Ross Stewart Outline of Lecture
Magnetic Neutron Scattering Ross Stewart Outline of Lecture
• How does the neutron see magnetism? • Magnetic form factors • Magnetic neutron diffraction • Magnetic inelastic neutron scattering How do the neutrons see magnetism? Magnetic “dipoles”
Magnetic Dipole Moment, μ by definition from classical electromagnetism µ = IA → A
e → so in terms of orbital angular momentum, L e µ = L − 2m e Magnetic “dipoles”
Magnetic Dipole Moment, μ Dirac (1928) also postulated an intrinsic angular momentum with associated magnetic moment (seen experimentally) → S
e e Except that this time: µ = S − m e For particle with spin and orbital contributions: e µ = g J “g-factor” − 2m e J = L + S Magnetic moment defs.
Angular momenta, are measured in units of ℏ, so we can write e µ = gµBJ where µB = − 2me Bohr Magneton The gyromagnetic ratio, is defined as the ratio of the magnetic dipole moment, to the total angular momentum e γ = g − 2m
Therefore gµB = γ and µ = γ J − Magnetic properties of the neutron For neutrons, we assume a spin-only angular momentum, with eigenvalues of ±ℏ/2 z 1 “Spin-Up” = 1/2 | ↑ 0 s = 1/2
ms = ± 1/2 0 “Spin-Down” = -1/2 | ↓ 1 Magnetic properties of the neutron According to quantum mechanics, the “classical” angular momentum is replaced by an angular momentum operator, σ
µ = γσ
The components of σ for a spin-1/2 particle (the neutron) are 01 0 i 10 σ = 1 σ = 1 σ = 1 x 2 10 y 2 i −0 z 2 0 1 −
Pauli spin matrices Magnetic properties of the neutron Some gyromagnetic ratios - (all spin-1/2) Electron: 1.76 x 105 MHz / T Proton: 267 MHz / T Neutron: 183 MHz / T So the neutron moment is around 960 times smaller than the electron moment e Nuclear magnetons: µN = 2mp μp = 2.793μN μn = 1.913μN So we have for the neutron µ = γ µ σ n − n N where γn = 1.913 Neutron magnetic interaction Differential neutron cross-section The sum of all processes in which - the state of the scatterer changes from λ to λ` - the wavevector of the neutron changes from k to k` (where k` lies in solid angle dΩ) - the spin-state of the neutron changes from s to s` ∂2σ = Wk,λ,s k ,λ ,s ∂Ω∂E → k in dΩ 2 2 k mn = k, λ,s Vm k, λ,s δ(Eλ Eλ + ω) k 2π2 | | − “Fermi’s Golden Rule” Neutron magnetic interaction
Vm is the potential felt by the neutron due to a magnetic field → BL - field due to orbit
V = µ B m ·
→ Bs - field due to spin Neutron magnetic interaction Field due to spin and orbital moments is ˆ ˆ µ0 µe R 2µB p R B = Bs + BL = ×2 ×2 4π ∇× R − R Magnetic vector → potential A, of a dipolar Biot-Savart Law for a field due to electron single electron of linear → spin moment momentum, p Evaluating the spatial part of the transition matrix element i j iQ rj ˆ ˆ ˆ k Vm k e · Q sj Q + pj Q | | ∝ × × Q × → where ℏQ is the momentum transfer Q = (k k) − Neutron magnetic interaction Summing over all unpaired electrons, we find
j ˆ ˆ ˆ ˆ k Vm k Q M(Q) Q = M(Q) M(Q) Q Q = M (Q) | | ∝ × × − · ⊥ j where M⊥ is the perpendicular component of the Fourier iQ r transform of the magnetisation, in the sample M (Q )= M (r)e · dr Qˆ M
M Qˆ Qˆ · Qˆ M Qˆ × × M Qˆ × Neutrons only see the components of the magnetisation perpendicular to the scattering vector Average over neutron spin
Having calculated the spatial part of the matrix element we have to calculate the spin-part. If the neutrons are unpolarized - i.e. there are equal numbers of spin-up and spin-down, then we get
2 2 ∂ σ 2 k =(γr0) λ,s σ M λ,s δ(Eλ Eλ + ω) ∂Ω∂E k | · ⊥| − 2 k =(γr0) pλ λ M ∗ λ λ M λ δ(Eλ Eλ + ω) k | ⊥| · | ⊥| − λλ Collection of prefactors: 2 µ0 m µ0 e γµN 2µB 4π = γr0 where r0 = −4π 2π − 4π me
r0 is called the “classical electron radius” and has a value of 2.818 x 10-15 m Magnetic form factors Scattering from a crystal th Assume nucleus at position Rjd representing the d atomic position th th in the j unit cell. re is the position of the e unpaired electron relative to the nucleus r = R jd + re by the definition of M(Q) M(Q)= exp(iQ r )s · i i i = exp(iQ R ) exp(iQ r )s · ld · e e e ld The matrix element λ M λ now becomes | ⊥| F (Q ) λ exp(iQ R )S λ | · jd jd| Magnetic structure factor Magnetic form factor Form factors, example F (S)=c (j (S) + c (j (S) + c (j (S) + .... 1 0 2 2 1 4 Tabulated in International Tables of Crystallography, Vol C
spin only Dy3+ form factor orbital
π Form factors
• The magnetic form factor means that magnetic scattering is strongest at low Q (low angles for a cw neutron spectometer)
• Nuclear scattering has no form factor - since nuclei are point-like on the scale of the neutron wavelength Form factor measurements Crystal structure
