Lab course Measurement Technique

MARTIN-LUTHER-UNIVERSITÄT

HALLE-WITTENBERG

INSTITUT FÜR PHYSIKGRUNDPRAKTIKUM

Lab course

Measurement Technique

FOR

POLYMER MATERIAL SCIENCE

4TH EDITION (2011)

Preface

The lab course Measurements Methods is intended for those master students of AppliedPolymer Science who don't have the Bachelor's-degree in Physics. The successful completion ofthe course is certificated. The course consists of one introducing lecture (2 h) and nineexperiments (4 h each).

The subjects of the course are

(i) planing, performing and evaluating scientific experiments; record writing; estimation ofmeasurement errors,

(ii) working with modern measurement technique (viscometer, ultrasonic device, powersupplies, amplifier, electrical multimeter, oscilloscope, function generator, optical spectrometer,X-ray device, G.M.-counter tube, digital counter, computer),

(iii) selected physical topics (mechanical properties of materials, radioactivity and X-rays, ACcurrent and electrical signals, light and optical spectra).

The introduction chapters of this booklet describe all general aspects of the laboratory course(safety in the lab, requirements to protocol writing and estimation of measurement errors,literature and software). The main part describes shortly the physical basics of each experimentand gives detailed instructions to experimenting and evaluating the results. The questions at theend of each 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

Lab Manager: Dr. Mathias Stölzer

4rd edition Halle, September 2011

Contents

INTRODUCTION

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

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

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

Error calculation and statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

English Literature on Basic Experimental Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Software in the Basic Laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

MECHANICS

M 14 Viscosity (falling ball viscometer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

M 19 Ultrasonic pulse-echo methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

THERMODYNAMICS

W12 Humidity (dew point hygrometer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

ELECTRICITY

E 37 Transistor amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

E 40 RLC Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

OPTICS AND RADIATION

O 6 Diffraction spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

O 10 Polarimeter and Refractometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

O 16 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

O 22 X-ray methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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 Procedure of a Laboratory Course

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 Do not throw used matches into the waste-paper bin!

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

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

29 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 protocoland 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 the

tutor before putting them into operation.

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

Introduction Guidelines to Writing a Protocol

3

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

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 9 experi-ments. At least 80 % (i.e. 7) have to bemarked with the 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 errorcalculation 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(mark and subscription of the tutor) if theexperiment was completely performed andevaluated.

Error Calculation 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 “errors calculation” 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.

Error (of meassurement):

Measurement result minus true value. Theerror consists of a random and a system-atic part. Generally, the measurement


5

2

1

( )

.1

n

i

i

x x

sn

=

−=

−

∑ (2)

.x

ss

n= (3)

1

1 .n

i

i

x xn =

= ∑ (1)

error can not be known exactly becausethe true value is not exactly known. The“error calculation” results in an estima-tion 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 ∆x und according to the new inter-national standard GUM u(x).

Example for m = 2.041 g:

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

The true value is with high probabilityexpected in the interval (m!∆m, m+∆m).

Complete measurement result:

Measurement result ± uncertainty, e.g.m = 2.041 g ± 0.002 gm = (2.041 ± 0.002) gm = 2.041 g and ∆m/m = 0.1 %

2 Determination of uncertainties

2.1 Calculation of meas. uncertainties in

the case of random errors

A quantity x is measured n times. The individ-ual measured values scatter around the aver-age

If only random errors occur in that series ofmeasurements, the distribution of the values isa normal (or GAUSS) distribution with thewell-known bell-shaped distribution curve.The scattering of the values is characterizedby the standard deviation

Within the interval ± s are 68.3 % of allxvalues. In other words: The probability to finda single measured value in that interval is68.3 %. If more series of n measurements are taken,the respective averages are normal distributedas well. The standard deviation of theseaverages is then

If n $ 10 and any systematic errors can beneglected, the standard deviation s' is equal to


6

y a b x= + ⋅ . (5)

[ ]

F a b y

y a bx

i

n

i ii

n

( , )

( ) min.

=

= − + →

=

=

∑

∑

∆ 2

1

2

1

(6)

( )( )

ax y x x y

n x x

ny b x

i i i i i

i i

i i

=−

−

= −

∑ ∑ ∑ ∑∑ ∑

∑ ∑

2

22

1(7a)

( )b

n x y x y

n x x

i i i i

i i

=−

−

∑ ∑ ∑∑ ∑2

2 , (7b)

∆ x

x x

n n

i

i

n

=−

⋅ −=∑ ( )

( ).

2

1

1

(4)

the (standard) measurement uncertainty:

If the measurement is a counting of randomevents x = N (e.g. radioactive decay events),the results are POISSON-distributed. In thatspecial case the standard deviation is the rootof the mean value and hence the measure-

ment uncertainty is ∆x = (see experimentN

O16).

2.2 Manufacturer guaranteed measure-

ment accuracy

The manufacturer of a measuring deviceusually specifies the measurement accuracywithin certain environmental conditions.(Examples: 1.5 % of full scale if 5°C # T #40°C; 0.1 % of full scale + 2 digit) Often theso-called accuracy class is specified which isthe guarantied accuracy in percent of fullscale or of the value of the material measure.A voltmeter with an accuracy class of 1.5 anda measurement range of 30 V has an uncer-tainty of ∆U = (1,5 % of 30 V) = 0,45 V.

2.3 Estimation of measurement errors

In there is no information about the accuracy,the error is to be estimated:S rule of thumb for reading scales: ∆x =

(0.5 ... 1) division,S Vernier caliber: ∆l = 0.1 mm,S time measurement with a stopwatch:

∆T = 0.2 s,S the uncertainty of a digital device is at

least 1 digit, in most cases more.

3 Regression (fit) of a function to a

series of measurements

3.1 Linear regression (linear fit)

Frequently, different measured quantities x

and y are linearly related or such a relation issupposed to exist:

Example:Thermal expansion of metals. The length of ametal rod depends on the temperature accord-ing to l = l0 + α@l0@∆T, where α is the coeffi-cient of linear expansion and l0 is the length at∆T=0.

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 unavoid-able random errors. The task is now to findthe straight line that “fits best” the measuredpoints. In this consideration it is assumed for simpli-fication that only the values yi are inaccurate.The deviation between the measured point(xi, yi) and the straight line at xi is

∆y = yi - y(xi) = yi - (a+bxi).

