[Paper Review] T-odd correlations in neutron reflectometry experiments

V. K. Ignatovich, Yu. V. Nikitenko|ArXiv.org|Jun 15, 2009

Nuclear Physics and Applications4 references3 citations

TL;DR

This paper demonstrates that T-odd correlations—terms proportional to $\mathbf{\sigma} \cdot [\mathbf{B}_1 \times \mathbf{B}_2]$—emerge in neutron transmission through magnetic multilayers with noncollinear magnetizations, even when time-reversal symmetry is preserved. Despite the appearance of such T-odd terms, the authors show that T-invariance is not violated due to the system's overall time-reversal symmetry under complex conjugation and field reversal, while detailed balance is violated in noncollinear magnetic configurations.

ABSTRACT

It is shown that transmission amplitudes of magnetic systems with noncollinear magnetization contain T-odd correlations. Relation of these T-odd correlations to T-invariance and detailed balance is discussed.

Motivation & Objective

• To investigate the origin and implications of T-odd correlations in neutron transmission through magnetic multilayers with noncollinear magnetization.
• To clarify the apparent conflict between T-odd terms and the preservation of T-invariance in systems with absorption.
• To examine the violation of detailed balance in neutron scattering from noncollinear magnetic fields.
• To demonstrate that time-reversal symmetry is preserved despite the presence of T-odd terms in the transmission amplitude.

Proposed method

• Derivation of the transmission matrix $T_t$ for a two-layer magnetic mirror using spin-dependent transmission amplitudes $T_i(\mathbf{\sigma} \cdot \mathbf{B}_i)$, incorporating complex wave vectors and reflection coefficients.
• Expansion of the transmission matrix into symmetric and antisymmetric parts using $f^{(\pm)} = [f(B) \pm f(-B)]/2$, isolating the T-odd term proportional to $\mathbf{\sigma} \cdot [\mathbf{b}_1 \times \mathbf{b}_2]$.
• Use of the identity $(\mathbf{\sigma} \cdot \mathbf{b}_1)(\mathbf{\sigma} \cdot \mathbf{b}_2) = (\mathbf{b}_1 \cdot \mathbf{b}_2) + i\, \mathbf{\sigma} \cdot [\mathbf{b}_1 \times \mathbf{b}_2]$ to explicitly identify the T-odd term in the transmission amplitude.
• Analysis of time-reversal transformation properties of the wave function and scattering amplitudes, showing that the full scattering process remains time-reversible under $k \to k^*$, $\mathbf{B} \to -\mathbf{B}$, and $\mathbf{\sigma} \to \mathbf{\sigma}^*$.
• Verification of detailed balance violation by checking that $R^*(k,u)T(k,u^*) + T^*(k,u)R(k,u^*) \neq 1$ in noncollinear configurations, indicating asymmetric scattering probabilities.
• Numerical and symbolic verification of wave function continuity and amplitude matching across interfaces, confirming consistency of the scattering solution under time reversal.

Experimental results

Research questions

• RQ1 How do T-odd correlations arise in neutron transmission through noncollinear magnetic multilayers?
• RQ2 Why do T-odd terms not imply a violation of T-invariance in systems with absorption?
• RQ3 Does the presence of noncollinear magnetic fields lead to a breakdown of detailed balance in neutron scattering?
• RQ4 Can time-reversal symmetry be preserved in scattering systems with T-odd terms due to complex conjugation and field reversal?
• RQ5 What is the role of spin-orbit coupling and magnetic field geometry in generating observable T-odd effects?

Key findings

• The transmission matrix amplitude contains a T-odd term proportional to $\mathbf{\sigma} \cdot [\mathbf{B}_1 \times \mathbf{B}_2]$, arising from the noncommutativity of spin-dependent transmission through two noncollinear magnetic layers.
• Despite the presence of T-odd terms, the full scattering process remains time-reversal invariant under the combined transformation of complex conjugation, spin reversal, and magnetic field reversal.
• The reflection matrix does not contain T-odd terms, but the system violates the principle of detailed balance due to noncollinear magnetization.
• Detailed balance is most strongly violated in systems with helicoidal magnetization, where the asymmetry in forward and backward scattering becomes most apparent.
• The scattering amplitudes satisfy $R^*(k,u)T(k,u^*) + T^*(k,u)R(k,u^*) = 0$, indicating a $\pi/2$ phase shift between reflection and transmission amplitudes, which breaks detailed balance.
• The time-reversal symmetry of the system is preserved because the full wave function transformation under time reversal yields a consistent solution to the Schrödinger equation, even with absorption.