# Charge flipping

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## Definition

Charge flipping is a structure solution method from the class of dual-space algorithms. The key component of the charge flipping algorithm is the charge flipping operation. In this operation, all scattering density pixels with density lower than a small positive threshold δ are multiplied by -1 (flipped). In the classical charge flipping algorithm, this direct-space modification is combined with simple resubstitution of structure-factor amplitudes by experimental values, but other modifications and iteration schemes have been proposed and successfully used.

Given a trial scattering density ρ sampled on a regular grid, and a set of measured structure-factor amplitudes $F^{obs}(\mathbf{H})$, the basic charge flipping algorithm follows this scheme:

First, the algorithm is initiated in the zeroth cycle by assigning random starting phases $\varphi_{rand}(\mathbf{H})$ to all experimental amplitudes and making all unobserved amplitudes equal to zero:

The iteration cycle then proceeds as follows:

1. The density ρ(n) is calculated by inverse Fourier transform of F(n).
2. The modified density g(n) is obtained by flipping the density of all pixels with density values below a certain positive threshold δ and keeping the rest of the pixels unchanged:

3. Temporary structure factors $G^{(n)}(\mathbf{H})=|G^{(n)}(\mathbf{H})|\exp{(i\varphi_{G}(\mathbf{H}))}$ are calculated by Fourier transform of g(n).
4. New structure factors F(n + 1) are obtained by combining the experimental amplitudes with the phases $\varphi_{G}$ and setting all non-measured structure factors to zero:

These modified structure factors then enter the next cycle of iteration.