alpha decay equation calculator alpha decay equation calculator

Your Mobile number and Email id will not be published. Alpha Decay. There are 5 different types of radioactive decay. So he talks about the three types of radioactive decay, but how do you know what kind of decay say, Uranium, for instance, would give off? Take 4 away from the mass number (210-4 = 206) Take 2 away from the atomic number (84-2 = 82). In practice given some reagents and products, \(Q\) give the quality of the reaction, i.e. We measure it using detectors. How to calculate the kinetic energy of an alpha particle - We are assuming that the energy released in this decay, given by the disintegration energy, all goes . Alpha decay or -decay is a type of radioactive decay in which the atomic nucleus emits an alpha particle thereby transforming or decaying into a new atomic nucleus. This is the best app to help you with math problems. What else do we make? This relationship enables the determination of all values, as long as at least one is known. Alpha decay formula can be written in the following way . In -decay, the mass number of the product nucleus (daughter nucleus) is four less than that of the decaying nucleus (parent nucleus), while the atomic number decreases by two. 17.3: Types of Radioactivity: Alpha, Beta, and Gamma Decay - decay is the radioactive emission of an -particle which is the nucleus of 4 This formula applies to a potential barrier of constant height U0, Required fields are marked *, \(\begin{array}{l}_{Z}^{A}\textrm{X}\rightarrow _{Z-2}^{A-4}\textrm{Y}+_{2}^{4}\textrm{He}\end{array} \), \(\begin{array}{l}_{Z}^{A}\textrm{X} \textup{ is the parent nucleus}\end{array} \), \(\begin{array}{l}_{Z-2}^{A-4}\textrm{Y} \textup{ is the daughter nucleus}\end{array} \), \(\begin{array}{l}_{2}^{4}\textrm{He} \textup{ is the released alpha particle}\end{array} \), \(\begin{array}{l}_{92}^{238}\textrm{U} \textup{ to thorium } _{90}^{234}\textrm{Th} \textup{ with the emission of a helium nucleus } _{2}^{4}\textrm{He}.\end{array} \), \(\begin{array}{l}_{92}^{238}\textrm{Ur}\rightarrow _{90}^{234}\textrm{Th}+_{2}^{4}\textrm{He}\end{array} \), \(\begin{array}{l}_{93}^{237}\textrm{Np}\rightarrow _{91}^{233}\textrm{Pa}+_{2}^{4}\textrm{He}\end{array} \), \(\begin{array}{l}_{78}^{175}\textrm{Pt}\rightarrow _{76}^{171}\textrm{Os}+_{2}^{4}\textrm{He}\end{array} \), \(\begin{array}{l}_{64}^{149}\textrm{Gd}\rightarrow _{62}^{145}\textrm{Sm}+_{2}^{4}\textrm{He}\end{array} \). A-4 \\ So we need 90 positive charges. Further, take for example Francium-200 (\({ }_{87}^{200} \mathrm{Fr}_{113}\)). Though the alpha particles are not very penetrating, the substance that undergoes alpha decay when ingested can be harmful as the ejected alpha particles can damage the internal tissues very easily even if they have a short-range. Whenever it rearranges into a low energy level, a high energy photon is shooted out which is called the gamma-ray. Step 1: Select a Radionuclide Select radionuclide: You can't. The phenomenon of alpha decay is also found in rare earth elements ranging from neodymium, which has atomic number 60, to lutetium, which has atomic number 71. Application of matrix and matrix algebra Best video camera for online content Calculate pka from ka Calculator soup investment calculator Create and share an absolute value equation with only one solution Create math questions Examples of perpendicular lines Faraday's law equation Find f a f t 8t 24 t 7 Find function from points wolfram alpha . Let's do beta decay. If the isotope that you wish to decay is not on the drop down list, check the 'not listed' check-box and manually enter the isotope name and its half-life to perform the calculation. Radioactivity is the phenomenon exhibited by the nuclei of an atom as a result of nuclear instability. Determine mathematic equation. something else is produced. On the other hand, a Carbon-11 nucleus . Alpha-decay energies are most precisely measured in magnetic spectrometers. What is the relevant momentum \(\hbar \kappa \) here? However, with practice and perseverance, it is possible to improve one's skills in this area. not part of this video, so we'll just ignore it for now. An alpha particle, which is the same as a helium nucleus, is represented by 4 2H e. How would you complete the equation for this reaction? . What is the mechanism behind the phenomenon of alpha decay? Therefore, such nuclei accelerate the stability by reducing their size results in alpha decay. More specifically, the decrease in binding energy at high \(A\) is due to Coulomb repulsion. Using the above equations, it is also possible for a