Muonium - Home | TRIUMF · Muonium — a light isotope of hydrogen ... Muonium Isotope Effects The chemistry of an atom depends primarily on ... The muon (proton) and the electron

Muonium

Paul Percival

TRIUMF Summer Institute 2011

Lecture 9

Paul Percival TRIUMF Summer Institute, August 2011

2

Hydrogen-like Atoms

µ+

e−

p+

e− e−

e+

H Mu Ps He+µ−

µ−

e−

He++

In the Bohr (planetary) model, the electrostatic attraction is balanced by the centrifugal acceleration of the orbiting electron

Assuming the orbital angular momentum is quantized,

2e

20

( )(4 )

Ze e m vr r

−− =

πε

2 2 421

2 2 2 2 20 04 2 (4 )n

Ze mZ eE T V mvr n n

= + = − = − = −πε πε

1, 2, 3L mvr n n= = =

Rydberg constant⇒


3

Muonium — a light isotope of hydrogen

reduced mass of Mu = 0.995 mr(H)

ionization potential = 13.539 eV

Bohr radius = 0.532 Å

μ+

e−

The properties of a single electron

atom are determined by mr

r N e e

1 1 1 1m m m m

= + ≈

1p9m mμ =

Quantum mechanics gives the same result as the Bohr model

but if the coordinate system is defined by the centre of mass we need the reduced mass mr

2 4r

2 2 20

e 12 (4 )n

Z mEn

= − ⋅πε

20

0 2r

(4 )e

am

πε=


4

Muonium Isotope Effects

The chemistry of an atom depends primarily on

the ionization potential How easy is it to remove an electron?

the radius How are the electrons distributed?

For Mu these are almost the same as for H

However, for molecular vibrations involving Mu,

∴ = ≈υ υ υMuXHX

MuXHX HX

mm

3Xr

X

m mm m

m mμ

μμ

= ≈+

⇒ Vibrational frequencies involving Mu are higher than for H

if mX >> mµ

Mu―X


5

Muonium is a two-spin system like Hydrogen

e–

µ+

e–

p+

The muon (proton) and the electron have spins and magnetic moments.

The interaction between them is termed the hyperfine interaction.


6

The H atom has no net dipolar interaction

2

dipolar I S3(1 3cos )E

r− θ

= μ μ

Averaging over the spherical distribution of an electron in an s orbital

(in its ground state)

2 22 0 0

2

0 0

dipolar

cos sin d d 1cos3sin d d

0E

π π

π π

θ θ θ φθ = =

θ θ φ

=

∫ ∫∫ ∫

xy

z

The dipolar interaction determines the anisotropic part of the hyperfine interaction (off-diagonal components of the hyperfine tensor.


7

Isotropic hf constants are due to the “contact” interaction

Fermi (1930)

( ) 2N N

8 03

a g gπ= β β ψ

For a single electron

Only s electrons have density at the nucleus.

For the H atom

( )0 21s 33

00

/1 1e 0r a

aa−ψ = ψ =

ππψ(r)

r/a0

AH = 1420 MHz

AMu = 4463 MHz


8

Matrix Representation of Spin Operators

S :z

Consider a simple spin-½ system, such as an electron, a proton, or a muon.

