Behaviour of criticality eigenvalues of one-speed transport operator with linearly anisotropic scattering

Influence of anisotropic scattering on the size of time-dependent systems in monoenergetic neutron transport

“…As mentioned in section 1, the fundamental decay problem is mathematically equivalent to the critical problem [1][2][3][4][15][16][17][18][19][20][21][22][23][24]. We compute the c eigenvalue from equation (7) for a fixed value of and then obtain the critical thickness d t by solving equation ( 4).…”

“…when |b 1 | 1/3 the Krein-Rutman [3] theory shows that there exists a real positive c eigenvalue whose magnitude is smaller than all others [3,4]. If the scattering is limited to linearly anisotropic scattering, then the c eigenvalues are all real for negative values of b 1 , irrespective of the system size [1,3,4]. However the dependence of the c eigenvalues on b 1 is more complicated.…”

“…The positivity of the operator and the presence of a scattering function ensure the existence of a simple, real positive fundamental eigenvalue. Negative or complex cross sections are frequently found in computing the time eigenvalues [3][4][5][6][7][8]. We note that the fundamental time eigenvalue is defined as the smallest number λ with a time behaviour of the form exp(−λt) for the neutron distribution [1,14].…”

“…solving an eigenvalue problem. The criticality eigenvalue problem is, of course, essential in determining the size of a reactor [1][2][3][4][5][6][7][8][9][10][11][12][13], in other words, obtaining the critical conditions and calculating the decay constant. Usually for some physical assembly, the conditions that criticality eigenvalues exist and that they are all real may not be realized.…”

“…Thus, the criticality problem can often be best approached by introducing auxiliary eigenvalues. Moreover, one usually deals with some approximation of the stationary transport operator [1][2][3][4]14]. Deviations in the eigenvalues, both the fundamental and the higher ones, yield information on the range of validity of an approximate operator [1,4].…”