Geochemistry & Isotope Studies

Geochemistry is the study of the chemical composition and processes within Earth and other planetary bodies. It's like having a superpower that allows geologists to read the chemical history written in rocks, telling us when they formed, where the material came from, and what conditions existed during their formation.

Chemical Composition of Earth's Layers

Major Element Chemistry

Earth's composition varies dramatically from core to crust, following a pattern of differentiation based on density and chemical behavior:

Where Mg# (magnesium number) indicates the degree of differentiation, with higher values indicating more magnesium-rich compositions.

The major element composition of different Earth layers:

• Core: ~85% Fe, ~5% Ni, ~10% light elements (S, Si, O)
• Mantle: ~45% O, ~21% Si, ~23% Mg, ~5% Fe
• Continental crust: ~47% O, ~28% Si, ~8% Al, ~5% Fe
• Oceanic crust: ~50% O, ~25% Si, ~12% Mg, ~8% Fe

Trace Elements

Trace elements are elements that occur at concentrations less than 1000 ppm (0.1%) but are incredibly important for understanding geological processes. They're divided into:

• Compatible elements: Prefer to stay in the solid phase during melting (e.g., Cr, V, Sc)
• Incompatible elements: Prefer to enter the melt during melting (e.g., Rb, Ba, Th, U)

The compatibility of an element is quantified by the partition coefficient:

For example, D_Nd ≈ 1.8 for garnet suggests Neodymium is moderately compatible, while D_Ba ≈ 0.03 for plagioclase shows Barium is strongly incompatible.

Radiometric Dating

Radioactive Decay Principles

Radioactive decay follows first-order kinetics:

Where:

• $N(t)$ = number of parent atoms at time $t$
• $N_0$ = initial number of parent atoms
• $\lambda$ = decay constant
• $t$ = time

The decay constant is related to half-life by:

Common Geochronometers

Uranium-Lead System

The U-Pb system is the gold standard for dating ancient rocks:

The concordia diagram plots radiogenic Pb ratios and provides internal consistency checks.

Potassium-Argon and Argon-Argon Systems

The Ar-Ar system uses neutron irradiation to convert $^{39}$K to $^{39}$Ar for more precise measurements.

Rubidium-Strontium System

The initial $^{87}$Sr/$^{86}$Sr ratio helps distinguish between different source rocks.

Stable Isotope Geochemistry

Isotope Fractionation

Stable isotopes don't decay but fractionate during chemical and physical processes due to mass differences:

Where $R$ is the ratio of the heavy isotope to the light isotope (e.g., $^{18}$O/$^{16}$O), and $\alpha$ is the fractionation factor between phases A and B.

The standard notation is:

Common Stable Isotope Systems

Oxygen Isotopes

Oxygen isotope ratios tell us about temperature and water-rock interaction:

Temperature dependence is given by:

Where T is in Kelvin, and A, B, C are constants specific to the mineral pair.

Carbon Isotopes

Carbon isotopes help distinguish biological and inorganic carbon sources:

• Organic carbon: typically $\delta^{13}$C ≈ -25‰ (biological fractionation)
• Carbonate rocks: $\delta^{13}$C ≈ 0‰ (marine)
• Methane: $\delta^{13}$C ≈ -60‰ (extremely light)

Hydrogen Isotopes

Hydrogen isotopes (D/H) are useful for studying water-rock interactions:

• The difference between liquid water and water vapor is about 90‰
• Clay minerals can preserve meteoric water signatures for millions of years

Trace Element Geochemistry

Rare Earth Elements (REE)

REE patterns are particularly diagnostic of geological processes. The chondrite-normalized pattern is standard:

Where chondrite represents the primitive solar system composition.

Interpretation of REE Patterns

• Light REE enrichment: Incompatible element behavior (e.g., garnet depletion)
• Heavy REE depletion: Garnet in the source
• Eu anomaly: Plagioclase fractionation or accumulation
• Ce anomaly: Redox conditions in the environment

High Field Strength Elements (HFSE)

HFSE like Zr, Hf, Nb, Ta are extremely incompatible during partial melting and are useful for tracing mantle processes.

