Lumens, Candelas, Lux, and Nits

In Chapter 14 (Atoms and Light) of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss photometry, the measurement of electromagnetic radiation and its ability to produce a human visual sensation. I find photometry interesting mainly because of all the unusual units.

Let’s start by assuming you have a source of light emitting a certain amount of energy per second, or in other words with a certain power in watts. This is called the radiant power or radiant flux, and is a fundamental concept in radiometry. But how do we perceive such a source of light? That is a question in photometry. Our perception will depend on the wavelength of light. If the light is all in the infrared or ultraviolet, we won’t see anything. If in the visible spectrum, our perception depends on the wavelength. In fact, the situation is even more complicated than this, because our perception depends on if we are using the cones in the retina of our eye to see bright light in color (photopic vision), or we are using rods to see dim light in black and white (scotopic vision). Moreover, our ability to see varies among individuals. The usual convention is to assume we are using photopic vision, and to say that a source radiating a power of one watt of light at a wavelength of 555 nm (green light, the wavelength that the eye is most sensitive to) has a luminous flux of 683 lumens.

The light source may emit different amounts of light in different directions. In radiometry, the radiant intensity is the power emitted per solid angle, in units of watt per steradian. We can define an analogous photometric unit for the luminous intensity to be the luman per steradian, or the candela. The candela is one of seven “SI base units” (the others are the kilogram, meter, second, ampere, mole, and kelvin). Russ and I mention the candela in Table 14.6, which is a large table that compares radiometric, photometric and actinometric quantities. We also define it in the text, using the old-fashioned name “candle” rather than candela.

Often you want to know the intensity of light per unit area, or irradiance. In radiometry, irradiance is measured in watts per square meter. In photometry, the illuminance is measured in lumens per square meter, also called the lux.

Finally, the radiance of a surface is the radiant power per solid angle per unit surface area (W sr⁻¹ m⁻²). The analogous photometric quantity is the luminance, which is measured in units of lumen sr⁻¹ m⁻², or candela m⁻², or lux sr⁻¹, or nit. The brightness of a computer display is measured in nits.

In summary, below is an abbreviated version of Table 14.6 in IPMB

Radiometry	Photometry
Radiant power (W)	Luminous flux (lumen)
Radiant Intensity (W sr−1)	Luminous intensity (candela)
Irradiance (W m−2)	Illuminance (lux)
Radiance (W sr−1 m−2)	Luminance (nit)

Where did the relationship between 1 W and 683 lumens come from? Before electric lights, a candle was a major source of light. A typical candle emits about 1 candela of light. The relationship between the watt and the lumen is somewhat analogous to the relationship between absolute temperature and thermal energy, and the relationship between a mole and the number of molecules. This would put the conversion factor of 683 lumens per watt in the same class as Boltzmann's constant (1.38 × 10⁻²³ J per K) and Avogadro's number (6.02 × 10²³ molecules per mole).

The Oxford English Dictionary

The Meaning of Everything,
by Simon Winchester.

I’m a fan of Simon Winchester, and I recently finished his book The Meaning of Everything: The Story of the Oxford English Dictionary. I enjoyed it immensely, and it motivated me to spend a morning browsing through the OED in the Oakland University library, which owns the 1989 twenty-volume second edition.

Rather than describe a typical OED entry, I’ll show ten examples using words drawn from Intermediate Physics for Medicine and Biology.

bremsstrahlung

In OED entries, the information right after the word in parentheses is the pronunciation based on the International Phonetic Alphabet, and the text within brackets is the etymology. Bremsstrahlung is German (G.; the OED uses lots of abbreviations). It has its own OED entry, so I guess it’s considered part of the English language too. The entry spans two columns, so I had to cut and paste photos of it. To my ear, bremsstrahlung is the oddest sounding word in IPMB.

candela

The origin of candela is from Latin (L.). IPMB and Wikipedia define the candela as lumen per steradian. I don’t see the solid angle connection listed in the OED.

chronaxie

Russ Hobbie and I spell chronaxie ending in -ie, which is the most common spelling, although some end it in -y. Chronaxie is from a French (F.) term that appeared in an 1909 article by Louis Lapicque, cited in IPMB.

cyclotron

My favorite part of an OED entry are the quotations illustrating usage. Several quotes are provided for cyclotron. The first is from a 1935 Physical Review article by Ernest Lawrence, the cyclotron’s inventor. XLVIII is the volume number in Roman numerals, and 495/2 means the quote can be found on page 495, column 2.

defibrillation

Two definitions of defibrillation exist. IPMB uses the word in the second sense: the stopping of fibrillation of the heart. Other forms of this medical (Med.) term are listed, with defibrillating being the participial adjective (ppl. a.) and defibrillator the noun. Carl Wiggers is a giant in cardiac electrophysiology, and the Lancet is one of the world’s leading medical journals.

electrotonus 

The OED’s definition of electrotonus is different from mine.

