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Structure of the Atom

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Title: The Nuclear Atom Author: ELENA Last modified by: Elena Flitsiyan Created Date: 9/13/2003 7:23:24 PM Document presentation format: On-screen Show (4:3) – PowerPoint PPT presentation

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Title: Structure of the Atom


1
Structure of the Atom
2
CHAPTER 4Structure of the Atom
  • The Atomic Models of Thomson and Rutherford
  • Rutherford Scattering
  • The Classic Atomic Model
  • The Bohr Model of the Hydrogen Atom
  • Successes Failures of the Bohr Model
  • Characteristic X-Ray Spectra and Atomic Number
  • Atomic Excitation by Electrons

Niels Bohr (1885-1962)
The opposite of a correct statement is a false
statement. But the opposite of a profound truth
may well be another profound truth. An expert is
a person who has made all the mistakes that can
be made in a very narrow field. Never express
yourself more clearly than you are able to think.
Prediction is very difficult, especially about
the future. - Niels Bohr
3
History
  • 450 BC, Democritus The idea that matter is
    composed of tiny particles, or atoms.
  • XVII-th century, Pierre Cassendi, Robert Hook
    explained states of matter and transactions
    between them with a model of tiny indestructible
    solid objects.
  • 1811 Avogadros hypothesis that all gases at
    given temperature contain the same number of
    molecules per unit volume.
  • 1900 Kinetic theory of gases.
  • Consequence Great three quantization
    discoveries of XX century (1) electric charge
    (2) light energy (3) energy of oscillating
    mechanical systems.

4
Historical Developments in Modern Physics
  • 1895 Discovery of x-rays by Wilhelm Röntgen.
  • 1896 Discovery of radioactivity of uranium by
    Henri Becquerel
  • 1897 Discovery of electron by J.J.Thomson
  • 1900 Derivation of black-body radiation formula
    by Max Plank.
  • 1905 Development of special relativity by
    Albert Einstein, and interpretation of the
    photoelectric effect.
  • 1911 Determination of electron charge by Robert
    Millikan.
  • 1911 Proposal of the atomic nucleus by Ernest
    Rutherford.
  • 1913 Development of atomic theory by Niels
    Bohr.
  • 1915 Development of general relativity by
    Albert Einstein.
  • 1924 - Development of Quantum Mechanics by
    deBroglie, Pauli, Schrödinger, Born, Heisenberg,
    Dirac,.

5

The Structure of Atoms
  • There are 112 chemical elements that have been
    discovered, and there are a couple of additional
    chemical elements that recently have been
    reported.
  • Flerovium is the radioactive chemical element
    with the symbol Fl and atomic number 114. The
    element is named after Russian physicist Georgy
    Flerov, the founder of the Joint Institute for
    Nuclear Research in Dubna, Russia, where the
    element was discovered.

Georgi Flerov (1913-1990)
The name was adopted by IUPAC on May 30, 2012.
About 80 decays of atoms of flerovium have been
observed to date. All decays have been assigned
to the five neighbouring isotopes with mass
numbers 285289. The longest-lived isotope
currently known is 289Fl with a half-life of 2.6
s, although there is evidence for a nuclear
isomer, 289bFl, with a half-life of 66 s, that
would be one of the longest-lived nuclei in the
superheavy element region.
6
The Structure of Atoms
  • Each element is characterized by atom that
    contains a number of protons Z, and equal number
    of electrons, and a number of neutrons N. The
    number of protons Z is called the atomic number.
    The lightest atom, hydrogen (H), has Z1 the
    next lightest atom, helium (He), has Z2 the
    third lightest, lithium (Li), has Z3 and so
    forth.

7
The Nuclear Atoms
  • Nearly all the mass of the atom is concentrated
    in a tiny nucleus which contains the protons and
    neutrons.
  • Typically, the nuclear radius is approximately
    from 1 fm to 10 fm (1fm 10-15m). The distance
    between the nucleus and the electrons is
    approximately 0.1 nm100,000fm. This distance
    determines the size of the atom.

