Atoms and Light

Abstract

This chapter describes the biological properties of infrared, visible, and ultraviolet light. Photons are emitted or absorbed when atoms or molecules change energy levels, leading to the techniques of infrared spectroscopy and Raman scattering. Scattering and absorption are described by the cross section. When scattering dominates, light is governed by the diffusion equation. Photons can be absorbed and emitted in a continuous range of frequencies, leading to thermal radiation. Blue and ultraviolet light are used for therapy, and can also be harmful to the skin and eyes. Lasers heat tissue, a process that is modeled by the bioheat equation. The chapter closes with a discussion of vision.

Appendices

Symbols Used in Chapter 14

Table 10

Problems

14.2.1 Section 14.1

Problem 1

The velocity of light $C$ depends on the parameters $\epsilon _{0}$ and $\mu _{0}.$ Use dimensional analysis to find what the dependence must be. Insert numerical values to obtain $c.$

Problem 2

An einstein is 1 mol of photons. Derive an expression for the energy in an einstein as a function of wavelength. Express the answer in kilocalories and the wavelength $\lambda$ in nanometers.

14.2.2 Section 14.3

Problem 3

Use Eq. 14.8 to derive Eq. 14.9.

Problem 4

(a) Starting with Eq. 14.8, derive a formula for the hydrogen atom spectrum in the form

$$ \frac{1}{\lambda}=R\left[ \frac{1}{n^{2}}-\frac{1}{m^{2}}\right] , $$

where $n$ and $m$ are integers. $R$ is called the Rydberg constant. Find an expression for $R$ in terms of fundamental constants.

(b) Verify that the wavelengths of the spectral lines a–d at the top of Fig. 14.3 are consistent with the energy transitions shown at the bottom of the figure.

Problem 5

The Lyman series, part of the spectrum of hydrogen, is shown at the top of Fig. 14.3 as the line labeled a and the band of lines to the left of that line. Create a figure like Fig. 14.3, but which shows a detailed view of the Lyman series. Let the wavelength scale at the top of your figure range from 0 to 150 nm, as opposed to 0–2$~\upmu$m in Fig. 14.3. Also include an energy level drawing like at the bottom of Fig. 14.3 and indicate which transitions correspond to which lines in the Lyman spectrum. Indicate the shortest possible wavelength in the Lyman spectrum, show what transition that wavelength corresponds to, and determine how this wavelength is related to the Rydberg constant.

Problem 6

The left side of Fig. 14.1 shows the emission of a photon during a transition from an initial state with energy $E_i$ to a final one with energy $E_f$. Usually the Boltzmann factor ensures that the population of the initial state is less than the final state. In some cases however, when the initial state is metastable, one can create a population inversion. Photons with energy $h\nu$ corresponding to the energy difference $E_i - E_f$ can produce stimulated emission of other photons with the same energy, a type of positive feedback. Lasers work on this principle. Suppose a laser is made using two states having an energy difference of 1.79 eV. What is the wavelength of the output light? What color does this correspond to? Lasers have many uses in medicine (Peng et al. 1993).

14.2.3 Section 14.4

Problem 7

Estimate $\hslash ^{2}/2I$ for an HCl molecule. What would the spacing of rotational levels be?

Problem 8

An inulin molecule has a molecular weight of 4000 dalton (that is, 1 mol has a mass of 4000 g). Assume that it is spherical with a radius of 1.2 nm. What is the angular frequency $\omega$ of a photon absorbed when its rotational quantum number changes from 10 to 11? The moment of inertia of a sphere rotating about an axis through its center is $I=(2/5)mR^{2}$.

Problem 9

The rotational spectrum of HCl contains lines at 60.4, 69.0, 80.4, 96.4, and 120.4$~\upmu$m. What is the moment of inertia of an HCl molecule?

Problem 10

Consider a combined rotational–vibrational transition for which $R$ goes from 1 to 0 while $V$ goes from $V$ to $(v-1)$. Find the frequencies of the photons emitted in terms of the moment of inertia of the molecule $I$, the angular frequency of vibration of the atoms in the molecule $\omega$, and the quantum number $V$.

