Chapter 30: The Nature of the Atom
• Rutherford scattering and the nuclear atom –!the nucleus is really " small!
• Atomic line spectra – wavelengths characteristic of each element
• Bohr model of the hydrogen atom –! " quantization of angular momentum ! quantization of energy
• x-rays
• The laser
• Omit 30.5, 6 – quantum mechanical picture of H-atom, Pauli ! Exclusion Principle, periodic table
Monday, April 2, 2007 1
Early model of the atom
• About 10-10 m in size.
• Positively charged (“pudding”), with negatively charged electrons " (“plums”) embedded in the pudding – “plum pudding model”
Stability of the plum pudding was questionable.
Where would the characteristic spectral lines come from?
#
Monday, April 2, 2007 2 Geiger+Marsden: Fall of the plum pudding
Alpha (!) particles – nuclei of 4He atom, emitted by some radioactive nuclei – were scattered from a thin gold foil and observed on a screen as flashes of light.
Far more were scattered at large angle than would be possible with the weak electric field inside a “plum pudding” atom.
Rutherford: realized that the positive charges of the atom had to be contained in a very small volume – the nucleus –!electric " planetary model of the atom with electrons field very strong close to it. " in orbit around nucleus
Monday, April 2, 2007 3
Very schematic picture of an atom
Size of the nucleus $ 10-15 – 10-14 m
Size of the atom $ 10-10 m
Equal amounts of + and – charges in the neutral atom.
Electrons are in orbit around the nucleus in a planetary model of the atom.
(charge +Ze)
Monday, April 2, 2007 4 Problem with a Planetary Model of the Atom
• The electrons suffer centripetal acceleration in their orbital motion.
• Accelerated charges should radiate electromagnetic energy.
" The electrons should lose energy and spiral into the nucleus in very " " little time.
" A planetary atom should not be stable!
" Classical theory does not explain the structure of the atom.
" Small systems, such as atoms, must behave differently from ! large.
30.1 Monday, April 2, 2007 5
Fraunhofer absorption lines from the sun http://www.harmsy.freeuk.com/fraunhofer.html 1 Å (Angstrom) = 0.1 nm • Fraunhofer lines – due to absorption of "sunlight " by elements in the atmosphere of the sun • chemical elements emit and absorb light at ! the same wavelengths • Models of the atom need to explain this
The sun: blackbody radiation for T $ 6000 K
Monday, April 2, 2007 6 Line spectrum of the hydrogen atom
n = 2 n = 4 n = 6 n = 3 n = 5 n = 4
m = 1 m = 2 m = 3 n = & Ultraviolet Visible series limit Infrared + other series Balmer found by trial and error a simple formula to calculate the wavelength of all lines of the hydrogen atom: m Series 1 1 1 = R m = 1,2,3... 1 Lyman ultraviolet m2 − n2 ! 2 Balmer visible ! " n = m + 1,m + 2,m + 3... 3 Paschen infrared R = Rydberg constant = 1.097 % 107 m-1 4 Brackett infrared
Monday, April 2, 2007 7
Spectrum of H-atom 1 1 1 = R m = 1,2,3... ! m2 − n2 ! " n = m + 1,m + 2,m + 3... R = Rydberg constant = 1.097 % 107 m-1 n = & n = 2 Example: Lyman series, m = 1:
For n = &: 1 1 1 m = 1 = 1.097 107 = 1.097 107 m–1 ! × 12 − " × ! " ! = 91.2 nm (series(serieslimit) limit, shortest wavelength in Lyman series) For n = 2: 1 1 1 = 1.097 107 = 0.823 107 m–1 ! × 12 − 22 × ! " ! = 121.5 nm (longest wavelength in Lyman series)
Monday, April 2, 2007 8 Bohr model of the hydrogen atom
Now superseded by more modern quantum mechanical ideas, but gives the correct answers. E = hf
= Ei – Ef Assume: • A planetary model with electron " in orbit around the nucleus. • There are certain electron orbits “Stationary” " that are stable (“stationary states”). states " This is contrary to classical theory " and is not explained in this model. • Light is emitted or absorbed when an electron changes state. • Energy is conserved, so the energy of the photon is the difference in " energy between the initial and final states:
" E = hf = Ei – Ef " – what are the allowed energies?
