Ionic bonding is characterized by electron transfer; one of the reacting atoms loses one or more electrons, and the other atom gains one or more electrons. For example, consider sodium chloride (NaCl, otherwise known as table salt). Sodium is an alkali metal and chlorine is a halogen. To get a complete shell, a sodium atom gives its outer 3selectron to a chlorine atom which otherwise would be one electron short of a complete shell. As a result the sodium ion (Na+) acquires a 1+ charge and the chlorine ion (Cl-) acquires a 1- charge. Positive ions are called cations and negative ions are called anions. We say that chlorine is very electronegative. Electronegativity is a measure of the ability of an atom in a molecule to attract (or hog) electrons to itself.
In covalent bonding electrons are not transferred but are shared;
a covalent bond consists of a pair of electrons (with
opposite spins in a singlet state) shared by two atoms.
An example is the covalent bond formed by two hydrogen atoms in an
H2 molecule. The electrons are attracted by the positive nuclear charge
to the region between the nuclei. The 1s orbitals
overlap to form a sigma bonding orbital (). The spatial
part of the sigma bonding orbital
is a symmetric linear combination of the 1s orbitals:
(1) |
There is also
a sigma antibonding orbital (
) which is spatially an antisymmetric
linear combination of the 1s orbitals:
(2) |
Notice that the number of single electron states is conserved in bonding. We started with 2 individual atoms, each with a 1s orbital. That makes a total of 4 single electron states, counting spin. Covalent bonding leads to 2 sigma orbitals ( and ) which have a total of 4 single particle states. If we try to form a helium molecule, He2, we would put the two 1s orbitals together and form the and orbitals. Then we would fill them with the 4 helium electrons. The energy gained by putting two electrons in the sigma bonding orbital is lost by putting the other two electrons in the sigma antibonding orbital. So He2 is unstable and no bond is formed. Helium is a noble gas and doesn't need other electrons to complete its atomic orbital shell.
The molecular orbitals derived from the overlap of 2p atomic orbitals are slightly more complicated. The three 2p orbitals of an atom are directed along the Cartesian coordinates x, y, and z. If we consider that a diatomic molecule is formed by the atoms approaching each other along the x axis, the px atomic orbitals approach each other head on and overlap to produce bonding and antibonding molecular orbitals. All sigma orbitals are symmetrical about the internuclear axis; the and orbitals have shapes that are similar to those of the and orbitals.
In the formation of a diatomic molecule, the pz atomic orbitals approach each other side to side and produce pi () bonding and antibonding molecular orbitals. orbitals are not symmetrical about the internuclear axis. The electron density of a bonding orbital is zero in a plane that includes both nuclei of the molecule, but charge is concentrated in two regions that lie above and below this plane and between the two nuclei. The orbital reduces electronic density in the internuclear region.
The py orbitals also approach each other sideways. Consequently another set of and molecular orbitals, which lie at right angles to the set resulting from the overlap of pzorbitals, can be produced. The two orbitals are degenerate (have equal energy), and the two orbitals are degenerate. Six molecular orbitals, therefore, arise from the two sets of porbitals-one , one , two , and two .
In forming a diatomic molecule, the and orbitals are lower in energy than the orbitals formed from the 2p orbitals. So the and orbitals will be occupied before the or orbitals are.
Pure ionic bonding and pure covalent bonding are seldom encountered.
Most bonds have intermediate character, although many bonds are
predominantly either ionic or covalent. In heteronuclear, diatomic
molecules the clouds of the bonding orbitals are distorted so that more
electronic charge is located around the more electronegative atom than
around the other atom; antibonding orbitals have their largest
electron density in the regions close to the less electronegative
atom. The molecular orbital wavefunction has the form
(3) |
In some molecules which consist of more than 2 atoms, there can be multicenter or delocalized bonding in which some of the bonding electrons bond more than 2 atoms. One example is the bonding found in benzene which consists of 6 carbon atoms in a ring. A carbon atom has 4 valence electrons, so it wants to form 4 bonds to have a filled shell. In benzene each carbon atom forms one bond with a hydrogen atom and 3 bonds with its neighboring 2 carbon atoms.
