629 lines
22 KiB
Fortran
629 lines
22 KiB
Fortran
!
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!=======================================================================
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!
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MODULE DFUNCT_STAN_DYNAMIC
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!
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USE ACCURACY_REAL
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!
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!
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CONTAINS
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!
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!
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!=======================================================================
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!
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! Transverse Dielectric Functions
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!
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!=======================================================================
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!
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SUBROUTINE DFUNCT_DYNAMIC(X,Z,D_FUNCT,EPSR,EPSI)
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!
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! This subroutine computes the transverse dynamic
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! dielectric functions
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!
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!
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! Input parameters:
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!
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! * X : dimensionless factor --> X = q / (2 * k_F)
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! * Z : dimensionless factor --> Z = omega / omega_q
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! * D_FUNCT : type of transverse dielectric function
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!
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!
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! Output parameters:
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!
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! * EPSR : real part of the dielectric function
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! * EPSI : imaginary part of the dielectric function
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!
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!
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 19 Jun 2020
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!
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!
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USE MATERIAL_PROP, ONLY : DMN
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!
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IMPLICIT NONE
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!
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CHARACTER (LEN = 4) :: D_FUNCT
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!
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REAL (WP) :: X,Z,RS,T,TAU
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REAL (WP) :: EPSR,EPSI
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!
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IF(DMN == '3D') THEN !
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CALL DFUNCT_DYNAMIC_3D(X,Z,D_FUNCT,EPSR,EPSI) !
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ELSE IF(DMN == '2D') THEN !
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CALL DFUNCT_DYNAMIC_2D(X,Z,D_FUNCT,EPSR,EPSI) !
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ELSE IF(DMN == '1D') THEN !
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CONTINUE !
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END IF !
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!
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END SUBROUTINE DFUNCT_DYNAMIC
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!
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!=======================================================================
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!
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! 1) 3D case
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!
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!=======================================================================
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!
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SUBROUTINE DFUNCT_DYNAMIC_3D(X,Z,D_FUNCT,EPSR,EPSI)
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!
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! This subroutine computes the transverse dynamic
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! dielectric functions in 3D
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!
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!
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! Input parameters:
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!
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! * X : dimensionless factor --> X = q / (2 * k_F)
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! * Z : dimensionless factor --> Z = omega / omega_q
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! * RS : Wigner-Seitz radius (in units of a_0)
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! * T : temperature in SI
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! * D_FUNCT : type of transverse dielectric function (3D)
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! D_FUNCT = 'RPA1' random phase approximation
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! D_FUNCT = 'RPA2' random phase approximation
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! D_FUNCT = 'LVLA' linearized Vlasov
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! D_FUNCT = 'MER1' Mermin
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! D_FUNCT = 'BLTZ' Boltzmann
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!
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!
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! Output parameters:
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!
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! * EPSR : real part of the dielectric function
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! * EPSI : imaginary part of the dielectric function
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!
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!
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 16 Jun 2020
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!
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!
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!
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IMPLICIT NONE
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!
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CHARACTER (LEN = 4) :: D_FUNCT
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!
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REAL (WP) :: X,Z,RS,T,TAU
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REAL (WP) :: EPSR,EPSI
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!
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IF(D_FUNCT == 'RPA1') THEN !
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CALL RPA1_EPS_D_TR_3D(X,Z,RS,EPSR,EPSI) !
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ELSE IF(D_FUNCT == 'RPA2') THEN !
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CALL RPA2_EPS_D_TR_3D(X,Z,RS,T,EPSR,EPSI) !
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ELSE IF(D_FUNCT == 'LVLA') THEN !
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CALL LVLA_EPS_D_TR_3D(X,Z,RS,T,EPSR,EPSI) !
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ELSE IF(D_FUNCT == 'MER1') THEN !
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CALL MER1_EPS_D_TR_3D(X,Z,RS,TAU,EPSR,EPSI) !
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ELSE IF(D_FUNCT == 'BLTZ') THEN !
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CALL BLTZ_EPS_D_TR_3D(X,Z,RS,TAU,EPSR,EPSI) !
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END IF !
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!
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END SUBROUTINE DFUNCT_DYNAMIC_3D
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!
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!=======================================================================
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!
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SUBROUTINE RPA1_EPS_D_TR_3D(X,Z,RS,EPSR,EPSI)
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!
