The document discusses exchange interactions, their origins, classifications, and effects on magnetic properties in quantum mechanics. It elaborates on direct and indirect exchange, outlining the mechanisms of super exchange, RKKY exchange, double exchange, and anisotropic exchange interactions. These interactions primarily influence behavior in magnetic materials, emphasizing the significance of orbital overlaps and the role of different bond angles in determining coupling strengths.

1. Topic: - Exchange Interactions and Their Consequences
Presented by :-
Name – Subhajit Pramanick
Roll No – 22PH91R16
Department of Physics
IIT Kharagpur
Name – Anannya Sahaw
Roll No – 22CY91R01
Department of Chemistry
IIT Kharagpur

2. Contents
 What is Exchange Interaction?
 Origin of Exchange Interaction
 Classification of Exchange Interaction
 Direct Exchange
 Indirect Exchange
 Super Exchange
 RKKY Exchange
 Double Exchange
 Anisotropic Exchange Interaction

3. What is Exchange Interaction?
 Purely QM effect, occurs only between identical
particles.
 Boson and fermion both can experience it. For
fermions, it is Pauli repulsion, related to Pauli
exclusion principle whereas for bosons, it is one
type of effective attraction which makes them
close to each other, as in Bose-Einstein
Condensation.
 In case of two fermions, if they are parallel, they
remain far apart (Pauli Exclusion Principle) whereas
if they are antiparallel, they may come closer
together such that their wave functions may overlap
(see figure).
 Charges of same sign cost energy when they are
close together and save energy when they are far
apart.

4. Origin of Exchange Interaction
f
 Consider a simple model with just two electrons (𝐬𝟏= 𝐬𝟐=
𝟏
𝟐
).
So, total spin, S =
1
2
−
1
2
, (
1
2
+
1
2
) = 0 (singlet),1 (triplet)
 Singlet State :
S = 0. So, 𝐦𝐬 = 0 → only one state
Total wave function, 𝚿𝐬 = 𝛗𝐬 𝛘𝐬
𝛗𝐬 →symmetric, 𝛘𝐬 →antisymmetric (since 𝚿𝐬 →antisymmetric).
Here, 𝛗𝐬 =
𝟏
𝟐
[ 𝝋𝒂(𝑟𝑎) 𝝋𝒃(𝑟𝑏) + 𝝋𝒂(𝑟𝑏) 𝝋𝒃(𝑟𝑎) ] and
𝛘𝐬 =
𝟏
𝟐
[ ↑ ↓ - ↓ ↑ ]
Energy: 𝐄𝐒 = 𝚿𝐬
∗
𝐇 𝚿𝐬 d𝐫𝟏 d𝐫𝟐 and 𝐬𝟏.𝐬𝟐 = -
𝟑
𝟒
 Triplet States :
S = 1. So, 𝐦𝐬 = -1,0,+1 → total three states
Total wave function, 𝚿𝐓 = 𝛗𝐓 𝛘𝐓
𝛗𝐓 →antisymmetric, 𝛘𝐓 →symmetric (since 𝚿𝐓 →antisymmetric).
Here, 𝛗𝐓 =
𝟏
𝟐
[ 𝝋𝒂(𝑟𝑎) 𝝋𝒃(𝑟𝑏) - 𝝋𝒂(𝑟𝑏) 𝝋𝒃(𝑟𝑎) ] and
𝛘𝐓 = ↑ ↑ , ↓ ↓ ,
𝟏
𝟐
[ ↑ ↓ + ↓ ↑ ]
Energy: 𝐄𝐓 = 𝚿𝐓
∗
𝐇 𝚿𝐓 d𝐫𝟏 d𝐫𝟐 and 𝐬𝟏.𝐬𝟐 =
𝟏
𝟒

5. Origin of Exchange Interaction
 So, 𝐄𝐒 - 𝐄𝐓 = 2 𝝋𝒂
∗
(𝒓𝒂) 𝝋𝒃
∗
(𝑟𝑏) 𝐇 𝝋𝒂(𝒓𝒃) 𝝋𝒃(𝑟𝑎) d𝐫𝟏 d𝐫𝟐
Now, Hamiltonian can be written as : 𝐇 =
𝟏
𝟒
(𝐄𝐒 + 3𝐄𝐓) – (𝐄𝐒 - 𝐄𝐓) 𝐬𝟏.𝐬𝟐
Define, Exchange Integral:
J =
𝐄𝐒 − 𝐄𝐓
𝟐
= 𝝋𝒂
∗
(𝒓𝒂) 𝝋𝒃
∗
(𝑟𝑏) 𝐇 𝝋𝒂(𝒓𝒃) 𝝋𝒃(𝑟𝑎) d𝐫𝟏 d𝐫𝟐
So, the spin-dependent term in the Hamiltonian becomes:
𝐇𝒔𝒑𝒊𝒏 = - 2J 𝐬𝟏.𝐬𝟐
This term in the Hamiltonian is the origin of exchange interaction
between two identical particles.
 If J>0, 𝐄𝐒> 𝐄𝐓 then triplet state S=1 is favoured (Ferromagnetism). If J<0, 𝐄𝐒<𝐄𝐓
then singlet state S=0 is favoured (Anti-ferromagnetism).
 In many electron system for ferromagnetism or anti-ferromagnetism, considering
exchange interaction Heisenberg gave the simplest model in which,
𝐇 = - 𝐢𝐣 𝐉𝐢𝐣 𝐬𝐢 . 𝐬𝐣 or 𝐇 = - 2 𝐢>𝐣 𝐉𝐢𝐣 𝐬𝐢 . 𝐬𝐣
sometimes, 𝐇 = - 𝐢𝐣 𝐉𝐢𝐣 𝐬𝐢 . 𝐬𝐣
more simply, 𝐇 = - J 𝐢𝐣 𝐬𝐢 . 𝐬𝐣

