# IIT JEE / CET – Main / Advanced Physics 2018 & 2019 : Free Physics Notes > Atomic Physics

The bleeding edge of atomic research – compact muon solenoid detector at CERN

ATOMIC  PHYSICS    –   MIND   MAP

Atomic physics – click to zoom

A T O M I C    P H Y S I C S

Def  :  The scientific study of the structure of the atom, its dynamical properties, including energy states, and its interactions with particles and fields.

Discovery of Electrons

In 1897 , English physicist Joseph. J. Thomson discovered the electron through his work on cathode rays . This he did by experimenting with a Crookes tube – a sealed glass container in which two electrodes are separated by a vaccum. On application of  a voltage across the electrodes, cathode rays are generated, creating a glowing patch when they strike the glass at the opposite end of the tube. Through experimentation, Thomson discovered that the rays could be deflected by an electric feild (in addition to magnetic feilds, which was already known). He concluded that these rays, rather than being a form of light, were composed of very light negatively charged particles , which  he called as”corpuscles” (later  renamed electrons by other scientists) .

Crookes tube – As the cathode ray (blue line) passed through the electric field (yellow ) , it gets deflected.

The ‘mouth watering’  plum pudding model – In order to explain the overall neutral charge of the atom, he proposed that the corpuscles ( electrons ) were embedded  and distributed in a uniform sea of positive charge likes the plums in a pudding . In this model , unlike plums , the electrons were not stationary . According to J.J.Thomson –

• Atom is a positively charged sphere .
• Instead of having a nucleus at the centre of the atom , the positive charges and the related mass was spread throughout the atom.
• The atom was electrically neutral as the no of electrons ( and the charge therein ) was equal to that of the positive charges.

The above model was proved wrong by Ernest Rutherford . His team directed a beam of alpha particles at a thin gold foil and found that they were deflected. If  Thomson’s model was correct , the paticles would have passed straight as according to him , positive charge was spreadout uniformely in the atom.

Contributions :  Ernest Rutherford is often referred to as the ‘ father of nuclear physics’ . [ Nobel Prize for Chemistry- 1908]

Ernest Rutherford

Rutherford is known for his studies in radioactivity and credited for the discovery of the atomic nucleus . He discovered and named alpha and beta radiation and with Frederick Soddy proposed a theory of radioactive transformation of atoms; and for this work he was awarded the 1908 Nobel Prize in Chemistry. On the basis of experiments with alpha rays carried out under his direction , he was led to a description of the atom as a small, heavy nucleus surrounded by orbital electrons; this nuclear model of the atom was taken by Niels Bohr and combined with the new quantum theory to provide for the basic description of the atom accepted today. In the course of his researches, Rutherford produced hydrogen by bombarding atoms of various elements, e.g., nitrogen, with helium nuclei (alpha rays); these results were the first evidence of artificially produced splitting of atomic nuclei.

Rutherford’s model had two drawbacks –

• About the Atom’s stability – Electron moving in a circular orbit around a nucleus is accelerating and based on electromagnetic theory , it should be emitting radiation continuously and hence should be losing energy. This means that the ordital radius should be decreasing leading to the electron falling into the nucleus . But , atoms doesn’t collapse.
• About the Line spectrum – The changing radii of the circular electron orbits means that the frequency of revolution of the electron should be changing and therefore electrons ought to be radiating electromagnetic waves of all frequencies . Hence their spectrum should be continuous – but that’s not the case and atomic spectra are line spectrums.
line spectra of neon