Monday, April 2, 2007 9
Energies of hydrogen atom in the Bohr model
The electron is kept in orbit by the attractive A hydrogen-like atom or ion with just Coulomb force between it and the nucleus. one electron in orbit kZe2 mv2 F = = r2 r kZe2 mv2 = = 2 KE (A) r The potential energy is: m kZe2 PE = − r And so the total mechanical energy, E = KE + PE, is: kZe2 kZe2 kZe2 E = = (B) 2r − r − 2r But, what are the “good” values of r corresponding to the stationary states?
Monday, April 2, 2007 10 Energies of hydrogen atom in the Bohr model
Assertion (Bohr): the angular momentum of the electron around the nucleus can have only certain, quantized, values: nh Ln = mvr = n = 1, 2, 3... 2! Combine with (A): mv2r = kZe2 kZe2 nh 2 m So, v2 = = mr 2!mr ! " 2 2 1 nh 11 n r = = (5.29 10− m) n = “quantum number” kZe2m 2! × Z ! " kZe2 And then, substituting into (B), the total mechanical energy is: E = − 2r ! " 2!2mk2e4 Z2 Z2 E = , n = 1,2,3... E = 13.6 eV n − h2 n2 n − n2 ! "
Monday, April 2, 2007 11
Bohr Model for H and H-like Atoms
• A planetary model in which the electron defies the normal laws of " electromagnetism and does not radiate when in certain “stationary” " states.
• The angular momentum of the electron in the stationary states is: nh L = mvr = , n = 1, 2, 3 . . . n 2π " n is a “quantum number”. Z2 • The energy of the n’th quantum state is: E = 13.6 eV n − n2 • Light is emitted or absorbed when the atom changes energy state.
• The energy of the photon is the difference in energy of the two states " of the atom.
Monday, April 2, 2007 12 Emission of a photon
E = E + h f E = hf i f
Absorption of a photon
E f + h f = Ei
Emission and absorption occur at the same wavelengths (as seen in the E = hf Fraunhofer absorption lines of sunlight).
Monday, April 2, 2007 13
Spectrum of hydrogen-like atoms and ions (Just one electron in orbit around the n nucleus) i
E = hf
= Ei – Ef
nf Z2 Energy levels: E = 13.6 eV Z = 1 for hydrogen n − n2
hc 2 1 1 Energy of photons: E = Ei Ef = = 13.6Z 2 2 eV − λ !nf − ni " In agreement with Balmer’s formula 19 (13.6 1.602 10− J) 7 1 R = × × = 1.097 10 m− hc × ! "
Monday, April 2, 2007 14 Energy levels of the hydrogen atom – Bohr Free electron # # 13.6 Origin of the lines of the gy gy En = eV − n2 hydrogen atom spectrum Ener Ener (Z = 1)
Ground state
Monday, April 2, 2007 15
Free electron Prob. 30.-/7: It is possible to use electromagnetic radiation to ionize atoms. To do so, the atoms must absorb the radiation, the photons of which must have enough energy to remove an electron from an atom.
What is the longest radiation wavelength #
that can be used to ionize the ground state gy of the hydrogen atom? Ener
The least energy to remove a ground state electron is E& – E1.
Ground state, n = 1
Monday, April 2, 2007 16 Free electron
First excited state, n = 2 #
gy Prob. 30.7/8: The electron in a hydrogen atom is in the first excited state, when the electron acquires an
Ener additional 2.86 eV of energy.
What is the quantum number, n, of the state into which the electron moves?
Ground state
Monday, April 2, 2007 17
Prob. 30.12/14: A hydrogen atom is in the ground state. It absorbs energy and makes a transition to the n = 3 excited state. The atom returns to the ground state emitting two photons. What are their wavelengths?
Monday, April 2, 2007 18 Justification of Bohr’s condition on the angular momentum nh Ln = mvr = n = 1, 2, 3... 2!
Substitute for the de Broglie wavelength of the electron: h This does NOT show mv = p = the path of the ! electron, but indicates n = 4 hr nh there is a wave moving So, mvr = = around the orbit –!take ! 2" with a pinch of salt! Therefore, 2!r = n"
" The circumference of the orbit is an integer number of wavelengths.
