Sunday, July 24, 2011

Radioactive Decay


Radioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles (ionizing radiation). The emission is spontaneous, in that the atom decays without any interaction with another particle from outside the atom (i.e., without a nuclear reaction). Usually, radioactive decay happens due to a process confined to the nucleus of the unstable atom, but, on occasion (as with the different processes of electron capture and internal conversion), an inner electron of the radioactive atom is also necessary to the process.

Radioactive decay is a stochastic (i.e., random) process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a given atom will decay. However, given a large number of identical atoms (nuclides), the decay rate for the collection is predictable, via the Law of Large Numbers.

The decay, or loss of energy, results when an atom with one type of nucleus, called the parent radionuclide, transforms to an atom with a nucleus in a different state, or a different nucleus, either of which is named the daughter nuclide. Often the parent and daughter are different chemical elements, and in such cases the decay process results in nuclear transmutation. In an example of this, a carbon-14 atom (the "parent") emits radiation (a beta particle, antineutrino, and a gamma ray) and transforms to a nitrogen-14 atom (the "daughter"). By contrast, there exist two types of radioactive decay processes (gamma decay and internal conversion decay) that do not result in transmutation, but only decrease the energy of an excited nucleus. This results in an atom of the same element as before but with a nucleus in a lower energy state. An example is the nuclear isomer technetium-99m decaying, by the emission of a gamma ray, to an atom of technetium-99.

Nuclides produced as daughters are called radiogenic nuclides, whether they themselves are stable or not. A number of naturally occurring radionuclides are short-lived radiogenic nuclides that are the daughters of radioactive primordial nuclides (types of radioactive atoms that have been present since the beginning of the Earth and solar system). Other naturally occurring radioactive nuclides are cosmogenic nuclides, formed by cosmic ray bombardment of material in the Earth's atmosphere or crust. For a summary table showing the number of stable nuclides and of radioactive nuclides in each category, see Radionuclide.

The SI unit of activity is the becquerel (Bq). One Bq is defined as one transformation (or decay) per second. Since any reasonably-sized sample of radioactive material contains many atoms, a Bq is a tiny measure of activity; amounts on the order of GBq (gigabecquerel, 1 x 109 decays per second) or TBq (terabecquerel, 1 x 1012 decays per second) are commonly used. Another unit of radioactivity is the curie, Ci, which was originally defined as the amount of radium emanation (radon-222) in equilibrium with one gram of pure radium, isotope Ra-226. At present it is equal, by definition, to the activity of any radionuclide decaying with a disintegration rate of 3.7 × 1010 Bq. The use of Ci is presently discouraged by the SI.

Alpha particles may be completely stopped by a sheet of paper, beta particles by aluminum shielding. Gamma rays can only be reduced by much more substantial barriers, such as a very thick layer of lead.
Different types of decay of a radionuclide. Vertical: atomic number Z, Horizontal: neutron number N.

The dangers of radioactivity and radiation were not immediately recognized. Acute effects of radiation were first observed in the use of X-rays when electrical engineer and physicist Nikola Tesla intentionally subjected his fingers to X-rays in 1896.[2] He published his observations concerning the burns that developed, though he attributed them to ozone rather than to X-rays. His injuries healed later.

The genetic effects of radiation, including the effects on cancer risk, were recognized much later. In 1927, Hermann Joseph Muller published research showing genetic effects, and in 1946 was awarded the Nobel prize for his findings.

Before the biological effects of radiation were known, many physicians and corporations had begun marketing radioactive substances as patent medicine, glow-in-the-dark pigments, and radioactive quackery. Examples were radium enema treatments, and radium-containing waters to be drunk as tonics. Marie Curie spoke out against this sort of treatment, warning that the effects of radiation on the human body were not well understood (Curie later died from aplastic anemia, which was likely caused by exposure to ionizing radiation). By the 1930s, after a number of cases of bone necrosis and death in enthusiasts, radium-containing medical products had nearly vanished from the market.

