Half life m&m lab answers

Half-Life

The half-life (τ1/2) is the time required for the decay rate of a sample of unstable nuclei to decrease by a factor of 2. If a0 is the initial decay rate, then the decay rate after the elapse of one half-life is a0/2. Substituting into Eq. 43.21 we have

(43.23)ao2 =aoe−λτ1/212=e −λτ1/2In12=−λτ1/2 τ1/2=0.693λor

The half-life can be determined by an inspection of the experimental data. Or if λ is determined by fitting Eq. 43.22 to the data, the half-life can be calculated from Eq. 43.23. There are very few physical parameters in nature spanning a greater range of values than the half-life for nuclear disintegration. The reported half-life of only 2 × 10−21 s for 5He is not much greater than the time required for a light beam to travel a distance equal to the diameter of a nucleus An extremely important nucleus in nuclear technology, 239Pu, has a half-life of 24,390 yr. A sampling of alpha decay half-lives is presented in Table 43.2.

Table 43.2. Some natural alpha-particle emitters and their half-lives *

Example 7

Experimental Determination of the Half-Life for Nuclear Decay

The radioactive element 137Ba has a relatively short half-life and can be easily extracted from a solution containing radioactive cesium (137Cs). We want to determine the half-life for the decay of 137Ba by using the experimental data presented in Figure 43.11

The logarithmic plot shown in Figure 43.11 is fitted by

Ina=8.44−0.262t

Thus the decay constant λ is 0.262 per minute and the half-life is

τ1/2=.6930λ=0.6930.262 =2.64min

The reported half-life for 137Ba is 2.55 min. The difference reflects experimental uncertainties.

X Metabolism and Clearance

The half-life of hPL in the mother, as estimated by the rate of disappearance after delivery of the placenta, is 10–20 minutes (Spellacy et al., 1966; Pavlou et al., 1972). As with other biological molecules the disappearance is biphasic with an initial rapid fall (yielding the half-life stated above) followed by a period of less rapid decrease. The differences in half-lives between individual subjects cannot be explained by observational error alone (Pavlou et al., 1972). Each subject has her own characteristic half-life and this can be attributed to differences in the mode of delivery and the physiological state of the patient.

The rate of synthesis of hPL has been estimated at 1–12 g/day (Beck et al., 1965; Kaplan et al., 1968). This is much greater than that of any other protein hormone, an interesting observation in the light of the dispute as to whether hPL has any function.

Maternal levels of hPL show no nyctohemeral rhythm (Beck et al., 1965; Samaan et al., 1966; Pavlou et al., 1972; Lindberg and Nilsson, 1973a). However, the variation over a 24-hour period is greater than that which can be attributed to the assay itself (Pavlou et al., 1972). Thus a single sample taken from an individual will not be representative of all samples taken from the individual, and the diagnostic significance of hPL levels is increased if serial levels are examined. The factors responsible for the variation are unknown. Posture has no effect (Ylikorkala et al., 1973), nor does strenuous physical exercise (Lindberg and Nilsson, 1973a; Pavlou et al., 1973). There is no change following the infusion of amino acids or ingestion of a protein meal (Tyson et al., 1971). Some have claimed that there are wide fluctuations in levels during normal labor (e.g., Cramer et al., 1971) but this has not been confirmed in other studies (Gillard et al., 1973).

Hindrance factor

The half-life for α decay, t12 α, is empirically related to the Q-value of the α decay Qα, by the Keller–Munzel relationship14:

(1.38)log10t12α =a(ZQα−Z23)+b

with Qα in MeV, and a and b are best-fit constants for t12 (in seconds):

It is obvious that the lowest half-life, hence highest rate of decay, is achieved in even-even nuclides. The α decay of such a nuclide is a transition to another even–even nuclide. Since the total angular momentum for even–even nuclide is zero, and their parity is even, the ground state to ground state (gs–gs) α transitions are 0+ → 0+ transitions. The half-life of this type of decay can be theoretically calculated using the so-called “one-body” model of α particles, which assumes that the α particle is pre-formed (as one body) within the nucleus before penetrating the Coulomb potential barrier by the tunneling effect. In this model, the probability of decay (i.e. release of the pre-formed α particle) in the nucleus depends on the probability of the particle hitting the Coulomb barrier times the probability of penetrating the barrier. The ratio between the measured partial half-life of a particular transition and that calculated using the one-particle model for an even-even nuclei at the energy of α decay is known as the hindrance factor (hf).