Problem 11

A rotating molecule emits photons when the rotational quantum number changes by $1$. Find the ratio of the angular frequency of the photons, $\omega _{\text {phot}}$, to the angular frequency of rotation of the molecule, $\omega _{\text {rot}}$, as a function of the rotational quantum number $R$.

14.2.4 Section 14.5

Problem 12

A beam with 200 particles per square centimeter passes by an atom. The particles are uniformly and randomly distributed in the area of the beam.

• (a) Fifty particles are scattered. What is the total scattering cross section?

• (b) Ten particles are scattered in a cone of 0.1 sr solid angle about a particular direction. What is the differential cross section in m$^{2}\,$sr$^{-1}$?

Problem 13

The differential scattering cross section for a beam of x-ray photons of a certain energy from carbon at an angle $\theta$ is $50\times 10^{-30}$ m$^{2}\,$sr$^{-1}$. A beam of $10^{5}$ photons strikes a pure carbon target of thickness 0.3 cm. The density of carbon is $2~$g cm$^{-3}$, and the atomic weight is 12. The detector is a circle of 1 cm radius located 20 cm from the target. How many scattered photons enter the detector?

Problem 14

Photochemists often use the extinction coefficient $e$, defined by $\mu _{a}=eC$, where $C$ is the concentration in moles per liter. This assumes the substance being measured is dissolved in a completely transparent solvent.

• (a) What are the units of the extinction coefficient?

• (b) What is the conversion between the extinction coefficient and the absorption cross section?

Problem 15

Suppose that the absorption coefficient in some biological substance is 5 m$^{-1}$. Make the very crude assumption that the substance has the density of water and a molecular weight of 18. What is the absorption cross section?

Problem 16

For blue light ($\lambda =470~$ nm), the attenuation coefficient in air is about $2\times 10^{-5}~$m$^{-1}$, and the attenuation coefficient in pure water is about $5\times 10^{-3}~$m$^{-1}$. Calculate the distance that blue light must pass through air and water before the intensity is reduced to 1% of the original intensity. Compare these distances to the thickness of the atmosphere and the depth of the ocean. Do you think that aquatic plants can use photosynthesis effectively at the bottom of the ocean? For more on the differences between the optical properties of air and water, see Denny (2007).

14.2.5 Section 14.6

Problem 17

(a) Find the slope of $\log R$ versus $T$ in Eq. 14.30. What is its value for large times?

(b) What can be determined from the time when $R$ has its maximum value? (Hint: $R$ has a maximum when $\log R$ has a maximum.)

Problem 18

The result of one set of infrared measurements in human calf (leg) muscle gave a total scattering coefficient $\mu _{s}=8.3~$cm$^{-1}$ and an absorption coefficient $\mu _{a}=0.176~$cm$^{-1}$.

• (a) What fraction of the photons have not scattered in passing through a layer that is 8 $\upmu$m thick? (This corresponds roughly to the size of a cell.)

• (b) On average, how many scattering events take place for each absorption event?

• (c) What is the cross section for scattering per molecule? For this estimate, assume the muscle consists entirely of water, with molecular weight 18 and density $10^{3}~$kg m$^{-3}$.

Problem 19

Consider light with fluence rate $\varphi _0$ continuously and uniformly irradiating a half-infinite slab of tissue having an absorption coefficient $\mu _a$ and a reduced scattering coefficient $\mu ^\prime _s$. Divide the photons into two types: the incident ballistic photons that have not yet interacted with the tissue, and the diffuse photons undergoing multiple scattering. The diffuse photon fluence rate, $\varphi$, is governed by the steady state limit of the photon diffusion equation (Eq. 14.27). The source of diffuse photons is the scattering of ballistic photons, so the source term in Eq. 14.27 is $s = \mu ^\prime _s \exp (-z/\lambda _{\text {unatten}})$, where $z$ is the depth below the tissue surface. At the surface ($z=0$), the diffuse photons obey the boundary condition $\varphi = 2 D d\varphi /dz$.¹⁷

• (a) Derive an analytical expression for the diffuse photon fluence rate in the tissue, $\varphi (z)$.