Types of decay

As for types of radioactive radiation, it was found that an electric or magnetic field could split such emissions into three types of beams. For lack of better terms, the rays were given the alphabetic names alpha, beta, and gamma, still in use today. While alpha decay was seen only in heavier elements (atomic number 52, tellurium, and greater), the other two types of decay were seen in all of the elements.

In analyzing the nature of the decay products, it was obvious from the direction of electromagnetic forces produced upon the radiations by external magnetic and electric fields that alpha rays carried a positive charge, beta rays carried a negative charge, and gamma rays were neutral. From the magnitude of deflection, it was clear that alpha particles were much more massive than beta particles. Passing alpha particles through a very thin glass window and trapping them in a discharge tube allowed researchers to study the emission spectrum of the resulting gas, and ultimately prove that alpha particles are helium nuclei. Other experiments showed the similarity between classical beta radiation and cathode rays: They are both streams of electrons. Likewise gamma radiation and X-rays were found to be similar high-energy electromagnetic radiation.

The relationship between types of decays also began to be examined: For example, gamma decay was almost always found associated with other types of decay, occurring at about the same time, or afterward. Gamma decay as a separate phenomenon (with its own half-life, now termed isomeric transition), was found in natural radioactivity to be a result of the gamma decay of excited metastable nuclear isomers, in turn created from other types of decay.

Although alpha, beta, and gamma were found most commonly, other types of decay were eventually discovered. Shortly after the discovery of the positron in cosmic ray products, it was realized that the same process that operates in classical beta decay can also produce positrons (positron emission). In an analogous process, instead of emitting positrons and neutrinos, some proton-rich nuclides were found to capture their own atomic electrons (electron capture), and emit only a neutrino (and usually also a gamma ray). Each of these types of decay involves the capture or emission of nuclear electrons or positrons, and acts to move a nucleus toward the ratio of neutrons to protons that has the least energy for a given total number of nucleons (neutrons plus protons).

Shortly after discovery of the neutron in 1932, it was discovered by Enrico Fermi that certain rare decay reactions yield neutrons as a decay particle (neutron emission). Isolated proton emission was eventually observed in some elements. It was also found that some heavy elements may undergo spontaneous fission into products that vary in composition. In a phenomenon called cluster decay, specific combinations of neutrons and protons (atomic nuclei) other than alpha particles (helium nuclei) were found to be spontaneously emitted from atoms, on occasion.

Other types of radioactive decay that emit previously seen particles were found, but by different mechanisms. An example is internal conversion, which results in electron and sometimes high-energy photon emission, even though it involves neither beta nor gamma decay. This type of decay (like isomeric transition gamma decay) did not transmute one element to another.

Rare events that involve a combination of two beta-decay type events happening simultaneously (see below) are known. Any decay process that does not violate conservation of energy or momentum laws (and perhaps other particle conservation laws) is permitted to happen, although not all have been detected. An interesting example (discussed in a final section) is bound state beta decay of rhenium-187. In this process, an inverse of electron capture, beta electron-decay of the parent nuclide is not accompanied by beta electron emission, because the beta particle has been captured into the K-shell of the emitting atom. An antineutrino, however, is emitted.

Radioactive decay rates

The decay rate, or activity, of a radioactive substance are characterized by:

Constant quantities:

half-life — symbol t1/2 — the time taken for the activity of a given amount of a radioactive substance to decay to half of its initial value.
mean lifetime — symbol t — the average lifetime of a radioactive particle.
decay constant — symbol ? — the inverse of the mean lifetime.

Although these are constants, they are associated with statistically random behavior of populations of atoms. In consequence predictions using these constants are less accurate for small number of atoms.

Time-variable quantities:

Total activity — symbol A — number of decays an object undergoes per second.
Number of particles — symbol N — the total number of particles in the sample.
Specific activity — symbol SA — number of decays per second per amount of substance. (The "amount of substance" can be the unit of either mass or volume.)

