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Dr Roger Bennett Rm. 23 Xtn Lecture 19

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The Boltzmann Distribution This is the Boltzmann distribution and gives the probability that a system in contact with a heat bath at temperature T should be in a particular state. r labels all the states of the system. At low temperature only the lowest states have any chance of being occupied. As the temperature is raised higher lying states become more and more likely to be occupied. In this case, in contact with the heat bath, all the microstates are therefore not equally likely to be populated.

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The Boltzmann Distribution Usually there are huge numbers of microstates that can all have the same energy. This is called degeneracy. In this case we can do our summations above over each individual energy level rather than sum over each individual microstate. The summation is now over all the different energies U r and g(U r ) is the number of states possessing the energy U r. The probability is that of finding the system with energy U r.

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Entropy in ensembles Our system embedded in a heat bath is called a canonical ensemble (our isolated system on its own from Lecture 16 is termed a microcanonical ensemble). When isolated the microcanonical ensemble has a defined internal energy so that the probability of finding a system in a particular microstate is the same as any other microstate. In a heat bath the energy of the system fluctuates and the probability of finding any particular microstate is not equal. Can we now calculate the entropy for such a system and hence derive thermodynamic variables from statistical properties?

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Entropy in the canonical ensemble Embed our system in a heat bath made up of (M-1) replica subsystems to the one were interested in. Each subsystem may be in one of many microstates. The number of subsystems in the i th microstate is n i. The number of ways of arranging n 1 systems of µstate 1, n 2 systems of µstate 2, n 3 ….

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Entropy in the canonical ensemble This is the general definition of entropy and holds even if the probabilities of each individual microstate are different. If all microstates are equally probable p i = 1/W (microcanonical ensemble) Which brings us nicely back to the Boltzmann relation

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Entropy in the canonical ensemble The general definition of entropy, in combination with the Boltzmann distribution allows us to calculate real properties of the system.

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Helmholtz Free Energy Ū is the average value of the internal energy of the system. (Ū – TS) is the average value of the Helmholtz free energy, F. This is a function of state that we briefly mentioned in earlier lectures. It is central to statistical mechanics. The Partition function Z has appeared in our result –it seems to be much more than a mere normalising factor. Z acts as a bridge linking the microscopic world of microstates (quantum states) to the free energy and hence to all the large scale properties of a system.

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Helmholtz Free Energy F is a state function. Can we now calculate the thermal properties of our system? Lets ignore Ū is an average and assume U = Ū. We can now use our thermodynamics knowledge to define our thermodynamic variables. For a small reversible change:

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Helmholtz Free Energy Employing partial differentials we find:- We have intimately related the energies of the microstates of the system to the pressure and entropy.

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Response functions –2 nd derivatives of free energy Moduli of elasticity are the stress/strain or force/unit area divided by the fractional deformation (Lecture 8):- Most usefully the heat capacity at constant volume:-

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Mean Energy Ū is the average value of the internal energy of the system. The actual internal energy fluctuates because we have defined the temperature of the heat bath. How big are the fluctuations? – are they important?

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Fluctuations in Internal Energy We measure departures from the mean value by use of standard deviations - as we would with any distribution.

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Fluctuations in Internal Energy

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We have calculated the variance – the relative fluctuation U/ Ū is of most use:- But Ū and C V are extensive properties proportional to the size of the system ~ N, the number of particles in the system. T intensive and is size independent.

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Fluctuations in Internal Energy For typical macroscopic systems with ~10 23 particles fluctuations U/ Ū ~ Fluctuations are tiny and hence U and Ū can be considered identical for all practical purposes. Macroscopic systems in a heat bath effectively have their energy determined. Similar relationships can be found for other relative fluctuations of properties of macroscopic systems.

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Dr Roger Bennett Rm. 23 Xtn Lecture 20

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Summary – Statistical Mechanics Microstate – The state of a system defined microscopically – a complete description on the atomic scale. Macrostate – The state of a stystem of macroscopic size specified by a few macroscopically observable quantities only. Statistical Weight (W or ) of a macrostate – is the number of microstates compising the macrostate. Postulate of equal a priori probabilities – for an isolated system in a definite macrostate, the W microstates comprising this macrostate occur with equal probability. Equilibrium Postulate – for an isolated macroscopic system, defined by U,V,N (which are fixed) and variable parameters, equilibrium corresponds to those values of for which the statistical weight W(U,V,N, ) attains its maximum.

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Summary – Statistical Mechanics Boltzmann definition of the Entropy Temperature definition Pressure definition General definition of the Entropy

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Summary – Statistical Mechanics Boltzmann Distribution – p i is the probability that a system at temperature T is in the state i with energy U i. Partition Function – summed over all microstates Mean Energy Helmholtz free energy F = U - TS

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Paramagnetic materials This is a simple model to develop our understanding of stat. Mechanics but it proves to be very significant. Paramagnets contain atoms which have magnetic dipole moments ( ). These do not interact with each other but can respond to an applied external magnetic (B) field. The dipoles can be thought of as independent (atomic) bar magnets arranged in a crystal lattice.

