# BLACK BODY RADIATION Object that is HOT (anything BLACK BODY RADIATION Object that is HOT (anything > 0 K is considered hot) emits EM radiation For example, an incandescent lamp is red HOT because it emits a lot of EM wave, especially in the IR region 1 Thermal radiation

An ordinary object can emit and absorb electromagnetic radiation. Particles that constitute an object are constantly in thermal motion These particles interact with pervasive electromagnetic fields energy is constantly exchanged between the object and the electromagnetic (EM) field The interchange is assumed to be an equilibrium process occurring at a certain temperature 2

Frequency spectrum is temperature dependent At equillibrium, the rate of radiation energy absorbed by the body and and that emitted by the body equals There is correlation between the distribution of frequency of the EM radiation and the temperature, T At different T, the wavelength spectrum is different The frequency spectrum is quantified by M(T), called spectral radiant emittance.

T Thermal radiation at equillibrium with an object 3 at a fixed temperature M (T), spectral radiant emittance 4 Radiant emittance (radiancy, emissive power), M(T) M (T) refers to the total energy radiated

at T per unit time per unit area of the object It is a temperature-dependent function Unit in W/m2. M T M d 0 5 The wavelength distribution of the electromagnetic radiation spectrum, 6 Attempt to understand the origin of radiation from hot bodies from

classical theories In the early years, around 1888 1900, light is understood to be EM radiation Since hot body radiate EM radiation, hence physicists at that time naturally attempted to understand the origin of hot body in terms of classical EM theory and thermodynamics (which has been well established at that time) 7 All hot object radiate EM wave of all wavelengths

However, the energy intensities of the wavelengths differ continuously from wavelength to wavelength (or equivalently, frequency) Hence the term: the spectral distribution of energy as a function of wavelength 8 Spectral distribution of energy in radiation depends only on temperature The distribution of intensity of the emitted radiation from a hot body at a given wavelength depends on the temperature 9

Radiance In the measurement of the distribution of intensity of the emitted radiation from a hot body, one measures dI where dI is the intensity of EM radiation emitted between and +dd about a particular wavelength . Intensity = power per unit area, in unit if Watt per m2. Radiance R() is defined as per dI = R() d R() is the power radiated per unit area (intensity) per unit wavelength interval at a given wavelength and a given temperature T.

Its unit could be in Watt per meter square per m or W per meter square per nm. 10 Total radiated power per unit area The total power radiated per unit area (intensity) of the BB is given by the integral I T R , T d 0 For a blackbody with a total area of A, its total power emitted at temperature T is

P T AI T Note: The SI unit for P is Watt, SI unit for I is Watt per meter square; for A, the SI unit is meter square 11 Introducing idealised black body In reality the spectral distribution of intensity of radiation of a given body could depend on the type of the surface which may differ in absorption and radiation efficiency (i.e.

frequency-dependent) This renders the study of the origin of radiation by hot bodies case-dependent (which means no good because the conclusions made based on one body cannot be applicable to other bodies that have different surface absorption characteristics) E.g. At the same temperature, the spectral distribution by the exhaust pipe from a Proton GEN2 and a Toyota Altis is different 12 Emmissivity, e As a strategy to overcome this non-generality, we introduce an idealised black body which, by definition,

absorbs all radiation incident upon it, regardless of frequency Such idealised body is universal and allows one to disregard the precise nature of whatever is radiating, since all BB behave identically All real surfaces could be approximate to the behavior of a black body via a parameter EMMISSIVITY e (e=1 means ideally approximated, 0< e < 1 means poorly approximated) 13 Blackbody Approximation A good approximation of a black body is a small hole leading to the inside of a

hollow object The HOLE acts as a perfect absorber The Black Body is the HOLE 14 Any radiation striking the HOLE enters the cavity, trapped by reflection until is absorbed by the inner walls The walls are constantly absorbing and emitting energy at thermal EB The nature of the radiation leaving

the cavity through the hole depends only on the temperature of the cavity and not the detail of the surfaces nor frequency of the radiation 15 Essentially A black body in thermal EB absorbs and emits radiation at the same rate The HOLE effectively behave like a

Black Body because it effectively absorbs all radiation fall upon it And at the same time, it also emits all the absorbed radiations at the same rate as the radiations are absorbed The measured spectral distribution of black bodies is universal and depends only on temperature. In other words: THE SPECTRAL DISTRIBUTION OF EMISSION DEPENDS SOLELY ON THE TEMPERATURE AND NOT OTHER DETAILS. T BB at thermodynamic equilibrium at a fixed temperature

