ClosedOrbitFinder.h 27.2 KB
Newer Older
1 2 3 4 5 6 7 8 9 10 11
/**
 * @file ClosedOrbitFinder.h
 * The algorithm is based on the paper of M. M. Gordon: "Computation of closed orbits and basic focusing properties for
 * sector-focused cyclotrons and the design of 'cyclops'" (1983)
 * As template arguments one chooses the type of the variables and the integrator for the ODEs. The supported steppers can be found on
 * http://www.boost.org/ where it is part of the library Odeint.
 *
 * @author Matthias Frey
 * @version 1.0
 */

12 13 14 15 16 17
#ifndef CLOSEDORBITFINDER_H
#define CLOSEDORBITFINDER_H

#include <array>
#include <cmath>
#include <functional>
adelmann's avatar
adelmann committed
18
#include <limits>
19
#include <numeric>
adelmann's avatar
adelmann committed
20
#include <string>
21
#include <utility>
22 23 24 25 26 27 28 29 30 31 32 33 34 35
#include <vector>

#include "error.h"
#include "physics.h"
#include "physical_error.h"

#include "MagneticField.h" // ONLY FOR STAND-ALONE PROGRAM


#include <fstream>

// include headers for integration
#include <boost/numeric/odeint/integrate/integrate_n_steps.hpp>

36
/// @brief Finds a closed orbit of a cyclotron for a given energy
37 38 39
template<typename Value_type, typename Size_type, class Stepper>
class ClosedOrbitFinder
{
40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185
    public:
        /// Type of variables
        typedef Value_type value_type;
        /// Type for specifying sizes
        typedef Size_type size_type;
        /// Type of container for storing quantities (path length, orbit, etc.)
        typedef std::vector<value_type> container_type;
        /// Type for holding state of ODE values
        typedef std::vector<value_type> state_type;

        /// Sets the initial values for the integration and calls findOrbit().
        /*!
         * @param E is the energy [MeV] to which the closed orbit should be found
         * @param wo is the nominal orbital frequency (see paper of Dr. C. Baumgarten: "Transverse-Longitudinal Coupling by Space Charge in Cyclotrons" (2012), formula (1))
         * @param N specifies the number of splits (2pi/N)
         * @param accuracy specifies the accuracy of the closed orbit
         * @param maxit is the maximal number of iterations done. Program stops if closed orbit not found within this time.
         * @param Emin is the minimum energy [MeV] needed in cyclotron
         * @param Emax is the maximum energy [MeV] reached in cyclotron
         * @param fieldmap is the location of the file that specifies the magnetic field
         */
        ClosedOrbitFinder(value_type, value_type, size_type, value_type, size_type, value_type, value_type, const std::string&);

        /// Returns the inverse bending radius (size of container N+1)
        container_type& getInverseBendingRadius();

        /// Returns the step lengths of the path (size of container N+1)
        container_type& getPathLength();

        /// Returns the field index (size of container N+1)
        container_type& getFieldIndex();

        /// Returns the radial and vertical tunes (in that order)
        std::pair<value_type,value_type> getTunes();

        /// Returns the closed orbit (size of container N+1)
        container_type& getOrbit();

        /// Returns the relativistic factor gamma
        value_type getGamma();

        /// Returns the average orbit radius
        value_type getAverageRadius();

        /// Returns the phase
        value_type getPhase();

        /// Returns true if a closed orbit could be found
        bool isConverged();

    private:
        /// Computes the closed orbit
        /*!
         * @param accuracy specifies the accuracy of the closed orbit
         * @param maxit is the maximal number of iterations done for finding the closed orbit
         */
        bool findOrbit(value_type, size_type);

        /// Fills in the values of h_m, ds_m, fidx_m. It gets called by in by constructor.
        void computeOrbitProperties();

        /// This function is called by the function getTunes().
        /*! Transfer matrix Y = [y11, y12; y21, y22] (see Gordon paper for more details).
         * @param y are the positions (elements y11 and y12 of Y)
         * @param py2 is the momentum of the second solution (element y22 of Y)
         * @param ncross is the number of sign changes (\#crossings of zero-line)
         */
        value_type computeTune(const std::array<value_type,2>&, value_type, size_type);

        value_type christian_computeTune(const std::array<value_type,2>&, value_type, size_type);

