10.The Structure of the Atom - Inside Minesinside.mines.edu/~fsarazin/phgn310/PDFs/Slides/10.The...

PHGN300/310: The structure of the atomFred Sarazin ([email protected])Physics Department, Colorado School of Mines

The Structure of the Atom


• ~400 BCGreek Philosopher Democritus believed that each kind ofmatter could be subdivided into smaller and smaller bits untilone reached the very limit beyond which no further divisionwas possible (“atomos” = “that cannot be cut”).

• As of 1900, about 70 different “atoms” are known (elements àdifferent chemical properties). It is argued that this number is too large to really be an elementary constituent of matter

• Hints:• Atoms and electromagnetic radiations (not understood

but…)• Chemistry: the problem of valence. How molecules are

formed ? Why some combinations of atoms just don’t bind as molecules ?

• New phenomena: X-rays, Radioactivity (1896),…

Henri Becquerel 1852-1908, Nobel Prize in Physics 1903

Atoms?


• Scientists are struggling to understand the atom and reproduce experimental results, such as the spectral lines. [Physics of the electrons inside the atom à Atomic Physics]

• Also puzzled with a new observed phenomenon: radioactivity. [Physics of the nucleus à Nuclear Physics]

• But, around 1900, nobody knows the structure of the atom: a puzzle of evidences that needs to be put together.

Inside an atom?


1895: X-rays are interpreted as a chemical process similar to phosphorescence and fluorescenceà An external source is required to “trigger” the emission of X-rays

Henri Becquerel’s idea:• Look for X-ray emission in known phosphorescent/fluorescent substances.

Experimental procedure:• Wrap a photographic plate with thick black paper, place the substance to be tested

on the paper and then expose to sunlight for several hours à Sunlight = external source.

If X-rays were emitted, they would pass through the paper and fog the plate.Tests are all negative, except for a Uranium Salt.

Sunlight

Substance

Photographic Plate

Black paper

Discovery of radioactivity (1896)


• Reproducibility: an experimental result should be reproducible

• End of February, Henri Becquerel is ready to repeat the experiment. But it is cloudy over Paris !!! Becquerel puts his experimental setup in a drawer till March 1st.

• When Becquerel develops the plate (not exposed to sunlight), he finds that the fogging is just as intense as when the uranium salt had been exposed to sunlight.

à No need for external energy source !!!à The energy is already available/stored in the material.

Becquerel experiment


Pierre (1859-1906) and (1867-1934) Marie Curie

• Isolation of Radium and Polonium (1898), Nobel Prize in Physics in 1903

• Marie Curie (1867-1934): Got her PhD in 1903. She also get a Nobel Prize in Chemistry (1911)

Search for elements of similar properties

• Radium is so active that it shines brightly in its pure form

• Marie Curie about Radium: “it’s active and it radiates”àRADIOACTIVITY

• Marie Curie dies of Leukemia at 67 (very likely due to exposure to radiations)


Part I: the atomic models of Thomson and Rutherford

J.J.Thomson1856-1940

Nobel Prize in Physics 1906

Ernerst Rutherford1871-1937

Nobel Prize in Physics 1908


• Atoms are neutral

• Electrons are much less massive than the atom

• Number of electrons Ne- corresponds to about half the atomic mass number. Example:– Carbon: Atomic Mass Number A = 12; Ne-=6– Oxygen: Atomic Mass Number A = 16; Ne-=8

• Size of the atom ~ 10-10m

• The atoms can emit and absorb electromagnetic radiations

Known facts (~1900)


• Thomson’s “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 Thomson’s view, when the atom is heated, the electrons could vibrate about their equilibrium positions, thus producing electromagnetic radiation.

• The model fails: cannot reproduce the spectral lines of the Hydrogen atom

Positively charged, so that the whole atom is neutral

Thomson’s atomic model


• 1898: Pierre and Marie discover the Radium à Emission of a-particle

• 1900: Rutherford and Royds determine the nature of the a-particles, they are charged Helium atoms

Radioactivity?


Ernest Rutherford 1871-1937 Nobel Prize in Physics 1908

Ernest Rutherford


• 1909: Rutherford, Geiger and Marsden conceive a new technique to probe the structure of matter by scattering a-particles from atoms

• Geiger shows that many a-particles are scattered from thin gold-leaf targets at backward angles greater than 90º

Geiger and Marsden experiments


• In contradiction with J.J.Thomson’s model:– At best, a-particles should only

be slightly deflected

• Large deflections ? (see example 4.1, p129-130)– 99.95% of the MASS of the atom

lies in a hard, dense nucleusoccupying only ~10-15m of the atomic volume.

