The Interaction of Light and Matter

Learning Objectives Structure of the Atom:

Discovery of Electrons, and Thomson’s Model of the AtomDiscovery of Atomic Nuclei, and Rutherford’s Model of the Atom

Spectral Lines of Hydrogen:Balmer seriesRydberg’s formula for hydrogen spectra

Bohr’s Semiclassical Model for the Atom:Application of Classical PhysicsApplication of Quantum Physics

Theoretical Explanation for Kirchhoff’s Laws Hydrogen lines in Astronomy:

Lyman, Balmer, and Paschen seriesInterstellar and Intergalactic absorption

Learning Objectives Structure of the Atom:

Discovery of Electrons, and Thomson’s Model of the AtomDiscovery of Atomic Nuclei, and Rutherford’s Model of the Atom

Spectral Lines of Hydrogen:Balmer seriesRydberg’s formula for hydrogen spectra

Bohr’s Semiclassical Model for the Atom:Application of Classical PhysicsApplication of Quantum Physics

Theoretical Explanation for Kirchhoff’s Laws Hydrogen lines in Astronomy:

Lyman, Balmer, and Paschen seriesInterstellar and Intergalactic absorption

Discovery of the Electron In 1897, the British physicist Joseph John Thomson announced the

discovery of negatively charged elementary particles (later named electrons) and measured their charge-to-mass ratio through a series of experiments using cathode ray tubes.

Thomson used a “cold” cathode, whereby a high voltage was applied between two metal plates to attract gas ions in the tube (ionized by natural radioactivity) to the cathode. The accelerated ions collide with and ionize other gas atoms in the tube along the way to the anode, creating an avalanche of electrons (then known as cathode rays) that are accelerated towards the anode.

Joseph John Thomson, 1856-1940

Discovery of the Electron “Hot” cathodes comprise thin wire filaments through which an electric current is passed

to increase the random (heat) motion of the filament atoms, which through collisions with other filament atoms knock electrons out of the atoms at the surface of the filament. The electrons are accelerated by the anode.

By applying an electric and/or magnetic field in a direction orthogonal to the electron beam, the path of the electron beam can be deflected.

Discovery of the Electron In this way, Thomson measured the charge to mass (e/m) ratio of the electron. He found

e/m to be independent of the type of gas used, and therefore established the existence of a common particle with negative charge for matter.

See notes posted on course website for Thomson’s Nobel Prize lecture on the discovery and measurement of the charge to mass ratio of the electron. Note that it was left to Robert A. Millikan to make (at the time) the most accurate measurement of the charge of the electron, and from Thomson’s measurement of e/m therefore also mass of the electron, from his famous oil drop experiment. The results of this experiment were published in 1913.

Thomson’s model of the Atom Because bulk matter is electrically neutral, atoms were deduced to comprise negative-

charged electrons along with an equal positive charge of uncertain distribution. Thomson proposed that atoms comprised electrons in a cloud of positive charge.

Thomson’s plum pudding model of the atom

Plum pudding

Discovery of the Atomic Nucleus In the years leading up to the publication of his seminal paper on the structure of the atom in 1911, Ernest Rutherford* and his research assistant, Hans Geiger, had been conducting experiments to study the structure of the atom by directing high-speed alpha particles (which he showed were doubly ionized – positively charged – helium atoms) produced by radioactive thorium onto thin metal foils.

(Alpha particles are produced by the radioactive decay of elements such as uranium, thorium, etc.)

Ernest Rutherford, 1871-1937

* Rutherford was Thomson’s student.

Discovery of the Atomic Nucleus At the time, it was known that alpha particles were much more massive than the electron. What do you think Rutherford expected to find based on Thomson’s model for the atom, the prevailing model at the time?