2.5.2 Large Sources

The half-life of radioactive sources and the penetration depth of β particles place an upper limit on the intensity per unit area of a positron source. For example, pure 64Cu of a thickness equal to the positron range [70] of 28 cm2g−1 would yield at most 1013 slow e+ sec−1 cm−2. If the isotope is made by the (n,γ) reaction (Table III) only 0.1% of the Cu would be radioactive given a flux of 5 × 1015 n cm-2 sec−1. To obtain a source of 1013 slow e+ sec−1 would require a 103-cm−2 source area. Such a large area source can be made generally useful only by some form of phase-space compression, i.e., brightness enhancement [61].

17.2 Physical Characteristics of Radionuclides

The half-life is one of the most important characteristics that determine the possibility of operating with nuclides and the danger that can be expected from them. Radionuclides with a very long lifetime have insignificant specific activity. For example, metal uranium can be picked up by hand, kept on a desk, drugs of uranium and thorium salts are widely used in chemistry and are in open sale. Depleted uranium (235U content is 0.2%–0.4%, in natural mixture 0.71%), formed as waste when 235U is separated from the natural mixture, is a radioactive isotope 238U, and it has a wide range of applications. Having a high density (19.1 g/cm³), it is used as counterweights in airplanes and missiles in various versions of radiation protection. The military industry uses it to produce armor sheets and armor-piercing subcaliber projectiles.

Nuclides with a short lifetime (<1 s) are very difficult to use, and they do not have time to bring significant harm. But short-lived nuclides with a lifetime of the order of tens of minutes and of several hours now attract a lot of interest and are increasingly used in medical diagnostics.

So, nuclides with a long lifetime have a small initial activity, and although they emit a long time, for human life, they are less dangerous than nuclides with intermediate half-lives. From practical use of radionuclides, it is clear that the most dangerous nuclides with T1/2–30 years are 137Cs and 90Sr.

Other important characteristics of the nuclide are the decay mode, the type of radiation, and the set of particle and quantum energies. In most cases, nuclides experience one of the variants of beta decay with transition to the excited state of the daughter nucleus and with the subsequent emission of gamma quanta. This can be seen, for example, in the energy diagrams of the decay shown in Fig. 17.1. In decays, beta-plus and beta-minus particles of different energies and a whole gamma-ray spectrum can be emitted. Heavy nuclides (A > 140) mainly experience alpha decay.

For adjustment and calibration of equipment, it is convenient to have nuclides emitting either only alpha or only beta particles. For practical applications, it is desirable that at the subsequent gamma transitions one or at most two particle lines are emitted. Such sources are called model spectrometric. The number of pure alpha, and beta emitters is quite small.

The characteristic maximum energies of beta decay are in the order of magnitude from tens of keV to several MeV.

Excited nuclei are formed either in the processes of alpha or beta decay, or in nuclear reactions. Therefore, there are no pure gamma emitters among radioactive sources, and all of them simultaneously emit alpha or beta particles. However, because the range of alpha or beta particles is much smaller than the penetration depth of gamma radiation, the particles can stop in the walls of the source, and gamma quanta can come out of it. Thus, the packed nuclide becomes a “pure” gamma emitter.

Usually, the energies of the gamma quanta accompanying alpha decay do not exceed 0.5 MeV, and the energy of the gamma quanta accompanying beta decay can be higher. Gamma quanta of higher energy are emitted by 56Co (T1/2 = 77 days). The decay scheme of this nuclide, which proceeds both by beta-plus decay or electron capture, gives rise to a complex spectrum of gamma quanta, whose energy extend up to 3.55 MeV.