Large Ion Lithophile Elements (LILE)

LILE like Rb, Ba, K, Pb are strongly incompatible and mobile during alteration processes.

Applications in Geological Processes

Mantle Heterogeneity

Isotope geochemistry reveals that the mantle is not homogeneous:

• Depleted MORB Mantle (DMM): Low $^{87}$Sr/$^{86}$Sr, high $^{143}$Nd/$^{144}$Nd
• Enriched Mantle (EM): High $^{87}$Sr/$^{86}$Sr, low $^{143}$Nd/$^{144}$Nd
• HIMU (High U/Pb): High $^{206}$Pb/$^{204}$Pb ratios

Crustal Processes

Trace element patterns record the history of magmatic differentiation:

• Fractional crystallization: Creates systematic element depletion patterns
• Partial melting: Produces incompatible element enrichment
• Assimilation: Contamination with continental crust shows specific trace element signatures

Ore Formation

Element mobility during hydrothermal processes is controlled by:

• Temperature and pressure
• pH and oxidation state
• Ligand availability
• Redox conditions

Real-World Application: Isotopic Signatures in Ocean Island Basalts

Let's examine how geochemistry reveals the deep Earth's secrets through the study of ocean island basalts (OIBs), which represent plumes from the deep mantle.

Mantle Source Characterization

# Isotopic ratios for a typical OIB sample
isotope_data = {
    '87Sr/86Sr': 0.7035,
    '143Nd/144Nd': 0.5130,
    '206Pb/204Pb': 18.6,
    '207Pb/204Pb': 15.5,
    '208Pb/204Pb': 38.3
}

# Compare with different mantle reservoirs
mantle_reservoirs = {
    'DMM': {'Sr': 0.7025, 'Nd': 0.5132, 'Pb206': 18.2},
    'EM1': {'Sr': 0.7045, 'Nd': 0.5127, 'Pb206': 18.9},
    'EM2': {'Sr': 0.7050, 'Nd': 0.5126, 'Pb206': 19.1}
}

# Calculate deviation from MORB values (Mid-Ocean Ridge Basalt)
MORB_values = {'Sr': 0.7025, 'Nd': 0.5130}

Sr_enrichment = isotope_data['87Sr/86Sr'] / MORB_values['Sr']
Nd_depletion = MORB_values['Nd'] / isotope_data['143Nd/144Nd']

print(f"Sample Sr isotope ratio: {isotope_data['87Sr/86Sr']:.4f}")
print(f"Sample Nd isotope ratio: {isotope_data['143Nd/144Nd']:.5f}")
print(f"Enrichment in Sr relative to MORB: {Sr_enrichment:.3f}")
print(f"Depletion in Nd relative to MORB: {Nd_depletion:.3f}")

# Determine source characteristics
if Sr_enrichment > 1.02 and Nd_depletion > 1.02:
    source_type = "Enriched mantle (EM-type)"
elif Sr_enrichment < 1.01 and Nd_depletion < 0.99:
    source_type = "Depleted mantle (DMM-type)"
else:
    source_type = "Mixed or transitional source"

print(f"Interpreted source type: {source_type}")

Trace Element Patterns

Ocean island basalts help us understand deep mantle processes and the long-term evolution of our planet.

Your Challenge: Radiometric Dating Analysis

Analyze the age of a granite sample using multiple radiometric systems to test for consistency.

Goal: Calculate the age of a granite sample using U-Pb, K-Ar, and Rb-Sr systems and evaluate the consistency of the results.

