Step 1: Understanding the Concept:
The question asks to identify a radiometric dating method with a very wide effective range, from thousands to billions of years. Each method has a specific range based on the half-life of the radioactive isotopes involved.
Step 2: Detailed Explanation:
Let's evaluate the ranges of the given dating methods:
- Carbon-14 (Radiocarbon) dating: This method is used to date organic materials. The half-life of Carbon-14 is about 5,730 years. It is effective for dating samples up to about 50,000 to 60,000 years old. This range is too short for samples that are billions of years old.
- Potassium-Argon (K-Ar) dating: This method is used to date volcanic rocks and minerals. It is based on the decay of Potassium-40 ($^{40}$K) into Argon-40 ($^{40}$Ar). The half-life of $^{40}$K is very long, about 1.25 billion years. This long half-life makes it suitable for dating very old geological samples, from as young as a few thousand years to billions of years old. It is a key method used to date early hominin sites in East Africa.
- Uranium Series dating: This is a group of methods (like Uranium-lead and Uranium-thorium). Uranium-lead dating is used for extremely old samples (billions of years), while Uranium-thorium is used for younger samples (up to 500,000 years). While the "series" covers a wide range, K-Ar is a single method known for this specific broad application in paleoanthropology.
- Thermoluminescence (TL) dating: This method is used to date crystalline materials that have been fired, such as pottery or burnt flint. Its effective range is from a few hundred years up to about 500,000 years. This range is too short for the upper limit required.
Step 3: Final Answer:
Comparing the options, Potassium-Argon (K-Ar) dating is the most appropriate single method that covers the vast time span from thousands of years up to billions of years.