P. Javorsky et al., Phys. Rev. B 67 (2003) 224429! UPtAl!
Derived from Braggcrystal peakstructure positions/intensities! Magnetisation density
Derived from inverse Fourier transform of F(Q)
magnetic moment density! Form factors - moments 4 Lee et. a;. arXiv:1001.3658v13 Givord, et. al. J. Appl. Phys. 50(3), March 1979 µB to 0.879 µB (at the optimized theoretical minimum at larger and more difficult to measure scattering vectors. energy z As position). Table II shows the Fe d-orbital decomposed charges and moments. Although there are magnetic amplitudes deduced at 300 K andThe 4.2 similarityK are with bcc Fe is particularly evident in the variations among the occupied electronic orbitals, mag- reported in table II and shown in Fig. totalI(a) and 3d I(b)charge density within the muffin-tin sphere. This netic moments are fairly well distributed across all of the respectively. The difference between the two sets of orbitals. This would not be expected if one or more of data is striking. is 5.95 electrons for the experimental atomic positions in the orbitals were “localized”. Indeed, all of orbitals con- In the case of samarium, it is knownSrFe that2As the2, 6.02 electrons for the corresponding calculated tribute to a broad density of states. orbital and spin parts of the magnetization are opposite and nearly cancel out. As theirenergy spatial optimized AsB position and 5.83 electrons for bcc TABLE II: Fe d-orbital electron occupation and magnetic mo- distributions are very different,it resultsFe calculated in regions with the same muffin-tin radius. ment orbital decomposition. of space with a positive magnetization density and regions with a negative one. Consequently, the magnetic spin up spin dn Magnetic Moment :7 We have determined6 the magnetic form factor of Fe 10 C) (electrons) (electrons) (µB) structure factors FM' which are the Fourier coeffici- ents of the magnetization density, a maximum '::' ,,:\ exp zAs opt zAs exp zAs opt zAs exp zAs opt zAs in SrFe2As2 by neutron diffraction experiments and de- d 3.810 3.450 2.142 2.574 1.668 0.876 non located at sin 8/), = 0 (sin 8/), "'0.4 A-I), duced the Fe magnetic moment as 1.04(1)µB. We also dz2 0.719 0.618 0.495 0.513 0.224 0.105 leading to an unusual magnetic form factor. Moreover, , T 042 K dx2 y2 0.757 0.684 0.402 0.491 0.355 0.193 in smCos the very large exchange field andcalculated the crystal the magnetic form0 factor by first principles − 0 dxy 0.769 0.700 0.502 0.568 0.267 0.132 field mix the excited multiplets J=7/2 and J=9/2 into M electronic structure methods using both the experimen- 0 dxz 0.767 0.718 0.354 0.476 0.413 0.242 the ground multiplet J=5/2. This leads to large changes 00 6 0 - 0<'<-0 C'l.,..:. MN 0 ., 0 ;;; C'1NCO)'
Downloaded 05 Mar 2011 to 130.246.132.178. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions Measuring form factors
• Form factors are often measured using polarized neutrons
• This is due to the fact that the magnetic intensity and the nuclear intensity often fall at the same positions - esp. in ferromagnets Magnetic Neutron diffraction Elastic magnetic scattering
For elastic scattering k = k and ω =0 | | | | and the scattering cross-section becomes
prefactors form factor Debye-Waller factor dσ γr 2 = 0 g2F 2(Q )exp( 2W ) dΩ 2 − β (δ Q Q ) exp(iQ R )
• e.g. flux line lattice λ =2d sin θ
MgB2 is a superconductor below λ = 10 Å 39K, and expels all magnetic field d =~2 425°! Å lines (Meisner effect). Above a critical field, flux lines penetrate the sample.
