Physics 290 - Lecture Notes on Atoms and Atomic Models

Physics 290 12.1 INTRODUCTION

  • By the nineteenth century, enough evidence had accumulated in favor of the atomic hypothesis of matter.

  • In 1897, experiments on electric discharge through gases conducted by English physicist J. J. Thomson (1856 – 1940) revealed that:

    • Atoms of different elements contain negatively charged constituents (electrons) that are identical for all atoms.

    • Atoms, as a whole, are electrically neutral. Hence, they must also contain some positive charge to neutralize the negative charge of the electrons.

  • The arrangement of the positive charge and the electrons inside the atom leads us to the question: What is the structure of an atom?

  • The first model of the atom was proposed by J. J. Thomson in 1898.

    • According to this model:

    • The positive charge of the atom is uniformly distributed throughout its volume.

    • Negatively charged electrons are embedded in this positive charge like seeds in a watermelon.

    • This model is referred to as the plum pudding model of the atom.

  • Subsequent studies on atoms have shown that the distribution of electrons and positive charges differs significantly from this model.

  • It is known that condensed matter (solids and liquids) and dense gases at all temperatures emit electromagnetic radiation, characterized by a continuous distribution of several wavelengths, albeit with different intensities.

  • The radiation is attributed to the oscillations of atoms.

ATOMS

  • Atoms and molecules are governed by the interactions of each atom or molecule with its neighbors.

  • In contrast, light emitted from rarefied gases heated in flames or excited electrically (e.g., in neon signs or mercury vapor lights) only has certain discrete wavelengths.

  • The spectrum appears as a series of bright lines, indicating that in such gases, the average spacing between atoms is large. Consequently, the radiation emitted can be attributed to individual atoms rather than the interactions between atoms or molecules.

  • In the early nineteenth century, it was established that:

    • Each element has a characteristic spectrum of radiation.

    • For example, hydrogen emits a set of lines with fixed relative positions.

  • This observation suggested a relationship between the internal structure of an atom and its emitted radiation spectrum.

HISTORICAL CONTEXT

  • In 1885, Johann Jakob Balmer (1825 – 1898) derived a simple empirical formula that provided the wavelengths of a group of lines emitted by atomic hydrogen.

  • Given hydrogen is the simplest of elements, its spectrum will be examined in detail.

  • Ernst Rutherford (1871–1937), a former research student of J. J. Thomson, investigated experiments involving α-particles emitted from radioactive elements.

  • In 1906, Rutherford proposed a classic experiment to scatter these α-particles by atoms for atomic structural investigation.

  • This experiment was ultimately performed around 1911 by Hans Geiger (1882–1945) and Ernst Marsden (1889–1970, who was only 20 years old and had not yet completed his bachelor's degree).

    • The results led to the creation of the Rutherford's planetary model of the atom.

    • This model posits that:

    • The entire positive charge and most of the mass of the atom is found in a small volume called the nucleus.

    • Electrons revolve around the nucleus, similar to planets orbiting the sun.

  • Despite its advancements, Rutherford’s model could not explain why atoms emit light of only discrete wavelengths.

  • How could a simple atom like hydrogen, with a single electron and proton, produce a complex spectrum of specific wavelengths?

  • Classically, an electron revolves around a nucleus analogous to a planet orbiting the sun, but there are difficulties in accepting such a model.

ALPHA-PARTICLE SCATTERING AND RUTHERFORD'S NUCLEAR MODEL

The Geiger-Marsden Experiment

  • In 1911, H. Geiger and E. Marsden conducted experiments proposed by Ernst Rutherford.

  • Experimental setup:

    • A beam of 5.5 MeV α-particles emitted from a 214 83Bi radioactive source was directed at a thin gold foil.

    • The α-particles were collimated into a narrow beam using lead bricks.

    • The beam was allowed to strike a gold foil with a thickness of 2.1imes10−7extm2.1 imes 10^{-7} ext{ m}.

    • Scattered α-particles were detected via a rotatable detector consisting of a zinc sulfide screen and a microscope.

    • Scintillations generated from scattered α-particles upon striking the screen could be viewed through a microscope.

    • A graph representing the total number of scattered α-particles at various angles was plotted, demonstrating the data points with a solid curve representing theoretical predictions based on dense positively charged nuclei.

  • Results:

    • Most α-particles passed through the foil without collisions (indicating empty space in the atom).

    • Approximately 0.14% of the incoming α-particles scattered more than 1°.

    • Around 1 in 8000 α-particles deflected by over 90°.

  • Rutherford deduced:

    • A large repulsive force must have acted to deflect an α-particle backward, indicating the presence of a concentrated mass and charge at the center, the nucleus.

  • This strong correlation supported the nuclear atom hypothesis and resulted in Rutherford being credited with discovering the atomic nucleus.

RUTHERFORD'S NUCLEAR MODEL

  • In Rutherford’s model:

    • The positive charge and most mass of the atom are concentrated in the nucleus, with electrons moving at a distance.

    • The size of the nucleus is proposed to be between 10−1510^{-15} m to 10−1410^{-14} m, while the atom itself is in the range of 10−1010^{-10} m.

    • Electrons are situated at a distance approximately 10,000 to 100,000 times that of the nucleus, illustrating that most of an atom is empty space.

  • Such a configuration explains why most α-particles pass through thin metal foils unscathed.

  • An α-particle nearing the nucleus may experience scattering due to the intense electric field at the nucleus.

