)
Einstein resolved these puzzles (1905) using an idea originated by Max Planck in 1900 from a completely different phenomenon (we'll talk about it soon) - he postulated the Quantum Theory of Light. Namely, despite all the successes and experimental evidence that light is a wave, this seems irreconcilable with the photoelectric effect.
Planck had been studying the spectrum of light emitted by glowing, hot objects (think of metal, glowing red, then yellow, then white as it gets hotter) in fact, all objects radiate (even if they're cool) - which frequencies they radiate depends on the temperature, not the material. (cool objects emit infrared which is invisible to our eyes) We'll learn more soon about this "blackbody radiation". Planck discovered a "trick" to correctly predict the spectrum. The trick assumed that radiation of the object is emitted in discrete bursts, or bundles, called quanta. The energy of a bundle associated with frequency nu must always be the same (proportional to nu), i.e. E=(h nu). From experiment, he got the SAME h as above (6.626*10^-34 J s) This was weird, made no sense at all, and Planck was not a happy camper. He tried hard to create "pictures" of what was going on that left the radiation as a plain old wave. Einstein, however, ran with the idea. If light is emitted in bundles, it should be able to cruise through space as a bundle too, rather than a wave. These bundles have acquired the name PHOTONS.
So, back to the photoelectric effect. Recall, we had KE_max = h (nu-nu_0), i.e.
(h nu) = KE_max + h nu_0. Interpretation? (h nu) = energy of a bundle of
incoming light. KE _Max = maximum energy of electron. (h nu_0) = energy
required to dislodge an electron from the metal surface. The bundles
apparently deliver their energy to a single electron. The last term in the
equation is called the work function of the metal, and is often called phi (
)
So now we see that the energy (h nu) of the photons has nothing to do with the intensity of the light beam, but only the frequency. Brighter light has more photons, but each one has the same energy (h nu). Dim light hitting a metal still dumps (h nu) into individual electrons, just not so many. That's why the photoelectric effect happens so fast - even one single photon can produce an electron.
If an electron gets hit with a photon, absorbing (h nu) of energy, and then it
has to do "phi" amount of work to break free of the crystal it's trapped in, it
will have a maximum of (h nu - phi) left over for kinetic energy. (It could
have *less*, of course, if it interacts with neighbors on the way out, but
won't be likely to ever have *more*) So, KE_max = h nu - phi. I.e, the equation
we had above makes sense, if you buy that light comes "bundled". The electron
energy depends only on the frequency of the light, not the intensity, and grows
linearly with nu. By the way, Einstein predicted (1905) the linear relation
*before* it was experimentally seen in photoelectric experiments. Millikan
spent 10 years, till 1915, trying to DISprove this!
Millikan in 1916: "We are confronted by the astonishing situation that these facts were correctly and exactly predicted 9 years ago by a form of quantum theory which has now been generally abandoned". He also refers to Einstein's "bold, not to say reckless, hypothesis of an EM light corpuscle of energy (h nu) which flies in the face of the thoroughly established facts of interference".
And then, later in his life, Millikan in 1949: "[Einstein] ignored and indeed seemed to contradict all the manifold facts of interference and thus to be a straight return to the corpuscular theory of light which had been completely abandoned since the time of Young and Fresnel... I spent 10 years of my life testing the 1905 equation of Einstein's, and, contrary to all my expectations, I was compelled in 1915 to assert its unambiguous experimental verification in spite of all its unreasonableness since it seemed to violate everything we know about the interference of light." Summarizing how the idea of photons explains the photoelectric effect:
1) EM wave energy is concentrated in photons, not in a spread out wave front. So, there's no time delay in emission of photoelectrons.
2) All photons of frequency (
)
have the same energy,
,
so the change in intensity of monochromatic light changes the number of
photoelectrons, but not their energies.
3) Higher frequency means more energy (
)
in each photon, which means the photoelectrons get more energy too.
Each metal has a different crystal structure, different electron affinity, and thus a different work function phi. Typically, it's a couple of eV's for any metal. (the same scale as ionizing free metal atoms)
Visible light is (4 -> 7 )*10^14 Hz which corresponds to 2 -> 3 eV, so the photoelectric effect usually involves visible or ultraviolet light.
