Thermodynamic theory of saturation of the impulse magnon excitation

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Thermodynamic theory of saturation of the impulse magnon excitation

Yu.D. Kalafati, V.L. Safonov

To cite this version:

Yu.D. Kalafati, V.L. Safonov. Thermodynamic theory of saturation of the impulse magnon exci- tation. Journal de Physique, 1989, 50 (10), pp.1157-1161. �10.1051/jphys:0198900500100115700�.

�jpa-00210986�

Short Communication

Thermodynamic theory

impulse

Yu.D. Kalafati and V.L. Safonov

Kurchatov Institute of Atomic Energy, 123182 Moscow, ^U.S.S.R.

(Reçu ^{le 16} janvier 1989, accepté ^{le 17 mars} 1989)

Resume. 2014 On développe ^{une théorie} statistique quantique self-consistante de magnons excités pa-

ramétriquement par ^une impulsion puissante et courte. Dans cette théorie, ^{la fonction de} distribution des magnons excités contient deux parties: 1) ^le "condensat", ^la distribution des magnons ^cohérente

avec la pompe ^externe 2) ^le "surcondensat", ^la distribution thermique ^des quasiparticules ^{dans le} systeme de coordonnées en rotation.

Abstract. ²⁰¹⁴ Self-consistent quantum statistical theory ^of magnons parametrically ^excited by ^{a short} powerful impulse ^is developed. According ^{to this} theory the distribution function of excited magnons

can be written in two parts : 1) "condensate", ^the distribution of magnons, ^{coherent with the exter-} nal pump ; 2) "overcondensate", the thermal distribution of quasiparticles ^{in the} rotating system ^of coordinates.

Classification

Physics ^Abstracts

76.50 - 05.30J

Introduction.

The study ^{of the} parametric excitation of magnons by ^a microwave field h.cos wpt yields ^im- portant information on the properties ^of spin systems in ferro- and antiferromagnets. ^{Above the}

threshold h,, the number of parametric ^{magnons of the} frequency cvp/2, ^{with equal and} oppo-

sitely ^{directed wave vectors, increases} exponentially with time until a balance is reached between the energy supplied ^{to the} system ^{from the} pump and the loss of energy ^{due to the} relaxation of the excited magnons. Zakharov et al. [1] developed ^{a model} theory of turbulence of parametric spin

waves a number of the conclusions of which permit ^a satisfactory interpretation ^{of many observed}

effects for the region hlh, - ^{1 1.}

In this paper ^{we consider a} theoretical model to study ^{the behavior of} magnons parametrically

excited by ^{a shot} powerful (h ^» h,) ^microwave impulse (see ^also [2]). ^The impulse ^duration (T)

should satisfy ^the following conditions : Tm ^T Tph, ^{where Tm,} Tph ^{are the} characteristic times of magnon-magnon ^and magnon-phonon interactions respectively. The fact that in a coor-

dinate system rotating with the half frequency ^{of the} pumping ^{field the} Hamiltonian of parametric

magnons ^is independent of time enables us to use thermodynamic ^methods.

Here we obtain self-consistent equations ^which determine the distribution of excited magnons. ^{Then we} study ^{the exact} solutions of these equations ^and analyze the collective os-

cillations of steady ^state.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500100115700

1158

Principal equations.

The Hamiltonian of magnons ^{in a} coordinate system rotating with the half frequency ^{of the} pumping ^{field is} (see ^Ref. [3]) :

where Ek ^{is the} magnon spectrum ; ak ^{and ajb are the Bose} operators; Yk ^{is the} coupling coefficient of the magnon system ^{with the} pump field ; V ( ... ) ^{is the} four-magnon interaction amplitude.

The diagonalization ^{of the} Hamiltonian (1) by ^{means of} Bogolyubov’s transformation

gives

Here

is the energy of the coherent magnon ^state (condensate),

is the spectrum ^{of new} quasiparticles, ^{and H4 is the} four-quasiparticle interaction (all ^notations

will be explained below).

