.4.3 HeatCapacity 57 2.4.4 InternalDegrees of Freedom 57 2.5 ProcessesinIdealGas 60 Contents 2.5.1 IsothermalProcess 62 2.5.2 IsobaricProcess 62 2.5.3 IsochoricProcess 63 2.5.4 Isentropic Process 63 2.6 CarnotHeatEngine 64 2.7 Limits Imposed Upon the Efficiency 66 2.8 ProblemsSet2 71 2.9 SolutionsSet2 79 3. Bosonic and Fermionic Systems 97 3.1 ElectromagneticRadiation 97 3.1.1 ElectromagneticCavity 97 3.1.2 PartitionFunction 100 3.1.3 CubeCavity 100 3.1.4 AverageEnergy 102 3.1.5 Stefan-Boltzmann Radiation Law 103 3.2 PhononsinSolids 105 3.2.1 OneDimensionalExample 105 3.2.2 The3DCase 107 3.3 FermiGas 110 3.3.1 OrbitalPartitionFunction 110 3.3.2 PartitionFunctionof theGas 110 3.3.3 EnergyandNumberofParticles 112 3.3.4 Example: ElectronsinMetal 112 3.4 Semiconductor Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.5 ProblemsSet3 115 3.6 SolutionsSet3 117 4. Classical Limit of Statistical M echanics 127 4.1 ClassicalHamiltonian 127 4.1.1 Hamilton-Jacobi Equations 128 4.1.2 Example 128 4.1.3 Example 129 4.2 Density Function 130 4.2.1 EquipartitionTheorem 130 4.2.2 Example 131 4.3 NyquistNoise 132 4.4 ProblemsSet4 136 4.5 SolutionsSet4 138 5. Exam Wint er 2010 A 147 5.1 Problems 147 5.2 Solutions 148 Eyal Buks Thermodynamics and Statistical Physics 6 Con tents 6. Exam Wint er 2010 B 155 6.1 Problems 155 6.2 Solutions 156 References 163 Index 165 Eyal Buks Thermodynamics and Statistical Physics 7 1. The Principle of Largest Uncertainty In this chapter w e discuss relations between information theory and statistical mechanics. We show that the canonical and grand canonical distributions can be obtained from Shannon’s principle of maximum uncertainty [1, 2, 3]. Moreover, the tim e evolution of the entropy of an isolated system and the H theorem are discussed. 1.1 Entropy in Information Theory The possible states of a given system are denoted as e m ,wherem =1, 2, 3, , and the prob ability that s tate e m is occupied is denoted by p m . The normal- ization condition reads X m p m =1. (1.1) For a giv en probability distribution {p m } the entropy is defined as σ = − X m p m log p m . (1.2) Below we show that this quantity characterizes the uncertaint y in the knowl- edge of the state of the syste m. 1.1.1 Example - Two States System Consider a system which can occupy either state e 1 with probability p,or state e 2 with probability 1 − p,where0≤ p ≤ 1. The entropy is given by σ = −p log p −(1 − p)log(1− p) . (1.3) Chapter 1. The Principle o f Largest Un certainty 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.2 0.4 0.6 0.8 1 x −p log p − (1 − p)log(1− p) As expected, the entropy vanishes at p =0andp = 1, since in bo th cases there is no uncertainty in wha t is the state which is occupied by the system. The largest uncertainty is obtained at p =0.5, for which σ Chapter 1. The Principle of Largest Uncertainty where l =0, 1, 2, L. A stationary point of σ occurs iff for every sm all change δ¯p, which is orthogonal to all vectors ¯ ∇g 0 , ¯ ∇g 1 , ¯ ∇g 2 , , ¯ ∇g L one has 0=δσ = ¯ ∇σ · δ¯p. (1.39) This condition is fulfilled only when the ve ctor ¯ ∇σ belongs to the subspace spanned by the vectors © ¯ ∇g 0 , ¯ ∇g 1 , ¯ ∇g 2 , , ¯ ∇g L ª [see also the discussion be- low Eq. (1.12) above]. In other words, only when ¯ ∇σ = ξ 0 ¯ ∇g 0 + ξ 1 ¯ ∇g 1 + ξ 2 ¯ ∇g 2 + + ξ L ¯ ∇g L , (1.40) where the num bers ξ 0 ,ξ 1 , , ξ L , which are called Lagrange m ultipliers, are constan ts. Using Eqs. (1.2), (1.5) and (1.37) the condition (1.40) can be expressed as −log p m − 1=ξ 0 + L X l=1 ξ l X l (m) . (1.41) From Eq. (1.41) one obtains p m =exp(−1 −ξ 0 )exp à − L X l=1 ξ l X l (m) ! . (1.42) The Lagrange multipliers ξ 0 ,ξ 1 , , ξ L can be determined from Eqs. (1.5) and (1.37) 1= X m p m =exp(−1 −ξ 0 ) X m exp à − L X l=1 ξ l X l (m) ! , (1.43) hX l i = X m p m X l (m) =exp(−1 −ξ 0 ) X m exp