Health and Radiation Physics

Heiko Timmers

A lecture course

for 2nd year students of physics

University of New South Wales

Australian Defence Force Academy

July 2003

Contents

1 Bones and body mechanics 11.1 Mechanical representation of the human skeleton . . . . . . . . . . . 21.2 Standing, bending, lifting . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Walking and running . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Bone: A good material choice? . . . . . . . . . . . . . . . . . . . . . 9

2 The eye and vision 132.1 Cornea, iris, lens and retina . . . . . . . . . . . . . . . . . . . . . . 142.2 Colour perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Exploring vision with simple experiments . . . . . . . . . . . . . . . 20

3 Hearing 213.1 Hearing sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Structure of the ear . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Outer ear and middle ear . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Inner ear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Alpha-decay 33

5 Beta-decay 435.1 The story of a carbon atom . . . . . . . . . . . . . . . . . . . . . . 445.2 Carbon-14 dating . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6 Gamma-decay 556.1 Nuclear medicine: history and modern practice . . . . . . . . . . . 566.2 Gamma-rays in nuclear medicine . . . . . . . . . . . . . . . . . . . 596.3 The equivalence of energy and mass: . . . . . . . . . . . . . . . . . 606.4 Scintillation detectors . . . . . . . . . . . . . . . . . . . . . . . . . . 656.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7 Appendix 677.1 First in-class test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

References 85

iii

Chapter 1

Bones and body mechanics

1

2

Figure 1.1: The three lever classes and schematic examples of each in the body. W is aweight, F is the reaction force at the Fulcrum point and M is the muscle force.

The adult human body has 206 bones, forming the skeleton, which gives thebody static rigidity and in conjunction with joints, ligaments and muscles allowsdynamic motion. Insights into fundamental body actions, such as walking, running,or lifting loads, which are generally so familiar that we rarely analyze them, can begained by applying mechanics principles. This can explain for example, why eachperson has a different advantageous step frequency, which surfaces require shortstrides, or why certain lifting techniques pose a health risk.

Bone is a complex, living composite-material. It appears better suited than anysynthetic material for its purpose, so that it is interesting to explore how our bonematerial provides the functionality required to ensure survival.

1.1 Mechanical representation of the human skeleton

The complexity of the human skeleton can be reduced by distinguishing threeclasses of lever action, see Fig. 1.1. The classes may be defined as follows:

• Class 1 The fulcrum F is between the load W and the point where the muscletendons are attached M .

• Class 2 W is between M and F .

• Class 3 M is between fulcrum F and load W .

For an equilibrium situation, for example holding a weight in your hand orbending over, each of the three forces (W,F, and M) can be calculated, if one ofthe other forces and the geometrical dimensions are known. This is possible, sincefor a static equilibrium, both, the vector sum of all external forces ~Fex, and the

Bones and body mechanics 3

Figure 1.2: The forearm. (a) The muscle and bone system. (b) Forces and dimensions: R isthe reaction force of the humerus on the ulna joint. (c) The weight of the arm H is includedat its centre of gravity.

vector sum of all external torques ~τex (about any point) are zero, i.e.∑ ~F = 0 (1.1)∑

~τ = 0 (1.2)

Exercise 1 Which lever class represents the forearm, when holding a weight? Forthe forearm shown in Fig. 1.2 calculate the muscle force M required to hold a weightW = 1 kg in the hand. What reaction force R acts on the ulna bone at the jointwith the humerus bone (fulcrum) for this weight. For a first estimate ignore themass of the arm. Then include the mass of the arm (1.5 kg) in your calculation.

Exercise 2 An exercise device is used to strengthen the leg muscles, see Fig. 7.1.Calculate the force M exerted by the muscle in the upper leg, when moving the footforward to lift the weight. What is the reaction force exerted onto the joint?

Experience tells that it is more difficult to hold a weight when the forearm isat angle α below or above the horizontal and that a weight can be sustained thelongest, when forearm and body are at a right angle. This suggests that the muscle

4

Figure 1.3: Mechanical representation of an exercise to strengthen the leg muscles [Mc-Cormick & Elliot 2001].

force M is angle dependent. Indeed, for the torque τ at the joint of humerus andulna it follows, using the same weight and dimensions as in Fig. 1.2, that

τ = (30 cm · 10 N + 14 cm · 15 N) cos α (1.3)

The torque thus increases with angle, however, since

M =τ

4 cm · cos α(1.4)

the muscle force required remains constant with angle α. The apparent difficulty inholding a weight off the horizontal must therefore have a different reason. Indeed, itis found that this is a consequence of muscle physiology, which gives most strengthat the muscle resting length, in between the two extremes of a fully stretchedmuscle and that of a contracted muscle.

1.2 Standing, bending, lifting

The mechanics of the spine may be investigated by treating the spine approximatelyas a rigid beam. When a person stands erect, the weight of the upper body W

is directly over the legs and little force is exerted by the back and leg muscles.This changes, and considerable muscle work is then required, for an overweight orpregnant condition, since the person needs to tilt slightly backwards.

It can be estimated that a disk between vertebrae in the spine column is likelyto be damaged or even ruptures, when it experiences pressures of the order of 107

Pa, which is about 100 at.

Bones and body mechanics 5

Figure 1.4: Bending at an angle of 60◦ to the vertical with and without load L; W is theupper body weight, M is the force of the back muscles, exerted on the spine at a distance dfrom its base [McCormick & Elliot 2001].

Exercise 3 A person has a body mass of 75 kg and an upper body length of d, seeFig. 1.4. (a) Estimate the force M required from the back muscles, when the personbends to an angle of 60◦ to the vertical. (b) What pressure does the lumbrosacraldisk at the base of the spine experience? (c) How do these values change, whenthe person lifts a weight of 20 kg? (d) For what bending angles with respect to thevertical is the pressure on the spinal disks the largest?

The exercise shows that lifting heavy objects by bending over can increase thepressure on the disks between the vertebrae of the spine to values, which comeclose to their mechanical strength of about 107 Pa. This is illustrated in moredetail in Fig. 1.5.

1.3 Walking and running

For a human standing still, the body weight is balanced by the reaction force ~G

of the ground pointing in the opposite direction than the force associated with theweight, ~W = −~G. When walking or running, an additional force comes into play,

6

Figure 1.5: (top) The average pressure on disks of the spine for different lifting situations.(bottom) The pressure as a function of time for the two extreme cases.

Bones and body mechanics 7

material pair µst µsl

steel on steel 0.15 0.10 - 0.05steel on ice 0.027 0.014leather on metal 0.6 0.4oak on oak 0.58 0.48blocked tire on dry street 0.8blocked tire on wet street 0.5blocked tire on ice 0.05

Table 1.1: Coefficients µst and µsl for static and sliding friction, respectively, for somematerial pairs.

the frictional force associated with the ground ~F , which is directed horizontally,either in the direction of motion or opposite to it. The magnitude of the frictionforce is found to be a constant fraction of the normal force ~N

|~F | = µst| ~N | (1.5)

where µst is the coefficient of static friction, which is low for slippery surfaces andhigh for ‘firm’ ground. For a horizontal surface the normal force is equal to theweight ( ~N = ~W ), so that

|~F | = µst| ~W | (1.6)

The coefficient of sliding friction µsl is defined equivalently. Table 1.1 gives somecharacteristic values.

It is interesting that the coefficient for sliding friction is always smaller than thatfor static friction. This explains for example, why it is usually difficult to avoid afall on a slope, once your boots have lost grip.

Exercise 4 Using a wooden board, sheet metal, a ruler and a weight, measure thestatic and sliding coefficients of friction for your shoe or boot on wood and on themetal. Give a rough estimate of the experimental uncertainty.

The reaction force ~G and the friction force ~F combine to give a net force, whicheither stops or propels our strides, see Figure 1.6 The net force points more upward,the smaller the friction force is. Thus for slippery surfaces, it pays off to makeshort strides, so that the momentum of the foot is directed downward rather thanforward.

Figure 1.7 illustrates that walking legs are like two pendulums, which swing backand forth, and that most of the energy is used to move the mass associated withthe legs forward and backward, rather than for lifting the feet of the ground. Sincethe least energy is required, when a pendulum swings with its eigen-frequency, itis advantageous to walk steady and move the legs at the eigen-frequency of ourmotoric system. Speed can then be controlled by varying the length of the strides.

When running, the number of strides per second has to be increased and moreenergy is needed. In addition, we lean forward and launch the body into brief leaps.

8

Figure 1.6: Forces acting on the foot when walking.

Figure 1.7: Action of the leg muscles in walking [McCormick & Elliot 2001].

Bones and body mechanics 9

Figure 1.8: Action of the leg muscles in running [McCormick & Elliot 2001].

Still, the centre-of-mass of our body barely moves in the vertical and thus littleenergy is consumed to account for the slight changes in potential energy associatedwith such movements, see Fig. 1.8

1.4 Bone: A good material choice?

The theory of evolution would suggest that our natural bone material is optimizedfor its purpose. The question arises, if this can be supported with physics argu-ments. The following properties may be considered important for bones:

• Strength, to sustain large forces

• Elasticity, to avoid damage and thus immobility, which to a prehistoric humanbeing was equivalent to death

• Low weight, to reduce the energy, and thus the food, required to carry thebones around

• Self-repair, to counter wear, achieve longevity, and regain mobility followingdamage

It is instructive to compare the properties of bone with those of other commonmaterials. This shows that bones are by far not the strongest materials available.The compressive breaking stress of trabecular bone is 2.2 N/mm2, which is two or-ders of magnitude lower than than of granite (145 N/mm2) and steel (552 N/mm2).However, as demonstrated earlier, our bones are strong enough to sustain pressuresof tens of atmospheres.

