6 where the summation is over the spin variable s = ± 1 and all the states related to the quasiparticles trajectories with | α | ≤ α$_{c}$ in Eq. (4). Then, we obtain the following expression for the free energy per unit length L While deriving the expression for a current [1], the au- thor assumed that α$_{c}$ = π/ 2 (the case of a plane). This means account for the contribution non-Andreev states ( α > α$_{c}$ ). For the calculation of thermodynamic po- tential in equation (8) we shall use the actual magnitude of the parameter α$_{c}$ ( sin ( α$_{c}$ ) = R/ ( R + d ) ). The ap- proximation d << R was used by us in the derivation of the spectrum (4) only. In other words, the neglect of the curvature of cylindrical samples (i.e. the path length of quasiparticles was choosen as d/ cos ( α ) (3)) does not entail the need of account for the states with α > α$_{c}$ for the cylindrical NS of structures ( Fig.1 ). We proceed from the expression for the free energy of a NS contact: C. Analytical estimation of the magnetic moment of the NS structure Ω = − k$_{B}$T ∑ n,q,α,s ln ( 1 + e − E n ( q,α ) k B T ) (11) Ω (Φ) = − R · k$_{B}$T π ℏ 2 n = ∞ ∑ n = $_{−∞}$∫ α$_{c}$ − αc ∫ p$_{F}$ − p$_{F}$ ln ( 1 + exp ($_{−}$ E$_{n}$ ( q, α, Φ) k$_{B}$T )) √ p 2 F − q $^{2}$cos ( α ) dqdα (12) dε$_{n}$ d Φ = − v · √ 1 − ε 2 n u + 1 ε 2 n (17)     u = σ · √ 1 − ( q p$_{F}$ ) 2 · cos ( α ) v = σ · √ 1 − ( q p$_{F}$ ) 2 · sin ( α ) . (13) M $^{∗}$(Φ) = c$_{3}$ n = ∞ ∑ n =0 ∫ ∫ S uv tanh ( ε$_{n}$ 2 c$_{2}$ · T ) √ 1 − ε 2 $_{n}$dudv ( u + √ 1 − ε 2 n √$_{σ}$ 2 − u 2 − v 2 (18) ε$_{n}$ = [ nπ + arccos ( ε$_{n}$ )] · u − Φ · v , (14) Ω (Φ) = c$_{1}$T n = ∞ ∑ n =0 ∫∫ S u ln ( 2 cosh ( ε$_{n}$ 2 c$_{2}$T )) dudv √$_{σ}$ 2 − u 2 − v 2 (15) M ( T, h) ≃ M$_{0}$ n$_{0}$ ∑ n =0 ln cosh ( T$_{A}$ · ˜ n 2 T ) ˜ n 3 [ 1 + ( Φ( T, h) π · ˜ n 2 ] 3 2 (19) ε$_{−}$$_{|}$$_{n}$$_{|}$ ( u, v, Φ) = − ε$_{|}$$_{n}$$_{|}$$_{−}$$_{1}$ ( u, − v, Φ) . (16) Making use of the relation where the energy E$_{n}$ ( q, α, Φ) is given by the exact ex- pression for the spectrum in Eq. (4). For simplification, we introduce the dimensionless quantities ε$_{n}$ = E$_{n}$ ( q,α, Φ) Δ , σ = ℏ · p$_{F}$ 2 d · Δ · m $_{∗}$, − 1 ≤ ε$_{n}$ ≤ 1, ( Δ is the superconduct- ing gap ) and perform the change of variables { q, α } → { u, v } : we evaluate the derivative of the free energy with respect to the flux M $^{∗}$(Φ) = − d Ω d $_{Φ}$: The spectrum and the free energy become where c$_{3}$ = R · Δ π ( p$_{F}$ σ · ℏ ) $^{2}$. Eqs. (9), (8) and (18) fully de- termine the non-linear magnetic response of a cylindrical NS structure to an externally applied magnetic field H. The integral expression of Eq. (18) suggests that M $^{∗}$(Φ) is the odd function of the flux Φ: M $^{∗}$(Φ) = − M $^{∗}$( − Φ). A linear term of the function M $^{∗}$(Φ) has been determined from an approximate estimation of the integral in Eq. (18). This calculation is similar to that in the Attachment of [9]. The final expression for the magnetic moment is where c$_{1}$ = − 2 R · Δ · c$_{2}$ π · ( p$_{F}$ σ · ℏ ) $^{2}$, c$_{2}$ = k$_{B}$ Δ , 0 ≤ u ≤ σ , − σ sin ( α$_{c}$ ) ≤ v ≤ σ sin ( α$_{c}$ ), ε$_{n}$ def = ε$_{n}$ ( u, v, Φ), an inte- gration domain S is a sector of a circle of radius σ . In the expression (15) we also took into account the symmetry of the spectrum in Eq. (14):