institut fÜr physik rundpraktikumany measurement of a physical quantity is imperfect. if a quantity...

MARTIN-LUTHER-UNIVERSITÄT

HALLE-WITTENBERG

INSTITUT FÜR PHYSIKGRUNDPRAKTIKUM

Lab course

Measurement Technique

FOR

APPLIED POLYMER SCIENCE

3RD EDITION (2008)

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; consideration 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 introducing chapters of this booklet describe all general aspects of the laboratory course(safety in the lab, requirements to protocol writing and consideration 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

3rd edition Halle, October 2008

Contents

INTRODUCTION

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

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

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

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

Available 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

THERMODYNAMICS

W12 Humidity (dew point hygrometer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

ELECTRICITY

E 20 Oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

E 37 Transistor amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

OPTICS AND RADIATION

O 6 Diffraction spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

O 10 Polarimeter and Refractometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

O 16 Radioactivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

O 22 X-ray methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Introduction Laboratory Rules

1

Laboratory Rules

General Rules

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

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

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

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

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

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

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

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

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

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

Working with electrical circuits

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

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

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

14 Energized systems are to be supervisedpermanently.

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

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

Working with chemicals

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

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

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

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

Introduction 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 The radioactive preparations used in theexperiment O16 (74 kBq Co-60) are allowedto be handled by students. The radiationexposure during the experiment is 100...1000times lower compared to a standardroentgenization.

23 Nevertheless avoid any needless expo-sure! Do not carry the preparation in yourhand if not necessary! Keep a distance of0.5 m to the preparation during the experi-ment!

24 It is prohibited to remove the radioactivepreparation from the surrounding plexiglassblock.

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 toextinct 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 in 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 examine 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. Please

have 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 has to sign your records (seeabove).

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 (1...5) that takes account ofyour preparation (your knowledge and theprepared protocol), your experimental workand 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 notebookof the size 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

• Each record contains the following at thebeginning:- 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 “calculation of errors” isto determine the best estimation of the truevalue (the “measurement result”) and anestimation of the deviation of that result fromthe true value (the “measurement error”).

1 Definitions

Measured quantity:

The physical quantity to be measured;e.g. voltage U, current I, mass m

Measured value:

A single value measured including theunit; e.g. U = 230 V, I = 2 A, m = 2 kg

Measurement result:

A result calculated from several measuredvalues; e.g. P = U@I = 230 V @ 2 A = 460 W

Measurement error:

Difference between a measured value or ameasurement result and the true value.There are systematic errors and randomerrors (statistical errors) to distinguish.Generally, the measurement error is notexactly known because the true value isnot exactly known.

Random (or statistical) errors:

They occur irregularly, with varyingmagnitude and sign. They arise fromuncontrollable variations of the experi-mental and environmental conditions,from physical limits of observation (e.g.noise, quantum effects) and from the


5

s

x x

n

ii

n

=

−

−=

∑ ( )

.

2

1

1

(2)

′ =ss

n. (3)

∆ x

x x

n n

ii

n

=

−

⋅ −=

∑ ( )

( ).

2

1

1

(4)

xn

xii

n

==

∑1

1

. (1)

limits of human senses. Random errorscan be minimized by multiple measuringand calculation of the average accordingto eq. (1).

Systematic errors:

Under the same conditions they affect themeasurement in the same manner. Theyarise from imperfections of the devices,calibers and measurement proceduresused and from systematic changes in theexperimental conditions. Generally, theyconsist of a known and an unknown part.The measurement result is to be correctedby a known systematic error.

Measurement uncertainty:

Estimation of the measurement error. Itdefines a range (or interval) within whichthe true value of the measured quantitycan be found with high probability. Themeasurement uncertainty is estimated onthe basis of the measured values (bystatistical methods) and the knowledgeabout systematic errors.

Example for U = 230 V: ∆U = 2,4 V (absolute m. uncertainty)∆U/U = 1,1% (relative m. uncertainty)

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

Complete measurement result:

Measurement result ± uncertainty, e.g.U = 220,0 V ± 2,4 VU = (220,0 ± 2,4) VU = 220,0 V and ∆U/U = 1,1 %

2 Determination of meas. uncertainties

2.1 Calculation of meas. uncertainties in

the case of random errors

A quantity x is measured n times. The indi-vidual measured values scatter around theaverage

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 %. In the interval ± 2s are 95.5 % and in thexinterval ± 3s are 99.7 % of all values.xIf 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' can betaken as the measurement uncertainty:

If the measurement is a counting of arbitraryincidents x = N (e.g. radioactive decay inci-dents), the results are POISSON-distributed. Inthat special case the standard deviation is theroot of the mean value and hence the measu-

rement uncertainty is ∆x = (see experi-N

ment 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 #


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)

( )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)

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 atleast 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 lengthat ∆T=0.

