# Presentation: 478255

## Inelastic Neutron Scattering and Applications

*A Short Course on Neutron Scattering in Earth Sciences*
*Dec 7-8, 2006*
*Hilton Garden Inn, Emeryville, CA*

*Chun Loong, IPNS*

## An Outline

- The Technique of Inelastic Neutron Scattering (INS)
- Double differential cross section
- Instruments: Triple-axis vs. TOF chopper spectrometers
- Applications of INS in Earth Sciences
- Lattice dynamics: phonon dispersion & density of states
- Magnetic scattering: rare-earth energy levels structures
- Examples:
- Xenotime (RPO4)
- Spinels
- Nanostructured bone minerals (hydroxyapatite)

## What is Inelastic Neutron Scattering (INS)?  Scattering processes which involve energy and momentum exchange between the neutron & the scatterer

**Notes:**
Inelastic neutron scattering (INS) in general refers to scattering processes which involve energy and momentum exchange between the neutron and the scatterer. It is widely utilized for characterization of materials in basic and applied research across many disciplines.
The energy and momentum transfer for a neutron which has undergone inelastic scattering are defined in the yellow text box.
E is the energy, Q and k are the wavevectors (the product of the Planck’s constant and Q is the momentum transfer). Subscribes 0 and 1 denote quantities pertaining to the incident and scattered neutrons, respectively. Positive and negative energy transfer E corresponds to, respectively, scattering with neutron energy loss and neutron energy gain, or, by analogy of Raman scattering, the Stokes and anti-Stokes lines. However, in case of light scattering, the wavevector  is essentially zero hence Raman and IR spectroscopy measure only the Brillouin-zone-center modes of crystallite solids, where neutrons probe wave vectors spanning the entire Brillouin zone. 
The goal of INS is to measure precisely the double differential cross section defined in the blue textbox, where sigma, OMEGA, N, ms, are the scattering cross-section, solid angle, number of scattering units in the scatterer, and neutron mass, respectively.  Xi(0) and xi(1) denote the state of the scatterer before and after scattering, respectively. V is the scattering potential. In principle, the neutron may also undergo a change of its spin state after scattering but for simplicity we do not consider the scattering of a polarized neutron beam here. INS effectively aims at the determination of the scattering function S(Q,E) that contains the matrix element <.|.|.> characterized by the nature of many-body interactions within the scatterer. For example, it can be shown that the scattering function, depending on coherent or incoherent scattering, is related to respectively the inter-particle or self-particle space-time correlation functions of the scatterer under study. According to linear response theory, they are also linked to the Green functions and the dynamic susceptibilities of the scatterer, thereby enabling a direct connection between experimental result and theoretical prediction. 

## What Does an INS Instrument Do? Performs Accurate Measurements of S(Q,E) in Absolute Units

- In order to differentiate the net energy change for a scattering event, an ***energy filter***, which selects neutrons with a narrow distribution of energies and/or spins over a collimated solid angle, has to be inserted in the incident or scattered beam, sometimes in both places.

- For fixed incident energy + variable scattered energies: ***direct geometry***
- For variable incident energy + fixed scattered energy: ***inverse geometry***

**Notes:**
The goal of an INS instrument is to quantitatively measure the scattering events, in terms of counts of the scattered neutrons as a function of energy recorded by well-defined detectors (with known pixel resolution and position, efficiency , etc.), from which a scattering function,  pertinent to the sample in the environment under study, can be accurately constructed. 
In this figure, the sample, whose orientation defined, for example, by its crystal orientation or magnetic quantization axis relative to the neutron beam, is seen to scatter the incident neutrons to a detector, characterized by a pixelized solid angle delta-Omega and neutron-detection efficiency eta, at a scattering angle .

## How to Define the Energy of a Neutron Beam? Crystals & Choppers

**Notes:**
The mechanism for neutron beam definition employed by an energy filter varies from transmission, diffraction, reflection, and refraction in materials as well as other means of neutron optics including manipulation of the neutron spins. Two most important energy filters, as illustrated in this Figure, are crystal monochromators (M) for incident neutrons or analyzers (A) for scattered neutrons and choppers. In the case of chopper utilization, the transmitted beam yields narrow bursts of neutrons whose speeds (hence energies) are subsequently determined by the flight time over a known distance, i.e., by the time-of-flight (TOF) method that is particularly suitable to spallation sources owing to the inherent time structure of the neutron fluxes. The energy resolution, E/E, is governed, among many factors, mainly by the crystal Bragg angle, X, X=M or A, for the crystal case and by the rotational speed and slit-package configuration for the chopper case. From the Bragg law, we find ll ~ cotX. This fact accounts for the requirement of high monochromator ‘kick-off’ angle or nearly ‘backscattering’ geometry for high- resolution instruments. For high-resolution choppers, a rotation speed of 36,000 rotations per minute is routinely applied. Additionally, TOF instruments must deal with the effects of uncertainties and broadening (due to the finite size of various components) in flight times and distances on the resolutions. The Q-resolution, often coupled with that of energy, is more complicated. Time-focusing is an important aspect in designing pulsed-source instruments.