Take as basis set, the eigenfunctions of 1 12 2

ˆ ˆS Sz zα = α β = − β

The matrix elements are 1 12 2

12

ˆ ˆS Sˆ ˆS 0 S 0

z z

z z

α α = β β = −

α β = α β = β α =12

1 0S

0 1z⎛ ⎞

= ⎜ ⎟−⎝ ⎠⇒

( )2 34

2 34

S 1

S

s sα = + α = α

β = β⇒ 2 3

4

1 0S

0 1⎛ ⎞

= ⎜ ⎟⎝ ⎠

2 238

1 0ˆ ˆ ˆ ˆS S S S0 1z z

⎛ ⎞= =⎜ ⎟

⎝ ⎠2ˆ ˆS ,S 0z⎡ ⎤ =⎣ ⎦

Similarly,

1 12 2

1 12 2

ˆ ˆS Sˆ ˆS S

x x

y yi i

α = β β = α

α = β β = − α1 12 2

0 1 0ˆ ˆS S1 0 0x y

ii

−⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

α and β are noteigenfunctions of andˆ ˆS Sx y

1 14 4

0 1 0 1 0ˆ ˆS S1 0 0 0 1x y

ii

i−⎛ ⎞⎛ ⎞ ⎛ ⎞

= =⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠ ⎝ ⎠

14

1 0ˆ ˆS S0 1y x i−⎛ ⎞

= ⎜ ⎟⎝ ⎠

12

1 0ˆ ˆ ˆS ,S S0 1x y zi i⎛ ⎞⎡ ⎤ = =⎜ ⎟⎣ ⎦ −⎝ ⎠

⇒


9

Matrix Diagonalization of a Spin Operator

12

0 1S

1 0x⎛ ⎞

= ⎜ ⎟⎝ ⎠

122 1

412

0det 0

0− λ⎛ ⎞

= λ − =⎜ ⎟− λ⎝ ⎠

To find the coefficients of the eigenvectors, write out the simultaneous equations for each eigenvalue. Solve for coefficients and normalize

( )( )

1 111 122 2

1 111 122 2

0 0

0 0

c c

c c

+ + =

+ + =12λ = −For 2 2

11 12 1c c+ =11 12c c= − 111 122

c c= = −

( )( )

1 121 222 2

1 121 222 2

0 0

0 0

c c

c c

− + =

+ − =12λ =For 21 22c c= 2 2

21 22 1c c+ = 121 222

c c= =

1

2

1 111 12

ψ − α⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟ψ β⎝ ⎠⎝ ⎠⎝ ⎠

1

2

1 111 12

ψ α⎛ ⎞ ⎛ ⎞⎛ ⎞=⎜ ⎟ ⎜ ⎟⎜ ⎟ψ − β⎝ ⎠⎝ ⎠⎝ ⎠

Set up the secular equation: and solve:

12λ = ±

or

1 1 0 1 1 1 1 01 11 1 1 0 1 1 0 14 2

− −⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠

1 1 0 1 1 1 1 01 11 1 1 0 1 1 0 14 2

⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞=⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟− − −⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠

arbitrary labelling of wave functions Sx is diagonal in the new basis


10

H Atom Spin States − 1also muonium!

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆx x y y z z= + +S I S I S I S Ii

( )( )

12

12

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆx y x

x y y i

i

i+ + −

− + −

⎫⎫= + = +⎪ ⎪⇔⎬ ⎬= − = −⎪ ⎪⎭ ⎭

S S S S S S

S S S S S S

( )12

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆz z+ − − += + +S I S I S I S Ii

( )( )

14

14

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆx x

y y

+ + + − − + − −

+ + + − − + − −

= + + +

= − − − +

S I S Ι S Ι S Ι S Ι

S I S Ι S Ι S Ι S Ι

S I, , and ,m mαα αβ βα β ≡βConsider a basis set of product spin functions

( )1e p 0 02

1 1 1e p 02 2 4

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ

0z z z z+ − − +αα = ω αα − ω αα + ω + αα + ω αα

= ω αα − ω αα + + ω αα

H S I S I S I S I

is an eigenfunctionαα

( )( )

1e p 0 02

1 1 1 1e p 0 02 2 2 4

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ

0z z z z+ − − +αβ = ω αβ − ω αβ + ω + αβ + ω αβ

= ω αβ + ω αβ + ω + βα − ω αβ

H S I S I S I S I

is not an eigenfunction

αβ

e p 0ˆ ˆˆ ˆ ˆ

z z= ω − ω + ωH S I S IiSpin Hamiltonianelectron nuclear

Zeeman interactionshyperfine interaction(isotropic case)

in units of


11

H Atom Spin States − 2

1 1 11 e p 02 2 4

1 1 14 e p 02 2 4

1

4

E

E

= ω − ω + ω = αα

= − ω + ω + ω = ββ

Evaluating all matrix elements…

The central block must be diagonalized to find E2 and E3.

