###
Simulation of the distance between gold nanoparticles when they form crystal structure and make structural colors.

The particle’s plane in the three-dimensional crystal structure strengthens the light of the incident wavelength by interference of the light by Bragg reflection. We put d = the spacing between the planes in the nanoparticle’s lattice, n = reflective index, θ= the angle between the incident ray and the scattering planes, λ= the wavelength of incident wave, so we can express the equation of λas follows:

(m is any integer)

**Fig. 1. **“d” is the distance between I and II.(111) planes.

Now, we define m=1. The crystal structure of gold nanoparticles we made is a face-centered cubic. In this case, since the distance between the particle’s planes is the biggest on the (111) planes (Fig.1), the reflection with (111) planes gives the longest Bragg wavelength. When nanoparticles are small and a volume fraction of the particle among the overall volumes of crystal structure is small enough, we can hypothesize that the refractive index n is almost equal to a refractive index of a dielectric buried around a particles.

Thus, when n is approximated by the value (n=1.33) of the refractive index of the water, we express the equation of Bragg reflection as follows:

(d = the spacing between the planes in the nanoparticle’s lattice(distance between I and II ), θ= the angle between the incident ray and the scattering planes)

This equation shows that when a value of λ is 400nm<λ<800nm, λ depends on d andθ.

Then, how does the crystal structure of gold nanoparticles show the structural color. In the case of face-centered cubic, the distance of (111) plane is the biggest. So, when we put a = the lattice constant, we can express d as follows:

With the face-centered cubic, the distance of the particles “D” expressed as follows:

We express the equation by rearranging it as follows:

When we put R= the radius of a gold nanoparticle, B= the number of bases, d is
￼￼
Fig. 1. “d” is the distance between I and II.(111) planes.
￼￼￼￼￼￼￼￼￼
expressed as follows:

￼￼￼(0.255 is the length of DNA per one base)
And the radium of the gold nanoparticles which we used is 50nm. So, equation4 is expressed as follows:

By substituting equation 4 into equation of Bragg reflection( 5 ), equation 5 is expressed as follows:

We made the graph of this equation.