Question

The gap between valence and conduction bands in silicon is 1.12 eV. A nickel nucleus in an excited state emits a gammaray photon with wavelength 9.31 \(\times\) 10\(^-$$^4\) nm. How many electrons can be excited from the top of the valence band to the bottom of the conduction band by the absorption of this gamma ray?

Short answer

Approximately \( 1.20 \times 10^{8} \) electrons can be excited.

Step-by-step solution

Convert Wavelength to Energy

To find the energy of the gamma-ray photon, we use the formula for energy of a photon: \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ Js} \), \( c \) is the speed of light \( 3.00 \times 10^8 \text{ m/s} \), and \( \lambda \) is the wavelength. Convert the wavelength from nanometers to meters: \( 9.31 \times 10^{-4} \text{ nm} = 9.31 \times 10^{-13} \text{ m} \). Substituting the values, we get \( E = \frac{(6.626 \times 10^{-34} \text{ Js})(3.00 \times 10^8 \text{ m/s})}{9.31 \times 10^{-13} \text{ m}} \). Calculating, \( E \approx 2.14 \times 10^{-11} \text{ J} \).

Convert Energy from Joules to Electronvolts

To convert the energy from joules to electronvolts, use the conversion factor \( 1 \text{ eV} = 1.602 \times 10^{-19} \text{ J} \). Thus, \( E = \frac{2.14 \times 10^{-11} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} \approx 1.34 \times 10^{8} \text{ eV} \).

Calculate Number of Electrons Excited

Each electron transition requires 1.12 eV. To find how many electrons can be excited, divide the photon energy by the energy gap: \( \frac{1.34 \times 10^{8} \text{ eV}}{1.12 \text{ eV}} \approx 1.20 \times 10^{8} \). Therefore, approximately \( 1.20 \times 10^{8} \) electrons can be excited.

Key concepts

Band Gap

The band gap is a fundamental property of semiconductors. It is the energy difference between the valence band (where electrons are normally found) and the conduction band (where electrons are free to move, conducting electricity). In semiconductors like silicon, this gap is about 1.12 electronvolts (eV).
To make the semiconductor conduct electricity, electrons need to gain energy equivalent to the band gap to jump from the valence band to the conduction band.

• The size of the band gap determines the electrical properties of the material.
• A larger band gap means the material is less conductive at room temperature.
• To conduct current, electrons need external energy to transition from the valence to the conduction band.

In photovoltaic cells, light provides this energy. In our example's gamma-ray photon, the energy is significantly higher than the band gap, thus exciting many electrons across this gap.

Valence Band

The valence band is the highest range of electron energies in a solid where electrons are normally present at absolute zero temperature. These electrons are tightly bound to atoms and have low energy.
When an electron absorbs energy, it can jump from the valence to the conduction band, leading to electrical conductivity.

• The valence band is filled with valence electrons.
• Valence electrons are involved in chemical bonding with neighboring atoms.
• Electrons must reach the energy level of the conduction band to participate in conduction.

In silicon, energy around 1.12 eV is required to promote an electron from the top of the valence band to the bottom of the conduction band, initiating conduction.

Conduction Band

The conduction band is the range of electron energy higher than the valence band. Electrons in this band are not tied to any specific atom and can move freely through the material.

• This free movement of electrons allows them to carry electric current.
• Transitioning requires external energy to push valence electrons to the conduction band.
• In metals, conduction bands overlap with the valence bands, allowing for free electron flow without added energy.

In semiconductors, energy input like heat or light is necessary for this transition, as in the case of the gamma-ray photon providing much more energy than silicon's band gap.

Gamma Ray

Gamma rays are a form of electromagnetic radiation with the shortest wavelength and highest energy. They have substantial energy capable of penetrating most materials and exciting electrons in the process.

• Wavelengths are incredibly short, below 0.01 nm, contributing to their high energy.
• They are often emitted by radioactive substances during nuclear reactions.
• Contact with gamma rays can lead to significant electron excitations due to their high energy.

In our example, a gamma-ray photon emitted by a nickel nucleus possessed more than enough energy to excite numerous electrons from silicon's valence band to its conduction band.

Electron Transition

Electron transition describes the process when an electron moves between different energy levels within an atom, often between the valence and conduction bands in the context of solid-state physics.

• External energy must match or exceed the band gap for transition.
• Upon absorbing energy, an electron transitions from the valence band to the conduction band, resulting in electrical conductivity.
• The reverse process involves emitting energy as the electron drops back to the lower energy level.

In the provided scenario, the gamma-ray photon supplied a significantly larger amount of energy than the band gap (1.12 eV), facilitating the excitation of a vast number of electrons in silicon, enabling conductivity.