Jim Rhyne

Deputy DirectorLujan Neutron Scattering Center

Los Alamos National Laboratory

What's Cool About Neutron Scattering -- the Basics with a bias toward Magnetism

Summer Student Lecture Series June 8, 2007

LA-UR-06-4041

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Magnetic Materials and Devices (a realization of the technological potential of magnetism that has only been speculated about by others in the past)

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Magnetism Solves All Your Problems

New Physics Here!

Ref. Sharper Image, Nov. 2002

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On to Neutron Scattering PhenomenaOutline -- References

• Neutron Sources• General Concepts of Scattering• Diffractometers and Diffraction• Magnetic Diffraction• Reflectometry• Inelastic Scattering

• References:– Neutron Diffraction, G.E. Bacon, 5th edition, Oxford Press, 1975– Theory of Neutron Scattering From Condensed Matter, S.W.

Lovesey, Oxford Press 1984– Introduction to the Theory of Neutron Scattering, G.L. Squires,

Dover, 1996.– Solid State Physics, N.W. Ashcroft, N.D. Mermin, Holt, Rinehart &

Winston, 1976

What Can Neutrons Do?

• Diffraction (the momentum [direction] change of the neutron is measured)

– Atomic Structure via nuclear positions

– Magnetic Structure(neutron magnetic moment interacts with internal fields)

– Disordered systems - radial distribution functions

– Depth profile of order parameters from neutron reflectivity

– Macro-scale structures from Small Angle Scattering (1 nm to 100 nm)

• Inelastic Scattering (the momentum and energy change of the neutron is measured)

– Dispersive and non-dispersive phonon and magnon excitations

– Density of states

– Quasi-elastic scattering

Neutrons measure the space and time-dependent correlation function of atoms and spins – All the Physics!

What do we need to do neutron scattering?

• Neutron Source – produces neutrons

• Diffractometer or Spectrometer– Allows neutrons to interact with sample

– Sorts out discrete wavelengths by monochromator (reactor) or by time of flight (pulse source)

– Detectors pick up neutrons scattered from sample

• Analysis methods to determine material properties

• Brain power to interpret results

Sources of neutrons for scattering?

• Nuclear Reactor– Neutrons produced from fission of 235U

– Fission spectrum neutrons moderated to thermal energies (e.g. with D20)

– Continuous source – no time structure

– Common neutron energies -- 3.5 meV < E < 200 meV

• Proton accelerator and heavy metal target (e.g., W or U)

– Neutrons produced by spallation

– Higher energy neutrons moderated to thermal energies

– Neutrons come in pulses (e.g. 20 Hz at LANSCE)

– Wider range of incident neutron energies

Lujan Neutron Scattering Center

WNR Facility

Proton Radiography

800 MeV Proton Linear Accelerator

Isotope ProductionFacility

Proton Storage Ring

High-FluxIsotope Reactor

Spallation Neutron Source (first neutrons in

May -- operational instruments late in 2006)

(1000 kW)

Intense Pulsed Neutron Source(7 kw)

Manuel Lujan Jr. Neutron Scattering

Center(100 kW)

National User National User FacilitiesFacilitiesHFIR 1966 HFIR 1966 NCNR 1969 NCNR 1969 IPNS 1981 IPNS 1981 Lujan 1985 Lujan 1985 (SNS 2006) (SNS 2006)

Local/Regional Local/Regional FacilitiesFacilities(University (University Reactors)Reactors)MITMITMissouriMissouri……

NIST Center NIST Center for Neutron for Neutron ResearchResearch

There are four National User Facilities for neutron scattering in the US

Neutron scattering machines

• Spectrometers or diffractometers– typically live in a beam room

– are heavily shielded to keep background low and protect us

– receive neutrons from the target (or reactor)

– correlate data with specific neutron wavelengths by time of flight

– accommodate sample environments (high/low temperature, magnetic fields, pressure apparatus)

Neutron Scattering’s Moment in the Limelight

Source

Restelli

What is neutron scattering all about?

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General Properties of the Neutron The kinetic energy of a 1.8 Å neutron is equivalent to T = 293K

(warm coffee!), so it is called a thermal neutron. The relationships between wavelength (Å) and the energy (meV),

and the speed (m/s, mi/hr) of the neutron are:

e.g. the 1.8 Å neutron has E = 25.3 meV and v = 2200 m/s = 4900 mi/hr

The wavelength if of the same order as the atomic separation so interference occurs between waves scattered by neighboring atoms (diffraction).