8
Nuclear Structure
An atom consists of an extremely small,
positively charged nucleus surrounded by a cloud
of negatively charged electrons. Although
typically the nucleus is less than one
ten-thousandth the size of the atom, the nucleus
contains more than 99.9 of the mass of the atom!
9
     The number of protons in the nucleus, Z, is
called the atomic number. This determines
what chemical element the atom is. The number of
neutrons in the nucleus is denoted by N. The
atomic mass of the nucleus, A, is equal to Z N.
A given element can have many different
isotopes, which differ from one another by the
number of neutrons contained in the nuclei. In a
neutral atom, the number of electrons orbiting
the nucleus equals the number of protons in the
nucleus.
10
Structure of the Atom
  • Evidence in 1900 indicated that the atom was not
    a fundamental unit
  • There seemed to be too many kinds of atoms, each
    belonging to a distinct chemical element (way
    more than earth, air, water, and fire!).
  • Atoms and electromagnetic phenomena were
    intimately related (magnetic materials
    insulators vs. conductors different emission
    spectra).
  • Elements combine with some elements but not with
    others, a characteristic that hinted at an
    internal atomic structure (valence).
  • The discoveries of radioactivity, x-rays, and the
    electron (all seemed to involve atoms breaking
    apart in some way).

11
The Nuclear Atoms
  • We will begin our study of atoms by discussing
    some early models, developed in beginning of 20
    century to explain the spectra emitted by
    hydrogen atoms.

12
Atomic Spectra
  • By the beginning of the 20th century a large
    body of data has been collected on the emission
    of light by atoms of individual elements in a
    flame or in a gas exited by electrical discharge.

Diagram of the spectrometer
13
Atomic Spectra
Light from the source passed through a narrow
slit before falling on the prism. The purpose of
this slit is to ensure that all the incident
light strikes the prism face at the same angle so
that the dispersion by the prism caused the
various frequencies that may be present to strike
the screen at different places with minimum
overlap.
14
  • The source emits only two wavelengths, ?2gt?1.
    The source is located at the focal point of the
    lens so that parallel light passes through the
    narrow slit, projecting a narrow line onto the
    face of the prism. Ordinary dispersion in the
    prism bends the shorter wavelength through the
    lager total angel, separating the two wavelength
    at the screen.

15
  • In this arrangement each wavelength appears as a
    narrow line, which is the image of the slit. Such
    a spectrum was dubbed a line spectrum for that
    reason. Prisms have been almost entirely replaced
    in modern spectroscopes by diffraction gratings,
    which have much higher resolving power.

16
  • When viewed through the spectroscope, the
    characteristic radiation, emitted by atoms of
    individual elements in flame or in gas exited by
    electrical charge, appears as a set of discrete
    lines, each of a particular color or wavelength.
  • The positions and intensities of the lines are
    a characteristic of the element. The wavelength
    of these lines could be determined with great
    precision.

17
  • Emission line spectrum of hydrogen in the
    visible and near ultraviolet. The lines appear
    dark because the spectrum was photographed. The
    names of the first five lines are shown. As is
    the point beyond which no lines appear, H8 called
    the limits of the series.

18
Atomic Spectra
  • In 1885 a Swiss schoolteacher, Johann Balmer,
    found that the wavelengths of the lines in the
    visible spectrum of hydrogen can be represented
    by formula
  • Balmer suggested that this might be a special
    case of more general expression that would be
    applicable to the spectra of other elements.

19
Atomic Spectra
  • Such an expression, found by J.R.Rydberg and W.
    Ritz and known as the Rydberg-Ritz formula, gives
    the reciprocal wavelengths as
  • where m and n are integers with ngtm, and R is
    the Rydberg constant.

20
Atomic Spectra
  • The Rydberg constant is the same for all
    spectral series.
  • For hydrogen the RH 1.096776 x 107m-1.
  • For very heavy elements R approaches the value
    R8 1.097373 x 107m-1.
  • Such empirical expressions were successful in
    predicting other spectra, such as other hydrogen
    lines outside the visible spectrum.

21
Atomic Spectra
  • So, the hydrogen Balmer series wavelength are
    those given by Rydberg equation
  • with m2 and n3,4,5,
  • Other series of hydrogen spectral lines were
    found for m1 (by Lyman) and m3 (by Paschen).

22
Hydrogen Spectral Series
  • Compute the wavelengths of the first lines of
    the Lyman, Balmer, and Paschen series.

23
  • Emission line spectrum of hydrogen in the
    visible and near ultraviolet. The lines appear
    dark because the spectrum was photographed. The
    names of the first five lines are shown, as is
    the point beyond which no lines appear, H8 called
    the limits of the series.

