A high-quality diamond may be quite clear and colorless, transmitting all visible wavelengths with little absorption. Explain how it can sparkle with flashes of brilliant color when illuminated by white light.

When light travels from one medium to another, its speed changes, which causes the light to bend or refract. The refractive index of a material quantifies this change, denoting the ratio of the speed of light in a vacuum to its speed in the material. A high refractive index, such as that of a diamond, indicates a significant reduction in the speed of light, leading to more pronounced bending of light rays when they enter the diamond from air.

This concept is crucial to understanding why a diamond sparkles with color when light is shone upon it. Due to the diamond's high refractive index, light rays are significantly refracted as they enter and exit the gem, contributing to the diamond's ability to separate white light into its colorful components and creating a dazzling display.

White light is not a single color but a combination of many colors that can be observed in a rainbow or a spectrum. The visible spectrum is the portion of the electromagnetic spectrum that is visible to the human eye, ranging from violet with the shortest wavelengths to red with the longest wavelengths.

When discussing diamond dispersion, it helps to know that each color within the visible spectrum has a different wavelength. This property is what makes dispersion in diamonds possible, as every color is refracted by a slightly different amount when passing through the diamond, which is what leads to the array of colors seen in the sparkle of a well-cut diamond.

Total internal reflection is a phenomenon that occurs when a light ray traveling within a denser medium, like diamond, hits the interface with a less dense medium, such as air, at an angle greater than a certain 'critical angle'. Instead of refracting out, all the light is reflected back inside the diamond. This is a pivotal concept in understanding a diamond's brilliance.

Diamond cutters use this knowledge to their advantage by fashioning diamonds with exact angles and proportions to maximize the total internal reflection within the stone. This results in light rays bouncing around inside the diamond, emphasizing the stone's fire and brilliance before finally exiting and reaching the observer's eye.

White light is composed of an array of colors—each with distinct wavelengths—that together appear white to our eyes. This array includes red, orange, yellow, green, blue, indigo, and violet. Each component of white light is separated when it passes through a prism-like structure, due to differences in refraction for each wavelength.

In the context of diamonds, these components explain the characteristic sparkle we see. When white light enters the diamond, each of its components is refracted differently due to their varying wavelengths, with shorter wavelengths bending more than longer ones. After multiple reflections within the diamond, these colors emerge as flashes of color, contributing to the diamond's characteristic fire.

Light dispersion is the separation of white light into its constituent colors, a phenomenon elegantly demonstrated by diamonds. Diamonds act as prisms, bending and spreading out the different wavelengths of light at different angles. The high refractive index of diamonds exacerbates this effect, amplifying the dispersion of light.

When we observe a sparkling diamond, what we perceive are the results of this light dispersion, combined with the internal reflections. The myriad of faces and the expert cut of the diamond manipulate the paths of light, causing the spectrum of visible colors to emerge more vividly. It's the interaction of light dispersion with total internal reflection that creates the fiery dance of colors that makes diamonds so mesmerizing.