Discovery of the Atomic Nucleus Rutherford developed a way of counting individual alpha particles by coating a screen, mounted at the end of a microscope, with zinc sulfide. The screen would emit a flash of light each time it was hit by an alpha particle. This could only be reliably seen by dark-adapted eyes (after half an hour in complete darkness) and one person could only count the flashes accurately for one minute before needing a break, and counts above 90 per minute were too fast for reliability. The experiment accumulated data from hundreds of thousands of flashes.

Consistent with expectations, they found that the scattering angle was small, on the order of 1 degree.

Hans Wilhelm Geiger, 1882-1945

Discovery of the Atomic Nucleus Geiger later invented a device to detect and count alpha particles. This device later became known as the Geiger counter, and is widely use today to detect ionization radiation.

Discovery of the Atomic Nucleus In 1909, Ernest Marsden, then an undergraduate student, was being trained by Hans Geiger. In the words of Rutherford: I had observed the scattering of alpha-particles, and Dr. Geiger in my laboratory had examined it in detail. He found, in thin pieces of heavy metal, that the scattering was usually small, of the order of one degree. One day Geiger came to me and said, "Don't you think that young Marsden, whom I

Ernest Marsden, 1889-1970

training in radioactive methods, ought to begin a small research?" Now I had thought that, too, so I said, " Why not let him see if any alpha-particles can be scattered through a large angle?" I may tell you in confidence that I did not believe that they would be, since we knew the alpha-particle was a very fast, massive particle with a great deal of energy, and you could show that if the scattering was due to the accumulated effect of a number of small scatterings, the chance of an alpha-particle's being scattered backward was very small. Then I remember two or three days later Geiger coming to me in great excitement and saying "We have been able to get some of the alpha-particles coming backward …" It was quite the most incredible event that ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you."

Rutherford’s model of the Atom About 1 in 8000 alpha particles were deflected by angles larger than 90 degrees. Rutherford reasoned that the alpha particles experienced “collisions” with, during

which they were repulsed by, the massive and positively-charged particles of the atom. The angle through which the alpha particles were deflected depended on how close they came to these massive and positively-charged particles.

By carefully measuring the fraction of alpha particles deflected through different angles, Rutherford estimated that the size of this massive positively-charged particles is less than about 10-14 m. Today, we know atoms to have sizes of about 10-10 m.

In 1920, Rutherford named the positively-charged particles of the atom protons, which reside in a nucleus (latin for nut) at the center of the atom. Electrons were assumed to orbit the positively-charged nucleus.

Learning Objectives Structure of the Atom:

Discovery of Electrons, and Thomson’s Model of the Atom Discovery of Atomic Nuclei, and Rutherford’s Model of the Atom

Spectral Lines of Hydrogen:Balmer seriesRydberg’s formula for hydrogen spectra

Bohr’s Semiclassical Model for the Atom:Application of Classical PhysicsApplication of Quantum Physics

Theoretical Explanation for Kirchhoff’s Laws A simple application in astrophysics:

Why can we see stars and galaxies?

Spectral Lines of Hydrogen By the 1880s, the (four) spectral lines of hydrogen in the visible region of the

electromagnetic spectrum had been precisely determined.

Johann Balmer, 1825-1898

In 1885, a Swiss school teacher, Johann Balmer (PhD in Mathematics), proposed through trial and error an empirical formula that reproduced the wavelengths of these (four) spectral lines:

where n = 2, h = 3.6456 × 10−7 m (known today as the Balmer constant), and m = 3, 4, 5, and 6. Today, the spectral lines corresponding to this formula are known as the Balmer lines or series.

656.3 nm486.1 nm410.2 nm434.0 nm

Spectral Lines of Hydrogen At the same time, Swedish physicist Johannes Rydberg was trying to

find an empirical formula that reproduced the wavelengths emitted by alkali metals (elements in the first column of the periodic table: Li, Na, Rb, Cs, and Fr).

The simplest formula he could find required expressing the wavelengths in terms of wavenumbers 1/ (number of waves occupying a unit length). Hearing about Balmer’s formula, Rydberg realized that Balmer’s formula was just a specific case of his own more general formula.

