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Neutron Scattering EG0600179 2nd Conference on Nuclear and Particle Physics, 13 -17 Nov. 1999, Cairo, Egypt INSTRUMENTS FOR NEUTRON SCATTERING M.Adib Reactor Physics Department, NRC, AEA, Egypt The wide variety of science that can be presented with neutron scattering essentially boils down to determine two vectors, the momentum k of the neutron before it hits the sample and after it leaves the sample. All experimentally obtainable information is contained in the probability distribution W(k,k') of a neutron to undergo scattering takes it from k to k'. In this paper we will introduce the principles and concepts to understand what one is doing if one perform an experiment on a certain instrument We will describe the components of which almost all neutron scattering instruments are made up and their functions and show how these components can be combined to contribute in the best possible way to the solution of questions in a large number of scientific areas. 1 Introduction For the purpose of discussing the basic features of a neutron scattering instrument, we will use the representation shown in Fig.(l) PS ss PO' PO" SD DP PS = primary source 55 = spectrum shifter PO = (selective) phase space operator SE = Sample environment S = Sample SD - Signal detector DP - Data processing Figure 1: representation of a neutron scattering experiment. 109 Neutrons when released from nuclei in the primary source have an energy spectrum that cannot be used for neutron scattering in condensed matter science. The spectrum must be shifted to the desired range by collisions with suitable moderating material. The combination of primary source and spectrum shifter is usually called "neutron-source" All neutron sources radiate more or less isotropically in real space and extend over a certain range in momentum space. In order to select the desired momentum vector k to impinge on the sample, a phase space operation (limitation of direction, magnitude and uncertainty) is needed to define k. After the scattering event, another phase space operation is required to define k' weutron detectors are not sensitive to the energy of the neutron!). With a suitable detector the probability of the transition from k to k' is recorded and the data must be analyzed to obtain the desired information. In this paper we will introduce the principles and concepts necessary to understand what one is doing if one performs an experiment on a certain instrument. We will describe the components of which almost all neutron scattering instruments are made up and their functions and show how these components can be combined to contribute in the best possible way to the soluti6n of questions in a large number of scientific areas. 2. Fundamentals of Experimental Neutron Scattering 2.1 The neutron scattering experiment The neutron has a mass m = 1.675.10'24 g. At thermal energies it moves with a velocity V of the order of km/sec, its momentum is rj = m.v = tik 2 2 P m 2 ti k its energy is E = — v = 2 1m with the wave vector k= —.— " * M and the de Broglie wave-length X of the order of A. 110 Upon scattering the neutron is transferred from an incident wave vector k to a final wave vector &_'. As said before, the experiment aims at determining the probability distribution for. this transition, w( k, k[) by recording the count rate in a detection system and restricting the uncertainty in k and k' within the necessary limits. The analysis determines the momentum transfer Q = k-k^ (2.1) h2k2 h2k'2 energy transfer A£ = = ha> (2.2) 2m 2m and deduces the scattering law S(Q,a>) = —W(k,k). (2.3) h It then remains to explain the scattering law in terms of the physics of the sample and to assess the precision of the data. 2.2 The scattering triangle The vector sum Q = Jc- k' can be represented as a triangle as shown in Fig.2. Fig.2 the scattering triangle Q = k_- kl for inelastic scattering with energy transfer hco 111 h.co k-i 1 + - + P - 2 2m (2.4) -1 + -w + P 2m which can be satisfied by an infinite number of combinations. The 2-dimensional vector P which is perpendicular to Q(PQ- 0) results from the transition from the 6-dimensional k, k[-space to the 4 dimensional Q, 0) -space. 