Muon Spin Relaxation Functions

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1 Muon Spin Relaxation Functions Bob Cywinski Department of Physics and Astronomy University of eeds eeds S 9JT Muon Training Course, February 005

2 Introduction Positive muon spin relaxation (µsr) is a point-like magnetic probe in real space - similar to NMR and ESR, but probing time-space rather than frequency-space The time evolution of the muon spin in µsr can be measured in ero applied magnetic field via the radioactive decay of the muon - NMR and ESR measurements are generally performed in high applied fields and resonating RF fields In this lecture we shall look at muon spin relaxation under typical conditions in both ero field and applied field, and in the presence of fluctuating internal fields

3 Muon implantation Implantation is rapid and occurs without loss of muon polarisation Each muon spin therefore starts its time evolution with an initial spin polarisation of 100% The average spin polarisation of an ensemble of muons at time t after implantation is defined as the muon spin relaxation function, G(t) ~1-3 mm

4 Muon decay ifetime:.19714µs Decay asymmetry: W(θ) 1+a 0 cosθ a o ~0.5 Gyromagnetic ratio: x10 8 xπ s -1 T -1

5 Measuring the relaxation process F(t) R(t) F(t) + B(t) B(t)

6 Relaxation... µ % ]Ã RU% ) % R F(t) B(t) (t) F(t) + B(t) a o G (t)

7 Muon precession A individual muon at any specific site will generally experience a finite magnetic field along an arbitrary (-) direction The expectation values of the muon spin along the x and y directions, <S x > and <S y >, will precess at the armor frequency, ω. <S > is time independent In a sample without long range magnetic order, the magnetic field varies in both direction and magnitude from site to site. So, for ensemble of muons distributed over many sites we must account for a distribution of armor frequencies

8 Muon depolarisation in a static gaussian field distribution The local internal field responsible for the muon spin precession at each muon site originates from a dipolar interaction 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 static nuclear dipoles the probability distributions of the x, y and components of resultant internal fields, P(B i ) are Gaussian: G 1 P (H i) exp( Bi ) ( i x,y,) ) π Similarly the distribution of the magnitudes of the internal fields 3 ( ) G 1 P (H) exp B 4πB π

9 Gaussian fields 0.5 Probability distribution of x,y and components of internal field B i P(H i ) ix,y, mT Internal field, H i, ix,y, (mt) Probability distribution of the magnitude of the internal field B P( H ) mT Internal field H, mt

10 Precession in Gaussian fields: R(t)cos(γ µ Bt)

11 Relaxation in Gaussian fields: If we assume at t0 all muons are polarised along the - direction, then on average 1/3 will sense a net field directed along the x-, the y- and the -directions The 1/3 sensing a field along the -direction will not precess Averaged over all muons the resulting relaxation function is: x x1/3 y x1/3 x1/3

12 Relaxation in Gaussian fields θ cos θ sin θ µ B The -component of the muon spin polarisation s (t) has a time-independent component, proportional to cos θ and a sin θ component precessing at a frequency γ µ B s (t) cos θ + sin θcos( γ µ The relaxation function is given by the statistical average of s (t) G (t) s (t)p(b x )P(B y )P(B )db giving, for a Gaussian field distribution, the famous static Gaussian Kubo-Toyabe function : G G (t) (1 σ 3 t )exp x db y 1 ( σ t ) db (eg Hayano et al PRB 0 (1979) 850) Bt) ( γ ) σ µ

13 Relaxation in Gaussian fields and an external field If an external magnetic field, B ext, is applied along the -axis, B i should be replaced by B + B ext before the statistical average is taken: G σ 1 G (t, ω ) 1 ( 1 cos( ωt)exp( σ t )) ω σ + 3 ω 4 t 0 sin( ω τ)exp 1 ( σ τ ) dτ with ω γ µ B ext Note that this calculation assumes that the external field does not reorient the dipoles which give rise to the internal fields at the muon sites