Also, the energy is of same order as that of lattice vibrations (phonons) or magnetic excitations (magnons) and thus creation of annihilation of a lattice wave produces a measurable shift in neutron energy (inelastic scattering).

/3960 and /89.81 2 vE

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COMPARATIVE PROPERTIES OF X-RAY AND NEUTRON SCATTERING

Property X-Rays Neutrons

Wavelength Characteristic line spectra such as Cu K

= 1.54 Å

Continuous wavelength band, or single = 1.1 0.05 Å separated out from Maxwell spectrum by crystal monochromator or chopper

Energy for = 1 Å 1018 h 1013 h (same order as energy of elementary excitations)

Nature of scattering by atoms

ElectronicForm factor dependence on [sin]/Linear increase of scattering amplitude with atomic number, calculable from known electronic configurations

Nuclear, Isotropic, no angular dependent factor Irregular variation with atomic number. Dependent on nuclear structure and only determined empirically by experiment

Magnetic Scattering Very weak additional scattering ( 10-5) Additional scattering by atoms with magnetic moments (same magnitude as nuclear scattering) Amplitude of scattering falls off with increasing [sin ]/

Absorption coefficient

Very large, true absorption much larger than scattering abs 102 - 103

increases with atomic number

Absorption usually very small (exceptions Gd, Cd, B …) and less than scattering abs 10-1

Method of Detection Solid State Detector, Image Plate Proportional 3He counter

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Golden Rule of Neutron Scattering

We don’t take pictures of atoms!

Job preservation for neutron scatterers – we live in reciprocal space

Atoms in fcc crystal

Inte

nsit

y

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How are neutrons scattered by atoms (nuclei)?

Short-range scattering potential:

The quantity “b” (or f) is the strength of the potential and is called the scattering length – depends on isotopic composition

Thus “b” varies over N nuclei – can find average defines coherent scattering amplitude leads to

diffraction – turns on only at Bragg peaks But what about deviations from average? This defines the

incoherent scattering

Incoherent scattering doesn’t depend on Bragg diffrac. condition, thus has no angular dependence – leads to background (e.g., H)

bbcoh b

)(2

)(2

rbm

rV

2/122 bbbinc

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Scattering of neutrons by nuclei

A single isolated nucleus will scatter neutrons with an intensity (isotropic)– I = I0 [4b2]

where I0 = incident neutron intensity, b = scattering amplitude for nucleus

What happens when we put nucleus (atom) in lattice?– Scattering from N neuclei can add up because they are on a lattice– Adding is controlled by phase relationship between waves scattered

from different lattice planes– Intensity is no longer isotropic Bragg law gives directional dependence

– Intensity I (Q, or ) is given by a scattering cross-section or scattering function

sin2d

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Observed Coherent Scattering Intensity of diffracted x-ray or neutron beam produces series of peaks at discrete

values of 2 [or d or K (also Q)]Note: d = /(2 sin) or K = 4sin/ = 2/d are more fundamental since values are independent of and thus characteristic only of material.

Benzine Pattern (partial) Note: Inversion of scales - 2 f(1/d)

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Scattering Factors f, cont’d For x-rays the magntude of f is proportional to Z For neutrons nuclear factors determine f, thus no regular with Z (different

isotopes can have different f s)

Shaded (negative) --> phase changeFor neutrons conventionally f = b (Scattering length - constant for an element)

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Magnetic Powder Diffraction

Neutron has a magnetic moment -- will interact with any magnetic fields within a solid, e.g., exchange field

Magnetic scattering amplitude for an atom (equivalent to b)

where g = Lande “g” factor, J = total spin angular momentum, f = magnetic electrons form factor

Magnetic scattering comes from polarizedspins (e.g., 3d [Fe] or 4f [RE]) not fromnucleus -- Therefore scattering amplitudeis Q-dependent (like for x-rays) via f

at Q = 0 for Fe = gJ = 2.2 Bohr magnetonsp = 0.6 (comparable to nuclear b = 0.954)all in units of 10-12cm

Refinement gives moment magnitudes oneach site and x,y,z components(if symmetry permits)

)(10269.02

122

2

cmfgJxfgJmc

ep

Mn+2

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Form Factors

Experimental Calculated

LessLocalizedMoment

MoreLocalizedMoment

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Magnetic Powder Diffraction II In diffraction with unpolarized neutrons (polarized

scattering is a separate topic) the nuclear and magnetic cross sections are independent and additive:

q2 is a “switch” reflecting fact that only the component of the magnetic moment scattering vector K (or Q) contributes to the scattering

magnnuclmagnnucl SSSqSQSd

d 22 cos1)(

K

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Basic Types of Magnetic Order and Resulting Scattering

