The Bose Einstein quantum statistics
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1 Page 1 The Bose Einstein quantum statistics 1. Introduction Quantized lattice vibrations Thermal lattice vibrations in a solid are sorted in classical mechanics in normal modes, special oscillation patterns that do not interfere with each other (the orthogonality property from which their name derives). They are useful because an arbitrary classical oscillation can be written as a sum over these modes (the completeness property). Similarly, in QM normal modes give us a good basis set of quantum states to represent an arbitrary quantum state αα of lattice vibrations in a solid. Their completeness property is applied to correctly count all the vibration states in a solid, which is useful in statistics. In a second step (sometime called second quantization ), the normal mode wavefunctions are quantized. As for one SHO, a transition between two levels of a normal mode is associated with creation aa + and annihilation aa operators. This procedure is advantageous because commutation or anti-commutation relations on aa + and aa operators insure that the required multi-particle state permutation symmetry is satisfied (for lattice modes with zero spin, the statistics is Bose Einstein and commutation relations are applied). One may then associate the eigenvalue nn + 1 of the number operatornn = aa + aa with the number nn of phonons of the 2 normal mode. When commutation relations are applied, the energy distribution is given by the Bore Einstein quantum statistics as nn(ee ii ) = 1. This is the case of quantized normal ee EE ii kkkk 1 vibration modes, as well as quantized light modes. Decreasing temperatures TT leave higher energy states less populated [one may say that fewer phonons are present (nn is smaller)]. As the lattice vibrations have energy, this can be observed with measurements of the specific heat cc vv = 9nnnn TT θθdd 3 θθ DD xx4 ee xx dddd TT 0 (ee xx 1) 2 where cc vv,eeee cc vv,llllllll (you will see this later, in a thermopower experiment that probes the electronic specific heat) and cc vv,llllllll cccccccccc above θθ DD. The dependence of cc vv at low and high TT is shown in figure 1(a). We will observe this in practice with a different, conductivity measurement. Electron states in a lattice have periodic wave functions (Bloch theorem). This would predict an infinite conductivity. The finite conductivity observed in practice must then be due to variations of the lattice for a perfectly periodic case and to impurity scattering. The 2 nd case gives a resistance independent of TT. The 1 st case (also called phonon scattering) gives a TT dependent resistance because the lattice oscillations depend on TT, as shown quantitatively by the Debye function. The resistivity can be calculated with a scattering model dddd ddω = 2ππ ii VV ff 2 DDDDDD(EE FF ). At high temperatures (TT > θθ DD ) the lattice vibrations give a contribution which is TT because the
2 number of phonons increases with T as nn(qq, ωω = qqqq) = 1 qqqq eekkkk 1 TT when qqqq kkkk. We must satisfy both the conservation of energy and momentum in a scattering process. A phonon of the right energy will not scatter an electron if it does not have the right momentum as well. Using this it can be shown that at low TT we obtain a variation ρρ TT 5 (figure 1(b)). cccccccccc. ln ρρ TT TT 3 TT 5 θθ DD = 180 KK θθ DD ln TT (a) (b) Fig. 1: (a) Specific heat of a solid is described by the Debye function DDDD(xx) = 1 tt3 dddd, xx 3 0 ee tt 1 showing the gradual freezing-out of lattice modes below θθ DD and the saturation to the classical value at high TT. (b) Asymptotes of the resistance of a solid. The power-laws give linear dependences on the double-logarithmic plots. 2. Using the lock-in amplifier Connect a 1kkΩ resistor across the current source (CS) red and black leads. Leave the green lead unconnected, but make sure it is not touching a conducting surface. Connect two leads across the resistor in-between the CS leads and send to the LIA input AA and oscilloscope channel 2. Connect the LIA reference out ( 1.5 VV amplitude) to the Voltage input of the CS and in the channel 1 of the oscilloscope (figure 2) Keep the measurement frequency, set by the large knob that adjusts the LIA internal reference, to less than 5 kkkkkk. This is because the effective circuit changes at higher frequency due to loading the CS and LIA away from their limited range of operating conditions determined by their output and input impedances, which distorts the results. Set CS gain to 10 μμμμ/vv, and into the Fast, Return, Float configuration Connect handheld voltmeter to the CS monitor output Turn on the CS voltage input and its output xx Page 2