relationship to be derived between t1/2, , and . The 5 different types of radioactive decay are alpha decay, beta decay, gamma emission, positron emission (+ decay), and electron capture. Since all particles possess some rest energy in the form of mass, which we can find through the mass-energy equivalence method, we can then find the change in energy by comparing them. Helps other - Leave a rating for this alpha decay (see below). The spontaneous decay or breakdown of an atomic nucleus is known as Radioactive Decay. Also, note that because this was an alpha reaction, one . The decay probability has a very strong dependence on not only \(Q_{\alpha} \) but also on Z1Z2 (where Zi are the number of protons in the two daughters). Gamma Decay: The nucleus has orbiting electrons which have some energy, when an electron jumps from a level of high energy to low energy, there is an emission of a photon. around the world. When an alpha decay takes place, the electric charge in the daughter nucleus bemomes smaller than in the parent nucleus. See more. This disruptive electromagnetic force is proportional to the square of its number. The steps to find the radioactive decay are given here. A general equation for this type of alpha decay radiation is: AZXZ-4Z-2Y+42He Z A X Z - 2 Z - 4 Y + 2 4 He . Recall that in the case of a square barrier, we expressed the wavefunction inside a barrier (in the classically forbidden region) as a plane wave with imaginary momentum, hence a decaying exponential \( \psi_{i n}(r) \sim e^{-\kappa r}\). Here, a high-energy radioactive nucleus can lower its energy state by emitting electromagnetic radiation. 4. Enter your queries using plain English. 14964Gd undergoes decay to form one nucleus of Sm. Calendar pickers are included for date entry convenience. Notice that its no coincidence that its called \(Q\). Then \(\log \left(P_{T}\right)=\sum_{k} \log \left(d P_{T}^{k}\right)\) and taking the continuous limit \(\log \left(P_{T}\right)=\int_{R}^{R_{c}} \log \left[d P_{T}(r)\right]=-2 \int_{R}^{R_{c}} \kappa(r) d r\). Note: this video is aimed at GCSE Physics students, so does not discuss beta-positive decay or neutrinos/antineutrinos. So I need an atomic number of 91. This is also equal to the total kinetic energy of the fragments, here Q = T X + T (here assuming that the parent nuclide is at rest). little bit more detail. Continuing learning nucleur physics - read our next physics tutorial. Updated: 08/20/2021 Create an account Isotopes and Nuclear Symbols Example Problem. In this article, you will study alpha decay in detail. going to affect our numbers, so if we start with nucleons, we have 99 nucleons on the left, we're going to have 99 Learn about radioactive decay (alpha, beta, & gamma), how to balance nuclear equations for nuclear decay, and how to predict the products of nuclear reactions. Below are shown three equivalent formulas describing exponential decay: If an archaeologist found a fossil sample that contained 25% carbon-14 in comparison to a living sample, the time of the fossil sample's death could be determined by rearranging equation 1, since Nt, N0, and t1/2 are known. According to this law, those isotopes which are short-lived emit more energetic alpha particles as compared to those isotopes which are long-lived. On the other hand, 210Pb nucleus has 82 protons and 124 neutrons, thereby resulting in a ratio of 82/124, or 0.661. Direct link to Gray Can's post He didn't mention positro, Posted 6 years ago. And also actually, - [Voiceover] Let's look at three types of radioactive decay, and we'll start with alpha decay. To understand what is meant by multiplicity, take, for example, . resulting in a Q-value of: Lets model each type of decay through equations. The deflection of alpha decay would be a positive charge as the particles have a +2e charge. So the equation is So here's our electron and an electron ejected from the nucleus Todd Helmenstine is a science writer and illustrator who has taught physics and math at the college level. An alpha particle has the same composition as a helium nucleus. In some cases, linear algebra methods such as Gaussian elimination are used, with optimizations to increase speed and reliability. To avoid ambiguous queries, make sure to use parentheses where necessary. If you look at the periodic table, and you find the atomic number of 91, you'll see that this is protactinium. Then: \[Q_{\alpha}=B\left(\begin{array}{c} A \\ This decay leads to a decrease in the mass number and atomic number, due to the release of a helium atom. For example, any of uranium isotopes such as (23892U) may emit an alpha particle