The ultimate in delocalized bonding is metallic bonding. In a metal the atoms contribute their valence electrons to the collective whole. As a result the electrons do not belong to any specific atom, but have wavefunctions that extend throughout the whole system. These conduction electrons are responsible for holding the metal together. This is metallic bonding.
I want to mention 2 models of the nucleus. One is called the liquid drop model and the other is called the nuclear shell model. The liquid drop model is based on two properties that are common to all except the lightest nuclei, (1) the interior mass densities are approximately the same and (2) the binding energy per nucleon is approximately 8 MeV. The liquid droplet model views the nucleus as a tiny incompressible drop of liquid with the constituents being nucleons. In physics a ``gas'' is used to describe a bunch of noninteracting particles whose energy consists only of kinetic energy, and a liquid is a ``gas'' of interacting particles. The liquid droplet model views the nucleons largely as classical particles.
(4) |
(5) |
(6) |
Besides predicting the magic numbers, the shell model can also
predict the total angular momentum I of the ground states of
almost all the nuclei. (The total angular momentum of the
nucleus is often referred to as its spin.)
It turns out that pairs of nucleons of the
same type (e.g., pairs of protons) strongly tend to combine with one
another to yield zero angular momentum. Thus all even-even nuclei
have I=0. ``Even-even nuclei'' means nuclei with even numbers
of protons and even numbers of neutrons. Nucleons of the same kind
tend to pair because it increases their overlap and enables them
to take advantage of the short range attractive strong force.
A simple example is the fact that 2 fermions in a singlet state,
which is antisymmetric in spin space, have a symmetric spatial wavefunction
which has more overlap than its antisymmetric spatial counterpart. Now
consider the case of odd-A nuclei which have an even number
of nucleons of one kind and an odd number of the other. All the
nucleons pair to yield zero angular momentum except the odd nucleon.
The angular momentum of the odd-A nucleus should just be that
of the odd nucleon, and indeed this is the case for virtually all
odd-A nuclei whose spin is known. Further the same argument applied
to (presumably unstable)
odd-odd nuclei suggests that their angular momentum ought to be
some combination of the angular momenta of the two odd nucleons.
Again this is observed to be the case, but, of course,
this test is not nearly as definitive as the first one.
Odd-odd nuclei have an odd number of neutrons and an odd number of
protons. An odd-odd nucleus should be
unstable because it can become an even-even
nucleus of lower energy by having one of its neutrons decay into
a proton. A neutron can decay into a proton, an electron, and an
electron antineutrino in a process called beta decay.
(7) |
The shell model has also met with considerable success in providing a semiquantitative description of many of the properties of the low-lying excited states of most odd-A nuclei.
Another interesting phenomenon for which the shell model provides an illuminating semiquantitative explanation is that of nuclear isomerism. In the study of the excited states of nuclei it is found that certain nuclei can exist for quite extended periods of time in an excited state, from which they often finally decay by -ray emission. Such states may have decay lifetimes ranging up to many days. The nuclei which can exist in these so-called metastable states tend to be those having a nearly closed shell configuration of nucleons. The explanation of the long lifetimes of these metastable states lies in the fact that the ground state and the first excited state differ in angular momentum by several units of . Thus, many of the lower-order transitions, such as electric dipole, magnetic dipole, and electric quadrupole, are forbidden, so that a transition of very high multipole order may be required. The probability of transition rapidly becomes smaller with increasing multipole order, so that a long lifetime for the state results.
Another nuclear property for which the shell model provides an explanation is the parity of the ground state. Recall that parity refers to whether the spatial part of the wavefunction changes sign when . By the arguments used in connection with angular momenta, the ground state parity of even-even nuclei should be even, and that of odd-A nuclei should correspond to where is the orbital angular momentum of the odd nucleon. This rule is observed to lead to the correct parities for virtually all cases.
So we can see that the nuclear shell model is quite good at explaining many of the properties of nuclei.
