What is Q Value of Nuclear Reaction? Derive the Q- value equation.

Q Value of Nuclear Reaction

Definition:

                  " The difference between sum of kinetic energies of the product and sum of kinetic energies of the reactant in a nuclear reaction is called Q-value of a nuclear reaction. "

                   Consider a projectile nucleus "a" having kinetic energy  E_a and mass m_a is incident on rest target nucleus "X" having mass m_x . The target nucleus "X" has no kinetic energy because it is at rest.

                   The product nucleus "Y" is generated after nuclear reaction has kinetic energy E_y and mass m_y. The emerging nucleus "b" is generated after nuclear reaction has kinetic energy E_b and mass m_b. 

 

                  The law of mass energy is given as 

                                    ∑ (KE + mc² ) = Constant

                  The nuclear reaction is 

          

                                     a + X  ⟶  Y + b

                

                 (m_a c²  + E_a) + (m_x c²  + 0 )   =  (m_y c²  + E_y ) + (m_b c²  + E_b)

                 (m_a + m_x)c² - (m_y+ m_b )c²   =  E_y + E_{b -} E_a

                 (m_a + m_x)c² - (m_y+ m_b )c²   = Q

               Where Q-value of the nuclear reaction is defined as

                                Q =  E_y + E_b  - E_a   

  

_{Q -Value Equation}

                   

                The Q -value equation is used to establish a relationship between Q-value of a nuclear reaction and kinetic energy of projectile nucleus, kinetic energy of emerging nucleus and their masses. 

               Consider a projectile nucleus "a" having kinetic energy "E_a" and mass  "m_a"  is incident on a target nucleus "X" at rest having mass  "m_a".

    

                The projectile and target merge into a compound nucleus after reaction. The compound nucleus breaks into residual nucleus "Y" and emerging nucleus "b" after an interval of time 10^-16 s. Now residual nucleus "Y" has kinetic energy "E_y" and mass "m_y"  while emerging nucleus "b" has kinetic energy "E_b" and mass "m_b". 

                The daughter nucleus "Y" scatters at angle φ while emerging nucleus "b" gets scattered at angle Ө as shown in diagram below.

           Now redraw the diagram in terms of momentum vectors.

    Momentum conservation along horizontal direction is

                               P_a + 0 = P_yCosφ + P_bCosӨ

                                P_yCosφ =  P_a  - P_bCosӨ    - - - - - - - - - - -(1)

   Momentum conservation along vertical direction is 

                                0 = P_ySinφ - P_bSinӨ

                               P_ySinφ = P_bSinӨ               - - - - - - - - - - -(2)

  Square eq(1) and eq(2) and add 

  P_y²Cos²φ  +  P_y²Sin²φ  =  ( P_a  - P_bCosӨ)² + (  P_bSinӨ)²

      P_y² ( Cos²φ + Sin²φ) =  P_a² + P_b² Cos²Ө - 2P_a P_b CosӨ + P_b² Sin²Ө

                          P_y²  =  P_a² + P_b² - 2P_a P_b CosӨ

    Use relation P² = 2mE derived from KE = E= 1/2 mV² = P²/2m

    2m_yE_y  =  2m_a E_a  + 2m_bE_b -2`\sqrt{4m_aE_am_bE_b}`CosӨ

    m_yE_y   =   m_aE_a  + m_bE_b - `\sqrt{4m_aE_am_bE_b}`CosӨ   - - - - - - -  - - - (3)

  The Q-value of a nuclear reaction X(a,b)Y 

                            Q =  E_y + E_b  - E_a   

                             E_y =  Q - E_b  + E_a    

        Put this value in eq(3)

     

           m_y( Q - E_b + E_a) =  m_aE_a + m_bE_b - `\sqrt{4m_aE_am_bE_b}`CosӨ 

           m_yQ - m_yE_b + m_yE_a =  m_aE_a + m_bE_b - `\2sqrt{m_aE_am_bE_b}`CosӨ

     

          m_yQ = m_yE_b - m_yE_a + m_aE_a + m_bE_b - `\2sqrt{m_aE_am_bE_b}`CosӨ

              Q = E_b(1 + m_b/m_y ) + E_a( m_a/m_y - 1) - `\frac{2\sqrt{m_aE_am_bE_b}\}{m_y} `CosӨ

          This is called Q-value of a nuclear reaction. 