Cubitt et. al. Phys. Rev. Lett. 90 157002 (2003) Magnetic scattering in MnO
MnO, 80 K, antiferromagnetic
MnO, 300 K, paramagnetic λ =2d sin θ
New magnetic peaks
C. G. Shull & J. S. Smart, Phys. Rev. 76 (1949) 1256 Magnetic scattering in MnO
a 2a
Nuclear unit cell Magnetic unit cell reciprocal space Magnetic scattering in MnO
How do we know that the moments lie in the (111) plane?
reciprocal space - (111) (311)
- - _1 _1 _1 _3 _1 _1 _5 _3 _1 ( 2 2 2 ( ( 2 2 2 ( ( 2 2 2 ( - (220)
moment direction
fcc lattice, hkl all even/odd
H. Shaked et al., Phys. Rev. B 38 (1988) 11901 Magnetism in chromium Longitudinal body centre atoms Corner atoms Spin density wave
Corner atoms body centre atoms
Cr moment not constant bcc
Itinerant electron magnet Other sorts of elastic magnetic scattering
• Magnetic diffuse scattering • Magnetic reflectometry • Magnetic small-angle scattering (000 Bragg peak) Magnetic inelastic neutron scattering Inelastic scattering
The full inelastic cross-section is given by 2 ∂ σ k 2 2 1 ∞ iωt = (r0γn) F (Q) S (0) S (t) e− dt ∂Ω∂E k | | 2π ⊥ · ⊥ −∞ Magnetic fluctuations in a material are fixed by the Schrödinger equation Ψ = EΨ H where the fluctuation energies are eigenvalues of the Hamiltonian of the material
Thermal neutrons can measure the fluctuation energies - which gives information on the Hamiltonian Dynamic spin correlations Magnetic neutron scattering depends on the time/space correlations between magnetic spins α β ˆα ˆβ S0 Sn = S0 Sn δαβ Q Q ⊥ · ⊥ − αβ The magnetic dynamical structure factor is defined as
αβ N ∞ i(Q r ωt) α β S (Q, ω)= e · n− S S dt 2π 0 n n −∞ which allows us to write the neutron cross-section as 2 2 ∂ σ k (r0γn) 2 αβ = F (Q) (δαβ QαQβ) S (Q, ω) ∂Ω∂E k | | − αβ So the dynamical structure factor gives us all the physical information about the magnetic fluctuations (space and time) without “probe bias” Spin-waves
A simple Hamiltonian describing a ferromagnet is = J Si Sj H − · i,j where i and j are near neighbours, and J gives the energy scale of the magnetic exchange simple ferromagnet possible excited state
Spin-waves have a frequency (ω) and a wavevector (q)
The relation between ω and q is defined by the Hamiltonian, and directly measurable with neutrons Spin-waves
ω =4JS(1 cos(2qa)) − =2Dq2 for qa << 1 2 2π a
(4JS) 1
ω ℏ
0 0 0.5 1 Q (2π/a)
Brillouin Zones Example - Rb2MnF4 Magnetic susceptibility A very powerful result in non-equilibrium thermodynamics is the Fluctuation-Dissipation Theorem of Kubo N 1 S (Q, ω)= 2 ω/k T χ (Q, ω) π (gµ ) 1 e− B B − =[n(ω) + 1]
1 kBT To first order we have [n(ω) + 1] = ω = 1 (1 ) ω − − kB T so in high temperature limit when kBT >> ħω N χ (Q, ω) S (Q, ω)= 2 kBT π (gµB) ω Complex susceptibility Dynamical susceptibility is the response function of a system to a magnetic field - which can vary in both space and time - which are common in nature. i(Q r ωt) H (Q, ω)=H0e · − The susceptibility is then M (MQ, ω(Q) , ω) χχ(Q(Q, ω,)ω=)= H (QH, ω(Q) , ω) Any phase lag between M and H is described by an out-of-phase component to the susceptibility
χχ(Q(Q, ,ωω)=) = χχ((QQ,, ω))++χχ ((QQ,,ωω) )
Out-of-phase response In-phase response (magnetic relaxation - dissipative) Kramers-Krönig Assuming that the system response is linear (doubling the perturbing force doubles the response) and that the response is causal (no response before perturbation) it can be shown that: 1 ∞ χ (Q, ω) χ (Q, ω)= P dω −π ω ω 1 ∞−∞χ (Q,−ω) Of particular importanceχ (Q, ω) =is the staticP susceptibility at dωω = 0 −π ω ω 1 1 ∞−∞χ∞ (Qχ, ω(−Q) , ω) χχ(Q(Q) )== P P dω dω π π ω ω −∞−∞ so from the FD theorem, integrating under S(Q,ω) gives
N N S (Q)= kBT χ (Q) S (Q) = 2 kB2T χ (Q) (g(µgµB)B) gives direct comparison between SANS (Q ~ 0) and bulk χ Example - Mn metal
3.0
15K #-Mn 159K In this case, the magnetic 302K 2.5 fluctuations follow a damped harmonic oscillator model )
- ) 1 1 - 2.0 m o t χ (Q, ω) Γ(Q)
a
n M = χ(Q) - M 1 2 2 n
-1 ω Γ (Q)+ω a t V o m e
m 1.5 - 1 e m
V -1 t s b d m ( E
"
/ ( i.e. Lorentzian response,
d 1.0 dE b ! 2
" s d t d / !