• (b) Plot $\varphi (z)$ versus $z$ for $\mu _a=0.08~$mm$^{-1}$ and $\mu ^\prime _s=4~$mm$^{-1}$.

• (c) Evaluate $\lambda _\text {unatten}$ and $\lambda _\text {diffuse}$ for these parameters.

14.2.6 Section 14.7

Problem 20

Carry out the averages leading to Eq. 14.31.

Problem 21

If yellow light from a source has a coherence time of $10^{-8}\operatorname {s}$, how many cycles are there in the wave?

Problem 22

What coherence time is needed for a spatial resolution of $1\upmu {\rm m}$?

Problem 23

An infrared transition involves an energy of 0.1 eV. What are the corresponding frequency and wavelength? If the Raman effect is observed with light at 550 nm, what will be the frequencies and wavelengths of each Raman line?

Problem 24

A Raman spectrum has a line at 500 nm with subsidiary lines at 400 and 667 nm. What is the wavelength of the corresponding infrared line?

14.2.7 Section 14.8

Problem 25

Sodium is introduced into a flame at 2500 K. What fraction of the atoms are in their first excited state? In their ground state? (Remember that the characteristic sodium line is yellow.) If the flame temperature changes by 10 K, what is the fractional change in the population of each state? Which method of measuring sodium concentration is more stable to changes in flame temperature: measuring the intensity of an emitted line or measuring the amount of absorption?

Problem 26

(a) Show that the maximum of the thermal radiation function $W_{\lambda }(\lambda ,T)$ occurs at a wavelength such that $e^{x}(5-x)=5$, where $x=hc/(\lambda _{\text {max}}k_{B}T)$. Verify that $x=4.9651$ is a solution of this transcendental equation, so that

$$ T\lambda_{\text{max}}=\frac{hc}{4.9651k_{B}}. $$

(b) Similarly, show that

$$ \frac{\nu_{\text{max}}}{T}=\frac{2.82144k_{B}}{h} $$

and that $\lambda _{\text {max}}\nu _{\text {max}}=0.57c$.

Problem 27

Let $W_\nu (\nu ) = A \nu (\nu _0 - \nu )$ for $\nu <\nu _0$, and $W_\nu (\nu ) = 0$ otherwise.

• (a) Plot $W_\nu (\nu )$ versus $\nu .$

• (b) Calculate the frequency corresponding to the maximum of $W_\nu (\nu )$, called $\nu _\text {max}.$

• (c) Let $\lambda _0 = c/\nu _0$ and $\lambda _\text {max} = c/\nu _\text {max}.$ Write $\lambda _\text {max}$ in terms of $\lambda _0.$

• (d) Integrate $W_\nu (\nu )$ over all $\nu$ to find $W_\text {tot}$.

• (e) Use Eqs. 14.36 and 14.37 to calculate $W_\lambda (\lambda )$.

• (f) Plot $W_\lambda (\lambda )$ versus $\lambda$.

• (g) Calculate the wavelength corresponding to the maximum of $W_\lambda (\lambda )$, called $\lambda ^*_\text {max}$, in terms of $\lambda _0$.

• (h) Compare $\lambda _\text {max}$ and $\lambda ^*_\text {max}.$ Are they the same or different? If $\lambda _0$ is 400 nm, calculate $\lambda _\text {max}$ and $\lambda ^*_\text {max}$. What part of the electromagnetic spectrum is each of these in?

• (i) Integrate $W_\lambda (\lambda )$ over all $\lambda$ to find $W^*_\text {tot}.$ Compare $W_\text {tot}$ and $W^*_\text {tot}$. Are they the same or different?