and thus become a thorium isotope (23490Th). I have a bunch of confusion how the Gama ray decays. So thorium-234 is our other product. Any help would be appreciated, thanks! But inside the nucleus, the nucleons are bound to one another by the strong nuclear force, so you also get quantized energy levels for that smaller system. As a result, the binding energy in daughter nucleus is smaller too. ejected from the nucleus. negative charge here, so I have a negative one charge, and so I must need 91 positive charges, because 91 positive charges and one negative charge gives me 90 positive charges on the right. Retrieved from https://www.thoughtco.com/alpha-decay-nuclear-reaction-problem-609457. #""_2^4He# can be also represented as #alpha# as this is the sign for alpha. and giving that atomic masses of these three materials are 266 u, 261 u and 4.003 u respectively, we obtain for the mass defect of this process: Since this value corresponds to 1.66054 10-27 kg, we obtain for the binding energy of parent nucleus: This energy corresponds to energy released during the alpha decay; it is in the form of kinetic energy. Exponential decay refers to a process in which a quantity decreases over time, with the rate of decrease becoming proportionally smaller as the quantity gets smaller. You can then email or print this nuclear decay calculation as required for later use. If you did it would be great if you could spare the time to rate this physics lesson (simply click on the number of stars that match your assessment of this physics learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of physics and other disciplines. . The important thing is to be able to look at a nuclear equation, recognize it as beta decay, and be able to write everything in your nuclear equation. This app is a MUST HAVE. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Two protons are present in the alpha particle. Alpha decay: The nucleus splits into two chunks, a little chunk called an "alpha particle" (which is just two protons and two neutrons) and a daughter nucleus with a lower atomic number than the initial nucleus.The "radiation" here is the small chunk, which generally moves away from the nucleus at a pretty high speed. It can't take the place of an electron in a regular chemical reaction. Select the correct answer and click on the Finish buttonCheck your score and answers at the end of the quiz, Visit BYJUS for all Physics related queries and study materials. Arc Length S = R Discover Resources It is made of two down quarks (charge -1/3) and one up quark (charge 2/3). For example, in the alpha decay process shown below, the proton-neutron ratio of "parent" nucleus (Polonium, Po) is, and the proton-neutron ratio of "daughter" nucleus (Lead, Pb) is. He holds bachelor's degrees in both physics and mathematics. However \(\alpha\) decay is usually favored. How does alpha decay differ from beta decay? Direct link to Dhruv Pisharody's post No, a neutron is not made, Posted 8 years ago. We find that \(Q \geq 0\) for \(A \gtrsim 150\), and it is \(Q\) 6MeV for A = 200. A Uranium nucleus, 23892U undergoes alpha decay and turns into a Thorium (Th) nucleus. and diagnostic procedures, because we have ways of Can someone explain that or link to a video that better explains it? Well, that'd be 234 minus 91. This leads to the following observations: A final word of caution about the model: the semi-classical model used to describe the alpha decay gives quite accurate predictions of the decay rates over many order of magnitudes. The atoms involved in radioactive decay are referred to as isotopes. Can help answer any style question in detail. Get initial activity, decay time, and half-life. Alpha decay is schematically shown in the figure below. Generally an alpha particle is referred to a helium nucleus since it has 2 neutrons and 2 protons but no electrons. You may also back decay sources to find out the original activity (or for any date), Broyden's method for solving nonlinear equations, Cambridge checkpoint mathematics coursebook 6, Find the shaded area of a circle calculator, Find the value of the algebraic expression, How to find the transpose of a matrix in java, How to get rid of absolute value on both sides, How to solve number system problems in aptitude. and it's an electron, so a negative one charge, Direct link to jpkobrossi16's post When Thorium performs bet, Posted 6 years ago. This polynomial is considered to have two roots, both equal to 3. To calculate how long it would take for a sample of cesium137 to decrease in activity from 15 to 0 . Write a balanced nuclear equation for the alpha decay of thorium-232. When we think about what else is made, we know that nucleons are conserved, so we have one nucleon on the left, one nucleon on the right. And since the atomic For equation solving, Wolfram|Alpha calls the Wolfram Language's Solve and Reduce functions, which contain a broad range of methods for all kinds of algebra, from basic linear and quadratic equations to multivariate nonlinear systems. When Q > 0 energy is released in the . Therefore, the resulting Thorium nucleus should have 234 mass numbers and 90 atomic numbers. 