Elasticity may be characterized using Young’s modulus

Y =L

∆L· FA

(1.7)

where F is the force pulling a cylinder, A the cross-section area of the cylinder, andL is its length. An elastic material extends a fair distance ∆L for a given force F ,

10

whereas an inelastic material does not. Thus a low modulus corresponds to goodelasticity. Trabecular bone has a Young modulus of 0.76 N/mm2, which is muchlower than that of other strong biological materials such as oak (110 N/mm2), butnot as low as the modulus of rubber (0.01 N/mm2).

It is not immediately clear, why elasticity should be so important for bones.This may be illustrated using a car crashing into, say, a tree at about 50 km/h. Inthis case and without air bag and seat belt, the driver’s head impacts on the dashboard with a velocity of about 15 m/sec and 5 mm of skin are the only protectionfor the skull bone. The force to be sustained by the skull bone can be estimatedfrom the deceleration time

∆t =∆x

v=

5 · 10−3 m

15 m/s∼= 0.3 · 10−3 s (1.8)

The force F to be sustained in this case is therefore of the order of

F =∆p

∆t=

4 kg · 15 m/s

0.3 · 10−3 s= 200, 000 N (1.9)

where it has been assumed that the mass of the head is about 4 kg. The impactingforce is equivalent to a weight of 20 tons ! The elasticity of bone reduces this forcesomewhat by increasing the deceleration time ∆t. However, it can be estimatedwith Eqs 1.8 and 1.9 that the skull bone would have to give by almost one meterto reduce the impacting force to below 100 kg. Since this is not possible withoutlethal damage, the need for other crunch zones, as provided by the body of the caror an air bag, is emphasized.

With regard to ‘low weight’ compact bone actually performs poorly. The den-sity of compact bone (1.9 g/cm3) is larger than that of water, compared to only0.1 g/cm3 for balsa wood, another biological material, which might have been usedin its place, however, would lack sufficient strength. The large density of compactbone very much explains the prevalence of structured, trabecular bone in the body.‘Trabecular’ (Latin for ‘comprised of beams’) refers to the fact that most bone hasa ‘cathedral-like’ structure shown in Figure 1.9, which, as for the comparison, givesgreat strength with a minimum of material and weight.

It is interesting to note, in an age where materials science aims for so-called‘nano-structured materials’, that bone is a good example for a nano-structuredmaterial, since it consists of particles of bone mineral (Ca10PO4OH2) with dimen-sions of the order of about 10 nm, which are extremely hard, but form elasticpolymer tubes of collagen, which are typically about 200 nm large. The tubescombine to form trabecular cells, with diameters of about half-a-millimeter, whichgive bone its ‘spongy’ appearance.

During our life-time the structure of bone changes to adapt to changing require-ments (Wolf’s law). The elasticity required for babies and toddlers is reflected bylarge cells, whereas the more dense bone material of adults gives more strength atthe expense of elasticity. This is shown in Fig. 1.10

Bones and body mechanics 11

Figure 1.9: This micrograph shows the ‘spongy’, cathedral-like structure of trabecular bone.

Figure 1.10: Bone cross-sections at different stages of a human’s life. Baby bones (a, d) haverelatively large cells giving elasticity, compared to adult bones (b,c,e), which are structuredmore densely to increase strength.

12

In summary, it may be stated that there certainly exist many materials, whichexceed bone with regard to either strength, elasticity, or weight. However, noneof those reconciles these three properties better than trabecular bone and alsoallows for continuous restructuring to adapt to the changing human body, while inaddition offering the option of biological repair following damage.

Chapter 2

The eye and vision

13

14

Figure 2.1: Cross-section of the human eye [McCormick & Elliot 2001].

The capabilities of the human eye are remarkable, when compared to even advancedhigh-tech cameras:

• rapid automatic focussing

• wide angle of view, while simultaneously detailed vision at large distances

• brain transforms images from both eyes into three-dimensional perception

• operation over a large range of light intensities: from dark night to bright day(7 orders of magnitude!)

• some self-repair of local damage

2.1 Cornea, iris, lens and retina

Figure 2.1 show a cross-section through the human eye. Several properties of theeye can be understood by modelling it using the concepts of geometrical optics. Inthis approach the combination of cornea and lens may be viewed as a converginglens, while the iris determines how much light is focussed onto the retina, which isequivalent to an image screen.

It is interesting that the cornea breaks the incident light much more stronglythan the eye lens, which nevertheless is the active component of the lens systemand fine-tunes the focussing. The strong breaking power of the cornea results fromits relatively large refractive index of nco = 1.37. Table 2.1 compares this valuewith those for other media.

The eye and vision 15

refractive index ncornea 1.37eye lens 1.41vacuum 1air 1.003water 1.33glass 1.5 - 1.9diamond 2.42

Table 2.1: Refractive indices of some media.

Figure 2.2: Reflection and refraction of a light-ray at a plane glass surface [Halliday et al.1993].

A light-ray incident on the eye passes from air with a refractive index nair = 1.003into the watery medium of the cornea. As for any such transition between twotransparent media at the interface both reflection and refraction are observed.This is demonstrated for air and glass in Fig. 2.2.

The phenomenon is described within geometrical optics by the law of reflection

θ1 = θ′1 (2.1)

and Snell’s law of refraction

n2 sin θ2 = n1 sin θ1 (2.2)

16

Figure 2.3: Definition of focal length [McCormick & Elliot 2001]. (a) converging. (b)diverging.

and can be explained by a change in wave velocity which for every medium is givenby

v = c/n (2.3)

where c is the speed of light.When n2 > n1, it follows from Eq. 2.2 that θ2 < θ1 and the ray is refracted

toward the surface normal. The relatively large change of wave velocity at thecornea surface is thus responsible for the strong breaking power of the cornea.

Exercise 5 (a) Why is it easier under water to obtain a well focussed view withthe help of a face mask as compared to not wearing a mask? (b) Calculate thechange in direction for a light-ray incident on the cornea at an angle of 25◦.

Refraction can be exploited for focussing or de-focussing by curving the interfacebetween two media of different refractive indices and making a lens. Lenses areeither converging, e.g. convex lenses, or diverging, e.g. a concave lenses. The focallength of a lens f , which defines the lens power p, is equal to the distance betweenthe optical centre of the lens and the focal point for rays of light which are incidenton the lens and parallel to its optical axis. This is illustrated in Fig. 2.3. The lenspower is then given by

p =1

f [in m][in Dioptres] (2.4)

In reasonable approximation the combination of cornea and eye lens can be de-scribed as a thin converging lens with a power of 60 D - 70 D, depending on the lensmuscles being relaxed (unaccommodated) or tightened to focus on a near object(accommodated). For given power p the thin lens formula relates an object to therespective image as projected by the lens with

1

f=

1

s+

1

s′(2.5)

The eye and vision 17

where s and s′ are the distance of the object and the image from the optical centreof the lens as measured along its optical axis, respectively.

Exercise 6 The cornea has a power of about 50 Dioptres, while that of the eyelens is about 10 Didoptres, when unaccommodated. Calculated the combined focallength of cornea and unaccommodated eye lens. Is the result consistent with thephysical dimensions of the human eye?

A short-sighted person can see near objects clearly but distant objects appearblurred, because either the curvature of the cornea or the lens is too large or thedistance between eye lens and retina is too large. The farthest point still in focusis referred to as ‘far point’.

Equivalently a far-sighted person sees near objects blurred with the nearest pointin focus being called the ‘near-point’. It is usually caused by a flattening of theeye lens.

Exercise 7 (a) Suppose that a person is short-sighted with a far point at 0.2 m.The power of accommodation of that person is 4 Dioptres. Calculate the powerof the spectacle lenses that are required to see distant objects in focus. What isthe shape of those lenses? (b) Suppose that a person is far-sighted with the nearpoint at 1.0 m. For correct vision the near point should be at 0.25 m from the eye.Calculate the power of the spectacle lenses that are required to see an object at adistant of 0.25 m in focus. What is the shape of those lenses? (Approximate thedistance between the optical centre of the eye lens and the retina with 0.02 m.)

Another common eye defect in astigmatism, which can be detected with the eyetest shown in Fig. 2.4 (top). Astigmatism occurs, when the focal length of the eyeis different, say, in the horizontal plane, than in the vertical plane. This is causedby a distortion of the cornea. A person suffering from astigmatism perceives thelines at some angle α and those at α + 180◦ blurred and greyish, while the lines atthe other angles are perceived black and in focus. The defect can be correct witha cylindrical spectacle lens which thus de-focusses in one plane. The lens has tobe positioned, so that the de-focussing coincides with the plane defined by α andα + 180◦.

Exercise 8 A person has astigmatism and wears glasses to correct this eye defect.The person removes the glasses and holds them at some distance looking throughone of the lenses, while at the same time rotating the glasses. What happens to theimage seen through that lens?

2.2 Colour perception

Our colour perception is complex, but can be characterized employing three dif-ferent qualities:

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Figure 2.4: Eye test to detect astigmatism (top). Cylindrical lens correcting astigmatismbetween the vertical and horizontal planes (bottom) [McCormick & Elliot 2001].

The eye and vision 19

Figure 2.5: The colour circle showing all hues. It incorporates all spectral colours and thecolour triangle [McCormick & Elliot 2001]. The extraspectral hues result from combiningspectral hues of low and high wave lengths. The limits of this are indicated by the solid anddashed lines respectively. The colour triangle identifies the three primary colours blue, greenand red.