3 We expect an RMS voltage across the resistor of 1 μμμμ 1.5VV 10 2 VV 103 Ω 11 mmmm. Confirm that this is the amplitude of the voltage shown in the oscilloscope channel 2. Verify that both the handheld voltmeter and the LIA show this voltage (they both display the rms amplitude). Turn the CS output off and decrease its gain. Turn the CS on again. The voltage across the resistor will be correspondingly lower (see equation above). Observe how both the handheld voltmeter and the oscilloscope have lost this small signal. In contrast, the LIA can still detect it (you will have to adjust its sensitivity scale). We use the LIA because of its better sensitivity. V in Current source 11 kkωω Cu or Au wire Thermocouple Reference Lock-in amplifier 3. Experiments Fig.2: Diagram of a four-probe configuration. In the Johnson noise experiment, the LIA measurement is done directly across the resistor with no current source. 3.1 Quantum statistics Connect the instruments as shown in figure 2. Forcing a known current through the wire with the CS gives a voltage drop across it proportional to the resistance. We will measure σσ(tt) = 1, the dependence of electrical conductivity of two metals on ρρ(tt) temperature. We use Au and Cu wires, with a low and high θθ DD, respectively, to illustrate the dependence in the low-t and high-t ranges. Cool with liquid nitrogen Measure the temperature dependence of ρρ(tt) and obtain the TT 5 dependence below θθ DD (CCCC) = 344 KK in Cu and TT above θθ DD (AAAA) = 170 KK for the Au wire. Page 3
4 3.2 Classical statistics of Johnson noise Here no voltage is applied to the resistor. Use the TT = 0.1 ss time constant and the 12 dddd/oooooo setting at the SR810 giving a ENBW 1 8TT = 1.25 HHHH [see manual]. Set no input in the LIA reference and press the unlock setting. Use short test cables with clamps with a BNC adapter going directly in the A-input Connect resistors 50 Ω, 179 kkω, 403kkΩ, 827 kkω across clamps, one after another and observe the results. VV You are measuring the fluctuations (or noise ) in the voltage (note: units are in, HHHH not VV). One cannot assign these fluctuations to the instrument noise. To measure the instrument intrinsic noise, disconnect any input and observe that the LIA has a noise of 6 nnnn. This can explain the fluctuations at the lowest RR, but not at high RR. You may be surprised to find that you have measured the fluctuations of low-frequency 1D black-body radiation modes in the wires. The predicted noise is given by VV2 expression applies the classical statistics of 1D black body radiation modes. Note: you have seen when smoothing a plot that noise decreases with decreasing bandwidth BW BBBB = 4kkkkkk. This Calculate the RHS in the expression above and compare to your measurements and the BW used 4. Conclusion The modes of lattice vibrations are occupied according to the quantum Bose Einstein distribution. This statistics reduces to the classical Boltzmann statistics for light modes. In a later experiment you will see the other (Fermi Dirac) quantum distribution. Note: the QM of blackbody radiation arises from correctly counting the states, not from the distribution. Specifically, in classical mechanics one will have an infinite number of possible states, in particular at high momenta (that is, energy) because there are more and more ways to insert a wavefunction with a smaller and smaller period in a box of fixed dimensions. In QM, one introduces a cutoff at a certain momentum, determined by h [Planck], so that the number of states is not VV xxvv pp ττ3, where ττ is a small arbitrary quantity, that cancels in measurable variables as in classical statistics, but VV xxvv pp. h 3 Exercise: calculate the oscillation modes of two and three coupled oscillators. Extend your analysis to an infinite number of coupled oscillators and to the low and high momentum qq limits of the dispersion relations, corresponding to acoustic and optical modes. Page 4
5 Name Phys-601 Quantum Mechanics Laboratory The Bose Einstein quantum statistics lab report Dates of measurements: Page 5
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