Nuclear Fragmentation Reactions from Research to Applications

Fragment mass partitions {N A }
Euler's problem: find the number of ways, P(A 0 ,M), an integer A 0
can be represented as a sum of M integers, A 0 =A 1 +A 2 +A 3 +..., ∑N A=M
PAM ( , ) � PAM ( , �1) �PA ( �MM
, )
0 0 0
Total number of partitions, P(A), can be found from the generating function
� � 1
� � � �
N � �
A
Z x = � � c x = � = �P�A�x
1�
cx
A A
N
1
, N
2
, N
A
, �=
0 A= 1 A= 1 A= 0
A
At large A 0 the result is well approximated by Hardy-Ramanujan formula
1 � 2A �
� � 0
P A = exp ,
0
�π� 48A
3
0 � �
A A
1 3A 6A
0 0
M = ln
� �
� 2 �
π 2 � bπ �
where . Total number of partitions is huge for A0 ~100
P(50)=2*105 , P(100)= 2*108 , P(200)= 4*1012 b= 0.3150
Monte Carlo sampling is required - Markov Chain +Metropolis
A.S. Botvina, A.D. Jackson, I.N. Mishustin, Phys. Rev. E62 (2000) R64