Problem 28

Integrate Eq. 14.33 over all wavelengths to obtain the Stefan–Boltzmann law, Eq. 14.34. You will need the integral

$$ \int_{0}^{\infty}\frac{x^{3}dx}{e^{x}-1}=\frac{\uppi^{4}}{15}. $$

Problem 29

Two parallel surfaces of area $S$ have unit emissivity and are at temperatures $T_{1}$ and $T_{2}$ [$T_{1}>T_{2}$, panel (a)]. They are large compared to their separation, so that all radiation emitted by one surface strikes the other. Assume that radiation is emitted and absorbed only by the two surfaces that face each other. Let $P_{0}$ be the energy lost per unit time by body 1. A new sheet of perfectly absorbing material is introduced between bodies 1 and 2, as shown in panel (b). It comes to equilibrium temperature $T$. Let $p$ be the net energy lost by surface 1 in this case. Find $P/P_{0}$ in terms of $T_{1}$ and $T_{2}$.

Problem 30

The sun has a radius of $6.9\times 10^{8}\,$m. The earth is $149.5\times 10^{9}\,$m from the sun. Treat the sun as a thermal radiator at 5800 K and calculate the energy from the sun per unit area per unit time striking the upper atmosphere of the earth (the solar constant). State the result in W m$^{-2}$ and cal cm$^{-2}\,$min$^{-1}$.

Problem 31

If all the energy received by the earth from the sun is lost as thermal radiation (a poor assumption because a significant amount is reflected from cloud cover), what is the equilibrium temperature of the earth?

14.2.8 Section 14.9

Problem 32

Show that an approximation to Eq. 14.41 for small temperature differences is $w_{\text {tot}}=SK_{\text {rad}}(T-T_{s})$. Deduce the value of $K_{\text {rad}}$ at body temperature. Hint: Factor $T^{4}-T_{s}^{4}=(T-T_{s})(\cdots )$. You should get $K_{\text {rad}}=6.76~$W m$^{-2}~$K$^{-1}$.

Problem 33

What fractional change in $W_{\lambda }(\lambda ,T)$ for thermal radiation from the human body results when there is a temperature change of 1 K at 5$~\upmu$m? 9 $\upmu$m? 15$~\upmu$m?

14.2.9 Section 14.10

Problem 34

(a) Suppose that the threshold for erythema caused by sunlight with $\lambda =300~$nm is $30~$J m$^{-2}$. Does this suggest that the result is thermal (an excessive temperature increase) or something else, like the photoelectric effect or photodissociation? Make some reasonable assumptions to estimate the temperature rise.

(b) The energy in sunlight at all wavelengths reaching the earth is $2$ cal cm$^{-2}~$min$^{-1}$. Suppose that the total body area exposed is $0.6~$m$^{2}$. What would be the temperature rise per minute for a 60 kg person if there were no heat-loss mechanisms? Compare the rate of energy absorption to the basal metabolic rate, about $100~$W.

Problem 35

Suppose that the energy fluence rate of a parallel beam of ultraviolet light that has passed through thickness $x$ of solution is given by $\psi =\psi_0 e^{-{\upmu_a}x}$. (Scattering is ignored.) The absorption coefficient $\mu _{a}$ is related to the concentration $C$ (molecules cm$^{-3}$) of the absorbing molecules in the solution by $\mu _{a}=eC$. Biophysicists working with ultraviolet light define the dose rate to be the power absorbed per molecule of absorber. (This is a different definition of dose than is used in Chap. 15.) Calculate the dose rate for a thin layer $(\mu _{a}x\ll 1)$.

Problem 36

A beam of photons passes through a monatomic gas of molecular weight $A$ and absorption cross section $\sigma$. Ignore scattering. The gas obeys the ideal gas law, $pV=Nk_{B}T$.

• (a) Find the attenuation coefficient in terms of $\sigma$, $p$, and any other necessary variables.

• (b) Generalize the result to a mixture of several gases, each with cross section $\sigma _{i}$, partial pressure $p_{i}$, and $N_{i}$ molecules.

Problem 37

The attenuation of a beam of photons in a gas of pressure $p$ is given by $d\Phi =-\Phi (\sigma p/k_{B}T)\,dx$, where $\sigma$ is the cross section, $k_{B}$ the Boltzmann constant, $T$ the absolute temperature, and $x$ the path length. Suppose that the pressure is given as a function of altitude $y$ by $p=p_{0}e^{-mgy/k_{B}T}$. What is the total attenuation by the entire atmosphere?

