Physics 219 Help Session. Date: Wed 12/07, Time: 6:00-8:00 pm. Location: Physics 331

Transcription

1 Lecture 25-1 Physics 219 Help Session Date: Wed 12/07, Time: 6:00-8:00 pm Location: Physics 331

2 Lecture 25-2 Final Exam Dec :00-3:00pm in Phys. 112 Bring your ID card, your calculator and a soft pencil with you! Exam Calculator: When taking a Physics 219 Exam, there is only one type of calculator is accepted: CASIO fx-260 SLRSC FRACTION. NO OTHER BRAND or TYPE WILL BE ALLOWED!

3 Lecture 25-3 Alpha Decay Alpha decay is where a parent nucleus (P) decays into a daughter nucleus (D) plus an alpha particle (= 4 He nucleus of 2 protons and 2 neutrons). A Z A4 Z2 Sum of the masses of D and alpha particle < Mass of P The difference (disintegration energy) is converted into the kinetic energy of D and alpha, according to momentum conservation. 4 2 P D Example: 210 Po Pb? 4 2 (Former Russian spy Alexander Litvinenko was killed by this in 2006!)

4 Lecture 25-4 Alpha Decay Example A Z A4 Z2 4 2 P D Example: 210 Po Pb? u (with 84 electrons) u (with 82 electrons) u (with 2 electrons) m ( ) u E m c MeV

5 Lecture 25-5 Physics 219 Question 1 Dec decays by alpha decay. What are A and Z for the daughter nuclide? (Alpha decay emits.) Ra A. 222,86 (Rn) B. 222,84 (Po) C. 224,84 (Po) D. 226,88 (Ra) E. 224,86 (Rn)

6 Lecture 25-6 Beta Decay There are two forms of beta decays: In beta-minus decay, a neutron turns into a proton, emitting an electron (e - ) and an antineutrino. A Z P D e v A Z1 n p e Spontaneous by itself In beta-plus decay, a proton turns into a neutron, emitting a positron (e +, electrons s antiparticle) and a neutrino. A Z P D e v A Z1 p n e not possible by itself spontaneously, but can occur within a nucleus

7 Lecture 25-7 Beta Decay 2 Example: 14 N and 15 N are stable but 13 N is not. The latter has too few neutrons. So it decays by turning a proton into a neutron (plus positron and neutrino). N e 13 13? 7 6 C m ( m m ) m C e N Beta plus decay!! ( M 6 m m ) ( M 7 m ) C e e N u neutral atom masses Occurs spontaneously! Electron capture: Turns a proton into a neutron e p e n A A ZP e Z1 D v not by itself

8 Lecture 25-8 Gamma Decay Gamma decay is where an excited nucleus drops to a lower energy state, emitting a gamma ray (high energy photon). A * A ZP ZD Example: Th Th 208 *

9 Lecture 25-9 Radioactive Decay Rates Radioactive decays occur randomly; you only know the probability. = probability of a decay/unit time (decay constant) R = N (where N is the instantaneous number of the radioactive nuclei) is the average number of decays per unit time. (decay rate or activity) Activity R has a unit of decays/sec which is also called a becquerel (Bq): curie decay 1Bq 1 1s s 10 1 Ci Bq 1 = 1/ is expected time until the next decay (time constant or mean lifetime)

10 Lecture Mathematics of Radioactive Decay The number of undecayed nuclei N(t) varies as t / N() t N0e where = 1/ is the time constant. The half-life T 1/2 is the time required for half of the original number of the radioactive nuclei to decay: T ln / 2 Since the activity (decay rate) R = N where is constant, it also varies as t / R() t R0e

11 Lecture Radiocarbon Dating 12 C and 13 C are stable but 14 C is radioactive with the half-life of 5730 years. C N e Though essentially none of the 14 C that was present when the Earth was formed still remains today, new 14 C is continually formed in atmosphere by cosmic rays, absorbed by plants, eaten by animals, and incorporated into every living thing. Equilibrium ratio of 14 C to 12 C is known!! = 1.3 x When an organism dies, 14 C in it is no longer replenished from outside, and the ratio of 14 C to 12 C begins to decrease from the equilibrium value. By measuring the 14 C to 12 C ratio of a (dead) sample and comparing it to the equilibrium value, we can estimate the time since its death.

