Geophysical Principles: Elasticity, Magnetism, and Radiometric Dating Formulas

Fundamental Concepts in Earth Physics

Elastic Moduli and Seismic Waves

Elastic Moduli Definitions

• Young's Modulus (Y): Describes the fractional change in length of an object when subjected to a tensile stress. The formula is: Y = (F/A) / (ΔL/L₀)
• Bulk Modulus (K): Describes how a fractional change in volume depends on the applied pressure.
• Shear Modulus (G): Describes how an angle of shear depends upon a tangential stress.

Seismic Wave Types

• P-wave (Primary Wave): Characterized by compressional and expansion motion.
• S-wave (Secondary Wave): Characterized by shear motion.

Demonstration of Radiometric Age Dating

Derivation of the Decay Equation

If the decay rate is equal to $\lambda$, the probability that a given nucleus will decay in a time interval $dt$ is $\lambda dt$. Therefore, if at any time $t$ we have $P$ parent nuclei, the number that decay in the next moment ($dP$) is:

dP = -λ P dt

Separating variables and integrating:

dP/P = -λ dt

ln(P) = -λ t + C

Applying Boundary Conditions

The boundary condition is found because we know that when $t = 0$, the number of parent nuclei $P = P_0$ (initial number). Therefore, $C = \ln(P_0)$. We can write the equation as:

ln(P) = ln(P₀) - λt

Using logarithm properties, this yields the standard decay law:

ln (P/P₀) = -λt

P/P₀ = e⁻λt

P = P₀ e⁻λt

Calculating Daughter Nuclides and Age

As parent nuclides ($P$) diminish, the number of daughter nuclides ($D$) increases. By definition, $D$ is the difference between the initial parent nuclei ($P_0$) and the remaining parent nuclei ($P$):

D = P₀ - P = P₀ (1 - e⁻λt)

Since $P_0$ is often unknown, we reformulate the equation by eliminating $P_0$:

D = P (eλt - 1)

This equation can be solved with respect to time ($t$) to determine the age of the sample:

t = (1/λ) ln (D/P + 1)

Geophysical Methods and Magnetism

Electrical Methods

• Apparent Resistivity: A resistivity estimate based on assuming a half-space geometry, measured in ohm-meters ($\Omega \cdot m$).
• Magnetotellurics (MT): An electromagnetic geophysical method for inferring the Earth's subsurface electrical conductivity from measurements of natural geomagnetic and geoelectric field variations at the Earth's surface. Investigation depth ranges from 300 m (using higher frequencies) down to 10,000 m or deeper (using long-period soundings).

Types of Magnetization

Remanent Magnetization (Acquisition Mechanisms)

1. NRM: Natural Remanent Magnetization
2. TRM: Thermoremanent Magnetization
3. DRM: Depositional Remanent Magnetization
4. CRM: Chemical Remanent Magnetization
5. IRM: Isothermal Remanent Magnetization
6. VRM: Viscous Remanent Magnetization
7. ARM: Anhysteretic Remanent Magnetization

Magnetic Behavior Classification

1. Diamagnetism
2. Paramagnetism
3. Ferromagnetism

Induced vs. Remanent Magnetization

• Induced Magnetization ($M_{induced}$): Temporary magnetization induced by the ambient magnetic field ($B$), and is usually parallel to $B$.
• Remanent Magnetization ($M_{remanent}$): Magnetization acquired during a specific, often datable, process or event in the history of the rock, sediment, or artifact (e.g., TRM, CRM, DRM, IRM).

Resistance Calculation Example

Resistance Formula

The resistance $R$ of a conductor of length $L$ and cross-sectional area $A$, made of a material with resistivity $\rho$, is given by:

R = ρ (L/A)

Calculation for Slabs in Series

We consider the flow of current going through a path of arbitrary cross-sectional area $A$ normal to the interfaces of the given slabs. The total resistance $R$ is the sum of the resistances of the individual slabs ($R_1$ and $R_2$):

R = R₁ + R₂

Applying the formula to the specific slab parameters provided:

R = 4ρ (L/2)/A + (ρ/6) (L/2)/A

Simplifying the terms:

R = 2ρ (L/A) + (ρ/12) (L/A)

Combining the terms:

R = (25/12) ρ (L/A)