As far as the gamma radiation, e.g., of the 60Co nucleus, is considered, the gamma quanta are emitted not by the 60Co nucleus but by the daughter nucleus 60Ni. 60Co undergoes beta-minus decay, turning into a 60Ni nucleus in the excited state; the selection rules for beta decay forbid decay to the ground state. With an overwhelming probability of 99.88%, the transition is carried out to the level of 2.5 MeV, from which the gamma transition to the ground state is also forbidden. Therefore, in the 60Ni nucleus, a cascade transition occurs through the level of 1.33 MeV: two gamma quanta are emitted, first a quantum with the energy of 1.17 MeV and second comes a quantum with the energy of 1.33 MeV through an average lifetime equal to 0.73 ps (Fig. 17.1).

In the case of beta-plus decay, the emitting positron annihilates with some electrons of matter in the encapsulation around the source. As a result of annihilation, the original positron and electron disappear and their rest energy transfers into two oppositely directed gamma quanta with the energy of 0.511 MeV each. So, all beta-plus sources practically, simultaneously with their gamma radiation, emit annihilation radiation.

Physical characteristics of some nuclides are presented in Appendix A5.

TRANSIENT EQUILIBRIUM

If the half-life of the parent is longer than that of the daughter, but decay of the parent is detectable and significant in the time frame of the observer (starting, remember, with Ad = 0), the activity of the daughter will increase until it exceeds the activity of the parent and then the daughter will decay with a half-time that is the same as the half-time of the parent.

For small values of t, Eq. (3.11b) must be used to compute the activity of the daughter, Ad, but after a sufficiently long time, usually 10 half-lives or more, e−λdt will be negligible compared to e−λpt. Still assuming Ad to be zero at time zero, Eq. (3.11b) may be rewritten as

(3.14)Ad(t)=Ap(t0)λdλd−λpe−λpt.

Since Ap(t0)e−λpt is the activity of the parent at any time t, the activity of the daughter is a constant derived from the decay constants of the parent and daughter times the activity of the parent (after sufficient half-lives have passed):

(3.15)Ad(t)=Ap(t)λdλd−λ p.

When the conditions of Eq. (3.15) are achieved, the two nuclides are said to be in transient equilibrium. After equilibrium is achieved, the activities of parent and daughter are in a constant proportion which is determined by their respective half-lives. It must be remembered, however, that the activities of both parent and daughter are decreasing during the period of observation. This transient equilibrium condition is of particular importance to nuclear medicine, since it is possible to prepare a form of the parent attached in a nonremovable state to some substrate. The daughter is produced by decay of the parent, and the daughter can be removed separately from the parent by the use of a suitable eluting agent. One of the most widely used of these is a generator for the production of 99mTc from its parent, 99Mo.

The ratio of daughter activity to that of the parent at equilibrium is easily calculated from

(3.16)AdAp=λdλd−λp.

If this equation is rewritten in terms of half-lives rather than decay constants, the following expression is developed (the student may wish to do the simple algebra to show that it is correct):

(3.17) AdAp=TpTd−Td.

The derivation will not be given (see Suggested Additional Readings), but the time at which the maximum daughter activity is reached is given by the expression

(3.18)tmax=1.44TpTdTp−TdlnTpTd,

if the activity of the daughter, Ad(0), is taken to be zero at time zero. Note that the term 1.44TpTd is the product of the mean life, τ, of the parent and the daughter.

3.1.6 Short half-life materials

If the half life of the nuclide being measured is so short that the disintegration rate reduces significantly during the measurements, allowance must be made for the decay. Suppose the disintegration rate at arbitrary time zero (the start of the measurement) is N0 and at the end of the measurement, after time t, it is Nt. We know that Nt = N0 exp(—λt) and the number of disintegrations N in time t is N0 [1—exp(—λt)]/λ; the ratio of the total counts in time t, nβt nγt/nct is equal to N and hence N0 can be calculated. The situation is complicated if the count-rate dependent corrections are significant since the magnitude of these corrections is variable during the determination.

