Section 5.2 - 1

5.2 Magnetism: the basics Def.: “Magnetism” – the property of a material to be attracted to (paramagnetic

response) or repelled by (diamagnetic response) a magnetic field – These effects arise mainly from electrons (WHY?)

Source: “Physical Methods for Chemists” R. S. Drago, Surfside, Gainsville, FL, 1992. • Diamagnetism arises from the circulation of paired electrons within the species to

generate a weak magnetic dipole aligned parallel to the external magnetic field.

Source: Ege, “Organic Chemistry, 2nd Ed.”, D. C. Heath and Co., 1989, Lexington, MA • This is the same mechanism by which a ring current is induced in an aromatic

molecule in an applied magnetic field…observed as deshielding of peripheral protons in 1H NMR spectra.

Section 5.2 - 2

χ=HM!!

• Since all atom-based matter has some paired electrons, all atom-based matter will exhibit a diamagnetic response to an applied magnetic field…HOWEVER...that response is very small and will be overridden by any other magnetic properties that may arise if the matter also has unpaired electrons (i.e., is paramagnetic).

• The paramagnetic response of a material to a magnetic field is 2 to 3 orders of

magnitude greater than the diamagnetic response from the underlying paired electrons.

• All other types of magnetism (ferromagnetism, antiferromagnetism, ferrimagnetism,

superparamagnetism, etc.) are based on long range ordering of unpaired electrons and are critical phenomena … i.e., at some high temperature they revert to paramagnets.

Step-by-Step: Macroscopic (bulk) magnetic properties related to microscopic (atomic) principles I. All atom-based matter responds to an external applied magnetic field H …

i.e., internal magnetic dipole(s) are induced. II. The total magnetic induction B within a substance is proportional to the sum of

the applied magnetic field H and the magnetization M of the substance:

B = µo (H + M) where µ0 is the vacuum permeability µo = 1 in the cgsemu system of units (Note: µo = 4π10-7 newton/ampere2 in the SI system of units, but this convention makes the equations more difficult to work with, so cgsemu units are typically favoured.)

Another way of writing the above equation is:

The ratio of M/H is dependent on the material. It is basically a measure of how “magnetizable” a material is in the presence of an applied magnetic field…in other words the ratio of the induced magnetization to the applied magnetic field. M/H is called magnetic susceptibility (χ):

Using the volume magnetization (Mvol) gives χvolume. χvolume is unitless (i.e., dimensionless) but is conventionally expressed in emu/cm3 Thus the dimension of (electromagnetic units) emu is formally cm3!

⎟⎟⎠

⎞⎜⎜⎝

⎛+=HM

HB

!!

!!

10µ

Section 5.2 - 3

It is more useful to use the molar magnetic susceptibility (χ), related by dividing the volume magnetic susceptibility by the density of the material and multiplying by molecular weight. This is the magnetic susceptibility per mole:

χ is expressed in units of cm3 mol-1 or alternatively in emu/mol NOTE: From now on, we will consider the magnetization M to mean the molar magnetization and reference to magnetic susceptibility will imply molar magnetic susceptibility. It is worth noting that a more accurate representation of the relationship between molar magnetization and applied magnetic field is:

χ=∂∂HM!!

M is a vector, H is a vector, therefore χ is a second rank tensor. If the sample is magnetically isotropic, χ is a scalar. **If the magnetic field is weak enough, χ is independent of H and

M = χH is valid.**

III. A Note on Units (the cgsemu system) ...

• Strictly speaking, the unit of magnetic field H is the Oersted (Oe).

However, many authors express H in gauss (G) which is strictly speaking a unit of magnetic induction B created by the magnetic field. In a vacuum, B = µoH where µo = 1

so the two units are essentially interconvertible • We will see that χT is often the quantity of interest.