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! This subroutine computes the transvere
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! RPA dielectric function EPS(q,omega) for 3D systems.
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!
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! References: (1) Z. H. Levine and E. Cockayne,
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! J. Res. Natl. Inst. Stand. Technol. 113, 299-304 (2008)
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! (2) J. Solyom, "Fundamental of the Physics of Solids",
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! Vol3, Chap. 29, p. 117, Springer
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!
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! Notation: hbar omega_q = hbar^2 q^2 / 2m
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!
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! Input parameters:
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!
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! * X : dimensionless factor --> X = q / (2 * k_F)
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! * Z : dimensionless factor --> Z = omega / omega_q
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! * RS : Wigner-Seitz radius (in units of a_0)
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!
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! Intermediate parameters:
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!
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! * U : dimensionless factor --> U = omega / q v_F
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!
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! Output parameters:
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!
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! * EPSR : real part of the dielectric function
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! * EPSI : imaginary part of the dielectric function
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!
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! Note:
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!
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! The connection between the two reference is obtained through the relation:
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!
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! ( k_{TF}/q )^2 = 3 U^2 ( omega_p/omega )^2
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 16 Jun 2020
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!
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USE REAL_NUMBERS, ONLY : ZERO,ONE,TWO,THREE,EIGHT
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USE CONSTANTS_P1, ONLY : H_BAR
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USE FERMI_SI, ONLY : KF_SI,VF_SI
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USE PI_ETC, ONLY : PI
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USE PLASMON_ENE_SI
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!
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IMPLICIT NONE
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!
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REAL (WP) :: X,Z,U,RS
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REAL (WP) :: EPSR,EPSI
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REAL (WP) :: X1,X2
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REAL (WP) :: Q_SI,O_SI,COEF
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!
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U=X*Z ! omega / (q * v_F)
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!
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X1=X-U !
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X2=X+U !
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!
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Q_SI=TWO*X*KF_SI ! q in SI
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O_SI=U*Q_SI*VF_SI ! omega in SI
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!
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COEF=ENE_P_SI*ENE_P_SI/(H_BAR*H_BAR*O_SI*O_SI) !
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!
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! Real part
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!
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EPSR=ONE - COEF * ( & !
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THREE*(X*X + THREE*U*U + ONE)/EIGHT - & !
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THREE*( & !
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(ONE-X2*X2)**2 * & !
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DLOG(DABS((X2+ONE)/(X2-ONE))) + & ! eq. (7) ref. 1
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(ONE-X1*X1)**2 * & ! eq. (29.6.87) ref. 2
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DLOG(DABS((X1+ONE)/(X1-ONE))) & !
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) / (32.0E0_WP*X) & !
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) !
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!
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! Imaginary part
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!
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IF( U < DABS(ONE-X) ) THEN !
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!
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IF(X <= ONE) THEN !
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EPSI=0.75E0_WP*PI*COEF*U*(ONE-U*U-X*X) ! eq. (6) ref. 1
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ELSE !
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EPSI=ZERO ! eq. (29.6.89) ref. 2
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END IF !
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!
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ELSE IF( (DABS(ONE-X) <= U) .AND. (U <= (ONE+X)) ) THEN !
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!
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EPSI=THREE*PI*COEF * (ONE - (U-X)**2 )**2 / (32.0E0_WP*X) ! eq. (6) ref. 1
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!
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ELSE IF( (ONE+X) <= U ) THEN !
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!
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EPSI=ZERO !
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!
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END IF !
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!
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END SUBROUTINE RPA1_EPS_D_TR_3D
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!
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!=======================================================================
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!
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SUBROUTINE RPA2_EPS_D_TR_3D(X,Z,RS,T,EPSR,EPSI)
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!
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! This subroutine computes the transverse temperature-dependent
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! RPA dielectric function EPS(q,omega,T) for 3D systems.
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!
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! References: (1) H. Reinholz et al, Contrib. Plasma Phys. 43, 3-10 (2003)
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!
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! Notation: hbar omega_q = hbar^2 q^2 / 2m
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!
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! Input parameters:
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!
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! * X : dimensionless factor --> X = q / (2 * k_F)
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! * Z : dimensionless factor --> Y = omega / omega_q
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! * RS : Wigner-Seitz radius (in units of a_0)
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! * T : temperature (SI)
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!
