muon spin relaxation functions - isis neutron source · relaxation in gaussian fields θ z b µ...

Muon Spin RelaxationMuon Spin RelaxationFunctionsFunctions

Bob Cywinski

Department of Physics and AstronomyUniversity of Leeds

Leeds LS2 9JT

Muon Training Course, February 2005

MuSR -Relaxation Functions Muon Training Course Feb 2005

IntroductionIntroduction

The time evolution of the muon spin in µSR can be measuredin zero applied magnetic field via the radioactive decay of themuon

- NMR and ESR measurements are generally performed in high applied fields and resonatingRF fields

Positive muon spin relaxation (µSR) is a point-like magneticprobe in real space

- similar to NMR and ESR, but probing time-spacerather than frequency-space

In this lecture we shall look at muon spin relaxation undertypical conditions in both zero field and applied field, andin the presence of fluctuating internal fields

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Muon implantationMuon implantation

~1-3

mm

Implantation is rapid andoccurs without loss ofmuon polarisation

Each muon spin thereforestarts its time evolutionwith an initial spinpolarisation of 100%

The average spinpolarisation of an ensembleof muons at time t afterimplantation is defined asthe muon spin relaxationfunction, G(t)

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Muon decayMuon decay

Gyromagnetic ratio: 1.355342x108 x2π s-1T-1

Lifetime: 2.19714µs

Decay asymmetry: W(θ) = 1+a0cosθ

ao~0.25

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Measuring the relaxation processMeasuring the relaxation process

)t(B)t(F)t(B)t(F

)t(R+−=

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Relaxation...Relaxation... %]�

RU�% �

µ

)%

)t(Ga)t(B)t(F)t(B)t(F

)t(R zoz =+−=

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Muon precessionMuon precession

The expectation values of themuon spin along the x and ydirections, <Sx> and <Sy>, willprecess at the Larmorfrequency, ωL.

In a sample without long range magnetic order, themagnetic field varies in both direction and magnitudefrom site to site.

A individual muon at any specific site will generallyexperience a finite magnetic field along an arbitrary (z-)direction

So, for ensemble of muons distributed over many siteswe must account for a distribution of Larmor frequencies

<Sz> is time independent

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Muon Muon depolarisationdepolarisation in a static in a staticgaussiangaussian field distribution field distributionThe local internal field responsible for the muon spinprecession at each muon site originates from a dipolarinteraction with surrounding nuclear or electronic spins

(and contact hyperfine fields from the spin density at the muon site)

For a concentrated system of randomly oriented staticnuclear dipoles the probability distributions of the x, y and zcomponents of resultant internal fields, P(Bi) are Gaussian:

( ) ( ))z,y,xi2Bexp21

)H(P 22ii

G =∆−×∆π

=

Similarly the distribution of the magnitudes of the internalfields

( ) 2223

G B42Bexp21

)H(P π×∆−×

∆π=

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Internal field |H|, mT

0.0 0.1 0.2 0.3 0.4 0.5

0.0 0.1 0.2 0.3 0.4 0.5

P(|

H|)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Internal field, Hi, i=x,y,z (mT)

-0.4 -0.2 0.0 0.2 0.4

P(H

i) i=

x,y,

z

0.0

0.1

0.2

0.3

0.4

0.5

∆=0.1mT

∆=0.1mT

Gaussian fieldsGaussian fields

Probability distribution of x,yand z components ofinternal field Bi

Probability distribution ofthe magnitude of theinternal field |B|

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Precession in Gaussian fields:Precession in Gaussian fields:

R(t)=cos(γµBt)

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Relaxation in Gaussian fields:Relaxation in Gaussian fields:If we assume at t=0 all muons are polarised along the z-direction, then on average 1/3 will sense a net field directedalong the x-, the y- and the z-directions