According to GAUSS's method of least squa-res, the best straight line is found by minimis-ing the sum of squares of the ∆y:

This sum is a function of the two parameters aand b. The problem is solved by setting thepartial derivations MF/Ma and MF/Mb equal tozero. This way we obtain


7

∆ ∆ ∆ ∆

∆ ∆

yy

xx

y

xx

y

xx

yy

xx

nn

ii

i

n

= + + +

==∑

∂∂

∂∂

∂∂

∂∂

11

22

1

...

.

(9)

( )s

y

n

x

n x xa

i i

i i

2

2 2

222

=− −

∑ ∑∑ ∑

∆(8a)

( )s

y

n

n

n x xb

i

i i

2

2

222

=− −

∑∑ ∑

∆. (8b)

where all sums are taken from i = 1 to n. The line defined by (5) and (7) is called theregression line.If the statistical errors predominate the sys-tematic errors, the uncertainties (the “errors”)of the parameters a and b are given by theirstandard deviations: ∆a = sa and ∆b = sb

with

3.2 Regression analysis with other func-

tions

The method of least squares is not restrictedto straight lines as in eq. (5) but can be ap-plied 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-mented in many scientific computer programssuch as the programs Origin and CassyLabthat are available in the Basic Laboratory.Look for the keywords non-linear curve fit orfree fit in those programs.Some functions can easily be transformedinto a linear function. In this case, the linearfit may be performed on the transformedfunction.

Example:When radiation penetrates matter it is attenu-ated according to I = I0@e

-µx (I: intensity, x:thickness penetrated, I0: I at x=0, µ: attenua-tion coefficient). If several pairs of values(I, x) have been measured, µ may be deter-mined by linear regression according to ln I = ln I0 - µ.

3.3 Practical hints

You don’t need to keep in mind the formulas(7) and (8), 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 yourcalculator. The standard deviations sa and sb

are calculated by computer software only.In many cases (if suitable software or pocketcalculator is not available or not requird) it issufficient to determine the regression parame-ters a and b graphically in the following way:Plot the measured points into a coordinatesystem on graph paper and draw the best fitline according to visual judgement using atransparent ruler.

4 Uncertainties of measurement results

(error propagation)

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

4.1 The maximum error

For small uncertainties ∆xi the uncertainty ofthe result may be calculated as the completedifferential of y:

Here, My/Mxi means the partial derivation of yfor the measured quantity xi. With this kind of calculation it is assumedthat the influence of all measurement errorsadd to the error of the result, hence the maxi-mum error is calculated.


8

∆ ∆yy

xx

ii

i

n

=

=∑ ∂

∂

2

2

1

. (10)

∆ ∆ ∆y

yn

x

xm

x

x= +1

1

2

2

(15)

∆ ∆ ∆y

yn

x

xm

x

x=

+

2 1

1

2

2 2

2

2

. (16)

y y y y± ∆ ∆and . (17)y c x c x= +1 1 2 2 (11)

∆ ∆ ∆y c x c x= +1 1 2 2(12)

∆ ∆ ∆y c x c x= +12

12

22

22 . (13)

y c x xn m= ⋅ ⋅1 2 (14)

4.2 GAUSS's law of error propagation

If the single measured quantities are statisti-cally independent, their errors can be ex-pected to compensate each other partially.The mathematical treatment of this problemby C. F. GAUSS gives

Generally, the uncertainty of a measuringresult is to be calculated by this equation.According to GUM, the uncertainty (10) iscalled combined uncertainty uc(y).Eq. (9) is only applicable for raw estimationsand in case the measured quantities xi arestatistically dependent.

4.3 Simple cases

Often the function y = f(x1, x2, ..., xn) is verysimple. In two special cases the calculation oferror propagation according to eq. (9) or (10)can be very much simplified:

(i) If the function has the form

(c1, c2 are constants), we find by inserting(11) into (9) and (10), respectively, themaximum error

and the GAUSS's error

(ii) If the function has the form

(c real and n, m integer numbers), we find by

inserting (14) into (9) and (10), respectively,the relative maximum error

and the relative GAUSS's uncertainty (relativecombined uncertainty)

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 ∆d and ∆t, and a is to be calcu-lated, we get

( ) ( )2 2

22 , 2 .d a d t

aa d tt

∆ ∆ ∆= ⋅ = + ⋅

5 Presentation of measurement results

and uncertainties

Always present the complete measurementresult:

The uncertainty ∆y (which is commonlycalled „the error“, but see paragraph 1 forexact definitions) 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, ∆y/y = 0.43 %

U = (20.00 ± 0.15) V, ∆U/U = 0.12 %

R = 2.145 kΩ ± 0.043 kΩ, ∆R/R = 2.0 %

Introduction Available Literature on Basic Experimental Physics

9

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

10

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 8 is available on all computers in the lab and in the students computer pools.The university owns a campus licence that allows the use even on private computers, providedthere is a VPN connection to the university network (ask the staff in the lab for technicaldetails). Alternatively, there is the free Origin clone SciDAVis (http://scidavis.sourceforge.net/).This program runs on Windows, Linux and Mac OS and can read and write Origin files up toversion 7.5.Origin is not a part of your education in physics but is just an offer. You may evaluate yourexperimental data using a pocket calculator and graph paper only, or with the help of any othersoftware (e.g. Microsoft Excel).

Short introduction to Origin

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.

Introduction Software in the Basic Laboratory

11

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

• 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^2%&x 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 orclick . Right-click the layout to add graphs and worksheets.

