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PHYS 407 1
PHYS 407Senior Laboratory in Modern Physics
Techniques for recording, analyzing, and reporting data
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Outline
● Introduction● Experiments● Data Analysis● Presentations● Writing
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Introduction
● Scientific
● Logical
● Quantitative
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Being Scientific
● Acquaintance with the content of physics can be made by attending classes or studying books and journals
● Understanding physics comes from, among other things, doing physics━ Failures and frustrations are part of the process
● Good science exhausts the possibilities of a very limited problem; bad science deals superficially with broad or complicated matters
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Doing Physics?
There is no 'Scientific Method', but...
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Sound Scientific Practice
● Record every calculation and observation
━ Permanently and sufficiently ═►memory-proof
● Justify every step
● Anticipate contingencies
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Being Logical
● Formulate hypotheses
● Decide what constitutes evidence to gather
● Draw sound conclusions━ based on evidence━ incompatible with alternative conclusions
● Distinguish between facts and interpretations
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Being Quantitative
● Estimate with numerical approximations
● Provide values, uncertainties, and units for every relevant quantity
● Display information in graphs and tables
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Experiments I
● Investigate the properties of physical systems━ Aim to isolate a portion of nature, typically by
assuring it's sufficiently self-contained● Express properties as relations between
parameters━ First problem is to identify the relevant ones━ Those that are varied experimentally: variables
∘ Independent: adjusted∘ Dependent: assumes a value
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● Planned; based on a model━ Equipment━ Protocol
● Controlled● Recorded● Honest
━ Clear up doubtful results as questions arise
Experiments II
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Modeling
● Abstracts from the “real world”
● Includes only necessary features
● Exaggerates some aspects; ignores others
● Makes assumptions to simplify relationships
● Offers only limited, well-prescribed utility● May be adequate or inadequate, not right or
wrong
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Plan
● Identify the system and the model● Select the variables● Clarify the relationship● Determine ranges and increments
━ Coarse measurements over wide range, fine in regions of interest
● Consider the precision● Construct a measurement program
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Comparing Model and Nature
Properties of a model must be shown to correspond with the system under study before
proceeding to make conclusions.
Experiments answer the question:
Is the model good enough for our purposes at our level of precision?
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Experiments III
● Spectroscopy● Instrumentation
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Spectroscopy
● Spectrometer: source, sample, analyzer● Source particles incident on sample and
particles escaping after the interaction are analyzed
● Yields information about states of source or sample
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Nobel Prizes in Spectroscopy● Michelson 1907● von Laue 1914● Braggs 1915● Stark 1919● Siegbahn 1924● Lamb and Kusch 1955● Bloembergen and Schawlow, Siegbahn 1981● Brockhouse 1994● Hall and Hänsch 2005
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Spectroscopy Experiments
● Franck-Hertz● Zeeman Effect● Nuclear Magnetic Resonance● Electron Spin Resonance● Compton Scattering● Optical Pumping
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Instrumentation Experiments
● Coaxial transmission line● Waveguides● Optical fibers (??)● Laser spectra● Laser cavity modes (?)
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Franck-Hertz I
● James Franck and Gustav Hertz explored atomic energy levels, 1914
● Nobel Prize for both Franck and Hertz, 1925● Demonstrates that atoms absorb quantized
energy
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Franck-Hertz II
● Source: Electrons● Sample: Mercury vapor● Analyzer: Electrometer (ammeter)● Independent Variable: Accelerating voltage● Dependent Variable: Current● Controlled Parameters: Emission current,
ambient temperature, retarding voltage
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Frank-Hertz III
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Zeeman Effect I
● Pieter Zeeman discovers spectral line splitting in a strong magnetic field, 1896
● Nobel Prize 1902 (along with Lorentz)● Magnetic splitting of spectral lines● Fabry-Pérot technique named after Charles
Fabry and Alfred Pérot
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Zeeman Effect II
● Source and Sample: Light from excited mercury atoms
● Analyzer: Fabry-Perot optical resonator (etalon)● Independent Variable: Magnetic field● Dependent Variable: Bright line (energy level)
separation● Controlled Parameters: Spectrometer
configuration, meteorological conditions
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Zeeman Effect III
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Nuclear Magnetic Resonance I
● Molecular beam NMR described and measured by Isidor Rabi in 1938; solids and liquids by Felix Bloch and Edward Purcell in 1946
● Nobel prize for Rabi in 1944; for Bloch and Purcell in 1952
● Characterize non-zero spin nuclei in a magnetic field by their absorption and re-emission of electromagnetic radiation
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Nuclear Magnetic Resonance II
● Source: Radio waves● Sample: Any material with nuclear spin● Analyzer: Resonance detector (wire coil, eg.)● Independent Variable: Radio wave frequency or
time between source pulses● Dependent Variable: A characteristic of the
response (magnitude; time between peaks, etc.)● Controlled Parameters: Magnetic field,
temperature, etc.
