Transcript 投影片 1 - National Cheng Kung University

13. Magnetic Resonance
Nuclear Magnetic Resonance
Equations of Motion
Line Width
Motional Narrowing
Hyperfine Splitting
Examples: Paramagnetic Point Defects
F Centers in Alkali Halides
Donor Atoms in Silicon
Knight Shift
Nuclear Quadrupole Resonance
Ferromagnetic Resonance
Shape Effects in FMR
Spin Wave Resonance
Antiferromagnetic Resonance
Electron Paramagnetic Resonance
Exchange Narrowing
Zero-Field Splitting
Principle of Maser Action
Three-Level Maser
Ruby Laser
Notable Resonance Phenomena / Instruments:
NMR: Nuclear Magnetic Resonance
NQR:
Nuclear Quadrupole Resonance
EP(S)R: Electron Paramagnetic (Spin) Resonance
FMR:
Ferromagnetic Resonance
SWR:
Spin Wave Resonance
AFMR: Anti-Ferromagnetic Resonance
CESR: Conduction Electron Spin Resonance
Information gained:
• Fine struction of absorption: Electronic structure of individual defects.
• Change in line width: Motion of the spin or its surroundings.
• Chemical or Knight shift: Internal magnetic field felt by the spin.
• Collective spin excitations.
Prototype of all resonance phenomena is NMR.
Main applications of NMR:
• Identification & structure determination for organic / biochemical componds.
• Medical (MRI).
Nuclear Magnetic Resonance
Consider nucleus with magnetic moment μ and angular momentum  I.
μ 
In an applied field,
→
Ba  B0 zˆ
γ = gyromagnetic ratio
I
U  μ  Ba
U  z B0   
B0 I z   
B0 mI
mI  I , I 1,
,  I  1,  I
Resonance at
0   B0
I = ½ , mI =  ½
γ
[s-1
G–1
]
ν [MHz]
proton
2.675104
4.258 B0 [kG]
electron
1.759107
2799 B0 [kG]
Nuclear magneton
N 
p 
e
 5.0511024 erg / G
2M p c
gp
2
 N  2.793 N
g p  5.586
g n  3.826
Equations of Motion
dI
 μ  Ba
dt
Gyroscopic equation:
→
dμ
  μ  Ba
dt
In thermal equilibrium, M // Ba .
Ba  B0 zˆ
For I = ½ ,
→
M  M 0 zˆ
M 0   N1  N2  
N2
 e2  B0 / kB T  e  x
N1

with
→
dM
  M  Ba
dt
M  μ
N 2
C
B0
 B0 
T
3kB T
M 0  0 B0
where N1 is the density of population of the lower level
N1  N2  N
→
N
N1 
1  e x
e x N
N2 
1  e x
1  e x
x
 B0
M0 
N


N

tanh

N

tanh
1  e x
2
kB T
If system is slightly out of equilibrium,
the relaxation towards equilibrium can be described by
M  M0
d Mz
 z
dt
T1
T1 = spin-lattice (longitudinal) relaxation time
M  M0
d Mz
 z
dt
T1
→
 M z t   M 0 
t
ln 
  
T1
 M z 0  M 0 
Let an unmagnetized specimen be placed at t = 0 in field Ba  B0 zˆ
Then
so that
M z  0  0
M z    M 0
 M t   M 0 
t
ln  z