14 Relaxation in Gaussian fields and an external field Tσt

15 Relaxation in Gaussian fields and an external field If an external magnetic field, B ext, is applied along the -axis, B i should be replaced by B + B ext before the statistical average is taken: G σ 1 G (t, ω ) 1 ( 1 cos( ωt)exp( σ t )) ω σ + 3 ω 4 t 0 sin( ω τ)exp 1 ( σ τ ) dτ with ω γ µ B ext Note that this calculation assumes that the external field does not reorient the dipoles which give rise to the internal fields at the muon sites Note also that in the absence of an external field and for a unique internal field of magnitude B the directional average gives 1 G (t) cos( γ µ B t) 3 3

16 orentian field distributions Whilst Gaussian field distributions are appropriate for concentrated dipole moments, the field distribution for dilute dipole moments is better described by the orentian function 1 Λ P (H) i ( i x,y,) ) π Λ + B ( ) Taking a statistical average over the time-dependent - component of the muon spin then gives the static orentian Kubo Toyabe relaxation function i (t) G (1 at)exp( at) with aγ µ Λ Tat

17 orentian fields and an external field Again, with an external magnetic field, B ext, applied along the -axis, B i should be replaced by B + B ext before the statistical average is taken: G (t, ω a ) 1 ω a ω 1+ (j ( ω t)exp a ω j ( ω t)exp 1 o a t 0 ( at ) ( at ) 1) (j ( ω t)exp o ( at) dτ where j o and j 1 are spherical Bessel functions: j o ( ω t) sinω ω t t, j ( ω t) 1 sinωt ( ω t) + cosωt ω t

18 orentian fields and an external field Tat

19 Intermediate field distributions Whilst the Gaussian and orentian field distributions adequately describe the concentrated and dilute dipole moment limits respectively, the distiction between the two limits is rather arbitrary We can see that the ero field Kubo Toyabe rlaxation function can be generalised as (t) G 1 3 (1 ( λt) 3 α )exp( ( λt) α α) where >α>1 Crook and Cywinski showed that this generalisation interpolates between the concentrated and dilute limits, and corresponds to P(B i ) being Voigtian distributed Crook and Cywinski J Phys Condensed Matter 9 (1997) 1149

20 Dynamic muon spin relaxation functions Internal field dynamics, resulting either from the muon hopping from site to site or from the internal fields themselves fluctuating, can be accounted for within the strong collision approximation, ie it is assumed that the local field changes its direction at a time t according to a probability distribution p(t)exp(-νt), the field after such a collision is chosen randomly from the distribution P(B i ) and is entirely uncorrelated with the field before the collision

21 The strong collision model -fast fluctuations exp(-λt) σ0.1 µs -1 G (t) time (µs) The above curves have been calculated assuming that ν/σ5 This is within the fast fluctuation (motional narrowing) limit for which the relaxation envelope is well described by exp(-λt)

22 Dynamic relaxation The total muon polarisation at time t is the superposition of the polarisation of each muon at that time. So, the fraction that have not experienced a field change at time t is given by exp(-νt), and their contribution is g (0) (t) g (t) e A particular muon that has experienced one change at time t has a probability of remaining stationary until the further time t of exp(-ν(t-t )). The contribution to the total polarisation from all muons having had only one jump to time t is thus g (1) (t) ν 0 g (t )e νt νt ν( t t ) g (t t )e dt The higher order terms can be successively derived by the recurrence relation g (n) (t) ν t 0 g (0) (t )g (n 1) (t t )dt

23 Dynamic relaxation (cont( cont.) The total muon relaxation function can be written as the sum over all n of g (n) (t) G DKT (t) g g 0 (0) (0) g (n) (t) + (t) n 1 (t) + ν t 0 ν t 0 G g DKT (n 1) (t t ) g (t t ) g (0) (0) (t )dt (t )dt This expression can be evaluated by numerical integral for any internal field distribution (ie Gaussian, orentian* or Voigtian) with or without an external applied field The result is the dynamic Kubo-Toyabe function (eg Hayano et al PRB 0 (1979) 850)