        /// This function computes nzcross_ which is used to compute the tune in z-direction and the phase error
        void computeVerticalOscillations();

        /// Stores current position in horizontal direction for the solutions of the ODE with different initial values
        std::array<value_type,2> x_m; // x_m = [x1, x2]
        /// Stores current momenta in horizontal direction for the solutions of the ODE with different initial values
        std::array<value_type,2> px_m; // px_m = [px1, px2]
        /// Stores current position in longitudinal direction for the solutions of the ODE with different initial values
        std::array<value_type,2> z_m; // z_m = [z1, z2]
        /// Stores current momenta in longitudinal direction for the solutions of the ODE with different initial values
        std::array<value_type,2> pz_m; // pz_m = [pz1, pz2]

        /// Stores the inverse bending radius
        container_type h_m;
        /// Stores the step length
        container_type ds_m;
        /// Stores the radial orbit (size: N_m+1)
        container_type r_m;
        /// Stores the radial momentum
        container_type pr_m;
        /// Stores the field index
        container_type fidx_m;

        /// Counts the number of zero-line crossings in horizontal direction (used for computing horizontal tune)
        size_type nxcross_m;
        /// Counts the number of zero-line crossings in vertical direction (used for computing vertical tune)
        size_type nzcross_m; //#crossings of zero-line in x- and z-direction

        /// Is the energy for which the closed orbit should be found
        value_type E_m;
        /// Is the nominal orbital frequency
        value_type wo_m;
        /// Is the number of angle splits (2*pi/N) and the number of integration steps
        size_type N_m;
        /// Is the angle step size
        value_type dtheta_m;

        /// Is the relativistic factor
        value_type gamma_m;

        /// Is the average radius
        value_type ravg_m;

        /// Is the phase
        value_type phase_m;

        /// Boolean which tells if a closed orbit for this configuration could be found (get set by the function findOrbit)
        bool converged_m;

        /// Minimum energy needed in cyclotron
        value_type Emin_m;

        /// Maximum energy reached in cyclotron
        value_type Emax_m;

        /// Stores the last orbit value (since we have to return to the beginning to check the convergence in the findOrbit() function. This last value is then deleted from the array but is stored in lastOrbitVal_m to compute the vertical oscillations)
        value_type lastOrbitVal_m;

        /// Stores the last momentum value (since we have to return to the beginning to check the convergence in the findOrbit() function. This last value is then deleted from the array but is stored in lastMomentumVal_m to compute the vertical oscillations)
        value_type lastMomentumVal_m;
        
        /// Boolean which is true if computeVerticalOscillations() executed, otherwise false. This is used for checking in getTunes() and getPhase().
        bool vertOscDone_m;

        /// Location of magnetic field
        std::string fieldmap_m;

        /// Defines the stepper for integration of the ODE's
        Stepper stepper_m;

        /// ONLY FOR STAND-ALONE PROGRAM
        float** bmag_m;

        /// Christian's function for finding equilibrium orbit
        void christian_findOrbit(value_type, size_type);
186 187 188 189 190 191
};

// -----------------------------------------------------------------------------------------------------------------------
// PUBLIC MEMBER FUNCTIONS
// -----------------------------------------------------------------------------------------------------------------------

192
    template<typename Value_type, typename Size_type, class Stepper>
193
ClosedOrbitFinder<Value_type, Size_type, Stepper>::ClosedOrbitFinder(value_type E, value_type wo, size_type N, value_type accuracy,
194 195 196 197
        size_type maxit,value_type Emin, value_type Emax, const std::string& fieldmap)
    : nxcross_m(0), nzcross_m(0), E_m(E), wo_m(wo), N_m(N), dtheta_m(2.0*M_PI/double(N)),
    gamma_m(E/physics::E0+1.0), ravg_m(0), phase_m(0), converged_m(false), Emin_m(Emin), Emax_m(Emax),
    lastOrbitVal_m(0.0), lastMomentumVal_m(0.0), vertOscDone_m(false), fieldmap_m(fieldmap), stepper_m()
198
{
199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215

    if (Emin_m > Emax_m || E_m < Emin_m || E > Emax_m)
        Error::message("ClosedOrbitFinder",Error::invalid);
    
