Lab course Basic Physics

MARTIN-LUTHER-UNIVERSITÄT

HALLE-WITTENBERG

INSTITUT FÜR PHYSIK

GRUNDPRAKTIKUM

Lab course

Basic Physics

FOR

POLYMER MATERIALS SCIENCE

EDITION 2015

Preface

The lab course „Basic Physics and Physical Chemistry Lab” is intended for those masterstudents of Polymer Materials Science who don't have the Bachelor's-degree in Physics.

The subjects of the course are

• planing, performing and evaluating scientific experiments; record writing; estimation ofmeasurement uncertainties

• working with modern measurement technique

• selected physical topics of basic physics and physical chemistry

The course consists of

• one lecture (2 h) introducing to statistics and uncertainty analysis

• one tutorial experiment (4 h, exercise in statistics and uncertainty analysis and introduction tothe software Origin)

• fife experiments in physics (4 h each)

• four experiments in physical chemistry (4 h each)

This booklet describes the experiments in the Physics part of the course only.

The introduction chapters are related to all general aspects of the physics lab course (safety inthe lab, requirements to protocol writing and estimation of measurement uncertaunties, literatureand software). The main part describes shortly the physical basics of each experiment and givesdetailed instructions to experimenting and evaluating the results. The questions at the end ofeach experiment description are mainly intended for your self check. Depending on yourknowledge in Physics, you need to study the basic principles of an experiment using additionaltextbooks.

Martin Luther University Halle-WittenbergInstitute of PhysicsPhysics Basic Laboratory

http://www.physik.uni-halle.de/Lehre/Grundpraktikum

Editor:Martin Luther University Halle-WittenbergDepartment of Physics, Basic Laboratoryphone: 0345 55-25471, -25470fax: 0345 55-27300mail: [email protected]

Authors:K.-H. Felgner, H. Grätz, W. Fränzel, J.Leschhorn, M. Stölzer

Head of Physics Lab: Dr. Mathias Stölzer

Halle, October 2015

Contents

INTRODUCTION

Laboratory rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Procedure of a Laboratory Course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Guidelines to Writing a Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

uncertainty analysis and statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

English Literature on Basic Experimental Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Software in the Basic Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

EXPERIMENTS

E 40 RLC Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

M 14 Viscosity (falling ball viscometer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

O 6 Diffraction spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

O 10 Polarimeter and Refractometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

O 16 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

O 22 X-ray methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Introduction Laboratory Rules

1

Laboratory Rules

General Rules

1 When working in the laboratory do notendanger other persons and make sure notechnical devices or experimental arrange-ments get damaged.

2 The instructions given to you by thetutor or the laboratory staff and those writtenin this booklet regarding the use of devicesand experimental arrangements are strictly tobe observed.

3 Please report any troubles, irregularities,damages to or malfunctions of devices aswell as accidents to the tutor. You are notallowed to repair any devices by yourself!

4 You are to account for any damages ondevices or materials caused wilfully.

5 Use the equipment available at yourworkplace only. You are not allowed to useany equipment from other workplaces.

6 After finishing the experiment clean upyour workplace. Log off from any computeryou have used.

7 Eating and drinking is prohibited in thelaboratory rooms. The whole laboratorybuilding is a nonsmoking area.

8 The use of mobile phones in the labora-tory rooms is prohibited.

9 Laboratory courses start in time accord-ing to the timetable. If you are more than 15minutes late the staff may enjoin you fromstarting an experiment.

10 For finishing the course successfully youneed to perform all experiments. If for aserious reason (e.g. due to illness) you cannot attend the laboratory course, pleaseinform the laboratory staff and arrange anextra date to perform the missed experiment.Dates will be granted for the lecturing time ofthe current semester only.

Working with electrical circuits

11 Assemble and dismantle electricalcircuits with disconnected voltage (powersupplies off, batteries not connected, etc.)only. Clearly organize the circuit structure.

12 When working with measuring instru-ments, pay special attention to the correctpolarity, to the correct measuring range anduse the correct measuring inputs (danger ofoverloading and damaging).

13 You must have electrical circuits check-ed by the tutor before putting them intooperation.

14 Energized systems are to be supervisedpermanently.

15 Do not touch any components carryingelectric voltages. Dangerous voltages (>42V)are generally protected from being touched. Donot remove or short-circuit those protectiveequipments!

16 In case of an accident switch off the powerimmediately! (There is a yellow emergencyswitch in every room.) Report the accidentimmediately.

Working with chemicals

17 Work cleanly. If necessary use a funnel fortransferring liquids and absorbing pads forweighting chemicals.

18 Any safety materials (i.g. safety goggles)given to you with the experimental accessorieshave to be used!

19 In case of accident or spilt dangerouschemicals (e.g. mercury) inform the tutor orlaboratory staff immediately! Do not removethose spilt chemicals yourself!

20 All chemicals are stored in containersmarked with a content description. Make sure

Introduction Laboratory Rules

2

you always use the correct container, especiallywhen pouring the chemicals back into thecontainers after usage.

21 After finishing the experiment, carefullyrinse all used containers (except containersused for storing materials).

Working with radioactive material

22 A sealed radioactive radiation source(74 kBq Co-60) is used in the experimentO16. It is allowed to be handled by students.The radiation exposure during the experimentis 100 to 1000 times lower than during amedical X-ray examination.

23 Nevertheless avoid any needless expo-sure! Do not carry the radiation source inyour hand if not necessary! Keep a distanceof 0.5 m to the radiation source during theexperiment!

24 It is prohibited to remove the radiationsource from the surrounding plexiglass block.

Preventing fire

25 Place Bunsen burners or electric heaterssecurely so that neighbouring devices willnot catch fire. Permanently supervise openfire and heaters.

26 Take care when working with flammableliquids (for example ethanol)! Keep them awayfrom open fire!

27 In case of a fire, inform the supervisorassistant immediately and take first measures toextinguish the fire.

28 You are required to know where to findthe fire extinguisher, how to use it and whichescape routes and exits can be used.

Procedure of a Laboratory Course

1 Preparation

The subject of your next experiment is foundon the laboratory home page on the Internetor on the notice board in the corridor.

Prepare yourself at home. Study the physicalbasics of the experiment and prepare a proto-col (see Guidelines to Protocol Writing).

2 Starting a laboratory day

Be on time. Students who are more than 15minutes late may be excluded from perform-ing the experiment.

You are given the experimental accessoriesnecessary for your group on depositing a stu-dent card.

The tutor will inspect your prepared protocol

and question you shortly about the physicalbasics of the experiment. Students who arenot prepared are not allowed to work in thelaboratory at the present day.

3 Performing the Experiment

Experiments are carried out in groups of twostudents. Each student writes his or her ownprotocol.

Construct the experimental setup. Pleasehave electrical circuits checked by thetutor before putting them into operation.

Perform the measurements and keep recordsof the results, observations and notes. Ask thetutor if you need assistance.

The tutor will check your results and authen-ticate it with his short signature.

Basic Physics Lab Course WS14/15 Guidelines to Writing a Protocol

3

4 Finishing a laboratory day

Clean up your working place. Give the acces-sories back and receive your students card.

The tutor must sign your records (see above).

5 Evaluation of the experiment

You will need a pocket calculator, a ruler andpossibly graph paper. Graph papers (e.g.logarithmic paper) can be bought in thelaboratory.

Write (or at least start writing) the evaluationof the experiment during the laboratorycourse. You are expected to have completedthe experimental evaluation by the start ofyour next laboratory day.

6 Review of the record

Usually the tutor will review your recordduring the next laboratory day. You will begiven a mark (from 1 = excellent to 5 = fail )that takes account of your preparation (yourknowledge and the prepared protocol), yourexperimental work and your evaluation.

The mark is written into the protocol togetherwith the full signature of the tutor.

The completely evaluated protocol must bepresented not later two weeks after the dateof the experiment. For every additional weekyour mark is downgraded by one.

7 Finishing the laboratory course

You need to completely perform 5 experi-ments with grade 4 or better.

The successful finishing of the laboratorycourse is certificated.

Guidelines to Writing a Protocol

General

• Each student keeps his own record duringthe laboratory work. Please use a file ornotebook in the size of A4.

• Use ink or ball pen for writing the record.Write immediately into the protocol, donot use extra paper. If you've made amistake, mark the notes or values as beingwrong for a particular reason but do noterase them.

• Graphs are drawn by pencil on graph paperor printed by a computer, respectively.Label them with your name and date andinclude it into the protocol.

Preparation at home

• Prepare your record with the followingcontent:- Date, name of the experiment and the

exact task.- A short description of the experiment,

including the formulas necessary forunderstanding and evaluating the taskand a sketch (if applicable, i.g. an elec-trical circuit).

- Prepared tables for recording the mea-sured and (if required) the calculatedvalues.

• This part of the record will be supervisedat the beginning of your laboratory work.

Introduction Calculation of Errors

4

Recording during the experiment

• List all devices used in the experiment.

• Keep your record clear and readable.Distinguish the different parts of the ex-periment clearly.

• Introduce all physical quantities with theirname and symbol. In graphs and tables,write physical values with their symboland unit of measure.

• Write all measured values (before anycalculation is done) into the protocol.

• A protocol is complete, if even somebodyelse who did not perform the experimentcan understand it and evaluate it.

Evaluating the results

• All calculations should be comprehensible.(It has to be clear which result was calcu-lated from which data by which equation.)

• Graphs should be drawn clearly on graphpaper using a ruler or made by computer.The axes have to be labelled with thesymbol and the unit of measure.

• Estimate the experimental errors quantita-tively. In some experiments, an uncertaintyanalysis is required.

• Write your results and the errors estimatedin a whole sentence and discuss themcritically. If possible, compare your resultsto table data.

• The protocol will be supervised on thenext laboratory day. It will be certificated(grade and signature of the tutor) if theexperiment was completely performed andevaluated.

Uncertainty Analysis and Statistics

Any measurement of a physical quantity isimperfect. If a quantity is measured repeat-edly, the results will generally differ fromeach other as well as from the “true value”that is to be determined.The objective of the error analysis is todetermine the best estimation of the truevalue (the measurement result) and an esti-mation of the deviation of the result from thetrue value (the measurement uncertainty).

1 Definitions

Measurand:A particular physical quantity subject tomeasurement; e.g. the mass m of a givenbody, the voltage U of a battery at 20°C

Value (of a quantity):Magnitude of a particular physical quan-tity, expressed as a number multiplied bya unit (of the quantity), e.g. mass of thebody: 2.31 kg

True value (of a quantity):

Value consistent with the definition of agiven particular quantity. It would beobtained by a perfect measurement.

Result of a measurement:Value attributed to a measurand, obtainedby measurement. It may be considered tobe an estimation of the true value. In general, the result should not be givenwithout additional information about itsuncertainty and the way it was obtained.