The actual task of measurement is the deter-mination of the (constant) parameters a andb. In principle, a and b can be calculated fromtwo pairs of measured values (x, y). In mostcases, however, a whole series of measure-ments (n pairs of values (xi, yi), i = 1 ... n) istaken for verifying the linear relation. In agraphical representation the points (xi, yi) willscatter around a straight line, because of theunavoidable random errors. The task is nowto find the straight line that “fits best” the

measured points. 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

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


7

∆ ∆ ∆ ∆

∆ ∆

yy

xx

y

xx

y

xx

yy

xx

nn

ii

i

n

= + + +

==

∑

∂∂

∂∂

∂∂

∂∂

11

22

1

...

.

(9)

∆ ∆yy

xx

ii

i

n

=

=

∑∂∂

2

2

1

. (10)

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

Many scientific pocket calculators allowlinear regression, check the manual of yourcalculator. The standard deviations sa and sb

are only calculated by computer programs.In many cases (or if suitable software orpocket calculator is not available) it is suffi-cient to determine the regression parameters aand 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.

4.2 GAUSS's law of error propagation

If the single measured quantities are statisti-cally independent, their errors can be expectedto compensate each other partially. Themathematical treatment of this problem by C.F. GAUSS gives

Generally, the uncertainty of a measuringresult is to be calculated according to thisequation. Only if the statistical independencyof the single measured quantities is not sure,the uncertainty of the result should be calcu-lated according to eq. (9).


8

∆ ∆ ∆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)

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

and the relative GAUSS's error

Example:In an uniformly accelerated motion thedistance d depends on time t like d = a/2 @ t2.If d and t are measured with their correspond-ing errors ∆d and ∆t and a is to be calculated,we get

as

t

aa

ss

tt

= ⋅ =

+ ⋅

2 22

2 2

, .∆ ∆ ∆

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

Available Literature on Basic Experimental Physics

available at: Zweigbibliothek Technik Merseburg (Ha 55)Geusaer Straße 88

06217 Merseburghttp://www.bibliothek.uni-halle.de:80/zweigbib/zbha_55.htm

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

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

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

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

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

available at: Zweigbibliothek Physik (Ha 21) Friedemann-Bach-Platz 6

06108 Hallehttp://www.bibliothek.uni-halle.de:80/zweigbib/zbha_21.htm

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

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

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 7.5 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.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 you own (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 worksheet Data1 with one x and one y column for data

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

2. Worksheets

• Get more columns with .

• Give meaningful names to columns: Double-click the column head and type a Column

Label . The column label is automatically used in the legend.

• Denominating a column as x or y: Right-click the column head and choose Set As .

Introduction Software in the Basic Laboratory

11

• Calculations with columns: Right-click the column head 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. In the dialogueselect plottype, x column and y column and click the Add button.

• 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 .

• 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 more than one curves are present, selectthe right one in the Data menu. If only a part of the curve is to be fitted, define the range

before with the tool.

4. Printing graphs and worksheets

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

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

klick . 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

F F Fw l f= + . (1)

F r vf = 6π η , (4)

F r gl =43

32π ρ , (2)

F r gw =43

31π ρ , (3)

( )η ρ ρ= −29

2

1 2

rs

g 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 fluidsconsisting 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, thebuoyant force Fl and the friction force Ff areacting:

According to the principle of ARCHIMEDES,the buoyant force is equal to the weight of theliquid 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 ! Fa) 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

All invariant quantities are now combined to

Viscosity (falling ball viscometer) M 14

Mechanics M 14 Viscosity (falling ball viscometer)

13

( )η ρ ρ= ⋅ − ⋅K t1 2 . (8)

η η= ⋅0 e

E

k TA

. (10)

j e

E

k TE

~− (9)

the 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 hold on 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. As higher j is,as faster is that displacement. 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 the temperatureof which is controlled by the circulator. Thewhole arrangement can be turned by 180°into the measuring position and the roll-backposition, respectively. In your lab work youcan measure with sufficient accuracy in theroll-back position, 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).If a big air bubble obstructs the falling of theball please ask your supervisor for help. Youare not allowed to open the viscometer byyourself.

Mechanics M 19 Ultrasonic pulse-echo methods

14

λ =cf

. (1)

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 the circulator!). If the tem-perature 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 working place.

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 differ real from ideal liquids?

6.2 What is inner friction? How can it bemeasured?

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

1 Task

1.1 Determination of the sound velocity andthe wavelength of longitudinal waves inPolyethylene PE and calculation of Yang'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 with

another medium, through this couplingenergy is transferred from the oscillator to themedium. 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:

The mechanical waves in gases and liquidsoccur as longitudinal waves (direction ofoscillation in the direction of propagation),

Ultrasonic pulse-echo methods M 19


15

Z1 = ρ1@ c1

Z2 = ρ2@ c2

I0

IR

IT

Fig.1: Reflexion of ultrasound on an inter-

face between two materials with different

acoustic impedances

y yx= ⋅ − ⋅

0 e µ . (3)

cE

L =−

+ −ρ

ν

ν ν

1

1 1 2( )( )(2)

µ =−

⋅1

2 1

1

2x x

y

yln . (4)

Z c= ⋅ρ (5)

RI

I

Z Z

Z Z

R= =−

+0

1 2

1 2

(6)

I I IT R= −0 . (7)

because of the lacking shear elasticity. Insolid materials, besides the longitudinalwaves also transversely waves (direction ofoscillation perpendicular to the direction ofpropagation) as well as couplings betweenthem can occur.