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Nuclear Magnetic Resonance III
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Electron Spin Resonance I
● First observed (independently) by Yevgeny Zavoiski and Brebis Bleany in 1944
● Identify and characterize materials with unpaired valence electrons
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Electron Spin Resonance II● Source: Microwaves● Sample: Any paramagnetic material (small,
positive susceptibility to magnetic fields—atomic spins align with field)
● Analyzer: Microwave detector● Independent Variable: Magnetic field strength● Dependent Variable: Microwave signal strength● Controlled Parameters: Microwave frequency,
static magnetic field, temperature, etc.
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Electron Spin Resonance III
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Compton Scattering I
● Predicted and observed by Arthur Compton in 1923
● Nobel Prize 1927● Demonstrates the particulate nature of light by
showing it (inelastically) scatters off charged particles
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Compton Scattering II● Source: Gamma rays● Sample: Quasi-free electrons in aluminum● Analyzer: Scintillator-photomultiplier● Independent Variable: Angle between detector
and source-sample line● Dependent Variable: Photon (gamma ray)
energy or intensity● Controlled Parameters: Activity/flux of source,
photomultiplier voltage/gain
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Compton Scattering III
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Optical Pumping I
● Developed by Alfred Kestler, 1950s● Nobel Prize 1966● Light raises ("pumps") atomic or molecular
electron energy level
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Optical Pumping II
● Source: Radio frequency electromagnetic radiation
● Sample: Materials with single valence electron● Analyzer: Intensity detector● Independent Variable: Radiation frequency● Dependent Variable: Magnetic field strength● Controlled Parameters: Radiation polarization,
temperature, static field
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Optical Pumping III
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Data Analysis
● Describing the data
● Making inferences from the data
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Describing the Data
● Words (qualitative, descriptive)● Tables● Graphs● Numbers (measures of central tendency and
spread)● Trend lines and fits
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Words
● Say only what evidence supports
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Tables
● Informative title● Column and row labels including units● Typically (but not always) independent variable
on left, dependent variable along the top, with totals and averages to right and bottom.
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Graphs● Descriptive title● Independent variable on horizontal axis
(abscissa)● Dependent variable on vertical axis (ordinate)● Axes labeled, with units● Data points: small shapes, error bars preferred● Trend line must not obscure points● Theoretical line should be smooth, no points
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Correlated?
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Trends and Trend Lines
● Say no more, nor less, about a graph than the graph shows (don't call a trend exponential unless it is).
● Unless stating that trend line is a “guide for the eye”, show both equation and fit quality (R2)
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Only Straight Lines Can Be Identified
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Histograms
● Provide visual representations of distributions● Estimate probability distributions of continuous
variables● Height of bin = frequency
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Bin Number (n) and Width (w)
● No single standard
● Different binning can exhibit different features
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Binning Suggestions
For N measurements of x with SD s
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Descriptive Statistics
● Frequency Distributions● Measures of central tendency● Measures of dispersion or spread● Shape
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Probability
● Chance, expressed quantitatively● Assume an event is either a success or a
failure, the number of each is p and q● Probability of success:
● Probability of failure:
● P + Q = 1
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Alternative Definition
Imagine repeating the situation leading to an
event (infinitely) many times. Then P is the
fraction of successful events, and Q is the fraction
of failures.
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Combination of Events
The probability that a combination of events
occurs is the product of the probabilities of the
separate events, provided the events are
independent.
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Combinatorics
● Permutations
● Combinations
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Frequency Function
● Given a random variable x, f(x) is a frequency function when it gives the probably f(x0) that x=x0
● x discrete━ distribution function
● x continuous
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Probability of Measurement, z
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Single-Variable Frequency Functions
● A few parameters of a population frequency function can describe the entire population
● Moments of a histogram (with i bins)
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Mean
● First moment about the origin
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Variance
● Second moment about the mean
● Standard deviation
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RMS
● Root mean square deviation a minimum around average:
● Universal mean not sample average: rms not minimum; standard deviation better estimate
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Standard Deviation of the Standard Deviation
● Dispersion of sample standard deviation distribution
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Average Absolute Deviation
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● Easier to calculate but less meaningful than s
● Fractional (Relative) Standard/Average Deviation
● Fractional deviations unitless, but not significant
unless relative to a physically relevant zero
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Binomial (Bernoulli) Frequency Function
● Two possible outcomes━ p probability of one of the outcomes (A)━ q = 1 - p probability of the other outcome (B)
● In N trials, the probability of getting A x times:
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Poisson Frequency Function
● When N is large and p, the probability of a certain outcome, is small━ define y≡Np, which stays finite as N→∞, p→0━ the probability of finding x outcomes
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Gaussian (Normal) Distribution
h is the precision index
Total area under the curve is unity.
