Describe in detail what gives rise to the magnetic field in
materials?
Magnetism and Electrons
Magnetic materials, from lodestones to videotapes, are magnetic because of the electrons within them.
We have already seen one way in which electrons can generate a magnetic field: Send them
through a wire as an electric current, and their motion produces a magnetic field around the wire. There
are two more ways, each involving a magnetic dipole moment that produces a magnetic field in the
surrounding space.
Loop Model for Electron Orbits
For the purposes of deriving the orbital magnetic moment we will assume
that an electron moves along a circular path with a radius that is much larger
than an atomic radius (hence the name “loop model”). However, the
derivation does not apply to an electron within an atom (for which we
need quantum physics).
We imagine an electron moving at constant speed v in a circular path
of radius r, counterclockwise as shown at right. The motion of the negative
charge of the electron is equivalent to a conventional current i (of
positive charge) that is clockwise, as also shown. The magnitude of the orbital
magnetic dipole moment of such a current loop is obtained from the magnetic dipole moment of a
current coil as  = NiA with N = 1:
where A is the area enclosed by the loop. The direction of this magnetic dipole moment is, from the
right-hand rule, downward in the figure above.
To evaluate the equation, we need the current i. Current is, generally, the rate at which charge passes
some point in a circuit. Here, the charge of magnitude e takes a time T = 2r/v to circle from any
point back through that point, so
Substituting this and the area A = r 2 of the loop into the equation for mangetic dipole moment gives
us
To find the electron’s orbital angular momentum Lorb , we use l = m (r x v). Because r and v are
perpendicular, Lorb has the magnitude
L orb = mrv sin 90° = mrv.
The vector L orb is directed upward in the figure above. Combine this equation with that for orb,
generalize to a vector formulation and write an expression for orb as a function of L orb.
Indicating the opposite directions of the vectors with a minus sign yields
Orbital Magnetic Dipole Moment
When it is in an atom, an electron has an additional angular momentum called its orbital angular
momentum Lorb. Associated with Lorb is an orbital magnetic dipole moment orb ; the two are related
by
The minus sign means that Lorb and orb have opposite directions. This was just dervived by “classical”
(nonquantum) analysis. You might wonder, seeing as this derivation gives the correct result for an
electron within an atom, why the derivation is invalid for that situation. The answer is that this line of
reasoning yields other results that are contradicted by experiments.
Orbital angular momentum Lorb cannot be measured; only its component along any axis can be
measured, and that component is quantized. The component along, say, a z axis can have only the
values given by
in which ml is called the orbital magnetic quantum number and “limit” refers to some largest allowed
integer value for ml. The signs the equation above have to do with the direction of L orb,z along the z axis.
The orbital magnetic dipole moment orb of an electron also cannot itself be measured; only its
component along an axis can be measured, and that component is quantized. Write the equation z
component  orb,z of the orbital magnetic dipole moment by substituting into our previous definition for
L orb,.
We introduce the Bohr magneton as:
eh / 4m
The Bohr magneton is the magnitude of the magnetic dipole moment of an orbiting electron with an
orbital angular momentum of one ħ. According to the Bohr model, this is the ground state, i.e. the state
of lowest possible energy.
In terms of the Bohr magneton, as
When an atom is placed in an external magnetic field Bext, an energy U can be associated with the
orientation of the orbital magnetic dipole moment of each electron in the atom. Its value is
where the z axis is taken in the direction of Bext. Although we have used the words “orbit” and “orbital”
here, electrons do not orbit the nucleus of an atom like planets orbiting the Sun. How can an electron
have an orbital angular momentum without orbiting in the common meaning of the term? Once again,
this can be explained only with quantum physics.
Spin Magnetic Dipole Moment
An electron has an intrinsic angular momentum called its spin angular momentum (or just spin) S;
associated with this spin is an intrinsic spin magnetic dipole moment s. (By intrinsic, we mean
that S and s are basic characteristics of an electron, like its mass and electric charge.) Vectors S and
s are related by
in which e is the elementary charge (1.60 * 10-19 C) and m is the mass of an electron (9.11 * 10-31 kg).The
minus sign means that s and S are oppositely directed.