Rutherford (1911): ”Considering the evidence as a whole, it seems simplest to suppose that the atom contains a central charge distributed through a small volume, and that a large single deflections are due to the central charge as a whole, and not to its constituents”.

Analysis / conclusions


Impact parameter b:

Note: the nucleus is 105 times smaller than the atom, large deflections do not occur often

Rutherford scattering

𝑏 =𝑍$𝑍%𝑒%

8𝜋𝜀*𝐾cot

𝜃2

where 𝐾 =12𝑚𝑣*

%


Part II: the Bohr model (of the Hydrogen atom)


• Force applied on the electron:

• From which one deduce:

• Total energy* with 𝐾 = $%𝑚𝑣%:

Planetary model

�⃗�6 = −1

4𝜋𝜖*𝑒%

𝑟% �̂�< = −𝑚𝑣%

𝑟 �̂�<

𝑣 =𝑒

4𝜋𝜖*𝑚𝑟�

𝐸 = 𝐾 + 𝑉 =𝑒%

8𝜋𝜖*𝑟−

𝑒%

4𝜋𝜖*𝑟= −

𝑒%

8𝜋𝜖*𝑟

(*See derivation)


• Atom (neutral) = nucleus (+q) + q electrons

• Assuming the Hydrogen atom:

• The electron is attracted by the nucleus

• Even in circular motion around the nucleus, the electron loses energy:

• Radial acceleration: ar = v2/R

• Classical e.m. theory: an accelerating charge continuously radiates energy, r decreases…

The electron would eventually crash into the nucleus

Failure of the classical (planetary) atomic model


• Doomed because the electron radiates energy, while orbiting around the nucleus.

• But there must be some “truth” to it, because Rutherford was able to successfully describe the Coulomb scattering experiment using a nucleus much smaller than the size of an atom (= nucleus + electrons)

• In 1913, Niels Bohr 1885-1962 postulates that the electrons may be in stable (non-radiating) circular orbits, called stationary orbits

Failure of the classical (planetary) atomic model


• Existence of “stationary” states for the orbiting electrons

• Transitions between stationary states.Example: Δ𝐸 = ℎ𝜐 = 𝐸D − 𝐸%

• Classical laws of physics do not apply to transitions between stationary states

• The electron can only exist in orbits for which its angular momentum is given by: 𝐿 = 𝑟×�⃗� = 𝑚𝑣𝑟 = 𝑛ℎ/2𝜋 with n=1,2,3…

Niels Bohr’s postulates


• From Bohr’s postulate:

• Since:

• One can deduce* the diameter of the hydrogen atom for stationary states:

with a0, the Bohr radius

Bohr radius

𝑣 =𝑛ℏ𝑚𝑟

𝑣 =𝑒

4𝜋𝜖*𝑚𝑟�

(planetary model)

𝑟L =4𝜋𝜀*𝑛%ℏ%

𝑚𝑒% = 𝑛%𝑎*

(*See derivation)


with: h = 6.626 x 10-34 J.se0 = 8.85 x 10-12 C2/N.m2

m = 9.1 x 10-31 kge = 1.6 x 10-19 C

Exercise

• Calculate the value of the Bohr radius a0

𝑟L =4𝜋𝜀*𝑛%ℏ%

𝑚𝑒% = 𝑛%𝑎*


• Energy of the stationary states (with E0 = -13.6 eV)

• Emission of light:

• Using $N= O

P, one gets*:

Energy levels of the Hydrogen atom

𝐸L = −𝑒%

8𝜋𝜀*𝑟L= −

𝑒%

8𝜋𝜀*𝑎*𝑛%= −

𝐸*𝑛%

ℎ𝜐 = 𝐸QLQRQST − 𝐸UQLST

1𝜆 =

𝐸QLQRQST − 𝐸UQLSTℎ𝑐 = 𝑅Y

1𝑛UQLST% −

1𝑛QLQRQST%

with R∞, the Rydberg constant for Hydrogen

(*See derivation)


ninitial > nfinal:

• nfinal =1 & ninitial=2, 3,… Lyman Series

• nfinal =2 & ninitial=3, 4,… Balmer Series

• nfinal =3 & ninitial=4, 5,… Pashen Series

By 1913, some of these lines were not observed.Yet they were predicted by Bohr’s model.

Spectral line of Hydrogen


• When shining a white light (all visible frequencies), one observes a spectrum containing dark lines à Absorption

• These lines are the same than those observed in the Emission spectrum.