## How INS Instruments Work? Triple-Axis Spectrometers

**Notes:**
The triple-axis spectrometer, pioneered by B. N. Brockhouse, provides ingenious flexibility in exploring the reciprocal lattice space, it is used mainly for studying elemental excitations (e.g., phonons and magnons) and dynamics associated with phase transitions in solid-state physics, see the above Figures. Full polarization operation and analysis of the incident and scattered neutrons are readily achievable. It operates in a steady-state mode and normally ignores any time structure of the neutron beam, see for example . Thus triple-axis spectrometers are exclusively reactor instruments. Because it collects data point-by-point through the reciprocal space using a single or only a handful of detectors, data rate is relatively low. A reactor often is equipped with several triple-axis spectrometers in order to cover the cold and thermal neutron regimes.

## How Does a Triple-Axis Spectrometers Work? 

- Operation Modes:
- Fixed *E**0* or *E**1*
- May keep ***Q*** or *E *constant

***How Does a ******Triple-Axis Spectrometers****** Work? ***

**Notes:**
The three axes are rotation axes for the monochromator, sample, and analyzer. Usually Soller collimators are inserted in various segments of the incident and scattered beams in order to achieve better energy resolution. Notice that the vector equation of Q=K0-k1 defines the so-called ‘scattering triangle’. Because the wavevector also defines the neutron energy, the condition of conservation of energy AND momentum in general cannot be met for all the Q, E values. In other words, a scattering process has to satisfy the kinematic conditions. 
When a single-crystal specimen is mounted on the sample table, its orientation with respect to the incident and scattering beams defines a lattice zone on the scattering plane, as shown by the lattice points in the figure. The detector measures the scattering events at Q, which is the same of a lattice reciprocal vector and a reduced q-vector within the Brillouin zone boundaries. 
A triple-axis spectrometer allows measurements of scattering events along a q-locus strategically chosen by the experimenter. The operation can fix either the incident neutron energy (at a fixed monochromator angle) or the scattered (final) energy (fixed analyzer angle).

## How to Measure Phonons Using a Triple-Axis Spectrometer?

***How to Measure Phonons Using a ******Triple-Axis Spectrometer?***

**Notes:**
Note that the phonon dynamic structure factor, F, contains the scalar product of Q and the phonon polarization vector e. The Figure above shows the two different scans optimized for the measurement of the longitudinal and transverse phonons. During a constant-Q scan, the detector, represented by a resolution function, passes through a phonon dispersion curve. A the intersection, a peak called a phonon group is recorded, whose center defines the energy of the phonon at q.

Note also the square of the F is proportional to Q squared. Therefore, it is desirable to measure a phonon at a large Q vector in order to maximize intensity. The phonon dynamic structure factor shows complex dependency on the eigenvectors of phonon normal modes, thereby providing a means to test the correctness of a phonon model for the material under study. 

## How INS Instruments Work? Chopper Spectrometers

***How INS Instruments Work? ******Chopper Spectrometers***

**Notes:**
A chopper spectrometer employs a chopper, a rotating neutron collimator introduced by Fermi, to select a monochromatic incident neutron beam. The transmitted intensity profile is a trapezoid of narrow energy band whose value depends on the location and the opening time of the chopper window relative to the source pulse. 

## How Does a Chopper Spectrometers Work? 

***How Does a ******Chopper Spectrometers****** Work? ***

**Notes:**
A chopper spectrometer admits monochromatic bursts of neutrons incident on a sample and subsequently allows analysis of the scattered energy by TOF methods. For reactors, a first chopper has to be employed to ‘chop’ the steady-state beam into polychromatic pulses, then subsequently monochromized by a second chopper at a downstream position. For pulsed sources, pulsation is inherently built in by the source so only one chopper is in principle needed. Furthermore, a pulsed-source chopper spectrometer enables the coverage of a very wide dynamic range from cold to thermal to epithermal neutrons on the same instrument without interrupting the sample environment, see for example Loong et al. 1987.