30864 views Example Problem: Isotopes and Nuclear Symbols, How to Write the Nuclear Symbol of an Atom, Basic Model of the Atom and Atomic Theory, Nuclear Structure and Isotopes Practice Test Questions, Chemistry Vocabulary Terms You Should Know. ejected from the nucleus of a thorium-234, so we're The alpha particle is the same as a helium nucleus with 2 protons and 2 neutrons. in the previous video. Easy to use and type in problems, 5stars. All elements heavier than lead can undergo alpha decay. Now lets solve a real alpha decay equation, Atomic mass of seaborgium = 263 It is extremely unintuitive and I end up spending a inordinate amount of time typing it out, its an amazing app, this aap is so good. When \(Q\) > 0 energy is released in the nuclear reaction, while for \(Q\) < 0 we need to provide energy to make the reaction happen. An alpha particle is the same as a helium-4 nucleus . alpha particle in our nuclear equation, since an alpha particle has the same composition and we gained a proton. Thus, looking only at the energetic of the decay does not explain some questions that surround the alpha decay: We will use a semi-classical model (that is, combining quantum mechanics with classical physics) to answer the questions above. The daughter nucleus has two protons and four nucleons fewer than the parent nucleus. Step 2) Calculate the number of neutrons from the following equation In terms of charge, I know This relation also states that half-lives are exponentially dependent on decay energy, so that very large changes in half-life make comparatively small differences in decay energy, and thus alpha particle energy. That's 144 neutrons. If you're struggling with arithmetic, there's help available online. So this is just a visual representation of what's going on here, Describing Ionic Deal with math. Alpha decay, or alpha emission, is the release or emission of an alpha particle, which is a helium nucleus consisting of two protons and two neutrons. Step 3) After subtracting add the remaining protons and neutrons (4+6 = 10) 10 is the atomic weight of the new element nucleus. Alpha Decay Nuclear Reaction Example Problem. In order to study the quantum mechanical process underlying alpha decay, we consider the interaction between the daughter nuclide and the alpha particle. Alpha Decay Equation In -decay, the mass number of the product nucleus (daughter nucleus) is four less than that of the decaying nucleus (parent nucleus), while the atomic number decreases by two. Vedantu LIVE Online Master Classes is an incredibly personalized tutoring platform for you, while you are staying at your home. 0 92 238 U 0 90 234 Th + 2 4 He 0 92 238 U 0 90 234 Th + 2 4 Beta ( -) Decay A - particle is emitted. Then you must check out this page. The identity of the other product, just look it up here at our table, find atomic number of 90, and you'll see that's thorium here. 212 Po -10.3649 MeV. This decay in a nucleus causes the release of energy and matter from the nucleus. This is also equal to the total kinetic energy of the fragments, here \(Q=T_{X^{\prime}}+T_{\alpha} \) (here assuming that the parent nuclide is at rest). to eject an alpha particle, so an alpha particle is Geiger-Nutall law establishes a relation between the decay constant of a radioactive isotope and the energy of the emitted alpha particle. A particle may overcome obstacles that may seem impossible - this is known as the "tunnel effect". Then, the particles are inside a well, with a high barrier (as \(V_{\text {Coul }} \gg Q \)) but there is some probability of tunneling, since Q > 0 and the state is not stably bound. The mass of the alpha particles is relatively large and has a positive charge. This small change in the Z/N ratio is enough to put the nucleus into a more stable state (into the region of stable nuclei in the Chart of the Nuclides.). Alpha decay occurs in massive nuclei that have a large proton to neutron ratio. protactinium here, so Pa. What is happening in beta decay? (2021, July 29). Most often they will be annihilated by colliding with an electron eventually. If