Johannes Rydberg, 1854-1919

In 1888, Rydberg proposed the following empirical formula for the spectral lines of hydrogen:

where m and n are integers such that m < n, and RH = 1.09677583 x 107 ± 1.3 m-1 is the Rydberg constant for hydrogen (determined experimentally – to reproduce the observed wavelengths of hydrogen lines – through spectroscopy by Rydberg). Other spectral lines of hydrogen at wavelengths outside the optical spectrum were subsequently found in agreement with Eq. (5.8).

Learning Objectives Structure of the Atom:

Discovery of Electrons, and Thomson’s Model of the Atom Discovery of Atomic Nuclei, and Rutherford’s Model of the Atom

Spectral Lines of Hydrogen:Balmer seriesRydberg’s formula for hydrogen spectra

Bohr’s Semiclassical Model for the Atom:Application of Classical PhysicsApplication of Quantum Physics

Theoretical Explanation for Kirchhoff’s Laws Hydrogen lines in Astronomy:

Lyman, Balmer, and Paschen seriesInterstellar and Intergalactic absorption

Bohr’s Semiclassical Atom In 1913, the Danish physicist Niehls Bohr proposed a theoretical

model for the hydrogen atom that correctly reproduced the empirical formula for its spectral lines as proposed by Balmer and Rydberg.

Bohr’s model of the atom combined both classical physics and quantum mechanics, and has as its two central ingredients:

- electrons in circular orbits around the the nucleus (classical physics)

- the orbital angular momenta of electrons are quantized (quantum mechanics)

Electrons only absorb or emit electromagnetic waves when they make make a transition from one permitted orbit to another.

Niehls Borh, 1885-1962

* The notion that some property of electrons were quantized (could only take on discrete values) were popular in the early 1900s to explain the discrete spectral lines of atoms; e.g., as in an earlier model of the atom proposed by the British astronomer J. W. Nicholson.

Bohr’s Semiclassical Atom Both Balmer’s and Rydberg’s formulae for the spectral lines of

hydrogen are purely empirical formulations. In 1913, the Danish physicist Niehls Bohr proposed a theoretical model

for the hydrogen atom that correctly reproduced the formula for its spectral lines as proposed by Balmer and Rydberg.

Bohr’s model of the atom combined both classical physics and quantum mechanics, and has as its two central ingredients:

- electrons in circular orbits around the the nucleus (classical physics)

- the orbital angular momenta of electrons are quantized (quantum mechanics)

Electrons only emit electromagnetic waves when they make make a transition from one permitted orbit to another.

Niehls Borh, 1885-1962

* The notion that some property of electrons were quantized (could only take on discrete values) were popular in the early 1900s to explain the discrete spectral lines of atoms; e.g., as in an earlier model of the atom proposed by the British astronomer J. W. Nicholson.

Bohr’s Semiclassical Atom Consider a hydrogen atom comprising an electron in a circular orbit around the proton

(nucleus of the hydrogen atom). The electrical force exerted by the proton (charge q1)

on the electron (charge q2) is

where ε0 is the permittivity of free space and is a unit vector from the proton to the electron. (I keep the same notation as the textbook, but for mathematical accuracy I note that should point from e- to e+).

Bohr’s Semiclassical Atom Because the proton is much more massive than the electron, we can think of the electron

in a circular orbit around a stationary (non-accelerating) proton. This is almost but not exactly correct.

In actual fact, both the proton and electron are orbiting around the system’s center of mass.

Bohr’s Semiclassical Atom

m1

m2

at center of mass

We can transform this 2-body problem into an equivalent 1-body problem (which is much simpler to analyze) using the method of reduced mass (see §2 on Celestial Mechanics).

Bohr’s Semiclassical Atom We can transform this 2-body problem into an equivalent 1-body problem (which is

much simpler to analyze) using the method of reduced mass (see §2 on Celestial Mechanics).

