Complete description of Rutherford scattering experiment with quantitative analysis, Scattering formula

TERM PAPER

MODERN PHYSICS AND ELECTRONICS
PHY112



Topic: Complete description of Rutherford scattering experiment with quantitative analysis, Scattering formula








ACKNOWLEDGEMENT


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CONTENTS

 Introduction
 Thompson’s plum pudding model
 Rutherford’s model
 Failure of Rutherford’s model
 Bohr’s atomic model
 Shortcoming of bohr’s model
 Reference



INTRODUCTION


In chemistry and physics, atomic theory is a theory of the nature of matter, which states that matter is composed of discrete units called atoms, as opposed to the obsolete notion that matter could be divided into any arbitrarily small quantity. It began as a philosophical concept in ancient Greece and India and entered the scientific mainstream in the early 19th century when discoveries in the field of chemistry showed that matter did indeed behave as if it were made up of particles.
The word "atom" (from the Greek atomos, "indivisible"[1]) was applied to the basic particle that constituted a chemical element, because the chemists of the era believed that these were the fundamental particles of matter. However, around the turn of the 20th century, through various experiments with electromagnetism and radioactivity, physicists discovered that the so-called "indivisible atom" was actually a conglomerate of various subatomic particles (chiefly, electrons, protons and neutrons) which can exist separately from each other. In fact, in certain extreme environments such as neutron stars, extreme temperature and pressure prevents atoms from existing at all. Since atoms were found to be actually divisible, physicists later invented the term "elementary particles" to describe indivisible particles. The field of science which studies subatomic particles is particle physics, and it is in this field that physicists hope to discover the true fundamental nature of matter.

Thompson’s Plum pudding model



A schematic representation of the plum pudding model of the atom. In Thomson's mathematical model the "corpuscles" (or modern electrons) were arranged non-randomly, in rotating rings.
The plum pudding model of the atom by J.J. Thomson, who discovered the electron in 1897, was proposed in 1904 before the discovery of the atomic nucleus. In this model, the atom is composed of electrons (which Thomson still called "corpuscles," though G.J. Stoney had proposed that atoms of electricity be called electrons in 1894) [1] , surrounded by a soup of positive charge to balance the electron's negative charge, like negatively-charged "plums" surrounded by positively-charged "pudding". The electrons (as we know them today) were thought to be positioned throughout the atom, but with many structures possible for positioning multiple electrons, particularly rotating rings of electrons (see below). Instead of a soup, the atom was also sometimes said to have had a cloud of positive charge.
The model was disproved by the 1909 gold foil experiment, which was interpreted by Ernest Rutherford in 1911[2] to imply a very small nucleus of the atom containing a very high positive charge (enough to balance about 100 electrons in gold), thus leading to the Rutherford model of the atom, and finally (after Henry Moseley's work showed in 1913 that the nuclear charge was very close to the atomic number) to the Antonius Van den Broek suggestion that atomic number is nuclear charge. Eventually, by 1913, this work had culminated in the solar-system-like (but quantum-limited) Bohr model of the atom, in which a nucleus containing an atomic number of positive charge is surrounded by an equal number of electrons in orbital shells.
Thomson's model was compared (though not by Thomson) to a British treat called plum pudding, hence the name. It has also been called the chocolate chip cookie model or blueberry muffin model, but these mental pictures assume the particles as static, which they were not for Thomson.
Thomson's paper was published in the March 1904 edition of the Philosophical Magazine, the leading British science journal of the day. In Thompson's view:
... the atoms of the elements consist of a number of negatively electrified corpuscles enclosed in a sphere of uniform positive electrification, ...
In this model, the electrons were free to rotate within the blob or cloud of positive substance. These orbits were stabilized in the model by the fact that when an electron moved farther from the center of the positive cloud, it felt a larger net positive inward force, because there was more material of opposite charge, inside its orbit (see Gauss's law). In Thomson's model, electrons were free to rotate in rings which were further stabilized by interactions between the electrons, and spectra were to be accounted for by energy differences of different ring orbits. Thomson attempted to make his model account for some of the major spectral lines known for some elements, but was not notably successful at this. Still, Thomson's model (along with a similar Saturnian ring model for atomic electrons, put forward also in 1904 by Nagaoka after the Maxwell model of Saturn's rings), were earlier harbingers of the later and more successful solar-system-like Bohr model of the atom.
Rutherford model


