Which best scientist determine the age of fossils

They use several different methods. 
1. Dating the things found with the fossil. 
2. Checking what kind of fossils are in the same layer of sediment. (index fossils) 
3. Checking changes to the atoms in the rock. 

Here is some information from a good web site: 
Paleontologists use many ways of dating individual fossils in geologic time. 

* The oldest method is stratigraphy, studying how deeply a fossil is buried. Dinosaur fossils are usually found in sedimentary rock. Sedimentary rock layers or strata are formed episodically as earth is deposited horizontally over time. Newer layers are formed on top of older layers, pressurizing them into rocks. Paleontologists can estimate the amount of time that has passed since the stratum containing the fossil was formed. Generally, deeper rocks and fossils are older than those found above them. 
* Observations of the fluctuations of the Earth's magnetic field, which leaves different magnetic fields in rocks from different geological eras. 
* Dating a fossil in terms of approximately how many years old it is can be possible using radioisotope- dating of igneous rocks found near the fossil. Unstable radioactive isotopes of elements, such as Uranium-235, decay at constant, known rates over time (its half-life,which is over 700 million years). An accurate estimate of the rock's age can be determined by examining the ratios of the remaining radioactive element and its daughters. For example, when lava cools, it has no lead content but it does contain some radioactive Uranium (U-235). 

Over time, the unstable radioactive Uranium decays into its daughter, Lead-207, at a constant, known rate (its half-life). By comparing the relative proportion of Uranium-235 and Lead-207, the age of the igneous rock can be determined. Potassium-40 (which decays to argon-40) is also used to date fossils. 

The half-life of carbon-14 is 5,568 years. That means that half of the C-14 decays (into nitrogen-14) in 5,568 years. Half of the remaining C-14 decays in the next 5,568 years, etc. This is too short a half-life to date dinosaurs; C-14 dating is useful for dating items up to about 50,000 - 60,000 years ago (useful for dating organisms like Neanderthal man and ice age animals). 

Radioisotope dating cannot be used directly on fossils since they don't contain the unstable radioactive isotopes used in the dating process. To determine a fossil's age, igneous layers (volcanic rock) beneath the fossil (predating the fossil) and above it (representing a time after the dinosaur's existence) are dated, resulting in a time-range for the dinosaur's life. Thus, dinosaurs are dated with respect to volcanic eruptions. 
* Looking for index fossils - Certain common fossils are important in determining ancient biological history. These fossil are widely distributed around the Earth but limited in time span. Examples of index fossils include brachiopods (which appeared in the Cambrian period), trilobites (which probably originated in the pre-Cambrian or early Paleozoic and are common throughout the Paleozoic layer - about half of Paleozoic fossils are trilobites), ammonites (from the Triassic and Jurassic periods, and went extinct during the K-T extinction), many nanofossils (microscopic fossils from various eras which are widely distributed, abundant, and time- specific), etc.

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