Debye's theory of Atomic Heat Capacity of Solids - Important Notes

 The failure in Einsteins theory of specific heat at low temperature is due to assumption that the vibrations of all atoms are simple harmonic and have one and same frequency.

"Debye" improved the Einsteins theory by considering the atomic oscillators as a system of coupled oscillators having a continuous range of frequencies.

Essential difference between Debye & Einstein model:

Debye has considered the vibrational modes of a crystal as a whole, where as Einstein has considered the vibration of a single atom with the assumption that atomic vibrations are independent of each other.   

Debye has made following assumptions:

1. The solid is capable of vibrating elastically in many different modes.
2. The frequency of vibration is different for different modes.
3. The number of modes of vibration of waves solids are limited in number.
4. The maximum frequency is the fundamental frequency of solid. The maximum frequency is frequency of shorter waves which the solid can transmit. 

According to Debye, a solid can be treated as an elastic body in which vibrations of atoms generate "stationary waves" of both longitudinal & transverse types with velocities Vl & Vt respectively.

The velocities can be determined by elastic constants & densities of solids.

The frequencies range from zero to a definite upper limit.

The number of modes of longitudinal waves per unit volume with frequencies between '𝝂' & '𝝂+d𝝂 ' is represented by 4Π𝝂²d𝝂/(Vl)³.

The number of modes of transverse vibrations per unit volume with frequencies between '𝝂' & '𝝂+d𝝂 ' is represented by 8Π𝝂²d𝝂/(Vt)³. Here 8Π is taken in place of 4Π. Because transverse vibrations have two independent directions of vibration i.e. they are equivalent to two waves at right angles to each other.

So total number of modes of vibrations per unit volume with frequencies '𝝂' & '𝝂+d𝝂 ' is given by  

4Π[(1/Vl³)+(2/Vt³)]𝝂²d𝝂

The total number of independent modes of vibrations is given by  

4ΠV[(1/Vl³)+(2/Vt³)] ∫𝝂²d𝝂 ; V is volume of gm-mole of solid

Why U-235, as isotope of Uranium with only 0.7% abundance will undergo fission if it captures a thermal Neutron where as U-238 will not?

 From 'B/A' against A curve, we know that Binding Energy per Nucleon in Uranium region is about 7MeV, which is almost equal to height of potential barrier opposing fission. 

Now,U-238.in  when a Neutron is added to Nucleus of ₉₂U²³⁵, we get ₉₂U²³⁶ i.e. an Even-Even nuclide. The pairing term in mass formula for Uranium is about 0.6MeV. Thus, we see that Thermal Neutron capture in U-235 results in total energy release of 7.6 MeV and since to cross potential barrier, only about 7 MeV is required. So U-235 can undergo fission by Thermal Neutron capture.

On the other hand, U-238 nucleus, after Neutron capture becomes a U-239 nucleus which is an Even-Odd Nuclide. Due to absence of pairing term for U-239, the fission barrier cannot be crossed. So Thermal Neutrons will not induce fission.        

Useful Range of Materials for Vacuum Vessels and Pipes

 

Critical Energy required for fission of some target nucleus

 

Nuclear Physics - Quick Review for Graduate students

1) "Thorium" has highest half life period and "Polonium" has shortest half life period.

2) Elements lying beyond 92U238 in periodic table are called "Transuranic" elements. 

3) Effective Multiplication Factor Ke = P/(A+L); 'P' is Production of Neutrons; 'A' is Rate of absorption of Neutrons; 'L' is Rate of Leakage of neutrons.

Ke = 1   -----------------------chain reaction is "Critical"  

Ke > 1   -----------------------chain reaction is "Super-Critical"  

Ke < 1   -----------------------chain reaction is "Sub-Critical"  

Ke =NF/(A+L) as P=NF; F is rate at which Fission occur; N is average number of Neutrons emitted per Fission

Control Rods are used to keep Ke = 1

4) Slow Neutrons (Thermalized) are efficient in causing Nuclear Reaction.

5) Atoms having same 'Z' and 'A' but differ from one another in their Nuclear energy states and exhibit differences in internal structure. These Nuclei are distinguished by their different life times. Such Nuclei are called "Isomers".

6) Nuclei having same 'A' but with Proton & Neutron number interchanged  (i.e. No. of Protons in one is equal to No. of Neutrons in other) re called "Mirror nuclei". For instance, 4Be7 (Z=4 and N=3), 3Li7(Z=3 & N=4).

7) Law governing successive disintegration deals with quantity of daughter nuclei present at any instant. 

8) Formation of daughter element depends on decay constant of parent.

9) Decay of daughter element depends on decay constant of itself.

10) Condition for "Secular Equilibrium" is N₁ƛ₁ = N₂ƛ₂ ; this occurs when element-1 is long lived.

11) Condition for "Transient Equilibrium" is N₂/N₁ = ƛ₁/(ƛ₂ - ƛ₁); this occurs when element-2 has same half life as that of parent element-1 and also ƛ₁ < ƛ₂.

12) Let ‘A₀’ be the activity at time of death . Suppose this is reduced to ‘A’ after ‘t’ years. Then    

       t = 3.32 T₁/₂ log₁₀(A₀/A); T₁/₂ is 5568 yrs.

13) Age of rocks may be estimated with comparing quantities of various isotopes of Lead existing in sample.

14)  In “ GM Region",  ionization current is independent of initial ionization.

15)  35 eV is required for producing an ion pair.

14)  Proportional region is most suitable for proportional counters in which number of  pulses increases almost linearly with voltage.

15) Plateau region is most suitable for “ GM counter “ in which number of impulses remain constant. This region is function of Voltage , Resistance, nature of gas.

16) “quenching agent " should have “ Low Ionization Potential".

17)  “ Scintillation" counter depends on principle of “ fluorescence”.

18)  “ Solid state detector " depends on following principle : incoming particles into depletion region must lose all their Kinetic Energy.

19) Non integrating ionization chamber - a pulse type; Integrating Ion chamber works in  current mode.

20) "GM counter" cannot be used to detect ɑ- particles.

21)  “Cloud chamber “ cannot be used to detect ‘𝛾 ‘ rays.

22)  By counting drops in cloud track , specific ionization can be determined.

23)  Particle disintegration process ---- outgoing particle is material particle

24) Simple capture process -- outgoing particles are -γ -rays.

25) Particle disintegration process is more probable than simple capture process.

26) Nuclear Reaction Energy (Q)

Q= (M₀+M₁) - (M₂+M₃)

Q = ( E₂+E₃) - E₁

'M₀' is mass of target nucleus (A)

'M₁E₁' is Mass and Energy of projectile(B)
'M₂E₂' is Mass and Energy of product nucleus (P)
'M₃E₃' is Mass and Energy of outgoing particle(O)

CASE :

If (M₀+M₁) > (M₂+M₃), Q= +VE ;  Exothermic or Exoergic; ex:Li(P,𝛼)He⁴

If (M₀+M₁) > (M₂+M₃) , Q= -VE ;  Endothermic or Endoergic; ex: ₇N¹⁴(𝛼,P)₂He⁴

If (M₀+M₁) = (M₂+M₃), Q= 0 ;  Elastic Collision

27) Threshold energy(Eth) is minimum kinetic energy to initiate endoergic reaction.

Eth = [1+(M₁/M₀)]*Q ;  M₁ is mass of incident particle & M₀ is mass of target particle.

28)  When a parent Nucleus goes from its ground state to ground state of daughter nucleus, it emits an 𝛼-particle of maximum energy.

29) 𝛼-particle is preformed inside parent Nucleus.

Area - Conversion Factors