Condition for a “standing wave” that reinforces itself.
Monday, April 2, 2007 19
Bohr’s theory applies to hydrogen-like atoms and ions – that have just one electron in orbit around the nucleus. Examples: H atom, " Z = 1, 1 electron present He+ ion, " Z = 2, 1 electron removed Li2+ ion, " Z = 3, 2 electrons removed Be3+ ion," Z = 4, 3 electrons removed, etc
Prob. 30.16: Doubly ionized lithium Li2+ (Z = 3) and triply ionized beryllium Be3+ (Z = 4) each emit a line spectrum. For a certain series of lines in the lithium spectrum, the shortest wavelength if 40.5 nm. For the same series of lines in the beryllium spectrum, what is the shortest wavelength?
Both are hydrogen-like atoms, with all electrons removed but one, so Bohr’s formula applies for energies of stationary states:
" En = –13.6 Z2/n2 eV
Monday, April 2, 2007 20 An x-ray tube Spectrum of x-rays
Electrons: KE = eV
+
Vacuum –
• Sharp lines characteristic of the chemical " element of the target. • A smooth background (“Bremsstrahlung”) • A cutoff wavelength, $0, below which there are " no x-rays
Monday, April 2, 2007 21
Spectrum of x-rays • Bremsstrahlung: due to the " electrons being stopped " rapidly on hitting the target – " accelerated charges emit " electromagnetic radiation. #o = 0.03 nm • The shortest wavelength x-rays E = 41.4 keV " are produced when all of the V = 41.4 kV " kinetic energy of the electrons " is converted into x-rays:
" KE = eV = hf = hc/$0
• Characteristic x-rays: from electron transitions in the target atom. " Wavelengths depend on the target. “K-shell”
" K!: (n = 2) " (n = 1)" “L-shell” K%: (n = 3) " (n = 1) (Lyman)
! L!: (n = 3) " (n = 2)" L%: (n = 4) " (n = 2) (Balmer), etc Monday, April 2, 2007 22 x-ray spectra at different voltages
• The characteristic x-ray peaks " do not move – they are " properties of the target atoms.
• The cutoff wavelength, $0, & decreases as the accelerating " voltage increases:
hc & & $0 ∝ 1/V eV = !0 • The Bremsstrahlung " component also shifts as the " voltage is changed.
Monday, April 2, 2007 23
Wavelength of K! x-rays
Moseley found a quite accurate approximate result for K! x-rays for atoms with one electron removed from the K shell:
n = 2 L shell hc 2 1 1 EK = = 13.6(Z 1) eV K ! " − 12 − 22 ! ! "
The same as the Bohr formula, only with Z replaced by Z – 1. n = 1 K shell (there is a second electron in the K shell that partly shields the charge of the nucleus)
Found that some elements were in the wrong place in the periodic table. Identified other previously unknown elements.
30.34 Monday, April 2, 2007 24 Prob. 30.46/34: The atomic number of lead is Z = 82.
According to Moseley and the Bohr model, what is the energy of a K" x-ray photon?
Prob. 30.36/48: What is the minimum potential difference that must be applied to an x-ray tube to knock a K-shell electron completely out of an atom of copper (Z = 29) target?
Monday, April 2, 2007 25
A conventional x-ray picture, colour-enhanced
A picture of the shadows cast by bones and tissue.
The higher the accelerating voltage in the x-ray tube, the shorter the average wavelength of the x-rays and the more penetrating they are.
There is a trade-off between penetrating power of the x-rays and contrast of the image.
Lung tumour
Monday, April 2, 2007 26 CT scans (Computerized Tomography) fire x-rays into the subject and creates x-ray shadow pictures from many directions. Computer analysis is able to reconstruct three-dimensional images from the many pictures. X-ray detectors are used in place of film. In CAT scans (Computerized Axial Tomography) a narrow fan of x-rays is used to image a slice through the body.
Monday, April 2, 2007 27
CAT Scan
(movable)
Monday, April 2, 2007 28 http://en.wikipedia.org/wiki/Image:Venesreconstruction.JPG
Brain vessels reconstructed in 3D
Monday, April 2, 2007 29
The Laser Light Amplification by Stimulated Emission of Radiation
Spontaneous emission – the E = Ei – Ef electron in an excited state falls back to a lower state at a random time, emitting a photon of energy
Ei – Ef in a random direction.