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

Mechanics M 14 Viscosity (falling ball viscometer)

12

.w b f

F F F= + (1)

F r vf= 6πη , (4)

32

4 ,3b

F r gρ= π (2)

31

4 ,3w

F r gρ= π (3)

( )η ρ ρ= −29

2

1 2

r

sg t . (7)

Fig.1: Laminar flow in a tube

43

643

31

32π π πr g r v r gρ η ρ= + (5)

( )643

31 2π πη ρ ρr 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. Aliquid, the viscosity of which does not dependon the flow velocity, is called a NEWTONianliquid (also NEWTONian fluid or ideal viscousliquid). Most of the homogenous liquids (i.e.water, oil) behave like this, while fluids

consisting of different phases (i.e. ketchup,printing ink, blood) are non-NEWTONianfluids.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:

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

Viscosity (falling ball viscometer) M 14


13

( )η ρ ρ= ⋅ − ⋅K t1 2 . (8)

0 .EE

k Teη η= ⋅ (10)

~EE

k Tj e− (9)

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

3 Experimental setup

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 wholearrangement can be turned by 180° into themeasuring position and the roll-back position,respectively. In your lab work you can mea-sure with sufficient accuracy in the roll-backposition, too.The exact value of geometry constant K isgiven in a test certificate provided by themanufacturer.

4 Experimental procedure

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 five times at each temperature. If thefall time exceeds 2 min, you can use the halfmeasuring distance (upper and middle mark).


14

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. Thenmeasure the fall time (five 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.(10) this should result in a straight line.Calculate the excitation energy EE from theslope of that line.

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?

Mechanics M 19 Ultrasonic pulse-echo methods

15

λ =c

f. (1)

y yx= ⋅ − ⋅

0 e µ . (3)

cE

L=

−+ −ρ

νν ν1

1 1 2( )( )(2)

µ = − ⋅1

2 1

1

2x x

y

yln . (4)

Z c= ⋅ρ (5)

1 Task

1.1 Determination of the sound velocity andthe wavelength of longitudinal waves inPolyethylene PE and calculation of Young'smodulus (modulus of elasticity).

1.2 Determination of the attenuation coeffi-cient (damping constant) of ultrasonic wavesin PE at two different frequencies.

1.3 Determination of the positions of holesin a PE body and drawing a site map of theseholes.

2 Physical basics

If a mechanical oscillator is in contact withanother medium, energy is transferred fromthe oscillator to the medium through thiscoupling. This energy propagates as a me-chanic or elastic wave that is called soundwave. The arising periodical changes ofpressure and density in the medium propagatewith a phase velocity (sound velocity) c. Thewavelength λ in the medium is determined bythe frequency f of the sound source and thematerial dependent velocity of propagation c:

Because of the lack of shear elasticity, themechanical waves in gases and liquids occuras longitudinal waves (direction of oscillationin the direction of propagation), . In solidmaterials, in addition to longitudinal waves,transversely waves (oscillation perpendicularto the direction of propagation), as well ascouplings between them can occur.

In infinite, homogeneous, and isotropic solidsthe velocity of sound cL for longitudinalwaves can be obtained from the mechanical

properties of the medium as:

(E = YOUNG's modulus of elasticity; ρ =density; v= POISSON's ratio).

By inelastic interaction with the medium asound wave is damped. The amplitude ofoscillation y decreases according to theattenuation law.

Here, y0 is the amplitude at x = 0 and µ

the so-called attenuation coefficient, alsoreferred to as damping exponent. Consideringthe amplitudes y1 and y2 according to the twodifferent thicknesses x1 and x2, from eq. (3)follows

The damping is sometimes used to distin-guish different materials in ultrasonic mate-rial testing.

The product of the density ρ and the soundvelocity c of a material is referred to asacoustic impedance (acoustic characteristicimpedance, acoustic resistance) Z:

According to equation (2), the acousticimpedance characterizes the elastic propertiesof the material. A change or jump (saltus) inthe acoustic impedance (for example oninterfaces) along the direction of propagationresults in a partial reflexion of the acousticenergy and, additionally, in an attenuation inthe direction of propagation (“soundshadow”).For perpendicular incidence of a sound wave

Ultrasonic pulse-echo methods M 19


16

Fig.1: Reflexion of ultrasound on an inter-

face between two materials with different

acoustic impedances

Fig.2: Formation of A- und B-scan

cd

t=

2(8)

RI

I

Z Z

Z Z

R= =−+0

1 2

1 2

(6)

I I IT R= −0 . (7)

on a surface (see fig.1) the reflectivity R isgiven by:

(IR, I0: intensity of the reflected and incidentwave; Z1, Z2: characteristic sound impedanceof the neighbouring media). The intensitytransmitted through the interface is

The human audible range is approximately16 Hz – 16 kHz. Sound waves above thisrange are called ultrasound. Below 16 Hz isthe infrasonic range. Ultrasound waves are generated using piezo-electric ceramics as a mechanical oscillator.The oscillator is excited to oscillate with itsnatural frequency fR (the resonance fre-quency) that is defined by its material proper-ties and geometry.In the ultrasonic pulse-echo methods theoscillator (the “transducer”) is excited byvery short electric pulses to short-time thick-ness vibrations and the emission of ultra-sound impulses (reciprocal piezoelectriceffect). In the time interval between twopulses, ultrasound waves coming from thecoupled medium back to the same transducer(the “pulse-echos”) produce small deforma-

tions of the transducer, that will be trans-formed by the piezoelectric material intoelectric voltages (direct piezoelectric effect).Thus, the same transducer is used as transmit-ter as well as receiver.In the so-called A-mode (A-scan method) theAmplitude of the transmitted pulse and of thereceived and amplified echos are plotted as afunction of time on the screen of a cathoderay tube or a computer monitor. In this plotthe echos from structural boundaries withinthe medium where the acoustic impedancechanges occur as peaks (see fig.2). If thevelocity of sound c is known, the depth d of areflecting structure is measured according to

where t is the time interval between thetransmitted pulse and a received echo that istwice the running time of the acoustic pulsebetween transducer and reflective structure.With a suitable calibration of the time scale,the distance can be read directly on thedisplay.

In the B-mode (from Brightness), also calledimaging mode, the transducer is moved asshown in fig.2 along the surface of the speci-men investigated. The amplitude of the echosignal is mapped to brightness in a two-dimensional image of the cross-section of thespecimen. Modern B-scan devices use a


17

multi-element transducer that consists ofmany oscillator elements. In this case amovement of the transducer is no longerrequired. Instead, the single oscillator ele-ments are excited consecutively by the elec-tronics.