    // reserve storage for the orbit and momentum (--> size = 0, capacity = N_m+1)
    /*
     * we need N+1 storage, since dtheta = 2pi/N (and not 2pi/(N-1)) that's why we need N+1 integration steps
     * to return to the origin (but the return size is N_m)
     */
    r_m.reserve(N + 1);
    pr_m.reserve(N + 1);
    
    // reserve memory of N_m for the properties (--> size = 0, capacity = N_m)
    h_m.reserve(N);
    ds_m.reserve(N);
    fidx_m.reserve(N);
    
216
    // compute closed orbit
217
    converged_m = findOrbit(accuracy, maxit);
218 219 220 221 222 223 224
    
    // compute h, ds, fidx, rav (average radius)
    computeOrbitProperties();
}

template<typename Value_type, typename Size_type, class Stepper>
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::container_type& ClosedOrbitFinder<Value_type, Size_type, Stepper>::getInverseBendingRadius() {
225
    return h_m;
226 227 228 229
}

template<typename Value_type, typename Size_type, class Stepper>
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::container_type& ClosedOrbitFinder<Value_type, Size_type, Stepper>::getPathLength() {
230
    return ds_m;
231 232 233 234
}

template<typename Value_type, typename Size_type, class Stepper>
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::container_type& ClosedOrbitFinder<Value_type, Size_type, Stepper>::getFieldIndex() {
235
    return fidx_m;
236 237 238
}

template<typename Value_type, typename Size_type, class Stepper>
239 240 241
std::pair<Value_type,Value_type> ClosedOrbitFinder<Value_type, Size_type, Stepper>::getTunes() {
    // compute radial tune
    value_type nur = computeTune(x_m,px_m[1],nxcross_m);
242
    
243 244 245 246 247 248 249 250
    // compute nzcross_m
    if (!vertOscDone_m)
        computeVerticalOscillations();
    
    // compute vertical tune
    value_type nuz = computeTune(z_m,pz_m[1],nzcross_m);

    return std::make_pair(nur,nuz);
251 252 253 254
}

template<typename Value_type, typename Size_type, class Stepper>
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::container_type& ClosedOrbitFinder<Value_type, Size_type, Stepper>::getOrbit() {
255
    return r_m;
256 257 258 259
}

template<typename Value_type, typename Size_type, class Stepper>
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::value_type ClosedOrbitFinder<Value_type, Size_type, Stepper>::getGamma() {
260
    return gamma_m;
261 262 263 264
}

template<typename Value_type, typename Size_type, class Stepper>
inline typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::value_type ClosedOrbitFinder<Value_type, Size_type, Stepper>::getAverageRadius() {
265
    return ravg_m;
266 267 268
}

template<typename Value_type, typename Size_type, class Stepper>
269 270 271 272 273 274 275
typename ClosedOrbitFinder<Value_type, Size_type, Stepper>::value_type ClosedOrbitFinder<Value_type, Size_type, Stepper>::getPhase() {
    
    // if the vertical oscillations aren't computed, we have to, since there we also compuote the frequency error.
    if(!vertOscDone_m)
        computeVerticalOscillations();
    
    return phase_m;
276 277 278 279
}

template<typename Value_type, typename Size_type, class Stepper>
inline bool ClosedOrbitFinder<Value_type, Size_type, Stepper>::isConverged() {
280
    return converged_m;
281 282 283 284 285 286 287 288
} 

// -----------------------------------------------------------------------------------------------------------------------
// PRIVATE MEMBER FUNCTIONS
// -----------------------------------------------------------------------------------------------------------------------

template<typename Value_type, typename Size_type, class Stepper>
bool ClosedOrbitFinder<Value_type, Size_type, Stepper>::findOrbit(value_type accuracy, size_type maxit) {
289 290 291 292 293 294 295 296 297 298 299 300 301 302 303
    /* REMARK TO GORDON
     * q' = 1/b = 1/bcon
     * a' = a = acon
     */