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! Intermediate parameters:
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!
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! * U : dimensionless factor --> U = omega / q v_F
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!
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! Output parameters:
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!
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! * EPSR : real part of the dielectric function
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! * EPSI : imaginary part of the dielectric function
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 16 Jun 2020
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!
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USE REAL_NUMBERS, ONLY : ZERO,ONE,TWO
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USE CONSTANTS_P1, ONLY : H_BAR,M_E,K_B
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USE FERMI_SI, ONLY : KF_SI,VF_SI
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USE EXT_FUNCTIONS, ONLY : DAWSON
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USE PLASMON_ENE_SI
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!
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IMPLICIT NONE
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!
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REAL (WP) :: X,Z,U,RS,T
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REAL (WP) :: EPSR,EPSI
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REAL (WP) :: Q_SI,O_SI,XX
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!
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U=X*Z ! omega / (q * v_F)
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!
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Q_SI=TWO*X*KF_SI ! q in SI
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O_SI=U*Q_SI*VF_SI ! omega in SI
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!
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XX=O_SI*DSQRT(M_E/(TWO*K_B*T))/Q_SI !
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!
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EPSR=ONE+ENE_P_SI*ENE_P_SI*DAWSON(XX)/(H_BAR*H_BAR*O_SI*O_SI) ! ref. (1) eq. (10)
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EPSI=ZERO !
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!
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END SUBROUTINE RPA2_EPS_D_TR_3D
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!
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!=======================================================================
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!
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SUBROUTINE LVLA_EPS_D_TR_3D(X,Z,RS,T,EPSR,EPSI)
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!
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! This subroutine computes the transverse linearized Vlasov dynamical
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! dielectric function in 3D
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!
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! References: (1) S. Ichimaru, "Statistical Plasma Physics - Vol1",
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! CRC Press (2004)
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!
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! Notation: hbar omega_q = hbar q^2 / 2m
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!
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! Input parameters:
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!
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! * X : dimensionless factor --> X = q / (2 * k_F)
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! * Z : dimensionless factor --> Y = omega / omega_q
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! * T : temperature (SI)
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!
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! Intermediate parameters:
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!
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! * U : dimensionless factor --> U = omega / q v_F
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!
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! Output parameters:
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!
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! * EPSR : real part of the dielectric function
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! * EPSI : imaginary part of the dielectric function
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 16 Jun 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ONE,TWO,HALF
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USE CONSTANTS_P1, ONLY : H_BAR,M_E,K_B
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USE FERMI_SI, ONLY : KF_SI,VF_SI
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USE EXT_FUNCTIONS, ONLY : W
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USE PLASMON_ENE_SI
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!
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IMPLICIT NONE
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!
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REAL (WP) :: X,Z,RS,T,U,Y,Y2,V
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REAL (WP) :: EPSR,EPSI
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REAL (WP) :: ZZ,Q_SI,OM
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REAL (WP) :: FR
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!
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Y=X+X ! Y = q / k_F
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Y2=Y*Y !
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U=X*Z ! omega / (q * v_F)
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V=Z*Y2 ! omega / omega_{k_F}
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Q_SI=TWO*X*KF_SI ! q
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OM=V*HALF*H_BAR*KF_SI*KF_SI/M_E ! omega
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!
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ZZ=U*VF_SI/DSQRT(K_B*T/M_E) ! argument of PDF W(zz)
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FR=(ENE_P_SI/(H_BAR*OM))**2 ! (omega_p/omega)^2
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!
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EPSR=ONE - FR * (ONE - DREAL(W(ZZ))) ! ref. (2) eq. (4.76)
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EPSI=FR * (ONE - DIMAG(W(ZZ))) !
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!
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END SUBROUTINE LVLA_EPS_D_TR_3D
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!
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!=======================================================================
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!
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SUBROUTINE MER1_EPS_D_TR_3D(X,Z,RS,TAU,EPSR,EPSI)
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!
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! This subroutine computes the transverse Mermin dynamical
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! dielectric function in 3D
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!
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! References: (1) P.-O. Chapuis et al, Phys. Rev. B 77, 035441 (2008)
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!
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! Note: for TAU --> infinity, we should recover the RPA values
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!