• Hue is what we colloquially call colour. It is comprised of all the lightcolours associated with the electromagnetic spectrum, which can be identifieduniquely through their wavelength λ. In addition, it includes the extraspec-tral hues of magenta, which result from mixing violet and red. The varioushues are compiled in the colour circle in Fig. 2.5.

• Brightness is a qualitative measured of the light intensity. It is related tothe power reaching the retina per unit area.

• Saturation is the purity of the colour. The hues shown in Fig. 2.5 are sat-urated. When they are mixed with a neutral colour, such as white, black orgrey, they become less saturated.

The eye lens is opaque to wave length below 380 nm and thus limits our visibilityfor light of low wavelength. The long-wavelength limit is set by the sensitivity ofthe retina which is 760 nm. It is interesting to note that over this (narrow) range ofwavelengths atmospheric absorption is minimal so that the eye takes full advantageof the naturally accessible part of the electromagnetic spectrum.

Colour perception is made possible by the existence of three different types oflight-sensitive cells (cones) on the retina, which respond differently to the threeprimary hues of blue, green and red. In addition to the cones a second type ofreceptor cells exists on the retina (rods) which are only sensitive to light intensity,but not to colour. The colour sensitivity of the cones is due to three different

20

photosensitive pigments. Each one absorbs light over a range of wave lengths,with peaks at 445 nm, 535 nm, and 575 nm, respectively. The absorption of lightproduces a chemical change in the pigment molecule resulting in a nerve pulse,which is transmitted to the brain for processing.

When all three types of cones receive light of similar intensity and colour cor-responding to their characteristic hue (blue, green, red), the perception is that ofwhite light. The three combinations of two of the primary colours produce thesecondary colours:

• green + blue = cyan

• blue + red = magenta

• red+ green = yellow

2.3 Exploring vision with simple experiments

It is instructive to explore the eye with some simple experiments:

Exercise 9 Yellow spot and blood vessels In front of an intense light sourcelook at a pin hole in a piece of paper at a distance of about 10 cm. What can yousee ?

Exercise 10 Blind spot Try located the blind spot in you field-of-view. Estimatethis angle.

Exercise 11 Upward and downward rays on strong light sources Whenlooking at a strong light source, maybe a street light at night, it often appears thatlight-rays extend downwards and/or upwards from it, but not to the either side.Why?

Exercise 12 Curtains and Fechner’s law During the day, why do net curtainsobscure the view? Why does this change at night?

Chapter 3

Hearing

21

22

Figure 3.1: The sensitivity of the human ear as a function of sound frequency. The thresholdsensitivity (solid curve) is indicated. Dashed curves show the average threshold level, theaverage sound level where discomfort, and that where pain was felt, for a group of testpersons, respectively. The right ordinate axis gives the sound intensity in units of Watt permetre-squared on a logarithmic scale, while the left ordinate axis uses the specific units ofdecibel.

Sound is a longitudinal pressure wave, which is received by our ears and trans-formed into electrical signals, which can be processed by the brain. The ear thusacts as a ‘transducer’.

3.1 Hearing sensitivity

Our sense of hearing is astonishing. This is evident from Fig. 3.1, which demon-strates that the ear is sensitive over a frequency range of 20 − 20, 000 Hz, whichcorresponds to a change by a factor of 1000 or, in musical terminology, is equivalentto 8 octaves. For comparison, it may be noted that the frequency range visible byour eyes spans only 1 octave, ranging from 4 − 9 · 1014 Hz and is equivalent to achange of little more than a factor 2. The large range of our hearing appears evenmore remarkable, when remembering that musicians can tune their instruments tobetter than 1 Hz.

Figure 3.1 also shows that the sensitivity of the ear covers a range of 12 orders ofmagnitude. Sound intensity is power per area and is therefore measured in units ofWatt per metre-squared [W/m2]. Honouring Alexander Graham Bell, the quantity‘intensity level’ has been introduced with the units ‘decibel’ and ‘bel’ (1 dB =0.1 B). Intensity level is defined through

1 dB = 10 · log (I/I0) (3.1)

where I0 = 10−16 W/cm2, which is taken as the sensitivity threshold (comparewith Fig. 3.1). Some values for sound intensity and the corresponding intensitylevel have been compiled in Table 3.1.

Hearing 23

power intensity level (dB)threshold ≤ 1 nW 0conversation 7 µW 50pain > 1000 W 130

Table 3.1: Typical values for sound intensity I and the corresponding intensity level in dB.

Figure 3.2: The relation of intensity level and loudness as a function of frequency.

Exercise 13 At the nearest houses the traffic on a motorway produces noise withan intensity of 10−6 W/m2. An expansion of the motorway with additional lanesis expected to double the traffic. Calculate the intensity level in dB before and afterthe motorway expansion.

In the context of measuring and legislating environmental noise and its impacton humans, the quoting of intensity levels is not satisfactory, since the perceived“loudness” of a noise is frequency dependent and a subjective observation. Loud-ness perception decreases for low and high frequencies. In an attempt to quantify“loudness” the units “1 phon” has been introduced. For the frequency 1000 Hz1 phon = 1 dB. The loudness of a sound at a different frequency is determined bycomparing it with a sound at 1000 Hz until the loudness perception is the same.In practice frequency-dependent weights are used to simplify the measurement ofloudness. The change of loudness in units of “phon” with frequency is illustratedin Figure 3.2. Its astonishing sensitivity aside, it may also be noted that hearingis a 360◦, night-and-day sense and thus has been very useful in the evolution ofthe human species. The question arises how the ear achieves such extraordinaryperformance.

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Figure 3.3: Cut-through view of the human ear.

3.2 Structure of the ear

Figure 3.3 shows a cut-through view of the human ear. The important parts ofthe ear are the auricle (Pinna), the outer ear, the middle ear and the inner ear.Besides this the brain plays a very important part in hearing by processing digitalinformation received from the inner ear.

The human auricle does not have any role in hearing and may be considered anevolutionary relict1. However, cupping your hands behind the ears is equivalent toa 6-8 dB directional gain of sensitivity.

The outer ear comprises mainly a 2.5 cm long tube, the ear canal. Its mainpurpose is to transport the sound to the middle ear, which is thus located wellinside the body and protected. The length of the ear canal is optimized for thebest transport of sound waves with a frequency of 3300 Hz. This is the frequencywhere human hearing is most sensitive.

The middle ear comprises the eardrum and the three ossicles, referred to as ham-mer, anvil, and stirrup. The middle ear converts the sound waves into mechanicalvibration with an almost ‘static’ gain of a factor x 20. Relaxing muscles in themiddle ear, this gain factor can be reduced to protect middle and inner ear frombrief excessive noise.

The mechanical vibrations are then passed on to the inner ear, which houses thecochlea spiral, the complex hearing organ, which in many regards is still a mystery.In the cochlea the ‘analogue’ information contained in the mechanical vibrations ofthe ossicles is ‘dynamically’ amplified with large gain and ‘digitized’ using tiny hairbundles. The ‘digital’ information is then converted into electrical pulses, whichare transported to the brain for processing by the auditory nerves.

The hearing process thus involves the following steps:

1In some animals the auricle actually still aids the hearing sense.

Hearing 25

• (faint) sound

• longitudinal pressure wave

• mechanical motion and amplification

• ‘dynamic’ amplification

• digitization

• electric pulses

• brain processing and perception

The steps shown in italic are still poorly understood.

3.3 Outer ear and middle ear

The ear canal may be likened with a cylindrical pipe of 2.4 cm length, which isopen at one end and closed at the other by the 0.1 mm thin (paper thin) ear drum.The fact that the ear canal transports sound best in the most sensitive frequencyregion of the ear, and thus to some extend suppresses (background-)sounds at otherfrequencies, can be explained with wave physics. For clarity the longitudinal soundwave may be represented by a transversal wave, as it is shown in Fig. 3.4. For asingly-closed tube it holds that

ν =c

4`(3.2)

where ν is the sound frequency and c = 330 m/s is the velocity of sound in air.It therefore follows, using the dimensions of the ear canal, that the resonancefrequency νr of the ear canal, for which no destructive interference occurs, is givenby

νr =330 m/s

4 · 2.5 · 10−2 m= 3300 Hz (3.3)

The ear canal is thus tuned to sounds like the cracking of a twig or high-pitchspeech, such as that of an infant, which both must have had a particular importanceto the survival and evolution of the human species.

Figure 3.5 shows a cross-sectional view of the outer and middle ear illustratingthe arrangement of hammer, anvil and stirrup, which form a leaver system. Aschematic of this leaver system is displayed in Fig. 3.6. The physical motion ofthe eardrum is extremely small. At the hearing threshold the eardrum moves onlyabout 1 A. This motion is passed on by the ossicles to another membrane, theoval window. The leaver action of the ossicles ‘passively’ amplifies the mechanicalmotion. The amplification factor of about x 15 can be derived using the fact thatthe forces on both membranes are balanced and estimating the respective area ofeardrum and oval window, as it is shown in Fig. 3.6. The leaver arrangement itself

26

Figure 3.4: Illustration of the relation between tube length ` and resonance wave length λfor an open and singly-closed tube, respectively. For clarity the longitudinal sound wave hasbeen represented as transversal.

Hearing 27

Figure 3.5: Cross-sectional view of outer and middle ear.

Figure 3.6: Transformation of air pressure changes into mechanical motion in the middleear. This gives a mechanical amplification of a factor of 15.