The inability of the classical theory to explain the stability of an atom was dealt with by Neils Bohr,when  in 1913, he postulated that electrons move around the nucleus of the atom in restricted orbits and explained the manner in which the atom absorbs and emits energy. He thus combined the quantum theory with this concept of atomic structure. His assumption was that there were ‘stationary’ orbits for the electrons in which the electron did not radiate energy. He further assumed that such orbits occurred when the electron had definite values of angular momentum, specifically values h/2π, 2h/2π, 3h/2π, etc., where h is Planck’s constant. He used this idea to calculate energies E1, E2, E3, etc., for possible orbits of the electron. Further, he postulated that emission of light occurred when an electron moved from one orbit to a lower-energy orbit; absorption was accompanied by a change to a higher-energy orbit. In each case the energy difference produced radiation of energy hν, where ν is the frequency. In 1913 he realized that, using this idea, he could obtain a theoretical formula similar to the empirical formula of Johannes Balmer for a series of lines in the hydrogen spectrum. For this work ,Bohr was awarded the Nobel Prize for physics in 1922. He also made major contributions to the early development of quantum theory. The ‘correspondence principle’ is his principle that the quantum-theory description of the atom corresponds to classical physics at large magnitudes.

Niels Bohr with Albert Einstein (1925)

Bohr atomic model – figure shows states of electron with energy quantized by the number n. An electron dropping to a lower orbit emits a photon equal to the energy difference between the orbits.

Bohr model  >  details

• After incorporating Planck’s idea of quantized energy , he postulated that the electrons in a hydrogen atom could only have certain values of the total energy ( total energy = potential energy + electron kinetic energy ). These energy levels were specific for particular orbits , with larger orbits having larger total energies. Stationary orbits ( states ) referred to those electrons which didn’t emit electromagnetic radiations. This assumption was necessary because if the electron was accelerating and was radiating energy , then the atom wouldn’t be stable .
• Bohr ( by taking into account Einstein’s photon concept) postulated that electrons emit energy when they jump from orbits of higher energy to those of lower energy. The energy of an emitted photon could be calculated as Ei – Ef where Ei = energy related to an initial orbit of high energy and Ef = energy related to a final orbit of low energy.
• The magnitude of an electron’s angular momentum is quantized and is an integral multiple of  h/2π . Now, the magnitude of angular momentum of a particle with mass m moving with a speed v around a circle of radius r is L = mvr. Hence , mvr = nh/2π , where n=1,2,3,.

mv² / r    =   1 /4π ε0 x  (Ze) (e)/r²      – [eq.1]      also,  mvr  =  nh/2π         -[eq.2]

Eq.2 becomes   v = nh/ 2πmr

substituting this value in eq.1   rn = n²h²ε0/π me²Z  =  rn = (0.53) n²/Z  Å

Hence , for atoms like Hydrogen =  rn α n²/Z

Velocity of the electron in the nth orbit

v = Z e² / 2 ε0nh     or    v =  [e²/2ε0ch ] x [c Z/n ]  therefore, v = α [ cZ/n] , where α = e²/ 2hε0c and is the Summerfeild’s fine spectrum constant , which is a pure number whose value is 1/137. This means that the velocity of electron in Bohr’s first orbit of Hydrogen ( Z = 1 ) is c/137 and that of the second orbit is c / 274 and so on.

Orbital Frequency of the electron

f  =  v / 2π r  . Substituting the values of v and r ,  f  =  m Z² [ e²]² / 4 ε0² n³ h³  =>  f  α  Z² / n³ .

Energy of the electron in the nth orbit

Kinetic Energy –      mv²/r   =  1 /4 π ε0 x  Ze² / r²    i.e ,    1/2 mv²   =  Ze² / 8 π ε0r

hence , K.E  =  Ze² / 8 π ε0r

Potential Energy –     U   =   – 1/ 4π ε0 x  ( Ze) (e)/r     =    –  Ze² /4 π ε0r

Now , Total energy =  K.E + P.E    or E   =   Ze² / 8 π ε0r  –  Ze² /4 π ε0r    =  –  Ze² / 8 π ε0r .  [ The negative sign means that the electron is bound ( held in attraction ) to the nucleus , therein causing it to have a circular orbit around the nucleus ]

Therefore , for an electron ,  Total energy = – Kinetic energy  =  ½  Potential energy .