In infinite, homogeneous and isotropic solidsthe velocity of sound cL for longitudinalwaves can be obtained from the mechanicalproperties 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 µthe 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 saltus in theacoustic impedance (for example on inter-faces) along the direction of propagationresults in a partial reflexion of the acoustic

energy and, additionally, in an attenuation inthe direction of propagation (“soundshadow”).For perpendicular incidence of a sound waveon 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 two


16

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

cd

t=

2(8)

pulses, 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 appear as peaks (see fig. 1). 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. 1 along the surface of thespecimen investigated. The amplitude of the

echo signal is mapped to brightness in a two-dimensional image of the cross-section of thespecimen. Modern B-scan devices use multi-element transducer that consists of manyoscillator elements. In this case a movementof the transducer is no more required. Instead,the single oscillator elements are excitedelectronically time-controlled.

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 becomes better 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 it is formed,e.g. structures near the transducer cannotbe resolved.

- Sound shadow: Strong reflecting structuresreflect most of the sound energy. Objectsbehind such structures may be invisible.

- Multiple reflections between surface andstrong reflecting structures may causephantom images of these structures indouble distance from the surface.

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


17


3.0 Devices:- computer integrated pulse-echo device- 2 transducers (1 MHz, 2 MHz) with con-

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

3.1 The pulse-echo device allows an A-scanas well as a simple B-scan using the samesingle-oscillator transducer.For measuring time differences or distanceson the monitor of the A-scan there are twocoloured cursors that can be moved with themouse. The signal amplitude is measuredwith the standard mouse cursor. Controls onthe front panel of the computer are providedfor adjusting transmit power, receiver ampli-fication and the time-dependent amplifica-tion.


Help and instructions for operating the device(in particular the controls on the front panel)are provided by the tutor and by a shortmanual available at the workplace. Thetransducers can be plugged to a socket on thefront panel of the computer. The coupling ofthe transducer to the PE-body is made usingwater (you need only very little water).

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 PVC body. Adjust thetime-dependent amplification in order to getthe same constant amplification for bothdistances. Measure the amplitudes y1 and y2

of the back-plane echoes in the two direc-tions. Repeat these steps using the 2 MHztransducer.

4.3 There are 4 boreholes anywhere in the

PE 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 form 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 the settings for transmitpower, receiver amplification and timedependent amplification with respect to thefollowing aspects:

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

- The attenuation related to the increase ofpenetration should be counterbalanced.

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

When the settings are optimized switch to B-scan. Adjust minimum and maximum ampli-tude to the values observed in the A-scanbefore. Press 'Start' and move the transducerslowly and equably over the surface, thenpress 'Stop'. Adjust the settings and repeat thescan until all boreholes are visible. The scancan be printed out. (Please print it only oncefor each frequency and each student!)The B-scan with the given device is onlyquantitatively. The exact positions of theholes have to be measured using the A-scanfrom (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).

Thermodynamics W 12 Humidity

18

U TT = ⋅α ∆ (1)

Fig. 1:

Copper-constantan thermocouple

fm

Va

V= (2)

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 displayed

in A-scan and B-scan?

6.2 Why water is needed for cupelling 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?

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, the

ends 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 immediately beused 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 the

Humidity W 12


19

Fig. 2:

Experimental setup for

determination of humidity

fp

pr

V

S

= (3)

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)

actual 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 is established that de-pends only on the kind of liquid (here water)and on the temperature (about exponentially).The relative humidity is in that case 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 only on the amount of dissolvedparticles but not on the kind of substance:

Here, pS is the saturation vapour pressure ofthe pure solvent and pS,L that of the solution. xis the mole ratio of the dissolved substances,n1 the number of particles of the solvent andn2 the number of dissolved particles in mol.n2 accounts for the dissociation of the dis-solved molecules into ions. RAOULTs law isexactly valid only when the vapour pressureof the dissolved solid is negligible small and

for n2 « n1. At higher concentrations thelowering of vapour pressure is less. As aresult of Raoults law the humidity over asolution is always smaller than 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, the water vapourcondenses. A mirror surface will be coveredwith mist below this temperature. This isutilized in a dew point hygrometer. Thesaturation vapour pressure at dew point pS(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; cover)

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

welding point in a tube with Gallium - 2 beaker glasses- rectangular plastic dish- plastic cover