χT is expressed in emu K mol-1

• Molar magnetization M is expressed in units of emu G mol-1 Alternatively, M may be expressed in units of Nβ N = Avogadro’s number = 6.0221367 x 1023 mol-1 β = electronic Bohr magneton (also µB) 1 Nβ = 5585 emu G mol-1

ρχχ ÷×= MWvolume

Section 5.2 - 4

• In the cgsemu system of units, energy is expressed in erg (1 erg = 10-7 J):

Boltzmann constant kB = 1.3806580 x 10-16 erg K-1 Bohr magneton β = 9.27401549 x 10-21 erg T-1

• But it is more common to express energy in cm-1, so:

Boltzmann constant kB = 0.69503877 cm-1 K-1 Bohr magneton β = 4.66864374 x 10-5 cm-1 G-1

Using these common units, the constant Nβ2/3kB that appears in the Curie law (for future reference) is equal to 0.125048612 emu/mol ≈ 1/8

Section 5.2 - 5

Section 5.2 - 6

IV. Types of Magnetism ...

There are many different types of magnetism. Each corresponds to a unique set of characteristics that can be observed by measuring the magnetic susceptibility χ of a bulk sample as a function of temperature and applied magnetic field. It is important to note that molar magnetic susceptibility χ is a bulk measurable quantity. The most common types of magnetism are classified based on the measured magnetic susceptibility χ as a function of two factors:

1) temperature (T) 2) applied magnetic field (Ho)

Source: Drago

TC (Curie Temperature) – critical temperature for a ferromagnet TN (Néel Temperature) – critical temperature of an antiferromagnet

Section 5.2 - 7

Diamagnetism

• Diamagnetism is an underlying property of all atom-based matter.

• It is ALWAYS present, even when masked by paramagnetism, etc.

• In principle, the experimentally measured magnetic susceptibility (for any material above its critical temperature) is the algebraic sum:

χexp = χdia + χpara + χTIP

• Of these, χpara is typically the largest in magnitude and responsible for the

majority of the magnetic character of a material. (NOTE: Most authors assume that χpara is implied when discussing “χ”.)

• In order to extract χpara from χexp, we must first account for contributions from χdia. There are a number of acceptable ways of doing this.

1) The magnetic susceptibility of the empty (and usually diamagnetic)

sample holder is measured such that it can be subtracted from the experimental data. In addition...

2) The magnetic susceptibility of a structurally analogous diamagnetic

material is measured and this is used to estimate the diamagnetic contribution in a paramagnetic sample. (e.g., replace paramagnetic transition metal dications with Zn2+)

3) Use the rough approximation χdia = k x m.w. x 10-6 cm-1 mol-1

where k is a weighting factor k = (0.4 – 0.5)

4) Use Pascal’s constants to calculate a better estimate of χdia. Pascal’s constants are tabulated literature values: χdia = χdia,atom + χdia,bond (see table on following page) e.g. Using Pascal’s constants to calculate χdia for benzonitrile:

χdia,atom (x 106 cm3/mol) χdia,bond (x 106 cm3/mol) Caromatic 6 x -6.24 C≡N 1 x 0.8 H 5 x -2.93 C 1 x -6.00 N 1 x -5.57 Total: -63.66 +0.8

χdia = (-63.66 + 0.8) x 106 cm3/mol = -62.86 x 10-6 cm3/mol

C

N

Section 5.2 - 8

Source: Molecular Magnetism, Olivier Kahn, 1992. • χdia is always negative. • The value of χdia is substracted from χexp leaving χpara + χTIP. • χTIP may or may not be important. In many publications, it is assumed to be zero.

Section 5.2 - 9

V. Temperature independent paramagnetism (χTIP) is usually small – often the same order of magnitude as χdia but opposite in sign (i.e., positive).

What is the origin of temperature independent paramagnetism? • If the only thermally populated state of a molecule is a spin singlet without

first order angular momentum, then we expect that χpara = 0 and χexp = χdia. Therefore χexp is negative.

• In some cases, the diamagnetic (i.e., non-degenerate) ground state may couple

with a degenerate excited state (through a Zeeman perturbation...see Van Vleck equation ahead), as long as the energy gaps are not too large. THIS IS NOT A THERMAL POPULATION OF EXCITED STATES!

• This coupling of singlet ground state with degenerate excited state(s)

generates a (+)ve magnetic susceptibility that is temperature independent (χTIP).

• χTIP is not restricted to compounds with a diamagnetic ground state. The

coupling between a magnetic ground state and non-thermally populated excited state(s) may also give χTIP.

VI. Another form of temperature independent paramagnetism, called Pauli

paramagnetism (χPauli) is associated with species that display metallic conductivity.