it is a positron, it will not act like an electron because it has a positive charge, which will repel it from anything that an electron would interact with. Alpha, beta, and gamma decay are all ways that an unstable atom can decay into a more stable form. is the decay constant If an archaeologist found a fossil sample that contained 25% carbon-14 in comparison to a living sample, the time of the fossil sample's death could be determined by rearranging equation 1, since Nt, N0, and t1/2 are known. We have explained in the previous article that the proton-neutron ratio (or vice-versa) is an indicator on the nuclei stability. Please provide any one of the following to get the other two. Write the nuclear equation for the decay of Po-210 if it undergoes 2 consecutive The damage caused due to alpha particles increases a persons risk of cancer like lung cancer. We can do the same calculation for the hypothetical decay into a 12C and remaining fragment (\({}_{81}^{188} \mathrm{TI}_{ \ 107}\)): \[Q_{12} C=c^{2}\left[m\left(\begin{array}{c} way of thinking about it. Since there are two protons, the charge of an alpha write our proton here. By clicking Accept All Cookies, you agree to the storing of cookies on your device to enhance site navigation, analyze site usage, and assist in our marketing efforts. The general rule for decay is best written in the format Z A X N. If a certain nuclide is known to decay (generally this information must be looked up in a table of isotopes, such as in Appendix B ), its decay equation is (31.4.3) X Z A N Y N 2 Z 2 A 4 + He 2 2 4 ( d e c a y) 235 92 U ? Alpha decay is a nuclear change process which produces an alpha particle. the number of protons, so we have 90 protons on the left, how many neutrons do we have? The atomic mass number of the emitted Alpha particle is four. The radioactive disintegration of alpha decay is a phenomenon in which the atomic nuclei which are unstable dissipate excess energy by ejecting the alpha particles in a spontaneous manner. . Direct link to prajwalxdeval's post A beta particle is an ele, Posted 7 years ago. This ion consists of two protons and two neutrons and has a 2 + charge. total of 238 on the right, and so therefore nucleons For alpha decay equations, this Q-value is. Alpha particles were given this name prior to discovering what kind of particles they represent. Alpha decay nuclear equations calculator Learning Objectives Qualitatively compare the power of ionising and penetration of alpha particles \ (\left ( \alpha \right)\), beta particles \ (\left (\beta \right)\), and gamma rays \ (\left) \gamma \right)\). We saw in the previous video that you represent an electron, since it has a negative one charge, you put a negative one down here, it's not a proton, nor is it a neutron, so we put a zero here. As per this rule, short-lived isotopes emit more energetic alpha particles than long-lived ones. Then, \(\lambda_{\alpha}=1.6 \times 10^{-17} \mathrm{~s}\) or \(t_{1 / 2}=4.5 \times 10^{9}\) years, close to what observed. Alpha Decay - decay is the radioactive emission of an -particle which is the nucleus of 4 2 He, consisting of two protons and two neutrons. If we calculate \( Q_{\alpha}\) from the experimentally found mass differences we obtain \(Q_{\alpha} \approx 7.6 \mathrm{MeV}\) (the product is 196At). A = number of protons = 95 - 2 = 93. For example in the alpha-decay \( \log \left(t_{1 / 2}\right) \propto \frac{1}{\sqrt{Q_{\alpha}}}\), which is the Geiger-Nuttall rule (1928). Direct link to Andrew M's post The nucleus has nuclear e, Posted 3 years ago. The radiocative decay formula is A = A0 e-0.693t/T. Chapter 7 Alpha Decay. The radioactive decay is defined as the spontaneous breakdown of the nucleus, that results in the release of energy and also the matter from the atomic nucleus. How can we predict what type of radiation might happen to a certain isotope? Standard: Students know the three most common forms of radioactive decay (alpha, beta, and. For equation solving, Wolfram|Alpha calls the Wolfram Language's Solve and Reduce functions, which contain a broad range of methods for all kinds of algebra, from basic linear and quadratic equations to multivariate nonlinear systems. So for representing an Direct link to Andrew M's post Probably, but also probab, Posted 7 years ago. Two neutrons are present in the alpha particle. neutrons = 263 - 106 = 157, 106 - 2 = 104 protons If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. So we think about what's The energy Q derived from this decay is divided equally into the transformed nucleus and the Helium nucleus. We need to multiply the probability