5

1

1 .n

ii

x xn

(1)

Error (of meassurement):Measurement result minus true value.The error consists of a random and asystematic part. Generally, the measure-ment error can not be known exactlybecause the true value is not exactlyknown. The “error calculation” results inan estimation of the error.

Random (or statistical) error:Measurement result minus the mean thatwould result from an infinite number ofmeasurements of the same measurand.The random error varies in magnitudeand sign. It arises from uncontrollablevariations of the experimental andenvironmental conditions, from physicallimits of observation (e.g. noise, quantumeffects) and from the limits of humansenses. Random errors can be minimizedby taking multiple measurements.

Systematic errors:Mean that would result from an infinitenumber of measurements of the samemeasurand minus its true value.The systematic error is constant as longas the controllable conditions of theexperiment remain constant. It arisesfrom imperfections of the devices, cali-bers, measurement procedures, and fromsystematic changes in the experimentalconditions. It may consist of a known andan unknown part. The measurementresult has to be corrected by the knownpart of the systematic error (“correctedresult”).

Uncertainty (of meassurement):Parameter, associated with the measure-ment result, that characterizes the disper-sion of the values that could reasonablyattributed to the measurand. It may beconsidered to be an estimation of themeasurement error. The measurementuncertainty is estimated on the basis ofthe measured values (by statistical meth-ods) and the knowledge about systematicerrors.

The uncertainty of x is traditionally writ-ten Δx und according to the new inter-national standard GUM u(x).

Example for m = 2.041 g:

u(m) = 0.002 g (absolute uncertainty)u(m)/m = 0.1% (relative uncertainty)

The true value is with high probabilityexpected in the interval [m u(m), m+ u(m)].

Complete measurement result:Measurement result ± uncertainty, theuncertainty given with 1 or 2 digit accu-racy. Allowed notations are:m = 2.041 g ± 2 mgm = 2.041 g ± 0.002 gm = (2.041 ± 0.002) gm = 2.041(2) gm = 2.041 g and u(m)/m = 0.1 %

2 Determination of uncertainties

The determination of the accuracy ofmeasurements is based on different sources ofinformation: The statistics of measured val-ues, manufacturer's information and certifi-cates of measuring instruments and standardsused, or simple estimate. In every case theuncertainties should be characterized bystandard deviations - either calculated bystatistical methods (type A evaluation) orestimated by other means (type B evalua-tion). To underline this fact, they are oftencalled standard uncertainties.

2.1 Type A evaluation on uncertainty

A quantity x is measured n times. Because ofrandom errors, the individual measured valuesxi (i = 1 ... n) will scatter around an expecta-tion μ. In most cases the values are approxi-mately normal distributed (see fig.1). The bestapproximation of μ is then the arithmeticmean


6

2

1

( )

.1

n

ii

x x

sn

(2)

.x

ssn

(3)

2

1

( )

.( 1)

n

ii

x

x x

u x sn n

(4)

A measure of scattering of the values xi is thestandard deviation σ. The best estimation ofσ calculated on the basis of the xi is theexperimental standard deviation (sometimesdenoted as sample standard deviation)

Within the interval ± s are approximatelyx68 % of all normally distributed values. Inother words: The probability to find a singlemeasured value in that interval is 68 %. Forthe interval ± 2s this probability is aboutx95 %.

If random error are dominating (i.e. system-atic errors are negligible), the uncertainty of asingle measured value is u(x) = s.The mean of n single measurements isxmore accurate than one single measurement:Imagine that many series of n measurementsare taken. The respective means wouldslightly differ from each other. They arenormal distributed as well, and a measure oftheir scattering is the standard deviation ofmeans

If the result of a measurement is the mean xof a sufficient number (n 10) of singlemeassured values xi and systematic errors arenegligible, the (standard) uncertainty of this

result is:

More examples of type A evaluation ofmeasurement uncertainties will be given inchapter 3 (Regression Analysis).

2.2 Type B evaluation on uncertainty

If the calculation of a standard deviation is notpossible (e.g. if the uncertainty is dominatedby constant systematic errors or if only onemeasured value exists), the standard uncer-tainty is to be estimated on the basis of allgiven information.

2.2.1 Manufacturer guaranteed toleranceThe manufacturer of a measuring device usu-ally guarantees the measurement accuracywithin a certain tolerance or maximum error.Examples: 1.5 % of full scale; 0.1 % ofreading + 2 digit. Sometimes the so-calledaccuracy class is specified which is the guar-antied accuracy in percent of full scale or ofthe value of the material measure. Denoting the given tolerance of a measurandx by t(x), the standard uncertainty is due to

Explanation: The only known fact is that the measurementerror is not larger than t(x). Hence a uniformdistribution with the width 2 t can be attrib-uted to the measurand x. The standard devia-

tion of this distribution is .3t

2.2.2 Uncertainty of counter tube measure-mentsIf a measurement is consists of countingrandom events in a given time interval (e.gcounting the pulses of a Geiger counter formeasuring radioactivity), the standard uncer-tainty of a result N (without consideringsystematic errors) is

f(x)

x

Fig. 1: Normal distribution with the mean μand the standard deviation σ

3 .u x t x (5)

.u N N (6)


7

2

2

1

.n

c iii

yu y u

x

(8)

2 2

2 21 2

1 2

.cu y u un m

y x x

(12)

2 2 2 21 1 2 2 .cu y c u c u (10)

Explanation: See experiment O16.

2.2.3 Estimation of uncertaintiesWhen using very simple measurement de-vices (e.g. ruler, thermometer), there is oftenno information about the accuracy. In suchcases the standard uncertainty (not the maxi-mum error!) is to be estimated:S rule of thumb for reading scales: u(x)

0.5 division,S Length measurement with a Vernier cali-

per: u(l) = 1 division of the vernier scaleS time measurement with a stopwatch (by

hand): ΔT 0.1 s

2.3 Uncertainties of measurement results(uncertainty propagation)

We consider a measurement result y that isto be calculated from the measured values x1,x2, ..., xn with the respective uncertainties u1,u2, ..., un according to y = f(x1, x2, ..., xn).What is the uncertainty u(y) of that measure-ment result?

2.3.1 The maximum uncertaintyA small change Δxi of a value xi would cause

a change of in y, where i ii

yy x

x

i

y

x

denotes the partial derivative of the functiony = f(x1, x2, ..., xn) with respect to xi. If weassume the uncertainties ui to be small com-pared to the values xi, the maximum uncer-tainty u(y) (its upper limit) is the positive sumof the influences of all ui on y

This equation may only in simple cases (fewmeasurands) be used for estimating theuncertainty of a measurement result.

2.3.2 GAUSS's law of propagation of un-certaintyIn general the uncertainty components willnot always add as in eq. (7), they may alsopartially compensate each other. The exact

mathematical treatment of this problem byC. F. GAUSS, assuming statistical independ-ence of the measurands xi, results in the „lawof error propagation”

uc(y) is called combined standard uncer-tainty. Eq. (8), (10) or (12) are to be used forthe evaluation of uncertainties of measure-ment results in the Lab course in most cases.

2.3.3 Simple casesOften the function y = f(x1, x2, ..., xn) is verysimple. In two special cases the calculation ofthe combined uncertainty according to eq. (8)can be very much simplified:

case I: 1 1 2 2y c x c x (9)

(c1, c2 are constants)By inserting (9) into (8) we find:

case II: 1 2

n my c x x (11)

(c real and n, m integer numbers)Inserting (11) into (8) results in a simpleequation for the relative combined uncer-tainty:

Example:In an uniformly accelerated motion the dis-tance d depends on time t like d = a/2 t2. If dand t are measured with their correspondinguncertainties u(d) and u(t), and a is to becalculated, we get

.

2 2

22 , 2

u a u d u tdaa d tt

1

1 1

( )n

n i

n ii

y y yu y u u u

x x x

(7)


8

.y f x a b x (13)

F a b y

y a bx

i

n

i ii

n

( , )

( ) min.

2

1

2

1

(14)

ax y x x y

n x x

ny b x

i i i i i

i i

i i

2

22

1(15a)

b

n x y x y

n x x

i i i i

i i

2

2 , (15b)

s

y

n

x

n x xa

i i

i i

2

2 2

222

(16a)

s

y

n

n

n x xb

i

i i

2

2

222

. (16b)

3 Regression (fit) of a function to aseries of measurements

3.1 Linear regression (linear fit)

Frequently, different measured quantities xand y are linearly related or such a relation issupposed to exist:

Example:The length of a metal rod depends on thetemperature according to l = l0 + αl0ΔTwhere α is the coefficient of linear expansion,ΔT = T T0 and l0 is the length at T = T0.

The actual task of measurement is to deter-mine the (constant) parameters a and b. Inprinciple, a and b can be calculated from twopairs of measured values (x, y). In most cases,however, a whole series of measurements (npairs of values (xi, yi), i = 1 ... n) is taken forverifying the linear relation. In a graphicalrepresentation the points (xi, yi) will scatteraround a straight line, because of the un-avoidable random errors. The task is now tofind the straight line that „fits best” themeasured points (see fig. 2). The deviation between the measured point(xi, yi) and the straight line at xi is

Δy = yi - y(xi) = yi - (a+bxi).

According to GAUSS's method of least squa-res (where it is assumed for simplificationthat only the values yi are scattering), the beststraight line is found by minimising the sumof squares of the Δy:

This sum is a function of the two parameters aand b. The minimum is found by setting thepartial derivatives F/a and F/b equal tozero. In this way we obtain

where all sums are taken from i = 1 to n. The line defined by (13) and (15) is called theregression line.If the statistical errors predominate the sys-tematic errors, the uncertainties of the para-meters a and b are given by their standarddeviations: u(a) = sa and u(b) = sb with

3.2 Regression analysis with other func-tions

The method of least squares (14) is not re-stricted to straight lines as in eq. (13) but canbe applied to all functions with any number ofparameters. In general the problem cannot besolved analytically but must be solved numer-ically. Numerical methods for doing this„nonlinear regression analysis” are imple-

o u

o u

y yb

x x

xo

xu

yu

yo

yi

x

y

0

y = a + b·x

a

Fig. 2: Linear regression


9

and .y u y u y y (17)

mented in many scientific software such asthe programs Origin and CassyLab that areavailable in the Basic Laboratory. Look forthe keywords non-linear curve fit or free fit inthose programs.Some functions can easily be transformedinto a linear function. In this case, the linearfit may be performed on the transformedfunction.

