Magnetism I: Muons and NMRscuolaaimagn2016.fisica.unimi.it/lessons/De Renzi.pdf · A magnetic moment (spin) precesses around B B. 22 April 2016 RDR- - NMR and MuSR 6 NMR basics Internal

122 April 2016 RDR- - NMR and MuSR

Magnetism I: Muons and NMR

Outline - I• Un peu d'historie (Bloch, Purcell, Garwin, Ledermann)• NMR basics

o Precession and nutationo Effective fieldo Receptivity

• The muono Productiono Parity violationo Asymmetry

• Time scales

• Relaxationso Bloch equations

- Interactionso Magnetic dipoles (the muon, the nuclei, the electrons)o Fermi contact and pseudodipolar o Electric nuclear quadrupole

Roberto De Renzi– Parma (Italy)

e Sci en ze d e l l a T e rra

e Sci en ze d e l l a T e rra


Magnetism I: Muons and NMR

Outline - II• Magnetic phase transition (static, order parameter and dynamic, fluctuations)

o MnF2,CoF2 (19F NMR, muons)

o La1-xSrxMnO3 vs. La1-xCaxMnO3 (55Mn NMR and muons)

• Dilution of magnetic momentso La2Cu1-xMxO4 with M = Mg, Zn (139La NQR, muons)

• Frustration o Ca3Co2O6 (59Co NMR)

o Exchange in an Ising magnet

• Disordero Y1-zEuzBa2Cu3O6+x (muons)

o (Y:Ca)Ba2Cu3O6+x (muons)

• Superconductorso V (muons)o YBa2Cu3O6.9

- Molecular ringso Cr8Cd


Un peu d'histoire: NMR

3

I.I. Rabi et al. Phys. Rev. 53 (1938) 318Nobel Prize for the year 1944

F. Bloch et al. Phys. Rev. 69, 127 (1946)E. M. Purcell et al., Phys. Rev. 69, 37 (1946)Nobel Prize for the year 1952 with E.M. Purcell

The

Ste

rn-G

erla

ch a

ppro

ach

Rad

io freq

uen

cy resonan

ce


Un peu d'histoire: µSR

L. Garwin et al. Phys. Rev. 105, 1415 (1957)

Nobel Prize to Lee, Yang for parity violation (1957)


NMR cartoon

5

A magnetic moment (spin) precesses around B

B


NMR basics

6

Internal field

Precessing nuclear magnetization Mproduces an e.m.f. in a coil (Free Induction Decay, echoes, etc.)


NMR basics

7

Internal field

However M // B at equilibrium


NMR basics

8

A pulse of a small rf field at the Larmor frequency

makes M nutate until M ┴ B


NMR basics: the rf pulse

9

Nuclear magnetic moment

Larmor precessionLarmor frequency

Rotating frame at , apparent torque apparent field

effective field


NMR basics: the pulse in the rotating frame

B

Ba

Brf

Mn

Nutation

switch of the rf when M is in the xy plane


NMR basics:resonance and the effective field

In the frame rotating at the rf ω effective field: Beff = B – ω/γ + Brf

Beff ┴ B solo per B – ω/γ ≈ 0

Ba

Brf

Beff

sin(θ)= 1

√1+(B−ω/γ)2

Brf2


The probes

Receptivity

Group

=B

1γ=2π⋅42.8MHz /T


Receptivity

Amplitude A proportional to:

isotope abundance (natural or enriched)

nuclear magnetization M in a magnetic field B (Curie law)

Faraday induction M

BM=a

γ2ℏ2

3kBTB

A∝a

A∝dMdt

=ωM=aγ3ℏ2

3kBTB2 Brf


µSR: the muon

Charge µ- , µ+ (antiparticle)

Mass mµ = 0.1126 m

p = 206.8 m

e

Spin Iµ = ½

Gyromagnetic ratio γµ = 2π 135.5 MHz/T

Mean lifetime τµ = 2.197 µs

Production: π+ → µ+ + νµ (anti-muon, muon-neutrino)

Decay: µ+ → e+ + νe + ν

µ


µSR: basic aspectsproton accelerators

Pion production:

Impinge protons e.g. at E~800 MeV on solid target

π+ → µ+ + νµ

in 26 ns

http://www.fis.unipr.it/~roberto.derenzi/dispense/pmwiki.php?n=MuSR.MuSR

http://www.fis.unipr.it/~derenzi/dispense/pmwiki.php?n=MuSR.MuSR


µSR: basic aspectswhere?