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Paramagnetic materials The dipoles can be thought of as independent (atomic) bar magnets arranged in a crystal lattice. Think In a B field each dipole can exist in one of two states – aligned with the field (spin up) or anti aligned (spin down). Spin up dipoles have an energy - B, spin down + B. We want to find out how the magnetisation of the material depends on temperature and applied field.

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Paramagnetic materials As all the dipoles are independent of each other we really only need to look at the average properties of one dipole. We can use all the other dipoles as the heat bath – Canonical ensemble. We have the two possible microstates and energies already – get Partition Function Z 1 for our single dipole.

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Paramagnetic materials We can now calculate the probability of spin up versus spin down.

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Paramagnetic materials We can now calculate the mean magnetic moment of our individual dipole. And hence the mean energy of the individual

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Paramagnetic materials We have the values necessary for the individual dipole and because our dipoles do not interact all other dipoles must behave similarly. A solid of N dipoles therefore has and energy: And a mean magnetic moment (in the direction of the applied field) of: Note that U = -MB

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Paramagnetic materials The magnetisation L is the magnetic moment per unit volume In the limit of a weak filed or high temperature x<<1 and tanh x x so:

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Paramagnetic materials

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Paramagnetic materials –Curies Law The susceptibility is the magnetisation per applied field intensity which for small magnetisations is given by H = B/ 0. Is Curies law. It holds very well for paramagnetic materials with weakly interacting dipoles. So well that it can be used for temperature calibration for example Cerium Magnesium nitrate obeys curies lay to 0.01K!

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Dr Roger Bennett Rm. 23 Xtn Lecture 21

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Paramagnetic materials The dipoles can be thought of as independent (atomic) bar magnets arranged in a crystal lattice. Think In a B field each dipole can exist in one of two states – aligned with the field (spin up) or anti aligned (spin down). Spin up dipoles have an energy - B, spin down + B. We want to find out how the magnetisation of the material depends on temperature and applied field.

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Paramagnetic materials As all the dipoles are independent of each other we really only need to look at the average properties of one dipole. We can use all the other dipoles as the heat bath – Canonical ensemble. We have the two possible microstates and energies already – get Partition Function Z 1 for our single dipole.

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Paramagnetic materials We have the values necessary for the individual dipole and because our dipoles do not interact all other dipoles must behave similarly. A solid of N dipoles therefore has an energy: And a mean magnetic moment (in the direction of the applied field) of: Note that U = -MB

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Paramagnetic materials

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Hyperbolic Functions – Flap M4.6! Reminder – please revise

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Hyperbolic Functions – Flap M4.6! Use quotient rule to prove

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Heat Capacity Our paramagnetic solid has an energy that depends upon temperature – it therefore must have a magnetic heat capacity. Experimentally one measures the heat capacity at constant magnetic field intensity H. as our paramagnetic compound is weakly magnetic H = B/ 0 so B is also constant.

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Heat Capacity

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Schottky Heat Capacity This is in fact a general result for the heat capacity in any two level system. One example we have already encountered is the Schottky defect.

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Isolated Paramagnetic Solid Very similar problem to the one previously treated in a heat bath. Here we constrain (fix) the total energy U of the isolated system. N total dipoles, n spin-up aligned with the applied B field U is clearly a function of n. A given energy U(n) corresponds to a given number of n spin up atoms with statistical weight:-

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Isolated Paramagnetic Solid Hence Entropy is given by:- For large N (~10 23 ) can use Stirlings Approximation Should look familiar – its the same problem as the Schottky Vacancy formation.

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Isolated Paramagnetic Solid Solve for n, to find the density of spin up atoms:- This is identical to the probability of finding a spin up atom in a heat bath we started with last lecture!

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Negative Temperature From our previous derivation we had If n < N/2 then more than half the dipoles are anti- parallel and T becomes negative! What is a negative temperature? We know that as the temperature T the populations of spin-up and spin-down only become equal! A negative temperature state must therefore be hotter than T= as its is a more energetic state of the system.

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Negative Temperature For a negative temperature the entropy and statistical weight must be decreasing functions of E. This can happen if the system possess a state of finite maximum energy – such as our paramagnet with U=N B. No systems exist where this happens for all particular aspects (I.e. vibrational energies, electronic energies and magnetic energies). However, if one such aspect or subsystem is effectively decoupled from the others, so they do not interact, that subsystem may be considered to reach internal equilibrium without being in equilibrium with the others. This is the case for magnetic systems where the relaxation times between atomic spins is much quicker than the relaxation between spins and the vibrational modes of the lattice.

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Negative Temperature In the paramagnet the lowest possible energy is U=-N B and the highest U=+N B. These are both unique microstates so S=0. In between we can only reach states with positive energy with a negative temperature.

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