à − L X l=1 ξ l X l (m) ! X l (m) . (1.44) Using Eqs. (1.42) and (1.43) one finds p m = exp µ − L P l=1 ξ l X l (m) ¶ P m exp µ − L P l=1 ξ l X l (m) ¶ . (1.45) In terms of the partition function Z, which is defined as Eyal Buks Thermodynamics and Statistical Physics 10 1.2. Largest Uncertaint y Estimator Z = X m exp à − L X l=1 ξ l X l (m) ! , (1.46) one finds p m = 1 Z exp à − L X l=1 ξ l X l (m) ! . (1.47) Using the same arguments as in section 1.1.2 above [see Eq. (1 .16)] it is easy to show that at the stationary point that occurs for the proba bility distribution given by Eq. (1.47) the entropy obtains its largest value. 1.2.1 Useful Relations The expectation value hX l i can be expressed as hX l i = X m p m X l (m) = 1 Z X m exp à − L X l=1 ξ l X l (m) ! X l (m) = − 1 Z ∂Z ∂ξ l = − ∂ lo g Z ∂ξ l . (1.48) Similarly,  X 2 l ® can be expressed as  X 2 l ® = X m p m X 2 l (m) = 1 Z X m exp à − L X l=1 ξ l X l (m) ! X 2 l (m) = 1 Z ∂ 2 Z ∂ξ 2 l . (1.49) Using Eqs. (1.48) and (1.49) one finds that the variance of the variable X l is given by D (∆X l ) 2 E = D (X l − hX l i) 2 E = 1 Z ∂ 2 Z ∂ξ 2 l − µ 1 Z ∂Z ∂ξ l ¶ 2 . (1.50) However, using the following identity Eyal Buks Thermodynamics and Statistical Physics 11 Chapter 1. The Principle of Largest Uncertainty ∂ 2 log Z ∂ξ 2 l = ∂ ∂ξ l 1 Z ∂Z ∂ξ l = 1 Z ∂ 2 Z ∂ξ 2 l − µ 1 Z ∂Z ∂ξ l ¶ 2 , (1.51) one finds D (∆X l ) 2 E = ∂ 2 log Z ∂ξ 2 l . (1.52) Note that the above results Eqs. (1.48) and (1.52) are valid only when Z is expressed as a function of the the Lagrange multipliers, namely Z = Z (ξ 1 ,ξ 2 , , ξ L ) . (1.53) Using the definition of entropy (1.2) and Eq. (1.47) one finds σ = − X m p m log p m = − X m p m log à 1 Z exp à − L X l=1 ξ l X l (m) !! = X m p m à log Z + L X l=1 ξ l X l (m) ! =logZ + L X l=1 ξ l X m p m X l (m) , (1.54) thus σ =logZ + L X l=1 ξ l hX l i . (1.55) Using the above relations one can also evaluate the partial derivativ e of the entropy σ when it is expressed as a function of the expectation va lu es, namely σ = σ (hX 1 i , hX 2 i , , hX L i) . (1.56) Using Eq. (1.55) one has ∂σ ∂ hX l i = ∂ log Z ∂ hX l i + L X l 0 =1 hX l 0 i ∂ξ l 0 ∂ hX l i + L X l 0 =1 ξ l 0 ∂ hX l 0 i ∂ hX l i = ∂ log Z ∂ hX l i + L X l 0 =1 hX l 0 i ∂ξ l 0 ∂ hX l i + ξ l = L X l 0 =1 ∂ log Z ∂ξ l 0 ∂ξ l 0 ∂ hX l i + L X l 0 =1 hX l 0 i ∂ξ l 0 ∂ hX l i + ξ l , (1.57) Eyal Buks Thermodynamics and Statistical Physics 12 1.2. Largest Uncertaint y Estimator thus using Eq. (1.48) one finds ∂σ ∂ hX l i = ξ l . (1.58) 1.2.2 The Free En tropy The free entropy σ F is defined as the term log Z in Eq. (1.54) σ F =logZ = σ − L X l=1 ξ l X m p m X l (m) = − X m p m log p m − L X l=1 ξ l X m p m X l (m) . (1.59) ThefreeentropyiscommonlyexpressedasafunctionoftheLagrangemul- tipliers σ F = σ F (ξ 1 ,ξ 2 , , ξ L ) . (1.60) We have seen abo ve that the LUE maximizes σ for given values of expecta- tion values hX 1 i , hX 2 i , , hX L i. We show below that a similar result can be obtained 1.7. Problems Set 1 a) Sho w that α is given by α = a 2 · 1+tanh µ Fa 2τ ¶¸ . (1.135) b) Show that in the limit of high temperature the spring constant is given approximately by k ' 4τ Na 2 . (1.136) N α N α 19. A long elastic molecule can be modelled as a linear chain of N links. The state of each link is charac terized by two quantum numbers l and n.The length of a link is either l = a or l = b. The vibrational state of a link is modelled as a harmonic oscillator whose angular frequency is ω a for a link of length a and ω b for a link of length b. Thus, the energy of a link is E n,l = ½ }ω a ¡ n + 1 2 ¢ for l = a }ω b ¡ n + 1 2 ¢ for l = b , (1.137) n =0, 1, 2, The chain is held under a tension F . Show that the mean length hLi of the chain in the limit of high temperature T is given by hLi = N aω b + bω a ω b + ω a + N Fω b ω a (a −b) 2 (ω b + ω a ) 2 β + O ¡ β 2 ¢ , (1.138) where β =1/τ. 20. The elasticity of a rubber band can be described in terms of a one- dimensional model of N polymer molecules linked together end-to-end. The angle between successive links is equally likely to be 0 ◦ or 180 ◦ .The length of eac h polymer is d and the total length is L. The system is in thermal equilibrium at temperature τ. Show that the force f required to main tain a length L is given by f = τ d tanh −1 L Nd . (1.139) 21. Consider a sys tem which has two single particle states both of the same energy. When both states are unoccupied, the energy of the system is Eyal Buks Thermodynamics and Statist ical Physics 27 Chapter 1. The Principle of Largest Uncertainty zero; when one state or the other is occupied by one particle, th e energy is ε. We suppose that the energy of the system is much higher (infinitely higher) when both states are occupied. Show that in thermal equilibrium at temperature τ the av erage number of particles in the level is hNi = 2 2+exp[β (ε − µ)] , (1.140) where µ is the chemical potential and β =1/τ. 22. Consider an array of N tw o-lev el particles. Each one can be in o ne of two states, having energy E 1 and E 2 respectively. The n umbers of particles instates1and2aren 1 and n 2 respectively, where N = n 1 + n 2 (assume n 1 À 1andn 2 À 1). Consider a n energy exchange with a reservoir at temperature τ leading to population changes n 2 → n 2 −1andn 1 → n 1 +1. a) Calculate the en tropy change of the two-level system, (∆σ) 2LS . b) Calculate the entropy change of the reservoir, (∆σ) R . c) What can be said about th e relation bet ween (∆σ) 2LS and (∆σ) R in thermal equilibrium? Use your answer to express the ration n 2 /n 1 as a function of E 1 , E 2 and τ. 23. Consider a lattice containing N sites of one ty pe, which is denoted as A , and the same number of sites of another type, which is denoted as B. The lattice is occupied by N atoms. The number of atoms occupying sites of type A is denoted as N A , w hereas the number of atoms occupying atoms of type B is denoted as N B ,whereN A + N B = N.Letε be the energy necessary to remove an atom from a lattice site of type A to a lattice site of type B. The system is in thermal equilibrium at temperature τ . Assume that N,N A ,N B À 1. a) Calculate the entropy σ. b) Calculate the average number hN B i of atoms occupying sites of type B. 24. Consider a microcanonical ensem ble of N quantum harmonic oscillators in thermal equilibrium at temperature τ. The resonance frequency of all oscillators is ω. The quantum energy levels of each quantum oscillator is given by ε n = }ω µ n + 1 2 ¶ , (1.141) where n =0, 1, 2, is integer. The total energy E of the system is given by E = }ω µ m + N 2 ¶ , (1.142) where Eyal Buks Thermodynamics and Statist ical Physics 28 1.8. Solutions Set 1 m = N X l=1 n l , (1.143) and n l is state number of oscillator l. a) Calculate the number of states g (N,m) of the system with total energy }ω (m + N/2). b) Use this re sult to calculate the en tropy σ of the system with total energy }ω (m + N/2). Appro ximate the r esult by assuming t hat N À 2. Ideal Gas In this chapter w e study some basic properties of ideal gas of massive iden- tical particles. We start by considering a single partic le in a box. We then discuss the statistical pro perties of an ensemble of identical indistinguishable particles and introduce the concepts of Fermions and Bosons. In the rest of this chapter we mainly focus on the classical limit. For this case we derive expressions for the pressure, heat capacity, energy an d entropy and discuss how internal degrees of freedom may modify these results. In the last part of this chapter we discuss an example of an heat engine based on ideal gas (Carnot heat e ngine). We show t hat t he efficiency of such a heat engine, which employs a reversible process, obtains the largest possible value th at is allowed by the second law of ther mod ynamics. 