The quality of an ultrasonic scan is characte-rized by the resolution. This is defined as theinverse of the smallest distance between tworeflecting structures that can only just bedistinguished. In B-scan the vertical and the lateral resolu-tion (in the direction and perpendicular to thedirection of sound propagation, respectively)have to be distinguished. The vertical resolu-tion is mainly determined by the length of theultrasonic pulse that is limited by the wave-length. The lateral resolution is limited by thewavelength, too, but is also strongly affectedby the sound field geometry.Generally, the resolution improves withincreasing frequency. Simultaneously, how-ever, the damping of the ultrasonic waves inthe medium increases and hence the penetra-tion depth decreases.Known problems with pulse-echo methodsare:- Dead time: The ultrasonic pulse is made

by exciting the transducer to resonanceoscillations. During the decay time ofthese oscillations incoming echoes cannotbe registered. A dead-zone is formed, i.e.structures near the transducer cannot beresolved.

- Sound shadow: Strongly reflecting struc-tures reflect most of the sound energy.Objects behind such structures may beinvisible.

- Multiple reflections between the surfaceand strongly reflecting structures maycause phantom images of these structuresin double distance from the surface.

- Aberration may occur by refraction ofsound waves on structures with differentsound velocities.


3.0 Devices:

- GAMPT A-Scan device- 2 transducers (1 MHz, 2 MHz) with con-

necting cables- PE body with imperfections (drilled holes)- vernier calliper

3.1 The device allows an A-scan as well asa simple B-scan using one single-oscillatortransducer as transmitter and receiver asdescribed above (reflection mode). Addition-ally, transmission measurements with twoseparate probes for transmitting and receiv-ing ultrasonic pulses are possible but notutilized in the experiment. The REFLEC./

TRANS. switch must be set to reflection andthe left probe socket is used only. Additionalcontrols are provided for adjusting theTRANSMITTER output voltage, the RECEIVERgain and the parameters of the TIMER-TGC

(“Time-Gain-Control”). The computer software allows for measuringtime or depth differences and amplitudes inthe A-scan with the help of three colouredcursors that can be moved with the mouse.Advanced features of the software (HFdisplay, filtering, FFT) may be discussed withthe tutor.


The tutor will explain the device and itsoperation. Plug the transducers into the leftsocket on the front panel. Use water forcoupling the transducer to the PE-body (Onlyvery little water is needed!).

4.1 For the determination of the soundvelocity, determine the thickness d of the PEbody using the vernier calliper. Determine thetime between the initial pulse and the echofrom the back plane of the PE body for the1 MHz and the 2 MHz transducer.

4.2 For determining the attenuation coeffi-cient at 1 MHz, measure the thickness d1 andthe width d2 of the PE body. Adjust the TGC


18

in order to get the same constant amplifica-tion for both distances. Measure the ampli-tudes y1 and y2 of the back-plane echoes inthe two directions. Repeat these steps usingthe 2 MHz transducer.

4.3 There are 4 boreholes somewhere in thePE body, your task is to determine theirpositions.After the sound velocity has been calculated,it can be set in the options menu. Now youcan switch from time scale to distance scaleby clicking the button ‘Depth’. Verify that thedepth of the back plane echo is equal to themeasured thickness of the body! A-scan the lateral surfaces of the PE-bodywith the 1 MHz and the 2 MHz transducer.Adjust the device settings for transmitteroutput, receiver gain and the TGC controlswith respect to the following aspects:

- The desired echo should not be covered bythe initial echo.

- The attenuation of the signal with increas-ing depth should be counterbalanced.

- The echo signal should not be overampli-fied, for an accurate localization on thedisplay to be possible.

When the settings are optimized switch to B-scan. Press 'Start' and move the transducerslowly and steadily over the surface, thenpress 'Stop'. Invert and adjust the gray scaleuntil all boreholes are visible. Repeat the scanif necessary. The program window can beprinted out. (Please print it only once for eachfrequency and each student!)The B-scan with the given device is only

quantitative. The exact positions of the holeshave to be measured using the A-scan from(at least) two sides tilted by 90°.

5 Evaluation

5.1 Calculate the sound velocity accordingto eq. (8) and the wavelength λ for bothtransducers according to (1). The modulus ofelasticity is to be calculated using equation(2) with ν = 0.45 and ρ = 0.932 g cm-3.

5.2 Calculate the attenuation coefficientaccording to (4) for the two frequencies used.Because the sound echo travels twice throughthe PE body, in eq. (4) the denominator (x2-x1) has to be replaced by 2(d2-d1).

5.3 Draw a cross-section of the PE-body ongraph paper (1:1 scale) and mark the holesdetermined. Investigate the printouts of theB-scan with respect of sound shadows andmultiple reflections.

6 Questions

6.1 Which physical quantities are displayedin A-scan and B-scan?

6.2 Why is water needed for coupling thetransducer to the PE body?

6.3 The velocity of sound in PE is about2000 m/s. Which wavelength exhibits anultrasonic wave of (i) 1 MHz and (ii) 2 MHz?

Thermodynamics W 12 Humidity

19

U TT= ⋅α ∆ (1)

Fig. 1: Copper-constantan thermocouple

fp

pr

V

S

= (3)

fm

Va

V= (2)

1 Task

1.1 Calibrate a copper-constantan-thermo-couple.

1.2 Determine the relative humidity using adew point hygrometer.

1.3 Verify RAOULTs law (vapour pressuredepression of solutions) qualitatively.

2 Physical basics

2.1 Thermocouple: A temperature gradientalong an insulated electric conductor isalways accompanied by a small voltagegradient (electrons tend to leave the hot endbecause they move quicker than electronsfrom the cold end). This is called the absoluteSeebeck effect. To measure this voltage, theends of the conductor have to be joined toother conductors to construct an electriccircuit, as shown in fig.1. If all conductorswould consist of the same material, theabsolute Seebeck effects would compensateeach other and the resulting voltage would bezero. But if wires of different material areused (i.e. copper and constantan), a resultingvoltage difference UT appears that depends onthe two materials and on the temperaturedifference ∆T = T1 - T0 between the twojunctions:

This is called the relative Seebeck effect. Thecoefficient α that depends on both materialsis called the Seebeck-coefficient (alsothermopower). Thermocouples are widely used for tempera-ture measurement. They have the advantagethat the thermal voltage can be immediatelyused as an input signal for computers andcontrol devices.