24
The Limits of Series
  • Find the predicted by Rydberg-Ritz formula
    for Lyman, Balmer, and Paschen series.

25
  • A portion of the emission spectrum of sodium.
    The two very close bright lines at 589 nm are the
    D1 and D2 lines. They are the principal radiation
    from sodium street lighting.

26
  • A portion of emission spectrum of mercury.

27
  • Part of the dark line (absorption) spectrum of
    sodium. White light shining through sodium vapor
    is absorbed at certain wavelength, resulting in
    no exposure of the film at those points. Note
    that frequency increases toward the right ,
    wavelength toward the left in the spectra shown.

28
Nuclear Models
  • Many attempts were made to construct a model of
    the atom that yielded the Balmer and Rydberg-Ritz
    formulas.
  • It was known that an atom was about 10-10m in
    diameter, that it contained electrons much
    lighter than the atom, and that it was
    electrically neutral.
  • The most popular model was that of J.J.Thomson,
    already quite successful in explaining chemical
    reactions.

29
Knowledge of atoms in 1900
Electrons (discovered in 1897) carried the
negative charge. Electrons were very light, even
compared to the atom. Protons had not yet been
discovered, but clearly positive charge had to be
present to achieve charge neutrality.
30
Thomsons Atomic Model
  • Thomsons plum-pudding model of the atom had
    the positive charges spread uniformly throughout
    a sphere the size of the atom, with electrons
    embedded in the uniform background.
  • In Thomsons view, when the atom was heated, the
    electrons could vibrate about their equilibrium
    positions, thus producing electromagnetic
    radiation.
  • Unfortunately, Thomson couldnt explain spectra
    with this model.

31
  • The difficulty with all such models was that
    electrostatic forces alone cannot produce stable
    equilibrium. Thus the charges were required to
    move and, if they stayed within the atom, to
    accelerate. However, the acceleration would
    result in continuous radiation, which is not
    observed.
  • Thomson was unable to obtain from his model a
    set of frequencies that corresponded with the
    frequencies of observed spectra.
  • The Thomson model of the atom was replaced by
    one based on results of a set of experiments
    conducted by Ernest Rutherford and his student
    H.W.Geiger.

32
Experiments of Geiger and Marsden
  • Rutherford, Geiger, and Marsden conceived a new
    technique for investigating the structure of
    matter by scattering a particles from atoms.

33
  • Rutherford was investigating radioactivity
    and had shown that the radiations from uranium
    consist of at least two types, which he labeled a
    and ß.
  • He showed, by an experiment similar to that of
    Thompson, that q /m for the a - particles was
    half that of the proton.
  • Suspecting that the a particles were double
    ionized helium, Rutherford in his classical
    experiment let a radioactive substance decay and
    then, by spectroscopy, detected the spectra line
    of ordinary helium.

34
Beta decay
  • Beta decay occurs when the neutron to proton
    ratio is too great in the nucleus and causes
    instability. In basic beta decay, a neutron is
    turned into a proton and an electron. The
    electron is then emitted. Here's a diagram of
    beta decay with hydrogen-3

Beta Decay of Hydrogen-3 to Helium-3.
35
Alpha Decay
  • The reason alpha decay occurs is because the
    nucleus has too many protons which cause
    excessive repulsion. In an attempt to reduce the
    repulsion, a Helium nucleus is emitted. The way
    it works is that the Helium nuclei are in
    constant collision with the walls of the nucleus
    and because of its energy and mass, there exists
    a nonzero probability of transmission. That is,
    an alpha particle (Helium nucleus) will tunnel
    out of the nucleus. Here is an example of alpha
    emission with americium-241

Alpha Decay of Americium-241 to Neptunium-237
36
Gamma Decay
  • Gamma decay occurs because the nucleus is at too
    high an energy. The nucleus falls down to a lower
    energy state and, in the process, emits a high
    energy photon known as a gamma particle. Here's a
    diagram of gamma decay with helium-3

Gamma Decay of Helium-3
37
  • Rutherford was investigating radioactivity
    and had shown that the radiations from uranium
    consist of at least two types, which he labeled a
    and ß.
  • He showed, by an experiment similar to that of
    Thompson, that q /m for the a - particles was
    half that of the proton.
  • Suspecting that the a particles were double
    ionized helium, Rutherford in his classical
    experiment let a radioactive substance decay and
    then, by spectroscopy, detected the spectra line
    of ordinary helium.