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! Notation: hbar omega_q = hbar q^2 / 2m
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!
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! Input parameters:
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!
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! * X : dimensionless factor --> X = q / (2 * k_F)
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! * Z : dimensionless factor --> Y = omega / omega_q
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! * RS : Wigner-Seitz radius (in units of a_0)
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! * TAU : relaxation time (used for damping) in SI
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!
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! Intermediate parameters:
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!
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! * U : dimensionless factor --> U = omega / q v_F
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!
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! Output parameters:
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!
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! * EPSR : real part of the dielectric function
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! * EPSI : imaginary part of the dielectric function
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!
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!
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! Author : D. Sébilleau
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!
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! Last modified : 16 Jun 2020
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!
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!
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USE REAL_NUMBERS, ONLY : ONE,TWO,THREE,FOUR,EIGHT, &
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HALF
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USE COMPLEX_NUMBERS, ONLY : IC
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USE CONSTANTS_P1, ONLY : H_BAR
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USE FERMI_SI, ONLY : KF_SI,VF_SI
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USE MULTILAYER, ONLY : EPS_1
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USE PLASMON_ENE_SI
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!
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IMPLICIT NONE
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!
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REAL (WP) :: X,Z,U,RS,TAU
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REAL (WP) :: EPSR,EPSI
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REAL (WP) :: Q_SI,O_SI
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!
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REAL*8 EPS_INF
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!
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COMPLEX (WP) :: UU,FT_U,FT_0,FL_U,FL_0
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COMPLEX (WP) :: ZPU,ZMU
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COMPLEX (WP) :: COEF,EPS
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!
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U=X*Z ! omega / (q * v_F)
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Q_SI=TWO*X*KF_SI ! q in SI
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O_SI=U*Q_SI*VF_SI ! omega in SI
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!
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COEF=ENE_P_SI*ENE_P_SI/(H_BAR*H_BAR*(O_SI+IC/TAU)) !
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!
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UU=(O_SI+IC/TAU)/(Q_SI*VF_SI) !
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ZPU=X+UU !
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ZMU=X-UU !
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!
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FL_0=HALF + (ONE-X*X)*DLOG(DABS((X+ONE)/(X-ONE)))/(FOUR*X) ! ref (1) eq. (13)
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FL_U=HALF + (ONE-ZMU*ZMU)*CDLOG((ZMU+ONE)/(ZMU-ONE)) / & !
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(EIGHT*X) + & !
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(ONE-ZPU*ZPU)*CDLOG((ZPU+ONE)/(ZPU-ONE)) / & !
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(EIGHT*X) ! ref (1) eq. (11)
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!
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FT_0=0.375E0_WP*(X*X+ONE) - 0.1875E0_WP*(ONE-X*X)*(ONE-X*X)* &! ref (1) eq. (14)
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DLOG(DABS((X+ONE)/(X-ONE)))/X !
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FT_U=0.375E0_WP*(X*X+THREE*UU*UU+ONE) - & !
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0.09375E0_WP*(ONE-ZMU*ZMU)*(ONE-ZMU*ZMU)* & ! ref (1) eq. (12)
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CDLOG((ZMU+ONE)/(ZMU-ONE))/X - & !
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0.09375E0_WP*(ONE-ZPU*ZPU)*(ONE-ZPU*ZMU)* & !
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CDLOG((ZPU+ONE)/(ZPU-ONE))/X !
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!
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EPS=EPS_1 - COEF*( O_SI*(FT_U-THREE*X*X*FL_U) + & !
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IC*(FT_0-THREE*X*X*FL_0)/TAU & ! ref (1) eq. (11)
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) !
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!
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EPSR=DREAL(EPS) !
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EPSI=DIMAG(EPS) !
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!
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END SUBROUTINE MER1_EPS_D_TR_3D
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!
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!=======================================================================
|
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!
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|
SUBROUTINE BLTZ_EPS_D_TR_3D(X,Z,RS,TAU,EPSR,EPSI)
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|
!
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|
! This subroutine computes the transverse Boltzmann dynamical
|
|
! dielectric function in 3D
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|
!
|
|
! References: (1) R. Esquivel and V. B. Stetovoy, Phys. Rev. A 69, 062102 (2004)
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!