A stylised representation of the Rutherford model of a lithium atom (nuclear structure anachronistic)
The Rutherford model or planetary model is a model of the atom devised by Ernest Rutherford. Rutherford directed the famous Geiger-Marsden experiment in (1909), which suggested to Rutherford's analysis (1911) that the Plum pudding model (of J. J. Thomson) of the atom was incorrect. Rutherford's new model for the atom, based on the experimental results, had a number of essential modern features, including a relatively high central charge concentrated into a very small volume in comparison to the rest of the atom and containing the bulk of the atomic mass (the nucleus of the atom), and a number of tiny electrons circling around the nucleus like planets around the sun.
Experimental basis for the model
In 1911, Rutherford came forth with his own physical model for subatomic structure, as an interpretation for the unexpected experimental results. In it, the atom is made up of a central charge (this is the modern atomic nucleus, though Rutherford did not use the term "nucleus" in his paper) surrounded by a cloud of orbiting electrons. In this 1911 paper, Rutherford only commits himself to a small central region of very high positive or negative charge in the atom, but uses the following language for pictorial purposes:
"For concreteness, consider the passage of a high speed α particle through an atom having a positive central charge N e, and surrounded by a compensating charge of N electrons. [1]
From purely energetic considerations of how far α (alpha) particles of known speed would be able to penetrate toward a central charge of 100 e, Rutherford was able to calculate that the radius of his gold central charge would need to be less (how much less could not be told) than 3.4 x 10-14 metres (the modern value is only about a fifth of this). This was in a gold atom known to be 10-10 metres or so in radius--- a very surprising finding, as it implied a strong central charge less than 1/3000th of the diameter of the atom.
The Rutherford model didn't attribute any structure to the orbits of the electrons themselves, though it did mention the atomic model of Hantaro Nagaoka, in which the electrons are arranged in one or more rings.
The Rutherford paper suggested that the central charge of an atom might be "proportional" to its atomic mass in hydrogen mass units (roughly 1/2 of it, in Rutherford's model). For gold, this mass number is 197 (not then known to great accuracy) and was therefore modeled by Rutherford to be possibly 196. However, Rutherford did not attempt to make the direct connection of central charge to atomic number, since gold's place on the periodic table was known to be about 79, and Rutherford's more tentative model for the structure of the gold nucleus was 49 helium nuclei, which would have given it a mass of 196 and charge of 98. This differed enough from gold's "atomic number" (at that time merely its place number in the periodic table) that Rutherford did not formally suggest the two numbers might be exactly the same.
Key points of Rutherford model
• The electron clouds of the atom do not influence alpha scattering.
• A large number of the atom's charges, up to a number equal to about half the atomic mass in hydrogen units, are concentrated in very small volume at the center of the atom. These are responsible for deflecting both alpha and beta particles.
• The mass of heavy atoms such as gold is mostly concentrated in the central charge region, since calculations show it is not deflected or moved by the high speed alpha particles, which have very high momentum in comparison to electrons, but not with regard to heavy atoms (such as gold) on the whole. This suggests that much of the mass of atoms is concentrated in their centres.