In IPMB, Russ and I write

The simplest membrane model is one that obeys Ohm’s law. This approximation is valid if the voltage changes are small enough so the membrane conductance does not change, or if something is done to inactivate the normal changes of membrane conductance with voltage. It is also useful for myelinated nerves between the nodes of Ranvier. This is called electrotonus or passive spread.

IPMB says nothing about a constant current stimulus, and the OED says nothing about passive spread. I wonder if I’ve been using the word correctly? Wikipedia agrees with me.

The two vertical lines in the top left corner on the entry indicate an alien word (used in English, but from another language). I would have thought bremsstrahlung more deserving of this designation than electrotonus.

fluoroscope

Wilhelm Röntgen discovered x-rays in late 1895, so I’m surprised to see the term fluoroscope used only one year later. X-rays caught on fast. Nature is one of the best-known scientific journals.

leibniz

My PhD advisor John Wikswo and I are engaged in a quixotic attempt to introduce a new unit, the leibniz.

If I were going to append a new definition, it would look something like this:

2. A unit corresponding to a mole of differential equations. 2006 HUANG et al. Rev. Physiol. Biochem. Pharmacol. CLVII. 98 Avogadro’s number of differential equations may be defined as one Leibnitz. 2006 WIKSWO et al. IEE P-Nanobiotechnol. CLIII. 84 It is conceivable that the ultimate models for systems biology might require a mole of differential equations (called a Leibnitz). 2015 HOBBIE and ROTH Intermediate Physics for Medicine and Biology 53 In computational biology, a mole of differential equations is sometimes called a leibniz.

quatrefoil

Wikswo coined the term quatrefoil for four-fold symmetric reentry in cardiac tissue. Quatrefoil appears in the OED, but its definition is focused on foliage rather than heart arrhythmias. I guess Wikswo didn’t invent the word but he did propose a new meaning. I can’t complain that this sense of the word is missing from the OED, because quatrefoil reentry wasn’t discovered until after the second edition went to press. My proposed addition is:

3. A four-fold symmetric cardiac arrhythmia. 1999 LIN et al. J. Cardiovasc. Electrophysiol. X. 574 A novel quatrefoil-shaped reentry pattern consisting of two pairs of opposing rotors was created by delivering long stimuli during the vulnerable phase.

 tomography 

Godfrey Hounsfield built the first computed tomography machine in 1971. I didn’t realize that tomography had such a rich history before then. I don’t like the OED’s definition of tomography. I prefer something closer to IPMB’s: “reconstructing, for fixed z, a map of some function f(x,y) from a set of projections F(θ,x').”

Missing Words

Some words from IPMB are not in the OED; for example chemostat, electroporation, and magnetosome. Kerma is absent, but it’s an acronym and they aren’t included. Brachytherapy is absent, even from the long entry for the prefix brachy-. Sphygmomanometer doesn’t have its own entry, although it’s listed among the surprisingly large number of words starting with the prefix sphygmo-. Magnetocardiogram is included under the prefix magneto-, but the more important magnetoencephalogram is not. I was hoping to find the definition of bidomain, but alas it’s not there. Here’s my version.

bidomain (ˌbaɪdəʊ'meɪn). Phys. [f. BI- + -DOMAIN.] A mathematical description of the electrical behavior of syncytial tissue such as cardiac muscle. 1978 TUNG A Bi-domain Model for Describing Ischemic Myocardial D-C Potentials (Dissertation) 2 Bi-domain, volume-conductive structures differ from classical volume conductors (mono-domain structures) in that a distinction is made between current flow in the extracellular space and current flow in the intracellular space. 1983 GESELOWITZ and MILLER Ann. Biomed. Eng. XI. 200  The equations of the bidomain model are a three-dimensional version of the cable equations.