These are related as follows:

t_{1/2} = \frac{\ln(2)}{\lambda} = \tau \ln(2)
A = - \frac{dN}{dt} = \lambda N
S_A a_0 = - \frac{dN}{dt}\bigg|_{t=0} = \lambda N_0

where a0 is the initial amount of active substance — substance that has the same percentage of unstable particles as when the substance was formed.

Decay timing

The decay of an unstable nucleus is entirely random and it is impossible to predict when a particular atom will decay. However, it is equally likely to decay at any time. Therefore, given a sample of a particular radioisotope, the number of decay events -dN expected to occur in a small interval of time dt is proportional to the number of atoms present.

-\frac{dN}{dt} \propto N.

If N is the number of atoms, then the probability of decay (-dN/N) is proportional to dt:

\left(-\frac{dN}{N} \right) = \lambda \cdot dt.

Particular radionuclides decay at different rates, each having its own decay constant (?). The negative sign indicates that N decreases with each decay event. The solution to this first-order differential equation is the following function:

N(t) = N_0\,e^{-{\lambda}t} = N_0\,e^{-t/ \tau}. \,\!

Where N0 is the value of N at time zero (t = 0). The second equation recognizes that the differential decay constant ? has units of 1/time, and can thus also be represented as 1/t, where t is a characteristic time for the process. This characteristic time is called the time constant of the process. In radioactive decay, this process time constant is also the mean lifetime for decaying atoms. Each atom "lives" for a finite amount of time before it decays, and it may be shown that this mean lifetime is the arithmetic mean of all the atoms' lifetimes, and that it is t, which again is related to the decay constant as follows:

\tau = \frac{1}{\lambda}.

Simulation of many identical atoms undergoing radioactive decay, starting with either 4 atoms (left) or 400 (right). The number at the top indicates how many half-lives have elapsed. Note the law of large numbers: With more atoms, the overall decay is less random.

The previous exponential function, in general, represents the result of exponential decay. Although the parent decay distribution follows an exponential, observations of decay times will be limited by a finite integer number of N atoms and follow Poison statistics as a consequence of the random nature of the process.

A more commonly used parameter is the half-life. Given a sample of a particular radionuclide, the half-life is the time taken for half the radionuclide's atoms to decay. The half-life is related to the decay constant as follows:

t_{1/2} = \frac{\ln 2}{\lambda} = \tau \ln 2.

This relationship between the half-life and the decay constant shows that highly radioactive substances are quickly spent, while those that radiate weakly endure longer. Half-lives of known radionuclides vary widely, from more than 1019 years (such as for very nearly stable nuclides, e.g., 209Bi), to 10-23 seconds for highly unstable ones.

The factor of ln2 in the above relations results from the fact that concept of "half-life" is merely a way of selecting a different base other than the natural base e for the lifetime expression. The time constant t is the "1/e" life (time till only 1/e = about 36.8% remains) rather than the "1/2" life of a radionuclide where 50% remains (thus, t is longer than t½). Thus, the following equation can easily be shown to be valid.

N(t) = N_0\,e^{-t/ \tau} =N_0\,2^{-t/t_{1/2}}. \,\!

Since radioactive decay is exponential with a constant probability, each process could as easily be described with a different constant time period that (for example) gave its "1/3-life" (how long until only 1/3 is left) or "1/10-life" (a time period till only 10% is left), and so on. Thus, the choice of t and t½ for marker-times, are only for convenience, and from convention. They reflect a fundamental principle only in so much as they show that the same proportion of a given radioactive substance will decay, during any time-period that one chooses.
[edit] Example

A sample of 14C, whose half-life is 5730 years, has a decay rate of 14 disintegration per minute (dpm) per gram of natural carbon. An artifact is found to have radioactivity of 4 dpm per gram of its present C, how old is the artifact?

Using the above equation, we have:

N = N_0\,e^{-t/ \tau},

where: \frac{N}{ N_0} = 4/14 \approx 0.286,

\tau = \frac{T_{1/2}}{\ln 2} \approx 8267 years,
t = -\tau\,\ln\frac{N}{ N_0} \approx 10360 years.

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