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PHISICS PRESENTATION. ON. ATOMS AND NUCLEI. PRESENTER’S:. SUNIT: LOKESH: JASWANT. CHAPTER:-. 11. ATOMS. SOMThING ABOUT ATOMS. 1. Atom, as a whole, is electrically neutral and therefore contains equal amount of positive and negative charges.
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PHISICS PRESENTATION ON ATOMS AND NUCLEI PRESENTER’S: SUNIT: LOKESH: JASWANT
CHAPTER:- 11 ATOMS
SOMThINGABOUT ATOMS • 1. Atom, as a whole, is electrically neutral and therefore contains equal amount of positive and negative charges. • 2. In Thomson’s model, an atom is a spherical cloud of positive charges with electrons embedded in it. • 3. In Rutherford’s model, most of the mass of the atom and all its positive charge are concentrated in a tiny nucleus (typically one by ten thousand the size of an atom), and the electrons revolve around it.
ALPHA-PARTICLE SCATTERING AND RUTHERFORD’SNUCLEAR MODEL OF ATOM
At the suggestion of Ernst Rutherford, in 1911, H. Geiger and E. Marsden performed some experiments. In one of their experiments, they directed a beam of 5.5 MeV α-particles emitted from a 21483Bi radioactive source at a thin metal foil made of gold.
Alpha-particles emitted by a 214 83Biradioactive source were collimated into a narrow beam by their passage through lead bricks. The beam was allowed to fall on a thin foil of gold of thickness 2.1 × 10–7 m. The scattered alpha-particles were observed through a rotatable detector consisting of zinc sulphide screen and a microscope. The scattered alpha-particles on striking the screen produced brief light flashes or scintillations. These flashes may be viewed through a microscope and the of the number of scattered particles may be studied as a function of angle of scattering.
Rutherford’s nuclear model predicts the solid curve which is seen to be in good agreement with experiment
Experimental data points on scattering alpha-particles by a thin foil at different angles obtained by Geiger and Mardsen.
A typical graph of the total number of α-particles scattered at different angles, in a given interval of time. The dots in this figure represent the data points and the solid curve is the theoretical prediction based on the assumption that the target atom has a small, dense, positively charged nucleus. Many of the α-particles pass through the foil. It means that they do not suffer any collisions. Only about 0.14% of the incident α particles scatter by more than 1º; and about 1 in 8000 deflect by more than 90º. Rutherford argued that, to deflect the α-particle backwards, it must experience a large repulsive force. This force could be provided if the greater part of the mass of the atom and its positive charge were concentrated tightly at its centre. Then the incoming α- particle could get very close to the positive charge without penetrating it, and such a close encounter would result in a large deflection. This agreement supported the hypothesis of the nuclear atom. This is why Rutherford is credited with the discovery of the nucleus.
The trajectory traced by an α-particle depends on the impact parameter, b of collision. The impact parameter is the perpendicular distance of the initial velocity vector of the α-particle from the centre of the nucleus (Fig. 12.4). A given beam of α-particles has a distribution of impact parameters b, so that the beam is scattered in various directions with different probabilities (Fig. 12.4). (Ina beam, all particles have nearly same kinetic energy.) It is seen that an α-particle close to the nucleus (small impact parameter) suffers large scattering. In case of head- on collision, the impact parameter is minimum and the α-particle rebounds back ( θ ≅ π). For a large impact parameter, the α-particle goes nearly undedicated and has a small deflection ( θ ≅ 0). The fact that only , a small fraction of the number of incident particles rebound back indicates that the number of α-particles undergoing head on collision is small. This, in turn, implies that the mass of the atom is concentrated in a small volume. Rutherford scattering therefore, is a powerful way to determine an upper limit to the size of the nucleus.