Example (taken from the experiment O16):When radiation penetrates matter it is attenu-ated according to I = I0e

-μx (I: intensity, x:thickness penetrated, I0: I at x=0, μ: attenua-tion coefficient). The equation is transformedinto a linear equation by taking the logarithm:ln I = ln I0 μ x. If several pairs of values(I, x) have been measured, μ may be deter-mined by plotting ln I versus x. μ is thenfound as the negative slope of the regressionline.

3.3 Practical hints

You don’t need to keep the formulas (15) and(16) in mind, evaluations are usually done bysoftware. You need to know the basic princi-ple of regression analysis, and the meaning ofthe parameters a, b, sa and sb.Many scientific pocket calculators allowlinear regression, check the manual of your

calculator. The standard deviations sa and sb

are usually calculated by computer softwareonly.In many cases (if suitable software or pocketcalculator is not available or not required) itis sufficient to determine the regressionparameters a and b graphically in the follow-ing way:Plot the measured points into a coordinatesystem on graph paper and draw the best fitline according to visual judgement using atransparent ruler.

5 Presentation of measurement resultsand uncertainties

Always present the complete measurementresult:

The uncertainty u(y) has to be given with anaccuracy of one or two digits and the accuracyof the result y has to be chosen accordingly.

Examples:

y = (531.4 ± 2.3) mm, u(y)/y = 0.43 %

U = (20.00 ± 0.15) V, u(U)/U = 0.12 %

R = 2.145 kΩ ± 0.043 kΩ, u(R)/R = 2.0 %

Introduction Available Literature on Basic Experimental Physics

10

English Literature on Basic Experimental Physics

Library: ULB, Zweigbibliothek Heide-Süd, Von-Danckelmann-Platz 1

General Physics

Physics for Scientists and Engineers (Physics 5e)Paul A. Tipler, Gene P. Mosca

MODERN PHYSICS (Modern Physics 4e)Paul A. Tipler and Ralph A. Llewellyn

Physics for scientists and engineersPaul M. Fishbane. - 2. ed., extended. - Upper Saddle River, NJ : Prentice Hall, c 1996

Physics for scientists and engineersDouglas C. Giancoli. - 2. ed.. - Englewood Cliffs, N. J. : Prentice Hall, 1988

Thermodynamics for engineersKau-Fui Vincent Wong. - Boca Raton, Fla [u.a.] : CRC Press, c200

Basic optics for electrical engineersClint D. Harper. - Bellingham, Wash. : SPIE, 1997

Laboratory Work and Error Analysis

Practical PhysicsG. L. Squires. - Cambridge University Press, 2001

The art of experimental physicsDaryl W. Preston. - New York [u.a.] : Wiley, 1991

Experimentation and uncertainty analysis for engineersHugh W. Coleman. - New York [u.a.] : Wiley, 1989

GUM 2008: Guide to the Expression of Uncertainty in Measurementhttp://www.bipm.org/en/publications/guides/gum.html

Other Literature

Math refresher for scientists and engineersJohn R. Fanchi. - 2. ed. - New York, NY [u.a.] : J. Wiley, 2000

Physical properties of materials for engineersDaniel D. Pollock. - 2. ed. - Boca Raton, Fla. [u.a.] : CRC Press, 1993

Introduction Software in the Basic Laboratory

11

Software in the Basic Laboratory

All software used in the basic lab may be freely used (with some limitations) on privatecomputers.

Programs made by the educational systems manufacturer LD Didactic GmbH can bedownloaded from their website http://www.ld-didactic.com. These are:- CASSY Lab (used in W12 - humidity)- X-Ray Apparatus (used in O22 - x-ray methods)- Digital Counter (used in O16 - radioactivity)

For evaluating and plotting experimental results, the professional data visualisation and analysissoftware Origin 2015 (partly still Origin 8) is available on all computers in the lab and in thestudents computer pools. The university owns a campus licence that allows the use even onprivate computers, provided there is a VPN connection to the university network (ask the staffin the lab for technical details). Alternatively, there is the free Origin clone SciDAVis(http://scidavis.sourceforge.net/). This program runs on Windows, Linux and Mac OS and canread and write Origin files up to version 7.5 (unfortunately not compatible with the workbooksintroduced in Origin 8 ).

Short introduction to Origin (based on Origin 8)

1. General aspects

• All data, calculations and graphs are saved together in a project file. An empty project (atprogram start) contains only the workbook Book1 with one x and one y column for data

input. More columns can be added with Add New Column or , more Workbooks with

File - New or .

• The fastest way to get a graph: Select one or more y columns and choose Plot from the menu

or klick one of the buttons .

• All objects (e.g. column names and labels, axis labels, curve styles, legend) may be edited bydouble-clicking them.

• If the program starts in German language, switch to English via Hilfe - Sprache ändern... andrestart the program.

2. Workbooks

• Get more columns with .

• Give meaningful names to columns: Double-click the column header and enter Long Name

and Units. These are automatically used in graphs and legends.

• Denominating a column as x or y: Mark the column, right-click it and choose Set As .

Introduction Software in the Basic Laboratory

12

• Calculations with columns: Mark the column, right-click it and choose Set Column

Values... Syntax: column A column B write as col(A) col(B)

a b / (c + d) a * b / (c + d)x2 x^2x sqrt(x)ex exp(x)π pi

3. Graphs

• In Origin a coordinate system is called a layer. One graph may contain one or more layers.

• Refurbishing a graph: Double-click all things you want to change.

• Adding a curve to an existing graph:Way 1: Select the columns to plot in the worksheet, click into the graph (into the layer),choose from the menu Graph - Add Plot to Layer.

Way 2: Double.click the layer icon in the upper left corner of the graph and follow thedialogue.

• Adding a coordinate system or an axis to an existing graph: Choose Edit - New Layer(Axes)

or klick one of the buttons .

• Add a legend or refresh an existing legend: Choose Graph - New Legend or press .

• Write text to your graph with the tool. Use the format toolbar for Greek and indices.

• Read values from a graph with the Screen Reader or Data Reader .

• Drawing smooth curves through measurement points: Double-click the curve, choose theplot type Line+Symbol and select Line - Connect - Spline or B-Spline.

• Linear regression: Choose Analysis - Fit Linear. If there are more than one curve in thegraph, select the right one in the Data menu before. If only a part of the curve is to be fitted,

define the range before with the Data Selector and the Mask Tool .

4. Printing graphs and worksheets

• Check your graphs before printing (or have it checked by the tutor). Print only once for eachstudent. Wasting paper costs money and pollutes the environment. Do not print very largeworksheets (many pages).

• Combine several graphs and worksheets on one layout page: Choose File - New... Layout or

click . Right-click the layout to add graphs and worksheets.

• prints a graph or worksheet immediately on an A4 sheet.

Electricity E 40 RLC oscillator

13

1 Task

1.1 Learn how an oscilloscope is operatedfor measuring voltage, time and frequency.

1.2 Measure the natural frequency f0 and theattenuation α of a RLC-oscillator and recordthe resonance curve.

2 Physical Basics

2.1 The oscilloscope (or scope) is a veryversatile instrument with many applications,which allows the visualisation of rapidlychanging electrical signals on a screen (seefig.3 next page). In most applications, thevertical (Y) axis of the screen represents avoltage and the horizontal (X) axis the time.X may also represent another voltage. Theintensity or brightness of the display is some-times called the Z axis.

Usually, an oscilloscope is capable of show-ing (at least) two signals at the same time,which is well-suited for testing electriccircuits by comparing input and outputsignals.

A classical (analogue) oscilloscope basicallyconsists of a cathode ray tube (CRT, some-times also called BROWN's tube). In the CRT,an electron beam is formed by an electrodesystem. The electron beam is passing through

a deflection unit, consisting of two pairs ofmetal plates arranged in horizontal and verti-cal direction. When a voltage is applied to aplate pair, the electron beam is deflected bythe electric field. The beam then hits thescreen of the CRT, where it causes the lumi-nescent coating to glow. Since the deflectionangle is proportional to the applied voltage,the magnitude of that voltage can be mea-sured on the screen. In modern digitaloscilloscopes the voltages are measured byvery fast A/D converters and the results areshown on a computer screen. Operation, lookand feel is very similar to classical devices.

For measuring voltages over a wide range(from mV to V), the oscilloscope is equippedwith adjustable amplifiers. The amplificationis selected with the knob VOLTS/DIV (seefig.3) that controls the Y scaling factor inVolts per grid unit (1 cm).

For drawing an U(t) graph ( voltage vs. time),a so-called sweep voltage is applied to theX-plate pair. During a certain time period(the rise time or sweep time), this voltageconstantly increases and hence guides theelectron beam in the x-direction over thescreen with a constant rate. Subsequently, thevoltage drops to zero, and the beam thereforereturns to its starting position. The voltage tobe measured is applied to the Y-plate pair.

Fig.1: Cathode ray tube (CRT)

RLC oscillator E 40


14

Fig.2: Sweep voltage for X deflection withthe period Ts

Fig.3: Front panel of the HM303-6 oscilloscope. The square divisions on the screen are 1 cm ×1 cm

During the rise time, the electron beam writesthe graph of the function U(t) on the screen.This graph is refreshed every new period ofthe sweep voltage.

The time base knob (TIME/DIV) allows you tochange the sweep time over a wide range(2s...0.1μs). With this knob you select the Xscaling factor which is the time for a horizon-tal deflection by one grid unit (1 cm).

For obtaining a stagnant pattern from periodi-cally changing signals, one period of thesweep voltage must be an integer multiple ofone period of the measured signal. Thissynchronisation is performed by a componentcalled the trigger. The sweep pulse is trig-gered when the signal voltage reaches acertain level (which can be controlled by theLEVEL knob).

Like most oscilloscopes the HM303-6 isequipped with two identical input channels.Additional controls are provided for switch-ing between one Y-t graph (CH I/II), two Y-tgraphs (DUAL) or X-Y graph. In X-Y mode thetime deflection is disabled, the input CH I isapplied to the X-plates and the input CH II tothe Y-plates.

The front panel of the oscilloscope is clearlyorganised: there are groups of knobs andbuttons responsible for channel I (Y1 or X)and channel II (Y2) input, for the time deflec-tion, the trigger control, and for the operationmode. The Y (volts) and X (time) deflectionare adjusted by a rotary switch and a continu-ous rotary knob. The labels on the rotaryswitch are only valid when the rotary knob isin its rightmost position (position CAL). Theswitch AC-GND-DC at the signal inputs selectsthe coupling of the measured signal to thepre-amplifier: for direct coupling (DC), theentire signal is measured, for capacitivecoupling (AC), only the alternating voltagepart is measured, and for position GND, theinput is grounded and separated from thesignal.

