LEP5.2.41

-01Law of distance and absorption of gamma or beta rays

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 25241-01 1

Related topicsRadioactive radiation, beta-decay, conservation of parity, anti-neutrino, gamma quanta, half-value thickness, absorptioncoefficient, term diagram, pair formation, Compton effect,photoelectric effect, conservation of angular momentum, for-bidden transition, weak interaction, dead time.

PrincipleThe inverse square law of distance is demonstrated with thegamma radiation from a 60CO preparation, the half-valuethickness and absorption coefficient of various materialsdetermined with the narrow beam system and the corre-sponding mass attenuation coefficient calculated.

EquipmentRadioactive sources, set 09047.50 1Absorption plates for b-radiation 09024.00 1Unit-construction plate for radioactivity 09200.00 1Counter holder, magnet held 09202.00 1Source holder, magnet held 09201.00 1Plate holder for demonstration boardwith magnet 09204.00 1Vernier caliper 03010.00 1Screened cable, BNC, l = 750 mm 07542.10 1Counter tube, type A, BNC 09025.11 1

Geiger-Müller Counter 13606.99 1Absorption material, lead 09029.01 1Absorption material, Plexiglas 09029.04 1Absorption material, iron 09029.02 1Absorption material, concrete 09029.05 1Absorption material, aluminium 09029.03 1

Tasks1. To measure the impulse counting rate as a function of the

distance between the source and the counter tube.

2. To determine the half-value thickness d1/2 and the absorp-tion coefficient m of a number of materials by measuring theimpulse counting rate as a function of the thickness of theirradiated material. Lead, iron, aluminium, concrete andPlexiglas are used as absorbers.

3. To calculate the mass attenuation coefficient from themeasured values.

Set-upAccording to Fig. 1.The distance between the front edge of the source rod and thecounting tube window is approximately 4 cm; consequently,the absorption plates can be easily inserted into the radiationpath.

Fig. 1: Experimental set-up for measuring the half-value thickness of different materials.

LEP5.2.41

-01Law of distance and absorption of gamma or beta rays

25241-01 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen2

Theory and evalutionThe cobalt isotope 60

27Co has a half-life of 5.26 years; it under-goes beta-decay to yield the stable nickel isotope 60

28Ni– seeFig. 2.

As with most beta emitters, disintegration leads at first todaughter nuclei in an excited state, which change to theground state with the emission of gamma quanta. Whereasthe energy levels of the beta electrons can assume any valueup to the maximum because of the antineutrinos involved, thegamma quanta which participate in the same transition pro-cess have uniform energy, with the result that the gammaspectrum consists of two discrete, sharp lines (Fig. 2).

The impulse counting rate N.

(r) per area A around a point-source decreases in inverse proportion to the square of thedistance provided the gamma quanta can spread out instraight lines and are not deflected from their track by interac-tions.

The reason for this is that, as shown by Fig. 3, the area of asphere round the source through with a beam of rays passes,increases as the square of the distance r. In vacuum (in air),therefore

If we plot the counting rate N.(r) versus the distance r on a log-

log scale, we obtain a straight line of slope –2.

From the regression lines from the measured values in Fig. 4,applying the exponential expression

N.

(r) = a · rb,

we obtain the value

b = –2.07 ± 0.01

for the exponent.

This thus proves the applicability of the inverse square law.

The attenuation of the gamma rays when they pass through anabsorber of thickness d is expressed by the exponential law

N.

(d) = N.

(o) · e–Nd,

where N.(d) is the impulse counting rate after absorption in the

absorber, and N.

(o) is the impulse counting rate when no

N�1r 2

A �

N�1o 2

A ·

14 p

r�2

r2 � 2 r1 A2 � 4 · A1 � ar2

r1b

2

· A1

Fig. 2: Term diagram of 6027Co.

Fig. 3: Law of distance relating to rays which are propagatedin a straight line from a point source.

Fig. 4: Counting rate plotted against distance (log-log plot).

Law of distance and absorption of gamma or beta raysLEP

5.2.41-01

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 25241-01 3

absorption takes place: N is the absorption coefficient of theabsorber material and depends on the energy of the gammaquantum.

The absorption of the gamma rays is brought about by threeindependent effects – the Compton effect, the photelectriceffect and pairformation.

The relative contributions of these three effects to totalabsorption depends primarily on the energy of the quanta andon the atomic number of the absorber (Fig. 5).

We can see from the N/E curves in Fig. 6 that lead is particu-larly suitable as an absorber of gamma rays of low or highenergy.

The attenuation of gamma rays therefore takes place predom-inantly in the electron shell of the absorber atoms. Theabsorption coefficient N should therefore be proportional tothe number of electrons in the shell per unit volume, orapproximately proportional to the density S of the material.

The mass attenuation coefficient N/S is therefore roughly thesame for the different materials.

The half-value thickness d1/2 of a material is defined as thethickness at which the impulse counting rate is reduced byhalf, and can be calculated from the absorption coefficient inaccordance with

d1/2 = .

