Introduction to muon spin rotation and relaxation (SR)

Introduction to muon spin rotation and relaxation(µSR)

P. Dalmas de Reotier

CEA and University Joseph FourierGrenoble, France

November 2010

Slides available at

http://inac.cea.fr/Pisp/pierre.dalmas-de-reotier/introduction muSR.pdf

Outline

IntroductionWhat to do with muons ?The muon as an elementary particle

Principle of a µSR experimentMuon productionMuon implantation and decayThe Larmor equationTF and LF µSR experiments

The polarisation functionsField distribution approachQuantum approachThe Bloch equationsThe magnetic field at the muon site

Examples

Bibliography

What to do with muons ?

Probe the magnetic field inside matter at a microscopic level inorder to have an insight into its magnetic and electronic properties.

The muon as an elementary particle

Discovered in 1936 by Anderson and Neddermeyer who studiedcosmic radiations.

Mass 0.112 Mp

' 207 Me

Charge ±e

Spin Sµ 1/2

Gyromagnetic γµ 851.6ratio Mrad.s−1.T−1

Lifetime τµ 2.197 µs

Here we consider only the positive muon µ+.

Principle of a µSR experiment in shortMuon production, implantation and decay

A schematic view of a µSR facility

ISIS facility (UK).

View of an experimental hall

ISIS facility (UK).

Muon production and polarised beamsPions as intermediate particles

Protons of 600 to 800 MeV kinetic energy interact with protons orneutrons of the nuclei of a light element target (typically graphite)to produce pions (π+).

p + p −→ π+ + p + n (1)

p + n −→ π+ + n + n (2)

Pions are unstable (lifetime 26 ns).They decay into muons (and neutrinos):

π+ −→ µ+ + νµ (3)

Production of a beam of polarised muons

I Consider pions at rest,

I since the pion is spinless and the neutrino helicity is −1(i.e. its spin and linear momentum are antiparallel),

I conservation of linear and angular momenta dictates:

The muon beam is 100% polarisedwith Sµ antiparallel to Pµ.

Such a beam is called a surface muon beam.Muon kinetic energy: Eµ = 4.12 MeV.

Muon implantation, thermalisation and localisation

For decreasing kinetic energy of the muon:

I Inelastic scattering of the muon involving Coulomb interactionthrough atomic excitations and ionisations.

I µ+ picks up an electron to form a muonium atom Mu (i.e. aµ+-e− bound state) and releases it many times.

I Last stage of thermalisation through collisions between Muand atoms in the sample

I Eventually, dissociation of Mu into µ+ and a free e−, in mostcases. Exceptions are semiconductors, molecular materials. . .

This process takes place in 10−10 - 10−9 s.No loss of muon polarisation during thermalisation.Limited radiation damages: relatively few implanted µ+ (≈ 108).Muons finally localise in interstitial crystallographic sites.Implantation range for 4.12 MeV muons: 0.1 to 1 mm, dependingon material density.

The muon decay and its anisotropy

µ+ −→ e+ + νe + νµ (4)

Only the positron e+ is detected.

The direction of its emission is anisotropic with respect to Sµ:

Direction of positron emission.

Probability of emission:

W (θ) = 1 + a0 cos(θ) (5)

This curve is a cardiod.On average, a0 = 1

3

The evolution of the muon spinThe Larmor equation

Basic principle of mechanics:Time derivative of angular momentum is equal to the sum of thetorques:

d~Sµ(t)

dt= mµ(t)× Bloc(t). (6)

Sincemµ = γµ~Sµ, (7)

by definition of the gyromagnetic ratio, we have

dSµ(t)

dt= γµ Sµ(t)× Bloc(t). (8)

Consequences and solution of the Larmor equation

FromdSµ(t)

dt = γµ Sµ(t)× Bloc(t) we deduce:

I dSµ(t)dt · Sµ(t) = 0:

Sµ(t) is a constant of the motion, i.e. Sµ(t) = Sµ(0)

I dSµ(t)dt · Bloc(t) = 0:

this impliesdSµ(t)

dt is perpendicular to Bloc(t).

Assuming Bloc(t) = Bloc,

Sµ(t) = S‖µ(0)u + S⊥µ (0)[cos(ωµt) v − sin(ωµt)w],(9)

with ωµ = γµBloc. In particular, if S‖µ(0) = 0,

Sµ(t) = Sµ(0)[cos(ωµt) v − sin(ωµt)w], (10)

The precession frequency only depends on Bloc, noton the angle between Sµ and Bloc !

Principle of a transverse field µSR experiment

N(t) = N0 exp(−t/τµ)[1 + a0PexpX (t)] (11)

Principle of a zero or longitudinal field µSR experiment

NF,B(t) = N0 exp(−t/τµ)[1± a0PexpZ (t)] (12)

The ingredients for µSR spectroscopySummary

I Availability of polarised muon beamsI Anisotropy (or asymmetry) of muon decay (violation of parity

conservation)I The Larmor equation

Phys. Rev. 105, 1415 (1957).