Shorcomings of Rutherfords model
Rutherford naturally considered a planetary-model atom, the Rutherford model of 1911 – electrons orbiting a solar nucleus – however, said planetary-model atom has a technical difficulty. The laws of classical mechanics (i.e. the Larmor formula), predict that the electron will release electromagnetic radiation while orbiting a nucleus. Because the electron would lose energy, it would gradually spiral inwards, collapsing into the nucleus. This atom model is disastrous, because it predicts that all atoms are unstable.
Also, as the electron spirals inward, the emission would gradually increase in frequency as the orbit got smaller and faster. This would produce a continuous smear, in frequency, of electromagnetic radiation. However, late 19th century experiments with electric discharges through various low-pressure gasses in evacuated glass tubes had shown that atoms will only emit light (that is, electromagnetic radiation) at certain discrete frequencies.
T


Bohr model


Introduced by Niels Bohr in 1913, the model's key success lay in explaining the Rydberg formula for the spectral emission lines of atomic hydrogen. While the Rydberg formula had been known experimentally, it did not gain a theoretical underpinning until the Bohr model was introduced. Not only did the Bohr model explain the reason for the structure of the Rydberg formula, but it provided a justification for its empirical results in terms of fundamental physical constants.
The Bohr model is a primitive model of the hydrogen atom. As a theory, it can be derived as a first-order approximation of the hydrogen atom using the broader and much more accurate quantum mechanics, and thus may be considered to be an obsolete scientific theory. However, because of its simplicity, and its correct results for selected systems (see below for application), the Bohr model is still commonly taught to introduce students to quantum mechanics, before moving on to the more accurate but more complex valence shell atom. A related model was originally proposed by Arthur Erich Haas in 1910, but was rejected. The quantum theory of the period between Planck's discovery of the quantum (1900) and the advent of a full-blown quantum mechanics (1925) is often referred to as the old quantum theory.
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Origin
In the early 20th century, experiments by Ernest Rutherford established that atoms consisted of a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus. Given this experimental data, Rutherford naturally considered a planetary-model atom, the Rutherford model of 1911 – electrons orbiting a solar nucleus – however, said planetary-model atom has a technical difficulty. The laws of classical mechanics (i.e. the Larmor formula), predict that the electron will release electromagnetic radiation while orbiting a nucleus. Because the electron would lose energy, it would gradually spiral inwards, collapsing into the nucleus. This atom model is disastrous, because it predicts that all atoms are unstable.
Also, as the electron spirals inward, the emission would gradually increase in frequency as the orbit got smaller and faster. This would produce a continuous smear, in frequency, of electromagnetic radiation. However, late 19th century experiments with electric discharges through various low-pressure gasses in evacuated glass tubes had shown that atoms will only emit light (that is, electromagnetic radiation) at certain discrete frequencies.
To overcome this difficulty, Niels Bohr proposed, in 1913, what is now called the Bohr model of the atom. He suggested that electrons could only have certain classical motions:
1. The electrons can only travel in special orbits: at a certain discrete set of distances from the nucleus with specific energies.
2. The electrons do not continuously lose energy as they travel. They can only gain and lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a frequency ν determined by the energy difference of the levels according to the Planck relation:

where h is Planck's constant.
3. The frequency of the radiation emitted at an orbit of period T is as it would be in classical mechanics; it is the reciprocal of the classical orbit period:

The significance of the Bohr model is that the laws of classical mechanics apply to the motion of the electron about the nucleus only when restricted by a quantum rule. Although rule 3 is not completely well defined for small orbits, because the emission process involves two orbits with two different periods, Bohr could determine the energy spacing between levels using rule 3 and come to an exactly correct quantum rule: the angular momentum L is restricted to be an integer multiple of a fixed unit:

where n = 1, 2, 3, ... is called the principal quantum number, and ħ = h/2π. The lowest value of n is 1; this gives a smallest possible orbital radius of 0.0529 nm known as the Bohr radius. Once an electron is in this lowest orbit, it can get no closer to the proton. Starting from the angular momentum quantum rule Bohr[1] was able to calculate the energies of the allowed orbits of the hydrogen atom and other hydrogen-like atoms and ions.
Other points are:
1. Like Einstein's theory of the Photoelectric effect, Bohr's formula assumes that during a quantum jump a discrete amount of energy is radiated. However, unlike Einstein, Bohr stuck to the classical Maxwell theory of the electromagnetic field. Quantization of the electromagnetic field was explained by the discreteness of the atomic energy levels; Bohr did not believe in the existence of photons.
2. According to the Maxwell theory the frequency ν of classical radiation is equal to the rotation frequency νrot of the electron in its orbit, with harmonics at integer multiples of this frequency. This result is obtained from the Bohr model for jumps between energy levels En and En−k when k is much smaller than n. These jumps reproduce the frequency of the k-th harmonic of orbit n. For sufficiently large values of n (so-called Rydberg states), the two orbits involved in the emission process have nearly the same rotation frequency, so that the classical orbital frequency is not ambiguous. But for small n (or large k), the radiation frequency has no unambiguous classical interpretation. This marks the birth of the correspondence principle, requiring quantum theory to agree with the classical theory only in the limit of large quantum numbers.
3. The Bohr-Kramers-Slater theory (BKS theory) is a failed attempt to extend the Bohr model which violates the conservation of energy and momentum in quantum jumps, with the conservation laws only holding on average.
Bohr's condition, that the angular momentum is an integer multiple of ħ was later reinterpreted by de Broglie as a standing wave condition: the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:

Substituting de Broglie's wavelength reproduces Bohr's rule. Bohr justified his rule by appealing to the correspondence principle, without providing a wave interpretation.
In 1925 a new kind of mechanics was proposed, quantum mechanics in which Bohr's model of electrons traveling in quantized orbits was extended into a more accurate model of electron motion. The new theory was proposed by Werner Heisenberg. Another form of the same theory, modern quantum mechanics, was discovered by the Austrian physicist Erwin Schrödinger independently and by different reasoning.
Electron energy levels
The Bohr model gives almost exact results only for a system where two charged points orbit each other at speeds much less than that of light. This not only includes one-electron systems such as the hydrogen atom, singly-ionized helium, doubly ionized lithium, but it includes positronium and Rydberg states of any atom where one electron is far away from everything else. It can be used for K-line X-ray transition calculations if other assumptions are added (see Moseley's law below). In high energy physics, it can be used to calculate the masses of heavy quark mesons.
To calculate the orbits requires two assumptions:
1. Classical mechanics
The electron is held in a circular orbit by electrostatic attraction. The centripetal force is equal to the Coulomb force.

where me is the mass, e is the charge of the electron and ke is Coulomb's constant. This determines the speed at any radius:

It also determines the total energy at any radius:

The total energy is negative and inversely proportional to r. This means that it takes energy to pull the orbiting electron away from the proton. For infinite values of r, the energy is zero, corresponding to a motionless electron infinitely far from the proton. The total energy is half the potential energy, which is true for non circular orbits too by the virial theorem.
For larger nuclei, the only change is that kee2 is everywhere replaced by Zkee2 where Z is the number of protons. For positronium, me is replaced by its reduced mass (μ = me/2).
2. Quantum rule
The angular momentum L = mevr is an integer multiple of ħ:

Substituting the expression for the velocity gives an equation for r in terms of n:

so that the allowed orbit radius at any n is:

The smallest possible value of r is called the Bohr radius and is equal to:

The energy of the n-th level is determined by the radius:

An electron in the lowest energy level of hydrogen (n = 1) therefore has 13.6 eV less energy than a motionless electron infinitely far from the nucleus. The next energy level (n = 2) is −3.4 eV. The third (n = 3) is −1.51 eV, and so on. For larger values of n, these are also the binding energies of a highly excited atom with one electron in a large circular orbit around the rest of the atom.
The combination of natural constants in the energy formula is called the Rydberg energy (RE):

This expression is clarified by interpreting it in combinations which form more natural units:
is the rest mass energy of the electron (511 keV/c)
is the fine structure constant

Since this derivation is with the assumption that the nucleus is orbited by one electron, we can generalize this result by letting the nucleus have a charge q = Ze where Z is the atomic number. This will now give us energy levels for hydrogenic atoms, which can serve as a rough order-of-magnitude approximation of the actual energy levels. So, for nuclei with Z protons, the energy levels are (to a rough approximation):