Coaxial cables and BNC plugs/sockets areused for connecting a signal to the oscillo-scope (this is important when high frequentsignals are investigated). The core lead


15

Fig.5: Real RLC oscillator with AC genera-tor G for driving the circuit to forced oscilla-tionsFig.4: Ideal LC resonator

carries the signal and the metal sheath (theshield) is usually connected to ground. Whena coaxial cable is connected to a normal(bifilar) cable, the core is connected to the redlead and the shield to the black lead.Pay attention that the shield of all BNCsockets at the oscilloscope is internallyconnected to the protective earth conductor!

2.2 A capacitance C and an inductance L ina loop form an ideal LC circuit or LC resona-tor, see fig.4. If energy is brought into thecircuit (e.g. the capacitor is charged with thecharge q), electric oscillations occur. Theelectric energy flows back and forth betweenthe inductance and the capacitance.According to the Kirchhoff's loop rule, thevoltage drop U at the inductance and at thecapacitance is equal at any time. UsingC=q/U, the discharge current of the capaci-tance is

The voltage drop at the capacitance is

By differentiating (1) with respect to t,replacing dI/dt in (2) and rearranging we get

This is the differential equation of an undam-ped harmonic oscillator being solved by

with the angular frequency

The voltage oscillates with the natural fre-quency ω0 defined by the quantities L and C.The peak voltage Um depends on the startingconditions.

In a real circuit the resistance of the wires andthe coil is not zero. A real electric oscillatoris a RLC resonator. At the resistance R elec-tric energy is dissipated into heat. Thereforethe peak voltage decreases, the oscillationsare damped. Figure 5 shows a RLC resonator being sinu-soidally driven by an AC generator. In thiscircuit, two resistances have to be consideredseparately: The serial resistance RS (the sumof the inner resistance of the coil and theresistance of the wires), and the parallelresistance RP (the sum of the inner resistanceof the generator and the isolation resistanceof the capacitor). The higher RS and the lowerRP, the stronger the damping of the oscillator.With the directions of the three currents andthe voltage as shown in fig.5 we can write:

and

For simplification, we only consider the case

.dq dUI Cdt dt

(1)

.dIU Ldt

(2)

2

2

1 0 .d U ULCdt

(3)

0cosmU U t (4)

0

1 .L C

(5)

1 2 3

1

2

0 with

G

P

I I I

U UI

R

dUI Cdt

(6)

33 .S

dIU I R L

dt (7)


16

T

Un+1

Un

t

U

Fig.6: Damped oscillations according to (13)

RS = 0 in the following. This is approximatelytrue in the experiment. The circuit is thencalled a parallel RLC circuit.Let the generator voltage be

Solving equation (6) for I3, differentiating itwith respect to t and replacing dI3/dt in (7)with the result yields after rearrangement

This is the well-known differential equationfor forced damped oscillations

with the natural frequency ω0 according to eq.(5), the attenuation

and the factor

The complex term eiωt in (10) instead ofcosωt just simplifies the solution of eq. (9);only the real part has a physical meaning.The solution of the inhomogeneous lineardifferential equation (9) is the sum of thegeneral solution of the related homogeneous

equation (natural oscillation; K = 0) and theparticular solution of the inhomogeneousequation (forced oscillation).For our experiment, we only consider thecase ω0 > α (underdamped RLC circuit). Inthat case, the solution of the homogeneouseq. is

This is the damped natural oscillation thatdecays exponentially with the time constant1/α (see fig.6). You can easily determine the attenuation α inthe experiment via the logarithmic decrement

The particular solution of the inhomogeneouseq.(10) we are looking for is the steady-statesolution for αt 1. We find it with theapproach

By inserting (15) into (10) and taking intoaccount that A and K are real, we get

We finally replace the angular frequencies ωby the frequencies f that are more common ineveryday life and K by (12):

This is the amplitude (peak value) of thevoltage measured at the RLC circuit when itis driven by UG . If the driving frequency f isequal to the natural frequency f0, the ampli-

0 cos .GU U t (8)

2

02

1 1 cos .P P

d U dU U U tR C dt LC R Cdt

(9)

22

022 i td U dU U K e

dtdt (10)

12 PR C

(11)

0 .P

K UR C (12)

2 2

0

cos ,

.

t

mU U e t

(13)

1

1

1ln ln .n

n n

U UT

U n U

(14)

( )( ) .i tU t A e (15)

2 22 2

0

2 20

,

2

2arctan .

KA

(16)

0

22 2 2 2 2

0

.

4 4

PU f R CA

f f f

(17)


17

tude is maximum, i.e. resonance occurs.

The Q factor (quality factor) of a resonator is

with the upper and lower cut-off frequenciesfu and fl . These are the frequencies were

. For a parallel RLC circuit, themax 2A A

quality factor is

3 Experimental setup

3.0 Devices

- plug-in board 20×30 cm- plug-in elements: LC resonator circuit, 2

resistors, bridging plug- waveform generator HM8130- oscilloscope HM303-6 with manual- 3 BNC cables, BNC tee connector

3.1 The circuit is assembled with the givencomponents on the plug-in board. The data of the components are:L = 10 mH, C = 100 nF, R1 = 1.0 kΩ, R2 =5.6 kΩ, uncertainty of all values: ±5 %.

The voltage amplitudes displayed at thewaveform generator and measured with theoscilloscope are peak-peak values Upp = 2U0,if U0 denominates the maximum or peakvalue according to equation (8).

4 Experimental procedure

4.1 Examine the oscilloscope and discuss itwith the tutor.

4.2 Assemble the circuit according to fig.7,with the resistor R1 (1 kΩ), using the inputCH I at the oscilloscope. Additionally, connectthe generator output to the input CH II with thehelp of the tee connector.For observing the damped oscillations asshown in fig.6, apply a square wave voltageto the resonator circuit, with a large periodcompared to the decay time of the resonator.About 100 Hz and 3 V (peak-peak) arefavourable values. Observe both the oscillatorvoltage and the generator voltage, using thelast one as trigger source. Exchange R1 withR2 and see the differences. If you have acamera or mobile phone, you may takepictures. Measure the oscillation period and the heightof up to fife consecutive maxima for bothresistors.

For recording the resonance curves, switchthe waveform of the generator to sine; theamplitude remains at 3 V. Measure thevoltage at the resonator in the frequencyrange from 1 kHz to 15 kHz for both resis-tors. Hints: At first, search for the resonance fre-quency f0. Take about 20 measurements foreach curve. Choose smaller steps between thepoints in the vicinity of f0 and larger steps atthe ends of the frequency range.

Estimate the phase shift φ between theoscillator voltage and the driving voltage at1 kHz, at resonance and at 15 kHz.

5 Evaluation

5.2 Calculate the natural frequency f0 andthe two attenuations α from the given data ofthe components.

Calculate the natural frequency, the logarith-mic decrements and the attenuations from the

0 , u l

fQ f f f

f

(18)

.P

CQ RL

(19)

Fig.7: Measurement circuit. The two connec-tors on the right are not in use.


18

measured oscillation periods and the heightsof the oscillation maxima with the help ofeq.(14).

Plot the two resonance curves (U versus f) inone diagram. Compare the resonancefrequencies obtained from the curves with thenatural frequencies obtained from the dampedoscillations and from the component data.

Determine the upper and the lower cut-offfrequency fu and fl and the bandwidth Δf foreach resonance curve. Calculate the Q factorsof the two RLC circuits according to (18) and(19) and compare the results.

Facultative task for students with high skills

in computing (Origin): Determine α and f0 bynon-linear least squares fitting of eq.(17) tothe measured data. Use U0/(RPC) as a thirdfree parameter in you fitting model.

6 Questions

6.1 What is an oscilloscope used for? Whichquantities can be measured with it?

6.2 Explain how R, L and C behave in anAC circuit.

6.3 What is and which properties haveparallel and serial RLC resonators?

Mechanics M 14 Viscosity (falling ball viscometer)

19

.w b fF F F (2)

F r vf 6 , (5)

32

4 ,3bF r g (3)

31

4 ,3wF r g (4)

Fig.1: Laminar flow in a tube

43

643

31

32 r g r v r g (6)

1 Task

Determine the viscosity of ricinus oil as afunction of temperature using a HÖPPLER

viscometer (falling ball method).

2 Physical Basis

In real liquids and gases there are interaction-al forces between the molecules of onesubstance called “cohesion”, and between themolecules of different substances at aninterface (i.e. a liquid and the wall of thecontainer) called “adhesion”. When consider-ing ideal liquids or gases, these forces areneglected.If a real liquid flows through an inelastic tubewith a circular cross-section, in the case oflaminar flow a parabolic flow profile (that isflow velocity versus diameter) appears asshown in fig.1. Caused by the forces ofadhesion, the liquid adheres at the wall whilein the centre the velocity is maximum. Formathematical modelling, the flow is consid-ered as concentric cylindrical layers movingwith small velocity differences against eachother. In between the layers friction occurscaused by cohesion. The viscosity η is ameasure for this so-called inner friction. For a long thin tube of length l and radius r,the volume flow (volumetric current) at givenpressure difference Δp between the ends ofthe tube is described by the Hagen-Poiseuillelaw:

The unusual strong dependency on the radiusis caused by the inhomogeneous velositydistribution (fig.1).

A liquid, the viscosity of which does notdepend on the flow velocity, is called aNEWTONian liquid (also NEWTONian fluid orideal viscous liquid). Most of the homoge-nous liquids (i.e. water, oil) behave like this,while fluids consisting of different phases(i.e. ketchup, printing ink, blood) are non-NEWTONian fluids.

On a spherically shaped body (radius r,density ρ1) sinking within an ideal viscousliquid (density ρ2), the weight force Fw, thebuoyancy force Fb and the friction force Ff

are acting:

According to the principle of ARCHIMEDES,the buoyancy force is equal to the weight ofthe liquid that is displaced by the body:

and the weight Fw is

where g is the acceleration of fall.Because the friction force depends on thevelocity according to STOKES' law

after a short time of accelerated movement asteady state is reached (if Ff = Fw Fb) with aconstant falling speed. From eq. (1) follows:

4

.8

V r p

t l

(1)

Viscosity (falling ball viscometer) M 14


20

2

1 2

2.

9r

g ts

(8)

1 2 .K t (9)

0 .EE

k Te (11)

~EE

kTj e

(10)

31 2

46 .3

r v r g (7)

and

With eq. (6), the viscosity of a Newtonianliquid can be determined from the equilib-rium velocity of a sphere falling within theinfinite liquid. Replacing the velocity by theelapsed time t for moving a given distance s(v = s/t), we obtain

All invariant quantities are now combined tothe so-called geometry factor K:

In a HÖPPLER viscometer the ball does notfall within an infinite liquid but in a tube witha diameter only slightly larger than that of theball and tilted 10° against vertical. In thiscase the geometry factor is not calculated butdetermined experimentally.