28

Figure 3.7: Illustration of the cochlea. (a) The spiral duct of the cochlea is divided bythe flexible basilar membrane and is filled with a fluid. The hair cells, which sit on themembrane, connect directly to the auditory nerve. (b) When sound enters the cochlea (hereshown uncoiled) it agitates the fluid, causing a ripple to travel along the basilar membrane.This movement (which is grossly exaggerated here) is detected by sensory hair cells, supportedon the membrane.

adds another factor x 1.3 to the gain, so that the total amplification of the middleear amounts to a factor of η ∼= 20. Muscle action can reduce η briefly to protectthe inner ear from excessive noise.

Another important part of the middle ear is the Eustachian tube which equili-brates the pressure on either side of the eardrum. This is required, so that thevibration of the eardrum is not hindered by pressure differences. During a cold andwith sudden changes in external pressure (for example on board of an airplane) weexperience the effect of different pressures on either side of the eardrum. Our hear-ing becomes less sensitive and distorted. Also physical pain may be experienceddue to the permanent strain on ear drum and oval window.

3.4 Inner ear

On the other side of the oval window is the cochlea. Many aspects of the cochleaare still being researched, however, in general its functioning has been understood.A schematic illustration of the cochlea is shown in Fig. 3.7. It is a long, conical (i.e.the tube diameter decreases with length) tube which is coiled up towards the apex.The tube contains a fluid and it is separated into three chambers, referred to as

Hearing 29

Figure 3.8: A cross-sectional view of the cochlea tube.

tympanic chamber, cochlear duct, and vestibular chamber, as it is shown in Fig. 3.8.Between the tympanic chamber and the cochlea duct is the basilar membrane. Thestiffness of the basilar membrane is largest near the oval window and it becomesmore elastic towards the other end of the cochlea tube. This change in elasticityis quite dramatic, covering two orders of magnitude.

The mechanical motion of the oval window is transferred to the cochlea fluid.The fluid motion then causes ‘ripples’ of the basilar membrane, as it is illustratedin Fig. 3.7b. These ‘ripples’ pass along the membrane and cease at a certaindistance from the oval window as a consequence of a complex interplay of fluiddrag and the elasticity of the basilar membrane. Importantly, this distance iscorrelated with the frequency of the original sound. Thus the received soundfrequency is converted to a stimulus along the basilar membrane with a well definedlength. This is equivalent to a ‘digitization’ of the received frequency spectrum. Inprinciple this ‘digital’ information can then be passed on to the brain for processing.‘Ripples’ associated with high sound frequencies cease close to the oval window,whereas those corresponding to low sound frequencies travel a long distance alongthe basilar membrane.

Several question arise:

• How is the mechanical signal, or rather the distance travelled by this signalalong the basilar membrane, converted into an electrical nerve pulse?

• How can the extreme sensitivity of human hearing be achieved with such anarrangement?

• How well can neighbouring frequencies be distinguished?

30

Figure 3.9: The organ of corti.

It is clear from the outset that the outstanding performance of the human earcannot be achieved with a passive system, such as the leaver arrangement of theossicles in the middle ear. The interface between the basilar membrane and theauditory nerve which connects to the brain is the organ of corti. An illustration isshown in Fig. 3.9. The organ of corti consist of hair bundle cells which are locatedabove the basilar membrane and therefore move along with the ripples passingalong. Importantly, the hairs are not stationary but permanently in spontaneousoscillatory motion. The frequencies of this spontaneous motion are random, how-ever, very near to a certain eigenfrequency, which is characteristic for the celllocation. This allows for ‘dynamic’ amplification. The external stimulus of theripple, when of correct frequency, aligns the spontaneous oscillatory motion of allhairs at the characteristic eigenfrequency. Since the frequency of the hair motionis already very near this eigenfrequency, not much energy is required to achievethis. Thus a small stimulus achieves a large gain. This might explain the extremesensitivity of our hearing. The frequency spectrum (the Fourier transform) of thehair oscillations, sharpens drastically, once the hairs go from the permanent near-frequency motion to tuned motion, stimulated by the ripple. This is similar to thenon-linear behaviour of a so-called Hopf resonator. The non-linear response of ahair bundle is illustrated in Fig. 3.10 for a hair bundle from a frog ear.

The displacement of the hairs by their motion opens ion channels, which forexample allows K+ ions to move in, see Fig. 3.11. This creates a potential differenceand results in an electric nerve pulse through the auditory nerve to the brain.

Hearing 31

Figure 3.10: (a) When a frog hair bundle is shaken using a micro-needle, its response ischaracteristic of a noisy Hopf oscillator (top). The applied force (shown in addition for allother spectra), which is related to the amplitude of the displacement of the needle, progres-sively increases down the figure. When the bundle is shaken gently, the Hopf oscillator’s gain- its response divided by the input stimulus - is large. (b) The Fourier transform of the bundledisplacement has a peak at the stimulus frequency. The height of this peak grows as the cuberoot of the applied force. Thus a small stimulus force is amplified with larger gain than alarger stimulus.

32

Figure 3.11: (a) Microscopic image of a hair bundle cell. The bundle in this hair cell froma turtle is a pyramidal structure composed of stereocilia, which are connected by tip links.(b) The coordinated motion of the hair bundle opens ion channels (the gates are indicated bytwo red dots) and allows for the passage of K+ ions.

Chapter 4

Alpha-decay

33

34

Alpha-decay 35

36

Alpha-decay 37

38

Alpha-decay 39

40

Alpha-decay 41

Exercise 14 Depleted uranium (∼ 99.8 % 238U, ∼ 0.199 % 235U, 0.001 % 234U) is aby-product of the enrichment process for reactor fuel and therefore relatively cheap.It has the same chemical and materials properties as natural uranium (∼ 99.275 %238U, ∼ 0.720 % 235U, 0.005 % 234U). Uranium has 1.7 times the density of lead andit is pyrophoric, i.e. fine uranium particles ignite in air, so that is well suited forapplication in high-impact ammunition. It is also often used as ballast in aircraft.The three long-lived uranium isotopes decay predominantly by alpha-decay.

(a) Decide, if an area contaminated with depleted uranium is more radioactive, i.e.the overall activity is larger, than when it is contaminated with the same amountof natural uranium. Justify your answer.

(b) Can you establish a trend relating the activity −dN/dt and the mass numberA of the isotopic series of uranium nuclides, which decay by α-decay? Is this ageneral trend? Support your evidence for uranium with two other examples, i.e.two other chemical elements.

(c) Inhaling of fine uranium dust can cause toxic reactions. However, how does theradiological impact of uranium dust, i.e. the emission of energetic alpha particlesinside respiratory organs, compare with that of radium dust?

[The Table of Nuclides can be found at “http://www2.bnl.gov/ton/”. A PeriodicTable is available at “http://pearl1.lanl.gov/periodic/default.htm”.]

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Chapter 5

Beta-decay

43

44

5.1 The story of a carbon atom

Radioactive β-decay of 14C nuclei forms the basis of the powerful technique of C-14dating. In the story reprinted below [Levi 1975], Primo Levi beautifully narrateshow carbon atoms are constantly incorporated into plants as long as the plant isalive, thus replenishing an equilibrium ratio of the stable 12C and the radioactive14C carbon isotopes. The death of a plant then stops this process and starts theC-14 clock.

Born in Turin in 1919, Primo Levi graduated in chemistry shortly before the Fascistrace laws prohibited Jews like himself from taking university degrees. In 1943 hejoined a partisan group in northern Italy, was arrested and deported to Auschwitz.His expertise as a chemist saved him from the gas chambers, however. He was setto work in a factory, and liberated in 1945.

His memoir “The Periodic Table” takes its title from the table of elements, arrangedaccording to their atomic mass, which was originally devised by Dmitri Mendeleyevin 1869. Levi links each episode of his life to a certain element. But in the book’sfinal section, printed below, he sets himself to imagine the life of a carbon atom.This was, he says, his first ‘literary dream’, and came to him in Auschwitz.

“Our character lies for hundreds of millions of years, bound to three atoms ofoxygen and one of calcium, in the form of limestone: it already has a very longcosmic history behind it, but we shall ignore it. For it time does not exist, orexists only in the form of sluggish variations in temperature, daily or seasonal, if,for the good fortune of this tale, its position is not too far from the earth’s surface.Its existence, whose monotony cannot be thought of without horror, is a pitilessalternation of hots and colds, that is, of oscillations (always of equal frequency) atrifle more restricted and a trifle more ample: an imprisonment, for this potentiallyliving personage, worthy of the Catholic Hell. To it, until this moment, the presenttense is suited, which is that of description, rather than the past tense, which isthat of narration - it is congealed in an eternal present, barely scratched by themoderate quivers of thermal agitation.

But, precisely for the good fortune of the narrator, whose story could otherwisehave come to an end, the limestone rock ledge of which the atom forms a part lieson the surface. It lies within reach of man and his pickax (all honor to the pickaxand its modern equivalents; they are still the most important intermediaries in themillennial dialogue between the elements and man): at any moment - which I, thenarrator, decide out of pure caprice to be the year 1840 - a blow of the pickaxdetached it and sent it on its way to the lime kiln, plunging it into the worldof things that change. It was roasted until it separated from the calcium, whichremained so to speak with its feet on the ground and went to meet a less brilliant

Beta-decay 45

destiny, which we shall not narrate. Still firmly clinging to two of its three formeroxygen companions, it issued from the chimney and took the path of the air. Itsstory, which once was immobile, now turned tumultuous.

It was caught by the wind, flung down on the earth, lifted ten kilometers high.It was breathed in by a falcon, descending into its precipitous lungs, but did notpenetrate its rich blood and was expelled. It dissolved three times in the waterof the sea, once in the water of a cascading torrent, and again was expelled. Ittraveled with the wind, for eight years: now high, now low, on the sea and amongthe clouds, over forests, deserts, and limitless expanses of ice; then it stumbled intocapture and the organic adventure.