2 indicating exponential decay d
0.5 in time
0.0 -30 -20 -10 0 10 20 30 40 Energy Transfer (meV)
JRS, et. al. Phys. Rev. Lett. 89 (2002) 186403 Figure 8.13 The partial differential cross-section of pure !-Mn at Q = 1.4Å-1 at T = 15K, 159K and 303K. The solid lines are fits of eq. (8.16) plus an elastic lineshape given by an Ikeda- Carpenter function all convoluted with the instrumental resolution function. The energy gain side ("E < 0) illustrates the effects of detailed balance, while the energy loss side of the spectra shows remarkably little temperature dependence.
218 PAPPAS, MEZEI, EHLERS, MANUEL,ExampleAND CAMPBELL - AuMn spin-glassPHYSICAL REVIEW B 68, 054431 ͑2003͒ netization is always zero, theUsingfundamental neutronparameter spin-echois the mean spin autocorrelation function q(tϪtЈ)ϭ͗S (t) we can also probe thei •Si(tЈ)͘ where Si is the spin at a site i and the average runs over all sites and configurationsmagneticof the sample. responseCritical inbe- havior in the paramagnetic phase is seen in the nonlinear the time domain6–9 susceptibility ͑or ‘‘spin glass susceptibility’’͒. Below Tg the Edwards Anderson order parameter q(t ϱ)ϭ͗Si(t 1,13 → β ϭ0)Si(t ϱ)͘, becomes nonzero. Neutron spinx echo(Γt) → S(0)S(t) t− e− spectroscopy measures thescattering function∝ S(Q,t) and after normalization by S(Q,0) delivers a direct determination of the autocorrelation function q(t)ϭs(Q 0,t). NSE cov- ers a time domain ranging fromStrongly10Ϫ12 tonon-exponentialsome→ 10Ϫ8 s, i.e., from characteristic microscopic times up to times, which al- ready belong to the ‘‘long’’ dynamicstime relaxation - domain.commonThe first in NSE experiment ever performeddisorderedon a glassy magnetssystem was made on the reference spin glass CuMn 5% in 1979 ͑Ref. 10͒ FIG. 1. Temperature dependence of the normalized intermediate and the results strongly influenced subsequent thinking on scattering function s(Q,t) of Au0.86Fe0.14 . The spectra were col- ͑spin͒glass dynamics. It was shown that nonconventional dy- lected at Qϭ0.4 nmϪ1 with the neutron spin echo spectrometer namics is not limited to the spin glass phase but also extends IN15 ͑ILL͒ for Tϭ30.7 K ͑closed squares͒, 40.6 K (ϳTg , open into the paramagnetic phasePappaswell et al,above Phys. TRevg . BNonexponential 68 054431 (2003) squares͒, 45.7 K ͑closed circles͒, 50.8 K ͑open circles͒, and 55.8 K and Q-independent relaxation occurs in a large temperature ͑closed rhombs͒, respectively. The continuous lines are the best fits range up to 2–3 Tg , which can arguably be identified with to the data of a simple power law decay below Tg ͑ϳ41 K͒ and of 14 the Griffiths phase. For about TϾ1.2 Tg the relaxation can the Ogielski function above Tg ͑see text͒. be described by a broad distribution of Arrhenius activation energies. Closer to Tg , however, a more dramatic slowing measurements, which were made from 10 Hz up to 10 KHz down sets in, which can be interpreted as the footprint of a with the MAGLAB setup at the Physics and Astronomy De- phase transition with a critical region of usual extent. Here partment, University of Leeds. The spin glass temperature of we report a detailed analysis of s(Q,t) around Tg in spin Tgϭ41.0Ϯ0.3 K was determined from the maximum of the glasses, based on enhanced quality data obtained by using static susceptibility. new generation NSE spectrometers. The normalized intermediate scattering function s(Q,t) of Ϫ1 For an accurate determination of the NSE spectra we Au0.86Fe0.14 at Qϭ0.4 nm is shown in Fig. 1 plotted in a chose Au0.86Fe0.14 . AuFe is a classical metallic Heisenberg log-log scale. The