The viscous behaviour of a liquid (and someother properties too) can be understood withthe help of the interchange model. The parti-cles (atoms or molecules) are held in theirplaces by bonding forces. They performthermal oscillations around their places withconstantly changing kinetic energy (by inter-action with their neighbours). For moving toa nearby place, they have to overcome apotential barrier. That means, their kineticenergy must be higher than a certain excita-tion energy EE. The velocity of thermallyoscillating particles is MAXWELL-distributed.Therefore the number of place interchanges jmust be

(~ means proportional, k is BOLTZMANN'sconstant, T the temperature). A force applied to the liquid from outsidecauses a potential gradient. Place inter-changes in the direction of that potentialgradient are favoured - layers of the liquid are

displaced against each other. The higher j is,the faster the displacement is. Therefore theviscosity behaves approximately like

In contrast to liquids, in gases the viscosityincreases with rising temperature according

to .T


3.0 Equipment:- HÖPPLER viscometer- circulator thermostat - 2 stopwatches

3.1 The falling ball viscometer is a precisioninstrument. It consists of revolvable fallingtube filled with the liquid to be investigated.On the tube are three cylindric measuringmarks. The distance between the upper andthe lower mark is 100 mm and between upperand middle mark 50 mm. The falling tube issurrounded by a water-bath, it’s temperatureis controlled by the circulator. The instrumentcan be rotated by 180° into the measuringposition (thermometer upright) and the roll-back position, respectively, where it locks. Inyour lab work you can measure with suffi-cient accuracy in both directions.At the upper end of the falling tube is ahollow stopper that contains some air. This isneeded for preventing too high pressureswhen the liquid expands at elevated tempera-tures. The stopper should be always above(i.e. device in measuring position) when thetemperature is changed and when the experi-ment is finished. The viscometer is delivered together with aset of 6 balls for different viscosity ranges. Inthis experient, ball No. 4 is used. The exactvalue of geometry factor (the ball constant) Kis given in a test certificate provided by themanufacturer for each device separately.


21


Study the manuals of the viscometer and thecirculator. Do not power the heater of thecirculator before setting the working tempera-ture to a low value (cooling the bath circula-tor down again requires much time).The viscosity is to be measured at fife differ-ent temperatures between room temperatureand 50°C. At first, align the viscometerexactly horizontal with the help of the waterlevel on the base. Before the first measure-ment, let the ball fall trough the tube once toensure that the liquid is mixed well.For determining the viscosity, you have tomeasure the time it takes for the ball to coverthe distance between the upper and the lowermeasuring mark. Both students shell measurethis time independently: The first studentstarts and stops his watch when the balltouches the measuring marks, and the secondstudent starts and stops his watch when theball just leaves the marks. The values are tobe taken four times at each temperature (2students get independent measurements).If a big air bubble obstructs the falling of theball please ask your supervisor for help. Youare not allowed to open the viscometer byyourself.

The experiment is started at room tempera-ture. Power the circulator (after a while thedisplay should indicate OFF) and set theworking temperature T1 to 20°C or anytemperature below room temperature. Thenactivate the pump by pressing the start/stopkey. Observe the thermometer in theviscometer (not on the circulator!). If thetemperature remains constant, wait about fivemore minutes for the temperature of thericinus oil to take the same value. During thewaiting time, let the ball run through the tubeonce. Then measure the fall time four times.

Increase the temperature step by step (foursteps of 6…8 K) until 50°C is reached. Aftereach step wait for equilibration of tempera-ture as described above and measure the falltime.

5 Evaluation

Calculate the viscosity from the average ofthe measured fall times according to eq. (8)and plot it graphically as a function of tem-perature.

The density of ricinus oil is ρ2 = 0.96 g/cm3.

The density ρ1 of the ball and the geometryfactor K are to be taken from the test certifi-cate that is found at the lab station.

Discuss the experimental errors quantita-tively.

Plot ln(η) versus 1000/T (this is a verycommon plot type for thermal excited physi-cal and chemical processes). According to eq.(11) this should result in a straight line.Calculate the excitation energy EE from theslope of that line. Express EE in eV (electronvolt) and in kJ/mol and compare it withtypical ionisation energies and reactionenthalpies.

6 Questions

6.1 How do real liquids differ from idealliquids?

6.2 What is inner friction? How can it bemeasured?

6.3 How does inner friction influence theflow of a liquid through a tube?

Optics & Radiation O 6 Diffraction Spectrometer

22

Fig.1: Diffraction of a plane wave on aedge. Construction after Huygens-Fresnel.

Fig.2: Calculation of the path difference δ ofdiffracted light on a grating. a: width of a slit, b: grating constant, : angle of diffraction

k (2)

( )2 12

k (3)

b sin . (1)

1 Tasks

1.1 Adjust a diffraction spectrometer.

1.2 Determine the wavelengths of theHelium spectral lines.

2 Physical Basis

Diffraction means the deviation from the wayin which light propagates according to thelaws of geometrical optics. It can be under-stood only if light is considered a wave.Diffraction always appears when the freepropagation of a wave is obstructed, as forexample by an edge, by a single slit or bymany slits (grating). Diffraction is usually explained by means ofthe Huygens-Fresnel principle. According tothis, each point of a wave front is the originof a new elementary (spherical) wavelet. Thesum of these elementary wavelets forms thenew wave front. If a plane light wave hits anobstacle, the wave front behind cannot beformed completely because the elementarywaves from the opaque regions of the obsta-cle are missing. At an edge, the elementarywaves also propagate as spherical waveletsinto the geometrical shadow space (see fig.1).Fig.2 shows an optical diffraction grating

which is a plane two-dimensional periodicalarrangement of transparent (permeable tolight) and opaque zones. The distance be-tween these zones (the grating constant b) isof the order of magnitude of the light wave-length. If a plane wave reaches the grating,circular wavelets will appear behind each slit.While propagating, they will meet waveletsfrom the neighbouring slits. The superposi-tion of waves (i.e. the summation of theiramplitudes) is called interference. On obser-vation from a far distance, maxima andminima of light intensity occur by destructiveand constructive interference, respectively. For simplification, each slit in fig.2 is consid-ered to be the origin of only one elementarywavelet. The path difference δ between thewavelets coming from neighbouring slits is

Constructive interference occurs for a pathdifference

and destructive interference for a path differ-ence of

Diffraction Spectrometer O 6


23

Fig.3: sketch of a diffraction spectrometer

R N k (7)

sin

k

kb

(4)

b

kksin

. (5)

R

(6)

where k = 0, 1, 2, 3,… is called the diffrac-tion order. The undiffracted (straight ongoing) light is referred to as zeroth diffractionorder (k=0).From the equations (1) and (2) the angle k ofthe diffraction maxima follows:

The more wavelets constructively interfere inthis direction, the more intensive and sharpthese maxima are. This implies that thenumber of slits involved should be large.

Equation (4) shows that the diffraction angledepends on the wavelength. Thus, white lightcan be decomposed into its spectral colourswith a grating. By measuring the diffractionangle the wavelength of light can be deter-mined:

The capability of a spectrometer is character-ised by its resolution

where Δλ is the smallest resolvable wave-length difference. The theoretical resolutionof a diffraction grating is

where N is the total number of slits illumi-nated and contributing to the interferencepattern, and k is the diffraction order.The principle of a diffraction spectrometer isshown in Fig.3. The slit is located in the focalplane of the collimator lens, so that thegrating is illuminated by parallel light. The

parallel diffracted light (that apparentlycomes from infinity) is either observed by atelescope or focussed by a lens to a photo-graphic plate or a CCD sensor.


3.0 Devices:- Goniometer with collimator and telescope- Helium-lamp with power supply- Hand lamp with transformer- Auxiliary mirror.

3.1 The experimental arrangement is ac-cording to fig.3. The goniometer ERG3 isused for measuring the diffraction angle φ. Itconsists of the collimator with slit and lens, arotatable table with the grating, a moveabletelescope and an arrangement for measuringangles with an accuracy of 0.5 ' (angularminutes). The He-lamp is placed in front ofthe slit. The slit width, the collimator (dis-tance between slit and lens) and the telescopecan be adjusted.


At first, learn how the goniometer is oper-ated. Note the grating constant to your proto-col that is written on the grating.

4.1 Adjustment of the spectrometer:The goal is to lighten the grating with anexactly perpendicular incident beam ofparallel light and to see a sharp image of theslit and of the hair cross (reticle) in the tele-scope.

Telescope: Make sure the position of thefilter revolver is , and the lense slider is

shifted to the right. Press or pull the ocular tofocus the reticle. Then adjust the telescope toinfinity by autocollimation: Place the mirrorin the grating holder and align the telescopeapproximately perpendicular to the mirror.Illuminate the reticule with the GAUSS ocular.The light is reflected on the mirror, and you


24

k

right left

2(8)

can see the bright circular area of the tele-scope itself in the telescope. Find the blackreflection of the bright illuminated reticuleand bring it into focus. If you see it sharply,the telescope is adjusted to infinity.

Collimator: Remove the mirror. Place thetelescope opposite to the collimator (theyshare now the same optical axis) and lock it.You should see the slit through the telescopenow. Do not readjust the telescope at thispoint! If the slit is not sharp, carefully shiftthe whole slit into or out of the collimator inorder to focus it. Adjust the optimum slitwidth: As small as possible, but slit andreticule good visible.Lock the telescope.

Grating: Adjust the grating perpendicular tothe common axis of the telescope andcollimator. For this purpose use the mirroragain and illuminate the reticule as above.Bring the black reflex of the reticule incoincidence with the bright reticule. Now thetelescope is exactly perpendicular to themirror. When all adjustments are done, lock the gridtable, turn the light of the GAUSS ocular off,replace the mirror by the grating, and unlockthe telescope.

4.2 For measuring the diffraction angle φk,bring the reticule in coincidence with thespectral lines and read the corresponding

angles φ'. You have to measure 6 spectrallines in the first, second and third diffractionorder, respectively, on the left - as well as onthe right side relatively to the diffractionorder zero. The diffraction angles then followfrom:

5 Evaluation

Calculate the diffraction angles φk and thenthe wavelengths λ by means of eq. (8) and(5), respectively.Plot the wavelength versus the diffractionangle (the dispersion curves) for each diffrac-tion order in a diagram (all three curves inone diagram). Compare your results with the values given intables.

6 Questions

6.1 Which kind if interferences occur on anoptical grating?

6.2 For what is a diffraction spectrometerused? How does it work?

6.3 Explain the (wavelength) resolution of adiffraction spectrometer.

Optics & Radiation O 10 Polarimeter and Refractometer

25

Fig.1: Position of electric and magneticfield-strength vectors for a wave train

nc

c 0 . (2)

k l c , (1)

1 Tasks

1.1 Determine the concentration of a sugarsolution by means of a polarimeter.

1.2 Measure the refractive index of glycerol-water mixtures in dependence on the glycerolconcentration using a refractometer.