16 Experimentally measured curve of a BB 17 Stefans Law P = AeT4 P total power output of a BB A total surface area of a BB Stefan-Boltzmann constant = 5.670 x 10-8 W / m2 . K4 Stefans law can be written in terms of intensity

I = P/A = T4 For a blackbody, where e = 1 18 Wiens Displacement Law maxT = 2.898 x 10-3 m.K max is the wavelength at which the curve peaks T is the absolute temperature The wavelength at which the intensity peaks, max, is inversely proportional to the absolute temperature As the temperature increases, the peak wavelength max is displaced to shorter wavelengths. 19

Example This figure shows two stars in the constellation Orion. Betelgeuse appears to glow red, while Rigel looks blue in color. Which star has a higher surface temperature? (a) Betelgeuse (b) Rigel (c) They both have the same surface temperature. (d) Impossible to determine. 20 Intensity of Blackbody Radiation, Summary

The intensity increases with increasing temperature The amount of radiation emitted increases with increasing temperature The area under the curve The peak wavelength decreases with increasing temperature 21 Example

Find the peak wavelength of the blackbody radiation emitted by (A) the Sun (2000 K) (B) the tungsten of a light bulb at 3000 K 22 Solutions (A) the sun (2000 K) By Weins displacement law, 3 2.898 10

m K max 2000K 1.4 m (infrared) (B) the tungsten of a lightbulb at 3000 K max 3 2.898 10 m K 5800K 0.5 m

Yellow-green 23 Lord Rayleigh and James Jeans at 1890s try to theoretically derive the distribution based on statistical mechanics (some kind of generalised thermodynamics) and classical Maxwell theory (Details omitted, u will learn this when u study statistical mechanics later)

Radiance Why does the spectral distribution of black bodies have the shape as measured? 24 RJs model of BB radiation with classical EM theory and statistical physics Consider a cavity at temperature T whose walls are

considered as perfect reflectors The cavity supports many modes of oscillation of the EM field caused by accelerated charges in the cavity walls, resulting in the emission of EM waves at all wavelength These EM waves inside the cavity are the BB radiation They are considered to be a series of standing EM wave set up within the cavity 25 Number density of EM standing wave modes in the cavity The number of independent standing waves

G()d in the frequency interval between and +ddper unit volume in the cavity is (by applying statistical mechanics) 8 2 d G d c3 The next step is to find the average energy per standing wave 26 The average energy per standing wave, Theorem of equipartition of energy (a mainstay

theorem from statistical mechanics) says that the average energy per standing wave is = kT k 1.38 10 23 J/K, Boltzmann constant In classical physics, can take any value CONTINOUSLY and there is not reason to limit it to take only discrete values (this is because the temperature T is continuous and not discrete, hence e must also be continuous) 27 Energy density in the BB cavity Energy density of the radiation inside the BB cavity in the frequency interval between and +d ud(

v, T )dv = the total energy per unit volume in the cavity in the frequency interval between and +d d = the number of independent standing waves in the frequency interval between and +d dper unit volume, G()d the average energy per standing wave. 8 2 kTd u (v, T )dv = G()d = 3 c 28 Energy density in terms of radiance The energy density in the cavity in the frequency

interval between and +d d can be easily expressed in terms of wavelength, via c = 8 2 kTd 8 kT u v, T u ,T 4 d 3 c In experiment we measure the BB in terms of radiance R(,T) which is related to the energy density via a factor of c/4: R(,T) = (c/4)u() 2 ckT 4 29

Rayleigh-Jeans Law Rayleigh-Jeans law for the radiance (based on classical physics): 2ckTckT R T ,T T4 At long wavelengths, the law matched experimental results fairly well 30

At short wavelengths, there was a major disagreement between the Rayleigh-Jeans law and experiment This mismatch became known as the ultraviolet catastrophe Radiance Rayleigh-Jeans Law, cont. You would have infinite energy as the wavelength approaches zero 31 Max Planck

Introduced the concept of quantum of action In 1918 he was awarded the Nobel Prize for the discovery of the quantized nature of energy 32 Plancks Theory of Blackbody Radiation