Consider sodium metal: free electrons in the 3s band and positively charged, essentially diamagnetic ions with a [Ne] core make up the lattice points. • When a magnetic field is applied, there is tendency for the magnetic dipoles

associated with the intrinsic spin of the free electrons to align themselves parallel to the field, giving a positive magnetic moment.

• The electrons in a metal do not have a Boltzmann distribution (so we can’t use

the Langevin formula to calculate the magnetic susceptibility...see notes ahead). The Fermi-Dirac distribution function must be used instead...and since this varies little with temperature (Recall: it varies only with enormous changes in T), the susceptibility turns out to be practically independent of temperature!!

• Let’s derive an expression for the Pauli paramagnetism using a model metal in

which all electrons are spin paired (i.e., all energy levels are filled by two electrons of opposite spin) and the temperature is absolute zero (i.e., the Fermi surface is flat)...

Section 5.2 - 10

Note: In this diagram, W is energy, so dn/dW is the density of states (DOS). Here, the DOS is explicitly divided into electrons with spins parallel (+) to an applied field and spins anti-parallel (-) to the applied field. Source: “Electricity and Magnetism” Bleaney and Bleaney, Clarendon Press, Oxford, 1959.

When a magnetic field H is applied, an electron can only flip its spin magnetic dipole from anti-parallel to parallel if the decrease in its magnetic energy (2βH) is sufficient to supply the extra kinetic energy required to raise it to an empty translational energy level. (RECALL: β is the Bohr magneton...the intrinsic magnetic moment of an electron)

Another way of saying this is that the application of H lowers the energy of the spin parallel states by an amount 2βH. The total magnetic moment of the system is 2xβ, where x is the number of electrons transferred from the anti-parallel to the parallel orientation... ...note that it is 2x times the Bohr magneton because flipping a spin creates both one more spin parallel AND one less spin anti-parallel...so net effect is 2x.

The volume magnetic susceptibility is then Hxoµβ

χ2

=

So how do we find the value of x? It must be related to 2βH.

• We assume that the energy difference w between successive energy levels at

the top of the Fermi distribution is approximately constant. • To flip an electron’s spin from antiparallel to parallel requires that its kinetic

energy be increased by w, since we may take an electron from the topmost filled level and put it in the next highest level, which is vacant.

Section 5.2 - 11

• To flip the second electron requires an additional kinetic energy of 3w, since the next two levels with parallel orientation are already filled. The third electron must be given extra energy equal to 5w, and so forth.

• For the xth electron, the excess kinetic energy will be (2x-1)w.

• If x is very large (x >> 1), we can use 2xw as an approximation...therefore at

equilibrium 2xw = 2βH.

• Substituting this into the above equation, we get woµβ

χ22

=

• This still doesn’t help us much until we can estimate the value of w.

• Since two electrons with spins anti-parallel to one another can occupy each

kinetic energy level, the number of such levels in the range W to W + dW is ½(dn), where dn is the number of electrons in this range.

• Hence the energy separation w between successive levels at the top of the

Fermi distribution is dW / ½(dn) = 2 (dn / dW)F-1

(where WF is the Fermi energy)

Therefore Fo dW

dn⎟⎠

⎞⎜⎝

⎛=µβ

χ2

• Conveniently, the number dn of electrons per unit volume which have a

kinetic energy in the range W to W + dW has been worked out to be:

23

21

23

FWdWnWdn = (see a solid state physics text if you really want to

know how this was derived!)

• Thus FoW

nµβ

χ23 2

=

Pauli was the first to derive this expression. Notice that temperature doesn’t factor into the equation, hence the temperature independence of Pauli paramagnetism!

Section 5.2 - 12

• It was later shown by Landau that the effect of the magnetic field on the

translational motion of the free electrons gives a diamagnetic contribution to the susceptibility, whose value is just -1/3 of the Pauli paramagnetism originally calculated as above. Therefore the net volume Pauli paramagnetism effect is

Note: Eq. (20.16) in this table is the above equation. Source: Bleaney and Bleaney.

So now we have dealt with the diamagnetic contribution χdia and the possible types of temperature independent contributions, χPauli and χTIP (which we will look at in more detail when we derive the van Vleck equation)... ...now we get to heart of most molecular systems...the paramagnetic susceptibility χpara

FoPauli W

nµβ

χ2

=