The OED took decades to complete, mostly during the Victorian era. The effort was led by James Murray, the hero of Winchester’s book. He supervised a small group of assistants, plus a motley crew of contributors whose job was to search English literature for examples of word use. Winchester’s stories about this collection of oddballs and misfits is engrossing; they volunteered countless hours with little recognition, some contributing tens of thousands of quotations, each submitted on a slip of paper during those years before computers. I can think of only one modern parallel: those unsung heroes who labor over Wikipedia.

The Professor and the Madman,
by Simon Winchester.

If you like The Meaning of Everything, you’ll love Winchester’s The Professor and the Madman, also about the Oxford English Dictionary. In addition, Winchester has written several fine books about geology; my favorites are Krakatoa and The Map That Changed the World.

To close, I’ll quote the final paragraph of a speech that Prime Minister Stanley Baldwin gave in 1928 at a dinner celebrating the completion of the OED, which appears at the end of Winchester's Prologue to The Meaning of Everything.

It is in that grand spirit of devotion to our language as the great and noble instrument of our national life and literature that the editors and the staff of the Oxford Dictionary have laboured. They have laboured so well that, so far from lowering the standard with which the work began, they have sought to raise it as the work advanced. They have given us of their best. There can be no worldly recompense—expect that every man and woman in this country whose gratitude and respect is worth having, will rise up and call you blessed for this great work. The Oxford English Dictionary is the greatest enterprise of its kind in history.

Intermediate Physics for Medicine and Biology
nestled among volumes of the Oxford English Dictionary.

The SI Logo

Intermediate Physics for Medicine and Biology uses the metric system. On page 1, Russ Hobbie and I write

“The metric system is officially called the SI system (systeme internationale). It used to be called the MKS (meter kilogram second) system.”

In 2018, the International Bureau of Weights and Measures changed how the seven SI base units are defined. They are now based on seven defining constants. This change is summarized in the SI logo.

The SI logo, produced by the
International Bureau of Weights and Measures.

First let’s see where the seven base units appear in IPMB. Then we’ll examine the seven defining constants.

kilogram

The most basic units of the SI system are so familiar that Russ and I don’t bother defining them. The kilogram (mass, kg) appears throughout IPMB, but especially in Chapter 1, where density plays a major role in our analysis of fluid dynamics.

meter

We define the meter (distance, m) in Chapter 1 when discussing distances and scales: “The basic unit of length in the metric system is the meter (m): about the height of a 3-year-old child.” Both the meter and the kilogram are critical when discussing scaling in Chapter 2.

second

The second (time, s) is another unit that’s so basic Russ and I take it for granted. It plays a particularly large role in Chapter 10 when discussing nonlinear dynamics.

ampere

The SI system becomes more complicated when you add electrical units. IPMB defines the ampere (electrical current, A) in Section 6.8 about current and Ohm’s law: “The units of the current are C s⁻¹ [C is the unit of charge, a coulomb] or amperes (A) (sometimes called amps).”

kelvin

The unit for absolute temperature—the kelvin (temperature, K)—plays a central role in Chapter 3 of IPMB, when describing thermodynamics.

mole

The mole (number of molecules, mol) appears in Chapter 3 when relating microscopic quantities (Boltzmann’s constant, elementary charge) to macroscopic quantities (the gas constant, the Faraday). John Wikswo and I have introduced a name for a mole of differential equations (the leibniz), but the International Bureau of Weights and Measures inexplicably did not add it to their logo.

candela

Russ and I introduce the candela (luminous intensity, cd) in Section 14.12 of IPMB, when comparing radiometry to photometry: “The number of lumens per steradian is the luminous intensity, in lm sr⁻¹. The lumen per steradian is also called the candela.” The steradian (the unit of solid angle) used to play a more central role in the SI system, but appears to have been demoted.

Now we examine the seven constants that define these units.

Planck’s constant

In IPMB, the main role of Planck’s constant (h, 6.626 × 10⁻³⁴ J s) is to relate the frequency and energy of a photon. Quantum mechanics doesn’t play a major role in IPMB, so Planck’s constant appears less often than you might expect.

speed of light

Like quantum mechanics, relativity does not take center stage in IPMB, so the speed of light (c, 2.998 × 10⁸ m s⁻¹) appears rarely. We use it in Chapter 14 when relating the frequency of light to its wavelength, and in Chapter 17 when relating the mass of an elementary particle to its energy.

cesium hyperfine frequency

The cesium hyperfine frequency (Δν, 9.192 × 10⁹ Hz) defines the second. It never appears in IPMB. Why cesium? Why this particular atomic transition? I don’t know.

elementary charge

The elementary charge (e, 1.602 × 10⁻¹⁹ C) is used throughout IPMB, but is particularly important in Chapter 6 about bioelectricity.