The Rutherford nuclear model of the atom which involves classical concepts, pictures the atom as an electrically neutral sphere consisting of a very small, massive and positively charged nucleus at the centre surrounded by the revolving electrons in their respective dynamically stable orbits. The electrostatic force of attraction, Fe between the revolving electrons and the nucleus provides the requisite centripetal force (Fc ) to keep them in their orbits. Thus, for a dynamically stable orbit in a hydrogen atom: • Fe = Fc mv²/r = 1e²/4ΩEᵋr²
Thus the relation between the orbit radius and the electron velocity is r = e²/4πεmv² The kinetic energy (K) and electrostatic potential energy (U) of the electron in hydrogen atom are K= mv²/2 = e²/8πεr and U = e²/4πεr • (The negative sign in U signifies that the electrostatic force is in the –r direction.) Thus the total energy E of the electron in a hydrogen atom is E = K + U = e²/8πεr - e²/4πεr = - e²/8πεr • The total energy of the electron is negative. This implies the fact thatthe electron is bound to the nucleus. If E were positive, an electron will not follow a closed orbit around the nucleus.
Each element has a characteristic spectrum of radiation, which it emits. When an atomic gas or vapor is excited at low pressure, usually by passing an electric current through it, the emitted radiation has a spectrum which contains certain specific wavelengths only. A spectrum of this kind is termed as emission line spectrum and it consists of bright lines on the dark background. The spectrum emitted by atomic hydrogen .Study of emission line spectra of a material can therefore serve as a type of “fingerprint” for identification of the gas. When white light passes through a gas and we analyses the transmitted light using a spectrometer we find some dark lines in the spectrum. These dark lines correspond precisely to those wavelengths which were found in the emission line spectrum of the gas. This is called the absorption spectrum of the material of the gas.
We might expect that the frequencies of the light emitted by a particularelement • would exhibit some regular pattern. Hydrogen is the simplestatom and • therefore, has thesimplest spectrum. In theobservedspectrum,however, at first • sight, there does not seem to beany resemblance of order or regularity in • spectrallines. But the spacing between lines within certainsets of the hydrogen • spectrum decreases in aregular way . Each of these sets is calleda spectral series. • In 1885, the first such series wasobserved by a Swedish school teacher Johann • JakobBalmer (1825–1898) in the visible region of thehydrogen spectrum. This • series is called Balmerseries . The line with the longest wavelength, 656.3 nm in • the red is called Hα; thenext line with wavelength 486.1 nm in the bluegreen • is called Hβ, the third line 434.1 nm in theviolet is called Hγ; and so on. As the • wavelengthdecreases, the lines appear closer together and are weaker in • intensity. • Balmer found a simple empirical formula for the observed wavelengths • 1 = R 1 1 • λ 2² n² • where λ is the wavelength, R is a constant called the Rydberg constant, • and n may have integral values 3, 4, 5, etc. The value of R is 1.097 × 107 m–1. • This equation is also called Balmer formula.
Other series of spectra for hydrogen were subsequently discovered. These are known, after their discoverers, as Lyman, Paschen, Brackett, and Pfund series. These are represented by the formulae: L yman series: 1 = R( 1 1 ) λ ( 1² n²) n = 2,3,4... Paschen series: 1 = R( 1 1 ) λ ( 3² n²) n = 4,5,6... Brackett series: 1 = R ( 1 1 ) λ ( 4² n²) n = 5,6,7... Pfund series: 1 = R ( 1 1 ) λ ( 5² n²) n = 6,7,8…. The Lyman series is in the ultraviolet, and the Paschen and Brackett series are in the infrared region. The Balmer singly ionised helium, and doubly ionised lithium) whose spectra caformula may be written in terms of frequencies of the light, recalling that c = νλ or 1 = v λ c Thus, Eq. (12.5) becomes v = Rc(1 1 ) (2² n²) There are only a few elements (hydrogen, n be represented by simple formula like Esq..