The reduced mass

and the total mass

at center of mass

Bohr’s Semiclassical Atom The electrical force between the proton and electron produces the electron’s centripetal

acceleration as described by Newton’s 2nd law

implying that

With q1 = -q2 = e,

Bohr’s Semiclassical Atom From the previous equation we can solve for the kinetic

energy

We can derive the electrical potential energy

We define the electrical potential energy U = 0 when the reduced mass is at a distance r = ∞ from the central mass, and U < 0 when r < ∞.

The total energy of the system

* The electrical potential energy is the energy required to move the reduced mass from a distance r to a distance of ∞ from the central mass:

Bohr’s Semiclassical Atom There is a fundamental problem with Rutherford’s model in which an atom

comprises a very compact positively-charged nucleus orbited by electrons. According to Maxwell’s equations, an orbiting and therefore accelerating electron

emits electromagnetic waves. In this situation, will an electron emit at discrete frequencies thus producing spectral lines?

Bohr’s Semiclassical Atom We now introduce Bohr’s proposal that the angular momentum of the electron

(and consequently reduced mass) is quantized

The kinetic energy of the reduced mass

Substituting the kinetic energy derived classically in Eq. (5.10)

and solving for r we get

where a0 = 0.0529 nm is known as the Bohr radius (radius of the hydrogen atom).

What does quantizing the angular momentum of the electron imply?

Bohr’s Semiclassical Atom Compare the Bohr radius for the atom

a0 = 0.0529 nm ≈ 10-10 m

with the size of the nucleus estimated by Rutherford

diameter of nucleus ≈ 10-14 m

An atom, and therefore matter, is mostly empty space! If an atom was the size of the Earth, how big would the nucleus be?

Bohr’s Semiclassical Atom Recall the total energy of the system as derived in Eq. (5.11)

Substituting for the permitted radii of the reduced mass as derived in Eq. (5.13)

The integer n is known as the principal quantum number, and completely determines the characteristics of each orbit of the Bohr atom.

Spectral Absorption and Emission Lines Bohr proposed that an electron absorbs a photon of the appropriate energy to make a transition from a lower to higher orbit thus giving rise to an absorption line, and emits a photon to make a transition from a higher to lower orbit thus giving rise to an emission line.

Wavelengths of Spectral Lines The energy of the absorbed or emitted photon corresponds to the difference in energy

between the two corresponding orbits of the electron

or

which gives

Compare with Rydberg’s empirical formula of Eq. (5.8)

Wavelengths of Spectral Lines Bohr’s model therefore predicts a theoretical value for Rydberg’s constant of

in agreement with the value determined experimentally by Rydberg.

Learning Objectives Structure of the Atom:

Discovery of Electrons, and Thomson’s Model of the AtomDiscovery of Atomic Nuclei, and Rutherford’s Model of the Atom

Spectral Lines of Hydrogen:Balmer seriesRydberg’s formula for hydrogen spectra

Bohr’s Semiclassical Model for the Atom:Application of Classical PhysicsApplication of Quantum Physics

Theoretical Explanation for Kirchhoff’s Laws Hydrogen lines in Astronomy:

Lyman, Balmer, and Paschen seriesInterstellar absorption

Theoretical Explanation of Kirchhoff’s laws

Learning Objectives Structure of the Atom:

Discovery of Electrons, and Thomson’s Model of the AtomDiscovery of Atomic Nuclei, and Rutherford’s Model of the Atom

Spectral Lines of Hydrogen:Balmer seriesRydberg’s formula for hydrogen spectra

Bohr’s Semiclassical Model for the Atom:Application of Classical PhysicsApplication of Quantum Physics

Theoretical Explanation for Kirchhoff’s Laws Hydrogen lines in Astronomy:

Lyman, Balmer, and Paschen seriesInterstellar absorption

Lyman series Transitions from or to the ground state (n = 1) of hydrogen are known as the Lyman

series lines of hydrogen. In which part of the electromagnetic spectrum do we find the Lyman series lines of hydrogen?