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Probable Error, PE
Value of Z such that
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Standard Deviation
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Average Absolute Deviation
so
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Median
● A location parameter: the middle value in (linearly) ordered data
● Use with skewed data and with outliers
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Dispersion Around a Median
● Range● Interquartile Range● Average Absolute Deviation● Median Absolute Deviation
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Mode
● Most common value in a frequency distribution● Not necessarily unique (as is the mean and
usually the median)● Useful for nominal data
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Dispersion from a Mode
● Variation ratio, VR
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Skewness
● Measure of a frequency distributions asymmetry
● The sign indicates the direction of the tail
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Some Definitions● Error
━ Difference between measured and “true” value━ Estimated uncertainty in a result
● Discrepancy━ Difference between two measured values
● Statistical (Random, Experimental) Errors━ Judgment━ Fluctuations (in conditions)━ (Small) Disturbances━ Definition
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● Systematic (constant) Errors━ Mis-calibration━ Habits━ Conditions━ Technique
● Illegitimate Errors━ Blunders━ Computation━ Chaos (disturbances overwhelm random errors)
● Determinate (Indeterminate) Errors━ May be evaluated by some logical procedure
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● Corrections━ Compensation of determinate systematic or random
errors● Precision
━ Small random errors● Accuracy
━ Small systematic errors● Data Adjustment
━ Determining the most probable value from the data● Residuals (Deviations), δ
━ Differences between measured and most probable values
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Statistical Uncertainties● The arithmetic average of deviations from the
most probable value is zero● For directly measured quantities, we assume
the most probable value is the arithmetic average
● The sum of squared deviations about the average value is a minimum
● While plots of deviations give a qualitative sense of uncertainty, a characteristic quantity is preferable.
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Rejecting Data
● To first approximation: DON'T DO IT
● If data collection was disturbed, reject the data
—even if they seem reasonable.
● If P(|z| > Z) « 1/N, then measurement might be
questionable, but with small samples, the
Gaussian parameters are not very well known.
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Uncertainty Propagation● V = V(x, y), x and y measured with uncertainties
dx and dy, which may or may not be independent or correlated
● Nonindependent uncertainties: externally caused; incalculable with deviations
● Correlated uncertainties: associated with individual measurements in which independent and nonindependent parts are inseparable; calculable with correlation coefficient
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Correlation Coefficient
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Combining Independent and Correlated Uncertainties
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Combining Nonindependent Uncertainties
Completely correlated uncertainties reduce to this
Nonindependent and correlated uncertainties are independent of one another and combine in
quadrature
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Estimating Systematic Uncertainties
● Determinate━ Apply suitable correction to data━ Additional random uncertainty
● Indeterminate━ Guess or estimate magnitudes of all possible
sources━ Combine quadratically━ At least give the sign and the effect on the sign of
computed quantities
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Designing an Experiment
● Instruments and techniques the leave the uncertainty on V less than a certain value
● If K independent, nearly equally uncertain quantities contribute to V, then each must have
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● Typically, no more than 2 uncertainties dominate, and these are seldom equal:
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Weighted Averages
● Quantities with different uncertainties should not be simply averaged, but should be weighted by factors inversely proportional to the respective uncertainties squared:
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Counting Experiments● When the time to count N events is subject to
statistical fluctuations (e.g., radioactive decay)━ Counting rate:
━ Probability distribution, Poisson:
━ Average and standard deviation:
━ Counting rate standard deviation:
━ Fractional deviation:
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Counting with Backgrounds
●
●
● For t + tB fixed, the optimal time ratio with and
without the source (signal + background vs
background only):
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Determining the Frequency Function with Data
● Maximum Likelihood
● Least Squares
● χ2
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Maximum Likelihood: Determining a Parameter of a Frequency Function● Assuming
● Likelihood
● Log-likelihood
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Least -Squares: Determining the Relationship between Two Variables● y is related to x by a function with parameters
a1...aν
● Assume yi belong to a Gaussian population with standard deviation σi
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Again, Maximize Log-Likelihood
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χ2 Distribution
ν degrees of freedom
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χ2 Frequency Function
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Goodness of Fit
● Functional relations
● Classes of occurrences
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Comparing
● Values agree if their absolute difference is within the limits of uncertainty; otherwise they disagree
● For individual values, the central value(s) from experiment compared with reference to experimental (and theoretical) uncertainty
● For distributions, goodness of fit is given by a χ2
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Presentations
● Prepare adequately● Simplify and unclutter slides● No more than one point per slide● Graph and illustrate as in papers● Minimize mathematics; never derive equations● Define terms; never use jargon or acronyms● Progressively elevate sophistication
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Writing● Important points distinct from details● Fact separated from opinion and speculation; state
assumptions and approximations● Figures and tables numbered and captioned;
included only if referenced● Equations numbered; derivations in appendices● Consistent tense and person● No colloquialisms● Specific and quantitative (not vague and
qualitative); simple (not ornate or jargon-filled)
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Heading
TitleNameDate
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Abstract
● Brief summary of entire paper● Contains main points
━ Motivation━ Technique━ Results
● Contains no information not found in paper● References nothing outside of abstract
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Body
● Sectioned with headings● Pages numbered● Tables and graphs summarize data; axes and
column labels include units; labeled, captioned, and referenced in text
● Footnotes● Appendices● Bibliography
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