2.2 Humidity is the content of water vapourin air.The absolute humidity fa is the mass ofwater vapour mV per volume V of air:

The relative humidity fr is the ratio of theactual amount of water vapour to the amountat saturation, or the ratio of actual vapourpressure pV to the saturation pressure pS atthe actual temperature T.

The relative humidity is usually given inpercent. Consider a closed vessel containing purewater and air above the water. A part of thewater evaporates until the space above theliquid is saturated with water vapour. At thethermodynamic equilibrium the saturationvapour pressure pS appears that is only basedon the kind of liquid (here water) and on thetemperature (about exponentially). Therelative humidity in that case is 100 %.If other substances are dissolved in the water(the solvent), the saturation vapour pressureis decreased by ∆p. According to RAOULTslaw, the lowering of vapour pressure ∆p

depends entirely on the amount of dissolved

Humidity W 12


20

Fig. 2: Experimental setup for determination of humidity

fp T

p T

p

p Tr

V

S

S

S

= =( )

( )

( )

( )

τ(5)

∆p

p

p p

px

n

n nS

S S L

S

=−

= =+

,.2

1 2

(4)

particles but not on the kind of substance:

Here, pS is the saturation vapour pressure ofthe pure solvent, and pS,L that of the solution.x is the mole ratio of the dissolved sub-stances, n1 the number of particles of thesolvent, and n2 the number of dissolvedparticles in mol. n2 accounts for the dissocia-tion of the dissolved molecules into ions.RAOULTs law is exactly valid, only when thevapour pressure of the dissolved solid isnegligible small and for n2 n n1. At higherconcentrations the decrease of vapour pres-sure is less. As a result of Raoults law thehumidity over a solution is always smallerthan 100 %.

In the case that for a certain temperature T thehumidity in a room is smaller than 100 %, ahumidity of 100 % can be reached by lower-ing the temperature. At a certain temperatureτ, the so called dew point, water vapourbegins to condense. A mirror surface will becovered with mist below this temperature.This is utilized in a dew point hygrometer.The saturation vapour pressure at dew pointpS(τ) is equal to the vapour pressure at roomtemperature pV(T). The relative humidity fr

than results to

which can be evaluated from table data of thewell-known saturation vapour pressure ofwater.


3.0 Equipment:

- dew point hygrometer (aluminum bodywith a Peltier cooler, metal mirror, thermo-couple and photo sensor)

- control device for the photo sensor- power supply for Peltier cooler- copper-constantan thermocouple, one

welding point in a tube with Gallium - beaker glass and vacuum ice container- rectangular plastic dish- plastic cover for the hygrometer- bottle with 3 M CaCl2-solution- SensorCassy with µV-box (used as volt-

meter)- computer, CassyLab software

3.1 For calibration of copper-constantanthermocouples two fixed points are used: themelting point of water and the melting pointof gallium (Tm=29.5 °C). The thermal voltageis measured using the SensorCassy/µV-boxand computer.

3.2 The dew point hygrometer (fig. 2) has areflective metal surface (polished aluminumblock) that can be cooled down. The tempera-ture of the surface can be measured and theformation of mist (condensed water vapour)can be observed. A semiconductor coolingelement (utilising the PELTIER effect) is usedfor cooling. It is driven by a power supply(30V/1.5A) that is used as constant currentsource. The temperature is measured by acopper-constantan-thermocouple. One junc-tion of the thermocouple is placed in the


21

aluminum block and in the other one in anice-water-mixture at 0°C. The thermal volt-age is measured with the SensorCassy equip-ped with a µV-box and recorded by a com-puter. A photo sensor is used for reproducibleobservation of the condensation of water onthe cooled mirror. It is equipped with a statusLED and a relay controlled by the photosensor. The relay is used for constructing asimple temperature control to reach the dewpoint (see fig. 2). It switches the coolingcurrent automatically on and off dependingon the degree of mist cover on the mirror.

3.3 The dew point hygrometer is placedunder a plastic cover together with a dishcontaining water or 3M CaCl2-solution,respectively.


Ask the tutor when you need assistance withthe devices given. Instructions for operatingthe CassyLab software are provided in theonline help of the program. To switch thelanguage from German to English, selectEinstellungen ÷ Allgemein ÷ Sprache ÷English.4.1 Construct the wiring according to fig.1.For voltage measurement, the input channelof the SensorCassy equipped with the µV-box is used. Power the SensorCassy and startthe program CASSYLab-W12, if not yetdone. All options in the program (measuringrange and interval etc.) are already set ac-cordingly. The recording of a series ofmeasurements can be started and finished bypressing the key F9 or clicking to .

Fill the vacuum container with crushed iceand small amounts of water. The beaker glassis to be filled with hot water (at least 50°C).Dip one junction of the copper-constantan-thermocouple (the one without Gallium) intothe ice-water-mixture. For calibration, start recording a series ofmeasurements at the computer and dip the

other junction of the thermocouple enclosedin a tube with Gallium into the hot waterbath. At a temperature of T1 = 29.5 °C, theGallium melts. The voltage measured re-mains constant until all of the Gallium ismelted, than the voltage increases again. Nowdip the tube with Gallium into the beakerglass containing the ice-water. The voltagedecreases again and remains constant at29.5 °C while the Gallium is solidifying(possibly after some supercooling of theGallium melt).Calculate the Seebeck-coefficient α of thecopper-constantan-thermocouple from theaverage value of the voltages during themelting and the solidification of the Gallium,according to equation (1).

4.2 Construct the experimental setup ac-cording to fig.2. Dip the free junction of thethermocouple into the ice-water-mixture.Record the thermal voltage with the computeras described under 4.1. The status of the relay switch of the photosensor is displayed by an LED. You have toadjust the comparator threshold (labelled as'Komparatorschwelle') in a manner that theLED lights when the mirror is blank and itturns out when the mirror gets steamy.Breathe upon the mirror to test it.Switch on the power supply of the cooler,adjust the current to 1 A and measure thethermal voltage. The LED of the photo sensoris bright. When the temperature drops belowthe dew point, the LED turns out and thecooler is switched off by the relay. Thetemperature in the dew point hygrometerincreases again, the mist on the mirror sur-face evaporates. After several seconds theLED flashes up again and the cooler isswitched on automatically. In that mannerregular oscillations of the thermal voltageappear that are related to the dew point. Theregulation works at best when the coolercurrent is adjusted for the cooling rate beingthe same as the heating rate. (answer: Why?)Record about 10 oscillations, then stop.Determine he thermal voltage of the dew


22

n nV n M

M1 = =−

H2OCaCl2 CaCl2

H2O

ρ. (7)

ρ =+

=+

m m

V

n M n M

V

CaCl2 H2O

CaCl2 CaCl2 H2O H2O .