In 1900 Planck developed a theory of blackbody radiation that leads to an equation for the intensity of the radiation This equation is in complete agreement with experimental observations 33 Plancks Wavelength Distribution Function Planck generated a theoretical expression for the wavelength distribution (radiance) 2 2ckThc R T ,T 5 hc TkT T e

1 h = 6.626 x 10-34 J.s h is a fundamental constant of nature 34 Plancks Wavelength Distribution Function, cont. At long wavelengths, Plancks equation reduces to the Rayleigh-Jeans expression This can be shown by expanding the exponential 2 term hc 1 hc hc

e hc TkT 1 ... 1 TkT 2 ! TkT TkT in the long wavelength limit hc TkT At short wavelengths, it predicts an exponential decrease in intensity with decreasing wavelength This is in agreement with experimental results 35 Comparison between Plancks law

of BB radiation and RJs law Radiance R T ,T 2ckTckT T4 correctely fit by Planck's derivation 2 R T ,T T 5 2ckThc e hc TkT 1

36 How Planck modeled the BB He assumed the cavity radiation came from atomic oscillations in the cavity walls Planck made two assumptions about the nature of the oscillators in the cavity walls 37 Plancks Assumption, 1

The energy of an oscillator can have only certain discrete values En En = nh n =0,1,2,; n is called the quantum number h is Plancks constant = 6.63 x 10-34 Js is the frequency of oscillation the energy of the oscillator is quantized Each discrete energy value corresponds to a different quantum state This is in stark contrast to the case of RJ derivation according to classical theories, in which the energies of oscillators in the cavity must assume a continuous distribution 38

Energy-Level Diagram of the Planck Oscillator An energy-level diagram of the oscillators showing the quantized energy levels and allowed transitions Energy is on the vertical axis Horizontal lines represent the allowed energy levels of the oscillators The double-headed arrows

indicate allowed transitions 39 Oscillator in Plancks theory is quantised in energies (taking only discrete values) The energy of an oscillator can have only certain discrete values En = nh, n=0,1,2,3, The average energy per standing wave in the Planck oscillator is hf

e hf kT 1 (instead of =kT in classical theories) 40 Plancks Assumption, 2 The oscillators emit or absorb energy when making a transition from one quantum state to another The entire energy difference between the initial and final states in the transition is emitted or absorbed as a single quantum of radiation

An oscillator emits or absorbs energy only when it changes quantum states 41 Pictorial representation of oscillator transition between states A quantum of energy hf is absorbed or emitted during transition between states Transition between states Allowed states of the oscillators 42 Example: quantised oscillator vs classical oscillator

A 2.0 kg block is attached to a massless spring that has a force constant k=25 N/m. The spring is stretched 0.40 m from its EB position and released. (A) Find the total energy of the system and the frequency of oscillation according to classical mechanics. 43 Solution In classical mechanics, E= kA2 = 2.0 J The frequency of oscillation is 1 f 2 k

... 0.56 Hz m 44 (B) (B) Assuming that the energy is quantised, find the quantum number n for the system oscillating with this amplitude Solution: This is a quantum analysis of the oscillator En = nh = n (6.63 x 10-34 Js)(0.56 Hz) = 2.0 J n = 5.4 x 1033 !!! A very large quantum number, typical for macroscopin system

45 The previous example illustrated the fact that the quantum of action, h, is so tiny that, from macroscopic point of view, the quantisation of the energy level is so tiny that it is almost undetectable. Effectively, the energy level of a macroscopic system such as the energy of a harmonic oscillator form a continuum despite it is granular at the quantum scale 46 magnified view of the

energy continuum shows discrete energy levels allowed energies in quantised system discrete (such as energy levels in an atom, energies carried by a photon) allowed energies in classical system continuous (such as an harmonic oscillator, energy carried by a wave; total mechanical energy of an orbiting planet, etc.) 47

