Chapter 2 Experimental methods
2.9 Neutron scattering
The following theory of neutron scattering covers the diffraction process, i.e. elas- tically scattered waves, from both crystal (nuclear) and magnetic structures. This theory is applicable to the experiment conducted at GEM (ISIS), as presented in Chapter 5.
Neutron scattering is a tool which allows both the nuclear and the magnetic structure of a material to be examined. Unlike X-rays, neutrons have no electrical charge, and so do not strongly interact with the electron cloud of an atom. The result of this is that neutrons can probe deeper into a material than X-rays, and the effect of absorption does not increase with the size of the atom. These properties are particularly useful for defining the crystal structure of a compound, since the positions of relatively light atoms (e.g. oxygen) can be determined even when alongside much heavier rare earth atoms and transition metal ions.
Importantly, neutrons have a spin value of 1/2, allowing them to interact with
the magnetic moments present in a sample. The interaction between the intrinsic magnetic moment of the neutron, µN, and the atomic magnetic field, B (which
is due to the resultant spin and orbital angular momenta of the atom [67]) is de- scribed by a potential of the form −µN·B. Neutron scattering can therefore be
used to determine the magnetic structure of a compound, as well as its crystal structure. It is typical for neutron scattering experiments to complement labo- ratory measurements of the bulk magnetic properties of samples to give a more detailed picture of the magnetic properties of a system.
A disadvantage of using neutrons is that scattering is weak - only a small frac- tion of the incident neutron beam is scattered by the nuclei in the sample. This, combined with the fact that the incident neutron flux produced by a neutron source will be orders of magnitude smaller than the photon flux from an X-ray source means that, generally, much longer counting times are required in neutron scattering experiments than in X-ray scattering experiments. Also, samples are generally required to be large for magnetic measurements using neutrons, provid- ing a practical problem for single crystal work (ideally, crystals should be of the
order of 1-5 cm3).
It is necessary to go to central facilities to conduct neutron scattering experi- ments. At the Institute Laue-Langevin (Grenoble, France) neutrons are produced by a reactor source, by the nuclear fission of uranium. At ISIS (Oxfordshire, UK), a proton beam is accelerated to high energy before it collides with a heavy metal target, causing neutrons to be produced by spallation. The neutrons at ISIS are produced in bursts rather than in the continuous flow at the Institute Laue-Langevin. Experiments are generally carried out in evacuated conditions, to reduce/remove the effect of neutrons scattering with air particles.
2.9.1 Nuclear neutron scattering
In a neutron diffraction experiment, neutrons fired upon a sample have an incident wavevector ki, and scattered wavevector kf. The scattering vector, Q, is then defined as (assuming elastic scattering)
Q=ki−kf (2.19)
The effective cross section, σ, seen by the neutron beam is related to the nuclear scattering length, b, an atom (and isotope) dependent quantity:
σ = 4πb2 (2.20)
The differential cross section, dσ
dΩ, is defined as the number of neutrons scattered
into a solid angle dΩ per second, as a fraction of the incident flux (It is assumed that the distance between the sample and detector is large compared with the dimensions of the sample and detector, such that dΩ is well defined [67]). This value represents the probability of detecting a neutron over a solid angle range, and is dependent on the values of Q and b. For a neutron beam elastically scattered by an array of n atoms, the differential cross section is given by
dσ dΩ = X n bn exp (iQ·rn) 2 (2.21) where r is the vector describing the position of the nucleus the neutron interacts with. For a crystal with lattice parameters a, b and c:
Chapter 2. Experimental methods
Scattering from a crystal occurs when the Laue condition is satisfied:
Q=G=ha∗+kb∗+lc∗ (2.23)
where a∗, b∗ and c∗ are the reciprocal space lattice parameters (a∗ = 2π/a, etc), and h, k and l are the crystal Miller indices. The differential cross section in this case is given by
dσ dΩ =N
2|F
hkl|2 (2.24)
where N is the number of unit cells in the crystal. Fhkl is the nuclear structure factor, and is given by
Fhkl =
X
n
bnexp [2πi (hxn+kyn+lzn)] (2.25) The intensity,I, of the scattered beam is related to to the nuclear structure factor [67]:
I ∝ |Fhkl|2 (2.26)
2.9.2 Magnetic neutron scattering
As mentioned above, neutrons are also scattered by the magnetic moments in a crystal. Magnetic scattering only occurs from components of the magnetisation which are perpendicular to the scattering vectorQ. The magnetic structure factor, FM, is given by
FM =
X
j
fj(Q)µj exp (iQ·rj) (2.27)
whereµj is the magnetic moment at the jth site, and fj(Q) is the magnetic form factor (as given by the dipole approximation), which gives theQ-dependence of the magnetic scattering from an atom. The magnetic scattering intensity is dependent on FM, as with the nuclear scattering case [67]:
IM ∝ |FM|2 (2.28)
If a neutron powder diffraction experiment utilises measurements of the scattered intensity as a function of the scattering vector, Q or the crystal d-spacing, any
peaks found can then be related to the magnetic or crystal structure of the com- pound. In order to distinguish between nuclear peaks and magnetic peaks, mea- surements can be taken at temperatures in the paramagnetic phase, where no magnetic order (and thus no magnetic Bragg peaks) are expected. Models of the magnetic structure can then be determined by using Rietveld refinement of the data. The analysis of results in Chapter 5 was carried out using the Fullprof suite of programs [68,69].
Chapter 2. Experimental methods