2.1 A Particle in a Box Consider a particle having mass M in a box. For simplicity the b ox is assumed to have a cube shape with a volume V = L 3 . The corresponding potential energy is given by V (x, y, z)= ½ 00≤ x, y, z ≤ L ∞ else . (2.1) The quantum eigenstates and eigenenergies are determ ined by requiring that the wav efunction ψ (x, y, z)satisfies the Schr¨odinger equation − ~ 2 2M µ ∂ 2 ψ ∂x 2 + ∂ 2 ψ ∂y 2 + ∂ 2 ψ ∂z 2 ¶ + Vψ= Eψ . (2.2) In addition, we require that the wa vefunction ψ vanishes on the surfaces of the bo x. The normalized s olutions are given by ψ n x ,n y ,n z (x, y, z)= µ 2 L ¶ 3/2 sin n x πx L sin n y πy L sin n z πz L , (2.3) where n x ,n y ,n z =1, 2, 3, (2.4) Chapter 2. Ideal Gas The corresponding eigenenerg ies are given by ε n x ,n y ,n z = ~ 2 2M ³ π L ´ 2 ¡ n 2 x + n 2 y + n 2 z ¢ . (2.5) For simplicity we consider the case where the particle doe sn’t hav e any in- ternal degree of freedom (such as spin). Later we will release this assumption and generalize the results for particles having internal degrees of freedom. The partition function is given by Z 1 = ∞ X n x =1 ∞ X n y =1 ∞ X n z =1 exp ³ − ε n x ,n y ,n z τ ´ = ∞ X n x =1 ∞ X n y =1 ∞ X n z =1 exp ¡ −α 2 ¡ n 2 x + n 2 y + n 2 z ¢¢ , (2.6) where α 2 = ~ 2 π 2 2ML 2 τ . (2.7) The follo wing relation can be employed to e stimate the dimensionless param - eter α α 2 = 7.9 × 10 −17 M m p ¡ L cm ¢ 2 τ 300 K , (2.8) where m p is the proton mass. As can be seen from the last r esult, it is often the case that α 2 ¿ 1. In this limit the sum can be approximated by an integral ∞ X n x =1 exp ¡ −α 2 n 2 x ¢ ' ∞ Z 0 exp ¡ −α 2 n 2 x ¢ dn x . (2.9) By changing the integration variable x = αn x one finds ∞ Z 0 exp ¡ −α 2 n 2 x ¢ dn x = 1 α ∞ Z 0 exp ¡ −x 2 ¢ dx = √ π 2α , (2.10) thus Z 1 = µ √ π 2α ¶ 3 = µ ML 2 τ 2π~ 2 ¶ 3/2 = n Q V, (2.11) where we have introduced the quantum density Eyal Buks Thermodynamics and Statistical Physics 46 2.1. A Particle in a Box n Q = µ Mτ 2π~ 2 ¶ 3/2 . (2.12) The par tition function (2.11) together with Eq. (1.70) allows evaluating the average energy (recall that β =1/τ ) hεi = − ∂ log Z 1 ∂β = − ∂ log µ ³ ML 2 2π~ 2 β ´ 3/2 ¶ ∂β = − ∂ log β −3/2 ∂β = 3 2 ∂ log β ∂β = 3τ 2 . (2.13) This result can be written as hεi = d τ 2 , (2.14) where d = 3 is the number of degrees of freedom of the pa rticle. As we will see later, this is an example of the equipartition theorem of statistical m echanics. Similarly, the energy va riance can be evaluated using Eq . (1.71) D (∆ε) 2 E = ∂ 2 log Z 1 ∂β 2 = − ∂ hεi ∂β = − ∂ ∂β 3 2β = 3 2β 2 = 3τ 2 2 . (2.15) Thus, using Eq. (2.13) the standard de viation is given by r D (∆ε) 2 E = r 2 3 hεi . (2.16) What is the physical meaning of the quantum density? The de Broglie wavelength λ of a particle having mass M and velocity v is giv en by Eyal Buks Thermodynamics and Statistical Physics 47 Chapter 2. Ideal Gas λ = h Mv . (2.17) For a particle having energy equals to the average energy hεi =3τ/2 one has Mv 2 2 = 3τ 2 , (2.18) thus in this case the de-Broglie wavelength, w hich is denoted as λ T (the thermal waveleng th) λ T = h √ 3Mτ , (2.19) and therefore one has (recall .. .Introduction to Radioactivity and Nuclear Physics and their decay emits radiations with energies millions of times greater than atomic energies Some of the mysteries... molten and how the sun produces its energy, are explained by nuclear phenomena The exploration of radioactivity and the nucleus revealed fundamental and previously unknown particles, forces, and. .. laws That exploration has evolved into a search for further underlying structures, such as quarks In this chapter, the fundamentals of nuclear radioactivity and the nucleus are explored The following