spectra span a dynamic range of three spin glass with significant local anisotropy15 and with strong orders of magnitude and by combining spectra collected at ferromagnetic correlations which amplify the magnetic scat- two wavelengths on IN15 and SPAN the time domain of the tering in the forward direction so improving the ratio be- observation is extended up to almost four decades ͑Fig. 2͒. tween the magnetic signal and all nonmagnetic ͑structural͒ The time dependence of the experimental s(Q,t) is impres- contributions, i.e., the signal to noise ratio. The sample was a sively similar to that of the numerical q(t) found in large polycrystalline disc 0.5 mm thick with a diameter of 37 mm scale Ising spin glass simulations, which revealed the exis- prepared by arc melting of the constituents. It was subse- tence of a phase transition in three-dimensional Ising spin quently cold worked, homogenized at 900 °C, annealed a glasses.20 From quite general scaling arguments,21 at a con- 550 °C and then quenched and kept in liquid nitrogen.16 tinuous phase transition relaxation must be of the form Ϫx Ϫz Given the vicinity to the percolation threshold xc t f ͓t/(T)͔ where (T) diverges as (TϪTg) , and f is a ͑ϳ15.5 at. % Fe͒, above which ferromagnetism sets in,17 the nonuniversal function to be determined for each system. For annealing and quenching procedure was repeated before ev- a spin glass the power law exponent x is related to the stan- ery series of measurements. The NSE data were collected dard critical exponents through xϭ(dϪ2ϩ)/2z.20 Here at the high resolution spectrometer IN15 of ILL18 at an in- is the Fisher or ‘‘anomalous dimension’’ exponent and z is coming wavelength of 0.8 nm for Qϭ0.4 and 0.8 nmϪ1, again the dynamical exponent. The Ising simulations showed respectively. These results were supplemented by measure- that, as Tg is approached from above, q(t) is strongly non- ments at the wide angle NSE spectrometer SPAN of exponential. Ogielski chose to represent f by the stretched BENSC19 at an incoming wavelength of 0.45 nm for exponential or KWW function, familiar in fragile glass dy- 0.6 nmϪ1рQр2.6 nmϪ1. We used the paramagnetic NSE namics. Excellent fits were obtained with q(t)ϰtϪx setup, which directly delivers the magnetic part of the NSE ϫexp͕͓Ϫt/(T)͔͖ and T dependent and . ͑The KWW  is signal and for this reason no background correction was re- not to be confused with the critical exponent .͒ He and quired. All NSE spectra were normalized against the resolu- others found a temperature dependent  tending to near 1/3 2,20,22,23 tion function of the spectrometers, determined with the at Tg and increasing with T. The most important point sample well below Tg , at 2 K, where the spin dynamics is for our data analysis is that precisely at Tg dynamic scaling completely frozen. A small part of the sample was taken out predicts a pure power law decay for the autocorrelation func- for dc susceptibility measurements with a commercial tion q(t)ϰtϪx. This rule is quite general; its functional form SQUID magnetometer at the HMI and for ac susceptibility does not depend on details such as the Ising character of the
054431-2 Other magnetic Hamiltonians
•Model magnetic systems (one, two and three dimensions) •Superconductivity •Giant and colossal magnetoresistance •Quantum magnetic fluctuations •Heavy fermion materials •Overdamped excitations in amorphous materials •Multiple magnon scattering •Fluctuations in Fractals and percolation theory •etc. etc. etc. Neutrons only see the components of the magnetisation perpendicular to the scattering vector