The acronym µSR

“µSR stands for Muon Spin Relaxation, Rotation, Resonance,Research or what have you. The intention of the mnemonicacronym is to draw attention to the analogy with NuclearMagnetic Resonance and Electron Spin Resonance, the range ofwhose applications is well known.”From J.E. Sonier, µSR brochure.

Advantages of µSR

I Implanted nuclear probe:little pertubation to the system,no requirement in the presence of isotopes.

I Full polarisation of the probe, including in zero field and atany temperature

I Extremely high sensitivity:detection of decay of (almost) all muons,detection of small magnetic field (large γµ).

I Only magnetic fields are probed.No sensitivity to electric fields.

I Local probe: independent determination of magnetic momentand magnetic volume fraction

Permanent facilities for µSR studies

A typical µSR spectrometer

Spectrometer at the Swiss Muon Source.

The measured physical quantities: PX (t) and PZ (t)

I The primary purpose of a µSR experiment is to determine theevolution of the polarisation of the implanted muons.

I The polarisation is defined as the average over the muonensemble of the muon normalised magnetic moment or spin.

I In fact, only the projection of the polarisation along a directionis measured. This is the direction of the positron detector.TF geometry: PX (t); LF or ZF geometry: PZ (t).

Axes definitions for thetwo geometries.

The polarisation functions from the field distributionStatic field at the muon site (1)

Recall solution of Larmor equation:

Sµ(t) = S‖µ(0)u + S⊥µ (0)[cos(ωµt) v − sin(ωµt)w].(13)

Then the projection of Sµ(t) along Sµ ≡ Sµ(t = 0)reads

Sαµ (t) = Sµ[cos2 θ + sin2 θ cos(ωµt)]. (14)


I. Assume all muons are submitted to the same field Bloc = B0,e.g. in the ordered phase of a magnet (crystal).

Case II.1: Sµ(t = 0) ⊥ B0, i.e. θ = 90◦.

PX (t) =SX

µ (t)

Sµ= cos(ωµt), (15)

with ωµ = γµB0.The precession frequency gives the magnitude of the local field.

Case II.2: Sµ(t = 0) ‖ B0, i.e. θ = 0◦.

PZ (t) =SZ

µ (t)

Sµ= 1. (16)


II. Assume now that the muons are submitted to a distributionDv (Bloc) of fields Bloc,

Pα(t) =Sα

µ (t)

Sµ=

∫[cos2 θ + sin2 θ cos(ωµt)]Dv (Bloc) d3Bloc.

(17)Applications:Case II.1: Dv (Bloc) = δ(Bloc − B0)/(4π2B2

0 ), e.g. in the orderedphase of a magnet (polycrystal):

PZ (t) =1

3+

2

3cos(ωµt). (18)


Case II.2: Dv (Bloc) =(

1√2π∆G

)3exp

(− B2

loc

2∆2G

), e.g. when the field

arises from muon neighbour nuclei;in zero external field

PZ (t) = PKT(t) =1

3+

2

3(1− γ2

µ∆2Gt2) exp

(−

γ2µ∆2

Gt2

2

), (19)

which is the so-called Kubo-Toyabe function.In an applied transverse field Bext,

PX (t) = exp

(−

γ2µ∆2

Gt2

2

)cos(ωµt), (20)

with ωµ = γµBext.


Polarisation function in magnets:

crystal. polycrystal.

Polarisation in disordered systems

Kubo-Toyabe function (zero field). Transverse field.

The polarisation functions from the field distributionDynamical field at muon site (1)

On the time scale of τµ = 2.2 µs, Bloc is dynamical.Assuming Bloc(t) to be a stochastic variable which proceeds injumps with a characteristic frequency νc and with a hoppingprobability which does not depend on the state before the jump,

Pα(t) =+∞∑`=0

Rα,`(t), (21)

where Rα,`(t) is the average contribution of the muons which haveexperienced ` jumps until time t. For example we have

Rα,0(t) = Pstatα (t) exp(−νct). (22)

Indeed the probability for Bloc(t) to be unchanged between times 0and t is exp(−νct).A recursion formula which links Rα,`+1(t) and Rα,`(t) can befound, and after some mathematics Pα(t) is found to be thesolution of an integral equation.

The polarisation functions from the field distributionDynamical field at muon site (2)

Results for the Gaussian field distribution:

PZ (t). Envelope of PX (t).

The number next to each

curve is the ratio νcγµ∆G

.

I Sensitivity to dynamics over several decades of νc.I When dynamics is fast, i.e. νc

γµ∆G> 1, PZ (t) and the envelope

of PX (t) are exponential functions (motional narrowing limit)

PZ (t) = exp(−λZ t) with λZ = 2γ2µ∆2

Gτc(23)

env. of PX (t) = exp(−λX t) with λX = γ2µ∆2

Gτc (24)

respectively; (τc ≡ 1/νc).I Increased sensitivity to slow dynamics in zero field.

Comparison of dynamical ranges accessible to differenttechniques

The polarisation functions from a quantum approachExample of longitudinal field experiments

At thermodynamical equilibrium, the populations of the two statesare equal since ~ωµ � kBT .Indeed, for Bloc = 1 T, ~ωµ = kBT with T = 6.5 mK.With this approach it is shown that

PZ (t) = exp(−λZ t), (25)

where the spin-lattice relaxation rate λZ can be computed usingFermi golden rule.