38
  • Schematic diagram of the Rutherford
    apparatus. The beam of a - particles is defined
    by the small hole D in the shield surrounding the
    radioactive source R of 214Bi . The a beam
    strikes an ultra thin gold foil F, and a
    particles are individually scattered through
    various angels ?. The experiment consisted of
    counting the number of scintillations on the
    screen S as a function of ?.

39
  • A diagram of the original apparatus as it appear
    in Geigers paper describing the results.

40
  • An a-particle by such an atom (Thompson model)
    would have a scattering angle ? much smaller than
    10. In the Rutherfords scattering experiment
    most of the a-particles were either undeflected,
    or deflected through very small angles of the
    order 10, however, a few a-particles were
    deflected through angles of 900 and more.

41
  • An a-particle by such an atom (Thompson model)
    would have a scattering angle ? much smaller than
    10. In the Rutherfords scattering experiment a
    few a-particles were deflected through angles of
    900 and more.

42
Experiments of Geiger and Marsden
  • Geiger showed that many a particles were
    scattered from thin gold-leaf targets at backward
    angles greater than 90.

43
Electrons cant back-scatter a particles.
  • Calculate the maximum scattering angle -
    corresponding to the maximum momentum change.

It can be shown that the maximum momentum
transfer to the a particle is
Determine qmax by letting Dpmax be perpendicular
to the direction of motion
too small!
44
Try multiple scattering from electrons
  • If an a particle is scattered by N electrons

N the number of atoms across the thin gold
layer, t 6 10-7 m
n
The distance between atoms, d n-1/3, is
N t / d
still too small!
45
  • If the atom consisted of a positively charged
    sphere of radius 10-10 m, containing electrons as
    in the Thomson model, only a very small
    scattering deflection angle could be observed.
  • Such model could not possibly account for the
    large angles scattering. The unexpected large
    angles a-particles scattering was described by
    Rutherford with these words
  • It was quite incredible event that ever
    happened to me in my life. It was as incredible
    as if you fired a 15-inch shell at a piece of
    tissue paper and it came back and hit you.

46
Rutherfords Atomic Model
even if the a particle is
scattered from all 79 electrons in each atom of
gold. Experimental results were not consistent
with Thomsons atomic model. Rutherford proposed
that an atom has a positively charged core
(nucleus!) surrounded by the negative
electrons. Geiger and Marsden confirmed the idea
in 1913.
Ernest Rutherford (1871-1937)
47
  • Rutherford concluded that the large angle
    scattering could result only from a single
    encounter of the a particle with a massive charge
    with volume much smaller than the whole atom.
  • Assuming this nucleus to be a point charge,
    he calculated the expected angular distribution
    for the scattered a particles.
  • His predictions on the dependence of scattering
    probability on angle, nuclear charge and kinetic
    energy were completely verified in experiments.

48
  • Rutherford Scattering geometry. The nucleus is
    assumed to be a point charge Q at the origin O.
    At any distance r the a particle experiences a
    repulsive force kqaQ/r2. The a particle travel
    along a hyperbolic path that is initially
    parallel to line OA a distance b from it and
    finally parallel to OB, which makes an angle ?
    with OA.

49
Rutherford Scattering
Theres a relationship between the impact
parameter b and the scattering angle q.
When b is small, r is small. the Coulomb force
is large. ? can be large and the particle can be
repelled backward.
where
50
Rutherford Scattering
  • Any particle inside the circle of area p b02
    will be similarly (or more) scattered.

The cross section s p b2 is related to the
probability for a particle being scattered by a
nucleus (t foil thickness) The fraction of
incident particles scattered is
51
  • Force on a point charge versus distance r from
    the center of a uniformly charged sphere of
    radius R. Outside the sphere the force is
    proportional to Q/r2, inside the sphere the force
    is proportional to qI/r2 Qr/R3, where qI
    Q(r/R)3 is the charge within a sphere of radius
    r. The maximum force occurs at r R

52
  • Two a particles with equal kinetic energies
    approach the positive charge Q Ze with impact
    parameters b1 and b2, where b1ltb2. According to
    equation for impact parameter in this case ?1 gt
    ?2.

53
  • The path of a particle can be shown to be a
    hyperbola, and the scattering angle ? can be
    relate to the impact parameter b from the laws of
    classical mechanics.
  • The quantity pb2, which has the dimension of
    the area , is called the cross section s for
    scattering.

