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MIT OpenCourseWare http://ocw.mit.edu 8.13-14 Experimental Physics I & II Junior Lab Fall 2007 - Spring 2008 For information about citing these materials or…

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MIT OpenCourseWare http://ocw.mit.edu 8.13-14 Experimental Physics I & II Junior Lab Fall 2007 - Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.14 Junior Lab Data analysis Assignment #2 2/6/2008 Due: at the start of session #4 Objective: Distinguish Physical phenomena from resolution effects Analysis Exercise: Gaussian or Lorentzian ? In the experiments of Dopplerfree, Moessbauer, QIP, and Zeeman you will have to fit line-shapes (or dips). From the Uncertainty Principle we know: ΔE Δt ≥ h(bar)/2, for a wave-packet, which translates into: Γ τ ≥ h for a resonant line with width Γ (FWHM) from a de-excitation exponential lifetime τ. There is an interesting relation between Γ and τ, by the fact, that the energy Fourier transform of an exponential decay in time results into a Lorentzian non-relativisticaly (as mostly the case in 8.14). An easy derivation follows: Emission of a Spectral line is described as a damped oscillator with ω0 represented by a time dependent amplitude: ∞ ∞ C2 f (t ) = C ⋅ e iω 0 t e−γ t with ∫ f (t ) dt = C 2 ∫ e−2γt dt = 2 = 1 hence C = 2γ −∞ −∞ 2γ Complete Amplitude: f (t ) = 2γ e iω 0 t e−γ ⋅ t ∞ γ γ i Fourier transform: F(ω ) = ⋅ ∫ e− iωte iω te −γt dt = 0 π −∞ π ω 0 − ω + iγ γ ∫ 1 F (ω ) = ( ) 2 2 Spectral intensity I(ω) with F ω dt = 1 π (ω 0 − ω ) + γ 2 2 Inserting E= hν =h(bar)ω and Γ= 2γ h(bar) we obtain the probability/energy, the Γ /2π Lorentzian I (E ) = I 0 (E0 − E ) + Γ2 / 4 2 Compare this line shape to a Gaussian of equivalent: FWHM = 2.354 ⋅ σ 1 ⎡ 1 ⎛ E − E ⎞ 2⎤ I (E,E0,σ ) = exp⎢− ⎜ 0 ⎟⎥ you will see a substantial difference. σ 2π ⎣ 2⎝ σ ⎠ ⎦ Bevington, page 33. Also read chapter 9. What is the point? Well, the physically interesting quantity is the “Natural Line Width Γ”, whereas resolution effects like Doppler broadening and/or stochastic errors will make the line-shape appear more Gaussian; – recall the Central Limit Theorem. So, if you want to claim that you measured a Natural Line, better prove that the shape is right. Then you also may state the lifetime of the excited state. Problem: 1) Fit the dataset ‘lineshape1.txt’ available from http://web.mit.edu/8.13/www/handouts.shtml under Ulrich Becker to both a Lorentzian PDF and a Gaussian PDF. Compare the fit results and chi‐squared values to determine the correct fit hypothesis. 2)Fit the dataset ‘lineshapedata.txt’ also available from http://web.mit.edu/8.13/www/handouts.shtml to both a lorentzian and Gaussian PDF’s, this time with the possible addition of DC and linear background terms. What are the chi‐squared values? Which fit hypothesis is justified ? For each case: 1. Produce a plot of the data with error bars, assuming Poisson statistics. 2. Make an educated guess for initial values for a fit to Gaussians plus background. 3. Perform the fit and retain all values. Give a plot with the fit. 4. Make a subtraction plot Data – Fit, to show the residuals. Comments? 5. As a check repeat with different starting values. Compare the results. 6. Optional: Can you give confidence limits for each hypothesis?

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