    // READ IN MAGNETIC FIELD: ONLY FOR STAND-ALONE PROGRAM
    int nsc = 8, nr = 141, Nth = 1440, nth = 1440 / 8; value_type r0 = 1.8, dr = 0.02;
    bmag_m = MagneticField::malloc2df(Nth,nr);
    MagneticField::ReadSectorMap(bmag_m,nr,Nth,1,fieldmap_m,0.0);
    MagneticField::MakeNFoldSymmetric(bmag_m,Nth,nr,nth,nsc);
    value_type bint, brint, btint;

    // velocity: beta = v/c = sqrt(1-1/(gamma*gamma))
    if (gamma_m == 0)
        PhysicalError::message("ClosedOrbitFinder::findOrbit",PhysicalError::undefined);
304
    
305 306 307
    // resize vectors (--> size = N_m+1, capacity = N_m+1), note: we do N_m+1 integration steps
    r_m.resize(N_m+1);
    pr_m.resize(N_m+1);
308
    
309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380
    // store acon and bcon locally
    value_type acon = physics::acon(wo_m);               // [acon] = m
    value_type invbcon = 1.0 / physics::bcon(wo_m);        // [bcon] = MeV*s/(C*m^2) = 10^6 T = 10^7 kG (kilo Gauss)

    // helper constants
    value_type p2;                                      // p^2 = p*p
    value_type pr2;                                     // squared radial momentum (pr^2 = pr*pr)
    value_type ptheta, invptheta;                       // Gordon, formula (5c)
    value_type invdenom;                                // denominator for computing dr,dpr
    value_type xold = 0.0;                              // for counting nxcross

    // index for reaching next element of the arrays r and pr (no nicer way found yet)
    size_type idx = 0;
    // observer for storing the current value after each ODE step (e.g. Runge-Kutta step) into the containers of r and pr
    auto store = [&](state_type& y, const value_type t)
    {        
        r_m[idx] = y[0];
        pr_m[idx] = y[1];

        // count number of crossings (excluding starting point --> idx>0)
        nxcross_m += (idx > 0) * (y[4] * xold < 0);
        xold = y[4];
        
        ++idx;
    };

    // define the six ODEs (using lambda function)
    std::function<void(const state_type&, state_type&, const double)> orbit_integration = [&](const state_type &y, state_type &dydt, const double theta){
        pr2 = y[1] * y[1];
        if (p2 < pr2)
            PhysicalError::message("ClosedOrbitFinder::findOrbit",PhysicalError::negative);
            
        // Gordon, formula (5c)
        ptheta = std::sqrt(p2 - pr2);
        invptheta = 1.0 / ptheta;

        // intepolate values of magnetic field
        MagneticField::interpolate(&bint,&brint,&btint,theta * 180 / M_PI,nr,Nth,y[0],r0,dr,bmag_m);
        bint *= invbcon;
        brint *= invbcon;

        // Gordon, formula (5a)
        dydt[0] = y[0] * y[1] * invptheta;
        // Gordon, formula (5b)
        dydt[1] = ptheta - y[0] * bint;
        // Gordon, formulas (9a) and (9b)
        for (size_type i = 2; i < 5; i += 2) {
            dydt[i] = (y[1] * y[i] + y[0] * p2 * y[i+1] * invptheta * invptheta) * invptheta;
            dydt[i+1] = - y[1] * y[i+1] * invptheta - (bint + y[0] * brint) * y[i];
        }
    };

    // define initial state container for integration: y = {r, pr, x1, px1, x2, px2}
    state_type y(6);

    container_type err(2);                                      // difference of last and first value of r (1. element) and pr (2. element)
    container_type delta = {0.0, 0.0};                          // correction term for initial values: r = r + dr, pr = pr + dpr; Gordon, formula (17)
    value_type error = std::numeric_limits<value_type>::max();  // amplitude of error; Gordon, formula (18) (a = a')
    size_type niterations = 0;                                  // if niterations > maxit --> stop iteration

