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2D lattice

Lattice geometry

We consider a 2D Bravais lattice that is specified by lattice constants $a$ and $b$, by the angle $\alpha$ between the lattice vectors $\mathbf{a}$ and $\mathbf{b}$, and by the angle $\xi$ between $\mathbf{a}$ and the $x$ axis of the instrument frame.

Structure constructors

The structure for an infinite 2D lattice is created from a particle and a lattice object:

lattice = ba.BasicLattice2D(length_a, length_b, alpha, xi)
structure = ba.Crystal2D(particle, lattice)

To account for finite-size effects, and to give the simulated Bragg peaks a finite width, attach a peak profile as described below.

The finite 2D lattice model is computed by explicit summation over contributions from all unit cells. The lattice is constructed by one of the following:

ba.BasicLattice2D(length_a, length_b, alpha, xi)
ba.SquareLattice2D(length, xi)
ba.HexagonalLattice2D(length, xi)

Then the finite structure is constructed by

structure = ba.FiniteCrystal2D(particle, lattice, size_a, size_b)

where the arguments $\textrm{size}_a$ and $\textrm{size}_b$ are the crystal sizes in lattice directions $\mathbf{a}$ and $\mathbf{b}$. Each size is rounded to the nearest integer number of unit cells; the crystal is a parallelogram spanned by the two lattice vectors.

Decay function (peak profile)

The decay function accounts for the finite size of the sample, the crystalline domain, or the scattering volume. It contributes a multiplicative factor $h(\mathbf{r})$ to the correlation function in real space that lets it decay for $r$.

Equivalently, its Fourier transform $H(\mathbf{q})$ is convolved with the structure factor of the ideal, infinite lattice,

$$S_0(\mathbf{q}) = \rho \sum_i c_i \delta(\mathbf{q} - \mathbf{q}_i),$$

where the sum runs over reciprocal lattice points, resulting in

$$S(\mathbf{q}) = \rho \sum_i c_i H(\mathbf{q} - \mathbf{q}_i).$$

That is, each Bragg peak gets the same smooth profile $H(\mathbf{q})$ instead of the infinitely sharp delta function.

Decay functions for the one-dimensional case are explained in the grating section. Decay functions for two-dimensional lattices work similarly, but require two length parameters and an orientation. The lengths $\omega_x$ and $\omega_y$ refer to the principal axes of an ellipsis that is rotated by an angle $\gamma$ with respect to the coordinate axes $x$, $y$.

BornAgain supports three two-dimensional decay functions:

ba.Profile2DCauchy(omega_x, omega_y, gamma)
ba.Profile2DGauss(omega_x, omega_y, gamma)
ba.Profile2DVoigt(omega_x, omega_y, gamma, eta)

The Cauchy distribution is in physics better known as the Lorentzian profile. The Voigt distribution is computed as the true convolution of a Gaussian and a Lorentzian, using our numeric library libcerf. The dimensionless parameter eta=$\eta$ may take values between 0 (Lorentian) and 1 (Gaussian).

Decay lengths should be much larger than the lattice constants. Otherwise the model becomes physically questionable, and computation times increase a lot. Typical values for meaningful decay lengths are in the micrometer range (≥ 1000 nm).

Setting the decay function

The decay function is attached to a lattice structure by the setDecayFunction method:

lattice = ba.BasicLattice2D(20*nm, 20*nm, 120*deg, 0)
structure = ba.Crystal2D(particle, lattice)
profile = ba.Profile2DCauchy(1000*nm, 1000*nm, 0*deg)
structure.setDecayFunction(profile)

Particle density

The particle density is determined from the 2D lattice parameters. There is no separate particle-density setting for Crystal2D or FiniteCrystal2D.

Examples and extensions