Carbon, in fact, is a singular element: it is the only element that can bind itselfin long stable chains without a great expense of energy, and for life on earth (theonly one we know so far) precisely long chains are required. Therefore carbon is thekey element of living substance: but its promotion, its entry into the living world,is not easy and must follow an obligatory, intricate path, which has been clarified(and not yet definitively) only in recent years. If the elaboration of carbon werenot a common daily occurrence, on the scale of billions of tons a week, whereverthe green of a leaf appears, it would by full right deserve to be called a miracle.

The atom we are speaking of, accompanied by its two satellites, which main-tained it in a gaseous state, was therefore borne by the wind along a row of vinesin the year 1848. It had the good fortune to brush against a leaf, penetrate it,and be nailed there by a ray of the sun. If my language here becomes impreciseand allusive, it is not only because of my ignorance: this decisive event, this in-stantaneous work a tre - of the carbon dioxide, the light, and the vegetal greenery- has not yet been described in definitive terms, and perhaps it will not be for along time to come, so different is it from the other organic chemistry which is thecumbersome, slow, and ponderous work of man: and yet this refined, minute, andquick-witted chemistry was invented two or three billion years ago by our silentsisters, the plants, which do not experiment and do not discuss, and whose temper-ature is identical to that of the environment in which they live. If to comprehend isthe same as forming an image, we will never form an image of a happening whosescale is a millionth of a millimeter, whose rhythm is a millionth of a second andwhose protagonists are in their essence invisible. Every verbal description must heinadequate, and one will be as good as the next, so let us settle for the followingdescription.

Our atom of carbon enters the leaf, colliding with other innumerable (but hereuseless) molecules of nitrogen and oxygen. It adheres to a large and complicatedmolecule that activates it, and simultaneously receives the decisive message fromthe sky, in the flashing form of a packet of solar light: in an instant, like an insectcaught by a spider, it is separated from its oxygen, combined with hydrogen and(one thinks) phosphorus, and finally inserted in a chain, whether long or short does

46

not matter, but it is the chain of life. All this happens swiftly, in silence, at thetemperature and pressure of the atmosphere, and gratis: dear colleagues, when welearn to do likewise we will be sicut Deus [like God], and we will have also solvedthe problem of hunger in the world.

But there is more and worse, to our shame and that of our art. Carbon dioxide,that is, the aerial form of the carbon of which we have up till now spoken: thisgas which constitutes the raw material of life, the permanent store upon which allthat grows draws, and the ultimate destiny of all flesh, is not one of the principalcomponents of air but rather a ridiculous remnant, an ’impurity’, thirty times lessabundant than argon, which nobody even notices. The air contains 0.03 percent;if Italy was air, the only Italians fit to build life would be, for example, the fif-teen thousand inhabitants of Milazzo in the province of Messina. This, on thehuman scale, is ironic acrobatics, a juggler’s trick, an incomprehensible display ofomnipotence-arrogance, since from this ever renewed impurity of the air we come,we animals and we plants, and we the human species, with our four billion dis-cordant opinions, our milleniums of history, our wars and shames, nobility andpride. In any event, our very presence on the planet becomes laughable in geomet-ric terms: if all of humanity, about 250 million tons, were distributed in a layer ofhomogeneous thickness on all the emergent lands, the stature of man would not bevisible to the naked eye; the thickness one would obtain would be around sixteenthousandths of a millimeter.

Now our atom is inserted: it is part of a structure, in an architectural sense; ithas become related and tied to five companions so identical with it that only thefiction of the story permits me to distinguish them. It is a beautiful ring-shapedstructure, an almost regular hexagon, which however is subjected to complicatedexchanges and balances with the water in which it is dissolved; because by now itis dissolved in water, indeed in the sap of the vine, and this, to remain dissolved,is both the obligation and the privilege of all substances that are destined (I wasabout to say ’wish’) to change. And if then anyone really wanted to find out whya ring, and why a hexagon, and why soluble in water, well, he need not worry;these are among the not many questions to which our doctrine can reply with apersuasive discourse, accessible to everyone, but out of place here.

It has entered to form part of a molecule of glucose, just to speak plainly: a fatethat is neither fish, flesh, nor fowl, which is intermediary, which prepares it for itsfirst contact with the animal world but does not authorize it to take on a higherresponsibility: that of becoming part of a proteic edifice. Hence it travels, at theslow pace of vegetal juices, from the leaf through the pedicel and by the shoot tothe trunk, and from here descends to the almost ripe bunch of grapes. What thenfollows is the province of the winemakers: we are only interested in pinpointingthe fact that it escaped (to our advantage, since we would not know how to putit in words) the alcoholic fermentation, and reached the wine without changing its

Beta-decay 47

nature.It is the destiny of wine to be drunk, and it is the destiny of glucose to be

oxidized. But it was not oxidized immediately: its drinker kept it in his liver formore than a week, well curled up and tranquil, as a reserve aliment for a suddeneffort; an effort that he was forced to make the following Sunday, pursuing a boltinghorse. Farewell to the hexagonal structure: in the space of a few instants the skeinwas unwound and became glucose again, and this was dragged by the bloodstreamall the way to a minute muscle fiber in the thigh, and here brutally split into twomolecules of lactic acid, the grim harbinger of fatigue: only later, some minutesafter, the panting of the lungs was able to supply the oxygen necessary to quietlyoxidize the latter. So a new molecule of carbon dioxide returned to the atmosphere,and a parcel of the energy that the sun had handed to the vine-shoot passed fromthe state of chemical energy to that of mechanical energy, and thereafter settleddown in the slothful condition of heat, warming up imperceptibly the air movedby the running and the blood of the runner. ‘Such is life’, although rarely is itdescribed in this manner: an inserting itself, a drawing off to its advantage, aparasitizing of the downward course of energy, from its noble solar form to thedegraded one of low temperature heat. In this downward course, which leads toequilibrium and thus death, life draws a bend and nests in it.

Our atom is again carbon dioxide, for which we apologize: this too is an obliga-tory passage; one can imagine and invent others, but on earth that’s the way it is.Once again the wind, which this time travels far; sails over die Apennines and theAdriatic, Greece, the Aegean, and Cyprus: we are over Lebanon, and the danceis repeated. The atom we are concerned with is now trapped in a structure thatpromises to last for a long time: it is the venerable trunk of a cedar, one of the last;it is passed again through the stages we have already described, and the glucoseof which it is a part belongs, like the bead of a rosary, to a long chain of cellulose.This is no longer the hallucinatory and geological fixity of rock, this is no longermillions of years, but we can easily speak of centuries because the cedar is a tree ofgreat longevity. It is our whim to abandon it for a year or five hundred years: letus say that after twenty years (we are in 1868) a wood worm has taken an interestin it. It has dug its tunnel between the trunk and the bark, with the obstinateand blind voracity of its race; as it drills it grows, and its tunnel grows with it.There it has swallowed and provided a setting for the subject of this story; then ithas formed a pupa, and in the spring it has come out in the shape of an ugly graymoth which is now drying in the sun, confused and dazzled by the splendor of theday. Our atom is in one of the insects thousand eyes, contributing to the summaryand crude vision with which it orients itself in space. The insect is fecundated,lays its eggs, and dies: the small cadaver lies in the undergrowth of the woods,it is emptied of its fluids, but the chitin carapace resists for a long time, almostindestructible. The snow and sun return above it without injuring it: it is buried

48

by the dead leaves and the loam, it has become a slough, a ’thing’, but the deathof atoms, unlike ours, is never irrevocable. Here are at work the omnipresent,untiring, and invisible gravediggers of the undergrowth, the microorganisms of thehumus. The carapace, with its eyes by now blind, has slowly disintegrated and theex-drinker, ex-cedar, ex-wood worm has once again taken wing.

We will let it fly three times around the world, until 1960, and in justificationof so long an interval in respect to the human measure we will point out that itis, however, much shorter than the average: which, we understand, is two hundredyears. Every two hundred years, every atom of carbon that is not congealed inmaterials by now stable (such as, precisely, limestone, or coal, or diamond, orcertain plastics) enters and reenters the cycle of life, through the narrow doorof photosynthesis. Do other doors exist? Yes, some syntheses created by man;they are a title of nobility for man-the-maker, but until now their quantitativeimportance is negligible. They are doors still much narrower than that of thevegetable greenery; knowingly or not, man has not tried until now to compete withnature on this terrain, that is, he has not striven to draw from the carbon dioxidein the air the carbon that is necessary to nourish him, clothe him, warm him, andfor the hundred other more sophisticated needs of modern life. He has not done itbecause he has not needed to: he has found, and is still finding (but for how manymore decades?) gigantic reserves of carbon already organicized or at least reduced.Besides the vegetable and animal worlds, these reserves are constituted by depositsof coal and petroleum: but these too are the inheritance of photosynthetic activitycarried out in distant epochs, so that one can well affirm that photosynthesis isnot only the sole path by which carbon becomes living matter, but also the solepath by which the sun’s energy becomes chemically usable.