of tunneling PT by the frequency \(f\) at which \( {}^{238} \mathrm{U}\) could actually be found as being in two fragments \({ }^{234} \mathrm{Th}+\alpha \) (although still bound together inside the potential barrier). Although \(Q\) > 0, we find experimentally that \(\alpha\) decay only arise for \(A \geq 200\). Direct link to Ryley's post So he talks about the thr, Posted 8 years ago. Thus, you can see that the mass number and the atomic number balances out on both sides of this equation. We can approximate the finite difference with the relevant gradient: \[\begin{align} So a neutron has turned into a proton, and we're also getting a beta particle ejected from the nucleus. nucleus, uranium-238. No, a neutron is not made of a proton, electron and antineutrino. Here, we are offering simple steps to solve the half-life and radioactive decay of the substance. So I go ahead and draw Try the free Mathway calculator and problem solver below to . A beta particle is an electron. To return to a stable state, these nuclei emit electromagnetic radiation in the form of one or multiple gamma rays. We provide you year-long structured coaching classes for CBSE and ICSE Board & JEE and NEET entrance exam preparation at affordable tuition fees, with an exclusive session for clearing doubts, ensuring that neither you nor the topics remain unattended. Go through the step by step procedure to find the radioactive decay and half-life of the substance. In analyzing a radioactive decay (or any nuclear reaction) an important quantity is Q, the net energy released in the decay: Q = ( m X m X m ) c 2. New Resources Infinite limits Knight's tour (with draggable start position) Subtraction up to 20 - ? Alpha Decay Calculations Author: Barb Newitt Enter the exact values for the parent and daughter nucleus mass to calculate the energy released in one alpha decay event. Geiger-Nuttall law is used in nuclear physics and it relates the energy of the alpha particle emitted to the decay constant of a radioactive isotope. As you enter the specific factors of each nuclear decay calculation, the Nuclear Decay Calculator will automatically calculate the results and update the Physics formula elements with each element of the nuclear decay calculation. The element which has 259 as the atomic weight is rutherfordium. This element is also the object that undergoes radioactivity. The radioactive or nuclear decay takes place, when an unstable atom loses its energy by emitting radiation like alpha, beta, and other particles. I recall learning about an N/Z ratio (using the belt of stability), but I'm really confused about it. Which elements can undergo alpha decay? Since the strong force is much stronger than the electric force at subatomic range, the energy levels in the nucleus are much larger than those for the atom, and this is why the energy released in nuclear reactions is so much greater than the energy released in chemical reactions (eg a nuclear electric power facility produces energy from a lot less fuel than a similarly powerful coal-fired electric power facility). You're also going to I have a zero here, so So lets take metal #'x'# as the metal and its atomic weight as 14 and number of protons is 6 We already talked about The exponent is thus a large number, giving a very low tunneling probabily: \(e^{-2 G}=e^{-89}=4 \times 10^{-39}\). Example The atomic number of such nuclei has a mass that is four units less than the parent and an atomic number that is two units less than the parent. between the parent and daughter element? Henderson Hasselbalch Equation Calculator, Linear Correlation Coefficient Calculator, Partial Fraction Decomposition Calculator, Linear Equations in Three Variables Calculator. Therefore, we may use the law of conservation of momentum to determine how fast the daughter nucleus and helium nucleus will move after an alpha-decay process does occur. particle is two plus. . The electromagnetic force is a disruptive force that breaks the nucleus apart. There are two protons in the helium nucleus and two neutrons. charge is also conserved. In the alpha decay of \(\ce{^{238}U}\) (Equation \(\ref{alpha1}\)), both atomic and mass numbers are conserved: mass number: \(238 = 4 + 234\) atomic number: \(92 = 2 + 90\) Confirm that this equation is correctly balanced by adding up the reactants' and products' atomic and mass numbers. Now you can even download our Vedantu app for easier access to online study material and interactive classes. On the other side, the Coulomb energy at this separation is \(V_{C o u l}=e^{2} Z^{\prime} Z_{\alpha} / R=28 M e V \gg Q_{\alpha}\) (here Z' = Z 2 ).