24 Dynamic Gaussian Kubo-Toyabe function 1.0 R R0 0.6 R10 G DKT (t) R R R1 R0 R T The dynamic Kubo-Toyabe function plotted as a function of the dimensionless parameters, Tσt and Rν/σ. For R>1 approximate forms for G DKT (t) can be used

25 Dynamic Gaussian Kubo-Toyabe function in an applied longitudinal field Tσt

26 Some approximations For a gaussian field distribution in the fast fluctuation limit (Rν/σ>>1) in ero field G (t) exp( σ t ν) exp( λt) whilst in an applied field σ τc λ τc 1/ ν (1+ ω τ ) c For a gaussian field distribution in the intermediate fluctuation limit (Rν/σ>1) in ero field we have the so-called Abragam form σ t G(t) exp (exp( νt) 1+ νt) ν For very slow fluctuations R<1 only the 1/3 Kubo-Toyabe tail is affected. The form of this tail becomes G G (t, ν) 1 exp( (/ 3) νt) 3

27 Dynamic orentian Kubo-Toyabe function *Note that for the orentian case a hopping muon and fluctuating fields do not necessarily give the same result

28 Special cases: dilute magnetic alloys Uemura (PRB31 (1985) 546) assumed that the fluctuation of magnetic impurity moments in dilute spin glasses leads to a time modulated field at the muon site. The dynamic range of this field modulation will depend on the proximity of the muon to its neighbouring spins µ + Au P(H) The dynamic variable range of the local fields at each muon site is approximated by a Gaussian distribution of width σ/γ µ, G γ µ γµ Bi P (B i) exp σ π σ Fe, i H P(H) µ + H x, y,

29 dilute magnetic alloys (cont( cont) The probability, ρ(σ j ), of choosing a muon site j for which the width of the dynamic range is σ j must satisfy the condition G P (B ) P (B, σ ) ρ( σ )dσ i 0 where P (B i ) is the original orentian field distribution Uemura showed that ρ( σ j ) (a π σ i j ) exp( a Assuming the fast fluctuation limit with a unique fluctuation rate ν we calculate G (t,a, ν) 0 G G j j σ (t, σ, ν) ρ( σ)dσ Hence, in the fast fluctuation limit, we find the root-exponential form G (t,a, ν) exp 4a t ν ( ) )

30 Distributed relaxation rates In many spin glasses the relaxation function is found to be not root-exponential but stretched exponential β G (t) exp( ( λt) ) Moreover, β itself is temperature dependent often decreasing from 1 at high temperatures (4T g ) to 1/3 at T g Ag-5at%Mn b T g T g Campbell, et al PR 7 (1994) 191 (eg Ogielski PRB 3 (1985) 7384) Whilst the muon relaxation is measured in aplace time, this functional form mirrors exactly the Kohlrausch relaxation predicted theoretically for spin-spin correlations in real time

31 Distributed relaxation rates This behaviour can be modeled quite simply by assuming the rapid fluctuation limit but with a broad distribution of muon spin relaxatio rates, P(λ), so G(t) 0 P( λ)e λ t d We find that for any (broad) P(λ) leads to the form λ β ( ( λt ) G (t) exp ) As P(λ) becomes extremely broad (as predicted for a spin glass approaching T g ) β approaches 1/3 asymptotically. Cywinski et al (unpublished)

32 Rotation. µ %[Ã! ) % R x (t) F(t) F(t) + B(t) B(t) a o G x (t) cos( ω t)

33 Transverse field µsr In a high applied fields B T transverse to the initial muon spin (-)direction, the muons will precess around the vector sum of B T and the internal field B For a Gaussian internal field distribution for which B T >>, the direction of the local field is almost parallel to B T In this situation the magnitude of the field at the muon site is approximately B T + B i where i is the component in the field direction (say ix) The muon precession is therefore of the form R(t) cos( ωtt)exp( σ t / ) and G G (t) exp( σ t / ) x Note that nuclear dipoles may also precess in an applied field thereby reducing the effective s by a factor of 5 compared to ero field