ISIS Chilton UK

TRIUMFVancouverCanada

JPARK TsukubaJapan

PSI Villigen CH


µSR: basic aspectsparity violaton

The mirror image does not exixt in nature

E.g. only negative helicity (chirality) neutrinos exist.

At work with angular momenum conservation...


Spin polarized muon production

@ ISIS:

2. transmission target(1 cm thick C)pions (π+) production and decay into polarized µ+

3. magnetic transport optics

1. primary proton beam


µ decay

4. μ+ stops in the sample. μ decay violates parity as well

e+ preferentially emitted along Sμ

probability lobe


Longitudinal asymmetry

θ = 0θ = π

Decadimento nel tempo

Asymmetry

5. More counts for θ = 0 , less for quello θ = π


%

Spin and emission probability precess at the Larmor frequency

Statistics on million events, eachrecorded vs time from its

own implantation

c


Continuous vs. pulsed beams

(nearly) continuous: PSI Villigen TRIUMF Vancouver

Cockroft-Walton Proton syncro-cyclotrone590 MeV β=0.8 2mA cfr. LHC 20 pA, ESRF 200mA (e-)

~ 15 m

Injector IIring

fascio primariop+


Continuous beams

μ+ extracted randomly from

Each μ lifetime measured individually

- Start when μ+ stops in the sample

- Stop when e+ is emitted

One at a time, τμ = 2.2 μs, up to 5τ

μ

→ not more than 50000 ev/s

~20 ns

typical resolution 80 - 1000 ps


Pulsed beams

ISIS: Fascio primario (~0.3 mA):

70 ns

300 ns 20 ms

~ 103 μ+/pacchetto

100 ns

400 ns 40 ms

JPARK-MUSE (~0.3 mA)

~ 5 103 μ+/pacchetto


Comparison

Polarization 100% Probe implanted nucleus (interstitial) in the compoundMaximumtime 50 μs 1-100 swindow

Mininum time 2-3 ns 2-3 μswindow

Detection broad band narrow band

M=γ

2ℏ

2

3k BTB

NMRμSR


Time windows

10-17 10-15 10-13 10-11 10-9 10-7 10-5 10-3 10-1 101 103

μSR

INS NSE

NMR

Macroscopic techniques

time (s)

electronic excitations molecular motions, diffusion


Relaxations: Bloch equations

Nuclear magnetization

When the equation of motion is

With a relaxation mechanism

Decay of transverse coherence

Decay towards thermodynamicequilibrium M

z0

Precession


Spin interactions

Magnetic interactions, between classical dipoles

S

S

Mn3+

Mu

O

Iμr

With electrons and positive nuclei, muons, finite probability that r = 0,

leading to Fermi contact interaction

eg

t2g

eg

t2g

Mn4+


Spin interactions: muon case, simple 1s hydrogen-like

Distant dipole field, dominant

S

S

Mn3+

Mn4+

Mu

O

Iμr

Fermi contact interaction

isotropic, Bc || S, transferred


Spin interactions: nuclear casemultielectron

Distant dipole field

S

S

Mn3+

Mn4+

O

139I

La

Fermi contact interaction

isotropic, Bc || S, core polarized

plus anisotropic pseudodipolar, e.g.