≤ 91.2 nm

≤ 365 nm≤ 820 nm Ionized state

1000

Transitions from or to the ground state (n = 1) of hydrogen are known as the Lyman series lines of hydrogen. The Lyα line is one of the brightest and most important spectral lines in astronomy, but for objects in the nearby Universe can only be detected from space.

≤ 91.2 nm

≤ 365 nm≤ 820 nm Ionized state

Lyman series

At z ≥ 2.3, the Lyα line is redshifted into the optical part of the electromagnetic spectrum, and can therefore be detected by ground-based telescopes.

Spectrum of a galaxy at z = 3.92. The brightest line in the measured spectrum is the Lyα line.

Lyα line from Distant Galaxies

32”

What kind of stars can excite hydrogen so as to produce the Lyman series lines?

≤ 91.2 nm

≤ 365 nm≤ 820 nm Ionized state

Lyman series

Balmer series Transitions from or to the 1st excited state (n = 2) of hydrogen are known as the Balmer

series lines of hydrogen. In which part of the electromagnetic spectrum do we find the Balmer series lines of hydrogen?

1000

≤ 91.2 nm

≤ 365 nm≤ 820 nm Ionized state

Spectrum of the Orion Nebula in an approximately 5 x 6 arcminute region centered on the Trapezium region of the nebula. Notice that the Balmer lines of hydrogen are among the brightest lines in the optical spectrum.

Balmer lines from star-forming regions

What kind of stars can excite hydrogen from the ground state so as to produce the Balmer series lines?

≤ 91.2 nm

≤ 365 nm≤ 820 nm Ionized state

Lyman series

Paschen series

1000

≤ 91.2 nm

≤ 365 nm≤ 820 nm Ionized state

Transitions from or to the 2nd excited state (n = 3) of hydrogen are known as the Paschen series lines of hydrogen. In which part of the electromagnetic spectrum do we find the Paschen series lines of hydrogen?

Interstellar Absorption Ultraviolet spectrum of the dwarf nova Z Camelopardalis. Why does the light intensity

drop to zero at wavelengths shortwards of about 910 Å?

Interstellar Absorption Lyman absorption lines and the Lyman limit. The Lyman limit corresponds to a

wavelength of 912 Å; at this and shorter wavelengths, photons are able to ionize hydrogen atoms from their ground state.

Interstellar Absorption Ultraviolet/Optical spectrum of the star HD 86986. Why does the light intensity drop

steeply at wavelengths shortwards of about 3650 Å?

Interstellar Absorption The space between stars (interstellar space) is filled with hydrogen gas with a density of

~1 atom/cm3 (interstellar medium). The closest stars (the α Centauri system) lie at a distance of 1.34 pc = 4 x 1018 cm, making it highly likely that starlight will encounter many hydrogen atoms on the way to the Earth.

What does our ability to see stars and galaxies in the optical tell us about the excitation state of the vast majority of hydrogen atoms in the interstellar medium?

Intergalactic Absorption Spectra of distant quasars. Why does the

light from distant quasars exhibit a sudden decrease in intensity at wavelengths just shortwards of their Lyα lines?

Wavelength (Å)

Intergalactic Absorption Spectra of distant quasars. Could the

absorption shortwards of Lyα be caused solely by hydrogen gas in the galaxy hosting the quasar?

Wavelength (Å)

≤ 91.2 nm

≤ 365 nm≤ 820 nm Ionized state

Intergalactic Absorption Illustration of absorption in the Lyα line by intergalatic hydrogen gas clouds at different

redshifts.

Intergalactic Absorption Lyα absorption lines in the spectrum of the quasar QSO 1422+23 at z = 3.62 by

intergalatic hydrogen gas clouds at different redshifts.

Intergalactic Absorption Intergalactic absorption in the Lyman series is used to identify distant galaxies: by taking

images through narrow-band filters, galaxy redshifts can be quite accurately estimated in a technique known as photometric redshifts.