To summarise Classical BB presents a ultraviolet catastrophe The spectral energy distribution of electromagnetic radiation in a black body CANNOT be explained in terms of classical Maxwell EM theory, in which the average energy in the cavity assumes continuous values of <> = kT (this is the result of the wave nature of radiation) To solve the BB catastrophe one has to assume that the energy of individual radiation oscillator in the cavity of a BB is quantised as per En = nh This picture is in conflict with classical physics because in

classical physics energy is in principle a continuous variable that can take any value between 0 One is then lead to the revolutionary concept that ENERGY OF AN OSCILLATOR IS QUANTISED 48 Cosmic microwave background (CMBR) as perfect black body radiation 49 1965, cosmic microwave background was first detected by Penzias and Wilson Nobel Prize 1976 Pigeon Trap Used Penzias and Wilson thought the static their radio antenna was

picking up might be due to droppings from pigeons roosting in the antenna horn. They captured the pigeons with this trap 50 and cleaned out the horn, but the static persisted. CMBR the most perfect Black Body Measurements of the cosmic microwave background radiation allow us to determine the temperature of the universe today. The brightness of the relic radiation is measured as a function of the radio frequency. To an excellent approximation it is described by a thermal of blackbody distribution with a temperature of T=2.735 degrees above absolute zero. This is a dramatic and direct confirmation of one of the

predictions of the Hot Big Bang model. The COBE satellite measured the spectrum of the cosmic microwave background in 1990, showing remarkable agreement between theory and experiment. 51 microwave Far infrared The Temperature of the Universe Today, as implied from CMBR The diagram shows the results plotted in waves per centimeter versus intensity. The theoretical

best fit curve (the solid line) is indistinguishable from the experimental data points (the point-size is greater than the experimental errors). 52 COBE The Cosmic Background Explorer satellite was launched twenty five years after the discovery of the microwave background radiation in 1964. In spectacular fashion in 1992, the COBE team

announces that they had discovered `ripples at the edge of the universe', that is, the first sign of primordial fluctuations at 100,000 years after the Big Bang. These are the imprint of the seeds of galaxy formation. 53 Faces of God 54 The faces of God: a map of temperature variations on the full sky picture that COBE obtained.

They are at the level of only one part in one hundred thousand. Viewed in reverse the Universe is highly uniform in every direction lending strong support for the cosmological principle. 55 The Nobel Prize in Physics 2006 "for their discovery of the blackbody form and anisotropy of the cosmic microwave background radiation" John C. Mather George F. Smoot 56

New material pushes the boundary of blackness http://www.reuters.com/article/scienceNews/idU SN1555030620080116?sp=true 57 By Julie Steenhuysen CHICAGO (Reuters) - U.S. researchers said on Tuesday they have made the darkest material on Earth, a substance so black it absorbs more than 99.9 percent of light. Made from tiny tubes of carbon standing on end, this material is almost 30 times darker than a carbon substance used by the U.S. National Institute of Standards and Technology as the current benchmark of blackness. And the material is close to the long-sought ideal black, which could absorb all colors of light and reflect none.

"All the light that goes in is basically absorbed," Pulickel Ajayan, who led the research team at Rice University in Houston, said in a telephone interview. "It is almost pushing the limit of how much light can be absorbed into one material." The substance has a total reflective index of 0.045 percent -- which is more than three times darker than the nickel-phosphorous alloy that now holds the record as the world's darkest material. Basic black paint, by comparison, has a reflective index of 5 percent to 10 percent. The researchers are seeking a world's darkest material designation by Guinness World Records. But their work will likely yield more than just bragging rights. Ajayan said the material could be used in solar energy conversion. "You could think of a material that basically collects all the light that falls into it," he said. It could also could be used in infrared detection or astronomical observation. THREE-FOLD BLACKNESS Ajayan, who worked with a team at Rensselaer Polytechnic Institute in Troy, New York, said the material gets its blackness from three things. It is composed of carbon nano-tubes, tiny tubes of tightly rolled carbon that are 400 hundred times smaller than the diameter of a strand of hair. The carbon helps absorb some of the light. These tubes are standing on end, much like a patch of grass. This arrangement traps light in the tiny gaps between the "blades." The researchers have also made the surface of this carbon nano-tube carpet irregular and rough to cut down on reflectivity. "Such a nano-tube array not only reflects light weakly, but also absorbs light strongly," said Shawn-Yu Lin, a professor of physics at Rensselaer, who helped make the substance. The researchers have tested the material on visible light only. Now they want to see how it fares against infrared and ultraviolet

light, and other wavelengths such as radiation used in communications systems. "If you could make materials that would block these radiations, it could have serious applications for stealth and defense," Ajayan said. The work was released online last week and will be published in an upcoming issue of the journal Nano Letters. The Indianborn Ajayan holds the 2006 Guinness World Record as co-inventor of the smallest brush in the world. (Editing by Maggie Fox and Xavier Briand)

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