Spin S is different from the classical angular momenta of in two respects:
1. Spin S itself cannot be measured. However, its component along any axis can be measured.
2. A measured component of S is quantized, which is a general term that means it is restricted to
certain values. A measured component of S can have only two values, which differ only in sign.
Let us assume that the component of spin S is measured along the z axis of a coordinate system. Then
the measured component S z can have only the two values given by
where ms is called the spin magnetic quantum number and h is the Planck constant, the ubiquitous
constant of quantum physics. The signs given in the equation above have to do with the direction of Sz
along the z axis. When S z is parallel to the z axis, ms is + ½ and the electron is said to be spin up. When
S z is antiparallel to the z axis, ms is - ½ and the electron is said to be spin down.
The spin magnetic dipole moment s of an electron also cannot be measured; only its component along
any axis can be measured, and that component too is quantized, with two possible values of the same
magnitude but different signs. We can relate the component  s,z measured on the z axis to Sz by
rewriting the equation for s in component form for the z axis as
Substituting for S z from the equation abvoe then gives us
where the plus and minus signs correspond to s,z being parallel and antiparallel to the z axis,
respectively.
The quantity on the right side of the equation is a constant called the Bohr magneton B :
Spin magnetic dipole moments of electrons and other elementary particles can be expressed in terms of
B . For an electron, the magnitude of the measured z component of s is
(The quantum physics of the electron, called quantum electrodynamics, or QED, reveals that s,z is
actually slightly greater than 1B , but we shall neglect that fact.)
When an electron is placed in an external magnetic field B , an energy U can be associated with the
orientation of the electron’s spin magnetic dipole moment just as an energy can be associated with the
orientation of the magnetic dipole moment  of a current loop placed in Bext. The orenz tation energy
for the electron is
where the z axis is taken to be in the direction of Bext.
If we imagine an electron to be a microscopic sphere (which it is not), we can
represent the spin S, the spin magnetic dipole moment s, and the associated
magnetic dipole field as the figure at right .Although we use the word “spin”
here, electrons do not spin like tops. How, then, can something have angular
momentum without actually rotating?
Protons and neutrons also have an intrinsic angular momentum called spin and
an associated intrinsic spin magnetic dipole moment. For a proton those two
vectors have the same direction, and for a neutron they have opposite
directions. We shall not examine(yet) the contributions of these dipole
moments to the magnetic fields of atoms because they are about a thousand times smaller than that
due to an electron.
Some Properties of Atoms
1. Atoms are stable. Essentially all the atoms that form our tangible world have existed
without change for billions of years. What would the world be like if atoms continually changed
into other forms, perhaps every few weeks or every few years?
2. Atoms combine with each other. They stick together to form stable molecules and stack up to
form rigid solids. An atom is mostly empty space, but you can stand on a floor — made up of
atoms — without falling through it.
These basic properties of atoms can be explained by quantum physics, as can the three less apparent
properties that follow.
3. Atoms Are Put Together Systematically
The figure above shows an example of a repetitive property of the elements as a function of their
position in the periodic table (Appendix G). The figure is a plot of the ionization energy of the elements;
the energy required to remove the most loosely bound electron from a neutral atom is plotted as a
function of the position in the periodic table of the element to which the atom belongs. The
remarkable similarities in the chemical and physical properties of the elements in each vertical column
of the periodic table are evidence enough that the atoms are constructed according to systematic rules.
The elements are arranged in the periodic table in six complete horizontal periods (and a seventh
incomplete period): except for the first, each period starts at the left with a highly reactive alkali metal
(lithium, sodium, potassium, and so on) and ends at the right with a chemically inert noble gas (neon,
argon, krypton, and so on). Quantum physics accounts for the chemical properties of these
elements. The numbers of elements in the six periods are 2, 8, 8, 18, 18, and 32.
Quantum physics predicts these numbers.