It is possible to demonstrate that this completely arbitrary story is neverthelesstrue. I could tell innumerable other stories, and they would all be true: all literallytrue, in the nature of the transitions, in their order and data. The number ofatoms is so great that one could always be found whose story coincides with anycapriciously invented story. I could recount an endless number of stories aboutcarbon atoms that become colors or perfumes in flowers; of others which, fromtiny algae to small crustaceans to fish, gradually return as carbon dioxide to thewaters of the sea, in a perpetual, frightening round-dance of life and death, in whichevery devourer is immediately devoured, of others which instead attain a decoroussemi-eternity in the yellowed pages of some archival document, or the canvas ofa famous painter; or those to which fell the privilege of forming part of a grainof pollen and left their fossil imprint in the rocks for our curiosity; of others stillthat descended to become part of the mysterious shapemessengers of the humanseed, and participated in the subtle process of division, duplication, and fusionfrom which each of us is born. Instead, I will tell just one more story, the mostsecret, and I will tell it with the humility and restraint of him who knows from the

Beta-decay 49

Figure 5.1: Oetzi, the well preserved iceman and oldest known mummy, who was datedwith the radiocarbon technique using accelerator mass spectrometry to have died sometimebetween 3360 - 3100 BC.

start that his theme is desperate, his means feeble, and the trade of clothing factsin words is bound by its very nature to fail.

It is again among us, in a glass of milk. It is inserted in a very complex, longchain, yet such that almost all of its links are acceptable to the human body. It isswallowed; and since every living structure harbors a savage distrust toward everycontribution of any material of living origin, the chain is meticulously broken apartand the fragments, one by one, are accepted or rejected. One, the one that concernsus, crosses the intestinal threshold and enters the bloodstream: it migrates, knocksat the door of a nerve cell, enters, and supplants the carbon which was part of it.This cell belongs to a brain, and it is my brain, the brain of the me who is writing;and the cell in question, and within it the atom in question, is in charge of mywriting, in a gigantic minuscule game which nobody has yet described. It is thatwhich at this instant, issuing out of a labyrinthine tangle of yeses and nos, makesmy hand run along a certain path on the paper, mark it with these volutes thatare signs: a double snap, up and down, between two levels of energy, guides thishand of mine to impress on the paper this dot, here, this one.”

5.2 Carbon-14 dating

The death of the icemen “Oetzi”, whose mummy is shown in Fig. 5.1, has beendated to have occurred between 3360 − 3100 BC using radiocarbon dating basedon the isotope 14C. Figure 5.2 is a picture of the mountain range in the Alps wherethe mummy was found, right on the border between Austria and Italy, sparking adispute about the right to exhibit the remains.

The isotopic composition of atmospheric carbon is 99% stable 126 C, 1% stable 13

6 Cand only 1.2× 10−10% radioactive 14

6 C. The latter decays with a half-life of 5730 a,however, it is constantly reproduced in the atmosphere and then incorporated intoliving matter via photosynthesis due to the impact of cosmic rays on the earth

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Figure 5.2: The ice covered mountain ridge between Italy and Austria where Oetzi wasfound.

atmosphere, see Fig. 5.3. The balance of decay and reproduction accounts for thestable concentration quoted. The death of an organism, however, terminates theincorporation of new carbon atoms and the amount of C-14 isotopes thus graduallydecreases due to radioactive decays. The C-14 decay is a prominent example ofbeta-decay. The daughter product is a 14

7 N nucleus which is the result of theemission of a fast electron. The decay is thus referred to as a β−-decay, in contrastto a β+-decay, where a positively charged positron is emitted. The positron isotherwise similar to an electron, however, being an anti-particle it annihilates withthe nearest electron resulting in the emission of two 511 keV, ‘back-to-back’ γ-rays.

Curiously the energy spectrum of β-decay is a continuous distribution of electron(or positron) energies, as it is shown in Fig. 5.4. This is in stark contrast to energyspectra for α-decay. A typical example is shown in Fig. 5.5. Four discrete linescan be distinguished, which reflect the states of the daughter nucleus (also shownin the figure).

The question arises why the energy spectra of the emitted particle is funda-mentally different for two seemingly similar processes. Wolfgang Pauli proposedthat a continuous energy spectrum can only be in agreement with the conservationlaws for energy and momentum, if the momentum and energy are not only sharedamong two particles as in α-decay, namely the α-particle and the recoil parentnucleus, but among three particles. He proposed that the third particle would beneutral and without rest mass, and named it appropriately ‘neutrino’. Indeed, a‘neutrino’ is being observed in β+-decay, while an anti-neutrino is emitted in β−-decay. As electrons, neutrinos are leptonic particles. Combining a particle with ananti-particle results in a lepton number 0, so that the conservation law of leptonconservation is also fulfilled in both β+- and in β−-decay.

The complete decay equation is thus for 14C:

146 C −→ 14

7 N + e− + ν + ∆E (5.1)

An important example of β+-decay is the decay of the radioisotope 189 F, which is

Beta-decay 51

Figure 5.3: The isotopic composition of carbon and the processes which lead to a stable C-14 concentration in the atmosphere and through photosynthesis in all living plants. (1) GeVprotons and other highly energetic particles from cosmic rays produce (2) particle showers inthe atmosphere. (3) Neutrons from these showers undergo neutron/proton exchange reactionswith 14N nuclei, thus forming radioactive 14C. (4) As carbon dioxide the 14C is (5) incorporatedinto plants. (6) Following the death of the plant, photosynthesis and the incorporation of 14Cstop, and the concentration of this isotope gradually decreases through radioactive β−-decay.

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Figure 5.4: Typical continuous energy spectrum for electrons from β-decay.

Figure 5.5: Energy spectrum for α-particles from the α-decay of 241Am.

Beta-decay 53

Figure 5.6: Schematic of a mass-spectrometer as envisaged by Wilhelm Wien and oftenreferred to as ‘Wien’-filter. The balance of forces can be used to express the mass-chargeratio as a function of measurable quantities, as it is shown below.

often used in nuclear medicine. In this case the decay equation is:

189 F −→ 18

8 O + e+ + ν + ∆E (5.2)

In accelerator mass spectrometry and other mass-spectrometric techniques, iso-topes with different masses are separated from each using combinations of electricand magnetic fields. This is illustrated in Fig. 5.6

Exercise 15 In positron emission tomography (PET) bio-molecules, such as glu-

54

cose, labelled with β-emitters are introduced into the body. The intake of themolecules by different organs or tissues varies, so that physiological phenomenacan be imaged and studied. The positron from the β-decay of the radionuclide an-nihilates with a nearby electron and two 511 keV γ-photons are emitted in oppositedirections, so that the location of the labelled molecules can be detected.

(a) Write down the decay equations and the half-lives for the following radioiso-topes used for PET imaging: 15O, 13N , 11C, and 18F .

(b) Why do these radionuclides have to be produced on-site using for example acyclotron facility in the hospital.

(c) Show that using a mass spectrometer the radioactive 15O can be separated fromthe stable 16O. What magnetic field is required to select 1 MeV 15O 3+ ionswith this mass spectrometer, when the electric field E is perpendicular to themagnetic field B and has an electric field strengths of E = 103 kV/m ?

Chapter 6

Gamma-decay

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6.1 Nuclear medicine: history and modern practice

Gamma-photons are extensively used in nuclear medicine for diagnosis and therapy.A typical radionuclide for such applications is 99Tc which is well suited, since ithas a meta-stable, excited state which gamma-decays with a half-life of 6 hours.

Nuclear medicine using gamma-rays and other ionising radiation has a longhistory. An account given by Perkins [Perkins 1995] is reproduced here:

“During the first half of the twentieth century radioactivity was used for a widevariety of purposes. The main radionuclide was radium-226 which was available insmall quantities, being formed continuously by the decay of uranium-238 in pitch-blende ore. Radium-226 has a half-life of 1600 years and decays by the emission ofalpha particles and gamma radiation (4.8 MeV alpha particles, 0.186-0.601 MeVgamma rays). This was used as the activating agent in luminous watches, clockfaces and dials produced up to the early 1960s. In some cases the radiation dosesresulting from such items could be significant, for example the skin dose directlyunder the face of a pocket watch could be as high as 1.65 Gy. Radium was alsoused for instrument panels in aeroplanes, scientific instruments and survey metersmany of which were in widespread use by the military. Nowadays many of theseitems such as clocks and watches ore highly collectable items and are commonlyfound by antique dealers at auctions and markets. Buffing and polishing of suchitems by dealers to polish brass ornamental instruments could result in contami-nation of radioactivity to surrounding surfaces and objects. Today radioactivityis still used for a number of industrial and domestic purposes. Radionuclides areused for several industrial applications such as in industrial radiography and non-destructive testing. Gamma ray sources are used for the sterilization of medicalinstruments, thickness gauging, automated production control lines, and tracer ap-plications. The main household items familiar to most people ore smoke detectorswhich mostly contain Americium-241 (241Am).