Example: UV lights hit potassium, K. The Area of plate is 1 cm^2, wavelength of the light is 350 nm = 3500 Angstroms, intensity of light is
1 W/m^2, and the work function of K is 2.2 eV.
1) What is e V_0, i.e. the maximum energy of the emitted electrons?
A:
2) Suppose 1% of the incident light actually makes photoelectrons. How many electrons are emitted?
A: E of each photon is h nu = 3.6 eV/photon,
So I/E = 6.25*10^14/3.6 photons/(s cm^2) => 1.7*10^14 photons hit/sec
1% means 1.7*10^12 are absorbed and "become" photoelectrons.
More evidence: Heating a metal also gives energy to electrons (thermionic emission) One can compute the energy required to let electrons escape, and its the same as the work function phi. Here, heat provides the energy.
So, what's going on? Is light a wave, or is it a bunch of photons (particles)? I guess the answer is sort of both. Or maybe it's neither. It depends on the experiment... In 1900, Planck's blackbody theory (that said radiating objects give off energy in separate quanta) raised a few eyebrows. But this wasn't yet really in conflict with a wave picture of light. Einstein's proposal, that light travels as distinct, particular, photons, was outrageous! Waves spread out, like ripples in a pond. Photons are small, so small they can apparently be absorbed by single electrons. Worse yet, E = h nu involves nu, the "wave frequency", in a formula about particle-like properties of light. It's an odd mish-mash. Here, we need two different theories to explain light. It's not like relativity, where one theory is "right" in one regime, and another theory is right in another. (Newtonian mechanics is an approximation of relativity, but wave theory of light is not an approximation of the quantum theory, really.)
Let's look at some more experiments that demonstrated the existence of photons - particles of light. We've just seen how photons can transfer energy to electrons. How about the reverse? In the late 1800's Roentgen discovered penetrating radiation ("X-rays") emitted when fast electrons hit matter. These rays pass through materials, travel in straight lines, are unaffected by E and B fields, expose photos, make some materials glow... They are EM waves. Maxwell predicts that stopping electrons should make Bremsstrahlung ("Braking radiation") The spectrum (distribution of frequencies) is continuous and predictable. How to show they are EM waves? Diffract them through a grating, just like light! How to make a grating with lines close enough together? (one expects wavelengths of the order of Angstroms for the electron energies I'm about to consider) Use a crystal, the atoms themselves will be the grating![* ] In 1913, this experiment was done, with electrons being stopped and producing X-rays with wavelengths from 0.013 to 0.048 nm, i.e. 10^4 times more energetic than visible photons.
BUT, something unexpected occurred.
(Graph for a tungsten target)
This figure shows the intensity of outgoing X-rays as a function of the
wavelength of the X-ray, and shows a sharp lower cutoff in wavelength! The
Maxwellian bremsstrahlung predictions gives all wavelengths. Since
,
a minimum wavelength implies a maximum nu, i.e. a maximum energy.
Duane (the same one as our Physics building is named after) and Hunt discovered (1918) that
And, they found the (constant) was numerically = e/(hc). I.e, the maximum energy of the X-ray = h nu = e V_0 = KE of incident electron. This is precisely the reverse of the photoelectric effect - now the KE of the electron gets converted into the energy of photons. (The work function is now tiny compared to these energies, so it's negligible)
Another weird thing discovered was that for some materials, the spectrum looked like:
We will understand this later, but in any case, it was totally nonclassical, and incomprehensible from Maxwell's equations alone.
One more experiment, that really helped clinch the idea that light can be though of as photons, was the Compton Effect (1923) This effect demonstrates the dynamics of individual photons. Photons are not just some abstract idea having to do with quantization of energy. They really do behave like particles. Photons can collide with electrons, or even other photons. Waves certainly don't do that!
We have a little problem, in that these kinds of collisions involve intrinsically relativistic particles, the photons are traveling at c, and the electron velocities are also plenty big. So, we'll have to use some relativity. First, the qualitative result:
If you think of a billiard ball collision (which is non-relativistic), energy conservation says
Since the second term on the RHS above must be >=0, the final photon kinetic energy is less than the initial photon kinetic energy - i.e., the ball which bounces loses energy. Since here the "ball" is a photon, if it loses energy, its frequency will drop, and so its wavelength will GROW. This is a non-classical prediction. Maxwell's equations can describe light scattering from an electron, but would not predict this frequency shift.
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