20

- 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 amirroring metal surface (gold coated polishedcopper block) that can be cooled down. Thetemperature of the surface can be measuredand the cover with mist (condensed watervapour) can be observed. A semiconductorcooling element (utilising the PELTIER effect)is used for cooling. This is driven by a powersupply (30V/1.5A) that is used as constantcurrent source. The temperature is measuredby a copper-constantan-thermocouple. Onejunction of the thermocouple is placed in thecopper block and in the other one in an ice-water-mixture at 0°C. The thermal voltage ismeasured with the SensorCassy equippedwith a µV-box and a computer. A photosensor is used for reproducible observation ofthe condensation of water on the cooledmirror. It is equipped with a status LED and arelay controlled by the photo sensor. Therelay is used for constructing a simple tem-perature control to reach the dew point (seefig. 2). It switches the cooling current auto-matically on and off depending on the degreeof 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 the

language from German to English, selectEinstellungen ÷ Allgemein ÷ Sprache ÷English.4.1 Construct the wiring according to fig.1.For voltage measurement the input channel ofthe SensorCassy equipped with the µV-box isused. Power the SensorCassy and start theprogram CASSYLab-W12, if not yet done.All options in the program (measuring rangeand interval etc.) are already set accordingly.The recording of a series of measurementscan be started and finished by pressing thekey F9 or clicking to .

Fill one of the beaker glasses with crashed iceand small amounts of water. The second glassis to be filled with hot water (at least 50°C).Dip one junction of the copper-constantan-thermocouple (that one without Gallium) intothe ice-water-mixture. For calibration, start recording a series ofmeasurements at the computer and dip theother 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 ismolten, than the voltage increases again.Now dip the tube with Gallium into thebeaker glass containing the ice-water. Thevoltage decreases again and remains constantat 29.5 °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 the


21

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)

LED 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 dewpoint 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 identically to the room temper-ature. Place the hygrometer and the waterdish together under the plastic cover. Thehumidity under the cover will slowly increaseup to nearly 100 %. After waiting 20...30min, switch on the cooler and start operatingthe dew point hygrometer as described under4.2. Record the thermal voltage until thedewpoint remains constant (circa 5 min). Forminimizing the temperature oscillationsaround the dew point, the cooler current canbe lowered to 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 the

bottle 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 whether phe-nomena like rain, fog, dew.

6.2 What is a thermocouple? How does itwork?

6.3 At which temperature boils salt water?

Electricity E 20 Oscilloscope

22

Fig.1 Cathode ray tube (CRT)

1 Tasks

1.1 Adjust the electron beam of an oscillos-cope.

1.2 Measure various direct (DC) and alter-nating (AC) voltages (battery, wall powersupply, waveform generator), using the oscil-loscope and a voltmeter.

1.3 Determine the frequency of a tuningfork by measuring the cycle duration and bycomparing it to a frequency generator.

2 Physical Basis

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 representsthe time. X may also represent another volt-age. The intensity or brightness of the displayis sometimes called the Z axis.Usually, an oscilloscope is capable to show(at least) two signals at the same time, whichis best suited for testing electric circuits bycomparing input and output signals. An oscilloscope basically consists of a cath-ode ray tube (CRT, sometimes also called

BROWN's tube, see fig.1), two deflection unitsfor the X and Y direction, and time basegenerator. In the CRT, electrons are emittedby a hot cathode, and subsequently acceler-ated in the electrical field between cathodeand anode. The intensity control of the elec-tron beam is performed by a cylindricallyshaped electrode (WEHNELT's cylinder),which sits right next to the hot cathode.Additional electrodes situated betweenWEHNELT's cylinder and the anode are re-sponsible for focussing the beam and henceserve as a contrast control.The electrons are passing through the cylin-drically shaped anode and reach the deflec-tion unit, which consists of two pairs ofplates arranged at 90 degrees to each other.By applying a voltage to a plate pair, anelectrical field is generated between them,which deflects the electron beam. The elec-tron beam then hits the screen of the CRT,where it causes the luminescent coating toglow. Since the deflection angle is propor-tional to the applied voltage, the magnitudeof that voltage can be measured on thescreen.For measuring voltages over a wide range(from mV to V), the oscilloscope equippedwith adjustable amplifiers. The amplificationis selected at the knob VOLTS/DIV (seefig.3) that controls the Y scaling factor inVolts per grid unit (1 cm).

Oscilloscope E 20


23

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

Fig.3:

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

For drawing an U(t) graph (i.e. voltage vs.time), a so-called sweep voltage is applied tothe X-plate pair. During a certain time period(the rise time or sweep time), this voltageconstantly increases and hence guides theelectron beam in x-direction over the screenwith a constant rate. Subsequently, the volt-age drops to zero, and the beam thereforereturns to its starting position. The voltage tobe measured is applied to the Y-plate pair.During the rise time, the electron beamtherefore writes the graph of the function U(t)on the screen. This graph is refreshed everynew period of the sweep voltage.