## How to Measure Phonons/Magnons Using a Chopper Spectrometer?

***How to Measure Phonons/Magnons Using a ******Chopper Spectrometer?***

**Notes:**
Chopper spectrometers are equipped with large detector banks, often with position-sensitive detectors, subtending a large solid angle. Thus voluminous data are collected in one setting, surveying a sizeable region of the (Q, E) space but sorting out data along specific Q directions is difficult because the fixed-detector configuration limits the flexibility in assessing Q. Therefore, traditionally the majority of samples used are polycrystalline or disordered materials. Nowadays, however, a state-of-the-art pulsed-source chopper spectrometer such as the MAP instrument of ISIS is also capable of conducting single-crystal experiments for studying collective excitations, much like the triple-axis spectrometers at reactors.

## Applications of INS: 1. Lattice Dynamics of MineralsPhonon Dispersion Relations

## Phonon Density of States (DOS), g()

***Phonon Density of States****** (DOS), g(************)***

## Phonon DOS & Thermodynamic Properties

## Phonons & Mechanical Properties: The Continuum Limit

## Important References

- Bibliography & Compilation of Phonon Spectra
- _Scattering of Thermal Neutrons, A Bibliography (1932-1974)_, Complied by A. Larose and J. Vanderwal (Plenum, New York 1974).
- H. Bilz and W. Kress, _Phonon Dispersion Relations in Insulators_ (Springer-Verlag, Berlin, 1979).

## Prerequisite for Phonon Experiments Using INS

***Prerequisite for Phonon Experiments Using INS***

- Prepare your samples: single crystals (the larger the better) for phonon dispersion and/or polycrystalline sample (the purer the better) for DOS measurements

## Monazite and Xenotime: Rare-Earth Orthophosphates RPO4

- J. C. Nipko et al., *Phys. Rev.* **B 56 **(18), 11584-11592 (1997), & C.-K. Loong et al, *Phys. Rev.* **B 60 **(18), R12549-R12552 (1999).

***Chemistry***
- High melting points (>2000°C)
- Not attacked by water, organic solvents and common acids
- Resistant to radiation damage
- High-temperature components, Medium for nuclear waste storage
***Optics***
- High density, Mohr hardness ~5.5
- Rare-earth activated luminescence
- Phosphors, Lasers, Scintillators
***Magnetism***
- Antiferromagnetic phase transitions
- Cooperative Jahn-Teller effects
- Magnetoelastic effects
- Rare-earth spin-lattice coupling
- Magnetic refrigerants, Sensors

## Zircon-type Structure of Nonmagnetic LuPO4 Xenotime

- Body-centered tetragonal structure *I4**1**/amd*     Z=2

- The Brillouin Zone

**36 phonon branches along each direction**

## LuPO4 Lattice Dynamics: Group Theoretical Analysis

## Single-Crystal, Triple-Axis Measurements of Phonon Dispersion Curves

***Single-Crystal, Triple-Axis Measurements of ******Phonon Dispersion****** Curves***

- LuPO4 Room Temperature

## Polycrystalline, Chopper Spectrometer TOF Measurements of Phonon DOS

## Magnetic INS: The Ground-State Wavefunction of a Magnetic Ion

***Magnetic INS******: The Ground-State Wavefunction of a Magnetic Ion***

***Magnetic Scattering***

- Dipole Approximation of Crystal-Field Transitions of Non-interacting Rare-Earth Ions in a Crystalline Host

**Notes:**
The neutron carries an intrinsic magnetic moment of -1.91 nuclear magneton, which can interact with the magnetic electrons, e.g., the incompletely filled d and f orbitals of transition metal compounds. A simple example is magnetic INS of neutrons from a lattice of non-interacting rare-earth ions. Here the ground-state multiplet of the rare-earth ion is split by the crystalline electric field (crystal field) according to the point symmetry of the rare-earth site. Neutrons can be scattered elastically within a degenerate state are be excited or de-excited by absorbing or emitting a crystal-field transition, respectively. 
The reader should note that the scattering functions of magnetic scattering include the square of the magnetic form factor, f(Q), as a multiplying factor. The form factor arises from the fact that the magnetic orbital within an atom in general has a finite size, of which the Fourier transform yields a form factor peaking at small Q and vanishing at high Q values. Consequently, magnetic scattering intensity occurs mainly at low Q or small scattering angles. Therefore, studying the Q-dependence of the scattered intensity provides a means, albeit qualitative, to differentiate magnetic scattering from nuclear scattering. 