12 Lecture Physics 219 Question 2 Dec C has an activity of 0.25 Bq per gram of carbon for living organisms. Let s say the half-life of 14 C is about 6000 years. If a sample of charcoal from an archeological dig yields a 14 C activity of Bq per gram of carbon, approximately how old is the material? A. 3,000 years B. 4,500 years C. 6,000 years D. 12,000 years E. 24,000 years

13 Law of Radioactive Decay (From G.R.R. page 1062) Radioactive decay is a quantum mechanical process that can only be describe in terms of probability. Given a collection of independent and identical nuclides, they do not all decay at the same time and there is no way to determine which one decays when. The decay probability for one nucleus is independent of its past history and of other nuclei. Each radioactive nuclei has a common decay probability per unit time, written (no relation to wavelength). The decay probability per unit time is also called the decay constant. The decay constant = probability of decay unit time The probability that a nucleus decays during a short time interval tis t

14 In a collection of a large number N of identical and independent nuclei, each one has the same decay probability per unit time. Since the nuclei are independent, the decay of one has nothing to do with the decay of an other. Since the decays are independent, the average number that decay during a short time interval t is just N times the probability that any one decays: N Nt (Eq.1) The above equation is valid for short time interval because it assumes that the number of nuclei is a constant N. Thus after rearranging Eq. 1 N N t

15 Differential Equation and its Solution for Rad. Decay (L.G.) We go to the lim t 0 N t dn dt dn dt N and obtain (Eq.2) The function which is a solution of the Eq.2 differential equation: t 0 0 N N e N e where is the time constant 1 t/ (Eq.3) (Eq.4) an N 0 is the number of nuclei at t=0.

16 The time t which has to elapse to obtain any given ratio can be obtained as After rearrangement: () ln N t t N 0 t () ln Nt N0 N( t) / N 0 (Eq.5) Definition of Half Life T 1/2 In nuclear physics is expressed in terms of half life: T1/2 The relation between can be obtained from Eq. 5 When half of the nuclei decayed thus in Eq.3 t T 1/2 T 1/2 and

17 Thus we can express T 1/2 Thus 14 N( t) N / 2 and t =T 1/2 0 1/2 in terms of T 1/2 1/ 2N 1 ln 0 ln N 2 0 T = 1/ (Eq.6) From C decay T is measured to be: 5730yr 5730yr Thus Eq. 7 becomes = (Eq.7) (Eq.8)

18 Substituting Eq. 8 into Eq. 5 we obtain t N( t) 5730 yr ln N (Eq.9) Relation Between Radio Active and Activity Decay Laws From Cutnell and Johnson VIII, Vol.2, page The activity R is the number of disintegration per second l x 0 im N dn t dt N N( t) N( t) R( t) Thus N(0) N(0) R( 0)

19 and thus t R( t) 5730 yr ln R(0) (Eq.10) The unit of R(t) is 1Bq.=1 disintegration per sec Since source of is cosmic rays whose intensity has not changed for years, thus we know R(0) for living things: R(0) = 0.25 Bq/gram of carbon The measured activity of 1gr of carbon of the Iceman Ötzi was measured: R(t) = 0.131Bq/gram of carbon Substituting these values into Eq. 10 we finally have t t 14 C Bq / gr 5730yr ln 0.25 Bq / gr year (Eq.11)

20 Relation between the Lifetime and Mass Uncertainties of Radioactive Particles The wave function ( xt, ) of a radioactive particle, rest at the origin is: ( ie ) / ( /2) / R t ier t 2 ( x, t) (0) e (0) e The observable time distribution of the radioactive decay law is: N() t x t/ e The corresponding wave function ( p, E) g(0) g( E) where ge ( ) ( E E ) i / 2 R K

21 The observable energy dependence of the particle mass distribution is: g( E) g( E) * 2 /4 2 2 ( E E ) / 4 R Thus an exponential time decay distribution with time constant predicts a Breit-Wigner mass distribution with a width, where in agreement with the uncertainty relation: / as shown on the figure below.