Ferromagnet (parallel spins)– Single Magnetic site (e.g., Fe, Co,

Tb)

– Scattering only at Bragg peak positions (adds to nuclear), but not necessarily all (q2 switch)

– Multi Site Ferromagnet

(e.g. Y6Fe23 (4 distinct Fe sites) -- no new peaks in scattering

Antiferromagnet (parallel spins with alternate sites reversed in direction)

– equivalent to new magnetic unit cell doubled in propagation direction of AFM

– Purely magnetic scattering peaks at half Miller index positions (e.g., 1,1,1/2)

– Overall net magnetic moment adds to 0 [job security for neutrons!!]

c

a

Polarized Neutron Reflectometry

Detector

SampleAl-Coil Spin Flipper

Spin Polarizing Supermirror

Specular Reflectivity

Incident Polarized Neutrons

Al-Coil Spin Flipper

]n1)[4/k()z( 22i

22 dz)ikzexp()z()z(

iQ

4)Q(r

cospN2

1n

sinpbN2

1n

2

2

• index of refraction: sensitive to

• scattering length density: used to model reflectivity

• reflectivity: measured quantity

spin-flip

non spin-flip

Ga1-xMnxAs

• Dilute ferromagnetic semiconductor

• Spintronics applications

• Annealing increases magnetization & Tc

• Interstitial Mn go to the surface! – K. W. Edmonds et al., PRL, 92, 37201, (2004) - Auger

• Depth-dependence of chemical order and magnetization determined Polarized-Beam Neutron Reflectivity

• Compared similar as-grown and annealed films

– T = 13 K, H = 1 kOe (in plane)

J. Blinowski et al, Phys. Rev. B 67, 121204 (2003)

Ga1-xMnxAs As-Grown & Annealedt = 110 nm, x = 0.08, TC = 50 K, 120 K

0.01 0.02 0.03 0.04 0.0510-10

10-9

10-8

Q (Å-1)

Re

flect

ivity

x Q

4 (Å

-4)

As-Grown Film measured R

++ fit to R

++

measured R- -

fit to R- -

Annealed Film measured R

++ fit to R

++

measured R- -

fit to R- -

0.01 0.02 0.03 0.04 0.05

-0.1

0.0

0.1

0.2

Annealed Film measured SA fit from SLD model

Q (Å-1)

Spi

n A

sym

met

ry

0.01 0.02 0.03 0.04 0.05

-0.1

0.0

0.1

0.2

Q (Å-1)

As-Grown Film measured SA fit from SLD model

Sp

in A

sym

met

ry

0 200 400 600 800 1000 12000

2

4

6

8

290

300

310

320

As-Grown Film

Depth (Å)

Mavg = 17 emu cm-3

moment per Mn = 1.2 B

m

-2

nuc

mag

0

10

20

30

substratesurface

M (e

mu

cm-3)

0 200 400 600 800 1000 12000

5

10

15

290

300

310

320

nuc

mag

Mavg = 48 emu cm-3

moment per Mn = 3.4 B

(m

-2)

substratesurface Depth (Å)

Annealed Film

0

20

40

60

M (em

u cm

-3)

• Measured reflectivities & fits– Spin up & spin down splitting due to

sample magnetization– Spin up reflectivities are different– “Slope” at high Q different – Fits are good

• Magnetic signal: spin asymmetry– SA = (up – down) / (up + down)– Larger amplitude for annealed film– Better defined for annealed film

• SLD Models (mag. & chem.)– As-grown M doubles near surface – M increases and more uniform for

annealed film– Both films show magnetic depletion at

surface– Drastic chemical change at annealed

film’s surface– Interstitial Mn have diffused to

surface! (combined with N2 during annealing)

Inelastic Scattering

• Inelastic Scattering (the momentum and energy change of the neutron is measured)

– Dispersive and non-dispersive phonon and magnon excitations

– Density of states

– Quasi-elastic scattering

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Triple Axis Neutron Scattering Spectrometer

Want Thermal neutronse.g., E=14mev, =2.4Å

i = 2dmsin m |ki| = 2/i

n

ii m

kE

2

22

f = 2dasina

|kf| = 2/f

dEd

dcountsI

m

kE

n

ff

2det

22

)(

2

kTEE e

kT

E

kT

nn /

2

1

02

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MURR Triple Axis Neutron Spectrometer (TRIAX)