4. Atoms Emit and Absorb Light
We have already seen that atoms can exist only in discrete quantum states, each state having a certain
energy. An atom can make a transition from one state to another by emitting light (to jump to a lower
energy level E low ) or by absorbing light (to jump to a higher energy level E high ). Light is emitted or
absorbed as a photon with energy
Thus, the problem of finding the frequencies of light emitted or absorbed by an atom reduces to the
problem of finding the energies of the quantum states of that atom. Quantum
physics allows us—in principle at least—to calculate these energies.
5. Atoms Have Angular Momentum and Magnetism
The figure at right shows a negatively charged particle moving in a circular orbit around a fixed center.
As we discussed earlier in 4B, the orbiting particle has both an angular momentum L and (because its
path is equivalent to a tiny current loop) a magnetic dipole moment  . As the figure shows, vectors L
and  are both perpendicular to the plane of the orbit but, because the charge is negative, they point
in opposite directions.
The model above is strictly classical and does not accurately represent an electron in an atom. In
quantum physics, the rigid orbit model has been replaced by the probability density model, best
visualized as a dot plot. In quantum physics, however, it is still true that in general, each quantum state
of an electron in an atom involves an angular momentum L and a magnetic dipole moment  that
have opposite directions (those vector quantities are said to be coupled).
The Einstein–de Haas Experiment
In 1915, well before the discovery of quantum physics, Albert Einstein and Dutch physicist W. J. de Haas
carried out a clever experiment designed to show that the angular momentum and magnetic moment of
individual atoms are coupled.
Einstein and de Haas suspended an iron cylinder from a thin fiber, as shown at right. A solenoid was
placed around the cylinder but not touching it. What happens when a current is placed in the
solenoid?
Initially, the magnetic dipole moments of the atoms of the cylinder point in random directions,
and so their external magnetic effects cancel
(a). However, when a current is switched on in
the solenoid (b) so that a magnetic field B is
set up parallel to the long axis of the cylinder, the
magnetic dipole moments of the atoms of the
cylinder reorient themselves, lining up with that
field. If the angular momentum L of each atom
is coupled to its magnetic moment  , then this
alignment of the atomic magnetic moments must
cause an alignment of the atomic angular
momenta opposite the magnetic field.
No external torques initially act on the cylinder; thus, its angular momentum must remain at its initial
zero value. However, when B is turned on and the atomic angular momenta line up antiparallel to B ,
they tend to give a net angular momentum L to the cylinder as a whole (directed downward in b). To
maintain zero angular momentum, the cylinder begins to rotate around its central axis to produce an
angular momentum L in the opposite direction (upward in b).
The twisting of the fiber quickly produces a torque that momentarily stops the cylinder’s rotation and
then rotates the cylinder in the opposite direction as the twisting is undone. Thereafter, the fiber will
twist and untwist as the cylinder oscillates about its initial orientation in angular simple harmonic
motion.
Observation of the cylinder’s rotation verified that the angular momentum and the magnetic dipole
moment of an atom are coupled in opposite directions.
Moreover, it dramatically demonstrated that the angular momenta associated with quantum states of
atoms can result in visible rotation of an object of everyday size.
Electron Spin
As we discussed in 4B, whether an electron is trapped in an atom or is free, it has an intrinsic spin
angular momentum S often called simply spin. (If you didn’t do this in 4B be sure to read the section
in your text on Spin Magnetic Dipole Moments)
Recall that intrinsic means that S is a basic characteristic of an electron, like its mass and electric
charge. As we shall discuss in the next section, the magnitude of S is quantized and depends on a spin
quantum number s, which is always ½ for electrons (and for protons and neutrons). In addition, the
component of S measured along any axis is quantized and depends on a spin magnetic quantum number
ms , which can have only the value + ½ or – ½
The existence of electron spin was postulated on an empirical basis by two Dutch graduate students,
George Uhlenbeck and Samuel Goudsmit, from their studies of atomic spectra. The quantum physics
basis for electron spin was provided a few years later, by British physicist P. A. M. Dirac, who
developed (in 1929) a relativistic quantum theory of the electron.