1 1 1e p 02 2 4

1 1 1 1e p 0 02 2 4 2

1 1 1 10 e p 02 2 2 4

1 1 1e p 02 2 4

0 0 00 0ˆ0 00 0 0

ω − ω + ω⎛ ⎞⎜ ⎟ω + ω − ω ω⎜ ⎟=⎜ ⎟ω − ω − ω − ω⎜ ⎟⎜ ⎟− ω + ω + ω⎝ ⎠

H

αααββαββ

( )

( )

1/ 22 21 12 e p 0 02 4

1/ 22 21 13 e p 0 02 4

2

3

E c s

E c s

⎡ ⎤= ω + ω + ω − ω = αβ + βα⎣ ⎦

⎡ ⎤= − ω + ω + ω − ω = βα − αβ⎣ ⎦2 2 1c s+ =

( ) ( ) 21 1 1 1 1e p 0 e p 0 02 4 2 4 4 0E Eω + ω − ω − − ω + ω − ω − − ω =⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦


12

Summary of Muonium Energies and States

( )

( )

1 1 11 e 02 2 4

1/22 21 12 e 0 02 4

1/22 21 13 e 0 02 4

1 1 14 e 02 2 4

1

2

3

4

E

E c s

E c s

E

μ

μ

μ

μ

= ω − ω + ω = αα

⎡ ⎤= ω + ω + ω − ω = αβ + βα⎣ ⎦

⎡ ⎤= − ω + ω + ω − ω = βα − αβ⎣ ⎦= − ω + ω + ω = ββ

2 2 1c s+ =( )( )

( )( )

1/2

e1/222

0 e

1/2

e1/222

0 e

1 12

1 12

c

s

μ

μ

μ

μ

⎧ ⎫ω + ω⎪ ⎪

= +⎨ ⎬⎡ ⎤⎪ ⎪ω + ω + ω⎣ ⎦⎩ ⎭

⎧ ⎫ω + ω⎪ ⎪

= −⎨ ⎬⎡ ⎤⎪ ⎪ω + ω + ω⎣ ⎦⎩ ⎭

At zero field,12

c s= =

12

c s→ ←At low field,

1, 0c s→ →At high field,

The mixing coefficients (c and s) govern the curvature in the Breit Rabi plot.

Beware!

This is not the usual numbering of these two states


13

Energy levels of a two spin-½ system

Breit-Rabi diagram

(αβ − βα)/√2

0.0 1.0 2.0 3.0

Magnetic Field / Hyperfine Constant

RelativeEnergy

(αβ + βα)/√2

αα

ββ

αα

ββ

αβ

βα

e p

long

. fie

ld

trans

ition

s tra

ns. f

ield

tra

nsiti

ons


14

Muonium Energies and States

11 0 –4

1/2 1/22 2 21 1 1 12 0 + 0 0 0 B4 4 4 2

13 0 –4

1/2 1/22 2 21 1 1 14 0 + 0 0 0 B4 4 4 2

1

2 1

3

4 1

E

c s E x

E

c s E x

= αα = ω + ω

⎡ ⎤ ⎡ ⎤= αβ + βα = − ω + ω + ω = − ω + ω +⎣ ⎦⎣ ⎦

= ββ = ω − ω

⎡ ⎤ ⎡ ⎤= βα − αβ = − ω − ω + ω = − ω − ω +⎣ ⎦⎣ ⎦

2 B1/22

B

2 B1/22

B

1 12 1

1 12 1

xcx

xsx

⎛ ⎞= +⎜ ⎟⎜ ⎟⎡ ⎤+⎣ ⎦⎝ ⎠⎛ ⎞= −⎜ ⎟⎜ ⎟⎡ ⎤+⎣ ⎦⎝ ⎠

( )( )

( )1– e2 e

B10+ e2

xμ μ

μ

ω = ω − ω ω + ω=

ωω = ω + ω

conventional numbering

0 1.0 2.0

12

3

4

xB

E


15

Muonium in Zero and Longitudinal Field

If the muon ensemble is initially polarized, but the electrons not, the initial state of the system is .