Stimulated emission – a photon
of energy Ei – Ef induces the emission of a photon of the same
E = Ei – Ef energy, resulting in two photons of the same energy travelling in E = E – E i f phase in the same direction
" amplification of the light.
Monday, April 2, 2007 30 The Laser E = Ei – Ef
Metastable state (~10-3 s)
For light amplification to occur: • There must be a large number of atoms in the excited state – a " “population inversion”.
• The excited state must be long-lived – a “metastable state”.
Then atoms in the metastable state stay there long enough for a photon of the right energy to come along and cause stimulated emission.
The result is a build up of photons of the same wavelength that are in phase – coherent light.
Monday, April 2, 2007 31
The Laser
Laser light is:
• Highly coherent. As the light waves are in phase, the combined " amplitude is large, and the intensity much greater than a source of " incoherent light, such as a lamp, that may emit the same number of " photons per second.
• Very sharply defined in wavelength, whereas an incandescent lamp " emits blackbody radiation, which is over a wide wavelength range, " including the invisible infra-red.
• Emitted in a narrow beam with divergence limited by diffraction.
Monday, April 2, 2007 32 Helium-Neon Laser (eg checkout scanner)
Low pressure He (15%) - Ne (85%) in a glass tube
A high voltage discharge in the gas produces a population inversion. Parallel mirrors at the ends reflect light back and forth. Light angled too far from the axis of the laser escapes through the sides. Only light travelling parallel to the axis is amplified.
Monday, April 2, 2007 33
Helium-Neon Laser
Population } inversion
A high voltage discharge excites helium atoms to the 20.61 eV metastable state. He and Ne atoms collide, the excitation is transferred to the Ne metastable level. The population inversion is between the 20.66 and 18.70 eV levels of Ne. The energy of the photons is 20.66 – 18.70 eV = hc/$ $ = 633 nm, red.
Monday, April 2, 2007 34 Lasers There are now many types of laser in power ranges from mW to MW, continuous beam and pulsed:
He-Ne, ruby, argon-ion, CO2, gallium arsenide solid state, chemical dye...
Uses: • playback/recording of CDs, DVDs • checkout scanners • welding metal parts • accurate distance measurement • telecommunications • study of molecular structure • medicine " – selective removal of tissue, example, shaping the cornea of the eye " – removal of “port wine stain” birth marks " – destroying tumours with light-activated drugs " – reattaching detached retinas
Monday, April 2, 2007 35
Holography A laser beam is split into two parts. The first, a reference beam, travels directly to the film. The second reflects from the object and recombines with the reference beam at the film, generating interference fringes there. Hologram (interference fringes, not an image)
Monday, April 2, 2007 36 Holography – the l interference fringes 0
For constructive l interference at P, the path m difference between the reference beam and the P beam that scatters from A should be an integer multiple of the wavelength of the light.
For the bright fringe of order m: lm – l0 = m$
r = l2 l2 = (l + m!)2 l2 m m − 0 0 − 0 ! ! rm = m!(m! + 2l0) 2ml0! " √m ! ∆m " !the fringes are close! together for large m ∆rm ∝ 2√m ! " Monday, April 2, 2007 37
Viewing the image by The fringes act as slits shining laser light on d of varying spacing the hologram
" for first-order sin! = maximum d (the brightest) d = spacing of the fringes
Monday, April 2, 2007 38 Summary of Chapter 30
• The positive charge of the the atom is concentrated in a nucleus that is " ~10-15 - 10-14 m in radius.
• The angular momentum of the electron in an atom is quantized, " leading to quantized energy levels in hydrogen-like atoms: " " " En = –13.6 Z2/n2 eV
• Photons are emitted and absorbed by atoms at the same wavelength " ! identification of elements in the “atmosphere” of stars, discovery of " " helium.
• The energies of K' x-rays can be calculated by replacing Z by Z–1 " ! identification of some chemical elements for the first time.
• Stimulated emission, the laser.
Monday, April 2, 2007 39