To explain the line spectra emitted by atoms, as well as the stability of atoms, Niel’s Bohr proposed a model for hydrogenic (single electron) atoms. He introduced three postulates and laid the foundations of quantum mechanics: • In a hydrogen atom, an electron revolves in certain stable orbits (called stationary orbits) without the emission of radiant energy. • The stationary orbits are those for which the angular momentum is some integral multiple of h/2πh. (Bohr’s quantization condition.) That is L = nh/2π, where n is an integer called a quantum number. • The third postulate states that an electron might make a transition from one of its specified non-radiating orbits to another of lower energy. When it does so, a photon is emitted having energy equal to the energy difference between the initial and final states. The frequency (ν) of the emitted photon is then given by: hν = Ei – Ef An atom absorbs radiation of the same frequency the atom emits, in which case the electron is transferred to an orbit with a higher value of n. Ei + hν = Ef
As a result of the quantisation condition of angular momentum, the electron orbits the nucleus at only specific radii. For a hydrogen atom it is given by r = (n²)(h)²4πe (m)(2π) e² The total energy is also quantised: En = - (me²)² 8n²ε²h² = –13.6 eV/n² The n = 1 state is called ground state. In hydrogen atom the ground state energy is –13.6 eV. Higher values of n correspond to excited states (n > 1). Atoms are excited to these higher states by collisions with other atoms or electrons or by absorption of a photon of right frequency.
The energy of an atom is the least (largest negative value) when its electron is revolving in an orbit closest to the nucleus i.e., the one for which n = 1. For n = 2, 3, ... The absolute value of the energy E is smaller, hence the energy is progressively larger in the outer orbits. The lowest state of the atom, called the ground state, is that of the lowest energy, with the electron revolving in the orbit of smallest radius, the Bohr radius, a0. The energy of this state (n = 1), E1 is –13.6 eV. Therefore, the minimum energy required to free the electron from the ground state of the hydrogen Atom is 13.6 eV. It is called the ionisation energy of the hydrogen atom. This prediction of the Bohr’s model is in excellent agreement with the experimental value of ionisation energy. At room temperature, most of the hydrogen atoms are in ground state. When a hydrogen atom receives energy by processes such as electron collisions, the atom may acquire sufficient energy to raise the electron to higher energy states. The atom is then said to be in an excited state. , for n = 2; the energy E2 is –3.40 eV. It means that the energy required to excite an electron in hydrogen atom to its first excited state, is an energy equal to E2 – E1 = –3.40 eV – (–13.6) eV = 10.2 eV. Similarly, E3 = – 1.51 eV and E3 – E1 = 12.09 eV, or to excite the hydrogen atom from its ground state (n = 1) to state (n = 3), 12.09 eV energy is required, and so on. From these excited states the electron can then fall back to a state of lower energy, emitting a photon in the process. Thus, as the excitation of hydrogen atom increases (that is as n increases) the value of minimum energy required to free the electron from the excited atom decreases.
Equation of the Rydberg formula Hνif = (me²)² ( 1 – 1 ) 8εh² ( n² n²i ) for the spectrum of the hydrogen atom. In this relation, if we take nf = 2 and ni = 3, 4, 5..., it reduces to a form similar to Eq for the Balmer series. The Rydberg constant R is readily identified to be R = (me²)² 8ε²c³ If we insert the values of various constants in Eq.we get R = 1.03 × 107 m–1 This is a value very close to the value (1.097 × 107 m–1) obtained from the empirical Balmer formula. This agreement between the theoretical and experimental values of the Rydberg constant provided a direct and striking confirmation of the Bohr’s model. hydrogen spectrum, the Balmer formulan corresponds to nf = 2 and ni = 3, 4, 5,etc. The results of the Bohr’s model suggested the presence of other series spectra for hydrogen atom–those corresponding to transitions resulting from nf = 1 and ni = 2, 3, etc.; nf = 3 and ni = 4, 5, etc., and so on. Such series were identified in the course of spectroscopic investigations and are known as the Lyman, Balmer, Paschen, Brackett, and Pfund series. The various lines in the atomic spectra are produced when electrons jump from higher energy state to a lower energy state and photons are emitted. These spectral lines are called emission lines. But when an atom absorbs a photon that has precisely Line spectra originate in transitions between energy levels. the same energy needed by the electron in a lower energy state to make transitions to a higher energy state, the process is called absorption. Thus if photons with a continuous range of frequencies pass through a rarefied gas and then are analysed with a spectrometer, a series of dark spectral absorption lines appear in the continuous spectrum. The dark lines indicate the frequencies that have been absorbed by the atoms of the gas.