Half of the muon ensemble is static, in the eigenstate ,the other half oscillates between the mixed states and at frequency

1 12 2αα + βα

αα2 4

0 1.0 2.0

12

3

4

xB

E

1/2224 0 B1 x⎡ ⎤ω = ω +⎣ ⎦

The muon polarization along the field direction is

2B 241

z 2 2B

cos11

x tPx

⎧ ⎫+ ω= +⎨ ⎬+⎩ ⎭

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

non-osc. Pz

Magnetic Field xB


16

Repolarization Curvessometimes described as decoupling curves

Contrary to common language in the field, the muon is never “decoupled” from the electron. As the external applied field increases, it surpasses the internal field from the hyperfine interaction. In energy terms, the Larmor term exceeds the hf term.

muon asymm.

In principle, the hyperfine constant can be found from the field at which half of the polarization has been recovered (or better, by fitting the whole curve).

( )0 e 1/2Bμω = γ + γ

the high-field (maximum) polarization may not be easily determined.

spin relaxation changes the shape of the repolarization curve.

But...

0.0 1.0 2.0 3.0

Magnetic Field

0.5 0.25

B1/2


17

Muonium in Transverse Field

( ) ( ){ }43 2312 142 212 e e e ei t i ti t i tP c sω ωω ω

⊥ = + + +

1/221 112 1 2 – 0 0 B2 2

1/221 143 4 3 – 0 0 B2 2

1/221 114 1 4 – 0 0 B2 2

1/221 123 2 3 – 0 0 B2 2

1

1

1

1

E E x

E E x

E E x

E E x

⎡ ⎤ω = − = ω + ω − ω +⎣ ⎦

⎡ ⎤ω = − = ω − ω − ω +⎣ ⎦

⎡ ⎤ω = − = ω + ω + ω +⎣ ⎦

⎡ ⎤ω = − = ω − ω + ω +⎣ ⎦

–

– 0

– 0

–

→ ω

→ ω − ω

→ ω + ω

→ ω

at low field

⎫⎬⎭

usually (?) too high to detect

Magnetic Field

1

2

3

E

low field12

3

4

E

Magnetic Field

high field


18

Muonium in supercritical water

Percival, Brodovitch, Ghandi et al., Phys. Chem. Chem. Phys. 1 (1999) 4999

400°C, 245 bar

0.0 0.5 1.0 1.5 2.0 2.5 3.0time / μs

-0.2

-0.1

0.0

0.1

0.2

Asym

met

ry

8 G

D signal

“triplet” precession


19

Muonium in supercritical water

-0.15

0.00

0.15

0.00 0.50 1.00 1.50 2.00 2.50 3.00

time / µs

Muo

n A

sym

met

ry400°C, 245 bar, 8 G

AMue-λt

Diamagnetic signal subtracted


20

Muonium in Moderate Transverse Field

1/221 112 1 2 – 0 0 B2 2

1/221 143 4 3 – 0 0 B2 2

1/221 114 1 4 – 0 0 B2 2

1/221 123 2 3 – 0 0 B2 2

1

1

1

1

E E x

E E x

E E x

E E x

⎡ ⎤ω = − = ω + ω − ω +⎣ ⎦

⎡ ⎤ω = − = ω − ω − ω +⎣ ⎦

⎡ ⎤ω = − = ω + ω + ω +⎣ ⎦

⎡ ⎤ω = − = ω − ω + ω +⎣ ⎦

( )1/2210 B2 1 1x⎡ ⎤Ω = ω + −⎢ ⎥⎣ ⎦Defining

12 –

43 – 0

14 – 0

23 –

ω = ω − Ω

ω = ω − Ω − ω

ω = ω + Ω + ω

ω = ω + Ω

a pair of frequencies “split” about ω−

Magnetic Field

1

2

3

E

The splitting can be used to determine ω0( ) ( )2 2

23 12 23 1210 2

23 12

2 μ⎡ ⎤ω + ω + ω − ω − ω⎢ ⎥ω =

ω − ω⎢ ⎥⎣ ⎦


21

Muonium in supercritical water at 196 G

400°C

245 bar

0.00 0.05 0.10 0.15 0.20

time / µs

Muo

n A

sym

met

ry

200 220 240 260 280 300 320 340

Frequency / MHz

Four

ier P

ower


22