So we can describe an evaluation of magnons ^{as a} relaxation of the Bose gas ^of quasiparticles

to an equilibrium ^state with the effective temperature ^{0 and the} density ^matrix of the form

The value of the chemical potential p is determined from the maximum of entropy : p ^{= min} (Ok)

for e > 0 and M ⁼ max (Ç2k) ^{for e 0. -}

Theequationsforuk, ^Vk can be written in terms of Nk ⁼ Sp(atak.p) ^{anduk =} Sp (aka_k.p) .

So they ^are

where

and nk ⁼ lexp [(S2k - J-l) 18] - 1}-i is the thermal population ^of quasiparticles.

The effective temperature ^{8 can} be determined from the initial conditions in the cases of adiabatic and abrupt switching ^{of the} pump impulse.

Exact solutions.

Equations (7, 8) ^{can be} exactly ^solved (Ref. [2]). ^{Here we consider a} simple ^{case Vx = V,} Tkq =- T and Sk9 - ^{S. Then Ak = 0 and Ek = Ek -} -k., where ko E Ko is defined from the condition of resonance

Solutions for uq ^and Nq at q ft ^{Ka are}

The respective ^{solutions on the resonance surface Ko have the form}

1160

where 1 is the measure of this surface.

One can write the equation ^{for ko} by substitution of Nq by Nko ⁱⁿ equation (9). The obtained

equation ^{can be} easily ^solved by ^{means of} iterations beginning ^from

ka)

The equation ^{for 0 in the case of} abrupt switching ^{has the form}

where _po, Oo ^are the initial chemical potential ^and temperature ^of magnons. For the adiabatic

switching ^of pumping the effective temperature ^of quasiparticles ^{is 0 N} 80.

We have found the equation ^{for the} spectrum À of collective oscillations of steady ^state

The frequency of collective oscillation in the single ^mode (A ⁼ 1) approximation ^is

One can also estabilish the criterion of dynamical stability of excited magnon system

Discussion.

The coming ^of quasiparticle system ^{to the} thermodynamical equilibrium ^can be observed

through ^the saturation of microwave power ^absorbed by ^the sample. ^{Such a} behavior of absorption

has been discovered in some experiments (Refs. [4, 5]) ⁱⁿ antiferromagnets. ^{However, this} picture

was unstable and periodically reiterated. According ^{to our} theory ^these parametric magnon sys- tems hâve oscillated in the vicinity ^{of the unstable} point ⁱⁿ k-space.

We suppose ^{that the} thermodynamic theory ^of parametric ^magnon excitation will permit ^{us to}

describe powerful impulse experiments ^{in real} magnetic systems. ^In principle, ^the thermodynamic

approach ^{in the} rotating system ^of coordinates ^can be generalized ^{to the cases of small energy}

flows. This will allow ^{us to} describe the phenomenon ^of parametric ^{magnon resonance in a wide} region of microwave pump.

Acknowledgment.

We wish to thank Dr. AK Hitrin for helpful discussions.

References

[1] ^{ZAKHAROV VE.,} L’VOV VS. and STAROBINETS S.S., Usp. ^{Fiz- Nauk 114} (1974) ⁶⁰⁹ (Sou Phys. Usp.

17 (1975) 896).

[2] ^KALAFATI Yu.D. and SAFONOV V.L., ^Fiz. Tverd. Tela 29 (1987) ³¹⁸⁹ (Sou Phys. Solid State 29 (1987) No.10).

[3] VINIKOVETSKY I.A., ^FRISHMAN A.M. and TSUKERNIK V.M., ^Zh. Eksp. Teor. Fiz. 76 (1979) ²¹¹⁰ (Sov. Phys. JETP (1979) No. 6).

[4] ANDRIENKO A.V. and PROZOROVA L.A., ^Zh. Eksp. ^{Teor. Fiz. 78} (1980) ²⁴¹¹ (Sov. Phys. ^{JETP 51} (1980)1213).

[5] GOVORKOV S.A. and TULIN V.A., ^Zh. Eksp. Teor. Fiz. 82 (1982) ¹²³⁴ (Sov. Phys. ^{JETP 55} (1982) 718).