1.3 Determine the concentration of a givenglycerol-water mixture.

2 Physical Basis

2.1 Light waves belong to the electromag-netic waves. Each light beam consists of avast number of separate wave trains. A wavetrain consists of an electric and a magneticfield which are both perpendicular to thedirection of propagation and perpendicular toeach other, see fig.1.If we consider natural, i.e. unpolarized light,the electric and magnetic fields can vibrate inarbitrary directions which, however, arealways perpendicular (transversal) to thedirection of propagation.Light is linearly polarized if all electric fieldsvibrate in only one transversal direction. Thedirection of the electric field-strength vector

is then called the direction of oscillation orthe polarizing direction.

2.2 Linearly polarized light may be genera-ted from natural light by (a) reflection at theBREWSTER angle, (b) by birefringence (dou-ble refraction in a NICOL prism) or (c) bymeans of polarizing filters on the basis ofdicroitic foils.Optically active materials are substances thatrotate the direction of oscillation when lin-early polarized light passes the substance.This optical activity may be caused by asym-metric molecule structures or by a screw-likearrangement of the lattice elements. Somesubstances like sugar have both a dextro-rotatory and a laevorotatory version.In solutions of optically active substances theangel of rotation depends on the kind ofsubstance, the thickness of the layer pene-trated by the light (i.e. the length l of thepolarimeter tube) and on the concentration cof the substance. Furthermore, there is awave-length dependence called rotary disper-sion: blue light is stronger rotated than redone. This effect is not considered here.It applies for the rotation angle φ:

where the material constant k is called spe-cific rotary power.

2.3 The refractive index n of a substance isdefined as the ratio of the vacuum velocity oflight to the velocity of light in the substance:

The refractive index depends on the materialand on the wave length of light (this effect iscalled dispersion). In a solution it also de-pends on the concentration (mixing ratio).Therefore, a measurement of the refractiveindex may be suitable for determining con-

Polarimeter and Refractometer O 10


26

Fig.2: Ray trace of refraction for n2 > n1.Left: general case, right: striping light entry

n n1 2 sin sin . (3)

sin .max n

n1

2

(4)

Fig.3: Ray trace at an ABBE refractometer

centrations.Applications, for example, are the determina-tion of the protein content in a blood serumor of the sugar degree of grape juice in awinery.During the transition of light from an opti-cally thinner medium with index n1 to anoptically denser medium with index n2 (n2 >n1) a light beam is refracted towards theperpendicular of incidence, see fig.2. With αand β as angle of entry and emergence, thelaw of refraction reads

For the largest possible angle of entryα = 90° (striping light entry) a maximumrefraction angle βmax can be obtained.

The path of rays in fig.2 can be inverted:From the optically denser medium (n2) to theoptically thinner medium (n1), angle of entryβ, angle of emergence α. For β > βmax nolight will be refracted into the opticallythinner medium because the law of refractioncannot be fulfilled. Instead, the light is com-pletely reflected at the interface of the twomedia. Therefore, βmax is called critical angleof total reflection. It results from eq. (3):

If the refractive index n2 (measuring prism ofrefractometer) is known, the refractive indexn1 of the other medium can be determined bymeasuring the critical angle of total reflec-tion.

Therefor the interface is lighted via a frostedglass plate with a rough surface, see fig.3. Inthis way the light beams enter at the interfacefrom all angles between 0° and 90°. So allrefraction angles between 0° and βmax arepossible. When looking at the interfacethrough a telescope at the angle βmax, a light-dark boundary can be seen which is used todetermine the refractive index of the sub-stance under investigation (as describedbelow).

3 Experimental Setup

3.0 Devices- polarimeter with sodium-spectral lamp- polarimeter tube (length 200 mm)- flask with sugar solution- ABBE refractometer - 2 burettes with glycerol 83 vol% and deio-

nized water- 3 beakers, funnel, pipette- flask with a glycerol-water mixture of

unknown concentration

3.1 The polarimeter consists of a monochro-matic light source (Na-D light, λ = 589.3 nm),the polarimeter tube, polarizer and a rotaryanalyser with an angular scale.If the polarizing directions of polarizer andanalyser are perpendicular to each other(“crossed position”), no light is transmittedand the visual field of the polarimeter is dark.After placing the polarimeter tube filled withan optically active medium between polarizerand analyser, the visual field is brightenedbecause the direction of oscillation of thelinearly polarized light has been rotated by an


27

Fig.4: Three-part visual field of thepolarimeter

1 0 . (5)

angle φ. Resetting the analyser by this angleyields the visual field becoming dark again.In this way the angel φ can be measured.An adjustment of the polarimeter to maxi-mum darkness or brightness without anyvisual comparison would be imprecise.Therefore, a three-part polarizer is used,resulting in a visual field according fig.4. Theinner part of the polariser is tilted against theouter parts by 10°. During the measurementthe analyser is adjusted to equal brightness ofall three parts in the visual field (half-shadepolarimeter). For a precise measurement ofangles, the scale is equipped with a vernierthat allows a read off with an uncertainty ofonly 0.05°.

3.2 The ABBE refractometer measures therefractive index nD (for Na-D light, 589 nm)using natural white light. The dispersionwhich is quite strong for liquids is compen-sated by a pair of AMICI prisms with directvision at 589 nm. Other refractometers areworking with monochomatic light. Suchinstruments would not need compensation ofdispersion but they are not able to measure it.

The device used consists of the followingessential parts:

- the lightning prism with a rough surface

- the measuring prism whose refractive indexn2 must be larger than the refractive indexn1 of the substance under investigation

- a tilting telescope for observing both themeasuring prism and an angular scalecalibrated according to eq. (4) for readingthe refractive index

- an equipment for compensating (or measur-

ing) the dispersion

The grazingly incident part of light (α 90°)is refracted at the critical angle βmax and canbe observed in the telescope as a light/darkboundary.A simple check of the device can be made bymeasuring the refractive index of pure water,which is nD = 1,3330 at 20°C and 1,3325 at25°C.

4 Experimental Procedure

4.1 Switch the sodium spectral lamp on atfirst; it needs about 5 minutes to reach itsmaximum brightness.Determine the zero position φ0 of the polari-meter by adjusting the visual field as de-scribed in 3.1, but without polarimeter tube.Take the reading of φ0 5 times and readjustthe polarimeter for every reading.If necessary clean the glass windows of thepolarimeter tube. The windows can be easilyremoved from the screw caps. When screw-ing the cap onto the tube, assure the rubberO-ring is between glass window and metalcap (not between window and glass tube). Donot tighten it too much!Fill the polarimeter tube completely withsugar solution. You may fill it on the wash-stand and dry it with paper towels. Theremust be no bubble in the beam path. A re-maining small bubble may be set into thebulge of the tube. Finally, put the tube intothe polarimeter.Now adjust the analyser again to equalbrightness of the visual field, and read thecorresponding angle φ1. This measurement isto be carried out 5 times, too.Then the rotation angel results as the differ-ence of the mean values of φ1 and φ0:

When finished, fill the sugar solution back tothe flask. Clean the polarimeter tube withwater and leave it open.

4.2 The two prisms of the refractometermust be on the right side, and the small


28

mirror for scale illumination (left hand side)must be open.Open the two prisms (measuring prism aboveand lightning prism below). If necessaryclean the prisms carefully with wet papertowel and dry it. Hold the lightning prismwith the rough surface about horizontally andput 1 or 2 drops of the sample liquid on thesurface. Ensure that there are no air bubblesin the liquid. Then close the prisms and lockthem gently.Look through the measuring eyepiece (theright one) and adjust it until the reticle issharply seen. Adjust the lightning mirror formaximum lightness. By turning the scaleadjustment knob (on the left) move along themeasuring range until the light/dark boundaryappears. Eliminate colour fringes by turningthe compensation knob (on the right) until theboundary appears black-and-white. Adjustthe centre of the reticle exactly to the light/dark boundary and read the correspondingrefractive index on the scale (left eyepiece).

Before taking measurements, check theadjustment of the device with de-ionizedwater. If the reading differs by more than onescale division from the reference value, askthe tutor to adjust the refractometer.

The refractive index is to be determined forthe following liquids:

- de-ionized water

- glycerol 83 vol%

- 5 glycerol-water mixtures:

4:1, 4:2, 4:4, 4:8, 4:16 and

- a glycerol-water mixture of unknown con-centration

For the mixture 4:1 take 4 ml glycerol 83%and 1 ml deion. water, and make the othermixtures by further dilution of this mixturewith water.Measure each refractive index 5 times (re-adjust the scale for each measurement).

Clean the prisms when changing the concen-tration and at the end of the measurements.

4.3 The refractive index of the glycerol-water mixture of unknown concentration is tobe measured 5 times as well.

5 Evaluation

5.1 Calculate the concentration c (in g/l) ofthe sugar solution according to the equations(1) and (5).The specific rotary power of saccharose(C12H22O11) at λ = 589.3 nm amounts tok = 0.66456 deg l m-1 g-1. The length of thepolarimeter tube is l = (200 ± 0.2) mm.Calculate the uncertainty of the concentrationusing Gauss’s error propagation law. u(φ) hasto be calculated from the standard deviations

of the means .0 1and

5.2 Calculate the volume concentrations(pure glycerol in water) of all mixtures. Plotthe refractive index via the volume concen-tration of glycerol in water.

5.3 Determine the concentration of theunknown glycerol-water mixture by means ofthe diagram from 5.2.; give the concentra-tions in terms of vol.% glycerol.

6 Questions

6.1 What is light?

6.2 How can linearly polarized light begenerated?

6.3 What is refraction? When does totalreflection occur?

6.4 Which influence has the dispersion onmeasurements with a refractometer?

Optics & Radiation O 16 Radioactivity

29

d .dNAt

(1)

dN N t d . (2)

N t Nt

( )

0 e (3)

.P ZI I I (4)

I Ix

0 e . (5)

1 Tasks

1.1 Measure the dependence of nuclearradiation on the distance to the radiationsource and verify the inverse-square law.

1.2 Determine the attenuation coefficientand the half-value thickness (HVT) of led(Pb) for the gamma radiation of Co-60.

1.3 Investigate the frequency distribution ofthe counts (counting statistics).

2 Physical Basis

Radioactivity is a property of atomic nucleihaving unfavourable proton-neutron ratios.Such nuclei transform spontaneously byemission of characteristic radiation into otheratomic nuclei or into nuclei of another energylevel (they are said to decay). Depending onthe kind of transformation, the radiationconsists of particles and high energeticelectromagnetic waves:α particles = He nuclei (2 protons, 2 neu-trons), β particles = electrons,β+ particles = positrons,γ quanta (electromagnetic radiation with aquantum energy >100 keV),neutrons and (rarely) protons.γ radiation arises when after a nuclear trans-formation the excited nucleus returns into itsbasic energy level.