  • Under certain assumptions, the trajectory of an α-particle can be computed using:

    • Newton’s second law of motion

    • Coulomb’s law for electrostatic repulsion.

EXAMPLE 12.1

  • To derive the force of interaction between the α-particle and nucleus: F=racZe24extπextε0r2F = rac{Ze^{2}}{4 ext{π} ext{ε}_{0} r^2}

    • Where:

    • rr is the distance between the α-particle and the nucleus.

    • The force changes continuously with the approach and receding of the α-particle relative to the nucleus.

ALPHA-PARTICLE TRAJECTORY

  • The trajectory of an α-particle is influenced by the impact parameter bb, defined as the perpendicular distance of the initial velocity vector from the nucleus's center.

    • As a beam of α-particles has various impact parameters, this leads to varying scattering angles.

    • Close encounters with the nucleus (small bb) impact significant scattering.

    • Larger impact parameters result in minimal deflections, indicating collisions between the α-particle and the nucleus are rare.

  • Rutherford scattering effectively estimates the nucleus's upper size limit as confirmed by geometric relationships in scattering experiments:

    • For example, the radius of the electron's orbit compared to the radius of the nucleus is rac10−1010−15=105rac{10^{-10}}{10^{-15}} = 10^{5}, indicating substantial amounts of empty space in atomic structures.

EXAMPLE 12.2

  • For a 7.7 MeV α-particle, find the closest approach to the nucleus:

  • Consider energy conservation in the system:
    E<em>i=E</em>fE<em>i = E</em>f

  • Initial energy is represented as the kinetic energy of the incoming α-particle, while final energy represents the electric potential energy of the system:
    d=racZe24extπKd = rac{Z e^{2}}{4 ext{π} K}

  • Substituting numerical values leads to results indicating that the distance of closest approach is approximately 3.0imes10−14extm3.0 imes 10^{-14} ext{ m}, less than the observed size of the gold nucleus (6 fm).

12.2.2 ELECTRON ORBITS

  • Rutherford’s nuclear model portrays

    • An electrically neutral sphere with a small, massive positive nucleus at the center surrounded by orbiting electrons in dynamically stable orbits.

  • The electrostatic force provides the centripetal force necessary to maintain this orbital stability: F<em>e=F</em>cF<em>e = F</em>c

    • Rearranging gives a relationship linking orbit radius and electron velocity as:
      2mv2=racZe24extπextε0r2mv^2 = rac{Ze^{2}}{4 ext{π} ext{ε}_{0} r}

12.3 ATOMIC SPECTRA

  • Each atom emits a characteristic spectrum of radiation.

  • Under excitation at low pressures, atoms emit spectra with specific wavelengths, defining an emission line spectrum that displays bright lines on a dark background.

  • The spectrum from atomic hydrogen has distinct features based on excitation states of the hydrogen atom when energy is provided to excite the electrons.

12.4 BOHR MODEL OF THE HYDROGEN ATOM

  • The Rutherford model led to observations suggesting that while the atom was stable in theory, it could not easily explain physical behavior observed empirically.

    • Classical electromagnetic theory stipulates an accelerating charged particle tends to emit radiation, losing energy, leading to spiraling toward the nucleus.

    • This results in the emission of continuous spectra in contradiction to atomic behavior.

BOHR'S POSTULATES

  • Niels Bohr (1885 – 1962), based on Rutherford’s model, proposed a theory in 1913 formulated via three postulates:

    • First Postulate: An electron in an atom revolves in specific orbits without emitting radiant energy, creating defined stable states called stationary states.

    • Second Postulate: The electron’s angular momentum in orbits is quantized as an integral multiple of rach2extπrac{h}{2 ext{π}}, hence:
      L=nrach2extπL = n rac{h}{2 ext{π}}

    • Third Postulate: An electron may transition from one non-radiating orbit to another lower-energy orbit, emitting a photon whose energy corresponds to the difference in energy between the initial and final states:
      E<em>i−E</em>f=h<br>uE<em>i - E</em>f = h<br>u

RADIUS OF ORBITAL STATES

  • Calculating the radius associated with each stable orbit is based on combining the second postulate and relationships defining energy states:
    rn=racn2h24extπ2extme2r_n = rac{n^2 h^2}{4 ext{π}^2 ext{m} e^2}

  • Total energy derived in stationary states can be defined as:
    En=−rac2extme4h2n2E_n = - rac{2 ext{m} e^4}{h^2 n^2}

  • Particularly, the minimum energy required to extract an electron from the hydrogen atom at the ground state equals 13.6 eV (the ionization energy).

12.4.1 ENERGY LEVELS

  • In these orbits, the energy progressively increases with radial distance from the nucleus.

  • Calculated energies of excited states indicate notable transitions between energy levels and their correlation to spectral emissions.

12.5 LINE SPECTRA OF THE HYDROGEN ATOM

  • Transitioning between the different energy levels produces radiative emissions, resulting in discrete frequencies forming line spectra corresponding to unique atomic transitions.

  • The process of absorption also creates dark lines in continuous spectra, indicating the electron’s transition between energies.

12.6 DE BROGLIE'S EXPLANATION OF BOHR'S POSTULATE

  • In 1923, Louis de Broglie offered insights into the quantization in Bohr's model by presenting the concept of electron wave-particle duality.

  • He proposed that electron orbits correspond to standing waves, defining conditions under which particular wavelengths persist as stable orbits: 2extπrn=nextλ2 ext{π}r_n = n ext{λ}

    • Where extλ=rachpext{λ} = rac{h}{p} represents the de Broglie