The early medical use of radioactivity was mainly for treatment. The firstrecorded medical use of a radioactive substance took place around 1901 whenDanlos and Block placed radium in contact with a tuberculous skin lesion. Thelong half-life of radium-226 (1600 years) provided on effectively constant rate ofgamma radiation which would penetrate deep into the body, This source of gammarays was more reliable than those which could be produced from the early x-raytubes. Radium also had the advantage that it could be introduced into the bodyin tubes or needles so as to cause intense local irradiation of tumours with littledamage to the skin, this often being the limiting factor with the therapeutic ap-plication of x-rays. The inability to provide uniform irradiation of tumours wasa particular problem, as was the necessity to perform a second operation to re-move the implant at the end of the period of treatment. Later treatments werecarried out using the gas radon-222 which is a daughter product from radium-

Gamma-decay 57

226. The gas was filled into small metal seeds or tubes and then inserted into thebody. The short 3.85 day physical half-life of radon meant that the tubes couldbe left in place permanently following the treatment period. The early successesof radiation therapy led to an almost religious belief in the therapeutic propertiesof radiation. Radioactivity became synonymous with good health and it becamefashionable to visit spa resorts in order to take the waters containing natural ra-dioactivity. An early American radiologist was quoted as saying radioactive waterwas a gift of God, curing practically all nervous disorders. Claims were made thatradioactivity was a cure for virtually all known ailments. The product descriptionof one preparation, ’Radiothor or perpetual sunshine’, stated-that radiation was’a physical means of re-energizing weak and inactive cells with millions of rays’. Itwas a common belief of the time that deep radium therapy utilized the destructiveproperties of rays whereas mild radium therapy was based on the stimulating prop-erties of small doses. A large number of devices became commercially available toproduce home brews of radium water. Such cures were claimed to be valuable inthe treatment of anaemia, arteriosclerosis, arthritis, catarrh, diabetes, goitre, highblood pressure, the menopause, menstrual disorders, nephritis, neuritis, nervousconditions, obesity, prostatitis, rheumatism, senility, sexual conditions and skindisorders. In some cases a number of bogus products were sold which claimed tocontain radioactivity but in fact contained common chemicals or samples of earth.One such preparation which contained no radioactivity was named Hearium withthe intention of curing ear problems. In some cases the makers and distributors ofthese fake remedies were prosecuted, fined and imprisoned because these remediesdid not contain radioactivity. A number of devices used for radium cures still turnup from time to time, in some cases causing concern when the nature of the con-tents are realized. For example a number of radium water siphons have been pickedup by antique dealers. Other unusual products include radioactive corsets for thetreatment of back ache and a more unusual example of a curative device was theQ-ray electro-compress. Such a compress was bought by a school teacher at a jum-ble sale in Mansfield, Nottinghamshire UK, in 1986. This dry compress containednatural uranium ore sewn into an electric blanket and was claimed to combine thenatural properties of radioactivity with heat. At the request of the local Policethis particular blanket was retrieved by the local Medical Physics Department atQueen’s Medical Centre, Nottingham, for disposal and was found boxed completewith manufacturer’s literature including medical testimonies and photographs ofthe compress in use at St Thomas’s Hospital London.

The early beliefs in the therapeutic properties of radiation lacked any real sci-entific basis. However as knowledge increased a number of different radionuclidesand radioactive compounds become available and the concept of targeting radi-ation to sites of disease in the body gradually became a reality. It is also nowapparent that the utilization of natural and artificial radionuclides has made an

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unparalleled contribution to the understanding of human physiology and pathol-ogy. In 1923 Hevesey introduced the use of radionuclides as biological tracers bystudying the absorption of radioactive lead in plants. He later used phosphate 32Pas a tracer to study the metabolism of phosphorus in the rat. Modern diagnosticnuclear medicine techniques are based on this concept. Unlike most other imag-ing modalities which provide information concerning human anatomy or structure,radionuclide counting and imaging techniques provide information on tissue andorgan function. Imaging techniques using a gamma camera are capable of mappingout the biodistribution of an administered radiolabelled compound providing infor-mation concerning organ and tissue and function. The early development of clinicalnuclear medicine in the 1940s was largely based on the use of radioisotopes of io-dine. Iodine-130 was first used for the investigation of patients with thyroid disor-ders. Artificially produced iodine- -131 subsequently became the main radionuclidefor the investigation and treatment of patients with thyroid disease. Iodine, withan atomic number of 53, is the heaviest element required for human metabolism.The body is unable to differentiate between the radioactive and non-radioactiveisotopes and they are therefore metabolized in an identical manner. Substitutionof a radioisotope of iodine for a naturally occurring atom provides a means of mon-itoring iodine distribution in the body and subsequently iodine metabolism in thethyroid gland. Iodine is therefore unique in this respect. There are very few bio-chemicals or drugs which contain an element suitable for isotopic substitution of agamma emitter suitable for gamma camera imaging. It is therefore necessary to in-corporate the radionuclide chemically without altering the biological properties ofthe material and ensuring that the compound remains stable after administration.In nuclear medical diagnosis we are principally concerned with gamma rays for ex-ternal imaging and detection within radiation sample counters although some betaemitters are administered for diagnostic purposes. Some in vitro diagnostic pro-cedures utilizing beta emitters are performed, and in particular there has recentlybeen an increase in the use of phosphorus-32 for autoradiographic studies suchas genetic mapping. In radiotherapy procedures the cell-killing properties of betaemitters are used, for example using iodine-131 for the treatment of thyrotoxico-sis, phosphorus-32 for the treatment of polycythaemia and strontium-89 for bonepain palliation. Alpha particles are seldom used in medicine and are restrictedto a few research applications. In each clinical case the administered activity isprescribed by the radiotherapist or oncologist and calculated by the medical physi-cist or therapy radiographer. Patients who receive therapeutic amounts of activityare admitted to specialized hospital suites and given special instructions for theduration of the period of active treatment.”

Gamma-decay 59

Figure 6.1: Single Photon Emission Tomography (left) and x-ray Computer Tomography(right) cross-sectional images of the brain of a patient diagnosed with Ischaemic Stroke. Thereduction in blood flow in the left part of the brain is clearly visible with Single PhotonEmission Tomography, whereas x-ray Computer Tomography cannot give an unambiguousdiagnosis.

6.2 Gamma-rays in nuclear medicine

The strength of Single Photon Emission Tomography (SPET) may be appreciatedwhen compared to the well known Computer Tomography (CT). Figure 6.1 showsas an example the tomography images obtained with 99mTc SPET and x-ray CTfor a patient diagnosed with ‘Ischaemic Stroke’, a condition where a blood clotblocks the blood circulation in parts of the brain. It is obvious that SPET canproduce much more definite evidence of this condition than CT might do. Whilethe tomographic image processing is similar, the two techniques differ in severalways.

1. In the case of SPET the source of the γ-photons is internal. This is achievedby injecting a bio-molecule such as glucose, which has been labelled with aradioisotope. CT employs an external source of x-rays.

2. While the energy of the γ-photon in SPET is well defined (for 99mTc it is143 MeV), CT uses a broad energy spectrum of x-rays.

3. The labelled bio-molecule is “selective” and accumulates in certain tissue ororgan, so that even physiological processes can be studied in real time. Incontrast, CT produces a “shadow”-images of the body, relying on differencesin x-ray absorption throughout the body. Since x-ray absorption generally

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changes little, when organs malfunction, CT is not very sensitive to suchpredicaments, while, for example, it is ideal for the identification of bonefractures, because of the large difference in x-ray absorption between boneand tissue.

4. X-ray CT is less complex than SPET, since the need to produce very activeshort-lived radioisotopes on-site and radiochemistry are not required.

A schematic illustration of the tomography procedure, as it is applied in thecase of SPET, is shown in Fig. 6.2. Figure 6.3 shows a picture of a typical SPETfacility.

In a hospital situation the 99m43 Tc is obtained from a technetium generator (see

Fig. 6.4) containing 9942Mo, which decays via β−-decay (t1/2 = 66 h) to 99

43Tc with81% of the decays populating an excited, meta-stable state of this nuclide with anexcitation energy of 143 MeV. The 143 MeV γ-rays employed in SPET imagingcorrespond to transitions from this excited state, 99m

43 Tc, to the long-living groundstate of 99

43Tc. Since neither mass- nor atomic-number change, this nuclear transfor-mation can only proceed via γ-decay, which releases energy as a single photon and,to conserve momentum and energy, as the kinetic recoil energy of the technetiumnuclide.

6.3 The equivalence of energy and mass:

It is not immediately obvious, from where the γ-ray photons emitted by 99mTcsource their energy Eγ = 143 MeV. Figure 6.5 shows the energy difference betweenthe mother nuclide 99Mo and its daughter product 99Tc including all intermediateexcited states of 99Tc, which are also populated by the β−-decay of 99Mo. It isapparent that 99Tc is energetically favoured compared to 99Mo. However, sinceenergy conservation has to be fulfilled, this energy difference has to be accountedfor. It is a fundamental outcome of Albert Einstein’s General Theory of Relativity,that the energy difference between 99Mo and 99Tc corresponds to a slight massdifference between these nuclides, and that energy is conserved, because of theprinciple of the equivalence of energy E and mass m, famously expressed as

E = mc2 (6.1)

which has general validity.Equation 6.1 can be applied to express masses in terms of energy per light

velocity-squared, since 1 atomic-mass-unit (1 amu or 1 u) equals 931.5 MeV/c2.In this form the masses of the unbound proton, neutron, and electron are given as

mp = 938.3 MeV/c2

mn = 939.6 MeV/c2 (6.2)

me = 0.511 MeV/c2

Gamma-decay 61

Figure 6.2: Principle of tomography as applied in SPET.

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Figure 6.3: A SPET facility.

Exercise 16 (a) Calculate the mass-difference of the nuclei 9942Mo and 99

43Tc inunits of MeV/c2. (b) How do the masses of 99

42Mo and 9943Tc compare with the

combined mass of 42 protons and 57 neutrons, and 43 protons and 56 neutrons,respectively ? Express the difference in units of MeV/c2. [According to the chart ofnuclides, “http://www2.bnl.gov/ton/”, the relevant atomic masses are m(99

42Mo) =98.90772 amu and m(99

43Tc) = 98.90626 amu, respectively.]