Problem 38

Consider a beam of photons incident on the atmosphere from directly overhead. The atmosphere contains several species of molecules, each with partial pressure $p_{i}$. The absorption coefficient is $\mu _{a}=(1/k_{B}T)\sum _{i}\sigma _{i}p_{i}$. If each constituent of the atmosphere varies with height $y$ as $p_{i}(y)=p_{0i}\exp (-m_{i}gy/k_{B}T)$, show that the fluence rate striking the earth is given by an expression of the form $e^{-\alpha }$ and find $\alpha$.

14.2.10 Section 14.11

Problem 39

Consider a tissue with a specific heat of 3.6 J kg$^{-1}\,$K$^{-1}$, a density of 1000 kg m$^{-3}$, and a thermal conductivity of 0.5 W m$^{-1}\,$K$^{-1}$. Assume the specific heat of blood is the same, and that the tissue perfusion is $4.17\times 10^{-6}~$m$^{3}$ kg$^{-1}\,$s$^{-1}$. Find the thermal diffusivity, the time for the heat to flow 1 cm, and the thermal penetration depth.

14.2.11 Section 14.12

Problem 40

Suppose that a sphere radiates uniformly from its surface according to Lambert’s cosine law: $L=L_{0}$. By considering area $dS=2\uppi r^2 \sin \theta \,d\theta$ on the surface of a sphere, find the power radiated per steradian in the direction of the $z$ axis and the total power radiated.

Problem 41

Show that the exitance, total power per unit area radiated from a surface obeying Lambert’s cosine law, is $W_r=\uppi L_0$.

Problem 42

How many photons per second correspond to 1 lm at 555 nm for photopic vision? At 510 nm for scotopic vision?

14.2.12 Section 14.13

Problem 43

A person is nearsighted, and the relaxed eye focuses at a distance of 50 cm. What is the strength of the desired corrective lens in diopters?

Problem 44

What is the distance of closest vision for an average person with normal vision at age 20? Age 40? Age 60?

Problem 45

A person of age 40 is fitted with bifocals with a strength of $+1~$diopter more than the correction for distance vision. What are the closest and farthest distances of focus without the bifocal lens and with it? By the time the person is age 50, what are they with and without the same lens?

Problem 46

You can make a rough measurement of your own eye’s properties. Tape a piece of paper with some pattern on it on the wall. Cover one eye. Move away from the wall until the pattern starts to blur. Measure the distance to the wall in meters. Calculate the vergence of the object, $U$. Assume that the $F$ of your relaxed eye is 59 diopters. Calculate $V$ for your eye. Now find the closest distance at which you can see the paper. Calculate the accommodation of your eye.

Problem 47

An object is placed 6 cm from a converging lens with a 5-cm focal length.

• (a) Use the thin-lens equation (Eq. 14.64) to calculate the image distance.

• (b) The magnification of the image is given by $m=-v/u$. (A negative magnification implies an inverted image.) What is the magnification for the image in part (a)? A value $|m|>1$ implies a“magnified” image. This is how a slide projector works.

Problem 48

An object is placed 15 cm from a converging lens with a focal length of 20 cm.

• (a) Use the thin-lens equation (Eq. 14.64) to calculate the image distance. Your value should be negative, corresponding to a “virtual image”.

• (b) The magnification of the image is again given by $m=-v/u$. What is the magnification for the image in part (a)? This is how a magnifying glass works.

Problem 49

Combine the results of Problems 682 and 683. Consider two lenses, the first with focal length 5 cm and the second with focal length 20 cm, separated by 45 cm. The object is 6 cm in front of the first lens. The image from the first lens is the object for the second.

• (a) Calculate the image distance and magnification of the image created by the first lens (called the objective).

• (b) Use the first image as the object for the second lens (called the eyepiece), and calculate the image distance and magnification of the second image.

• (c) The total magnification is the product of the magnifications of the objective and eyepiece. What is the total magnification? This is how the compound microscope works. The objective lens acts like a slide projector, and the eyepiece acts like a magnifying glass. Very large total magnifications can be obtained when the object is just to the left of the focal point of the objective, and the first image is just to the right of the focal point of the eyepiece.