The 1/3 sensing a field along the z-direction will not precess

Averaged over all muons the resulting relaxation function is:

x1/3

x1/3

x1/3

x

y

z

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Relaxation in Gaussian fieldsRelaxation in Gaussian fields

θz

B

µcos2θ

sin2θThe z-component of the muonspin polarisation sz(t) has atime-independent component,proportional to cos2θ and asin2θ component precessingat a frequency γµ|B|

)Btcos(sincos)t(s 22z µγθ+θ=

The relaxation function is given by the statistical average of sz(t)

zyxzyxzz dBdBdB)B(P)B(P)B(P)t(s)t(G ∫∫∫=

giving, for a Gaussian field distribution, the famous staticGaussian Kubo-Toyabe function :

( )222122G

z texp)t1(32

31

)t(G σ−σ−+=

(eg Hayano et al PRB 20 (1979) 850)

)( ∆γ=σ µ

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Relaxation in Gaussian fields and anRelaxation in Gaussian fields and anexternal fieldexternal fieldIf an external magnetic field, Bext, is applied along the z-axis,Bi=z should be replaced by Bz+ Bext before the statisticalaverage is taken:

( )( )

( ) ττσ−τωωσ+

σ−ω−ωσ−=ω

∫ dexp)sin(2

texp)tcos(12

1),t(G

t

0

2221

L3L

4

2221

L2L

2

LGz

with ωL=γµBext

Note that this calculation assumes that the external fielddoes not reorient the dipoles which give rise to the internalfields at the muon sites

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Relaxation in Gaussian fields and anRelaxation in Gaussian fields and anexternal fieldexternal field

T=σt

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Relaxation in Gaussian fields and anRelaxation in Gaussian fields and anexternal fieldexternal fieldIf an external magnetic field, Bext, is applied along the z-axis,Bi=z should be replaced by Bz+ Bext before the statisticalaverage is taken:

( )( )

( ) ττσ−τωωσ+

σ−ω−ωσ−=ω

∫ dexp)sin(2

texp)tcos(12

1),t(G

t

0

2221

L3L

4

2221

L2L

2

LGz

with ωL=γµBext

Note that this calculation assumes that the external fielddoes not reorient the dipoles which give rise to the internalfields at the muon sites

Note also that in the absence of an external field and for aunique internal field of magnitude |B| the directionalaverage gives

)tBcos(32

31

)t(Gz µγ−=

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Lorentzian Lorentzian field distributionsfield distributionsWhilst Gaussian field distributions are appropriate forconcentrated dipole moments, the field distribution for dilutedipole moments is better described by the Lorentzian function

( ) ( ))z,y,xiB

1)H(P

2i

2iL =

+ΛΛ

π=

Taking a statistical averageover the time-dependent z-component of the muon spinthen gives the static LorentzianKubo Toyabe relaxationfunction

with a=γµΛ

)atexp()at1(32

31

)t(Gz −−−=

T=at

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Lorentzian Lorentzian fieldsfields and an external fieldand an external fieldAgain, with an external magnetic field, Bext, applied alongthe z-axis, Bi=z should be replaced by Bz+ Bext before thestatistical average is taken:

( )

( )

( ) τ−ω

ω

+−

−−ω

ω

−

−ωω

−=ω

∫ datexp)t(j(aa

1

)1atexp)t(j(a

atexp)t(ja

1),t(G

t

0Lo

2

L

Lo

2

L

L1

L

LLz

where jo and j1 are spherical Bessel functions:

ttcos

)t(tsin

)t(j,t

tsin)t(j

L

L2

L

LL1

L

LLo ω

ω+ω

ω=ωω

ω=ω

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T=at

LorentzianLorentzian fields fields and an external fieldand an external field

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Intermediate field distributionsIntermediate field distributionsWhilst the Gaussian and Lorentzian field distributionsadequately describe the concentrated and dilute dipolemoment limits respectively, the distiction between the twolimits is rather arbitrary

We can see that the zero field Kubo Toyabe rlaxationfunction can be generalised as