It is tempting to account for electron spin by thinking of the electron as a tiny sphere spinning about
an axis. However, that classical model, like the classical model of orbits, does not hold up. In quantum
physics, spin angular momentum is best thought of as a measurable intrinsic property of the electron.
Previously, we briefly discussed the quantum numbers generated by applying Schrödinger’s
equation to the electron in a hydrogen atom. We can now extend the list of quantum numbers by
including s and m s , as shown in the table below .
This set of five quantum numbers completely specifies the quantum state of an electron in a hydrogen
atom or any other atom. All states with the same value of n form a shell. By counting the allowed values
of l and ml and then doubling the number to account for the two allowed values of m s , you can verify
that a shell defined by quantum number n has 2n2 states. All states with the same value of n and l form a
subshell and have the same energy. You can verify that a subshell defined by quantum number l has 2l (
l+1) states.
BREAK!!!
Angular Momenta and Magnetic Dipole Moments
Every quantum state of an electron in an atom has an associated orbital angular momentum and a
corresponding orbital magnetic dipole moment. Every electron, whether trapped in an atom or free,
has a spin angular momentum and a corresponding spin magnetic dipole moment. Let’s discuss these
quantities.
Orbital Angular Momentum and Magnetism
The magnitude L of the orbital angular momentum L of an electron in an atom is quantized; that is, it
can have only certain values. This comes from the solution of the Schrödinger equation for the hydrogen
potential. These values are
in which l is the orbital quantum number and hbar is h/2p. According to the previous table, l must be
either zero or a positive integer no greater than n - 1. For a state with n = 3, for example, only l=2,l=1
and l=0 are permitted.
As we discussed in 4B, a magnetic dipole is associated with the orbital angular momentum L of an
electron in an atom. This magnetic dipole has an orbital magnetic dipole moment orb , which is related
to the angular momentum by:
The minus sign in this relation means that orb is directed opposite L . Because the magnitude of L is
quantized as
, the magnitude of orb must also be quantized and given by
Neither orb nor L can be measured in any way. However, we can measure their components along a
given axis. Suppose that the atom is located in a magnetic field B, with a z axis extending in the direction
of the field lines at the atom’s location. Then we can measure the z components of orb and L along that
axis.
The components orb,z are quantized and given by
Here ml is the orbital magnetic quantum number from the table above and B is the Bohr
magneton:
(Bohr magneton)
where m is the electron mass.
The components L z of the angular momentum are also quantized, and they are
given by
The figure at right shows the five quantized components L z of the orbital
angular momentum for an electron with l=2, as well as the associated
orientations of the angular momentum L. However, do not take the figure
literally because we cannot detect L in any way. Thus, drawing it in a figure is
merely a visual aide. We can extend that visual aide by saying that L makes a
certain angle  with the z axis, such that
We can call  the semi-classical angle between vector L and the z axis
because  is a classical measure of something that quantum theory tells
us cannot be measured.
The allowed values of Lz for an electron in a
quantum state with l=2. For every orbital
angular momentum vector L in the figure,
there is a vector pointing in the opposite
direction, representing the magnitude and
direction of the orbital magnetic dipole
moment orb.
Spin Angular Momentum and Spin Magnetic Dipole Moment
The magnitude S of the spin angular momentum S of any electron, whether free or trapped, has the
single value given by
Where s = ½ is the spin quantum number of the electron.
As we discussed in 4B, an electron has an intrinsic magnetic dipole that is associated with its
spin angular momentum , whether the electron is confined to an atom or free. This magnetic dipole
has a spin magnetic dipole moment s , which is related to the spin angular momentum :
The minus sign in this relation means that s is directed opposite S. Because the magnitude of S is
quantized, the magnitude of s must also be quantized and given by
As with L and orb, Neither S nor s can be measured in any way. However, we can measure their
components along any given axis — call it the z axis. The components S z of the spin angular momentum
are quantized and given by
where s is the spin magnetic quantum number give in the previous table. That quantum number
can have only two values: s = + ½ (the electron is said to be spin up) and s = - ½ (the electron is said to
be spin down).