55I

55I


Electric quadrupole interactions

Nuclei with possess an electric quadrupole moment Q

, electric field gradient

I spin along majorprincipal axis


2I = 5 transitions

NMR + NQR, e.g. I = 5/2

B

z

+½

-½

E+3/2

-3/2

+5/2

-5/2

powders

Ix

single crystal


Continuous magnetic phase transitions

33

Order-disorder transition: critical behaviour at Tc

Tc

• Static aspects


MnF2 rutile

F-

eg

t2g

Mn2+

(spin-only, cubic Cristal Field)S=5/2 no single-ion anisotropy:

Heisenberg 3D?

Prototype localized antiferromagnet

34


MnF2 by 19F NMR:

P. Heller Phys. Rev. 146 (1966) 403

19 19 19ν γ B M Ts

35

0 333 3β . ( )


Ising 3D: muons in CoF2

TsMμBμγμν

36

Close to TN:

critical power-law behaviour

Ising

Heisenberg

1

0 33 2 , cfr.

0 327

0 367

β

μN

th

TB T

T

β . ( )

.β

.

R. De Renzi et al. Phys Rev. B 31 (1984) 186

Low T:

(spin wave excitations)

0

10

B TΔS μS Bμ


Ising 3D: muons in LaMnO3

Another antiferromagnet

Close to TN:

critical power-law behaviour

M. Cestelli et al. Phys Rev. B 64 (2001) 064414

37


Critical exponents

Critical exponents (statics)

Models

Ising 3D 1.2372(5) 0.6301(4) 0.0364(5) 0.110(1) 0.3265(3)

Ising 2D 7/4 1 1/4 1/8

Heisenberg3D

1.392(5) 0.71(1) 0.037(2) 0.367(3)

Heisenberg2D

Orders only at T=0 (see later)

XY 3D 1.317(1) 0.671(1) 0.037(1) -0.015(1)

XY 2D Kosterlitz-Thouless (not power laws)

Mean-field 1 1/2 0 0 1/2

γ

χ∝|t|−γ

ν

ξ ∝(.−t )−ν

η

G ∝|t|−d +2−η

α

C H∝|t|−α

β

|M|∝(−t )β

38

0c

c

T Tt

T

For vanish or diverge as power laws

A. Pelissetto, E. Vicari, Physics Reports 368 (2002 ) 549-727


Magnetic phase transitions

39

Order-disorder transition: critical behaviour at Tc

Tc

• Dynamic aspects (critical fluctuations)


F

µr

1µ

Mn

Interactions

Dipolar (distant dipoles)

contact (transferred)

40


F

µr

1µ

Mn

Interactions

Dipolar (distant dipoles)

contact (transferred)

41


Relaxations below and above Tc

T>Tc T<T

c

42

Different average, same fluctuations


NMR, muon relaxation vs neutrons

Magnetic neutron scattering cross-section

43

Relaxation rate of local probe coupled to

at the Larmor frequency!

In both the time Fourier transform of spin-spin correlations


NMR, muon relaxation


44

Simplest possible correlation: a decay


NMR, muon relaxation vs neutrons


45

For fast electronic dynamics ( )

ħ (eV)0

T (K)

A peak when


Dynamic critical behaviour

LaMnO3 T→T

N MnF2

(below TN) (above T

N)

Spin fluctuations slow down approaching the transition

Spin fluctuations slow down approaching the transition

46

2nν( z d η )

TN

0 67(7)n .

3D nIsing 0.717

Heisenberg 0.329

M. Cestelli et al. Phys Rev. B 64 (2001) 064414 Brown et al. Phil. Mag. Lett. 73 (1996) 195


La1-x

(CaySr

1-y)

xMnO

3:

FM double exchange

La5/8

Sr3/8

MnO3

La5/8

Ca3/8

MnO3

y

0

0.5

1

x0 0.5

optimal doping x = 3/8

Interplay of Double Exchange and bandwidth

Wide band

Narrow band


La1-x

(CaySr

1-y)

xMnO

3:

magnetic order parameter

La5/8

Sr3/8

MnO3

optimal wide band optimal narrow band

x

0 0.5 13/8 La

5/8Ca

3/8MnO

3y

Second order transition

First order truncated

First order truncated

Truncation by polaron transition

55Mn NMR


La1-x

(CaySr

1-y)

xMnO

3:

double exchange

La5/8

Sr3/8

MnO3

La5/8

Ca3/8

MnO3

a

c

Meskine Phys Rev B 64 094433

y

x

0 0.5 1

optimal doping x = 3/8

x=3/8

Sr-rich Ca-rich


1st vs. 2nd order, why?