Boltzmann’s constant

Boltzmann’s constant (k_B, 1.381 × 10⁻²³ J K⁻¹) appears primarily in Chapter 3 of IPMB, but also anytime Russ and I mention the Boltzmann factor.

Avogadro’s number

Like Boltzmann’s constant, Avogadro’s number (N_A, 6.022 × 10²³ mol⁻¹) shows up first in Chapter 3.

luminous efficacy

The luminous efficacy (K_cd, 683 lm W⁻¹) appears in Chapter 14 of IPMB: “The ratio P_v/P at 555 nm is the luminous efficacy for photopic vision, K_m = 683 lm W⁻¹.” I find this constant to be different from all the others. It’s a prime number specified to only three digits. Suppose a society of intelligent beings evolved on another planet. Their physicists would probably measure a set of constants similar to ours, and once we figured out how to convert units we would get the same values for six of the constants. The luminous efficacy, however, would depend on the physiology of their eyes (assuming they even have eyes). Perhaps I make too much about this. Perhaps the luminous efficacy merely defines the candela, just as Avogardo’s number defines the mole and Boltzmann’s constant defines the kelvin. Still, to me it has a different feel.

You can learn more about the SI units and constants in the International Bureau of Weights and Measures’ SI brochure. I’m fond of the SI logo, which reminds me of the circle of fifths. If you’re new to the metric systems, you might want to paste the logo into your copy of Intermediate Physics for Medicine and Biology; I suggest placing it in the white space on page 1, just above Table 1.1.

Page 1 of Intermediate Physics for Medicine and Biology,
with the SI Logo added at the top.

A Dozen Units from Intermediate Physics for Medicine and Biology

Medical and biological physics have their share of colorful and sometimes obsolete units. For the most part, Intermediate Physics for Medicine and Biology sticks with standard metric, or SI, units; mass, distance and time are in kilograms, meters, and seconds (mks). Some combinations of units are given special names, usually in honor of a famous physicist, such as the newton (N) for kg m s^-2. I have always found the units for electricity and magnetism difficult to remember. The coulomb (C) for charge is easy enough, but units such as the tesla (T) for magnetic field strength in kg s^-1 C^-1 are tricky. IPMB uses some common non-SI units, such as the liter (l) for 10^-3 m³, the angstrom (Å) for 10^-10 m, and the electron volt (eV) for 1.6 × 10^-19 J.

Let’s count down a dozen unfamiliar units discussed in Intermediate Physics for Medicine and Biology. We’ll start with the least important, and end with the one you really need to know.

12. The roentgen (R). Chapter 16 of IPMB states that the roentgen “is an old unit of [radiation] exposure equivalent to the production of 2.58 × 10^-4 C kg^-1 in dry air.” The unit’s name written out as “roentgen” begins with a lower case letter “r” even though Wilhelm Roentgen’s last name starts with an upper case “R.” It's always that way with units.

11. The diopter (diopter). The diopter is a nickname for m^-1, just as the hertz is a nickname for s^-1. It is used mainly when discussing the power, or vergence, of a lens, and appears in Chapter 14 of IPMB. The diopter does not have a symbol, you just write out “diopter” (“dioptre” if you are English, but that is so wrong).

10. The einstein (E). Homework Problem 2 of Chapter 14 defines the einstein as “1 mol of photons.” Units like the mole (mol) and the einstein are really dimensionless numbers: a mole is 6 × 10²³ molecules and an einstein is 6 × 10²³ photons. John Wikswo and I have proposed the leibniz (Lz) to be 6 × 10²³ differential equations. Some define the einstein as the energy of a mole of photons, so be careful when using this unit. I’ll let you guess who the unit was named for.

9. The poise (P). Chapter 1 of IPMB analyzes the coefficient of viscosity, which is often expressed in units of poise or centipoise. The poise is a leftover from the old centimeter-gram-second system of units, and is equal to a gram per centimeter per second. The viscosity of water at 20 °C is about 1 cP. The poise is named after Jean Leonard Marie Poiseuille (sort of), just as the unit of capacitance (the farad) is kind of named after Micheal Faraday. The mks unit of viscosity is the poiseuille (Pl), where 1 Pl = 10 P. The poiseuille is not used much, probably because no one can pronounce it.