The components s,z of the spin magnetic dipole moment are also quantized, and they are given by
The figure at right shows the two quantized components S z of the spin
angular momentum for an electron and the associated orientations of vector
S. It also shows the quantized components s,z of the spin magnetic dipole
moment and the associated orientations of s.
Orbital and Spin Angular Momenta Combined
For an atom containing more than one electron, we define a total
angular momentum J, which is the vector sum of the angular momenta of
the individual electrons — both their orbital and their spin angular momenta. Each element in the
periodic table is defined by the number of protons in the nucleus of an atom of the element. This
number of protons is defined as being the atomic number (or charge number) Z of the element. Because
an electrically neutral atom contains equal numbers of protons and electrons, Z is also the number of
electrons in the neutral atom, and we use this fact to indicate a J value for a neutral atom:
Similarly, the total magnetic dipole moment of a multielectron atom is the vector sum
of the magnetic dipole moments (both orbital and spin) of its individual electrons.
However, because of the factor 2 in
, the resultant magnetic dipole
moment for the atom does not have the direction of vector -J; instead, it makes a
certain angle with that vector. The effective magnetic dipole moment eff for the atom
is the component of the vector sum of the individual magnetic dipole moments in the
direction of -J (See figure at right). In typical atoms the orbital angular momenta and
the spin angular momenta of most of the electrons sum vectorially to zero. Then J and
eff of those atoms are due to a relatively small number of electrons, often only a single
valence electron.
An electron is in a quantum state for which the magnitude of the electron’s orbital angular
momentum L is 2 √ 3 hbar. How many projections of the electron’s orbital magnetic dipole moment
on a z axis are allowed?
The Stern–Gerlach Experiment
In 1922, Otto Stern and Walther Gerlach at the University of
Hamburg in Germany showed experimentally that the magnetic
moment of silver atoms is quantized. In the Stern–Gerlach
experiment, as it is now known, silver is vaporized in an oven, and
some of the atoms in that vapor escape through a narrow slit in the
oven wall and pass into an evacuated tube. Some of those escaping
atoms then pass through a second narrow slit, to form a narrow beam
of atoms (at right). (The atoms are said to be collimated—made into a
beam—and the second slit is called a collimator.) The beam passes
between the poles of an electromagnet and then lands on a glass detector plate where it forms a silver
deposit.
When the electromagnet is off, the silver deposit is a narrow spot. What should happen when the
magnetic field is turned on and why?
However, when the electromagnet is turned on, the silver deposit should be spread vertically. The
reason is that silver atoms are magnetic dipoles, and so vertical magnetic forces act on them as they
pass through the vertical magnetic field of the electromagnet; these forces deflect them slightly up
or down. Thus, by analyzing the silver deposit on the plate, we can determine what deflections the
atoms underwent in the magnetic field.
When Stern and Gerlach analyzed the pattern of silver on their detector plate, they found a surprise.
However, before we discuss that surprise and its quantum implications, let us discuss the magnetic
deflecting force acting on the silver atoms.
The Magnetic Deflecting Force on a Silver Atom
We have not previously discussed the type of magnetic force that deflects the silver atoms in a
Stern – Gerlach experiment. It is not the magnetic deflecting force that acts on a moving charged
particle, as given by F = qv x B. The reason is simple: A silver atom is electrically neutral (its net charge q
is zero), and thus this type of magnetic force is also zero.
The type of magnetic force we seek is due to an interaction between the magnetic field of the
electromagnet and the magnetic dipole of the individual silver atom. We can derive an expression for
the force in this interaction by starting with the energy U of the dipole
in the magnetic field. Recall from 4B that the energy U of a dipole in a
magnetic field is:
where  is the magnetic dipole moment of a silver atom. In the
apparatus shown, the positive direction of the z axis and the direction
of B are vertically upward. Thus, we can write the energy U in terms
of the component z of the atom’s magnetic dipole moment along
the direction of B:
Then, using (F = -dU/dx) for the z axis shown above, we obtain
This is what we sought — an equation for the magnetic force that deflects a silver atom as the atom
passes through a magnetic field.