The actual energy levels cannot be solved analytically for more than one electron (see n-body problem) because the electrons are not only affected by the nucleus but also interact with each other via the Coulomb Force. However, the analytic soltuion can be approximated using the Hartee-Fock method, which involves replacing "Z" with "Z - b" where b is constant representing electric field screening due to the inner-shell electron(s) (see Electron shell and the later discussion of the "Shell Model of the Atom" below).
When Z = 1/α (Z ≈ 137), the motion becomes highly relativistic, and Z2 cancels the α2 in R; the orbit energy begins to be comparable to rest energy. Sufficiently large nuclei, if they were stable, would reduce their charge by creating a bound electron from the vacuum, ejecting the positron to infinity. This is the theoretical phenomenon of electromagnetic charge screening which predicts a maximum nuclear charge. Emission of such positrons has been observed in the collisions of heavy ions to create temporary super-heavy nuclei.[citation needed]
For positronium, the formula uses the reduced mass. For any value of the radius, the electron and the positron are each moving at half the speed around their common center of mass, and each has only one fourth the kinetic energy. The total kinetic energy is half what it would be for a single electron moving around a heavy nucleus.
(positronium)

Shortcomings
The Bohr model gives an incorrect value for the ground state orbital angular momentum. The angular momentum in the true ground state is known to be zero. Although mental pictures fail somewhat at these levels of scale, an electron in the lowest modern "orbital" with no orbital momentum, may be thought of as not to rotate "around" the nucleus at all, but merely to go tightly around it in an ellipse with zero area (this may be pictured as "back and forth", without striking or interacting with the nucleus). This is only reproduced in a more sophisticated semiclassical treatment like Sommerfeld's. Still, even the most sophisticated semiclassical model fails to explain the fact that the lowest energy state is spherically symmetric--- it doesn't point in any particular direction.
In modern quantum mechanics, the electron in hydrogen is a spherical cloud of probability which grows denser near the nucleus. The rate-constant of probability-decay in hydrogen is equal to the inverse of the Bohr radius, but since Bohr worked with circular orbits, not zero area ellipses, the fact that these two numbers exactly agree, is considered a "coincidence." (Though many such coincidental agreements are found between the semi-classical vs. full quantum mechanical treatment of the atom; these include identical energy levels in the hydrogen atom, and the derivation of a fine structure constant, which arises from the relativistic Bohr-Sommerfield model (see below), and which happens to be equal to an entirely different concept, in full modern quantum mechanics).
The Bohr model also has difficulty with, or else fails to explain:
• Much of the spectra of larger atoms. At best, it can make predictions about the K-alpha and some L-alpha X-ray emission spectra for larger atoms, if two additional ad hoc assumptions are made (see Moseley's law above). Emission spectra for atoms with a single outer-shell electron (atoms in the lithium group) can also be approximately predicted. Also, if the empiric electron-nuclear screening factors for many atoms are known, many other spectral lines can be deduced from the information, in similar atoms of differing elements, via the Ritz-Rydberg combination principles (see Rydberg formula). All these techniques essentially make use of Bohr's Newtonian energy-potential picture of the atom.
• The theory does not predict the relative intensities of spectral lines; although in some simple cases, Bohr's formula or modifications of it, was able to provide reasonable estimates (for example, calculations by Kramers for the Stark effect).
• The existence of fine structure and hyperfine structure in spectral lines, which are known to be due to a variety of relativistic and subtle effects, as well as complications from electron spin.
• The Zeeman effect - changes in spectral lines due to external magnetic fields; these are also due to more complicated quantum principles interacting with electron spin and orbital magnetic fields.
• The model also violates the uncertainty principle in that it considers electrons to have known orbits and definite radius, two things which can not be directly known at once.


REFERENCE


 www.about.com
 www.wikipedia.com
 www.physicstoday.com
 Encarta encyclopedia
 Modern Physics- Arthurbeizer