(6)

n n2 3= ⋅ CaCl2 . (8)

point as the mean value over these oscilla-tions using the program CassyLab. Afterfinishing the measurement, you have toswitch off the cooler.

4.3 For the qualitative evaluation ofRAOULTs law, a dish with water is placednext to the hygrometer. The water tempera-ture should be equal to the room temperature.Place the hygrometer and the water dishtogether under the plastic cover. The humid-ity under the cover will slowly increase up tonearly 100 %. After waiting 20...30 min,switch on the cooler and start operating thedew point hygrometer as described under 4.2.Record the thermal voltage until the dewpointremains constant (circa 5 min). For minimiz-ing the temperature oscillations around thedew point, the cooler current can be loweredto 0.4…0.2 A.Repeat the experiment, with the dish beingnow filled with a 3M CaCl2 solution. Determine the thermal voltages at the dewpoints by calculating the mean values of theoscillations using the program CassyLab.

The CaCl2 solution is to be refilled into thebottle after the experiment is finished.

5 Evaluation

5.1 Print the diagram UT(t). Calculate theSeebeck coefficient α of the copper-constan-tan thermocouple according to equation (1).

5.2 Print the diagram UT(t). Determine theroom temperature and dew point from thediagram. Calculate the relative humidity fr

according to equation (5). The saturationvapour pressure is given in a table.

5.3 Determine the dew points and therelative humidities above water and 3MCaCl2 solution as under 5.2. Calculate theexpected humidity above the CaCl2 solutionusing RAOULTs law (4). The CaCl2 is fullydissociated, the density is 1.25 g cm-3.Compare the measured and the calculatedvalues.

Help for calculating the mole fraction:mCaCl2 und mH2O are the masses, nCaCl2 andnH2O the amounts of substances and MCaCl2

and MH2O the molar masses of the CaCl2 andthe H2O molecules in V = 1 l solution. Thedensity of this solution is then

Consequently, the amount of substance of thesolvent is

The amount of solved particles (i.e. Ca2+ andCl! ions)

6 Questions

6.1 Explain the formation of weather phe-nomena like rain, fog, dew.

6.2 What is a thermocouple? How does itwork?

6.3 At what temperature does salt waterboil?

Electricity E 37 Transistor amplifier

23

Fig. 1: Elementary amplifier

VU

UO

I

=∆∆

(1)

Fig. 2: Experimental circuit

1 Task

1.1 Record the transfer characteristic of anelementary transistor amplifier and calculatethe voltage gain at the operating point.

1.2 Determine the voltage gain of the basiccircuit as function of frequency.

1.3 Analyse the behaviour of the amplifierin case of wrong operating point setting andin case of overdriving.

2 Physical Basis

2.0 See chapter 2.1 of the experiment E40for a description of an oscilloscope.

2.1 Each electronic amplifier consists of anelementary amplifier and, depending on theapplication, additional components for theadjustment of the operating point, stabilisa-tion, degenerative feedback, or coupling andextraction of the signals.The elementary amplifier is composed of anelectric source (battery or power supply withthe operating voltage Uop) and a voltagedivider, consisting of a constant resistor Rop

(operating resistor or load resistor) and a con-trollable resistor - the transistor (see fig.1).The transistor is controlled by the voltage UI

at the input I of the amplifier. The voltage UO

at the output O of the amplifier is part of the

operating voltage Uop. The output power istaken from the power supply.The dependence of the output voltage UO onthe input voltage UI (at constant operatingvoltage) is represented by the transfer charac-teristic of the elementary transistor amplifier.The operating point of the amplifier is set onthe steep decaying part of this characteristic.Only in the vicinity of this point an optimalvoltage gain is possible. Small changes in theinput voltage cause large changes of theoutput voltage. The voltage gain G is givenby:

The voltage gain can be calculated from theslope of the tangent in the operating point.The elementary transistor amplifier can beextended to a RC-coupled basic circuit (seefig.2). Using a voltage divider, a constant DCvoltage of (0.7±0.1)V is applied to the inputfor the amplifier to works at the operatingpoint. The operating point is correctly ad-justed when the output voltage is about 50%of the operating voltage.The basic circuit includes an input capacitorthat blocks direct currents. A DC currentwould shift the operating point.The input capacitor, the resistors of thevoltage divider and the resistor Re at theemitter of the transistor together form a so-called RC high-pass filter. High frequencies

Transistor amplifier E 37

Electricity E 37 Transistor amplifier

24

Fig.3: frequency response of an amplifier

can pass this filter and low frequencies areblocked.The junction capacitance of the transistor andthe operating resistor Rop of the amplifierform a low-pass filter. Thus, AC voltageswith different frequencies are differentlyamplified.Plotting the gain G as a function of frequency f yields the frequency response curve likethat shown in Fig. 3. The abscissa (frequencyaxis) is logarithmically scaled.

The frequencies where G reaches Gmax / 2

are called cutoff frequencies (lower cutofffrequency fl, upper cutoff frequency fu). Theband width is given by the difference (fu - fl).


3.0 Devices:

Board with amplifierPower supply (stabilised)Oscilloscope (CRT) Function generator 81302 MultimetersCables

3.1. The amplifier is assembled on a board(Fig. 2).For determining the transfer characteristic,the input voltage UI and the output voltageUO (DC voltages) are measured by multi-meters.For determining the voltage gain of the basiccircuit, the AC input voltage is provided by a

function generator; the output voltage ismeasured using the oscilloscope (see experi-ment E20).


4.0 Read the paragraph about amplitudesetting in the manual of the function genera-tor HM8130 (page 23). Pay attention that thefunction generator displays the peak-to-peakvoltage UPP without load, while the multi-meters measure the effective value Ueff of anAC voltage. See experiment E20, chapter 2.2for explanation.