Tp polaron localization

Tc Curie temperature

G. Allodi et al. J. Phys. Cond. Mat., 26 266004


Dilution of the magnetic ions

Zn, Mg

La2Cu1-xMxO4

Carretta et al. Phys. Rev. B 83 (2011) 180411(R)

51

5% Zn T = 50 K = 0.2 TN



Zn, Mg

La2Cu1-xMxO4


52



Zn, Mg

La2Cu1-xMxO4


53

x=0 0.2 xc = 0.4

percolation



Zn, Mg

00 0.1 0.2

La2Cu1-xMxO4

x


54

0.3 0.4

Chernyshev Phys. Rev. B 65, 104407 dilution on a Heisenberg AF square lattice

Zn further reduction due to frustration



Zn, Mg

La2Cu1-xMxO4

55

Phys. Rev. B 83 (2011) 180411(R)

Also the order parameter M (staggered) is reduced


Frustration

56

Ising AF on a triangular lattice…

z

?


Frustration

57

Ising AF on a triangular lattice…

… a pyrochlore lattice

z

?

also Ising Ferromagnetis frustrated :

Ho2Ti2O7 SPIN ICE


Frustration

58

Geometric: Competing interactions AF-F Ising AF on a triangular lattice… generate frustration

… a pyrochlore lattice

z

?


A frustrated Ising chain systemCa

3Co

2O

6

Ferromagnetic (F) Ising chainswith AF interchain coupling

Two Co sites

Co3+

Co3+

G. Allodi et al. Phys Rev. B 89 104401G. Allodi et al. Phys Rev. B 83 104408 with Stefano Agrestini and Martin R. Lees


Determine J1 J

2 J

3

s=0s=2s=0s=2s=0

Ising chains(Ferromagnetic intra-chain coupling)

Ising AntiFerromagnetic inter-chain couplings

?

J2,3

< 0

geometricfrustration

J1

J2

J2


Aim: determine J1 J

2 J

3

s=0s=2s=0s=2s=0

Ising chains(F intra-chain coupling J

1)

Ising AF inter-chain couplings, J

2 and J

3

no spin waves, Glauber (activated) Isingdynamics

J1

J2

J2


Frustrated ground state(s)with M = 0

µ0H

determined by J1 J

2 J

3 from


Ground state with increasing µ0H

µ0H

FI

BFIM


Ground state with increasing µ0H

µ0H

FI

BFIM

F

BFM


FI regime

FI F

BFIM

BFM

Co3+

µ0H

sharp quadrupolar septets

in and

µ0H

2.8 TB

d

1.2T

59B = 1.6 T59γ = 10 MHz/T

νL = 16 MHz


FI majority and minority chainsFM

FI F

BFIM

BFM

Co3+

FIphase

majority

minority G. Allodi et al. Phys. Rev. B 83 104408


Calculate the cost of spin reversal from

G. Allodi et al. Phys Rev. B 83 104408


Calculate the cost of spin reversal from

Ferrimagneticmajority FI↑

Ferrimagneticminority FI↓

Ferromagnetic

G. Allodi et al. Phys Rev. B 89 104401


The clean cuprate:YBa

2Cu

3O

6+x

69

chains

also Y-Eu, Y-Nd isoelectronic Eu3+-Nd3+ for Y3+ (up to 8% disorder)

=0.6 µB

F. Coneri et al. PRB 81 104507


Y1-z

CazBa

2Cu

3O

6+x

70

Same Y site: disorder with charge (up to 8%)

dramatic widening of spin glass region with disorder

S. Sanna et al. PRB 82 100503(R)

Ca


YBa2Cu

3O

6+x the clean case

71

F. Coneri et al. PRB 81 104507

measures h = hp

similarly Tc(h) measures h

= hp

Remember!