))t(exp())t(1(32

31

)t(Gz αλ−λ−−= αα

where 2>α>1

Crook and Cywinski showed that this generalisationinterpolates between the concentrated and dilute limits,and corresponds to P(Bi) being Voigtian distributed

Crook and Cywinski J Phys Condensed Matter 9 (1997) 1149

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Dynamic muon spin relaxation functionsDynamic muon spin relaxation functions

Internal field dynamics, resulting either from the muonhopping from site to site or from the internal fields themselvesfluctuating, can be accounted for within the strong collisionapproximation, ie

it is assumed that the local field changes itsdirection at a time t according to a probabilitydistribution p(t)=exp(-νt),

the field after such a “collision” is chosenrandomly from the distribution P(Bi) and isentirely uncorrelated with the field before thecollision

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time (µs)

0 5 10 15 20 25 30 35 40

Gz(

t)

0.0

0.2

0.4

0.6

0.8

1.0σ=0.1 µs-1

The strong collision model -fast fluctuationsThe strong collision model -fast fluctuations

The above curves have been calculated assumingthat ν/σ=5

exp(-λt)

This is within the fast fluctuation (motional narrowing)limit for which the relaxation envelope is well describedby exp(-λt)

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Dynamic relaxationDynamic relaxationThe total muon polarisation at time t is the superpositionof the polarisation of each muon at that time.

So, the fraction that have not experienced a field changeat time t is given by exp(-νt), and their contribution is

tz

)0(z e)t(g)t(g ν−=

A particular muon that has experienced one change attime t’ has a probability of remaining stationary until thefurther time t of exp(-ν(t-t’)). The contribution to the totalpolarisation from all muons having had only one jump totime t is thus

∫∞

′−ν−′ν− ′′−′ν=0

)tt(z

tz

)1(z tde)tt(ge)t(g)t(g

The higher order terms can be successively derived bythe recurrence relation

‘dt‘)tt(g‘)t(g)t(g z)1n(

t

0

z)0(

z)n( −ν= −∫

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The total muon relaxation function can be written as thesum over all n of g(n)

z(t)

‘dt‘)t(g‘)tt(G)t(g

‘dt‘)t(g‘)tt(g)t(g

)t(g)t(G

)0(z

t

0

DKTz

)0(z

)0(z

t

0

)1n(z

1n

)0(z

0

)n(z

DKTz

∫

∫∑

∑

−ν+=

−ν+=

=

−∞

=

∞

This expression can be evaluated by numerical integral forany internal field distribution (ie Gaussian, Lorentzian* orVoigtian) with or without an external applied field

Dynamic relaxation (Dynamic relaxation (contcont.).)

The result is the dynamic Kubo-Toyabe function

(eg Hayano et al PRB 20 (1979) 850)

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Dynamic Gaussian Kubo-ToyabeDynamic Gaussian Kubo-Toyabefunctionfunction

0 1 2 3 4

GDKTz(t)

0.0

0.2

0.4

0.6

0.8

1.0

R=1

R=2

R=5

R=10

R=20

R=50

R=0.5

R=0

T

The dynamic Kubo-Toyabe function plotted as a functionof the dimensionless parameters, T=σt and R=ν/σ.

For R>1 approximate forms for GDKTz(t) can be used

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T=σt

Dynamic Gaussian Kubo-ToyabeDynamic Gaussian Kubo-Toyabefunction in an applied longitudinal fieldfunction in an applied longitudinal field

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Some approximationsSome approximationsFor a gaussian field distribution in the fast fluctuation limit(R=ν/σ>>1) in zero field

)texp()t2exp()t(G 2z λ−=νσ−=

For a gaussian field distribution in the intermediate fluctuationlimit (R=ν/σ>1) in zero field we have the so-called Abragamform