4.1 In order to record the transfer character-istic, apply the operating voltage (Uop =+10V) as follows (see fig. 3): The positivepole of the voltage source to the correspond-ing socket of the amplifier (+10V); the nega-tive pole of the voltage source to earth (z).Set the resistor Re to zero. Connect the twomultimeters for voltage measurements.Now vary the input voltage UI from 0 V tomaximum using the voltage divider andmeasure the corresponding output voltagesUO (at least 10 measuring points).

4.2 In order to determine the voltage gain asa function of frequency, adjust the operatingpoint of the amplifier (UO=5V) first. Discon-nect both multimeters and then attach thefunction generator to the input I, and theoscilloscope to the output O of the amplifier.On the Function generator, adjust an ACinput voltage (UI) of UPP = 40 mV. Adjustone after the other the following frequencies:30Hz; 100Hz; 300Hz; 1kHz; 3kHz; 10kHz;30kHz; 100kHz; 300kHz; 1MHz; 3MHz; 10MHz. Measure the AC output voltage UO

(peak-peak) for each frequency with the CRT.

4.3. The same circuit as in 4.2. is used. Setthe frequency to 10 kHz. Set the input cou-pling of the oscilloscope to DC.

4.3.1 Keep working at the correct operatingpoint and increase the input voltage UI to400 mV (peak-peak). Observe the overdrivenoutput voltage and draw an outline of UO(t).

Electricity E 40 RLC oscillator

25

4.3.2 Reduce the input voltage to 40 mVagain and shift the operating point of theamplifier up and down (on the characteristicline). Observe the resulting non-linear distor-tions and outline them.

4.3.3 Adjust the correct operating point andan input voltage of 200 mV. Change theresistance Re and observe the influence on theoutput voltage. Re causes a negative feedback.Can you explain this effect?

5 Evaluation

5.1 Plot the transfer characteristic. Calculatethe voltage gain G in the operating pointaccording to (1) from the slope of the curve.The slope should be determined at the linearpart of the transfer characteristic around UO =5V.

5.2 Calculate the voltage gain of the basic

circuit according to G = UO / UI and plot itas function of the frequency f. The abscissa(frequency axis) should be logarithmicallyscaled (compare with Fig. 2). Determine theupper and the lower cutoff frequencies fu andfl as well as the band width.

5.3. Discuss the sketches of the investiga-tions made in 4.3.

6 Questions

6.1 What is a bipolar transistor? Describethe principle of this important electroniccomponent.

6.2 What is the maximum value, the effec-tive value and the peak-to-peak value of anAC voltage?

6.3 Explain these items: amplification,transfer characteristic, cutoff frequency.

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. The

intensity 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 througha 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 the

RLC oscillator E 40


26

Fig.1: Sweep voltage for X deflection with

the period Ts

screen 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 to

be measured is applied to the Y-plate pair.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) deflection

Fig.2: Cathode ray tube (CRT)


27

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

Fig.4: Ideal LC resonator

are adjusted by a rotary switch and a continuous 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 leadcarries 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 between

the 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

.dq dU

I Cdt dt

= − = − ⋅ (1)

.dIU L

dt= ⋅ (2)

2

2

1 0 .d UU

LCdt+ = (3)

( )0cosm

U U tω ϕ= ⋅ + (4)


28

Fig.5: Real RLC oscillator with AC genera-

tor G for driving the circuit to forced oscilla-

tions

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 caseRS = 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 ofcos ωt 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 homogeneousequation (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 homogeneous

0

1 .L C

ω =⋅

(5)

1 2 3

1

2

0 with

G

P

I I I

U UI

R

dUI C

dt

+ + =

−=

= ⋅

(6)

33 .

S

dIU I R L

dt= + ⋅ (7)

0 cos .G

U U tω= ⋅ (8)

2

02

1 1 cos .P P

d U dUU U t

R C dt LC R Cdt

ω ω+ + =

(9)

22

022 i td U dU

U K edtdt

ωα ω+ + = ⋅ (10)

12

PR C

α = (11)

0 .P

K UR C

ω= (12)


29

T

Un+1

Un

t

U

Fig.6: Damped oscillations according to (13)

eq. 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 α@t o 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-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


3.0 Devices

- plug-in board 20×30 cm- LC resonator circuit in a housing- 2 resistors in a housing- 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).

2 20

cos ,

.

t

mU U e t

α ω

ω ω α

− ′= ⋅ ⋅

′ = −(13)

1

1

1ln ln .n

n n

U UT

U n Uδ α

+

= = = ⋅ (14)

( )( ) .i tU t A e

ω ϕ+= ⋅ (15)

( ) ( )2 22 2

0

2 20

,

2

2arctan .

KA

ω ω αω

αωϕ

ω ω

=− +

=−

(16)

( )

( )0

22 2 2 2 2

0

.

4 4

PU f R C

A

f f fπ α

⋅=

− +(17)

0 ,u l

fQ f f f

f= ∆ = −∆

(18)

.P

CQ R

L= (19)


30


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 the

points 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 themeasured 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 lowercut-off-frequency fu and fl and the bandwidth∆f for each resonance curve. Calculate the Qfactors of the two RLC circuits according to(18) and (19) and compare the results.

Facultative task for students with high skillsin 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?

Fig.7: Measurement circuit. The two connec-

tors on the right are not in use.

Optics & Radiation O 6 Diffraction Spectrometer

31

Fig.1: Diffraction of a plane wave on a

edge. 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, n: angle of diffraction

δ λ= ⋅k (2)

δ λ= + ⋅( )2 1

2k (3)

δ ϕ= ⋅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 gratingwhich is a plane two-dimensional periodical

arrangement 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

where k = 0, 1, 2, 3,… is called the diffrac-

Diffraction Spectrometer O 6

Optics & Radiation O 6 Diffraction Spectrometer

32

Fig.3: sketch of a diffraction spectrometer

R N k= ⋅ (7)

sinϕ λk

k

b=

⋅(4)

λϕ

=⋅b

kksin

. (5)

R =λλ∆

(6)

tion order. The undiffracted (straight ongoing) light is referred to as zeroth diffractionorder (k=0).From the equations (1) and (2) the angle nk 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 ∆λ 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. Theparallel diffracted light (that apparently

comes 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 isshifted 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 youcan see the bright circular area of the tele-

Optics & Radiation O 16 Polarimeter and Refractometer

33

ϕϕ ϕ

k

right left=

′ − ′2

(8)

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 correspondingangles φ'. You have to measure 6 spectral

lines 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.