2.1 The number of nuclei transforming in atime interval is proportional to the totalnumber of nuclei being present. The numberof decays per time within a sample of radio-active material is the called the activity A:

After the time interval dt the number of

radioactive nuclei is lowered by

λ is called the radioactive decay constant.From eq. (2) follows the law of radioactivedecay:

with N0 being the number of radioactivenuclei at the time t = 0.

If a γ quant (or α or β particle) is detected bya Geiger-Müller tube (GM tube), it triggers acurrent pulse. The pulses are counted, and thepulse rate I (the number of pulses per second)is proportional to the radiation intensity.Additionally, it depends on the characteristicsof the detector and possibly on the energy ofthe radiation. The pulse rate I caused by a radioactivepreparation is the difference of the pulse ratesmeasured with preparation IP and withoutpreparation IZ (zero rate):

The zero rate is caused by environmentalradiation (cosmic radiation and natural radio-activity) and by interfering pulses of thedetector.

2.2 If gamma radiation penetrates matter, itsintensity (measured as pulse rate I) reducesdepending on the penetrated thickness xaccording to the attenuation law

Here, I0 is the intensity of the incident radia-tion and I the intensity of the escaping radia-tion; μ is called the attenuation coefficient, itdepends on the material penetrated and on theenergy of the gamma quanta. Besides elastic scattering (μS), three differentabsorption effects are responsible for theattenuation: the photo effect (μPh), inelastic

Radioactivity O 16


30

S Ph C P . (6)

P nNn

p pN

n N n( )

1 (8)

1/ 2

ln 2 .x

(7)

e( )

!

n µµP n

n

(9)

( ) .n u n n (10)

2( )

21( )

2e .

n µ

µP nµ

(11)

scattering (Compton effect, μC) and the paircreation effect (μP):

The portion of these effects on the totalattenuation depends on the energy. At lowenergy elastic scattering predominates, and atvery high energy the pair creation is domi-nant.The half-value thickness (HVT) x1/2 of amaterial is the thickness required for theintensity to be attenuated to its half value.From eq. (5), it follows for I = ½ I0 :

2.3 The radioactive decay of a nucleus is aquantum process. The prediction of the exacttime of a decay is in principle impossible.Only the probability of the nucleus to decayin a certain time interval is known. Thereforethe number of counts measured is for funda-mental reasons (and not only because of themeasurement errors of the devices used) arandom number. This is particularly noticedwhen the counts measured are low.With N being the number of radioactiveatoms and p the probability of one atom todecay, the probability of n decays is

with the mean value (the expectation)µ = n p. In our experiment N is a hugenumber and p is very small. Passing to thelimits N and p0, the binomial distribu-tion (8) transforms into a POISSON distribu-tion

with the mean value µ. (Pay attention that themean value µ is not identical with the attenu-ation coefficient µ in section 2.2!)An important mathematical property of thePOISSON distribution is the equality of meanvalue µ and variance σ2 (square of the stan-

dard deviation σ). From that follows:If a number n of random events is measuredin a time period, the uncertainty of the mea-surement result is

The statistical uncertainty is approximatelyequal to the root of the measurement result.

Additionally, for large n the POISSON distri-bution can be approximated by a GAUSS

distribution


3.0 Devices:- radioactive preparation Co-60 (γ radiator

1.17 MeV and 1.33 MeV, A = 74 kBq 2010,t1/2 = 5.27 a)

- Geiger-Müller tube- digital counter- computer with program “Digitalzähler”- optical bench with measure- lead slabs of different thickness

3.1 The GM tube is an end-window counter.It is equipped with a thin mica window thatallows also for measuring low energetic γ andX radiation as well as β particles.The digital counter is both the rate meter andthe power supply for the GM tube. Thecounter automatically stores up to 2000measured values. For the statistical analysis itsends the counts to the computer.

The radioactive preparation and the GM tubeeach reside in a plexiglass block mounted ona sledge that can be shifted on the opticalbench. Radioactive preparation and GM tubeare facing to each other. A third sledge forcarrying the absorbing slabs can be mountedin between them. The distance between thepreparation and the counter tube is the dis-tance between the edges of the gray sledges+ 10 mm.


31

either

or e

ln ln

lg lg lg

I I x

I I x

0

0


The Co-60 radiator is an enclosed preparationwith an activity below the permitted limitaccording to the Radiation Protection Ordi-nance. Your radiation exposure in a 3 h labcourse is about 0.1% of the exposure causedby a medical radiogram.

4.1 At the digital counter, adjust an operat-ing voltage of 480 V for the GM tube.Choose rate measure, measuring interval60 s. Display the number of counts N. Allmeasurements shall be carried out fife times(5 minutes). The counts are stored every 60seconds in memory, after stopping the count-ing you can read the values.At first, measure the zero rate (five times).Put the preparation at least 1 m away fromthe counter tube for this measurement.Then place the Co-60 radiator in a distance of40, 50, 70, 100, 140, 190 and 250 mm fromthe GM tube and measure the pulse rate foreach distance.

4.2 For determining the attenuation coeffi-cient of Pb, put the third sledge between theradiator and the counter tube and place thepreparation in a distance of 70 mm from thecounter tube. This distance has to be keptconstant during the remaining experiment.Measure the pulse rates for the thicknessesx = 1, 2, 5, 10, 20 and 30 mm five times each.The measurement result for x = 0 mm can

be taken from task 4.1.

4.3 The measurements for the frequencydistribution may run unattendedly in thebackground while you are evaluating otherparts of the experiment at the same computeror during a discussion with the tutor.Delete all previously measured data at thecounter and start the program “Digitalzähler”.Press [F5] for the options dialogue, select the“Allgemein”-tab and change the languagefrom “Deutsch” to “English”. Select “Poisson”

from the predefined graph tabs.Place the radioactive preparation in a distanceof 10 cm from the counter tube. Switch thecounter to rate measurement with a gate timeof 1 s. Start the measurement either at thecounter device or at the program and recordat least 600 measurements (10 minutes).Evaluate or save this series of measurementsand record a second series with the distancebetween preparation and counter tube being5 cm.

5 Evaluation

Calculate the average of the five singlemeasurements in every part of the experi-ment. Correct the average pulse rates bysubtracting the zero rate according to eq. (4).

5.1 The inverse-square distance law is to beverified. (Answer: What's this law?) Plot thepulse rate I versus distance r on doublelogarithmic scales. Use either double-loga-rithmic graph paper or a computer in the labto do this. Fit a straight line to the measuringpoints and determine the slope. The slope sis the exponent in a distance law of the kindI(r) = C r

s. Compare your result with thetheoretical distance law.

5.2 By taking the logarithm of eq. (5) we get

with lg e = 1/ln10 = 0.434. For determiningthe attenuation coefficient μ of lead, plot the

Radiation Protection:

According to the German RadiationProtection Ordinance, every radioactiveexposure, also below the allowed limits,is to be minimised. Therefore: Do notcarry the preparation in your hand if notnecessary! Keep a distance of 0.5 m tothe preparation during the experiment! Itis not allowed to remove the Co-60preparation from its plexiglass block.


32

pulse rates I versus the total thickness x of theabsorbing Pb slabs on single logarithmiccoordinates (rate logarithmic, thicknesslinear). Alternatively, you can calculate thelogarithm of the rate ln(I) or lg(I) and plot itversus thickness on linear (“normal”) scales.(Although taking the natural logarithm iseasier here, in common scientific praxis thedecimal logarithm is preferred because thegraph is better readable.)In both cases the measuring points shouldfollow a straight line. Calculate the attenua-tion coefficient μ from the slope of this line.With the knowledge of μ, calculate the HVT.

5.3 For the two series of measurementscalculate the mean value and the standardndeviation σ. Check wether the prediction

is valid. nPlot the frequency distributions as bar graphand (in the same graph) the Poisson distribu-

tion and the normal distribution fitting thedata as curves. These tasks are easily done with the program“Digitalzähler”. Look for the menu item “Fit

function” in the context menu of the graph.

6 Questions

6.1 What is the difference between X-raysand γ-rays?

6.2 What is the half value thickness and thehalf life period?

6.3 How does the intensity of radiationdepend on the distance from the radiator?

6.4 A counter tube measures 10 000 pulses.How large is the uncertainty of this measure-ment?

Optics & Radiation O 22 X-ray methods

33

E h f hc

Ph

(2)

E e U h f hc

Ph maxmin

(3)

E e Um

v Em

ve

Ph

e 2 21

22

2 (1)

1 Tasks

1.1 Measure the X-ray emission spectra of amolybdenum anode using a LiF crystal anddetermine the maximum quantum energy ofthe X-radiation in dependence on the anodevoltage.

1.2 Determine the ion dose rate of the X-raytube within the apparatus.

1.3 X-ray examination and interpretation onseveral objects (bones, computer mouse, ...).

2 Physical Basis

2.1 X-ray radiation X-rays are electromagnetic waves (photons)with wavelengths between 0,01nm and 10nm.They are produced by bombarding an anodewith electrons the energy of which exceeds10 keV. At the impact two types of X-rayradiation are produced besides approx. 98%of heat:(i) Bremsstrahlung is produced by the suddenslowing down of incident electrons in thevicinity of the strong electric field of theatomic nuclei of the anode material. Afterthis interaction the electrons still have a partof their kinetic energy. The difference be-tween the kinetic energy before and after theinteraction is transformed into X-rays withthe frequency f. (see equation (2))With E being the kinetic energy of the elec-trons after acceleration through the voltage U,the following energy balance results:

withe: elementary charge of the electron

(e=1,602*10-19 C)U: anode voltage

me: electron massv1: velocity of the electron before the im-

pactv2: velocity of the electron after the impactEPh: photon energy (energy of an X-ray

quantum)

The energy of a radiation quantum is

h: PLANCKs constant (h = 6,625*10-34 Ws2)c: velocity of light in vacuum

(c = 2,998 108 m s-1)f: frequencyλ: wavelength

The bremsstrahlung has a continuous spec-trum with an edge at short wavelengths (seefig.1). This corresponds to those electronswhich transpose their whole kinetic energyinto an X-ray photon (total slowdown, v2=0).The photon has then a maximal energy, henceits wavelength is minimal in this case:

The energy in that context is usually countedin eV (electron volts). 1 eV is the energy thata particle with one elementary charge e getswhen accelerated through a voltage of 1 V.The energy in Joule is hence calculated bymultiplying the eV with e = 1.602 10-19 As.