2.3 Scattering from a Crystal 2.3.1 Elastic Bragg scattering Drawing k to be incident on the origin of the reciprocal crystal lattice, the locus for cell elastically scattering vectors |AT| = |A:'| is a sphere around the origin of k with radius k Bragg scattering occurs at points Ghki =h . bi + k .b2 +1. b3 is located on this sphere. Origin of r»c)proco) lattice 00 Intersection o Ewald sphen* with scattering plane Figure 3: Ewald's construction for elastic Bragg scattering in the reciprocal lattice 112 2.3.2 Inelastic "phonon" scattering It is one of the outstanding features of thermal neutrons that their kinetic energy is in the regime of the thermal vibration energies of a crystal (not surprising, since they are obtained by thermal equilibrium with a moderator). So, energy can be transferred to or from the lattice in the scattering event, leading to a large relative change in the neutron's energy. This goes along with an additional momentum transfer q within the Brilloum zone, as shown in Fig. 4. 000 Fig (4): Vector diagram for inelastic scattering with excitation wave vector 9. Q = a +Ghki 2.4 Resolution Since it is the goal of the measurement to determine the positions in Q, co-space with the maximum possible precision, it would be desirable to have k and k' defined as precisely as possible. Apart from this being impossible due to the way in which available phase space operators work , this would also result in an almost complete loss of intensity and hence poor counting statistics. It is therefore necessary, to retain a finite, but optimized volume 8k and dk' around k and k\ As can be seen from Fig.5 this results in an uncertainty of the value of Q, 8Q and, of course, also of co, 8co. 113 0,0.0 Fig.5 : Effect of uncertainties of k and &_' on Q The distribution function of 5Q and 8co is called the resolution function Scoj of the experiment (at < Q >, < co >). The count rate z in the measurement is then proportional to the convolution of this resolution function with the scattering law Z(<Q co> Figs.6 and 7 depict schematically, how the resolution function is moving across the scattering law in a measurement scan and how the width of the measured intensity depends on the shape and relative orientation of the two (3-dimensional) surfaces. Since Q = 2 n /R and co = 2 n /t where R is a distance in real space and t is the time, integrating over a certain range of Q or co, as one does within the resolution of the instrument, means an average over the corresponding real space-time range. In other words, phenomena that occur at shorter distances or times than the averaging interval cannot be resolved. At the same time it becomes obvious that, if an experiment is performed without energy analysis this means integration over all co or measuring the properties of the sample at t = 0 i.e., taking a "snapshot" of the sample. This would be strictly true under the condition that the integration is performed for constant Q, which is not the case in neutron scattering when measuring at fixed angles due to the kinematic conditions. On the other hand, if a measurement is made with energy analysis for co = 0, this amounts to taking a long time average, i.e. measuring the static properties or, for example, mean positions of the scattering centres. 114 scattering law Fig.(6) Scanning a wide resolution function Fig.(7) Intensity distributions for across a narrow scattering law essentially different scans with the same resolution reproduces the resolution function function Knowing the resolution of an experiment is therefore important in order to be able to interpret the measured data correctly. The shape of the resolution function is a result of the phase space operations performed on the vector fields {k} and {_£_'}. 2.5 Data analysis Since a probability distribution is measured in a scattering experiment, the statistical accuracy of a count rate N is -J~N . Hence there is no absolutely precise measurement. The precision can be increased by increasing the number of counts or by using neighboring points of a scan by ways of a curve fit to the points. Even if the resolution function of the measurement is known, it is normally not possible to deconvolute the measured curves to obtain the scattering law. A frequent practical procedure is to start from the best assumption about the scattering law, convolute it with the resolution function and compare the result with the measurement, refine the assumed scattering law and repeat the procedure until an acceptable degree of agreement between the calculated and the measured curves is obtained. This technique has been carried to an amazing power in powder diffraction measurements, where literally hundreds of peaks can be reproduced in a pulsed source time of flight spectrum. 115 2.6 Phase space operators 2.6.1 Beam holes and static collimators Since neutron sources radiate isotropically, they must be heavily shielded with only a few penetrations (beam holes) in the shield to allow neutrons to exit. The result is a limitation of the orientation of the vectors {k} in space Fig(8) The solid angle dQ = dkx.
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