From the regression lines from the measured values in Fig. 6we obtain the following values for N = b and for d1/2 and N/S,with the relevant standard errors, using the exponentialexpression

N.

= aebd.

Lead: (S = 11.34 gcm-3)

N = 0.62 cm-1, sN = 0.009 cm-1

d1/2 = 1.12 cm, sd1/2= 0.02 cm

= 0.055 cm2g-1; sN/S = 0.001 cm2g-1

Aluminium: (S = 2.69 gcm-3)

N = 0.15 cm-1, sN = 0.01 cm-1

d1/2 = 4.6 cm, sd1/2= 0.3 cm

= 0.056 cm2g-1; sN/S = 0.004 cm2g-1m

r

m

r

ln 2m

Fig. 5: Absorption of gamma rays by leads as a function of theenergy (NCo = fraction due to Compton effect, NPh =fraction due to photoelectric effect, NPa = fraction dueto pair formation). The total absorption coefficient(attenuation coefficient) is N = NCo + NPH + NPa

Fig. 6: Impulse counting rate N.

as a function of the thicknessd of the absorber.

LEP5.2.41

-01Law of distance and absorption of gamma or beta rays

25241-01 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen4

Iron: (S = 7.86 gcm-3)

N = 0.394 cm-1, sN = 0.006 cm-1

d1/2 = 1.76 cm, sd1/2= 0.03 cm

= 0.050 cm2g-1; sN/S = 0.001 cm2g-1

Concrete: (S = 2.35 gcm-3)

N = 0.124 cm-1, sN = 0.009 cm-1

d1/2 = 5.6 cm, sd1/2= 0.4 cm

= 0.053 cm2g-1; sN/S = 0.004 cm2g-1

Plexiglas: (S = 1.119 gcm-3)

N = 0.078 cm-1, sN = 0.004 cm-1

d1/2 = 8.9 cm, sd1/2= 0.5 cm

= 0.066 cm2g-1; sN/S = 0.003 cm2g-1

CommentThe procedure and evaluation are shown here in an exemplaryexperiment for g-quanta; however, they can also be performedin an analogous manner for electrons. In the latter case, theSr-90 source rod from the radioactive sources set (09047.50)and the absorption plate set for b-radiation (09024.00) mustbe used.

m

r

m

r

m

r

LEP5.2.41

-11Law of distance and absorption of gamma or beta rays with Cobra3

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 25241-11 1

Related topicsRadioactive radiation, beta-decay, conservation of parity, anti-neutrino, gamma quanta, half-value thickness, absorptioncoefficient, term diagram, pair formation, Compton effect,photoelectric effect, conservation of angular momentum, for-bidden transition, weak interaction, dead time.

PrincipleThe inverse square law of distance is demonstrated with thegamma radiation from a 60Co preparation, the half-value thick-ness and absorption coefficient of various materials deter-mined with the narrow beam system and the correspondingmass attenuation coefficient calculated.

EquipmentCobra3 BASIC-UNIT 12150.00 1Cobra3 Power supply 12151.99 1RS232 data cable 14602.00 1Cobra3 Radioactivity Software 14506.61 1Counter tube module 12106.00 1Unit-construction plate for radioactivity 09200.00 1Counter tube, magnet held 09201.00 1Source holder, magnet held 09202.00 1Plate holder for demonstration boardwith magnet 09204.00 1Counter tube, type A 09025.11 1Screened cable, BNC, l = 300 mm 07542.10 1

Vernier caliper 03010.00 1Radioactive sources, set 09047.50 1Absorption plates for b-radiation 09024.00 1Absorption material, lead 09029.01 1Absorption material, iron 09029.02 1Absorption material, aluminium 09029.03 1Absorption material, Plexiglas® 09029.04 1Absorption material, concrete 09029.05 1PC, Windows® 95 or higher

Tasks1. To measure the impulse counting rate as a function of the

distance between the source and the counter tube.

2. To determine the half-value thickness d1/2 and the absorp-tion coefficient N of a number of materials by measuring theimpulse counting rate as a function of the thickness of theirradiated material. Lead, iron, aluminium, concrete andPlexiglas are used as absorbers.

3. To calculate the mass attenuation coefficient from themeasured values.

Set-upAccording to Fig. 1.The distance between the front edge of the source rod and thecounting tube window is approximately 4 cm; consequently,the absorption plates can be easily inserted into the radiationpath.

Fig. 1: Experimental set-up.

LEP5.2.41

-11Law of distance and absorption of gamma or beta rays with Cobra3

25241-11 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen2

Procedure— Activate the ”Radioactivity” program module, and start the

measurement stand-by phase (cf. Fig. 2).— Measure the background radiation without the radiation

source. To do this, it is advisable to use a gate time ofmore than 500 s. The measured background rate remainsin the Cobra3’s memory until it is overwritten by a newbackground measurement.

— Diagram settings:y axis: 0 to 15x axis: 0 to 30 mm

— Activate measurement by clicking on <Continue>.— During the measurement the distance between the count-

ing tube and the source (Co-60) must not be changed.Initially, enter ”0” in the input field for the absorber layer (cf.Fig. 3) and click on <Measure>.RemarksImmediately after <Continue> has been clicked on in theParameter field (Fig. 2), the measurement process begins,i.e. one must wait until a gate time period has elapsedbefore a measuring result appears in the display.