54
  • The cross section s is thus defined as the
    number of particles scattered per nucleus per
    unit time divided by the incident intensity.

The total number of nuclei of foil atoms in the
area covered by the beam is nAt, where n is the
number of foil atoms per unit volume. A is the
area of the beam, and t is the thickness of the
foil.
55
  • The total number of particles scattered per
    second is obtained by multiplying pb2I0 by the
    number of nuclei in the scattering foil. Let n be
    the number of nuclei per unit volume
  • For a foil of thickness t, the total number of
    nuclei as seen by the beam is nAt, where A is
    the area of the beam. The total number scattered
    per second through angles grater than ? is thus
    pb2I0ntA. If divide this by the number of a
    particles incident per second I0A we get the
    fraction of a particles f scattered through
    angles grater than ?
  • f pb2nt

56
On the base of that nuclear model
Rutherford derived an expression for the number
of a particles ?N that would be scattered at any
angle ? All this predictions was
verified in Geiger experiments, who observe
several hundreds thousands a particles.
57
  • (a)Geiger data for a scattering from thin Au and
    Ag foils. The graph is in log-log plot to cover
    over several orders of magnitudes.
  • (b)Geiger also measured the dependence of ?N on
    t for different elements, that was also in good
    agreement with Rutherford
    formula.

58
  • Data from Rutherfords group showing observed a
    scattering at a large fixed angle versus values
    of rd
  • computed from for various
    kinetic energies.

59
The Size of the Nucleus
  • This equation can be used to estimate the size
    of the nucleus
  • For the case of 7.7-MeV a particles the
    distance of closest approach for a head-on
    collision is

60
The Classical Atomic Model
  • Consider an atom as a planetary system.
  • The Newtons 2nd Law force of attraction on the
    electron by the nucleus is

where v is the tangential velocity of the
electron
The total energy is then
This is negative, so the system is bound, which
is good.
61
The Planetary Model is Doomed
  • From classical EM theory, an accelerated
    electric charge radiates energy (electromagnetic
    radiation), which means the total energy must
    decrease. So the radius r must decrease!!

Electron crashes into the nucleus!?
62
  • According to classical physics, a charge e
    moving with an acceleration a radiates at a rate
  • Show that an electron in a classical hydrogen
    atom spirals into the nucleus at a rate

(b) Find the time interval over which the
electron will reach r 0, starting from r0
2.00 1010 m.
63
The Bohrs Postulates
  • Bohr overcome the difficulty of the collapsing
    atom by postulating that only certain orbits,
    called stationary states, are allowed, and that
    in these orbits the electron does not radiate. An
    atom radiates only when the electron makes a
    transition from one allowed orbit (stationary
    state) to another
  • 1. The electron in the hydrogen atom can move
    only in certain nonradiating, circular orbits
    called stationary states.
  • 2. The photon frequency from energy conservation
    is given by
  • where Ei and Ef are the energy of initial and
    final state, h is the Planks constant.

64
  • Such a model is mechanically stable , because
    the Coulomb potential
  • provides the centripetal force
  • The total energy for a such system can be
    written as the sum of kinetic and potential
    energy

65
The Bohrs Postulates
  • Combining the second postulate with the
    equation for the energy we obtain

where r1 and r2 are the radii of the initial
and final orbits.
66
The Bohrs Postulates

To obtain the frequencies implied by the
experimental Rydberg-Ritz formula,
it is evident that the radii of the stable orbits
must be proportional to the squares of integers.
67
The Bohrs Postulates
  • Bohr found that he could obtain this condition
    if he postulates the angular momentum of the
    electron in a stable orbit equals an integer
    times hh/2p. Since the angular momentum of a
    circular orbit is just mvr, the third Bohrs
    postulate is
  • 3.
    n1,2,3.
  • where hh/2p1.055 x 10-34Js6.582x10 -16eVs

68
The Bohrs Postulates
  • The obtained equation mvr nh/2pnh relates the
    speed of electron v to the radius r. Since we had

or
We can write
or
69
The Bohrs Postulates
  • Solving for r, we obtain

where a0 is called the first Bohrs radius
70
Bohrs Postulates
  • Substituting the expression for r in equation for
    frequency
  • If we will compare this expression with Z1 for
    fc/? with the empirical Rydberg-Ritz formula
  • we will obtain for the Rydberg constant

71
Bohrs Postulates
  • Using the known values of m, e, and h, Bohr
    calculated R and found his results to agree with
    the spectroscopy data.
  • The total mechanical energy of the electron in
    the hydrogen atom is related to the radius of the
    circular orbit

72
Energy levels
  • If we will substitute the quantized value of r
    as given by
  • we obtain

73
Energy levels
  • or
  • where
  • The energies En with Z1 are the quantized
    allowed energies for the hydrogen atom.