22 N 0 N(t) 0 t/ Ne 2 N(mc ) t 2 Mc 2 mc

23 Lecture Radiocarbon Dating The activity per gram of carbon is proportional to the relative abundance of 14 C nuclei in the sample. 14 C activity of a live organism = 0.25 Bq per gram of carbon. T ½ of 14 C is 5730 yrs. Question: If the 14 C activity of a sample is Bq per gram of carbon, then how long has it been since it was last alive? Answer: 5340 years. R() t R e 0 t/ Generally, Here, R(t)/R 0 =0.131/0.25= /2 Knowing that = t ln( R( t) / R0 ) T R Bq / gram R( t) Bq / gram Iceman found frozen in Italian Alps in 1991.

24 Lecture Induced Nuclear Reactions and Compound Nucleus Formation (H. R ) An induced nuclear reaction requires external action such as hitting the nucleus with another particle or radiation. Neutron Activation Compound Nucleus formation e p n e p n n N p C n N p C n Ag Ag Ag He N F O H Sum of daughter masses Sum of original masses

25 Lecture Biological Effects of Radiation Products of radioactive decays (alpha particles, electrons, positrons, and gamma rays) as well as other particles (protons and neutrons) can cause damage when they interact with cells. Typical energy of 1 MeV O(10000) times the energy required to ionize a molecule Ionized (i.e., charged) molecules are chemically active! The amount of (radiation) energy absorbed per unit mass of tissue Cell function disrupted, cell death, DNA damage, cancer growth, 1 J/kg = 1 Gy (gray) 0.01 Gy = 1 rad absorbed dose

26 Lecture Measuring Biological Effects of Radiation While rad measures the actual energy absorbed by a unit mass of tissue, its damaging effect is measured in rem. Extent of damage depends on the type of radiation and tissue exposed even for the same absorbed dose. Biologically equivalent dose (measured in rem) = Absorbed dose (in rad) x Quality factor QF is defined relative to the damage done by a 200 kev X ray.

27 Lecture Radiation Exposure and Effects On average you will receive about 0.3 rem of radiation per year due to natural sources: 0.2 rem from inhaled radon-222 gas rem from radioactive nuclei in what you eat ( 14 C and 40 K) rem from cosmic rays You will also receive another 0.06 rem due to human activities: the 0.06 rem is mostly from medical and dental x-rays about rem from past nuclear fallout (weapons and reactors) Flying in a jet for 40 hours will expose you to another 0.03 rem from cosmic rays. A single dose of 50 rem causes few effects, but one of 500 rem is fatal about half the time. A large single dose causes radiation sickness (nausea, diarrhea, vomiting, hair loss, ).

28 Lecture Penetration of alpha rays Alpha particles can only penetrate about mm of human tissue. They can be stopped by a few cm of air or by a very thin sheet of aluminum foil. Even though alpha s are potentially most dangerous because of their high energies, since they don t penetrate very far, those outside the body are not so worrisome. Alpha particles produced inside the body, however, are extremely dangerous. Inhaled radon gas exposes lung to alpha s and ingested radioactive iodine exposes thyroid.

29 Lecture Penetration and Effects of Beta Rays Beta particles (electrons and positrons) penetrate a few cm of human tissue, several meters of air or about 1 cm thick aluminum. So, although they are still more dangerous when produced inside the body, they are also dangerous coming from outside. Electrons ionize molecules and also emit X rays. Positrons quickly annihilate with an electron and produce 2 photons (cf. PET). Nuclear fission bomb fallout contains Strontium 90. It is similar to calcium chemically, and easily incorporated into bones and teeth. The radioactive Strontium 90 decays by a beta decay with a half life of 28 years. This increases chances of leukemia and other cancers.

30 Lecture Fission In large nuclei, the protons and neutrons are not as strongly bound together as in smaller nuclei. They may break up into two nuclei. fission Induced fission: e.g., 235 U, which decays by capturing a slow neutron, produces more new neutrons. So a chain reaction can be set up with large amounts of energy released.

31 Lecture Induced Fission of 235 U The incident neutron provides the necessary energy to compete with surface tension. Many different fission reactions are possible for a particular parent nucleus: for 235 U, some examples are 1 0n n U 236 U * U 236 U * Ba Xe Kr n 95 Sr n About 1 MeV per nucleaon of energy is released

32 Lecture Typical Decay Sequence of 238 U In the first step of the decay of a radioactive 238 U nucleus, the branching ratios are: alpha decay (almost 100%) spontaneous fission (5.45x10-5 %) (double) beta decay (2.2x10-10 %)

33 Lecture Fission Reactors

34 Lecture 25-34