We must now consider the corrections which arise from the effects of the finite resolving time of the coincidence unit and the finite dead times of the detectors (including subsequent amplifiers, discriminators etc.).

1.3.B The decay equation

Radioactive decay is a random process and has been observed to follow Poisson distribution (see chapter on Statistics). What this essentially means is that the rate of decay of radioactive nuclei in a large sample depends only on the number of decaying nuclei in the sample. Mathematically, this can be written as

(1.3.12)dNdt∝−Nord Ndt=−λdN.

Here dN represents the number of radioactive nuclei in the sample in the time window dt. λd is a proportionality constant generally referred to in the literature as the decay constant. In this book the subscript d in λd will be used to distinguish it from the wavelength symbol λ that was introduced earlier in the chapter. Conventionally, both of these quantities are represented by the same symbol λ. The negative sign signifies the fact that the number of nuclei in the sample decreases with time. This equation, when solved for the number N of the radioactive atoms present in the sample at time t, gives

(1.3.13)N=N0e−λ dt,

where N0 represents the number of radioactive atoms in the sample at t=0.

Equation (1.3.13) can be used to determine the decay constant of a radionuclide, provided we can somehow measure the amount of decayed radionuclide in the sample. This can be fairly accurately accomplished by a technique known as mass spectroscopy (details can be found in the chapter on Spectroscopy). If the mass of the isotope in the sample is known, the number of atoms can be estimated from

(1.3.14)N=NAw nmn,

where NA=6.02×1023 is the Avogadro number, wn is the atomic weight of the radionuclide, and mn is its mass as determined by mass spectroscopy.

Although this technique gives quite accurate results, it requires sophisticated equipment that is not always available. Fortunately, there is a straightforward experimental method that works almost equally well for nuclides that do not have very long half-lives. In this method the rate of decay of the sample is measured using a particle detector capable of counting individual particles emitted by the radionuclide. The rate of decay A, also called the activity, is defined as

(1.3.15) A=−dNdt=λdN.

Using this definition of activity, Eq. (1.3.13) can also be written as

(1.3.16)A=A0e−λdt ,

where A0=λdN0 is the initial activity of the sample.

Since every detection system has some intrinsic efficiency ϵ with which it can detect particles, the measured activity C would be lower than the actual activity by the factor ϵ.

(1.3.17)C=∈Α=∈[−dNdt]=∈ λdN

The detection efficiency of a good detection system should not depend on the count rate as this would imply nonlinear detector response and consequent uncertainty in determining the actual activity from the observed data. The above equation can be used to determine the count rate at t=0:

(1.3.18)C0=∈λdN0

The above two equations can be substituted in Eq. (1.3.13) to give

(1.3.19)C=C0e−λdt.

What this equation essentially implies is that the experimental determination of the decay constant λ is independent of the efficiency of the detection system, although the counts observed in the experiment will always be less than the actual decays. To see how the experimental values are used to determine the decay constant, let us rewrite Eqs. (1.3.16) and (1.3.19) as

(1.3.20)ln(A)=−λt+ln(A0)and

(1.3.21)ln(C)=−λt+ln(C 0).

Hence if we plot C versus t on a semi-logarithmic graph, we should get a straight line with a slope equal to −λ. Figure 1.3.2 depicts the result of such an experiment. The predicted activity has also been plotted on the same graph using Eq. (1.3.20). The difference between the two lines depends on the efficiency, resolution, and accuracy of the detector. Equation (1.3.13) can be used to estimate the average time a nucleus would take before it decays. This quantity is generally referred to as “lifetime” or “mean life” and is denoted by the symbol τ or T. In this book it will be denoted by the symbol τ. The mean life can be calculated from

(1.3.22)τ=1 λd.