The term dB/dz is the gradient of the magnetic field along the z axis. If the magnetic field does not
change along the z axis (as in a uniform magnetic field or no magnetic field), then dB/dz = 0 and
a silver atom is not deflected as it moves between the magnet’s poles. In the Stern – Gerlach
experiment, the poles are designed to maximize the gradient dB/dz, so as to vertically deflect the silver
atoms passing between the poles as much as possible, so that their deflections show up in the deposit
on the glass plate.
According to classical physics, the components z of silver atoms passing through the magnetic field
in the apparatus above should range in value from  (the dipole moment  is directed straight down
the z axis) to +  ( is directed straight up the z axis). Thus, from
there should be a range of forces on the atoms, and therefore a range of deflections of the atoms,
from a greatest downward deflection to a greatest upward deflection. This means that we should
expect the atoms to land along a vertical line on the glass plate.
The Experimental Surprise
What Stern and Gerlach found was that the atoms formed two distinct spots on the glass plate, one spot
above the point where they would have landed with no deflection and the other spot just as far below
that point. The spots were initially too faint to be seen, but they became visible when Stern happened to
breathe on the glass plate after smoking a cheap cigar. Sulfur in his breath (from the cigar) combined
with the silver to produce a noticeably black silver sulfide.
This two-spot result can be seen in the plots at right, which shows
the out-come of a more recent version of the Stern – Gerlach
experiment. In that version, a beam of cesium atoms (magnetic
dipoles like the silver atoms in the original Stern – Gerlach
experiment) was sent through a magnetic field with a large vertical
gradient dB/dz. The field could be turned on and off, and a
detector could be moved up and down through the beam.
When the field was turned off, the beam was, of course, undeflected and the detector recorded the
central-peak pattern shown above. When the field was turned on, the original beam was split vertically
by the magnetic field into two smaller beams, one beam higher than the previously undeflected beam
and the other beam lower. As the detector moved vertically up through these two smaller beams, it
recorded the two-peak pattern shown.
The Meaning of the Results
In the original Stern–Gerlach experiment, two spots of silver were formed on the glass plate, not a
vertical line of silver. This means that the component z along B (and along z) could not have any value
between - and + as classical physics predicts.
Instead, z is restricted to only two values, one for each spot on the glass. Thus, the original Stern–
Gerlach experiment showed z is quantized, implying (correctly) that  is also. Moreover, because the
angular momentum L of an atom is associated with  ,that angular momentum and its component L z
are also quantized.
With modern quantum theory, we can add to the explanation of the two-spot result in the Stern –
Gerlach experiment. We now know that a silver atom consists of many electrons, each with a spin
magnetic moment and an orbital magnetic moment. We also know that all those moments vectorially
cancel out except for a single electron, and the orbital dipole moment of that electron is zero. Thus, the
combined dipole moment  of a silver atom is the spin magnetic dipole moment of that single electron.
According to
, this means that z can have only two components along the z axis . One
component is for quantum number ms = + ½ (the single electron is spin up), and the other component
is for quantum number ms = -½ (the single electron is spin down). Substituting into the previous
equation gives:
Then substituting these expressions for z in
, we find that the force component Fz deflecting the silver atoms as they pass through the magnetic
field can have only the two values
which result in the two spots of silver on the glass.
In the Stern – Gerlach experiment , a beam of silver atoms passes through a magnetic field
gradient dB/dz of magnitude 1.4 T/mm that is set up along the z axis. This region has a length w of
3.5 cm in the direction of the original beam. The speed of the atoms is 750 m/s. By what distance d
have the atoms been deflected when they leave the region of the field gradient? The mass M of a silver
atom is 1.8 * 10 -25 kg.
Because the deflecting force on the atom acts perpendicular to the atom’s original direction of travel,
the component v of the atom’s velocity along the original direction of travel is not changed by the
force.Thus, the atom requires time t = w/v to travel through length w in that direction.
Seperation is then 2x this or 0.16 mm. This should be observablewith a simple magnifier!