Magnetism I: Muons and NMRscuolaaimagn2016.fisica.unimi.it/lessons/De Renzi.pdf · A magnetic...

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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

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

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

io freq

cy resonan

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

A magnetic moment (spin) precesses around B

NMR basics

Internal field

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

NMR basics

Internal field

However M // B at equilibrium

NMR basics

A pulse of a small rf field at the Larmor frequency

makes M nutate until M ┴ B

NMR basics: the rf pulse

Nuclear magnetic moment

Larmor precessionLarmor frequency

Rotating frame at , apparent torque apparent field

effective field

NMR basics: the pulse in the rotating frame

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

sin(θ)= 1

√1+(B−ω/γ)2

The probes

Receptivity

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

γ2ℏ2

A∝dMdt

=ωM=aγ3ℏ2

3kBTB2 Brf

µSR: the muon

Charge µ- , µ+ (antiparticle)

Mass mµ = 0.1126 m

p = 206.8 m

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

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

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

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

→ not more than 50000 ev/s

~20 ns

typical resolution 80 - 1000 ps

Pulsed beams

ISIS: Fascio primario (~0.3 mA):

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

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

INS NSE

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

Precession

Spin interactions

Magnetic interactions, between classical dipoles

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

leading to Fermi contact interaction

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

Distant dipole field, dominant

Fermi contact interaction

isotropic, Bc || S, transferred

Spin interactions: nuclear casemultielectron

Distant dipole field

Fermi contact interaction

isotropic, Bc || S, core polarized

plus anisotropic pseudodipolar, e.g.

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

powders

single crystal

Continuous magnetic phase transitions

Order-disorder transition: critical behaviour at Tc

• Static aspects

MnF2 rutile

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

Heisenberg 3D?

Prototype localized antiferromagnet

MnF2 by 19F NMR:

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

19 19 19ν γ B M Ts

0 333 3β . ( )

Ising 3D: muons in CoF2

TsMμBμγμν

Close to TN:

critical power-law behaviour

Heisenberg

0 33 2 , cfr.

β . ( )

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

Low T:

(spin wave excitations)

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

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

χ∝|t|−γ

ξ ∝(.−t )−ν

G ∝|t|−d +2−η

C H∝|t|−α

|M|∝(−t )β

For vanish or diverge as power laws

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

Magnetic phase transitions

Order-disorder transition: critical behaviour at Tc

• Dynamic aspects (critical fluctuations)

Interactions

Dipolar (distant dipoles)

contact (transferred)

Interactions

Dipolar (distant dipoles)

contact (transferred)

Relaxations below and above Tc

T>Tc T<T

Different average, same fluctuations

NMR, muon relaxation vs neutrons

Magnetic neutron scattering cross-section

Relaxation rate of local probe coupled to

at the Larmor frequency!

In both the time Fourier transform of spin-spin correlations

NMR, muon relaxation

Simplest possible correlation: a decay

NMR, muon relaxation vs neutrons

For fast electronic dynamics ( )

ħ (eV)0

A peak when

Dynamic critical behaviour

LaMnO3 T→T

N MnF2

(below TN) (above T

Spin fluctuations slow down approaching the transition

2nν( z d η )

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

(CaySr

FM double exchange

x0 0.5

optimal doping x = 3/8

Interplay of Double Exchange and bandwidth

Wide band

Narrow band

(CaySr

magnetic order parameter

optimal wide band optimal narrow band

0 0.5 13/8 La

3/8MnO

Second order transition

First order truncated

Truncation by polaron transition

55Mn NMR

(CaySr

double exchange

Meskine Phys Rev B 64 094433

0 0.5 1

optimal doping 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)

5% Zn T = 50 K = 0.2 TN

Zn, Mg

La2Cu1-xMxO4

Zn, Mg

La2Cu1-xMxO4

x=0 0.2 xc = 0.4

percolation

Zn, Mg

00 0.1 0.2

La2Cu1-xMxO4

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

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

Also the order parameter M (staggered) is reduced

Frustration

Ising AF on a triangular lattice…

Frustration

Ising AF on a triangular lattice…

… a pyrochlore lattice

also Ising Ferromagnetis frustrated :

Ho2Ti2O7 SPIN ICE

Frustration

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

… a pyrochlore lattice

A frustrated Ising chain systemCa

Ferromagnetic (F) Ising chainswith AF interchain coupling

Two Co sites

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

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

Ising chains(Ferromagnetic intra-chain coupling)

Ising AntiFerromagnetic inter-chain couplings

geometricfrustration

Aim: determine J1 J

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

Ising chains(F intra-chain coupling J

Ising AF inter-chain couplings, J

2 and J

no spin waves, Glauber (activated) Isingdynamics

Frustrated ground state(s)with M = 0

determined by J1 J

3 from

Ground state with increasing µ0H

FI regime

sharp quadrupolar septets

in and

2.8 TB

59B = 1.6 T59γ = 10 MHz/T

νL = 16 MHz

FI majority and minority chainsFM

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

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

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)

YBa2Cu

6+x the clean case

F. Coneri et al. PRB 81 104507

measures h = hp

similarly Tc(h) measures h

Remember!

Internal doping and disorderY

Eu3+ for Y3+ isoelectronic

0 0.5 1 O content, y

chain disorder vs chain order

doping linear in chain length

O content, y

0 0.5 1

Eu 1-z Y

(Å3 )

a series of samples at fixed O

Chain length ℓ

NQR spectroscopy of Cu(1)

chain interior

chain end

tetra ortho

→ doping linear in z

phase diagram

CuO2 hole density h

from increasing chain length

known h

p = 0.07

known h

p = 0.028

similar to O doping, but ...T

0 0.2 0.4 0.6 0.8 1.0

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

phase diagram

TN for the clean

system with

densities

predicts

density hp

similar to O doping, but ...

Correlates with disorder

Eu YY-Eu

Samuele Sanna

TN reduction with structural disorder

Type II superconductors

Type II:

Flux Lattice, incommensurate to the crystal lattice

muons are interstitials in the crystal lattice

→ dense random sampling

GL equations predict

B maxima

B minima

saddle point

Textbook case: V

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⟩=(

a⋅b )2

∑k≠0

K 02(k ξ)

(1+k 2λ2⏟k λ>1

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

open ring Cr8Cd exchange

anisotropy

dipolar

Zeeman

J = 1.32 meV

T. Guidi et al. Nature Comm. 2015

2 molecules per cell

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…

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...

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