M
T1
0


→
M z  t   M 0 1  e  t /T1 
Dominant relaxation mechanism:
phonon
emission
phonon
inelastic
scattering
phonon
absorption +
re-emission
For
Ba  B0 zˆ
M    M 0 zˆ
and in the presence of relaxation:
dMx
M
  B0 M y  x
dt
T2
dMy
dt
  B0 M x 
My
T 2 = transverse relaxation time
T2
M  M0
dMz
 z
dt
T1
U  M  Ba  M z B0
With initial conditions
we have
→ relaxation of Mx & My doesn’t affect U.
M x  0  m, M y  0  0, M z    M0
M x t   m et / T2 cos 0 t
M y t   m et / T2 sin 0 t
M z  t   M 0 1  e  t /T1 
where
0   B0
With an additional transverse rf field and setting d Mz / d t = 0, we have
Ba   B1 cos  t,  B1 sin  t, B0 
d Mx
M
  B0 M y  x   M 0 B1 sin  t
dt
T2
dMy
dt
  B0 M x 
My
T2
  M 0 B1 cos  t
The particular solutions are obtained by setting
i  m0 
0   B1 ( M x sin  t  M y cos  t ) 
m  M x  i M y  m0 ei  t
m0
 i  B0 m0  i  B1M z
T2
M  M0
i
0   B1 (m0  m0* )  z
2
T1
i
M z  M 0   B1 (m0  m0* ) T1
2
→
m0 
i  B1 T2 M z
1  i (   B0 )T2

 P   
2
2 /

Half-width =
i1T2 (1  i (  0 )T2 )
1  (  0 )2 T22  T1T212
  T2 M z
B2
2 2 1
1  (  0 ) T2
dM
dt

  1/2 
1
T2
dt H ·
0
 M0
M z  M0
T1
Line Width
Magnetic field seen by a magnetic dipole μ1 due to another dipole μ2 is
3  μ2  r12  r12  μ2 r122
B
r125
Bi 
→
 nearest neighbor interaction dominant:
For protons 2A apart,
Bi 
Bi 
1.4 1023 G cm3
 2 10
8
cm 
3

a3
 2G
H2 O

r3
Motional Narrowing
Li7 NMR
in metallic Li
rigid
lattice
τ = diffusion hopping time
Motion
narrowed

1
B
low
diffusion rate
high

1
B
Effect is more prominent in liquid,
e.g., proton line in water is 10–5 the width of that in ice.
(rotational motion)
T2 ~ time for spin to dephase by 1
radian due local perturbation Bi .
    Bi 
After t = n τ (random walk)
 2  n 
n
 Average number of steps for a spin to dephase by 1 radian is
→
T2  n  
 
H2O: τ ~10
̶ 10 s
1

Bi  
2
T20 
whereas
1
2
2
   Bi     0  
T2
0  105 s 1

1
 Bi
0
2
 n   Bi  
1

Bi  
2
for a rigid lattice
since

 0  105
0
0 
1
2
Hyperfine Splitting
Hyperfine interaction :
between μnucl = μI & μe
Contact hyperfine interaction:
when e is in L = 0 state
e




L=0→
e
B
Bohr magneton
2mc
current loop
Dirac: μB ~ circulation of e with v = c,
e m c2
Current in loop = I 

e
h
ec
Probability of e overlaping the nucleus :
Average field seen by nucleus:
Contact hyperfine
interaction:
e
 e 
 103 A
2
mc
Compton
wavelength
→ field at loop center = B  0  
2 I
e

2
Rc
e
R
P    0
2
3
e
B  B  0 P  e   0 
U  μI  B  μ I  μ B   0 
2

2
 e  2 B   0 
B   0 
2
2
I S  a I S
Uch f  a I  S
 a   erg
 a 
a in Gauss   
 B 
a in MHz   a in Gauss  
e
2
intersellar H
High field:
μ B B >> a
S=I=½
Selection rule for e:
ΔmS = 1
ΔmI = 0
  B B mS  a mI mS
Selection rule for nucl:
ΔmS = 0
nucl 
Number of hf splittings = (2I + 1) (2S+1)
a
mS
Examples: Paramagnetic Point Defects
F centers in Alkali Halides ( negative ion vacancy with 1 excess e )
K39 , I = 3/2
Vacancy surrounded by 6 K39 nuclei
→
I max  6 
3
9
2
Number of hf components:
2Imax +1 = 19
Number of ways to arrange the 6 spins:
( 2I + 1)6 = 46 = 4096
Donor Atoms in Silicon
P in Si (outer shell 3s23p3):
4e’s go diamagnetically into covalent bonding;
1e acts as paramagnetic center of S = 1/2.
motion narrowing due
to rapid hopping
Knight Shift
Knight (metallic) shift:
B0 required to achieve the same nuclear resonance ω for a given
spin depends on whether it is embedded in a metal or an insulator.
U    I
B0  a S z
For conduction electrons:
→