Another parameter that is extensively quoted and used is the half-life. It is defined as the time required by half of the nuclei in a sample to decay. It is given by

(1.3.23)T 1/2=0.693τ=ln(2)λd .

Since mean and half-lives depend on the decay constant, the experimental procedure to determine the decay constant can be used to find these quantities as well. In fact, whenever a new radionuclide is discovered, its half-life is one of the first quantities to be experimentally determined. The half-life of a radionuclide can range from a microsecond to millions of years. Unfortunately, this experimental method to determine the half-life does not work very well for nuclides having very long half-lives. The reason is quite simple: For such a nuclide the disintegration rate is so low that the difference in counts between two points in time is insignificantly small. As we saw earlier in this section, for such radionuclides, other techniques such as mass spectroscopy are generally employed.

Discussion

Van Lieshout: How well does the half-life of the missing transition fit with the M2 multipolarity?

Hollander: Well, half-lives of very low energy transitions are rather insensitive to energy, so I cannot really answer your question. It fits fine if you want to extrapolate the half-life energy curves as known, but as the radiative half-life gets very long, the conversion coefficient also gets very large and these two factors tend to cancel.

Van Lieshout: Well, the half-life for a low energy transition is almost independent of energy, is it not?

Hollander: If you want to look at it that way, it fits very well.

Rasmussen: Well, I would not know how to extrapolate a conversion coefficient down to threshold with confidence. In fact, I think this underscores the importance of our getting a good tabulation of threshold conversion coefficients. Perhaps this expansion that Dr. O'Connell spoke about giving the first order behavior as the energy goes just above threshold should be used to prepare numerical tables. I think we really need that sort of number for M2 and E3 to be able to make the analysis for the matrix elements in this case, based on the 24-sec half-live of the isomer.

Van Lieshout: This is the second elusive transition that in Ag110 still has not been found. I guess it is less than 250 eV.

Suppelmentary Questions and Examples

1.

The half-life of 31Si is 2.6 h. How long will it take the activity of a sample to decrease from 6.0 mCi to 0.75 mCi?

10.4 h

2.6 h

5.2 h

7.8 h

The amount of radioactive material in a certain sample has decreased to 1/32 of the amount that existed 225 minutes ago. What is the half-life of the substance?

The most concentrated ionization in matter is caused by the passage of

α particles.

β particles.

X rays.

electrons.

Radioactive isotopes can be obtained from all of the following except

radioactive wastes from nuclear reactors.

neutron capture reactions.

irradiation by X rays.

bombardment of material by high-speed particles in an accelerator.

Explain how chemical reactions can be induced by the use of radiation.

Irradiation by electrons is used to sterilize medical sutures. Could the same technique be used to sterilize surgical instruments? Explain.

List some ways that nuclear radiations are used in the chemical industry.

Neutron activation analysis depends on the fact that

each radioactive isotope has a characteristic half-life and type of emitted radiation.

all elements undergo neutron capture.

irradiation causes the disruption of chemical bonds.

radiation is absorbed as it travels through matter.

The two most significant sources of radiation exposure for average individuals are

X rays and fallout.

nuclear reactors and fallout.

X rays and natural sources.

fallout and cosmic rays.

The QF of X or γ rays is

1.

5.

10.

20.

Which is likely to produce greater radiation damage, a 100-rad whole-body dose of X rays or a 20-rem whole-body dose of α particles?

A dose of radiation that will cause inevitable death to human beings is

1 rem.

800 rem.

300 rem.

25 rem.

Long-term effects of radiation include

radiation sickness.

increased susceptibility to leukemia.

genetic mutation.

somatic effects.

Describe the radiation treatment given for a cancerous thyroid gland.

The unit rad specifies

the total amount of radiation received by an object.

the biological effectiveness of radiation.

the amount of energy absorbed per unit mass during irradiation.

the equivalent whole-body dose of radiation.

Draw the symbol used to indicate a radiation hazard area.