The polarisation functions from the Bloch equationsExample of transverse field experiments

Recall the Larmor equation

dSµ(t)

dt= γµ Sµ(t)× Bloc(t). (26)

Since in high transverse field, Bloc(t) ' Bext, andSµ(t) ⊥ Bext, we set phenomenologically

d〈SXµ (t)〉dt

= γµBext 〈SYµ (t)〉 − λX 〈SX

µ (t)〉d〈SY

µ (t)〉dt

= −γµBext 〈SXµ (t)〉 − λX 〈SY

µ (t)〉. (27)

The solution is

PX (t) =〈SX

µ (t)〉Sµ

= exp(−λX t) cos(ωµt). (28)

λX is the spin-spin relaxation rate.

The magnetic field at the muon site

Beside Bext, it is the dipolar field arising from localised magneticmoments mj (nuclei or electrons):

Bdip =µ0

4π

∑j

[−

mj

r3j

+ 3(mj · rj)rj

r5j

]. (29)

rj is the vector distance from the moment to the muon.

When a polarised electron density is present at the muon, anadditional contribution is present, the hyperfine field:

Bhyp =µ0

4π

∑j∈NN

Hjmj . (30)

Only the nearest neighbours (NN) to the muon usually contributeto this field.When both Bdip and Bhyp contribute to Bloc (i.e. in metals) theygenerally have the same order of magnitude.

Examples

µSR response in a paramagnetic phaseThe case of MnSi

A magnetic phase transition to a(nearly ferromagnetic)helimagnetic phase below Tc '30 K.

The sources of the magnetic field at the muon are:• the electronic moments of Mn,• the nuclear moments of 55Mn.Since these two sources are independent

PZ (t) = exp(−λZ t) PKT(t). (31)

PKT(t) is defined in Eq. 19.

Magnetic phase transition in a polycrystal

µSR spectra: a magnetic transition at

TC = 74.5 K.

Temperature dependence of B0.

Above TC,PZ (t) = exp(−λZ t). (32)

Below TC,

PZ (t) =1

3exp(−λZ t) +

2

3exp(−λX t) cos(ωµt). (33)

The spontaneous field is B0 = ωµ/γµ.

Magnetic phase transition in a crystal

I Spontaneous field parallel to c-axis.

I Large increase of λZ (Sµ ⊥ c) when T −→ T+N (TN = 57 K):

slowing down of magnetic fluctuations.

I Huge anisotropy of λZ (T ).

Detection of magnetic phase separation

Inhomogenous compound with regions of different magneticcharacters.

Neutron scattering intensity.

F. Bourdarot et al. Physica B 350, E179 (2004).

Muon spin rotation amplitude.

A. Amato et al. J. Phys.: Condens. Matter 16, S4403 (2004).

The muon is a local probe: it differentiates e.g. magnetic andnon-magnetic regions and provides their volume ratio.

Other fields addressed by the µSR techniquesand not mentioned here !

I Diffusion of a light interstitial particle in a crystal lattice(Brownian motion).

I Study of semiconductors: Mu is formed and it mimics ahydrogen impurity in the material. Study of its interactionswith the environment.

I Physical chemistry: Mu binds to a molecule to give aspin-labelled free radical. Study of chemical reactionmechanisms.

I Study of superconductors magnetic response.

I Study of thin materials with low-energy muons.

µSR study of superconductors

Type II superconductors submitted to a magnetic field:

Field (deviation) profile in the flux-line

lattice phase.

Associated field distribution.

I The shape of the field distribution gives access to the Londonpenetration depth λL and to the coherence length.

I From λL(T ), information on the symmetry of the orderparameter of the superconducting phase.

µSR with low energy muonsPossibility to decelerate the muons while keeping their polarisation.By controlling their final kinetic energy from the 100 eV to the20 keV ranges, the implantation depth can be varied from a fewnanometres to 100 nm.Applications of depth resolved µSR measurements:field profile near the surface in the Meissner state ofsuperconductors.

Examples of field profiles in Pb and (inset) a high Tc superconductor.

A. Suter et al. Phys. Rev. Lett. 92, 087001 (2004).

Bibliography

Introductory articles

I µSR brochure by J.E. Sonier (2002)http://musr.ca/intro/musr/muSRBrochure.pdf

I S.J. Blundell, Spin-Polarized Muons in Condensed MatterPhysics, Contemporary Physics 40, 175 (1999)

Books

I A. Yaouanc and P. Dalmas de Reotier, Muon Spin Rotation,Relaxation and Resonance: Applications to CondensedMatter, (Oxford University Press, Oxford, 2011)

I E. Karlsson, Solid State Phenomena, As Seen by Muons,Protons, And Excited Nuclei, (Clarendon, Oxford 1995)

I A. Schenck, Muon Spin Rotation Spectroscopy, (Adam Hilger,Bristol, 1985)

To appear in December 2010

Documents

Introduction to muon spin rotation and relaxation (SR)