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|
! Notation: hbar omega_q = hbar q^2 / 2m
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!
|
|
! Input parameters:
|
|
!
|
|
! * X : dimensionless factor --> X = q / (2 * k_F)
|
|
! * Z : dimensionless factor --> Z = omega / omega_q
|
|
! * RS : Wigner-Seitz radius (in units of a_0)
|
|
! * TAU : relaxation time (used for damping) in SI
|
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!
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|
! Intermediate parameters:
|
|
!
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|
! * U : dimensionless factor --> U = omega / q v_F
|
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!
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|
! Output parameters:
|
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!
|
|
! * EPSR : real part of the dielectric function
|
|
! * EPSI : imaginary part of the dielectric function
|
|
!
|
|
!
|
|
! Author : D. Sébilleau
|
|
!
|
|
! Last modified : 17 Jun 2020
|
|
!
|
|
!
|
|
USE REAL_NUMBERS, ONLY : ONE,TWO,THREE,HALF
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|
USE COMPLEX_NUMBERS, ONLY : IC
|
|
USE CONSTANTS_P1, ONLY : H_BAR
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|
USE FERMI_SI, ONLY : KF_SI,VF_SI
|
|
USE PLASMON_ENE_SI
|
|
!
|
|
IMPLICIT NONE
|
|
!
|
|
REAL (WP) :: X,Z,U,RS,TAU
|
|
REAL (WP) :: EPSR,EPSI
|
|
REAL (WP) :: Q_SI,O_SI
|
|
!
|
|
COMPLEX (WP) :: UU,U3,COEF
|
|
COMPLEX (WP) :: LLOG,FT
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|
!
|
|
U=X*Z ! omega / (q * v_F)
|
|
!
|
|
Q_SI=TWO*X*KF_SI ! q in SI
|
|
O_SI=U*Q_SI*VF_SI ! omega in SI
|
|
!
|
|
UU=Q_SI*VF_SI/(O_SI + IC/TAU) ! ref. (1) eq. (16)
|
|
U3=UU*UU*UU !
|
|
COEF=ENE_P_SI*ENE_P_SI/(H_BAR*H_BAR) * & !
|
|
ONE/(O_SI*O_SI + IC*O_SI/TAU) !
|
|
LLOG=CDLOG((ONE+UU)/(ONE-UU)) !
|
|
!
|
|
FT=THREE*HALF/U3 * ( UU - HALF*(ONE-UU*UU)*LLOG ) ! ref. (1) eq. (14)
|
|
!
|
|
EPSR=ONE - DREAL(COEF*FT) !
|
|
EPSI=DIMAG(COEF*FT) !
|
|
!
|
|
END SUBROUTINE BLTZ_EPS_D_TR_3D
|
|
!
|
|
!=======================================================================
|
|
!
|
|
! 1) 2D case
|
|
!
|
|
!=======================================================================
|
|
!
|
|
SUBROUTINE DFUNCT_DYNAMIC_2D(X,Z,D_FUNCT,EPSR,EPSI)
|
|
!
|
|
! This subroutine computes the transverse dynamic
|
|
! dielectric functions in 2D
|
|
!
|
|
!
|
|
! Input parameters:
|
|
!
|
|
! * X : dimensionless factor --> X = q / (2 * k_F)
|
|
! * Z : dimensionless factor --> Z = omega / omega_q
|
|
! * D_FUNCT : type of transverse dielectric function (2D)
|
|
! D_FUNCT = 'TRPA' random phase approximation
|
|
!
|
|
!
|
|
! Output parameters:
|
|
!
|
|
! * EPSR : real part of the dielectric function
|
|
! * EPSI : imaginary part of the dielectric function
|
|
!
|
|
!
|
|
!
|
|
!
|
|
! Author : D. Sébilleau
|
|
!
|
|
! Last modified : 12 Jun 2020
|
|
!
|
|
!
|
|
!
|
|
IMPLICIT NONE
|
|
!
|
|
CHARACTER (LEN = 4) :: D_FUNCT
|
|
!
|
|
REAL (WP) :: X,Z,RS
|
|
REAL (WP) :: EPSR,EPSI
|
|
!
|
|
IF(D_FUNCT == 'RPA1') THEN !
|
|
CALL RPA1_EPS_D_TR_2D(X,Z,RS,EPSR,EPSI) !