The time base knob (TIME/DIV) allows tochange the sweep time over a wide range(2s...0.1µs). At 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 (CH1/2), two Y-tgraphs (DUAL) or X-Y graph. In X-Y modethe time deflection is disabled, the input CH1is applied to the X-plates and the input CH2to the Y-plates.The front panel of the oscilloscope is clearlyorganised: there are groups of knobs andbuttons responsible for channel 1 (Y1 or X)and channel 2 (Y2) input, for the time deflec-tion, the trigger control and for the operationmode. The Y (volts) and X (time) deflectionare adjusted by a rotary switch and a continu-ously 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 inputsselects the coupling of the measured signal tothe pre-amplifier: for direct coupling (DC),the entire signal is measured, for capacitive


24

ω ππ

= ⋅ =22

fT

. (2)

U UPP = 2 0 (3)

P U IUR

I R= ⋅ = = ⋅2

2 , (4)

PT

UR

dtU

R

T

= =∫1 1

2

2

0

02

. (5)

U U I Ieff eff= =12

120 0, . (6)

U U t= ⋅ +0 sin( ) ,ω ϕ (1)

coupling (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 An alternating voltage (AC voltage) ismathematically described as harmonic oscil-lation:

where U0 is the maximum or peak voltageand φ a phase angle (which may be zerowhen the time scale is set accordingly). Theangular frequency ω is associated with thefrequency f and the period T by

With an oscilloscope mostly the peak-peakvoltage

is measured which is the difference betweena minimum and a maximum of U accordingto eq. (1). In contrast, voltmeters measurealways the effective voltage Ueff. The effec-tive voltage (or effective current) is definedas the value a DC voltage (or current) wouldhave that produces the same average thermalpower in a resistor than the AC voltage (orAC current, respectively). The power is

with eq.(1) the average over one period is

From that follows for the effective value of asinusoidal voltage (and for the current accor-dingly):


3.0 Devices

- oscilloscope HM303-6 with manual- battery- wall power supply- waveform generator HM8130 with manual- digital voltmeter- tuning fork- loudspeaker- BNC branch connection- connection cables

3.1 Instructions for operating the oscillo-scope and the waveform generator are to betaken from the supplied manuals for thesedevices. In case of doubt, ask the tutor. Thewaveform generator is used to supply asine-shaped alternating voltage with veryprecise frequency. The loudspeaker serves asa microphon for measuring the sound of thetuning fork. In addition to the listed tasks, theloudspeaker may be connected to the outputof the waveform generator to make thesupplied signal audible.


4.1 For adjusting the electron beam, disablethe time deflection by switching to X-Ymode, so that only the glowing point of thenon-deflected electron beam is visible on thescreen. Optimise beam intensity and contrastand adjust the beam to the centre of thescreen using the knobs for vertical and hori-zontal displacement. Now enable the time


25

fT

nt

= =1

. (7)

deflection again (X-Y off, mode CHAN1,time base 2 ms/cm), and if necessary,re-optimise intensity and contrast.

4.2 The voltages are to be measured at theleft input (CHAN1). Select a suitable inputrange at the rotary switch, while the continu-ously rotary knob is its rightmost position(see above). The input coupling has to beswitched to DC.

Battery:Connect the battery to the input of the oscil-loscope, and measure the voltage. If neces-sary, change the range (input amplification),for a precise reading. For comparison, mea-sure the battery voltage with the digitalvoltmeter, too.

Waveform Generator:Set up the waveform generator to signal form„sine“, an output voltage of 2V (peak-to-peak voltage, see manual), and initially afrequency of 50 Hz. On the digital voltmeter,choose a suitable measuring range for ACvoltage. Connect the output of the waveformgenerator to both the oscilloscope and thevoltmeter inputs by means of the BNC branchconnection piece. Now, the graph of the ACvoltage can be seen on the scope, and bychanging the time base this graph may bestretched or compressed.Measure the peek-to-peek voltage UPP at thescope, and the effective voltage Ueff at thedigital voltmeter. Repeat these measurementsat the frequencies 5 kHz, 500 kHz, 5 MHzand 5 Hz.

Wall Power Supply:The DC voltage of the wall power supply issuperposed by an interference (noise) voltage.The DC voltage is to be measured with boththe oscilloscope and the digital voltmeter,with a procedure similar to that used for thebattery.After that, set the input coupling selector ofthe oscilloscope to AC, so that only the ACpart of the voltage is measured. Choose asuitable range and determine the magnitude

of the interference voltage. The frequency ofthe interference voltage is also to be mea-sured, using the procedure described inSection 4.3.

4.3 To measure frequencies, the variableknob for the time base must be in the right-most position. Connect the cable of theloudspeaker to the oscilloscope input and putthe tuning fork in front of the loudspeaker.Strike the tuning fork, and choose a suitableY-amplitude and time base to display thesignal on the screen.If the duration of n oscillation periods t=n@T

is determined, the frequency can be calcu-lated to

Repeat this measurement three times fordifferent time bases.The second frequency measuring method isthe comparison with a frequency standard,the waveform generator.Connect the generator to the second channelof the oscilloscope, and select the two-beammode DUAL. A running graph of the genera-tor signal appears, as the sound of the tuningfork is used as a trigger. When the generatorsignal is changed to the same frequency asthe tuning fork, the graph stops moving. Readthe frequency at the function generator.

5 Evaluation

5.2 List all measured DC and AC voltagesas well as the nominal voltages (setting of thewaveform generator, labels of battery andwall power supply) in a table and comparethe results to each other. Calculate the effec-tive voltages from the peak-to-peak values.Discuss the differences between the valuesmeasured with different instruments andprocedures. Estimate the precision of themeasurements.The magnitude of the interference voltage isto be calculated as an absolute (in mV) and

Electricity E 37 Transistor amplifier

26

Fig. 1: Elementary amplifier

relative (in percent of the d.c. voltage) value.What is the cause of the interference voltage?