DE ROGLIE’S EXPLANATION OF BHOR’S SECOND POSTULATEOF QUANTISATION Standing waves on a circle for n = 4,5,6…
In 1913, Niels Bohr postulated that the angular momentum L of an orbiting atomic electron was quantized: its possible values are integral multiples of the reduced Planck constant: L = nℏ, n = 1,2,3.… Bohr’s postulate not only explained the stability of atoms but also accounted for the by then well-established fact that atoms absorb and emit electromagnetic radiation only at specific frequencies. It even enabled Bohr to calculate with remarkable accuracy the spectrum of atomic hydrogen — the particular frequencies at which this absorbs and emits light, visible as well as infrared and ultraviolet. Yet apart from his quantization postulate, Bohr’s reasoning at the time remained completely classical. He assumed that the orbit of the hydrogen atom’s single electron was a circle, and that the atom’s nucleus — a single proton — was at the center. He used classical laws to calculate the orbiting atom’s energy E, expressed E as a function of the classical expression for L, and only then replaced this expression by L = nℏ. In this way Bohr obtained a discrete sequence of values En. And since they were the only values the energy of the orbiting electron was “allowed” to take, the energy that a hydrogen atom could emit or absorb had to be equal to the difference between two of these values. The atom could “jump” from a state of lower energy Em to a state of higher energy En, absorbing a photon of frequency (En − Em)/h, and it could “jump” from a state of higher energy Em to a state of lower energy En, emitting a photon of frequency (Em − En)/h. Or so the story went. It took ten years, from 1913 to 1923, before someone finally found an explanation for the quantization of angular momentum. Planck’s radiation formula had implied a relation between a particle property (E) and a wave property (f or ω) for the quanta of electromagnetic radiation we now call photons. Einstein’s explanation of the photoelectric effect, published in 1905, established another such relation, between the momentum p of a photon and its wavelength λ: p = h/λ.
If electromagnetic waves have particle properties, Louis de Broglie reasoned, why cannot electrons have wave properties? Imagine that the electron in a hydrogen atom is a standing wave on a circle rather than some sort of corpuscle moving in a circle. (A standing wave does not travel. Its crests and troughs are stationary; they stay put. Such a wave has to satisfy the condition 2πr = nλ, n = 1,2,3.… In (other) words, the circumference 2πr of the circle must be an integral multiple of the wavelength λ. With ℏ = h/2π and de Broglie’s formula p = h/λ, pr = nℏ. But pr is just the angular momentum L of a classical particle moving in a circle of radius r. In this way de Broglie arrived at the quantization condition L = nℏ, which Bohr had simply postulated
CHAPTER:- 13 NUCLIEI
INTRODUCTION OF NUCLIE 1. An atom has a nucleus. The nucleus is positively charged. The radius of the nucleus is smaller than the radius of an atom by a factor of 104. More than 99.9% mass of the atom is concentrated in the nucleus. 2.The density of nuclear matter is independent of the size of the nucleus. The mass density of the atom does not follow this rule. 3. The radius of a nucleus determined by electron scattering is found to be slightly different from that determined by alpha-particle scattering. This is because electron scattering senses the charge distribution of the nucleus, whereas alpha and similar particles sense the nuclear matter.
ATOMIC MASSES AND COMPOSITION OF NUCLEUS • The mass of an atom is so small that it is inconvenient to express it in kilograms. • The unit in which atomic and nuclear masses are measured is called atomic mass unit (amu). • One amu is defined as 1/12th of the mass of an atom of isotope. Avogadro’s number = 6.023 × 1023 Mass of 6.023 × 1023 atoms of C12 = 12 g • Atomic masses can be measured using a mass spectrometer The mass of an atom is so small that it is inconvenient to express it in kilograms. • The unit in which atomic and nuclear masses are measured is called atomic mass unit (amu). • One amu is defined as 1/12th of the mass of an atom of isotope. Avogadro’s number = 6.023 × 1023 ∴Mass of 6.023 × 1023 atoms of C12 = 12 g Atomic masses can be measured using a mass spectrometer.
