# Quantitative Neutron Dark-Field Imaging of Milk: A Feasibility Study
## 2 Material and Method
### Dark-Field Contrast Imaging  
Figure 2 shows the principal set-up of a Talbot-Lau grating interferometer (TLI). The TLI consists of a source grating, G\({}_{0}\), a phase grating, G\({}_{1}\), and an analyzer grating, G\({}_{2}\). G\({}_{0}\) is an absorption grating that is positioned downstream of the source to generate multiple collimated beams, providing a partially coherent beam. G\({}_{1}\) is a phase grating that generates an interference pattern, called a Talbot pattern, further downstream. G\({}_{2}\) is another absorption grating that is positioned at a fractional Talbot distance and matches the period of the interference pattern. The geometry of G\({}_{0}\) and G\({}_{1}\) is such that the interference patterns of the individual coherent beams from G\({}_{0}\) add constructively at G\({}_{2}\). The imaging  
Figure 1: The deuterated milk samples. (**a**) Diluted milk powders in D\({}_{2}\)O solvent and (**b**) milk gels manufactured from diluted milk powders by acidification (marked “GA”) and rennet-induction (marked “GP”).  
detector system is positioned directly after G\({}_{2}\) and resolves the local interference pattern pixel-wise through the phase stepping of one of the gratings, typically G\({}_{0}\).  
Small-angle neutron scattering from a sample convolutes with the intensity modulation induced by the gratings. This results in the damping of the interference fringes at G\({}_{2}\), i.e., a loss of visibility, \(v_{s}\). The relative loss of visibility, \(v_{s}/v_{o}\), is referred to as the DFI contrast. The scattering distribution at G\({}_{2}\) varies with the sample distance \(L_{s}\) and the wavelength \(\lambda\), and it can be shown that these parameters, together with the beam modulation period \(p\), define the real-space length probed in the scattering structure [19]. This length is referred to as instrumental or probed correlation length, \(\xi\), in full analogy to the spin-echo length in SESANS. It can be written as:  
\[\xi=\lambda L_{s}/p, \tag{1}\]  
where \(p\) coincides with the period of G\({}_{2}\) in the TLI setup.  
Correspondingly, the DFI contrast can be described quantitatively with regards to the real-space correlation function \(G(\xi)\) of a scattering structure:  
\[DFI(\xi)=v_{s}(\xi)/v_{0}(\xi)=e^{\Sigma_{s}t(G(\xi)-1)}, \tag{2}\]  
where \(\Sigma_{s}\) is the total small-angle neutron scattering cross section, containing the volume fraction \(\varphi\), the nSLD contrast \(\Delta\rho\), the characteristic structure size \(\zeta\) (often referred to as the specific correlation length of a structure), and the utilized wavelength \(\lambda\), while \(t\) is the sample thickness [19]. The real-space correlation function, \(G(\xi)\), which is a one-dimensional projection of the three-dimensional scattering-length density correlation function of the scattering structure, is related to the small-angle scattering intensity \(I(q)\) of the structure through the Hankel transform [26]:  
\[G(\xi)=\frac{1}{2\pi\zeta}\int_{0}^{\infty}J_{0}(q\xi)I(q)qdq, \tag{3}\]  
where \(J_{0}\) is the zeroth-order Bessel function of the first kind, and \(q=(4\pi\sin\theta)/\lambda\) is the magnitude of the scattering vector, with \(\theta\) being the scattering angle. The characteristic structure size \(\zeta\) can be expressed as:  
\[\zeta=\frac{1}{2\pi\int_{0}^{\infty}I(q)dq. \tag{4}\]  
Figure 2: Schematics of the Talbot–Lau neutron grating interferometer.