    /*
     * Christian:
     * N = 1440 ---> N = 720 ---> dtheta = 2PI/720 --> nsteps = 721
     * 
     * 0, 2, 4, ... ---> jeden zweiten berechnene: 1, 3, 5, ... interpolieren --> 1440 Werte
     * 
     * Matthias:
     * N = 1440 --> dtheta = 2PI/1440 --> nsteps = 1441
     * 
     * 0, 1, 2, 3, 4, 5, ... --> 1440 Werte
     * 
     */
381
    
382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399
    // iterate until suggested energy (start with minimum energy)
    value_type E = Emin_m;

    // energy dependent values
    value_type en = E / physics::E0;                      // en = E/E0 = E/(mc^2) (E0 is kinetic energy)
    value_type p = acon * std::sqrt(en * (2.0 + en));     // momentum [p] = m; Gordon, formula (3)
    value_type gamma2 = (1.0 + en) * (1.0 + en);          // = gamma^2
    value_type beta = std::sqrt(1.0 - 1.0 / gamma2);
    p2 = p * p;                                           // p^2 = p*p
    value_type invgamma4 = 1.0 / (gamma2 * gamma2);       // = 1/gamma^4

    // set initial values for radius and radial momentum for lowest energy Emin
    // orbit, [r] = m; Gordon, formula (20)
    // radial momentum; Gordon, formula (20)
    container_type init = {beta * acon, 0.0};

    // store initial values for updating values for higher energies
    container_type previous_init = {0.0, 0.0};
400
    
401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462
    while (E <= E_m) {

        // (re-)set inital values for r and pr
        r_m[0] = init[0]; 
        pr_m[0] = init[1];
        
        // integrate until error smaller than user-define accuracy
        do {
            // (re-)set inital values
            x_m[0]  = 1.0;               // x1; Gordon, formula (10)
            px_m[0] = 0.0;               // px1; Gordon, formula (10)
            x_m[1]  = 0.0;               // x2; Gordon, formula (10)
            px_m[1] = 1.0;               // px2; Gordon, formula (10)
            nxcross_m = 0;               // counts the number of crossings of x-axis (excluding first step)
            idx = 0;                     // index for looping over r and pr arrays

            // fill container with initial states
            y = {init[0],init[1], x_m[0], px_m[0], x_m[1], px_m[1]};
            
            // integrate from 0 to 2*pi (one has to get back to the "origin")
            boost::numeric::odeint::integrate_n_steps(stepper_m,orbit_integration,y,0.0,dtheta_m,N_m,store);
            
            // write new state
            x_m[0] = y[2];
            px_m[0] = y[3];
            x_m[1] = y[4];
            px_m[1] = y[5];
            
            // compute error (compare values of orbit and momentum for theta = 0 and theta = 2*pi)
            // (Note: size = N_m+1 --> last entry is N_m)
            err[0] = r_m[N_m] - r_m[0];      // Gordon, formula (14)
            err[1] = pr_m[N_m] - pr_m[0];    // Gordon, formula (14)
            
            // correct inital values of r and pr
            invdenom = 1.0 / (x_m[0] + px_m[1] - 2.0);
            delta[0] = ((px_m[1] - 1.0) * err[0] - x_m[1] * err[1]) * invdenom; // dr; Gordon, formula (16a)
            delta[1] = ((x_m[0] - 1.0) * err[1] - px_m[0] * err[0]) * invdenom; // dpr; Gordon, formula (16b)
            
            // improved initial values; Gordon, formula (17) (here it's used for higher energies)
            init[0] += delta[0];
            init[1] += delta[1];
            
            // compute amplitude of the error
            error = std::sqrt(delta[0] * delta[0] + delta[1] * delta[1] * invgamma4) / r_m[0];
            
        } while (error > accuracy && niterations++ < maxit);
        
        // reset iteration counter
        niterations = 0;
        
        // reset correction term
        delta[0] = delta[1] = 0.0;
        
        // increase energy by one
        E += 1.0;
        
        // set constants for new energy E
        en = E / physics::E0;                     // en = E/E0 = E/(mc^2) (E0 is kinetic energy)
        p = acon * std::sqrt(en * (2.0 + en));    // momentum [p] = m; Gordon, formula (3)
        p2 = p * p;                               // p^2 = p*p
        gamma2 = (1.0 + en) * (1.0 + en);
        invgamma4 = 1.0 / (gamma2 * gamma2);
463 464
    }
    