Discovery of Neutron In 1932, James Chadwick bombarded beryllium (Be) with alpha a particles. He allowed the radiation emitted by beryllium to incident on a paraffin wax. It was found that protons were shot out form the paraffin wax. People began to look for what was in the "beryllium radiations". Finding Neutron Some people suggested that the radiations may be gamma radiation. However, Chadwick found that the radiation could not be gamma radiation since energy and momentum were not conserved in its production. He showed that all the observations could be explained if the radiation consisted of neutral particles of mass approximately equal to that of proton. This neutral particle was named neutron.
equation of the nuclear reaction This discovery of neutron marked the beginning of current theories of nuclear structure. Immediately, the neutron-proton model ( the Rutherford-Bohr model) of the nucleus was adopted: The nucleus is made up of protons and neutrons. These are bound together by a strong nuclear force. Electrons and protons carry equal but opposite charges. In a neutral atom, the number of electrons is the same as the number of protons. Electrons orbit the nucleus at certain fixed levels called shells.
SI`1ZE OF NUCLEUS Rutherford’s calculations are based on pure coulomb repulsion between the positive charges of the αparticle and the gold nucleus. From the distance at which deviations set in, nuclear sizes can be inferred. By performing scattering experiments in which fast electrons, instead of α-particles, are projectiles that bombard targets made up of various elements, the sizes of nuclei of various elements have been accurately measured. It has been found that a nucleus of mass number A has a radius R = R0 A1/3 (13.5) where R0 = 1.2 × 10–15 m. This means the volume of the nucleus, which is proportional to R3 is proportional to A. Thus the density of nucleus is a constant, independent of A, for all nuclei. Different nuclei are likes drop of liquid of constant density. The density of nuclear matter is approximately 2.3 × 1017 kg m–3. This density is very large compared to ordinary matter, say water, which is 103 kg m–3. This is understandable, as we have already seen that most of the atom is empty. Ordinary matter consisting of atoms has a large amount of empty space.
MASS - ENERGY Einstein showed from his theory of special relativity that it is necessary to treat mass as another form of energy. Before the advent of this theory of special relativity it was presumed that mass and energy were conserved separately in a reaction. However, Einstein showed that mass is another form of energy and one can convert mass-energy into other forms of energy, say kinetic energy and vice-versa. Einstein gave the famous mass-energy equivalence relation: E = mc² Here the energy equivalent of mass m is related by the above equation and c is the velocity of light in vacuum and is approximately equal to: 3×10 8 m s –1 .
Nuclei are made up of protons and neutron, but the mass of a nucleus is always less than the sum of the individual masses of the protons and neutrons which constitute it. The difference is a measure of the nuclear binding energy which holds the nucleus together. This binding energy can be calculated from the Einstein relationship: Nuclear binding energy = Δmc2 For the alpha particle Δm= 0.0304 u which gives a binding energy of 28.3 MeV.
The enormity of the nuclear binding energy can perhaps be better appreciated by comparing it to the binding energy of an electron in an atom. The comparison of the alpha particle binding energy with the binding energy of the electron in a hydrogen atom is shown below. The nuclear binding energies are on the order of a million times greater than the electron binding energies of atoms.
NUCLEAR FORCE The nuclear force (or nucleon-nucleon interaction or residual strong force) is the force between two or more nucleons. It is responsible for binding of protons and neutrons into atomic nuclei. The energyreleased causes the masses of nuclei to be less than the total mass of the protons and neutrons which form them. The force is powerfully attractive between nucleons at distances of about 1 femtometer(fm) between their centers, but rapidly decreases to insignificance at distances beyond about 2.5 fm. At very short distances less than 0.7 fm, it becomes repulsive, and is responsible for the physical size of nuclei, since the nucleons can come no closer than the force allows. The nuclear force is now understood as a residual effect of the even more powerful strong force, or strong interaction, which is the attractive force that binds particles called quarks together, to form the nucleons themselves. This more powerful force is mediated by particles called gluons . Gluons hold quarks together with a force like that of electric charge, but of far greater power. The concept of a nuclear force was first quantitatively constructed in 1934, shortly after the discovery of the neutron revealed that atomic nuclei were made of protons and neutrons, held together by an attractive force. The nuclear force at that time was conceived to be transmitted by particles called mesons, which were predicted in theory before being discovered in 1947. In the 1970’s, further understanding revealed these mesons to be combinations of quarks and gluons, transmitted between nucleons that themselves were made of quarks and gluons. This new model
The nuclear forces arising between nucleons are now seen to be analogous to the forces in chemistry between neutral atoms called vander Waals forces. Such forces between atoms are much weaker than the electrical forces that hold the atoms themselves together, and their range is shorter, because they arise from spontaneous separation of charges inside the atom. Similarly, even though nucleons are made of quarks and gluons that are in combinations which cancel most gluon forces, some combinations of quarks and gluons nevertheless leak away from nucleons, in the form of short-range nuclear force fields that extend from one nucleon to another close by. These nuclear forces are very weak compared to direct gluon forces inside nucleons, and they extend only over a few nuclear diameters, falling exponentially with distance. Nevertheless, they are strong enough to bind neutrons and protons over short distances, and overcome the electrical repulsion between protons in the nucleus.