Threshold Energy

Definition

              " The endothermic reactions in which Q < 0 requires a net energy input. The minimum amount of energy input required to start endothermic reaction is called threshold energy."

 

          The Q-value of a nuclear reaction can be written as 

    

                                      `\frac{m_y\+\m_b}{m_y}E_b\-\frac{m_y-\m_a}{m_y}E_a\-\frac{2\sqrt{m_am_bE_a}}{m_y}\Cos\theta\sqrt{E_b}\-\Q\=0`

                                        `(m_y+m_b)E_b-2\sqrt{m_am_bE_a}Cos\theta\sqrt{E_b}-\[(m_y-m_a)E_a+Qm_y\]=0`

          This is quadratic equation of the form ax² + bx + c = 0. Its solution is 

                                        `x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}`

                                        `\sqrt{E_b}=\frac{2\sqrt{m_am_bE_a}Cos\theta\pm\sqrt{4m_am_bCos^2\theta\+4(m_y+m_b)\[(m_y-m_a)E_a+m_yQ]}}{2(m_y+m_b)}`

`\sqrt{E_b}=\frac{\sqrt{m_am_bE_a}Cos\theta}{(m_y+m_b)} \pm\sqrt{\frac{m_am_bE_aCos^2\theta}{{(m_y+m_b)}^2}+\frac{(m_y-m_a)E_a+Qm_y}{(m_y+m_b)}}`

`\sqrt{E_b}=V\pm\sqrt{V^2+W}`

          The nuclear reaction is possible when `\sqrt{E_b}` is real and positive. The value of `\sqrt{E_b}` is real and positive when `\V^2+W` is zero.

                                                 `\V^2+W` = 0 

`\frac{m_am_bE_aCos^2\theta}{(m_y+m_b)^2}+\frac{(m_y-m_a)E_a+Qm_y}{(m_y+m_b)}` = 0

`\frac{m_am_bE_aCos^2\theta}{(m_y+m_b)}+(m_y-m_a)E_a+Qm_y` = 0

          Then θ goes to zero, E_a → E_th 

`\frac{m_am_bE_{th}(Cos0)^2}{(m_y+m_{b)}}+(m_y-m_a)E_{th}+Qm_y=0`

`\left[\frac{m_am_b+(m_y+m_b)(m_y-m_a)}{m_y+m_b}\right]E_{th}` = -Qm_y

`\left[\frac{m_am_b+m_ym_y-m_am_y+m_bm_y-m_am_b}{m_y+m_b}\right]E_{th}=-Qm_y`

`\left[\frac{m_y-m_a+m_b}{m_y+m_b}\right]m_yE_{th}=-Qm_y`

`\left[\frac{m_y-m_a+m_b}{m_y+m_b}\right]E_{th}=-Q`

`E_{th}=-Q\left[\frac{m_y+m_b}{m_y-m_a+m_b}\right]` - - - - - - - - -(4)

          The Q-value of a nuclear reaction is given as

                                          (m_a + m_x)c² - (m_y+ m_b )c²   = Q

                                                 m_a + m_x - m_y- m_b   = Q/c² 

                                                 m_a + m_x - m_y- m_b = 0 [ Use Q/ c² ≈   0]

                                                 m_x  =  m_y  - m_a + m_b      - - - - - - - - - - - - -(5)

          Put eq(5) in eq(4)

                            `E_{th}=-Q\left[\frac{m_y+m_b}{m_x}\right]`

          This is energy required to induce a nuclear reaction X(a, b)Y and called threshold energy.