(ii) Characteristic radiation: During the im-pact of electrons, atoms of the anode materialare ionised. If due to this a vacancy in theinnermost shell - the K-shell - arises, it willbe immediately occupied by L- and M-elec-trons, respectively, and the energy differenceswill be released in form of X-rays. Thephotons (energy quanta) which are emittedduring these electron jumps are called Kα andKβ photons, respectively. The correspondingwavelengths can be calculated from

X-ray methods O 22


34

Fig.1: typical X-ray spectrum revealingBremsstrahlung and characteristic radiation

Fig. 2 BRAGG reflexion

K

L KK

M K

h cE E

h cE E

(4)

2 1 2 3 d k ksin , , , , ... (5)

Eh c

dPh

2 sin(6)

EL-EK: the difference in electron energybetween the L- and K-shell

EM-EK: the difference in electron energybetween the M- and K-shell

Because this energy difference is characteris-tic of the material, the radiation is called”characteristic radiation”. This radiationexhibits a line spectrum.Fig. 1 shows a typical X-ray spectrum con-sisting of Bremsstrahlung and characteristicradiation. The spectrum of the Molybdenumanode used in this experiment has a similarshape.

X-ray diffraction:The wave length of X-rays may be deter-mined by means of diffraction on a crystallattice when the lattice distances are known(X-ray spectral analysis). Inversely, with X-rays of known wavelength the lattice dis-tances of crystals may be determined (X-raydiffraction analysis, BRAGG's method).According to the HUYGENS principle, eachatom of the crystal hit by X-rays can beconsidered as a source of an elementarywave. The atoms in the crystal can be sum-marized in multiple consecutive layers situ-ated parallel to the crystal surface. Thisplanes are called ”lattice planes”. In thesimplest case the diffraction of X-rays can bedescribed as reflection at the lattice planes of

a crystal. Each lattice plane acts on the inci-dent X-ray like a partial mirror, that reflects a(very small) part of the incident X-ray.Fig. 2 shows the fundamental processes ofthis so-called “BRAGG reflection”: The rays 1and 2 reflected on the planes A and B inter-fere with each other. Constructive interfer-ence (a so-called ”reflex”) appears only whenthe path difference 2 d sin β between the twowaves equals a multiple of wavelengths:

k is the order of diffraction and d is the latticeconstant (d = 0,201 nm for the LiF crystalused in that experiment). For the first order ofdiffraction (k=1), from equation (2) follows:

By rotating the crystal the incidence angle ofthe X-rays β and thus the path difference ofthe interfering rays can be varied so that thecondition for constructive interference (5) canbe fulfilled for different wavelengths of theprimary rays, respectively. While rotating thecrystal, also the radiation detector has to bemaintained at the Bragg angle, so that thereflection condition detector angle = 2 ×crystal angle is always fulfilled. In this waythe spectrum of the X-ray source can bedetermined.

2.2 Dosimetry is the measurement of theimpact that ionising rays (X-rays and radioac-tive rays) do have on matter. This impact canbe measured in two ways: by measuring thenumber of ions created within the matter orby measuring the amount of energy absorbed


35

JQm

. (7)

DEm

. (8)

H w D (9)

Fig.3: Measurement of the ion dose rate inan ionisation chamber

jQ

m t

I

mC

. (10)

-1

-1

Sv32,5 orAs kg

Sv32,5 .As kg

H J

h j

(11)

by the matter.The ion dose J is defined as the total chargeof ions ΔQ produced in a volume elementdivided by the mass Δm of that volumeelement:

The unit of measure of the ion dose is As/kgor C/kg.The absorbed dose D is defined as theenergy ΔE absorbed by a volume elementdivided by the mass of the radiated volumeelement Δm:

Its unit of measure is the Gray (Gy), 1 Gy =1 J/kg.The equivalent dose H characterises thebiological impact of ionising radiation and isdefined as

with the unit Sievert (Sv), 1 Sv = 1 J/kg. wis the radiation weighting factor, it is w = 1for X-ray, gamma and beta rays and w = 20for alpha rays.The effective intensity of ionising rays is thedose per time or dose rate. It may be given asion dose rate j (in A/kg), absorbed dose rate d(Gy/s) or equivalent dose rate h (Sv/s). 1 Sv/sis a very large unit (6 Sv are lethal to hu-mans), therefore mSv/h or μSv/h are morecommon units.The ion dose rate is usually measured with anionisation chamber, that is in principle alarge capacitor filled with air of the mass mas shown in fig.3. A voltage is applied to thecapacitor that is large enough for all ions toget to the plates. The radiation causes an ioncurrent IC that can be measured in the outercircuit. The ion dose rate is than

With the known mean ionisation energy of air

molecules the equivalent dose is calculatedfrom the ion dose according to


3.0 Devices- X-ray device with goniometer including

LiF crystal (d = 0,201 nm) and G.M.-counter.

- PC with program “Röntgengerät”- capacitor with X-ray aperture for ion dose

measurements (build into X-ray device)- power supply 0...450 V, Ri = 5 MΩ- measurement amplifier- electrical multimeter- cables- several objects for X-raying.

3.1 The X-ray device (see fig.4) consists ofa radiation shielding case that is separatedinto three chambers. The largest (right-handside) chamber is the experimental chamber. Itcontains either the goniometer (for diffractionmeasurements) or the capacitor (for dosemeasurements) or the objects for X-raying.The X-ray tube is placed in the middle cham-ber. The left chamber contains the micropro-cessor controlled electronics, the controls anddisplays.


36

Security declaration:

The device is constructed in a mannerthat X-ray is only created when the doorsof the chambers are closed. The radiationoutside of the case falls several times offthe admissible limit according to theGerman Radiation Protection Ordinance.According to the “Verordnung über denSchutz vor Schäden durch Röntgenstrah-len” the X-ray device is an admittedmodel. (admission symbol NW807/97Rö)

Fig.4: X-ray device with goniometer.

a Mains power panel, b Control panel, c Connection panel, d Tube chamber (with Mo tube),e Experiment chamber with goniometer, f Fluorescent scree, g Free channel, h Lock lever

The doors and windows consist of lead glass.This is a very soft material! Handle with care,do not scratch it!

3.2 The high voltage power supply exhibitsa very large output resistance. The contactsmay be touched without harm. For measuring the very small current anamplifier and the multimeter are used.


Please do not touch the LiF crystal fixedon the goniometer.

4.1 Use the X-ray device with the build-indiffractometer. Set up the following parame-ters for recording the X-ray spectra in theBRAGG arrangement:Tube current: I = 1,0 mAHigh voltage: U = 20…35 kVMeasuring time: Δt = 5 sStep width: Δβ = 0,1goniometer mode: coupledInitial angle: βmin = 4,0Final angle: βmax = 12,0

Start the computer program ”Röntgengerät”.You may change the program language fromGerman to English (press F5, choose Allge-

mein and change Sprache).The best way is to start with the maximumacceleration voltage (35 kV). The recording


37

00

0

,T p

m VT p

(12)

is started by pressing the SCAN button at theX-ray device. Record additional spectra at30 kV, 25 kV and 20 kV into the same graph.To increase the accuracy of the measuredvalues al low acceleration voltages, you canincrease the measuring time Δt. To save timeyou can reduce the measuring range (increaseβmin) as long as the edge of the spectrum isjust in the measuring range.

4.2 Use the X-ray device with the build-incapacitor for the ion dose measurement.Complete the wiring according to figure 3:Connect the coaxial cable from the lowercapacitor plate to the current input I of theamplifier. Interconnect the ground socket( 2 ) of the amplifier with the negative termi-nal and the upper capacitor plate with thepositive terminal of the power supply. Con-nect the multimeter to the output of theamplifier and select the range 10-9 A (1 Voutput is equivalent to IC = 1 nA). Measure the ion current IC at the maximumacceleration voltage of 35 kV and with tubecurrents of 1 mA, 0.8 mA, 0.6 mA, 0.4 mAand 0.2 mA. Record the air pressure p and thetemperature T in the X-ray device.

4.3 For X-raying of objects use the particu-lar X-ray device prepared for this task. Adjustthe maximum possible energy (U = 35 kV,I = 1 mA). The room has to be darkened.Observe the shade of the object under investi-gation on the screen. Investigate how theimage depends on the position of the objectin the chamber.X-ray the objects given and objects you own(pocket calculator, ball pen, ...) and recordthe observations to your protocol.After finishing this part of the experiment,the fluorescent screen has to be coveredagain.

5 Evaluation

5.1 Determine the wavelength and quantumenergies for the characteristic lines Kα and Kβ

of the Mo anode, using equation (5) and (6),respectively. The quantum energies areusually given in keV. Calculate the maximal quantum energy foreach value of the anode voltage U from theangles β of the corresponding short-waveedge, using equation (6). List the energies ina table and compare them with the kineticenergy E = eU of the electrons acceleratedby the voltage U.As part of your consideration of errors,estimate the wave length resolution of the X-ray device.

5.2 Calculate the ion dose rate according to(10) from the ion current IC and the mass m ofthe radiated air volume. This mass is given by

with V = 125 cm3, ρ0 = 1,293 kg/m3,T0 = 273 K and p0 = 1013 hPa.

Additionally, calculate the maximum equiva-lent dose within the X-ray device at I=1mA,in units of Sv/h by means of eq. (11) andcompare it with other values like the annualnatural dose and the lethal dose.

5.3 Record the observations made duringthe X-ray screening of the objects in yourprotocol.

6 Questions

6.1 Explain the spectrum of an X-ray tube.How is it influenced by tube current andacceleration voltage?

6.2 How is the biological effect of ionisingradiation measured?

6.3 Which X-ray methods for materialinvestigation do you know?

Physical constants

velocity of light in vacuum c = 2,997 924 58 108 m/s 300 000 km/s

gravitational constant γ = 6,674 1(3) 1011 N m2 kg2

gravity acceleration (local value) g = 9,812 03(2) m/s2

elementary charge e0 = 1,602 176 62(1) 1019 C

electron mass me = 9,109 383 6(1) 1031 kg

atomic mass unit u = 1,660 539 04(2) 1027 kg

electric field constant ε0 = 8,854 187 817 1012 A s V1 m1

(dielectric constant of free space)

magnetic field constant μ0 = 4 π 107 V s A1 m1

(permeability of free space) 1,256 637 ... 106 V s A1 m1

Planck constant h = 6,626 070 04(8) 1034 J s(quantum of action) = 4,135 667 66(3) 1015 eV s

Avogadro constant NA = 6,022 140 86(8) 1023 mol1

Boltzmann constant k = 1,380 648 5(8) 1023 J/K

molar gas constant R = 8,314 460(5) J mol1 K1

Faraday constant F = 9,648 533 29(6) 104 C/mol

Source and more data: http://physics.nist.gov/cuu/

Documents

Lab course Basic Physics