8. The torr (Torr). Pressure is measured in many units. The torr is nearly the same as a millimeter of mercury (mmHg), and is named after the Italian physicist Evangelista Torricelli. The SI unit for pressure is the pascal (Pa), a nickname for a newton per square meter. One Torr is about 133 Pa. The bar (bar) is 100,000 Pa, and is approximately equal to one atmosphere (atm). How confusing! All five units—torr, bar, atm, mmHg, and pascal—are used often, so you need to know them all.

7. The barn (b). The barn measures area and is 10^-28 m². It is equivalent to 100 fm² (the femtometer is also known as a fermi). Nuclear cross sections are measured in barns. By nuclear physics standards a barn is a pretty big cross section. The term barn comes from the idiom about “hitting the broad side of a barn.”

6. The debye (D). Homework Problem 3 in Chapter 6 of IPMB introduces the debye. It is defined as 10^-18 statcoulomb cm, where a statcoulomb is the unit of charge in the old cgs system. It is equivalent to 3.34 × 10^-30 C m. The debye is named after Dutch physicist Peter Debye, and measures dipole moment. The dipole moment of a water molecule is 1.85 D.

5. The candela (cd). Radiometry measures radiant energy using SI units. Photometry measures the sensation of human vision with its own oddball collection of units, such as lumens, candelas, lux, and nits. A candela depends on the color of the light; for green 1 cd is equal to a radiant intensity of about 0.0015 watts per steradian. A burning candle has a luminous intensity of about 1 cd.

4. The svedberg (Sv). The centrifuge is a common instrument in biological physics. A particle has a sedimentation coefficient equal to its sedimentation velocity per unit of centrifugal acceleration. The units of speed (m s^-1) divided by acceleration (m s^-2) is seconds, so sedimentation coefficient has dimensions of time. The svedberg is equal to 10^-13 s. IPMB gives the symbol as “Sv”, but sometimes it is just “S” (easily confused with a unit of conductance called the siemens and a unit of effective dose called the sievert). The unit is named after the Swedish chemist Theodor Svedberg, who invented the ultracentrifuge.

3. The curie (Ci). The curie is an older unit of radioactivity that is now out of fashion. It is named in honor of Pierre and Marie Curie, and it measures the activity, equal to the disintegration rate. The SI unit for activity is the becquerel (Bq), or disintegrations per second. The becquerel is named after Henri Becquerel, the French physicist who discovered radioactivity. One curie is 3.7 × 10¹⁰ Bq. The cumulated activity is the total number of disintegrations, and is a dimensionless number often expressed in Bq s (why bother?). An older unit for cumulated activity is the odd-sounding microcurie hour (µCi h).

2. The Hounsfield unit (HU). The Hounsfield unit is used to measure the x-ray attenuation coefficient µ during computed tomography. It is a dimensionless quantity defined by Eq. 16.25 in IPMB: H = 1000 (µ – µ_water)/µ_water (for some reason Russ Hobbie and I use H rather than HU). The unit is strange because everyone says the attenuation coefficient is so many Hounsfield units, including the word “units” (you never say a force is so many “newton units”). The attenuation coefficient of water is 0 HU. Air has a very small small attenuation coefficient, so on the Hounsfield scale it is -1000 HU. Many soft tissues have an attenuation coefficient on the order of +40 HU, and bone can be more than +1000 HU. The unit is named after English electrical engineer Godfrey Hounsfield, who won the 1979 Nobel Prize in Physiology or Medicine for developing the first clinical computed tomography machine.

and the winner is....

1. The sievert (Sv). The most important unusual unit in IPMB is the sievert. Both the sievert and the gray (Gy) are equal to a joule per kilogram. The gray is a physical unit measuring the energy deposited in tissue per unit mass, or the dose. The sievert is the gray multiplied by a dimensionless coefficient called the relative biological effectiveness and measures the effective dose. For x-rays, the sievert and gray are the same, but for alpha particles one gray can be many sieverts. An older unit for the gray is the rad (1 Gy = 100 rad) and an older unit for the sievert is the rem (1 Sv = 100 rem). The gray is named after English physicist Louis Gray, and the sievert after Swedish medical physicist Rolf Sievert.