U    I

Knight shift :

I
z
M z  g N B Sz
a S 
 B0 I z   I
g N B 
K 
a S
B

B0 g N B  I
  S B0

B 
B0 1 
 Iz
B0 


S
gN
  0
2

a  I
B   0 
2
aatom ≠ ametal → for Li,

| ψmetal(0)|2 ~ 0.44|ψatom(0)|2
Nuclear Quadrupole Resonance
Q in field
gradient
Q>0
Nuclei of spin I  1 have electric quadrupole moment.
Ref: C.P.Slichter,
“Principles of Magnetic
Resonance”, 2nd ed., Chap 9.
Q > 0 for convex (egg shaped) charge distribution.
HQ 
1
V  Q 

3!  , 
Wigner –Eckart Theorem:
Axial symmetry:
V 
2 V

 x  x
HQ 
Q   e
 3x
k protons
x    rk2 
k k
r 0
eQ
3
2
V
I
I

I
I


I



 
 
 


6I  2I  1  , 
2
e2 qQ
HQ 
3I z2  I 2 

4I  2I  1
e2 qQ
3m2  I  I  1
EQ 
4I  2I  1
Built-in field gradient (no need for H0 ) .
eq  Vzz
eQ  I I Qzz I I
Number of levels =  I  +
1
App. : Mine detection.
Ferromagnetic Resonance
Similar to NMR with S = total spin of ferromagnet.
Magnetic selection rule: Δ mS = 1.
Special features:
•
Transverse χ  & χ  very large ( M large).
•
•
Shape effect prominent (demagnetization field large).
Exchange narrowing
(dipolar contribution suppressed by strong exchange coupling).
•
Easily saturated (Spin waves excited before rotation of S ).
Shape Effects in FMR
Consider an ellipsoid sample of cubic ferromagnetic insulator
with principal axes aligned with the Cartesian axes.
Bi  B0  N  M
N
ij
 i j Ni
Lorenz field = (4 π / 3)M.
Exchange field = λ M.
M z  M0
→
M k  M k 0 e i  t →
FMR frequency:
( don’t contribute to torque)
dM
  M  Bi   M   B 0  N  M 
dt
Bloch equations:
B0  B0 zˆ
Bi = internal field .
B0 = external field.
N = demagnetization tensor
dMx
   B0   N y  N z  M 0  M y
dt
dMy
xˆ
yˆ
zˆ
Mx
My
M0
N M
N M
B N M
x
x
y
y
0
z
0
   B0   N x  N z  M 0  M x
dt

i
  B0   N y  N z  M 0    M x 0 


0
M
   B0   N x  N z  M 0 
  y0 
i




02   2  B0   N y  N z  M 0   B0   N x  N z  M 0 
uniform mode
02   2  B0   N y  N z  M 0   B0   N x  N z  M 0 
For a spherical sample,
Nx  N y  Nz
→
0   B0
For a plate  B0 ,
Nx  N y  0, N z  4
→
0    B0  4 M0 
For a plate // B0 ,
Nx  N z  0, N y  4
→
0  
B0  4 M 0
Shape-effect experiments
determine γ & hence g.
Fe
Co
Ni
g 2.10 2.18 2.21
Polished sphere of YIG
at 3.33GHz & 300K for
B0 // [111]
Spin Wave Resonance
Spin waves of odd number of half-wavelenths
can be excited in thin film by uniform Brf
Condition for long wavelength SWR:
0    B0  4 M0   Dk 2
For wave of n half-lengths:
 n 
0    B0  4 M 0   D 

