**Riki o Konno ^{1*}**

^{1}Kindai University Technical College, 7-1 Kasugaoka, Nabari-shi, Mie 518-0459, Japan.

** *Corresponding Author:**Riki o Konno, Kindai University Technical College, 7-1 Kasugaoka, Nabari-shi, Mie 518-0459, Japan, Tel:81-595-41-0111; Fax:81-595-41-0111

**Citation: **Riki o Konno (2024) Self-Consistent Renormalization Theory of Anisotropic Spin Fluctuations in Nearly Ferromagnetic Metals. *Nano Technol & Nano Sci J 6*: 160.

**Received**: July 10, 2024; **Accepted**: July 17, 2024; **Published**: July 20, 2024.

**Copyright: **© 2024 Rikio Konno, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

We investigated the temperature dependence of the inverse of the magnetic susceptibility, the nuclear magnetic relaxation rate, and the *T*-linear coefficient of the specific heat in nearly ferromagnetic metals by using the self-consistent renormalization theory of anisotropic spin fluctuations. At low temperatures, the inverse of the magnetic susceptibility has *T ^{2}*-linear dependence. In elevated temperatures, the inverse of the magnetic susceptibility has

The magnetic properties of nearly ferromagnetic metals have intrigued the interest of many experimental and theoretical researchers [1-14]. Recently, the anisotropic spin fluctuations were investigated in nearly antiferromagnetic metals beyond the random phase approximation [15, 16]. However, in the nearly ferromagnetic metals, the influence of the anisotropic spin fluctuations has not been resolved. Therefore, the self-consistent renormalization theory of anisotropic spin fluctuations in the nearly ferromagnetic metals is constructed beyond the random phase approximation in this paper. The inverse of the magnetic susceptibility is investigated. The nuclear magnetic relaxation rate is studied. The *T*-linear coefficient of the specific heat is examined. Throughout this paper, we use units of energy, such that *h=1, k _{B} = 1, and gμ_{B }*where

This paper is organized as follows: the formulation will be provided in section 2. The numerical results will be supplied in section 3. The conclusions will be given in section 4.

Let's begin the non-interacting dynamical susceptibility. By using Moriya's expression [14] based on the single band Hubbard model, the non-interacting dynamical susceptibility *x*_{0v }as follows:

*x*_{0v }(0)is the non-interacting magnetic susceptibility. *q*_{B} is the magnitude of the zone boundary wave vector. From Eq. (2), is is

where *Ψ(μ _{v}*) is the digamma function,

where *y _{v} (v* = || or ⊥) is the inverse of the reduced magnetic susceptibility.

From Eqs. (9) and (10), the equations of the inverse of the reduced magnetic susceptibility are obtained.

**Figure 1:** The temperature dependence of the inverse of the reduced magnetic susceptibility *y _{v} (v* = || or ⊥) when

Fig.1 shows the temperature dependence of *y _{v} (v* = ||, or ⊥) with

**Figure 2:** The temperature dependence of the inverse of the reduced magnetic susceptibility *y _{v} (v* = ||, or ⊥) when

The inverse of the reduced magnetic susceptibility has *T*-linear dependence from Fig.1 and Fig.2. At low temperatures t<< 1, we use the following asymtotic expansion of the digamma function in the integrand of Eqs. (11) and (12).

At low temperatures the inverse of the magnetic susceptibility has *T ^{2}*-linear dependence. In elevated temperatures, it has

The nuclear magnetic relaxation rate is studied by using the dynamical susceptibility in the nearly ferromagnetic metals. It is obtained:

where *T _{1v}* (

**Figure 3:** The temperature dependence of *y _{0||}* = 0.01 (the red line),

Fig. 3 shows the temperature dependence of the nuclear magnetic relaxation rate with *y _{0||}* = 0.01 (the red line),

**The ***T***-linear Coefficient of the Specific Heat**

The free energy of spin fluctuations is obtained as follows [5]:

** Figure 4: **The temperature dependence of when

where Γ_{qv} is the damping constant of spin fluctuations (*v*= || or ⊥) . The specific heat of spin fluctuations is

From Eq. (24), *γ _{m} *increases when

We have made the self-consistent renormalization theory of anisotropic spin fluctuations in three dimensional nearly ferromagnetic metals beyond the random phase approximation. We have investigated the temperature dependence of the inverse of the magnetic susceptibility, nuclear magnetic relaxation rate, and the *T*-linear coefficient of the specific heat in nearly ferromagnetic metals. We have found that the temperature dependence of the inverse of the magnetic susceptibility has *T*^{2}-linear behavior at low temperatures. With increasing temperatures, it has *T*-linear dependence. The nuclear magnetic relaxation rate has *t*/*y _{v}*-linear dependence. The anisotropy appears in the inverse of the magnetic susceptibility and the nuclear magnetic relaxation rate by anisotropic spin fluctuations.

This work is supported by the Kindai University Technical College grants. The author would like to thank Y. Takahashi, D. Legut, and S. Khmelevskyi for fruitfull discussions. He would like to also thank Y. Tokunaga, Y. Haga, H. von Lohneysen, M. Brando, F. Steglich, C. Geibel, J. Flouquet, A. de Visser, S. Murayama, E. Bauer, M. Grosche, K. Gofryk, P. Canfield, F. Honda,K. Ishida and V. Sechovsky for stimulating conversations.

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