Radiation from nuclear sources is distributed equally in all directions, obeying the inverse square law. Radioactivity refers to the particles which are emitted from nuclei as a result of nuclear instability. Because the nucleus experiences the intense conflict between the two strongest forces in nature, it should not be surprising that there are many nuclear isotopes which are unstable and emit some kind of radiation. The most common types of radiation are called alpha, beta, and gamma radiation, but there are several other varieties of radioactive decay. Radioactive decay rates are normally stated in terms of there half-lives , and the half-life of a given nuclear species is related to its radiation risk. The different types of radioactivity lead to different decay paths which transmute the nuclei into other chemical elements. Examining the amounts of the decay products makes possible radioactivedating.
Law of radioactive decay In any radioactive sample, which undergoes α, β or γ-decay, it is found that the number of nuclei undergoing the decay per unit time is proportional to the total number of nuclei in the sample. If N is the number of nuclei in the sample and ∆N undergo decay in time ∆t then ∆N/∆t = λN, where λ is called the radioactive decay constant or disintegration constant. The change in the number of nuclei in the sample* is dN = – ∆N in time ∆t. Thus the rate of change of N is (in the limit ∆t → 0) So this equation describes the situation for any brief time interval, dt. To find out what happens for all periods of time we simply add up what happens in each brief time interval. In other words we integrate the above equation. Expressing this more formally we can say that for the period of time from t = 0 to any later time t, the number of radioactive nuclei will decrease from N0 to Nt, so that:
This final expression is known as the Radioactive Decay Law. It tells us that the number of radioactive nuclei will decrease in an exponential fashion with time with the rate of decrease being controlled by the Decay Constant. • Before looking at this expression in further detail let us review the mathematics which we used above. First of all we used integral calculus to figure out what was happening over a period of time by integrating what we knew would occur in a brief interval of time. Secondly we used a calculus relationship that the • where lnx represents the natural logarithm of x. And thirdly we used the definition of logarithms that when
In alpha decay, shown in Fig, the nucleus emits a 4He nucleus, an alpha particle. Alpha decayoccurs most often in massive nuclei that have too large a proton to neutron ratio. An alpha particle, with its two protons and two neutrons, is a very stable configuration of particles. Alpha radiation reduces the ratio of protons to neutrons in the parent nucleus, bringing it to a more stable configuration. Many nuclei more massive than lead decay by this method. In alpha decay, the atomic number changes, so the original (or parent) atoms and the decay-product (or daughter) atoms are different elements and therefore have different chemical properties. In the alpha decay of a nucleus, the change in binding energy appears as the kinetic energy of the alpha particle and the daughter nucleus. Because this energy must be shared between these two particles, and because the alpha particle and daughter nucleus must have equal and opposite momenta, the emitted alpha particle and recoiling nucleus will each have a well-defined energy after the decay. Because of its smaller mass, most of the kinetic energy goes to the alpha particle.
Beta decay is one process that unstable atoms can use to become more stable. There are two types of beta decay, beta-minus and beta-plus. During beta-minus decay, a neutron in an atom's nucleus turns into a proton, an electron and an antineutrino. The electron and antineutrino fly away from the nucleus, which now has one more proton than it started with. Since an atom gains a proton during beta-minus decay, it changes from one element to other. Although the numbers of protons and neutrons in an atom's nucleus change during beta decay, the total number of particles (protons + neutrons) remains the same.
