# For the beta (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K−shell, is captured by the - Physics

Numerical

For the beta^+ (positron) emission from a nucleus, there is another competing process known as electron capture (electron from an inner orbit, say, the K−shell, is captured by the nucleus and a neutrino is emitted).

$\ce{e+ + ^A_Z X -> ^A_{Z - 1}Y + \text{v}}$

Show that if beta^+ emission is energetically allowed, electron capture is necessarily allowed but not vice−versa.

#### Solution

Let the amount of energy released during the electron capture process be Q1. The nuclear reaction can be written as:

$\ce{e+ + ^A_Z X -> ^A_{Z - 1}Y + \text{v} + Q_1}$   .....(1)

Let the amount of energy released during the positron capture process be Q2. The nuclear reaction can be written as:

$\ce{e+ + ^A_Z X -> ^A_{Z - 1}Y + \text{v} + Q_1}$ ...(2)

"m"_"N"(""_"Z"^"A" "X")   = Nuclear mass of ""_"Z"^"A" "X"

"m"_"N"(""_("Z"-1)^"A" "Y") = Nuclear mass of ""_("Z"-1)^"A" "Y"

"m"(""_"Z"^"A" "X") = Atomic mass of ""_"Z"^"A" "X"

"m"(""_("Z" - 1)^"A" "X")   = Atomic mass of ""_("Z" -1)^"A" "X"

me = Mass of an electron

c = Speed of light

Q-value of the electron capture reaction is given as:

"Q"_1 = ["m"_"N" (""_"Z"^"A" "X") + "m"_"e" - "m"_"N"(""_("Z"-1)^"A" "Y"))]"c"^2

= ["m"(""_"Z"^"A" "X") - "Zm"_"e" + "m"_"e" - "m"(""_("Z"-1)^"A" "Y") + ("Z" - 1)"m"_"e"]"c"^2

= ["m"(""_"Z"^"A"  "X") - "m"(""_("z" - 1)^"A" "Y")]"c"^2    ....(3)

Q-value of the positron capture reaction is given as:

"Q"_2 = ["m"_"N" (""_"Z"^"A" "X") - "m"_"N"(""_("z"-1)^"A" "Y") - "m"_"e"]"c"^2

= ["m"_"N"(""_"Z"^"A"  "X") - "m"_"N" (""_("z"-1)^"A" "Y") + ("Z" - 1)"m"_"e" - "m"_"e"]"c"^2

= ["m"(""_"Z"^"A" "X") - "m"(""_("z" - 1)^"A" "Y") - 2"m"_"e"]"c"^2   ...(4)

It can be inferred that if Q2 > 0, then Q> 0; Also, if Q1> 0, it does not necessarily mean that Q2 > 0.

In other words, this means that if beta^+ emission is energetically allowed, then the electron capture process is necessarily allowed, but not vice-versa. This is because the Q-value must be positive for an energetically allowed nuclear reaction.

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#### APPEARS IN

NCERT Physics Part 1 and 2 Class 12
Chapter 13 Nuclei
Exercise | Q 13.22 | Page 464
NCERT Class 12 Physics Textbook
Chapter 13 Nuclei
Exercise | Q 22 | Page 464
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