SCPY152
Lecture 21 Magnetic Resonances
Udom Robkob, Physics-MUSC
March 18, 2015
Udom Robkob, Physics-MUSC
SCPY152Lecture 21 Magnetic Resonances
Today Topics
Electron orbit an its magnetic moment
Magnetic interaction
Electron spin angular momentum
Electron spin magnetic moment
Electron spin resonance
Nuclear spin magnetic moment
Nuclear magnetic resonance
Udom Robkob, Physics-MUSC
SCPY152Lecture 21 Magnetic Resonances
Electron Orbit and Its Magnetic Moment
Let us determine an electron under circular motion with
~ = m~r × ~v .
corresponding angular momentum L
With electric charge −e of an electron, an orbiting electron
produces electric current i around the orbit but in opposite
direction as
i=
ev
eme vr
−e
=−
=−
T
2πr
2πme r2
Udom Robkob, Physics-MUSC
(1)
SCPY152Lecture 21 Magnetic Resonances
Electron Orbit and Its Magnetic Moment
The magnetic moment of current loop i is defined to be
~ A = πr2 − loop area
µ
~ = iA,
(2)
Then the magnetic moment of electron orbit is
µ
~ =−
eme vrˆ
n
e ~
=−
L
2m2
2me
(3)
~ = me vrˆ
where n
ˆ is direction of the orbitng plane and L
n=
me~r × ~v .
Udom Robkob, Physics-MUSC
SCPY152Lecture 21 Magnetic Resonances
Magnetic Interaction
The magnetic interaction of magnetic moment µ
~ with
~ = B zˆ is defined to be
uniform magnetic field B
~ =
U = −~
µ·B
eB
Lz .
2me
(4)
In quantum physics, this energy will quantized according
quantization of Lz to the values;
U =m
e~
B = mµb B, m = 0, ±1, ±2, ..., ±l.
2me
(5)
where µB = 5.79 × 10−5 eV/T is known as Bohr magneton.
Udom Robkob, Physics-MUSC
SCPY152Lecture 21 Magnetic Resonances
Magnetic Interaction
Diagram of magnetic interaction energy at l = 1 appear in
the following figure;
Exercise: Draw the energy digrams of magnetic interaction
at l = 0, 2, 3 for arbitrary energy E and magentic field B.
Udom Robkob, Physics-MUSC
SCPY152Lecture 21 Magnetic Resonances
Electron Spin Angular Momentum
Electron spin angular momentum is observed in
Stern-Gerlach expeiment.
Suppose µ
~ S is a magnetic moment of electron, interaction
energy with non-uniform magnetic field B(z)ˆ
z is
~ = −mz B(z)
U = −~
µS · B
Udom Robkob, Physics-MUSC
(6)
SCPY152Lecture 21 Magnetic Resonances
Electron Spin Angular Momentum
There will be force from magnetic non-unifomity in
z-direction acts on the magnetic moment;
dU
dB(z)
Fz = −
= µSz
(7)
dz
dz
The result of Stern Gerlach experiment made us to define
~ with
electron spin angular momentum, or spin, S
corrsesponding magnetic moment
gµB ~
µ
~S =
S.
(8)
~
~
where g ' 2 is known as g-factor. The quantization of S
~ as
will follow the same rule of L
S 2 = s(s + 1)~2 , Sz = ms ~.
(9)
where s is spin quantum number and ms is spin magnetic
moment quantum nimber and
ms = −s, −(s − 1), ..., (s − 1), s
Udom Robkob, Physics-MUSC
SCPY152Lecture 21 Magnetic Resonances
Electron Spin Angular Momentum
From Stern Gerlach experiment, we have to assign the
value of electron spin to be
1
1
s = , ms = ± .
2
2
(10)
Then we have
1
S =
2
2
√
1
3 2
2
~ = 3 ~,
+ 1 ~ = ~ → |S|
2
4
2
1
Sz = ± ~
2
(11)
(12)
Let us assign an abstract spin state χs,ms , then χ 1 , 1 is
2 2
known as ”spin-up” state and χ 1 ,− 1 is known as
2
2
”spin-down” state.
Udom Robkob, Physics-MUSC
SCPY152Lecture 21 Magnetic Resonances
Electron Spin Angular Momentum
The spin angular momentum orientation will appear as in
the following figure;
Magnetic interaction of electron spin magnetic moment
~ = B zˆ will be
with magnetic field B
~ = − gµB B Sz = ∓µB B.
U = −~
µS · B
~
Udom Robkob, Physics-MUSC
(13)
SCPY152Lecture 21 Magnetic Resonances
Electron Spin Resonance
Energy diagram of this interaction energy is;
The ”resonance” of photon energy to make transitions
between these two states will be
~ω =
hc
= ∆E = 2µB B
λ
Udom Robkob, Physics-MUSC
(14)
SCPY152Lecture 21 Magnetic Resonances
Nuclear Spin Magnetic Moment
Inside the atomic nucleus we have nucleons, proton p and
neutron n.
Nuclear spin magnetic moment comes from spin of the
nucleon.
Spin quantum number of proton and neutron is s = 1/2.
Their corresponding magnetic moments will be
µ
~ =g
µN ~
S
~
(15)
where g = 2.7928 for proton and g = −1.9130 for neutron.
e~
µN = 2m
= 3.153 × 10−8 eV/T is known as nuclear
p
magneton.
Udom Robkob, Physics-MUSC
SCPY152Lecture 21 Magnetic Resonances
Nuclear Spin Magnetic Moment
The alignments of proton and neutron magnetic moments
with respect their spin appear in the following figures.
Udom Robkob, Physics-MUSC
SCPY152Lecture 21 Magnetic Resonances
Nuclear Magnetic Resonance
Let us determine proton nuclear magnetic moment µ
~ in
~
uniform magnetic field B = B zˆ. The magnetic interaction
energy will be
~ =−
U = −~
µ·B
1
gµN B
Sz = ∓ gµN B
~
2
(16)
The corresponding emission or absorption photon will have
energy
hc
~ω =
= gµN B
(17)
λ
Udom Robkob, Physics-MUSC
SCPY152Lecture 21 Magnetic Resonances
Nuclear Magnetic Resonance
Example - calculate photon wavelength resonances to
proton spin magnetic moment energy in uniform magnetic
field of magnitude 1.5 tesla.
λ=
hc
1.25 × 10−6
=
= 9.42m.
gµN B
2.793 × 3.15 × 10−8 × 1.5
It is in the range of radiowaves
Exercise - calculate photon wavelength resonances to
electron spin magnetic moment energy in magnetic field of
magnitude 0.5 tesla.
Udom Robkob, Physics-MUSC
SCPY152Lecture 21 Magnetic Resonances