Question

The accompanying image shows photoluminescence from four different samples of CdTe nanocrystals, each embedded in a polymer matrix. The photoluminescence occurs because the samples are being irradiated by a UV light source. The nanocrystals in each vial have different average sizes. The sizes are \(4.0,3.5,3.2\), and \(2.8 \mathrm{~nm}\). (a) Which vial contains the \(4.0\)-nm nanocrystals? (b) Which vial contains the \(2.8\)-nm nanocrystals? (c) Crystals of CdTe that have sizes that are larger than approximately \(100 \mathrm{~nm}\) have a band gap of \(1.5 \mathrm{eV}\). What would be the wavelength and frequency of light emitted from these crystals? What type of light is this? [Sections \(12.7\) and 12.9]

Short answer

The vial with \(4.0\) nm nanocrystals will have red photoluminescence, while the vial with \(2.8\) nm nanocrystals will have blue photoluminescence. For crystals larger than approximately \(100 nm\), the emitted light has a wavelength of \(827 nm\) and a frequency of \(3.63 \times 10^{14} Hz\), corresponding to infrared light.

Step-by-step solution

Identify vials with \(4.0\) nm and \(2.8\) nm nanocrystals

Since smaller nanocrystals emit light with shorter wavelengths (higher energies) and the wavelengths increase as the nanocrystal size increases, we can identify the vials based on the color of the photoluminescence. (a) The vial with \(4.0\) nm nanocrystals will have the longest wavelength (lowest energy) light, which corresponds to red color. (b) The vial with \(2.8\) nm nanocrystals will have the shortest wavelength (highest energy) light, which corresponds to blue color.

Calculate the wavelength of the emitted light

We are given that CdTe crystals with sizes larger than approximately \(100 nm\) have a band gap of \(1.5 eV\). The energy band gap of the crystal can be related to the emitted light's wavelength, for which we will use the formula: \(E = \frac{hc}{\lambda}\) Where \(E\) is the bandgap energy, \(h\) is Planck's constant (\(6.626 \times 10^{-34} Js\)), \(c\) is the speed of light (\(3.0 \times 10^8 ms^{-1}\)), and \(\lambda\) is the wavelength of the emitted light. First, we need to convert the energy from electronvolt (eV) to joules (J): \(E = 1.5 eV \times \frac{1.6 \times 10^{-19} J}{1 eV} = 2.4 \times 10^{-19} J\) Now, we can use this energy value to find the wavelength: \(\lambda = \frac{hc}{E}\) \(\lambda = \frac{6.626 \times 10^{-34} Js \times 3.0 \times 10^8 ms^{-1}}{2.4 \times 10^{-19} J} ≈ 8.27 \times 10^{-7} m = 827 nm\) The wavelength of the emitted light from these crystals is approximately \(827 nm\).

Calculate the frequency of the emitted light

To calculate the frequency of the emitted light, we can use the formula: \(\nu = \frac{c}{\lambda}\) \(\nu = \frac{3.0 \times 10^8 ms^{-1}}{8.27 \times 10^{-7} m} ≈ 3.63 \times 10^{14} Hz\) The frequency of the emitted light is approximately \(3.63 \times 10^{14} Hz\).

Determine the type of emitted light

Now that we have the wavelength of the emitted light (\(827 nm\)), we can compare this value with the electromagnetic spectrum ranges to determine the type of light emitted. The electromagnetic spectrum contains the following ranges (as an example): - Ultraviolet: 10 nm - 400 nm, - Visible light: 400 nm - 700 nm, - Infrared: 700 nm - 1 mm. Comparing the wavelength of the emitted light with these ranges, we find that it falls within the infrared range. Therefore, the light emitted from CdTe crystals larger than approximately \(100 nm\) is infrared light.

Key concepts

CdTe Nanocrystals

Cadmium telluride (CdTe) nanocrystals are tiny semiconductor particles that exhibit extraordinary optical and electrical properties due to their quantum size. These properties are notably different from their bulk counterparts. Specifically, CdTe nanocrystals can absorb and emit light, a phenomenon known as photoluminescence. The color of the emitted light is directly related to the size of the nanocrystal. Smaller nanocrystals tend to emit light at shorter wavelengths, which corresponds to higher energy on the electromagnetic spectrum, while larger nanocrystals emit light at longer wavelengths.

This relationship between size and wavelength is crucial for applications in optoelectronics and biolabeling, where precise control over the emission wavelength is desired. By carefully crafting the size of CdTe nanocrystals, scientists can tailor them to release light in specific colors when excited by an external energy source like UV light.

Band Gap

The band gap of a semiconductor is a fundamental property that determines its electronic and optical behavior. In simple terms, the band gap is the energy difference between the top of the valence band (where the electron is before absorption) and the bottom of the conduction band (where the electron moves after absorption). The size of the band gap determines the energy of the photons that the material can absorb or emit.

For instance, a larger band gap means the material requires photons with higher energy to bridge the gap, resulting in the absorption or emission of light at shorter wavelengths. Conversely, a smaller band gap absorbs or emits at longer wavelengths. In the context of CdTe nanocrystals, the band gap increases as the crystal size decreases, thus altering the emitted photoluminescence color. This principle enables the tailoring of nanocrystal applications across various fields, including solar cells and LED technology.

Electromagnetic Spectrum

The electromagnetic spectrum encompasses all possible frequencies of electromagnetic radiation, ranging from very low frequency radio waves to very high frequency gamma rays. The spectrum is generally divided into specific regions, typically defined by their wavelengths, frequencies, and associated energy levels. These regions include, among others, radio waves, microwaves, infrared radiation, visible light, ultraviolet light, X-rays, and gamma rays.

Visible light is the portion of the spectrum that is visible to the human eye and is characterized by wavelengths in the range of approximately 400 to 700 nanometers (nm). When CdTe nanocrystals emit light within this range, they appear colored to the eye, corresponding to the visible spectrum. Light with wavelengths longer than 700 nm falls into the infrared range, which is not visible to the naked eye but can be detected as heat or with special instruments. Photoluminescence involving CdTe nanocrystals spans a portion of the spectrum, showcasing how these materials can be engineered to manipulate light within different regions of the electromagnetic spectrum.