ν+−ν−

νσ−= )t1)t(exp(

t2exp)t(G

2

2

z

whilst in an applied field

ν=ττω+

τσ=λ /1)1( c2

c

2

L

c2

For very slow fluctuations R<1 only the 1/3 Kubo-Toyabetail is affected. The form of this tail becomes

)t)3/2(exp(31

),t(GGz ν−=ν

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Dynamic Dynamic LorentzianLorentzian Kubo-Toyabe Kubo-Toyabefunctionfunction

*Note that for the Lorentzian case a hopping muon andfluctuating fields do not necessarily give the same result

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Special cases: dilute magnetic alloysSpecial cases: dilute magnetic alloysUemura (PRB31 (1985) 546) assumed that the fluctuation ofmagnetic impurity moments in dilute spin glasses leads toa time modulated field at the muon site.

The dynamic range of thisfield modulation willdepend on the proximity ofthe muon to itsneighbouring spins

µ+

H

Au

P(H)

P(H) H

Fe

µ+

The dynamic variablerange of the local fields ateach muon site isapproximated by aGaussian distribution ofwidth ∆ = σ/γµ,

z y, x, i ,2

Bexp

2)B(P

2

2

i

2

iG =

σ

γ−

πσγ

= µµ

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dilute magnetic alloys (dilute magnetic alloys (contcont))The probability, ρ(σj), of choosing a muon site j for whichthe width of the dynamic range is σj must satisfy thecondition

jj0

jiG

iL d)(),B(P)B(P σσρσ= ∫

∞

where PL(Bi) is the original Lorentzian field distribution

Uemura showed that

)2aexp()a(2

)( 222j σ−×σ×

π=σρ

Assuming the fast fluctuation limit with a unique fluctuationrate ν we calculate

σσρνσ=ν ∫∞

d)(),,t(G),a,t(G0

Gz

Lz

Hence, in the fast fluctuation limit, we find the root-exponentialform ( )ν−=ν ta4exp),a,t(G 2L

z

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Distributed relaxation ratesDistributed relaxation ratesIn many spin glasses the relaxation function is found to benot “root-exponential” but “stretched exponential”

( )βλ−= )t(exp)t(GLz

Moreover, β itself is temperature dependent oftendecreasing from 1 at high temperatures (4Tg) to 1/3 at Tg

Whilst the muon relaxation is measured in “Laplace” time,this functional form mirrors exactly the Kohlrauschrelaxation predicted theoretically for spin-spin correlationsin real time

(eg Ogielski PRB 32 (1985) 7384)

b

Tg

Ag-5at%Mn

Tg

Campbell, et al PRL 72 (1994) 1291

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Distributed relaxation ratesDistributed relaxation ratesThis behaviour can be modeled quite simply by assumingthe rapid fluctuation limit but with a broad distribution ofmuon spin relaxatio rates, P(λ), so

∫∞

λ− λλ=0

td)eP(G(t)

We find that for any (broad)P(λ) leads to the form

( )βλ−= )t(exp)t(GLz

As P(λ) becomes extremelybroad (as predicted for aspin glass approaching Tg)β approaches 1/3asymptotically.

Cywinski et al (unpublished)

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Rotation….Rotation….

)tcos()t(Ga)t(B)t(F)t(B)t(F

)t(R Lxox ω=+−=

µ

)%

%[�

!�

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Transverse field Transverse field µµSRSRIn a high applied fields BT transverse to the initial muonspin (z-)direction, the muons will precess around thevector sum of BT and the internal field B

For a Gaussian internal field distribution for which BT>>∆,the direction of the local field is almost parallel to BT

In this situation the magnitude of the field at the muonsite is approximately |BT| + |Bi| where i is thecomponent in the field direction (say i=x)

The muon precession is therefore of the form

)2/texp()tcos(R(t) 22T σ−ω=

and)2/texp((t)G 22G

x σ−=

Note that nuclear dipoles may also precess in an applied fieldthereby reducing the effective s2 by a factor of 5 compared tozero field