74
  • Energy level diagram for hydrogen showing the
    seven lowest stationary states. The energies of
    infinite number of levels are given by En
    (-13.6/n2)eV, where n is an integer.

75
  • A hydrogen atom is in its first excited state (n
    2). Using the Bohr theory of the atom,
    calculate (a) the radius of the orbit, (b) the
    linear momentum of the electron, (c) the angular
    momentum of the electron, (d) the kinetic energy
    of the electron, (e) the potential energy of the
    system, and (f) the total energy of the system.

76
Energy levels
  • Transitions between this allowed energies
    result in the emission or absorption of a photon
    whose frequency is given by
  • and whose wavelength is

77
Energy levels
  • Therefore is convenient to have the value of hc
    in electronvolt nanometers!
  • hc 1240 eVnm
  • Since the energies are quantized, the
    frequencies and the wavelengths of the radiation
    emitted by the hydrogen atom are quantized in
    agreement with the observed line spectrum.

78
  • (a) In the classical orbital model, the electron
    orbits about the nucleus and spirals into the
    center because of the energy radiated.
  • (b) In the Bohr model, the electron orbits
    without radiating until it jumps to another
    allowed radius of lower energy, at which time
    radiation is emitted.

79
?21hc / (E1-E2)
  • Energy level diagram for hydrogen showing the
    seven lowest stationary states and the four
    lowest energy transitions for the Lyman, Balmer,
    and Pashen series. The energies of infinite
    number of levels are given by En (-13.6/n2)eV,
    where n is an integer.

80
(No Transcript)
81
  • The spectral lines corresponding to the
    transitions shown for the three series.

82
?21hc / (E1-E2)
83
  • Compute the wavelength of the Hß spectral line
    of the Balmer series predicted by Bohr model.

84
  • A hydrogen atom at rest in the laboratory emits
    a photon of the Lyman a radiation. (a) Compute
    the recoil kinetic energy of the atom. (b) What
    fraction of the excitation energy of the n 2
    state is carried by the recoiling atom? (Hint
    Use conservation of momentum.)

85
  • In a hot star, because of the high temperature,
    an atom can absorb sufficient energy to remove
    several electrons from the atom. Consider such a
    multiply ionized atom with a single remaining
    electron. The ion produces a series of spectral
    lines as described by the Bohr model. The series
    corresponds to electronic transitions that
    terminate in the same final state. The longest
    and shortest wavelengths of the series are 63.3
    nm and 22.8 nm, respectively. (a) What is the
    ion? (b) Find the wavelengths of the next three
    spectral lines nearest to the line of longest
    wavelength.

86
  • A stylized picture of Bohr circular orbits for
    n1,2,3,4. The radii rnn2.
  • In high Z-elements (elements with Z 12),
    electrons are distributed over all the orbits
    shown. If an electron in the n1 orbit is knocked
    from the atom (e.g., by being hit by a fast
    electron accelerated by the voltage across an
    x-ray tube) the vacancy that produced is filed by
    an electron of higher energy (i.e., n2 or
    higher).

The difference in energy between the two orbits
is emitted as a photon, whose wavelength will be
in the x-ray region of the spectrum, if Z is
large enough.
87
(a)
(b)
(c)
  • Characteristic x-ray spectra. (a) Part the
    spectra of neodymium (Z60) and samarium
    (Z62).The two pairs of bright lines are the Ka
    and Kß lines. (b) Part of the spectrum of the
    artificially produced element promethium (Z61),
    its Ka and Kß lines fall between those of Nd and
    Sm. (c) Part of the spectrum of all three
    elements.

88
The Franck-Hertz Experiment
  • While investigating the inelastic
    scattering of electrons, J.Frank and G.Hertz in
    1914 performed an important experiment that
    confirmed by direct measurement Bohrs hypothesis
    of energy quantization in atoms.
  • The experiment involved measuring the plate
    current as a function of V0.