5.3 List the frequencies measured in a tableand estimate the precision of the two mea-surement methods.

Additional experimental tasks:

S The comparison of two frequencies may beperformed with Lissajous-patterns in X-Ymode.

S You may try to measure the frequency ofother sounds (whistling, musical instru-ments).

S With the loudspeaker connected directly to

the frequency generator, the frequencydependence of the human hearing abilitymay be tested by listening to the generatedsound.

6 Questions

6.1 Explain a cathode ray tube.

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

6.3 What is the task of the trigger in anoscilloscope?

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

Each electronic amplifier consists of anelementary amplifier and, depending on theapplication, of additional components for theadjustment of the operating point, for thestabilisation, for the degeneration or for thecoupling and the extraction 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 theoperating 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.

Transistor amplifier E 37


27

VU

UO

I

=∆

∆(1)

Fig. 2: Experimental circuit

Fig.3: frequency response of an amplifier

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 frequenciescan 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 afunction 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-


28

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 theCRT.

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).

4.3.2 Reduce the input voltage to 40 mVagain and shift the operating point of theamplifier up and down (on the characteristic

line). 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 basiccircuit 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.

Optics & Radiation O 6 Diffraction Spectrometer

29

Fig.1: Diffraction of a plane wave on a

edge. Construction after Huygens-Fresnel.

Fig.2: Calculation of the path difference δ of

diffracted light on a grating.

a: width of a slit, b: grating constant,

n: angle of diffraction

δ λ= ⋅k (2)

δλ

= + ⋅( )2 12

k (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 far distance maximums andminimums of light intensity occur by destruc-tive and constructive interference, respec-tively. For simplification each slit in fig.2 is consid-ered to be the origin of only one elementarywavelet. The path difference δ between thewavelets origin from neighbouring slits is

Constructive interference occurs for a pathdifference

and destructive interference for a path differ-ence of

Diffraction Spectrometer O 6


30

Fig.3: sketch of a diffraction spectrometer

R N k= ⋅ (7)

sinϕλ

k

kb

=⋅

(4)

λϕ

=⋅b

kksin

. (5)

R =λλ∆

(6)

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

The more wavelets constructively interfere inthis direction, the more intensive and sharpthese maximums 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 coloursby using a grating. By measuring the diffrac-tion angle the wavelength of light can bedetermined:

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 the

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


3.0 Devices:

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

3.1 The experimental arrangement is ac-cording to fig.3. The goniometer ERG3 isused for measuring the diffraction angle φ. Itconsists of the collimator with slit and lens, arotatable table with the grating, a moveabletelescope and an arrangement for measuringangles with an accuracy of 0.5 ' (angularminutes). The He-lamp is mounted on theslit. The slit width, the collimator (distancebetween slit and lens) and the telescope canbe adjusted.


At the beginning, learn how the goniometer isoperated. Note the grating constant to yourprotocol 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: Press or pull the ocular to focusthe reticle. Then adjust the telescope toinfinity. This is done when you see a sharpimage of objects situated very far awaycomparing to the distance to the grating. Youcan use e.g. the trees outside the window foradjusting the telescope.


31

ϕϕ ϕ

k

right left=

′ − ′

2(8)

Collimator: Place the telescope opposite tothe collimator (they share now the sameoptical axis) and lock it. You should see theslit through the telescope now. Do not read-just the telescope at this point! If the slit isnot sharp, carefully shift the whole slit to-gether with the mounted lamp in order tofocus it.

Grating: Adjust the grating perpendicular tothe common axis of the telescope andcollimator. For this purpose the reticule islightened by a GAUSS ocular. The light ispartially reflected at the grid, so that in thetelescope the bright reticule and its reflectedblack image can be observed. The blackimage of the reticule is very hard to seebecause the grid reflects only a small part ofthe light. For easier adjustment a mirror canbe temporarily mounted instead of the grid. Ifthe reflected image is not sharp, you canslightly readjust the telescope. Now, adjustthe grid table to bring the reticule and itsreflected image into coincidence. When all adjustments are done, lock the gridtable, switch the lighting of the GAUSS ocularoff and unlock the 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.

Optics & Radiation O 16 Polarimeter and Refractometer

32

Fig.1: Position of electric and magnetic

field-strength vectors for a wave train

nc

c= 0 . (2)

ϕ = ⋅ ⋅k l c , (1)

1 Tasks

1.1 The concentration of a water-sugarsolution is to be determined by means of apolarimeter.

1.2 The refractive index of glycerin-watermixtures is to be measured in dependence onthe concentration using a refractometer.

1.3 The concentration of a given glycerin-water mixture is to be determined.

2 Physical Basis

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

vibrate in only one transversal direction. The

direction of the electric field-strength vectoris then called direction of oscillation orpolarizing direction.