|
|
END IF !
|
|
!
|
|
END SUBROUTINE DFUNCT_DYNAMIC_2D
|
|
!
|
|
!=======================================================================
|
|
!
|
|
SUBROUTINE RPA1_EPS_D_TR_2D(X,Z,RS,EPSR,EPSI)
|
|
!
|
|
! This subroutine computes the transverse
|
|
! RPA dielectric function EPS(q,omega,T) for 2D systems.
|
|
!
|
|
! References: (1) R. Nifosi, S. Conti and M. P. Tosi,
|
|
! Phys. Rev. B 58, 12758 (1998)
|
|
!
|
|
! Notation: hbar omega_q = hbar^2 q^2 / 2m
|
|
!
|
|
! Input parameters:
|
|
!
|
|
! * X : dimensionless factor --> X = q / (2 * k_F)
|
|
! * Z : dimensionless factor --> Z = omega / omega_q
|
|
! * RS : Wigner-Seitz radius (in units of a_0)
|
|
!
|
|
! Intermediate parameters:
|
|
!
|
|
! * U : dimensionless factor --> U = omega / q v_F
|
|
!
|
|
! Output parameters:
|
|
!
|
|
! * EPSR : real part of the dielectric function
|
|
! * EPSI : imaginary part of the dielectric function
|
|
!
|
|
! Note:
|
|
!
|
|
! The dielectric function is obtained from the current-current susceptibility by
|
|
!
|
|
! eps = 1 - ( omega_p/omega )^2 * [ 1 + m/n * chi ]
|
|
!
|
|
!
|
|
! Author : D. Sébilleau
|
|
!
|
|
! Last modified : 12 Jun 2020
|
|
!
|
|
!
|
|
USE REAL_NUMBERS, ONLY : ZERO,ONE,TWO,SIX,THIRD
|
|
USE CONSTANTS_P1, ONLY : H_BAR
|
|
USE FERMI_SI, ONLY : KF_SI,VF_SI
|
|
USE PLASMON_ENE_SI
|
|
!
|
|
IMPLICIT NONE
|
|
!
|
|
REAL (WP) :: X,Z,U,RS
|
|
REAL (WP) :: EPSR,EPSI
|
|
REAL (WP) :: XP,XM,EP,EM,BP,BM
|
|
REAL (WP) :: Q_SI,O_SI,COEF
|
|
!
|
|
U=X*Z ! omega / (q * v_F)
|
|
!
|
|
XP=X+U !
|
|
XM=X-U !
|
|
!
|
|
Q_SI=TWO*X*KF_SI ! q in SI
|
|
O_SI=U*Q_SI*VF_SI ! omega in SI
|
|
!
|
|
COEF=ENE_P_SI*ENE_P_SI/(H_BAR*H_BAR*O_SI*O_SI) !
|
|
!
|
|
IF(XP.GE.ONE) THEN !
|
|
EP=SIGN(XP,(XP*XP-ONE)**1.5E0_WP) ! eq. (A7) ref. 1
|
|
BP=ZERO !
|
|
ELSE !
|
|
EP=ZERO !
|
|
BP=(ONE-XP*XP)**1.5E0_WP ! eq. (A4) ref. 1
|
|
ENDIF !
|
|
!
|
|
IF(XM.GE.ONE) THEN !
|
|
EM=SIGN(XM,(XM*XM-ONE)**1.5E0_WP) ! eq. (A7) ref. 1
|
|
BM=ZERO !
|
|
ELSE !
|
|
EM=ZERO !
|
|
BM=(ONE-XM*XM)**1.5E0_WP ! eq. (A4) ref. 1
|
|
ENDIF !
|
|
!
|
|
! Real part
|
|
!
|
|
EPSR=ONE-COEF*THIRD* ( & !
|
|
TWO*X*X*X + SIX*U*U*X - EP - EM & ! eq. (A7) ref. 1
|
|
) / X !
|
|
!
|
|
! Imaginary part
|
|
!
|
|
EPSI=-COEF*THIRD*(BP-BM)/X ! eq. (A4) ref. 1
|
|
!
|
|
END SUBROUTINE RPA1_EPS_D_TR_2D
|
|
!
|
|
END MODULE DFUNCT_STAN_DYNAMIC
|