Niels Bohr and John Wheeler produced a general theory of fission in 1939. They predicted that only the rare isotope of uranium, U-235, would be fissionable by slow neutrons. The reason is that U-235 has an odd number of neutrons. The incoming neutron binds with the odd neutron to form a neutron pair and the energy gained from this binding (of order an MeV) helps the nucleus to deform and break apart. However, U-238 has an even number of neutrons and doesn't gain the extra binding energy from forming a pair. So the energy to deform and fission the U-238 nucleus comes from the kinetic energy of the incoming neutron. Nier showed experimentally that indeed U-235 can be fissioned by slow neutrons while U-238 requires neutrons of about 1 MeV. Since the nuclear cross section goes as 1/E where E is the energy of the incoming neutron, slower neutrons have a bigger chance of hitting a nucleus than faster neutrons. (The cross section = decay rate/(incoming flux of particles).) So it's easier to cause fission in U-235 than in U-238. In nuclear reactors graphite is used to slow the fission neutrons, originally emitted at about 1 MeV energy, down to thermal energies which is less than 1 eV. Graphite is a form of carbon which absorbs very few neutrons. Heavy water works even better but is harder to come by. The heat from water used to cool the graphite is used to make steam which in turns runs turbines and generates electricity in a nuclear power plant. Neutron absorbers, like rods of boron, are used to control the reaction. The rods are inserted to slow the reaction and taken out if the reaction is too going to slowly.
To understand this in more detail, consider a
set of identical nuclei of spin I and magnetic moment
, where gnis the dimensionless nuclear g-factor (typically ),
is the nuclear magneton, and is the gyromagnetic ratio (
). (In
we just have the proton mass MP rather than the
mass of all the nucleons because it's usually the odd nucleon
that determines .) We start
with a simple quantum mechanical view of resonance theory. If we
apply a magnetic field , the Hamiltonian is
(8) |
(9) |
(10) |
lec7levels.eps
We can get transitions between the levels by applying an alternating
rf field perpendicular to the static field. (rf stands for radio
frequency.) We get a transition if
(11) |
(12) |
To see why the transitions are between adjacent levels, notice that
the rf field is given by
(14) |
(15) |
(16) |
(17) |
Note that Planck's constant has disappeared from the resonance condition, implying that a classical picture could also be used. We will describe such a picture, but first, some numbers. Note that and . Since , nuclear numbers differ from electronic numbers by about 103. Thus Hz/Tesla and Hz/Tesla. Electronic systems have a resonance at MHz ( cm which is in the microwave region), whereas nuclear systems typically resonate at 10 MHz (a radio frequency) for to 10,000 Gauss.
Classical Treatment A magnetic moment in a magnetic
field experiences a torque
. Thus
(18) |
(19) |
lec7precess.eps
Now let's examine the effect of the alternating magnetic field.
is most readily analyzed by breaking it
into two rotating components, each of amplitude Hox, one rotating
clockwise and the other counterclockwise. We denote the rotating
fields by and :
(20) |
lec7Hx.eps
Let's suppose that . This is the resonance condition. In the reference frame rotating with the magnetic moment of the nucleus, HL looks like a static field pointing along the x axis, while HR is rotating at and averages to zero. So we can ignore HR.
So in the rotating frame at resonance ( ), the magnetic moment sees a static magnetic field pointing along the x axis. So if initially points along the , then it will precess in the y-z plane about the x axis. If we turn on H1 for a short time tw, the moment precesses through an angle . If , the pulse inverts . This is called a ``180 degree pulse'' or `` pulse.'' If , is rotated until it lies parallel to . This is called a ``90 degree pulse'' or a `` pulse.'' After H1 is turned off, the moment remains at rest in the rotating frame, and hence precesses in the laboratory frame in a plane normal to the field. This suggests a way of observing magnetic resonance. Put the sample in a solenoid. Pulse the solenoid with an rf field that tips the nuclear spins by 90o. Then as the spins precess in the x-y plane, they produce a time-varying flux through the coil that induces an observable emf according to Faraday's law.
coil.eps
What we have suggested so far would indicate that the induced emf would persist indefinitely, but in practice, the interactions of the spins with their surroundings cause a decay. Let denote the relaxation time. Then milliseconds in liquids and s in solids. Even during that short time, there are many precession periods. What we have described is the observation of the ``free induction decay'' (i.e., free of H1).
T1decay.eps
We have been considering a uniform field . To give the system some spatial resolution, one can vary the static field spatially by putting on a field gradient. Then only those spins that are located where will give a strong signal.
spinecho.eps
If the spins initially are aligned along the z axis, the pulse puts them in the x-y plane where they dephase due to field inhomogeneities and spin-spin interactions. The pulse refocuses the spins and they produce an echo a time later.
rephase.eps