 L 
2
Permalloy
(80Ni20Fe)
at 9GHz
D = exchange constant
Antiferromagnetic Resonance
Consider a uniaxial antiferromagnet with spins on 2 sublattices 1 & 2.
Let
M1  BA zˆ
M2   BA zˆ
→
Exchange fields:
BA = anistropy field derived from
2K
BA 
M
U K  K sin 2 1
θ1 = angle between M1 & z-axis.
M  B1  B2
B1  ex    M2
For
B2  ex    M1
Ba  0
B1   M2  BA zˆ
B2   M1  BA zˆ
 0
With
M1z  M  M 2z
the linearized Bloch equations become:
d M1x
   M1y   M  BA   M   M 2y  
dt
d M 2x
   M 2y   M  BA    M    M1y  
dt
d M1y
   M   M 2x   M1x   M  BA  
dt
d M 2y
    M    M1x   M 2x   M  BA  
dt
M j  M jx  i M jy  ei  t
→
i M 1  i   M 1   M  BA   M 2   M  
i M 2  i   M 2   M  BA   M 1   M  
 BE
   BA  BE   
  M 1 

    0

B

B

B


 A E   M2 
E


02   2 BA  BA  2BE 
AFMR frequency
BE   M
exchange field
At 0K
BE  500kG
BA  8.8kG
MnF2 : TN = 67K
Electron Paramagnetic Resonance
Consider paramagnet with exchange J between n.n. e spins at T >> TC .
Observed line widths << those due to Udipole-dipole .
→ Exchange narrowing
Treating ωex  J /  as a hopping frequency 1/τ ,
the exchange induced motion-narrowing effect gives
 
2
  0  2
ex

Bi2
ex
(Δ ω)0 = dipolar half-width
For the paramagnetic organic crystal DPPH (DiPhenyl Picryl Hydrazyl),
also known as the g marker (used for H calibration),
Δ ω ~ 1.35G is only a few percent of (Δ ω)0
Zero-Field Splitting
Some paramagnetic ions has ground state crystal field splittings of 1010 - 1011 Hz (~MW).
E.g. Mn2+ as impurities gives splittings of 107 - 109 Hz.
Principle of Maser Action
Maser = Microwave Amplification by Stimulated Emission of Radiation
Laser = Light
Amplification by Stimulated Emission of Radiation
Transition rate per atom (Fermi’s golden rule):
P
2
f H i
2
  Brf  1
 


 
2
  l  u
Net radiated power:
  Brf  1
P 
  nu  nl 


 
2
Ambient: Brf at ω
thermal equilibrium → nu << nl .
stimulated emission → nu >> nl . (inversion)
In an EM cavity of volume V and Q factor Q, power loss is
Maser condition: P  PL
→ nu  nl 
V 
8  2 Q

V B
8  Q
Brf2 V 
PL 
8 Q
B 


line-width
Three-Level Maser
Population inversion is attained by pumping
n3 > n2
saturation: n3  n1
n2 > n1
d n2
 n2 P  2  1  n2 P  2  3  n3 P  3  2   n1 P 1  2 
dt
Steady state at saturation:
n2 n2 P  3  2  P 1  2
 
n1 n3 P  2  1  P  2  3
Er3+ are often used in fibre optics amplifiers ( n2 > n1 mode).
Signal: λ ~ 1.55 m, bandwidth ~ 41012 Hz.
n3  n2
 1


n2  n1
 1
Ruby Laser
Cr3+ in Ruby
Optical pumping by
xenon flash lamps
 nonrad  107 s
 rad  5 103 s
For 1020 Cr3+ cm−3 ,
stored U ~ 108 erg cm−3 .
→ High power pulse laser.
Efficiency ~ 1%.
Continuous lasing:
no need to empty G.S.
4 level Nd
glass laser