1 Tasks

1.1 Determine the concentration of a water-sugar solution 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 a

Polarimeter and Refractometer O 10


34

Fig.1: Position of electric and magnetic

field-strength vectors for a wave train

nc

c= 0 . (2)

ϕ = ⋅ ⋅k l c , (1)

n n1 2⋅ = ⋅sin sin .α β (3)

vast 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 vectoris 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-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-


35

Fig.2: Ray trace of refraction for n2 > n1.

Left: general case, right: striping light entry

Fig.4: Three-part visual field of the

polarimeter

sin .maxβ =n

n1

2

(4)

Fig.3: Ray trace at an ABBE refractometer

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- flask with sugar solution- ABBE refractometer - 2 burettes with glycerol and deion. 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 anangle 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°.


36

ϕ ϕ ϕ= −1 0 . (5)

3.2 The refractometer consists of the follow-ing essential parts:

- lightning prism with a rough surface

- measuring prism whose refractive index n2

must be larger than the refractive index n1

of the substance under investigation

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

- AMICI prisms, a device for compensatingthe dispersion (removing colour fringes)

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.

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.Measure the length of the polarimeter tube.

4.2 The two prisms of the refractometermust be on the right side, and the smallmirror 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.Look through the measuring eyepiece (theright one) and turn 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 measured value differs by morethan one scale division from the referencevalue 1.333, ask the tutor to adjust therefractometer.

The refractive index is to be determined forthe following liquids:

- de-ionized water

- pure glycerol

- 5 glycerol-water mixtures:

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

- glycerol-water mixture of unknown con-centration

Optics & Radiation O 16 Radioactivity

37

For the mixture 4:1 take 4 ml glycerol and1 ml deion. water, and make the other mix-tures by further dilution of this mixture withwater.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) amounts to k = 0.66456 deg m-1 l g-1 at λ = 589.3 nm. Perform an error calculation for the concen-tration. For the uncertainty of the rotary angle

take the sum of the standard deviations of φ0

and φ1, according to (5).

5.2 Make a plot of the refractive index viathe volume concentration of glycerol inwater.

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?

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.

Radioactivity O 16


38

d .dN

At

= − (1)

dN N t= − ⋅ ⋅λ d . (2)

µ µ µ µ µ= + + +S Ph C P

. (6)

N t Nt( ) = ⋅ − ⋅

0 e λ (3)

.P Z

I I I= − (4)

I Ix= ⋅ −µ⋅

0 e . (5)

( )P nN

np p

N

n N n

( ) =

−

−1 (8)

1/ 2ln 2 .xµ

= (7)

P nn

n

( )!

=−ν νe

(9)

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 ofradioactive 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 x

according 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, it

depends 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), inelasticscattering (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)ν = n@ p. In our experiment N is a huge num-ber and p is very small. Passing to the limitsN÷4 and p÷0, the binomial distribution (8)transforms into a POISSON distribution

with the mean value ν.An important mathematical property of the


39

.n nσ∆ = ≈ (10)

P nn

( )( )

.=−

−1

2

2

2

πν

ννe (11)

POISSON distribution is the equality of meanvalue ν and variance σ2 (square of the stan-dard deviation σ). From that follows:If a large number (n > 100) of random eventsis measured in a time period, the uncertaintyof the measurement result is

The statistical uncertainty is approximately

equal 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 radiator1.17 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 the

preparation and the counter tube is the dis-tance between the edges of the gray sledges+ 10 mm.


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 kept

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.


40

either

or e

ln ln

lg lg lg

I I x

I I x

= − ⋅

= − ⋅ ⋅0

0

µ

µ

constant 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 canbe 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 thepulse 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.σ ≈ 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 “Fitfunction” 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

41

E h f hc

Ph = ⋅ = ⋅ λ(2)

E e U h f hc

Ph maxmin

= ⋅ = ⋅ = ⋅ λ (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


42

Fig.1: typical X-ray spectrum revealing

Bremsstrahlung and characteristic radiation

Fig. 2 BRAGG reflexion

λ λα βK

L KK

M K

h c

E E

h c

E E=

⋅− =

⋅− (4)

2 1 2 3⋅ ⋅ = ⋅ =d k ksin , , , , ...β λ (5)

Eh c

dPh =⋅

2 sin β (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


43

JQ

m=

∆∆ . (7)

DE

m=

∆∆ . (8)

H w D= ⋅ (9)

Fig.3: Measurement of the ion dose rate in

an 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 ∆Q produced in a volume elementdivided by the mass ∆m of that volumeelement:

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

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. w

is 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 includingLiF crystal 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. The doors and windows consist of lead glass.


44

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

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 fixed

on 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: ∆t = 5 sStep width: ∆β = 0,1Egoniometer mode: coupledInitial angle: βmin = 4,0EFinal angle: βmax = 12,0EStart 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


45

m VT p

T p= ⋅ = ⋅ρ ρ ρ, 0

0

0

(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 ∆t. 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 = e@U 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,using eq. (11).

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 = 6,673 9 @ 1011 N m2 kg!2

elementary charge e0 = 1,602 177 33 @ 10!19 C

electron mass me = 9,109 389 7 @ 10!31 kg

atomic mass unit u = 1,660 277 @ 10!27 kg

electric field constant ε0 = 8,854 187 817 @ 10!12 A s V!1 m!1

(dielectric constant of free space)

magnetic field constant µ0 = 1,256 637 1 @ 10!6 V s A!1 m!1

(permeability of free space)

Planck constant h = 6,626 075 5 @ 10!34 J s(quantum of action) = 4,135 7 @ 10!15 eV s

Avogadro constant NA = 6,022 136 7 @ 1023 mol!1

Boltzmann constant k = 1,380 658 @ 10!23 J/K

ideal gas constant R = 8,314 510 J mol!1 K!1

Faraday constant F = 9,648 4 @ 104 As/mol

Documents

Lab course Measurement Technique