6. Classification of Exchange Interaction
Exchange
Interaction
Direct
Exchange
Indirect
Exchange
Super
Exchange
RKKY
Exchange
 There are mainly two types of exchange interactions:
 Besides this, there are many other exchange interactions like: Double Exchange interaction, Anisotropic
Exchange Interaction etc.

7. Direct Exchange
 Electrons of neighbouring magnetic atoms interact via
an exchange interaction. It don’t need any intermediary.
 Direct interaction between neighbouring atoms is due to the
spatial overlap of orbitals.
 The simplest model for this kind of interactions is the
Heisenberg model: 𝐇 = - J 𝐢𝐣 𝐬𝐢 . 𝐬𝐣
Depending on J values some of the metals are FM
and some are AFM (see, Bethe-Slater Curve).
 Very often direct exchange cannot be an important
mechanism in controlling the magnetic properties because
there is insufficient direct overlap between neighbouring
magnetic orbitals as in case of rare earths and transition
metals. Then it becomes necessary to consider indirect
exchange.
Bethe-Slater Curve

8. Indirect Exchange
 It is the coupling between magnetic moments over long distance and requires a mediator.
 It can be of two types (a) Super Exchange (b) RKKY Exchange
Indirect Exchange in Ionic Solid: Super Exchange
 In systems in which direct exchange cannot be realized due to insufficient overlap of magnetic orbitals,
magnetic coupling may be mediated by orbitals of a nonmagnetic ligand in between them. It is the super
exchange interaction, which is responsible for the magnetic properties of the most of magnetic materials,
especially nonmetallic compounds, for example, oxides or fluorides. Generally found in metal oxides
where the magnetic atoms are separated by non magnetic ions ( O2- ).

9.  Superexchange depends on the electron configuration of magnetic ions and 𝐌𝟏–O–𝐌𝟐 bond angle. The
rules given by Goodenough, Kanamori, and Anderson help us to predict the resulting coupling:
Indirect Exchange
 Goodenough-Kanamori rules
 Strong negative coupling when 𝐌𝟏–O–𝐌𝟐 angle
is equal to 180o: There is a strong
antiferromagnetic exchange interaction if the
half-filled orbitals of two cations overlap with
the same empty or filled orbital of the
intervening anion.
 Weak positive coupling when 𝐌𝟏–O–𝐌𝟐 angle is
equal 90o : There is a weaker ferromagnetic
exchange interaction if the half-filled orbitals of
two cations overlap with orthogonal orbitals of
the same intervening anion.

10. Indirect Exchange
Indirect Exchange in Metal: RKKY or Itinerant Exchange
 Another type of indirect exchange is active in metals, where
conduction electrons may mediate the interaction between
localized magnetic moments of metal ions known as
(Ruderman, Kittel, Kasuya, and Yosida interaction) or RKKY
exchange.
 This type of coupling applies mainly to lanthanides based
materials, in which the 4f shells are localized close to the
nucleus. The 4f moments polarize spins of the 5d or 6s electrons
and this polarization is transferred to the moment of the
adjacent metal ion.
 The RKKY mechanism depends on a density of states of
conduction electrons and works on a long range.
Exchange integral, JRKKY ∝
𝐜𝐨𝐬(𝟐𝑲𝑭 𝒓)
𝒓𝟑
Depending upon the distance between the localized
moments of two magnetic ions, it may be either FM or
AFM.

11. Double Exchange
 In compounds in which magnetic ion occurs in two oxidation states (mixed valency), for example Fe2+
and Fe3+ or Mn3+ and Mn4+, magnetic coupling may be realized by means of a real electron delocalization
to the empty orbital of the neighbour.
 The hopping of an extra electron of Fe2+ or of Mn3+ via the 2p orbital of oxygen proceeds without the
spin-flip of the hopping electron and results in the ferromagnetic coupling of the two centers. The
double exchange operates in Fe3O4, La1−x SrxMnO3.

12. Anisotropic Exchange Interaction
 This is also known as Dzyaloshinsky-Moriya (D-M) interaction. Here spin-orbit
interaction plays the role as the oxygen atoms act in super exchange.
 The excited state is produced by the spin-orbit interaction in one of the
magnetic ions and an exchange interaction occurs between excited state of one
ion and ground state of other ion.
 This interaction includes a term in the Hamiltonian, 𝐇𝑫𝑴 = 𝐃. 𝐬𝟏 × 𝐬𝟐
𝐃 will lie parallel or perpendicular to the line connecting two spins,
depending on the symmetry.
 This interaction is such that it tries to force 𝐬𝟏 and 𝐬𝟐 to be at right angles in a
plane perpendicular to 𝐃 in such an orientation as to ensure that energy is
negative. Its effect is therefore very often cant the spins by small angle.
 For this spin canting, antiferromagnetic materials show some non-zero
magnetic moment near absolute zero (Weak Ferrimagnetism).