C60Mu Radical Precession Frequencies in Low Magnetic Field

Magnetic Field

Ene

rgy 1

2

3

ν12

ν23

0

50

100

150

200

0 50 100 150 200Magnetic Field / G

Rad

ical

Fre

quen

cy /

MH

z

ν12

ν34

-0.0

15.0

30.0

0.0 25.0 50.0Field (G)

C60 Radical (Au=325MHz) Freq. Splitting vs. Field

ν23

− ν12

(MHz)


23

Muon spin precession in D2O crystal at 227 K

38 G

0.0 0.5 1.0 1.5 2.0

time /μs

Muo

n A

sym

met

ry

The high frequency precession is due to “triplet” (F=1) muonium.

The low frequency is due to muons in diamagnetic environments, such as MuOD and MuOD2

+

Why no Mu frequency splitting?


24

Mu diffuses along the c-axis channels of ice-Ih

Mu

side view view along c channel


25


0.0 0.2 0.4 0.6 0.8 1.0

time /μs

Muo

n A

sym

met

ry

11 G

At lower temperature the muonium atom diffuses more slowly. Thedipolar interaction between the muonium electron and the latticenuclei results in spin relaxation.


26


0.0 0.5 1.0 1.5 2.0

time /μs

Muo

n A

sym

met

ry

0.0 0.5 1.0 1.5 2.0

Muo

n A

sym

met

ry

23 G

11 G


27

Mu precession frequencies in low magnetic field

Magnetic Field

Ene

rgy

1

2

3

ν12

ν23

Magnetic Field

1

2

3

ν12

ν23

isotropic hyperfine case axial anisotropy


28

RF Transitions in Muonium at Low Magnetic Field

Magnetic Field

Ene

rgy

1

2

3

νrf

νrf 2νrf


29

RFµSR Spectrum of Mu@C60 in C60 Powder

120 130 140 150 160 170

Magnetic Field / G


30

Endohedral Muonium Mu@C60

Muonium in a universe of its own


31

The Curious Case of C60 (as studied by µSR)

Is it a bird? is it a plane?

0 50 100 150 200 250 300Frequency / MHz

Mu

ν12

ν34

ν23 ν14 →

First report: Ansaldo, Niedermayer et al., 1991.

The signals are characteristic of both muonium and a free radical. No other single phase material had shown such behaviour.


32

Muonium and Diamagnetic Fractions

1.00liquidCCl4

0.70.2liquidCyclohexane

0.50.5solidWater ice

0.60.2liquidWater

0.20.8gasNitrogen

0.40.6gasHydrogen

1.00gasHelium

PDPM

Why are the values so different? Do we care?

We start with high energy muons (MeV) but we measure these values after the muons have thermalized.

approximate values; they depend on temperature, pressure, …


33

History

1975 “final formation of stable, neutral muonium” by about 200 eVBrewer et al.

1978 Radiolysis effectsPercival et al.

1981 Arguments against a spur model for muonium formationWalker et al.

1988 Cyclic charge-exchange and Mu formation in gasesSenba et al.

1988 The reaction of muonium with hydrated electronsLeung, Percival et al.

1988 “…we conclude that the muon has no direct, persistent interaction with its ionization cloud on the time scale of a µSR experiment

Patterson1994 Electric field dependence of muonium formation

Storchak, Brewer et al.2009 Magnetic polaron (muon bound spin polaron) controversy

Storchak, Brewer et al.