• Hydrogen in its ground state:– Transition induced by

incoming photon hnà Absorption

– Then, the atom goes back to its ground state by emitting photon(s)à Emission

Emission & absorption


• The Correspondence Principle (Bohr): “In the limits where classical and quantum theories should agree, the quantum theory must reduce to the classical result.”

”Quantum” to classical mechanics


• The Bohr model was a great step of the new quantum theory, but it had its limitations:

– Works only to single-electron atoms.

– Could not account for the intensities or the fine structure of the spectral lines.

– Could not explain the binding of atoms into molecules.

Limitations


• The electron in a hydrogen atom in the n=2 state absorbs a photon of wavelengthl=327 nm. Does the hydrogen atom become ionized ?

Exercise


Henri G. J. Moseley1887 – 1915

Part III: X-rays


• While an electron is in the vicinity of a nucleus, it feels its electromagnetic field• Acceleration (v2/R) = Electromagnetic Radiation

• Bremsstrahlung (“Braking radiation”) effect à Emission of Photons (X-rays: l=0.01 to 1nm)

• Not a continuous process, since a photon can only be created with an energy E = hn

X-ray production


• When Bohr’s model appears (1913), people know about X-rays (Roentgen, 1895) obtained from the bombardment of e- into materials:

– Bremstrahlung (Continuum spectrum)

– Peaks (which are different for each element)

X-rays


• Historically:

– States were given letters:• n=1 K shell,• n=2 L shell,• n=3 M shell,• n=4 N shell, • etc…

– Transitions Dn between shells:• Dn=1 a,• Dn=2 b,• etc…

Shells


• Bohr’s model works for Hydrogen

• What about the other elements?– Z: Atomic number à Number of electrons– Examples:

• Z=1 Hydrogen• Z=2 Helium• Z=6 Carbon• Z=92 Uranium

– Approximation: we ignore the mutual repulsion of the electrons.

• The Coulomb force between one electron and the nucleus is: F = (Ze2)/(4pe0r

2)• Applying the same treatment than before:

-1.51 Z2 eV

-3.40 Z2 eV

-13.6 Z2 eV

…

10.2 Z2 eV

Many-electron atoms

𝐸L = 𝑍%𝐸$𝑛%

n=1

n=2

n=3


• 1913: H. G. J. Moseley (1887-1915)

• Catalogs the X-ray emission (K) of various elements

• Finds an empirical formula:

• Volunteered to fight in World war I, killed in action in 1915 (he was 27 years old)

Characteristic X-ray wavelengths

𝜐Z[ =3𝑐𝑅4 𝑍 − 1 %


• Transitions measured (K) correspond to the transition of the electrons deep in the atom. – Does not depend on Z à Always the closest– So why (Z-1) ?– At the time, nobody knows, but there can be 2 electrons in the inner orbit

à Charge screening effect

(Z-1)2 ? Why “-1” ?


• Show that it is possible to derive Moseley’s formula for 𝜈Z[ using the Bohr model if one take into account the charge screening effect mentioned earlier. What is the energy of a Ka photon for Uranium (Z=92)? Compare with the the Ka photon energy for Hydrogen (Z=1).

Exercise


• X-ray energy depends on Z, so one can determine Z by measuring the energy of the X-rays

• Some elements were missing !– Z=43 Technecium– Z=61 Promethium (1940) !– Z=75 Rhenium

• Element composition: X-ray fluorescence– Non-destructive

Moseley plot


• Max von Laue (1879-1960, Nobel prize in Physics 1914) suggests that if X-ray is a form of electromagnetic radiation, interference effects could be observed

• X-ray wavelength: l ~ 10-10-10-11 mà Distance between the “slits” to observe interferences

• Max von Laue: notices that interatomic distance in crystals ~ 10-10m à Idea: use the crystal as 3D-diffraction grating

X-ray scattering

Transmission method


• William Lawrence Bragg (1890-1971, Nobel prize in physics 1915 – with his dad !) interpreted the X-ray scattering as the reflection of the incident X-ray beam from a unique set of planes of atoms within the crystal.

Bragg’s law


• There are two conditions for constructive interference of the scattered X-rays:– The angle of incidence must equal the angle of reflection of the outgoing wave.– The difference in path lengths must be an integral number of wavelengths.

Bragg’s law

Bragg’s Law:nλ = 2d sin q(n = integer)


Bragg spectrometer


• Von Laue method (transmission of X-rays through a crystal): Study of the structure of crystal by analyzing the scattering of X-rays onto “unknown”material.

• Bragg spectrometer (diffraction of X-rays): measurement of X-ray wavelengths.

Applications


Structure of the DNA moleculeWatson and Crick (Nobel Prize Physiology / Medicine – 1962)

Structure of the DNA molecule

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