89
The Franck-Hertz Experiment
Schematic diagram of the Franck-Hertz experiment.
Electrons ejected from the heated cathode C at
zero potential are drawn to the positive grid G.
Those passing through the holes in the grid can
reach the plate P and contribute in the current
I, if they have sufficient kinetic energy to
overcome the small back potential ?V. The tube
contains a low pressure gas of the element being
studied.
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The Franck-Hertz Experiment
  • As V0 increased from 0, the current increases
    until a critical value (about 4.9 V for mercury)
    is reached. At this point the current suddenly
    decreases. As V0 is increased further, the
    current rises again.
  • They found that when the electrons kinetic
    energy was 4.9 eV or greater, the vapor of
    mercury emitted ultraviolet light of wavelength
    0.25 µm.

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Current versus acceleration voltage in the
Franck-Hertz experiment.
  • The current decreases because many electrons
    lose energy due to inelastic collisions with
    mercury atoms in the tube and therefore cannot
    overcome the small back potential.

Current, (mAmp)
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  • The regular spacing of the peaks in this curve
    indicates that only a certain quantity of energy,
    4.9 eV, can be lost to the mercury atoms. This
    interpretation is confirmed by the observation of
    radiation of photon energy 4.9 eV, emitted by
    mercury atoms.

Current, (mAmp)
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  • Suppose mercury atoms have an energy level 4.9
    eV above the lowest energy level. An atom can be
    raised to this level by collision with an
    electron it later decays back to the lowest
    energy level by emitting a photon. The wavelength
    of the photon should be

This is equal to the measured wavelength,
confirming the existence of this energy level of
the mercury atom.

Similar experiments with other atoms yield the
same kind of evidence for atomic energy levels.
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  • Lets consider an experimental tube filled by
    hydrogen atoms instead of mercury. Electrons
    accelerated by V0 that collide with hydrogen
    electrons cannot transfer the energy to letter
    electrons unless they have acquired kinetic
    energy
  • eV0E2 E110.2eV

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  • If the incoming electron does not have
    sufficient energy to transfer ?E E2 - E1 to the
    hydrogen electron in the n1 orbit (ground
    state), than the scattering will be elastic.

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  • If the incoming electron does have at least ?E
    kinetic energy, then an inelastic collision can
    occur in which ?E is transferred to the n1
    electron, moving it to the n2 orbit. The excited
    electron will typically return to the ground
    state very quickly, emitting a photon of energy
    ?E.

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  • Energy loss spectrum measurement. A
    well-defined electron beam impinges upon the
    sample. Electrons inelastically scattered at a
    convenient angle enter the slit of the magnetic
    spectrometer, whose B field is directed out of
    the page, and turn through radii R determined by
    their energy (Einc E1) via equation

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  • An energy-loss spectrum for a thin Al film.

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Reduced mass correction
  • The assumption by Bohr that the nucleus is
    fixed is equivalent the assumption that it has an
    infinity mass.
  • If instead we will assume that proton and
    electron both revolve in circular orbits about
    their common center of mass we will receive even
    better agreement for the values of the Rydberg
    constant R and ionization energy for the
    hydrogen.
  • We can take in account the motion of the
    nucleus (the proton) very simply by using in
    Bohrs equation not the electron rest mass m but
    a quantity called the reduce mass µ of the
    system. For a system composed from two masses m1
    and m2 the reduced mass is defined as

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Reduced mass correction
  • If the nucleus has the mass M its kinetic
    energy will be ½Mv2 p2/2M, where p Mv is the
    momentum.
  • If we assume that the total momentum of the
    atom is zero, from the conservation of momentum
    we will have that momentum of electron and
    momentum of nucleus are equal on the magnitude.

101
Reduced mass correction
  • The total kinetic energy is then
  • The Rydberg constant equation than changed to
  • The factor µ was called mass correction factor.

102
  • As the Earth moves around the Sun, its orbits
    are quantized. (a) Follow the steps of Bohrs
    analysis of the hydrogen atom to show that the
    allowed radii of the Earths orbit are given by
  • where MS is the mass of the Sun, ME is the mass
    of the Earth, and n is an integer quantum number.
    (b) Calculate the numerical value of n. (c) Find
    the distance between the orbit for quantum number
    n and the next orbit out from the Sun
    corresponding to the quantum number n 1
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