Now , using  r  =  n²h²e0/ pme²Z   ,  E  = – [  me4/8h²ε²0] Z/n ,  therefore ,    E  =  – ( 13.6 ) Z²/n² eV

Also , E =  – [  me4/8ch³ε²0] ch. Z²/n²     =    E  =  –  ( Rch )  Z²/n²

here    R    =      me4/8ch³ε²0 =  Rydberg’s constant   =   1.097  x   107m-1 .

Rydberg constant (   R ) is the most accurately measured of the fundamental constants, and is used in the formulas for wave numbers of atomic spectra and serves as a universal scaling factor for any spectroscopic transition . Moreover , it is an important cornerstone in the determination of other constants . Here , me is the electron mass, e is the electron charge, h is Planck’s constant, ε0 is the permittivity of vacuum, and c is the speed of light. To expresses the Rydberg constant in cgs units, the right-hand side must be multiplied by (4πε0)2. The subscript indicates that this is the Rydberg constant corresponding to an infinitely massive nucleus .  The latest scientifically accepted value of  R= 10,973,731.568,525 ± 0.000,073 m−1 .
Substituting the values of   m , e , ε0 , and h with n = 1 , the value for the least energy of the atom in the first orbit i.e ,  – 13.6 eV  is got .
Thus ,  E1  =  – 13.6 eV and  En  =  E1 / n² –  13.6 / n²  eV .  Therefore , by substituting  n = 2, 3 , 4 … the energies of the atom in various orbits are got.
Therefore , when n = 1 , E1 = – 13.6 eV ,  n = 2  E2 becomes  – 3.4 eV and so on . Note that when n = ∞ , E= 0 eV – this is the limiting case .
Energy level diagram

energy level diagram of the hydrogen atom

Energy level diagrams deal with fixed energy levels that a system like molecule , atom ,nucleus or electron can have , as calculated by quantum mechanics . In such a system , the atom can accept a quanta of energy to become excited , thereby raising an electron to the next higher permitted energy level. The lowest possible energy level is the ground state , and then is the first excited state . Note that between these two , there will not be other permissible energy states / levels [ i.e , there cannot be fractions of energy states ] . Consequently , the energy levels of  individual electrons in an atom will be lower than that for an arbitrary level for a free electron . In the case of molecules , there woul be energy levels for quantized rotational and vibrational motions .

In an energy level diagram , the lowest energy state is at the bottom [ ground state ] . As the levels go up , the energy value increases and becomes more positive .

When an electron jumps from a final / higher energy level / state , to a lower one , energy is released as radiation . Let Ef be the final state and Ei be the initial energy state . Then , the frequency of the emitted radiation is  –

hf = Ef – Ei ,   so f  =     Ef – Ei / h     =    – Z² Rc  [  1/ nf² –  1/ ni² ]

Let λ be the wavelength of emitted radiation , then ,  f =  c/λ  =   – Z² Rc  [  1/ nf² –  1/ ni² ]

and wave number is given by  v¯    =      1 / λ      =    – Z² R  [  1/ nf² –  1/ ni² ]

Note  : The above relation can be used for energy level transitions of hydrogen like atoms i.e , Hydrogen ( Z = 1 ) , Helium ( Z = 2 ) , Lithium ( Z = 3 ) and Beryllium ( Z = 4 ) .

Bohr Model and Energy levels

The properties of an electron as described by the Bohr model can be further studied by considering a potential energy feild which tends to influence an electron of the Bohr atom. Let this non-columbian , potential energy feild function be U = f (r) . Now , an orbiting electron will experience an inward force towards the centre of the orbital path.  This force   –

Force ,   F  =    | dU / dr |  =  | d / dr [ f ( r ) ] |

Now, consider a stable nth orbit where an electron is moving with a speed vn in an orbit of radius rn .  Here    –

| d / dr [ f ( rn ) ] |    =    m vn² / rn eq (1) ……… this equation could deal with the first postulate of the Bohr model .  Note that the equation dealing with the second postulate of the Bohr model is   m vn rn =  n h /2 π  …. eq (2) . This equation is based on the quantization of the angular momentum of electrons . Further derivations could be arrived at by combining equations (1) and (2) .

(   to continue …….. )