Internal doping and disorderY

1-zEu

zBa

2Cu

3O

6.35

72

Eu3+ for Y3+ isoelectronic

why?

TcT

N

0.3

0 0.5 1 O content, y

Y

Eu

b

a

chain disorder vs chain order

doping linear in chain length

Eu Y

Y Eu

O content, y

0 0.5 1

Eu 1-z Y

182

180

178

176

cell

volu

me

(Å3 )

a series of samples at fixed O

0.35

YEu

0.5


Chain length ℓ

NQR spectroscopy of Cu(1)

chain interior

chain end

Y

Eu

tetra ortho

→ doping linear in z


Y1-z

EuzBa

2Cu

3O

6.35

phase diagram

CuO2 hole density h

p

from increasing chain length

known h

p = 0.07

known h

p = 0.028

similar to O doping, but ...T

N

Tc

0 0.2 0.4 0.6 0.8 1.0

Y content, 1-z0.0 0.5 1.0 Y content, 1-z


Y1-z

EuzBa

2Cu

3O

6.35

phase diagram

TN for the clean

system with

these

densities

hp

hp

predicts

these

density hp

similar to O doping, but ...

Correlates with disorder

Eu YY-Eu


Y1-z

EuzBa

2Cu

3O

6.35

Samuele Sanna

TN reduction with structural disorder

Y-Eu


Type II superconductors

-M

H

Type II:

Hc

Hc1

Hc2

Flux Lattice, incommensurate to the crystal lattice

muons are interstitials in the crystal lattice

→ dense random sampling


GL equations predict

b

a

B maxima

B minima

saddle point


Textbook case: V

p(B

) arb

. un

its

0.16 T

0.20 T

0.24 T

0.29 T


Quick and dirty fits

Comparison with polycrystal data Second moment

B(r)exp(-r2/2σ2)

Is it worth while?

⟨ΔB 2⟩=(

Φ0

a⋅b )2

∑k≠0

K 02(k ξ)

(1+k 2λ2⏟k λ>1

)2∝

1λ4


σ(T): gap fits

La1.83

Sr0.17

CuO4 crystal

s-wave ns

YBCO6.95

crystal

Field cool, then shift field down(pinned flux lattice)

d-wave: lines of nodes

PRB42 8019 (1990)

YBCO6.95

powders (!)

0.05 T


Single molecule rings

Closed and open rings

spin singlets J ~ 15 K in zero field

Chemist can taylor couplings and play with topology

Cr8 Cr8Cd

Cr



open ring Cr8Cd exchange

anisotropy

dipolar

Zeeman

J = 1.32 meV

T. Guidi et al. Nature Comm. 2015

2 molecules per cell

B

θ


53Cr NMR at T = 1.4 K

nat. ab.

Iγ/2π

(MHz/T)53Cr 0.095 3/2 2.606

4 frozen inequivalent pairs

negligible quadrupole


53Cr resonances at 1.4 K

Field scan

Frequency scan


53Cr resonances at 1.4 K


Summarizing…

88

Local probes (nuclei, muons) can measure

• magnetic order parameters up to the transition• relaxation rates, i.e. excitations

With this tools we have seen:

• static critical exponents dynamic critical exponents• conventional magnetism: 3D Heisenberg,Ising,lower dimensions (2D,1D)• phase diagrams• magnetic dilution and frustration• exchange couplings and hyperfine couplings• superconducting penetration depth and gaps

Magnetism I: Muons and NMRscuolaaimagn2016.fisica.unimi.it/lessons/De Renzi.pdf · A magnetic moment (spin) precesses around B B. 22 April 2016 RDR- - NMR and MuSR 6 NMR basics Internal

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