Part (a) of the Exercise shows that 99Tc is lighter than 99Mo by ∆m = 1.87 MeV/c2.According to Einstein’s principle of the equivalence of energy and mass this massdifference accounts for the rest mass of the particles emitted during the β−-decayof 99Mo, an electron and an anti-neutrino, and the kinetic energies of these parti-cles plus the total energy of all γ-photons emitted subsequently until the groundstate of 99Tc has been attained. Since neutrinos do not have any rest mass, theonly mass removed is that of the β−-electron. The rest is taken off in the form ofenergy ∆E, which can thus be calculated as

∆E = (1.87 MeV − 0.511 MeV) ∼= 1.36 MeV (6.3)

The decay equation can be written, augmented by this energy difference ∆E, as

9942Mo −→ 99

43Tc + e− + ν + ∆E (6.4)

The energy difference ∆E = Q, which can be positive or negative, is commonlyreferred to as the Q-value of a nuclear decay or a nuclear reaction.

The exercise illustrates that the electron mass and the kinetic energy carried bythe e−, the ν, and the recoiling nuclide has its origin in the mass difference betweenthe two nuclei.

Gamma-decay 63

Figure 6.4: Cut-away view of a 99m43 Tc generator box.

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Figure 6.5: The energy differences between 99Mo and 99Tc including all intermediate excitedstates of 99Tc (Level diagram).

Part (b) of the Exercise shows that the total mass of the sum of all nucleonmasses, i.e. the sum of all proton and neutron masses, is larger for both nuclidesthan their actual mass. This difference is referred to as the mass defect ∆m. The‘defective’, or missing mass accounts for the nuclear binding energy B of the nu-cleus. For 99Mo it is BMo = 853.4 MeV/c2 and for 99Tc it is BTc = 855.3 MeV/c2.It is apparent that the technetium nucleus is bound more strongly than the molyb-denum nucleus with a difference of ∼ 1.9 MeV in binding energy, which is inagreement with the mass difference calculated in part (a) of the exercise.

The fact that the binding energy of the technetium nuclide is larger is consistentwith the observation that 99Mo decays to 99Tc, thus attaining more stability. Asother physical systems, nuclei ‘aim’ for minima of the potential energy.

Division by the nucleon number 99 shows that the binding energy per nucleonB/A is for both examples of the order of 8 MeV per nucleon. This value holdsapproximately for all isotopes with the exception of nuclei lighter than 16O. Agraph illustrating B/A as the function of mass number A is shown in Fig. 6.6.

Exercise 17 Apply the principle of the equivalence of energy and mass.

1. Calculate the Q-values for the following decay processes and explain why someof them are not observed: (a) α-decay of 235

92 U (b) p-decay of 23592 U

2. Calculate the Q-value for the β+-decay of 189 F and determine the mass defects

of 189 F and 18

8 O.

Gamma-decay 65

Figure 6.6: An illustration of the change of B/A with mass number A. B/A peaks at 58Fe.Superimposed on the smooth function shown is a fine structure due to nuclear shell effects.

3. The α-decay of 22688 Ra to 222

86 Rn involves four discrete α-lines. The most ener-getic α-particle has an energy approaching 4.785 MeV. Four discrete γ-lineswith energies of 0.601 MeV, 0.415 MeV, 0.186 MeV, and 0.262 MeV, respec-tively, are observed in coincidence with the α-decays or soon after. Draw alevel scheme.

6.4 Scintillation detectors

The observation of γ-rays in medical and other applications requires sophisticateddetection systems. An important example are scintillation detectors, which arebased on materials, such as sodium iodide, which scintillate, i.e. when they absorbγ-radiation, they emit optical photons. The number of photons emitted is propor-tional to the energy absorbed. Thus, when the photons are counted, the energy ofthe original γ-ray can be determined. This can be achieved using the photocath-ode of a photomultiplier by taking advantage of the photoelectric effect. Abovea certain energy, each emitted photo release an electron from the photocathode.Thus the number of electrons released from the photocathode is proportional tothe number of emitted photons, and therefore also proportional to the energy ofthe original γ-ray.

The number of electrons is too small that their combined charge could be mea-sured directly. Their number is therefore multiplied using a sequence of dynodes

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which are positively charged with increasing bias. Impact ionization results insecondary electrons, which in turn multiply at the next dynode. Eventually ameasurable signal can be obtained.

While the detection efficiency of scintillation detectors is excellent, their energyresolution is much poorer than that of semiconductor detectors. This due to thestatistical nature of the scintillation process. However, in medical applications theenergy of the important γ-rays is usually well known, so that energy resolution isnot of importance, whereas image quality improves with the number of γ-photonsdetected. Scintillation detectors are thus the system of choice for techniques suchas SPET or PET.

In order to achieve position-sensitivity and obtain two-dimensional images, alarge sodium iodide crystal is backed by an array of photomultiplier tubes. Foreach γ-ray detected the sum of the responses from all photomultiplier tubes isproportional to the energy of the γ-photon, while the tube with maximum responseindicates the location of its impact. For a specific energy observation over timethus produces a two-dimensional image of the γ-ray activity inside a patient andthe distribution of a particular radioisotope, such as 99mTc, can be mapped.

Such system is known as gamma camera, or Anger camera, named after the de-veloper Hal Anger. In order to ensure that gamma-rays are incident perpendicularto the crystal surface and thus ensure a reliable correlation between the origin ofthe γ-photon and the point of observation, the crystal is covered with a thick colli-mator (typically lead, because of its large absorption coefficient), which has up to35,000 parallel holes. This ensures that ‘stray γ-photons’, which leave the patientat angles much smaller or larger than 90◦, are suppressed. A magnification effectcan be achieved by using a special collimator, where the orientation of the channelsassociated with the holes is not parallel but such that they converge towards thepatient.

6.5 Summary

It may be summarized that nuclear transmutations such as α- and β-decay aredriven by increases in nuclear binding energy. The binding energy of a nucleusis equal to its mass defect, which is a consequence of the equivalence of energyand mass first suggested by Einstein and well-established experimentally. Nucleartransmutations do not necessarily lead directly to the ground state of a nuclide,but can populate excited states with varying life-times. Such states decay to theground state via γ-decay and the emission of a γ-photon. An important examplefor this process is the decay of 99mTc, which is used extensively in nuclear medicine,in particular for Single Photon Emission Tomography.

Chapter 7

Appendix

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68

7.1 First in-class test

In 20 minutes address the following questions. Calculators are permitted, however,no written documents or books are allowed.

1. [4 marks] An exercise device is used to strengthen the leg muscles, see Fig. 7.1.Calculate the force M exerted by the muscle in the upper leg, when movingthe foot forward to lift the weight. What is the reaction force exerted ontothe joint?

2. [4 marks] In about half-a-page comment on the material properties of ourbones. In what ways does bone out-perform other possible biological or syn-thetic material choices for the human skeleton?

3. [4 marks] (a) Sketch the human eye, label your drawing and briefly explain therole of important parts. (b) Calculate the change in direction for a light-rayincident on the cornea at an angle of 25◦ with respect to the surface normal(refractive indices: nair ' 1, ncornea ' 1.37).

4. [3 marks] At the nearest houses the traffic on a motorway produces noise withan intensity of 10−6 W/m2. An expansion of the motorway with additionallanes is expected to double the traffic. Calculate the intensity level in dBbefore and after the motorway expansion (Intensity level is defined through1 dB = 10 · log (I/I0), where I0 = 10−16 W/cm2).

Figure 7.1: Mechanical representation of an exercise to strengthen the leg muscles [Mc-Cormick & Elliot 2001].

Appendix 69

Exercise 1

70

Exercise 2

Appendix 71

Exercise 3

72

Appendix 73

Exercise 4

74

Exercise 7

Appendix 75

Exercise 9 Yellow spot and blood vessels When looking at a pin hole hole ina piece of paper at a distance of about 10 cm in front of an intense light source, theblood vessels of the retina are visible as an intricate black network. In the centreof the image a (black) spot can be seen, which is the yellow spot. The yellow spotis an area of the retina, about 1 mm in diameter, right on the optical axis, wherethe eye is most sensitive.

Exercise 10 Blind spot At the point where the optic nerve passes through theretina, it is not sensitive. The blind spot can be identified by concentrated lookingat a certain object with one eye closed. For the left eye the blind spot is about 15◦

to the left and for the right eye it is about 15◦ to the right of the nose, respectively.Any objects located at those angles disappear from view.

Exercise 11 Upward and downward rays on strong light sources Whenlooking at a strong light source (maybe a street light at night) it often appears thatlight-rays extend downwards and/or upwards from it, but not to the either side.The reason for this phenomenon is the diffraction of incident light in the meniscusof the tear liquid just underneath the upper eyelid (perception of upward rays) andjust above the lower eye lid (perception of downward rays).

Exercise 12 Why do curtains obscure the view one way, but not theother? Eye and brain can distinguish objects, when their brightness ratio is aboveabout 5% (Fechner’s law). When a curtain, during the day, is in bright sunshineall objects behind it reflect light with much lower absolute intensity, so that therelative differences are less than 5% on the scale defined by the bright curtain. Atnight, with light sources inside the room, the situation is somewhat reversed.

Exercise 13

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Exercise 14

Appendix 77

78

Appendix 79

Exercise 15

80

Appendix 81

Exercise 16

82

Appendix 83

Exercise 17

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References

[Halliday et al. 1993] David Halliday, Robert Resnick, Jearl Walker, Funda-mentals of Physics, John Wiley & Sons, USA, ISBN0-471-57578-x (1993)

[Levi 1975] Primo Levi, excerpt from The Periodic Table, in TheFaber Book of Science, ed. John Carey, Faber & FaberLtd, London, ISBN 0-571-16352-1, (1995) 338 - 344

[McCormick & Elliot 2001] Andrew McCormick and Alexander Elliot, HealthPhysics, Cambridge University Press, UK, ISBN 0-521-78726-2 (2001)

[Perkins 1995] A.C. Perkins, Nuclear Medicine: Science and Safety(1995)

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