Problem 50

Snell’s law, $n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2},$ gives an interesting result if light passes from a medium with a higher index of refraction to one with a lower index of refraction, $n_{1}>n_{2}.$ Assume light passes from glass $(n_{1}=1.5)$ to air $(n_{2}=1.0).$

• (a) If $\theta _{1}$is $30\operatorname {{{}^\circ }},$ what is $\theta _{2}$?

• (b) If $\theta _{1}$is $40\operatorname {{{}^\circ }}$, what is $\theta _{2}$?

• (c) If $\theta _{1}$is $50\operatorname {{{}^\circ }}$, what is $\theta _{2}$?

  This is really a tricky question, because for $\theta _{1}$ greater than some critical angle, $\theta _{c},~\theta _{2}$ exceeds $90\operatorname {{{}^\circ }}$, and light cannot pass into the second medium. Instead all the light is reflected and remains within the first medium.

• (d) Calculate the critical angle for total internal reflection from glass to air.

Total internal reflection allows thin glass fibers to act as fiberoptic “light pipes,” which can be used to transmit signals. Bundles of such optical fibers are used in endoscopes to see inside the body.

Problem 51

Table 14.7 shows that most of the converging power of the eye occurs at the air-cornea interface. When a person is under water, this must be supplied by the water-cornea surface. The index of refraction of the cornea is closer to that of water than to that of air. What are the implications for seeing under water? What are the implications for the vision of aquatic animals? (For more information on the difference between the eyes of aquatic and terrestrial animals, see Denny 2007.)

14.2.13 Section 14.14

Problem 52

How many photons per 0.1 s enter the eye from a 100 W light bulb 1000 ft away? Assume the pupil is 6 mm in diameter. How far away can a 100 W bulb be seen if there is no absorption in the atmosphere? Use a luminous efficiency of 17 lm W$^{-1}$ and then assume an equivalent light source at 555 nm.

Problem 53

The table below shows the radiance of some extended sources. Without worrying about obliquity factors (assume that all the light is at normal incidence), calculate the number of photons entering a receptive field of $0.17\operatorname {{{}^\circ }}$ diameter when the pupil diameter is 6 mm and the integration time is 0.1 s. Assume a conversion efficiency of 100 lm W$^{-1}$ and then assume that all the photons are at 555 nm.

$$ \begin{tabular} [c]{ll}Source & Radiance (lm~m$^{-2}$~sr$^{-1}$)\\ & \\[-10pt]White paper in sunlight & \multicolumn{1}{c}{25000}\\ Clear sky & \multicolumn{1}{c}{3200}\\ Surface of the moon & \multicolumn{1}{c}{2900}\\ White paper in moonlight & \multicolumn{1}{c}{0.03} \end{tabular} \ \ $$

Problem 54

A piece of paper is illuminated by dim light so that its radiance is 0.01 lm m$^{-2}~$sr$^{-1}$ in the direction of a camera. A camera lens 1 cm in diameter is 0.6 m from the paper. The sheet of paper is $10\times 10$ cm. The shutter of the camera is open for 1 ms. Assume all the light is at 555 nm. How many photons from the paper enter the lens of the camera while the shutter is open?

Problem 55

If three or more photons must be absorbed by a visual receptor field for the observer to see a flash, what fraction of the flashes are seen if the average number of photons absorbed in a receptor field per flash is four?

Problem 56

Assume that an average of $d$ photons are detected and that the photons are Poisson distributed. What must $d$ be to detect a signal that is a $1\,\%$ change in $d$, if the signal-to-noise ratio must be at least $5$?

Problem 57

Suppose that the average number of photons striking a target during an exposure is $m$. The probability that $x$ photons strike during a similar exposure is given by the Poisson distribution. What is the probability that an organism responds to an exposure of radiation in each of the following cases?

• (a) The response of the organism requires that a single target within the organism be hit by two or more photons.

• (b) The response of the organism requires that two targets within the organism each be struck by one or more photons during the exposure.