⇒ The muon is not in general an innocent probe of material


34

Muon Thermalization circa 1975

50 MeV

2.3 keV

200 eV

20 eV1 eV

MUON BEAM

SLOWING DOWN OF FAST MUONSEnergy loss through scattering with electrons

ELECTRON CAPTURE AND LOSS(many cycles)

EPITHERMAL SCATTERING OF MUONIUMBilliard-ball collisions with atoms and molecules

HOT ATOM REACTIONS

MUONIATED MOLECULES

THERMAL MUONIUM

based on Brewer, Crowe, Gygax and Schenck (1975)

Bethe-Bloch processes

muon E ~ electron orbital velocity

cf hot tritium reactions


35

The Spur Model for Mu Formation (in water) circa 1978based on Percival, Roduner and Fischer (1978)

eaq-

[ e ... µ ... ·OH ]- +

OH-

Mu MuOH

aq+µMu

muoniumdetectable

PMu

missingfraction

PL

diamagneticsignal

PD

][ eaq-Mu ..

terminal spur of radiation track


36

Muons are Peculiar

compared to conventional radiation chemistry

Walker et al published many papers arguing why the spur model must be wrong.

However…

We detect stopped muons.In gas-phase beam studies the mean free path between collisions is long.

The stopped muon is at the end of the radiation track. The extensive literature of radiation chemistry deals almost exclusively with radiolysis of the medium not the fate of the ionizing particle itself.

We detect muon spin polarization, not chemical fractions of muon species.We need to understand how muon spin polarization can be lost.


37

What Determines the Muonium Fraction? in a gas

Mu

+µ

+µ

+µ

Mu

Mu

Mu

+µ

+µ

Mu

Mu

Mu

+µ

+µ

Mu

+µ

+µ

Mu

+µ

+µ

Mu

Mu

Mu

+µ

Charge

Exchange

Region

Final

State

Emin

based on Senba (1988)

Energy time


38

Where does the Diamagnetic Signal come from?

Muonium has an ionization potential of 13.5 eV, greater than most molecules⇒ electron capture should occur down to thermal energy...? ??

Experimental evidence contradicts this.

The "hot fraction h has been a source of annoyance in the history of muonium chemistry, ...J.H. Brewer et al (1975)

There are two routes to stable diamagnetic compounds:

Muon attachment (molecular ion formation, followed by transfer)

Hot atom reactionsMuH + R*Mu + RH MuR + H

⎧→ ⎨

⎩

+

+2

+ RH RHMu

RHMu + RH RMu RH

+

+

μ →

→ +


39

Complications, complications, complications

The mean free path has the same order of magnitude as molecular dimensions.

Does the charge state of the muon have any meaning if the muon is travelling through the overlapping electron clouds of molecules?

The muon velocity is comparable to nuclear motion in molecules– a 3 eV muon takes 4 fs to traverse a molecular diameter of 3 Å, half the vibrational period of a typical C-H or O-H stretch.

In condensed matter the model of successive two-body collisions no longer holds. Consider the epithermal muon with a few eV kinetic energy:


40

Mu Formation in Water with Spin-dependent Chemistry based on Leung, Brodovitch, Percival, et al. (1987)

e -

Mu

Mu

muoniumdetectable

PMu

missingfraction

PL

diamagneticsignal

PD

+µH2O

Mu H 2O +

][ eaq-Mu ..3 ][ eaq

-Mu ..1

(Mu) MuH, MuOH

10 -12 s

10 s-9

10 s-6

time


41

History

1975 “final formation of stable, neutral muonium” by about 200 eVBrewer et al.

1978 Radiolysis effectsPercival et al.

1981 Arguments against a spur model for muonium formationWalker et al.

1988 Cyclic charge-exchange and Mu formation in gasesSenba et al.

1988 The reaction of muonium with hydrated electronsLeung, Percival et al.

1988 “…we conclude that the muon has no direct, persistent interaction with its ionization cloud on the time scale of a µSR experiment

Patterson1994 Electric field dependence of muonium formation

Storchak, Brewer et al.2009 Magnetic polaron (muon bound spin polaron) controversy

Storchak, Brewer et al.