2.2 Linearly polarized light may be genera-ted from natural light by (a) reflection at theBREWSTER angle, (b) by double refraction(NICOL prism) or (c) by means of polarizingfilters on the basis of dicroitic 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 dextrorota-ry and a levorotary 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 angel 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 index depends on the material and on thewave length of light (this effect is calleddispersion), in a solution it also depends onthe concentration (mixing ratio). Therefore, ameasurement of the refractive index may besuitable for determining concentrations.Applications, for example, are the determina-

Polarimeter and Refractometer O 10


33

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

Left: general case, right: striping light entry

n n1 2⋅ = ⋅sin sin .α β (3)

sin .maxβ =n

n1

2

(4)

Fig.3: Ray trace at an ABBE refractometer

tion of the protein content in a blood serumor of the sugar degree of grape juice in awinery.

During the transition of light from an opti-cally thinner medium with index n1 to anoptically denser medium with index n2 (n2 >n1) a light beam is refracted towards theperpendicular of incidence, see fig.2. With αand β as angle of entry and emergence, thelaw of refraction reads

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

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

For this the interface is lighted via a mat glassplate with a rough surface, see fig.3. In thisway 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 glycerin and aqua dest.- 3 beakers, funnel, pipette- flask with a glycerin-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 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 putting 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 rotateded byan angle φ. Resetting the analyzor by thisangle yields the visual field becoming dark


34

Fig.4: Three-part visual field in the

polarimeter

ϕ ϕ ϕ= −1 0 . (5)

again. In this way the angel φ can be mea-sured.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 (halfshadepolarimeter). For a precise measurement ofangles, the scale is equipped with a vernierthat allows a read off with an uncertainty ofonly 0.05°.

3.2 The 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 stripingly incident part of light (α .90°)is refracted at the critical angle βmax and canbe observed in the telescope as a borderbetween bright and dark areas.

4 Experimental Procedure

4.1 Switch the sodium spectral lamp on atfirst; after about 5 minutes the lamp reaches

its maximum 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 above)must be open.Open the two prisms (measuring prism aboveand lightning prism below) by turning thelocking knob. If necessary clean the prismscarefully with wet paper towel and dry it.Hold the lightning prism with the roughsurface about horizontally and put 1 or 2drops of the sample liquid on the surface.Ensure that there are no air bubbles in theliquid. Then close the prisms and lock it.Look through the measuring eyepiece (theright one) and turn it until the reticle is

Optics & Radiation O 16 Radioactivity

35

sharply seen. Adjust the lightning mirror formaximum lightness. By turning the scaleadjustment knob (on the left) move along themeasuring range until the light/dark borderappears. Eliminate colour fringes by turningthe compensation knob (on the right) until theborder appears black-and-white. Adjust thecentre of the reticle exactly to the light/darkline and read the corresponding refractiveindex with the eyepiece for scale.The refractive index is to be determined forthe following liquids:

- de-ionized water

- pure glycerin

- 5 glycerin-water mixtures:

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

- glycerin-water mixture of unknown con-centration

For the mixture 4:1 take 4 ml glycerin and1 ml aqua dest., and make the other mixturesby further dilution of this mixture with water.Measure each refractive index 5 times (read-just 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 glycerin-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 angletake the sum of the standard deviations of φ0

and φ1, according to (5).

5.2 Make a plot of the refractive index viavolume concentration of glycerin in water.

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

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 by

Radioactivity O 16


36

ANt

=dd

. (1)

dN N t= − ⋅ ⋅λ d . (2)

µ µ µ µ µ= + + +S Ph C P . (6)

N t Nt( ) = ⋅

− ⋅

0 e λ (3)

I I IS Z= − . (4)

I Ix

= ⋅−µ⋅

0 e . (5)

( )P nN

np pN

n N n

( ) =

−

−1 (8)

x1 2

2/

ln.=

µ (7)

emission 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).

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 radioactivepreparation 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.

Radiation detectors like the Geiger-Müllertube (GM tube) measure the pulse rate, i.e.the number of pulses per second. The pulserate I is proportional to the radiation inten-sity. Additionally, it depends on the energy ofthe radiation and on the characteristic of thedetector. The pulse rate I caused by aradioactive preparation is calculated as thedifference of the pulse rates measured withpreparation IS and without preparation IZ

(zero rate):

The zero rate is caused by environmentalradiation (cosmic radiation and natural radio-activity) and by interference 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 escapingintensity. µ is called the attenuation coeffi-cient, it depends on the material penetratedand on the energy 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.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 in principle impossible. Onlythe probability of the nucleus to decay in acertain time interval is known. Therefore thenumber of counts measured is for fundamen-tal 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 mathematicalexpectation) ν = n@ p. In our experiment N isa huge number and p is very small (the half-


37

P nn

n

( )!