Sample Data

import math

# Granite sample data
sample_data = {
    # U-Pb data
    'U238_Pb206_ratio': 0.892,  # Measured ratio
    'U235_Pb207_ratio': 0.064,  # Measured ratio
    
    # K-Ar data (in moles)
    'K40_moles_initial': 0.00025,  # Initial K-40
    'Ar40_moles_produced': 0.000042,  # Radiogenic Ar-40
    
    # Rb-Sr data (in ppm)
    'Rb87_concentration': 150,     # ppm Rb
    'Sr87_Sr86_ratio': 0.715,     # Measured ratio
    'Sr87_Sr86_initial': 0.700    # Estimated initial ratio
}

# Decay constants
decay_constants = {
    'lambda_U238': 1.55125e-10,   # 1/yr
    'lambda_U235': 9.8485e-10,    # 1/yr
    'lambda_K40': 5.543e-10,      # 1/yr
    'lambda_Rb87': 1.42e-11       # 1/yr
}

# Calculate ages using different systems
# U-Pb system (for a single zircon crystal)
age_U238_Pb206 = (1 / decay_constants['lambda_U238']) * math.log(1 + sample_data['U238_Pb206_ratio'])
age_U235_Pb207 = (1 / decay_constants['lambda_U235']) * math.log(1 + sample_data['U235_Pb207_ratio'])

# K-Ar system
# Formula: t = (1/lambda) * ln(1 + Ar40*/K40)
# Ar40* = Ar40 produced by K40 decay
K_Ar_age = (1 / decay_constants['lambda_K40']) * math.log(1 + 
                  sample_data['Ar40_moles_produced'] / sample_data['K40_moles_initial'])

# Rb-Sr system
# Formula: (Sr87/Sr86)present = (Sr87/Sr86)initial + (Rb87/Sr86) * (e^(lambda*t) - 1)
# For simplification, assume Sr87 concentration from Rb decay
Sr87_produced = sample_data['Sr87_Sr86_ratio'] - sample_data['Sr87_Sr86_initial']
# Assuming 100 ppm Sr total for a typical granite
Sr86_ppm = 100 * 0.56  # Approximate Sr86 abundance (about 56% of Sr)
Rb87_Sr86_ratio = sample_data['Rb87_concentration'] / Sr86_ppm

# Calculate age from Rb-Sr
Rb_Sr_age = (1 / decay_constants['lambda_Rb87']) * math.log(1 + 
                     (Sr87_produced / Rb87_Sr86_ratio))

Analyze the consistency of ages from different radiometric systems.

Hint:

• The different decay systems have different closure temperatures
• Discordant ages may indicate thermal events or loss of daughter products
• Concordant ages from multiple systems provide robust age estimates

# TODO: Calculate ages from each system and assess consistency
age_U_Pb_238 = 0  # Calculate U238-Pb206 age (years)
age_U_Pb_235 = 0  # Calculate U235-Pb207 age (years)
age_K_Ar = 0      # Calculate K-Ar age (years)  
age_Rb_Sr = 0     # Calculate Rb-Sr age (years)

# Calculate average age
average_age = 0   # Average of concordant systems

# Assess age consistency (within analytical uncertainty of ~1-2%)
ages = [age_U_Pb_238, age_U_Pb_235, age_K_Ar, age_Rb_Sr]
age_consistency = True  # Are all ages consistent within uncertainty?

# Print results
print(f"U238-Pb206 age: {age_U_Pb_238/1e6:.1f} Ma")
print(f"U235-Pb207 age: {age_U_Pb_235/1e6:.1f} Ma") 
print(f"K-Ar age: {age_K_Ar/1e6:.1f} Ma")
print(f"Rb-Sr age: {age_Rb_Sr/1e6:.1f} Ma")
print(f"Average age: {average_age/1e6:.1f} Ma")
print(f"Ages consistent: {age_consistency}")

# Interpretation
if age_consistency:
    interpretation = "Sample appears to have remained closed system since crystallization"
else:
    interpretation = "Sample may have experienced thermal disturbance or alteration"
print(f"Interpretation: {interpretation}")

What do the age relationships tell you about the thermal history of the granite?