# Basic principles of quantum mechanics demonstrated by neutron scattering

**Quantum Mechanics & Neutron Scattering - Part 1**

Source: Forschungszentrum Jülich/TRICKLABOR

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### Quantum Mechanics & Neutron Scattering - Part 2

Source: Forschungszentrum Jülich/TRICKLABOR

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## Background

*(Written by Herbert Capellmann)*

The basic laws of classical physics relied upon the principle "Natura non facit saltus" (nature does not make jumps), taken from ancient philosophy. Assuming that nature changes continuously in space and time, the classical laws were expressed in the form of differential equations or variational principles, where infinitesimally small changes of various physical variables are related to each other. When the microscopic world, the "quantum world", was explored at the beginning of the 20th century, this concept turned out to be invalid. Although the classical laws remained approximately valid for macroscopic averages, the microscopic "quantum world" was found to behave differently. Based on more and more experimental data obtained during the first quarter of the 20th century, Max Born found the keys to the understanding of the quantum world:

- on the microscopic level all elementary changes in nature consist of discrete and finite steps: "quantum transitions"
- the occurrence of these quantum transitions is not deterministic, but governed by probability laws.

Neutron scattering is particularly well-suited to demonstrate the basic principles of Quantum Mechanics. The interaction of the neutron with matter is weak and the individual neutron scattering process typically results from a single elementary quantum transition. Neutron scattering is able to provide direct and unambiguous information about the basic principles of Quantum Mechanics.

**A typical neutron scattering experiment**

Neutrons with well-defined initial momentum p are scattered by a crystal, undergoing a quantum transition to a final momentum p'.

**The principal observations:**

• Neutrons are scattered statistically into different directions, individual neutrons are always observed as particles.

• For special orientations of the crystal, neutrons are scattered preferentially into special directions: these are the Bragg-reflections or "Bragg peaks".

• There is always a uniform background of neutrons scattered with equal probability into all directions.

**Interference and background, general remarks **

A "wave-picture" is usually used to explain the origin of Bragg peaks, based on the definition of a wavelength for neutrons of momentum p according to de Broglie: λ = h/p. The same definition of "wavelength" is used for electrons and other particles.

The regular array of atoms in the crystal is viewed as consisting of parallel planes of atoms; the incident "neutron wave" is scattered preferentially into special directions, if the orientation of the crystal is such that the path difference from neighbouring planes is an integer multiple of the wavelength, allowing constructive interference.

Experimentally however, single neutrons are exclusively observed as particles, scattered statistically into different directions. Wave patterns, associated with Bragg peaks, may only result from averages of a large number of independent single particle scattering processes, statistically distributed. Wave patterns represent a fraction of the scattering events only, in addition to a uniform background. Already in 1905 Einstein recognized, that the same "problem" exists for light. Light, too, consists of particles, photons; individual photons interact with matter as point-like particles. Only averages over macroscopically large numbers of photons can be described as waves.

**Scattering events contributing to Bragg peaks **

In general, the quantities measured in neutron scattering are initial and final momenta of the neutron, which may be inferred from the direction and velocity of the neutron (e.g. if the "time of flight" method is used). The interaction of the neutron with the crystal provides a probability for a single quantum transition from initial neutron momentum p to final momentum p', the probability of the neutron undergoing multiple transitions typically can be neglected due to the weakness of the interaction. This is what makes neutrons so special compared to any other probe (electrons, photons etc.).

### Concerning Bragg scattering

All transitions contributing to Bragg peaks are purely elastic. While the neutron undergoes a single quantum transition, the crystal itself, however, remains unchanged! Under these conditions the crystal may be considered as a static scattering potential, and the transition amplitudes (or "matrix elements") have to be consistent with all symmetry requirements imposed by this static potential.

Whereas in free space, momentum has to be conserved due to full translational invariance, this is no longer true in the presence of the crystal potential. But there exists a new (lower) symmetry - discrete translational symmetry due to the periodicity of the crystal lattice - translations perpendicular to lattice planes by integer multiples of the spacing between equivalent planes. Under these conditions momentum p is no longer conserved, but "Quasi momentum", P = p + hQ, is conserved!

Q is a "reciprocal lattice vector", having the direction perpendicular to the lattice planes considered, the absolute value has to be an integer multiple of the inverse spacing between equivalent planes. The momentum transfer hQ during the scattering process is taken up by the rigid lattice. Furthermore, energy conservation requires that the absolute values of initial momentum p and final momentum p + hQ to be identical. These two symmetry requirements yield the "Bragg conditions".

The occurrence of Bragg peaks follows from general quantum mechanical laws and symmetry requirements, invoking particle properties of the neutron only. Furthermore, we learn that an additional condition has to be fulfilled for scattering events contributing to Bragg peaks: the elementary quantum transition of the neutron from initial momentum p to p + hQ must not cause a simultaneous transition in the crystal.

### Scattering involving nuclear spin transitions contributing to the uniform background

Elastic scattering contributions to Bragg peaks occur according to statistical laws only; there is always a finite probability for other scattering processes. The interaction of the neutron with the crystal may cause a combined transition involving the neutron and an individual lattice site. Most atoms possess nuclear spins and the interaction between the neutron spin with the nuclear spins of the lattice results in what is often the dominant source of the uniform scattering background. Nature provides us with a tool to identify scattering events involving nuclear spin transitions. Controlling the neutron spin before and after the scattering process, we can separate their contribution from those events, which contribute to Bragg scattering.

Again general symmetry arguments are helpful to understand the outcome of these scattering events. The transition of the neutron combined with the nuclear spin transition of a particular fixed lattice site breaks translational invariance completely. If we consider the nucleus to be point-like without an internal structure, the neutron may be scattered into all directions with equal probability; the momentum transfer is provided by the rigid lattice, serving as an anchor for the nuclear spin involved in the combined scattering process. Typically the nuclear spin transition does not cost any (or a negligibly small amount of) energy; therefore energy conservation requires that the absolute value of the neutron momentum remains unchanged in these scattering events. Neutrons causing a localized transition in the lattice are scattered with equal probability into all directions and cannot contribute to interference effects.

### In summary:

Neutron scattering is able to provide direct and quantitative information about elementary quantum transitions:

- statistical behaviour in space and time;
- single neutrons have particle properties and can only be observed as particles;
- interference effects and wave-like behaviour apparent in Bragg scattering result from averages over large numbers of individual particle processes;
- Bragg contributions must be purely elastic, without causing any simultaneous transition in the crystal;
- localized transitions in the lattice will cause scattering contributions into all directions with almost equal probability.

Further basic principles of Quantum Mechanics can be demonstrated in neutron scattering and will be discussed in other sections:

- transition probabilities are proportional to squares of transition amplitudes;
- interference effects due to different interactions of the neutron with matter;
- the quantum mechanical uncertainty relation for position can be measured.