=−ν νe (9)

∆ n n= ≈σ . (10)

P nn

( )( )

.=−

−1

2

2

2

πν

ννe (11)

life of Co-60 is 5.6 years). Passing to thelimits N÷4 and p÷0, the binomial distribu-tion (8) transforms into a POISSON distribu-tion

with the mean value ν.An important mathematical property of thePOISSON distribution is that the mean value νis equal to the variance σ2 (square of thestandard 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

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 1999)

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

3.1 The radioactive preparation and the GMtube each reside in a plexiglass block moun-ted on a sledge that can be shifted on theoptical bench. Radioactive preparation andGM tube are facing to each other. A thirdsledge that carries the absorbing slabs can bemounted in between them.The GM tube used 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 used as counter or ratemeter and power supply for the GM tube. The

counter automatically stores up to 2000measured values. For the statistical analysis itsends the counts automatically to the com-puter via a serial connection.


The Co-60 preparation is allowed for beingused in students experiments by Germanregulations. Your radiation exposure in a 3 hlab course is about 200 times lower as theexposure caused by 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 . Whenthe counting is started, the counts are storedevery 60 seconds in memory. After stoppingthe counting, you can read these values fromthe memory again.First, measure five times the zero rate (with-out Co-60 preparation). Put the preparation atleast 1 m away from the counter tube for thismeasurement.Then place the Co-60 preparation in a dis-tance of 40, 50, 70, 100, 140, 190 and 250mm from the GM tube and measure thecounts per minute five times at each distance.The distance is measured on the scale of theoptical bench between the inner edges of theblack sledges of the preparation and thecounter tube, respectively.

4.2 For determining the attenuation coeffi-cient of Pb, place the slab holder in between

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! Inno case it is allowed to remove the Co-60preparation from its plexiglass block.


38

either

or e

ln ln

lg lg lg

I I x

I I x

= − ⋅

= − ⋅ ⋅

0

0

µ

µ

the radioactive preparation and the countertube. The distance between these two shouldbe 100 mm and has to be kept constantduring the rest of the experiment.Measure the counts per minute for the thick-nesses x = 1, 2, 5, 10, 20 and 30 mm fivetimes each. The measurement result for x = 0mm can be taken from task 4.1.

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

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

5 Evaluation

Calculate the average of the five singlemeasurements in every part of the experi-ment. Correct the pulse rates (the averages)by subtracting 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 “Fit

function” in the context menu of the graph.

6 Questions

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

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

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

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

Optics & Radiation O 22 X-ray methods

39

E h f hc

Ph = ⋅ = ⋅λ

(2)

E e U h f hc

Ph maxmin

= ⋅ = ⋅ = ⋅λ

(3)

λ λα βK

L KK

M K

h cE E

h cE E

=⋅

−=

⋅−

(4)

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 voltageme: 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 differencewill be released in form of X-ray. The pho-tons (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


40

Fig.1: typical X-ray spectrum revealing

Bremsstrahlung and characteristic radiation

β

ββ

ββ

C C

A

B

1

2d

Fig. 2 BRAGG reflexion

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 a character-istic 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 crystalllattice 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-ray can be con-sidered as a source of an elementary wave.The atoms in the crystal can be summarisedin multiple consecutive layers situated paral-lel to the crystal surface. This planes arecalled ”lattice planes”. In the simplest casethe diffraction of X-ray can described asreflection at the lattice planes of a crystal.Each lattice plane acts on the incident X-raylike a partial mirror, that reflects a (verysmall) 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 @sinβ 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. 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 absorbedby the matter.The ion dose J is defined as the total charge


41

JQ

m=

∆

∆. (7)

DEm

=∆∆

. (8)

H q D= ⋅ (9)

Fig.3: Measurement of the ion dose rate in

an ionisation chamber

jQ

m t

I

mC= =

∆

∆ ∆. (10)

H J= ⋅32 5, .Sv

As kg-1 (11)of 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 energy dose D is defined as the energy∆E absorbed by a volume element divided bythe mass of the radiated volume element ∆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 of measure Sievert (Sv), 1 Sv =1 J/kg. q is the quality factor, it is q = 1 forX-ray, gamma and beta rays and q = 20 foralpha 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), energy 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.4. 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 airmolecules 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 consists of a radiationshielding case that is separated into threechambers. The largest (right-hand side)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.(Very soft material! Handle with care, do notscratch it!)

3.2 The high voltage power supply exhibitsa very large output resistance. The contactsmay be touched without harm.


42

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“ („Decree about protection againstdamages through X-rays”) the X-raydevice is an admitted model (admissionsymbol 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

For measuring the very small current anamplifier and the multimeter are used.


The experiment must not be started before

the instructions by the tutor.

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).


43

m VT p

T p= ⋅ = ⋅ρ ρ ρ, 0

0

0

(12)

The best way is to start with the maximumacceleration voltage (35 kV). The recordingis 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 terminaland the upper capacitor plate with the posi-tive terminal of the power supply. Connectthe multimeter to the output of the amplifierand select the range 10-9 A (1 V output isequivalent 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 energy possible (U=35kV,I=1mA). 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 objects of your belongings (pocketcalculator, ball pen, ...) and record theobservations 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

institut fÜr physik rundpraktikumany measurement of a physical quantity is imperfect. if a quantity...

Documents