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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

●

● 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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