London penetration depth formula

Explore the London penetration depth formula, its significance in superconductivity, and an example of calculating this crucial parameter.

Introduction to the London Penetration Depth Formula

The London penetration depth, often denoted as λ_L, is a key concept in the field of superconductivity. This parameter quantifies the depth at which an external magnetic field penetrates into a superconductor, and plays a vital role in understanding the behavior of superconducting materials. In this article, we will discuss the London penetration depth formula and its significance in the study of superconductors.

London Brothers’ Contribution to Superconductivity Theory

In 1935, Fritz and Heinz London proposed a set of phenomenological equations, known as the London equations, to describe the behavior of superconductors in the presence of an external magnetic field. The London penetration depth is derived from these equations and characterizes the exponential decay of the magnetic field inside a superconductor.

Formula for London Penetration Depth

The London penetration depth is given by the following formula:

λ_L = √(m_e / μ₀ n_s e²)

where:

• λ_L is the London penetration depth
• m_e is the effective mass of the superconducting charge carriers (typically electrons)
• μ₀ is the vacuum permeability, a fundamental physical constant
• n_s is the density of superconducting charge carriers
• e is the elementary charge, another fundamental physical constant

Significance of the London Penetration Depth

The London penetration depth is an essential parameter for characterizing superconducting materials. It indicates how well a superconductor can expel an external magnetic field, which is a phenomenon known as the Meissner effect. A shorter London penetration depth signifies a better shielding of the magnetic field by the superconductor.

Moreover, the London penetration depth is also used to classify superconductors into two categories:

1. Type I superconductors: These materials exhibit a complete Meissner effect and have a lower London penetration depth. They can completely expel an external magnetic field below a critical value (H_c).
2. Type II superconductors: These materials have a higher London penetration depth and allow partial penetration of the magnetic field through a mixed state. They can support magnetic flux vortices that enable the penetration of the magnetic field in a limited manner.

In summary, the London penetration depth formula is a crucial tool for understanding the behavior of superconductors in the presence of an external magnetic field. It helps researchers characterize superconducting materials and plays a significant role in the development of new applications in the field of superconductivity.

Example of London Penetration Depth Calculation

Let’s consider a hypothetical superconducting material with the following properties:

• Effective mass of superconducting charge carriers (m_e): 9.11 × 10^-31 kg (approximately equal to the mass of an electron)
• Density of superconducting charge carriers (n_s): 6.02 × 10²⁸ m^-3

We will now calculate the London penetration depth (λ_L) for this material using the formula:

λ_L = √(m_e / μ₀ n_s e²)

Recall that the vacuum permeability (μ₀) and the elementary charge (e) are fundamental physical constants with values:

• μ₀ = 4π × 10^-7 T m A^-1
• e = 1.6 × 10^-19 C

Substituting the values into the formula, we get:

λ_L = √((9.11 × 10^-31 kg) / (4π × 10^-7 T m A^-1 × 6.02 × 10²⁸ m^-3 × (1.6 × 10^-19 C)²))

After evaluating the expression, we obtain:

λ_L ≈ 5.29 × 10^-9 m

Thus, the London penetration depth for this hypothetical superconducting material is approximately 5.29 nm.

This example demonstrates how the London penetration depth formula can be used to calculate the depth at which an external magnetic field penetrates a superconducting material. The result helps us understand the material’s ability to shield against magnetic fields and can aid in the classification of the material as a Type I or Type II superconductor.