Analyzer Assembly

Monochromator Drum

Sample Table andGoniometer

Detector Shieldand Collimator

Beam Stop (pivots withdrum and sample)

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22

nf2

2

ni 2E

2E fi k

mk

m

2

222

ifN kk

mE

Qqkk if

2

P

0

i f

Inelastic Neutron Scattering ***

The measurement of the functional dependence of wave vector q and the energy E = h of an elementary excitation (e.g., magnon or phonon dispersion) Energy and Momentum Conservation Conditions: * ki = wave vector of incident neutron * kf = wave vector of scattered neutron

* Neutron energies

Change of EN creates (EN < 0) or annihilates (EN > 0) an excitation of energy

Neutron - excitation system

momentum conservation:

Constant q scan (Einc fixed)

Rotate P [kf,and ki] about 0

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P h y s i c a l P h e n o m e n a S t u d i e d w i t h I n e l a s t i c N e u t r o n S c a t t e r i n g * * * a c c e s s i b l e e n e r g y a n d m o m e n t u m

t r a n s f e r ( Q ) r a n g e s * * *

N o t e : C o l d a n d h o t s o u r c e s a t R e a c t o r s a n d p a r t i c u l a r l y S p a l l a t i o n N e u t r o n S o u r c e s h a v e g r e a t l y e x t e n d e d t h e u s e f u l r a n g e o f Q a n d E .

[ F r o m J . D . A x e , N e u t r o n s : T h e K i n d e r , G e n t l e r P r o b e o f C o n d e n s e d M a t t e r , M a t l s . R e s . S y m p . P r o c . 1 6 6 , 3 ( 1 9 9 0 ) ] .

310 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5

0

2 0

4 0

6 0

8 0

1 0 0

1 2 0

[ q q q ]

[ q q 0 ]

[ q 0 0 ]

En

erg

y (m

eV

)

q ( r l u )

D i s p e r s i o n i n a H e i s e n b e r g F e r r o m a g n e t

A t l o w q d i s p e r s i o n i s q u a d r a t i c ( t o l o w e s t o r d e r ) :

E ( q ) = ( q ) = + D q 2

D i s t h e s p i n w a v e s t i f f n e s s p a r a m e t e r ( m a y b e a n i s o t r o p i c i n q d i r e c t i o n

i s t h e g a p e n e r g y ( m a y b e 0 - - r e f l e c t s c r y s t a l f i e l d a n i s o t r o p y e n e r g y )

F u l l B r i l l o u i n z o n e d i s p e r s i o n f o r a s i n g l e n e a r e s t n e i g h b o r e x c h a n g e c o n s t a n t J 1 i s o f t h e f o r m :

q ) = + 4 J 1 S ( 3 – c o s q x a o – c o s q y a o – c o s q z a o )

w h i c h r e d u c e s a t s m a l l q ( c o s x 1 - x 2 / 2 ) t o ( 1 1 1 ) :

q 1 1 1 ) 6 J 1 S a 2 q 2 D q 2

F u l l z o n e d i s p e r s i o n f o r a f c c l a t t i c e

J 1 T c

I n m e a n f i e l d a p p r o x :

T c = 4 J 1 S ( S + 1 ) / k B

32

Exam ples of Inelastic Scattering D ata

Ferrom agnetic A m orphous m etallic glass Fe0.86B 0.14 Constant q scans left panel -

varying T right panel -

varying q) [Rhyne, Fish, and Lynn,

JAP 53 , 2316 (1982)]

Q uadratic dispersion (E

vs. q2) [J.A . Fernandez-Baca, J.W . Lynn, J.J.

Rhyne, and G .E. Fish, Physics B 136B, 53 (1986)

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Using Powder DiffractionInput Information -- Structure Determination

Know instrument-dependent scattering line-shape– Gaussian for fixed

– Convolution of rising and falling exponentials with Gaussian for TOF

– Sample distortions (pseudo Voigt) linear comb. of Lorentzian and Gaussian

Know or parameterize resolution and background functions

Know Space Group (or a limited # choices) [coordinates of atoms in cell - may be variables x,y,z]

2.00 2.040

1000

2000

3000

4000

5000

Gaussian

Inte

nsity

Q (Å-1)0.45 0.50 0.55

Lorentzian

Gaussian

2

22ln4

2

2ln4)(

oQQ

eQR

2

o

'

)Q-2(Q1

1 '

2)(

QR

)()()( zerfceyerfceNTR vu (VonDreele, Jorgensen & Windsor)