## Crystal-Field Excitation Spectra of TbPO4

**Notes:**
TbPO4 exhibits a Jahn-Teller phase transformation from tetragonal to monoclinic structure at 2.3K, which also induces an antiferromagnetic transition at 2.2 K. Above these temperatures, TbPO4 is paramagnetic. The magnetic INS of TbPO4 in the paramagnetic state reveals shape peaks corresponding to crystal-field transition of the Tb3+ ions, as shown above. At the temperature approaches to the low-temperature phase tansition, e.g., <4K, the crystal-field peaks broaden due to magnetic interaction between the Tb ions. 
Analysis of the neutron data using a crystal-field model permits the determination of the energy level structure (Right) and the wavefunctions of each crystal-field state in terms of a handful of crystal-field parameters.

## Crystal-Field Level Structure of TbPO4: The Magnetic Properties

- The magnetic INS measurement enables a characterization of the rare-earth ground- and excited states wavefunctions in terms of a handful of ***crystal-field parameters***. The model can then be applied for calculations of the magnetic properties of the material, e.g., susceptibility and magnetic specific heat.

**Notes:**
The f-electrons of a rare-earth ion in a crystal are influenced by the crystalline electric fields of the surrounding ions, resulted in lifting the degeneracy of the Hund’s-Rule ground multiplet into a level structure governed by the site symmetry. At temperatures comparable to the crystal-field (CF) splitting, all the levels are populated and the magnetic properties are relatively isotropic and weakly interactive. At low enough temperatures for the occupation of only the lowest states, the magnetization may be highly anisotropic thereby favoring strong coupling with the lattice and/or other rare-earth spins. In some cases, cooperative structural changes such as the Jahn-Teller transition or long-range magnetic ordering may occur. Therefore, it is paramount to characterize the rare-earth CF level structure for a better understanding of the magnetic and optical behavior of the materials. INS, in which neutrons are scattered via the interaction between the neutron spins and the magnetic moments of the rare-earth ions – neutron magnetic scattering -- is the best method to fulfill such task.

## Anomalous 4f-Electron Phonon Interaction in YbPO4 

- The coupling of the Yb3+ crystal-field states with the *E**g* phonon is very strong, with strengths much larger than those of any previously reported systems such as CeAl2, LnF3, LiTbF4 and Ln(OH)3.  The line widths change drastically with temperature.

## Dynamic Coupling of Crystal-Field and Phonon States in YbPO4

- The data suggest a large fluctuating component associated with the monopole term whereby coupling of the crystal field states, particularly the upper 6 and 7 doublets, with phonons of comparable strengths and energies.  The coupling to the monopole term does not require the compatible symmetry of specific phonon modes (such as in the case for CeAl2), and was observed throughout the Brillouin zone as long as the phonon energies and CF transition strengths are comparable.

## Spinels: From Gahnite (ZnAl2O4) to Nanostructured Li(H)Mn2O4 Adsorbent

- Koyanaka, Takeuchi, Loong, *Separ. Purif. Tech.* **43**, 9 (2005).
- Loong & Koyanaka, *J. Neutron Res.* **13**, 15 (2005).

- The classic spinel structure: (e.g., MgAl2O4) Cubic *Fd3m*, Z=2;  42 phonon branches

## Where are the Hydrogen Atoms in HMn2O4?

## Bone Minerals: Nanotechnology in Our Body

***Bone Minerals: Nanotechnology in Our Body***

***Apatite?***

## Evidence for the Lack of OH- Ions in Bone Crystals as Compared to Hydroxyapatite (HAp) Ca10(PO4)6(OH)2

## Concluding Remarks

- INS is capable of accounting for the detailed atomic/molecular motions and electronic/magnetic excitations -- individual or collective -- within a many-body system (e.g., minerals).
- Since microscopic motions or excitations may occur in vastly different time and length scales, typically ps to ms and sub-nm to m, the technique of INS necessitates an as wide as possible coverage in the energy (*E*) and wavevector (*Q*) space with good resolutions. *In situ* measurements under extreme sample environments (e.g., high T, high P) are highly desirable.
- INS is often flux-limited because of the weak intensity but this situation will be improved in the advent of new-generation high-flux neutron sources such as SNS.
- Interpretation of INS data can be a challenge facing experimentalists. Researchers nowadays have to apply methods of theoretical modeling and simulations that require high degree of sophistication and substantial amount of computing resources.

- Don’t worry, instrument scientists at neutron facilities will be glad to help you.