— After each measurement increase the layer thickness ofthe lead absorber by 5 mm, enter the new thickness valueof the absorber layer in the appropriate field and click on<Measure>. Continue in the same manner until the maxi-mum thickness of 30 mm has been reached. After the lastmeasurement has been made, click on the <Close> but-ton.

— Perform absorption measurements in the same mannerwith the following absorber materials:Iron, aluminium, Plexiglas®, concrete.

Results— Figure 3 shows the counting rate as a function of the

absorber layer thickness. The data (measured) points con-firm the approximately exponential decrease in the count-ing rate as a function the layer thickness.

— The g-quanta emitted by the source (Co-60) are absorbedin the lead layer to differing degrees depending on thelayer thickness d. Accordingly, the quantum flux I is atten-uated by the absorption layer compared to quantum flux I0in the air. The attenuation of the quantum flux occurs inaccordance with the absorption law; the quantum flux Idecreases approximately exponentially with increasinglayer thickness d.

I � I0 · e�md

Fig. 3. Typical display structure during the measurement;counting rate (Co-60) as a function of the absorberlayer thickness (Pb plates).

Fig. 4. Half-value thickness and the attenuation for leadabsorber plates.

Fig. 2. Measuring parameters.

Law of distance and absorption of gamma or beta rays with Cobra3LEP

5.2.41-11

PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen 25241-11 3

where m is the attenuation coefficient characteristic for thematerial (and the energy of the g-radiation).For a very specific layer thickness dH the initial quantumflux is reduced to half of its original value.

From this it follows that the half-value thickness dH isdetermined by the attenuation coefficient m.

or

The attenuation coefficient m characterises the absorptionbehaviour of the material with respect to g-quanta.

— Determination of the half-value thickness and the attenua-tion coefficient m.Both parameters are displayed if the exponential measure-ment curve in the active image can be seen and then theevaluation functions <Analysis>, <Half-value time /-layerthickness> are selected.Naturally, these parameters can also be manually deter-mined by initially calculating the natural logarithm of themeasured values with <Channel modification> and by sub-sequently fitting a straight line through the thus manipulat-ed measured values. The following is then true for the half-value layer thickness dH:

where m is the slope of the straight line.

In this exemplary measurement the half-value thickness oflead is dH = 1.416 ± 0.009 cm and the attenuation coeffi-cient is m = 0.5 ± 0.1 cm-1.

— In the second part of the experiment layers of materialexhibiting differing thickness d and different densities r arepositioned in the radiation field between the g-source Co-60 and the counter tube. In a manner analogous to theevaluation described above, determine the half-valuethicknesses for iron, aluminium, Plexiglas® and concrete,and calculate their attenuation coefficient:

The attenuation coefficient m increases approximately propor-tionally to the density r of the absorption material.

The proportionality factor, the mass attenuation coefficient mmfor lead is obtained from the attenuation coefficient m accord-ing to the following equation:

The mass attenuation coefficient mm is approximately thesame for all materials (if the energy of the g-quanta is fixed).Consequently, the attenuation law is also pragmatically writtenin the following form:

with

The mass coverage m” states which mass an absorption layerhas per unit surface. The mass coverage m” is a decisiveparameter for the attenuation of a g-flux.

Remarks— The counting rates measured depend on the source used

and on the age of the specimen.— The attenuation coefficients m are also a function of the

energy of the emitted g-quanta, which is relatively high forCo-60 (hard g-radiation). At lower energies (soft g-radia-tion) the attenuation coefficients exhibit different values.

— When using radioactive substances, conform absolutely tothe stipulations of the respective applicable radiation pro-tection regulations. Radioactive substances can be haz-ardous to your health! Always reduce the time spent han-dling radioactive substances to a minimum. Do not eat ordrink in the presence of radioactive substances andalways wash you hands after contact with radioactive sub-stances!

m'' � r · d .

I � I0 · e�mmm''

mm �m

r� 0.045

cm2

g .

m � mm · r

dH �ln 2m

,

m �ln 2dH

.

dH �ln 2m

12

I0 � I0 · e�mdH

Fig. 5. Attenuation coefficient m of different materials as afunction of the material density r (from left to right:Plexiglas®, concrete, aluminium, iron, lead).

Material Density Half-value (layer) Attenuationg/cm3 thickness coefficient

dH in cm m in cm-1

Lead 11.11 1.41(1) 0.50

Iron 7.68 2.3(4) 0.30

Aluminium 2.70 7(3) 0.09

Concrete 1.87 6(3) 0.12

Plexiglas® 1.19 35(82) 0.02

LEP5.2.41

-11Law of distance and absorption of gamma or beta rays with Cobra3

25241-11 PHYWE series of publications • Laboratory Experiments • Physics • © PHYWE SYSTEME GMBH & Co. KG • D-37070 Göttingen4