The muon is not in general an innocent probe of material


42


43

Perturbation Theorytime-independent theory for non-degenerate states

Used if the Schrödinger equation cannot be solved exactly, but the problem is similar to another whose solution is known.

0 0 00 0If whˆ ˆ ˆ ˆ ere n n nE= + ψ = ψH H h H

0 00 00 0 0

0 0

ˆ ˆh hh

j jk kj j j j

k j j k

E EE E≠

ψ ψψ ψ= + +ψ ψ

−∑zero order 1st-order

correction2nd-order correction

000 0

0 0wherh

e jkj j k k k

k j j k

c cE E≠

ψψψ = ψ + ψ =

−∑

Approximate energies

Approximate wave functions

The perturbed wave function has other eigenfunctions admixed.

The mixing coefficients ck depend on the strength of the perturbation relative to the energy separation 0 0

j kE E−

Does not work for 0 0 !j kE E=

If necessary, use linear combinations to avoid this problem.


44

Perturbation Theory − Example of H spin states

( )0 1 1e p 0 02ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ

z z z z + − − += ω − ω + ω = ω +H S I S I H S I S I

00 0 1 1 11 11 e p 02 2 4

00 0 1 1 12 22 e p 02 2 4

00 0 1 1 13 33 e p 02 2 4

00 0 1 1 14 44 e p 02 2 4

ˆ 1

ˆ 2

ˆ 3

ˆ 4

E

E

E

E

= = ω − ω + ω = αα

= = ω + ω − ω = αβ

= = − ω − ω − ω = βα

= = − ω + ω + ω = ββ

H

H

H

H

0th order:111

122

133

144

ˆ 0

ˆ 0

ˆ 0

ˆ 0

=

=

=

=

H

H

H

H

1st order:

1 1 1 1 1 1 112 13 14 24 34 23 021 1 1 1 1 1 121 31 41 42 43 32 02

ˆ ˆ ˆ ˆ ˆ ˆ0, 0, 0, 0, 0,ˆ ˆ ˆ ˆ ˆ ˆ0, 0, 0, 0, 0,

= = = = = = ω

= = = = = = ω

H H H H H H

H H H H H H

2nd order:

0 0 12 3 e p 02E E− = ω + ω − ω 0 0 1

3 2 e p 02E E− = −ω − ω + ω

Exact:21041 1 1

2 e p 02 2 4e p

E ω→ ω + ω − ω +

ω + ωfor ( )22

0 e pω ω + ω high field

21041 1 1

2 e p 02 2 4 1e p 02

E ω= ω + ω − ω +

ω + ω − ωPerturbation: to 2nd order


45

Spin Symmetry and Selection Rules

The transition moment is zero (“forbidden” transition) if the electronic (or nuclear) spin states (α, β) are orthogonal:

The simple rule breaks down when eigenstates have mixed spin functions.

The intensity of a stimulated electric dipole transition is proportional to the square of the transition dipole moment:

2 2 22 x y zm n m n m n m nI ∝ μ = μ + μ + μ where etc. ˆ x x

mn m nμ = μ

In magnetic resonance the relevant operator is the magnetic dipole.

For TF-µSR this involves muon-spin-flipping operators.

e p e p e e p p

e e p p

ˆ ˆP P

P if and

μ μ μ μ μ μ

μ μ μ

′ ′ ′ ′′ ′′ ′′ ′ ′′ ′ ′′ ′ ′′ψ ψ ψ ψ ψ ψ = ψ ψ ψ ψ ψ ψ

′ ′′ ′ ′′ ′ ′′= ψ ψ ψ = ψ ψ = ψ

1 10 0

i.e. andand

S S I I

S I

m m m mm m

′ ′′ ′ ′′= =

Δ = Δ =

0α β =

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