465 466 467 468 469
    /* store last entry, since it is needed in computeVerticalOscillations(), because we have to do the same
     * number of integrations steps there.
     */
    lastOrbitVal_m = r_m[N_m];           // needed in computeVerticalOscillations()
    lastMomentumVal_m = pr_m[N_m];       // needed in computeVerticalOscillations()
470
    
471 472 473
    // remove last entry (since we don't have to store [0,2pi], but [0,2pi[)  --> size = N_m, capacity = N_m+1
    r_m.pop_back();
    pr_m.pop_back();
474 475
    

476 477 478
    // returns true if converged, otherwise false
    return error < accuracy;
}
479 480 481

template<typename Value_type, typename Size_type, class Stepper>
Value_type ClosedOrbitFinder<Value_type, Size_type, Stepper>::computeTune(const std::array<value_type,2>& y, value_type py2, size_type ncross) {
482
    // Y = [y1, y2; py1, py2]
483
    
484 485
    // cos(mu)
    value_type cos = 0.5 * (y[0] + py2);
486
    
487 488
    value_type twopi = 2.0 * M_PI;
    value_type mu;
489
    
490 491
    // sign of sin(mu) has to be equal to y2
    bool negative = std::signbit(y[1]);
492
    
493
    bool uneven = (ncross % 2);
494
    
495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526
    if (std::fabs(cos) > 1.0) {
        // store the number of crossings
        value_type n = ncross;
        
        if (uneven)
            n = ncross - 1;
        
        // Gordon, formula (36b)
        value_type muPrime = -std::acosh(std::fabs(cos));      // mu'
        mu = n * M_PI + muPrime;
        
    } else {
        value_type muPrime = (uneven) ? std::acos(-cos) : std::acos(cos);    // mu'
        /* It has to be fulfilled: 0<= mu' <= pi
        * But since |cos(mu)| <= 1, we have
        * -1 <= cos(mu) <= 1 --> 0 <= mu <= pi (using above programmed line), such
        * that condition is already fulfilled.
        */
        
        // Gordon, formula (36)
        mu = ncross * M_PI + muPrime;
        
        // if sign(y[1]) > 0 && sign(sin(mu)) < 0
        if (!negative && std::signbit(std::sin(mu))) {
            mu = ncross * M_PI - muPrime;
        } else if (negative && !std::signbit(std::sin(mu))) {
            mu = ncross * M_PI - muPrime + twopi;
        }
    }
    
    // nu = mu/theta, where theta = integration domain
    return mu / twopi;
527 528 529
}

template<typename Value_type, typename Size_type, class Stepper>
530 531 532 533 534 535 536
void ClosedOrbitFinder<Value_type, Size_type, Stepper>::computeOrbitProperties() {
    /* 
     * The formulas for h, fidx and ds are from the paper:
     * "Tranverse-Longitudinal Coupling by Space Charge in Cyclotrons"
     * written by Dr. Christian Baumgarten (2012, PSI)
     * p. 6
     */
537
    
538 539 540
    // READ IN MAGNETIC FIELD: ONLY FOR STAND-ALONE PROGRAM
    int nsc = 8, nr = 141/*, Nth = 1440*/, nth = 1440 / 8; value_type r0 = 1.8, dr = 0.02;
    value_type bint, brint, btint; // B, dB/dr, dB/dtheta
541

542 543 544 545 546 547
    value_type invbcon = 1.0 / physics::bcon(wo_m);
    value_type en = E_m / physics::E0;                                  // en = E/E0 = E/(mc^2) (E0 is kinetic energy)
    value_type p = physics::acon(wo_m) * std::sqrt(en * (2.0 + en));    // momentum [p] = m; Gordon, formula (3)
    value_type p2 = p * p;
    value_type theta = 0.0;                                             // angle for interpolating
    value_type ptheta;
548

549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572
    // resize of container (--> size = N_m, capacity = N_m)
    h_m.resize(N_m);
    fidx_m.resize(N_m);
    ds_m.resize(N_m);

    for (size_type i = 0; i < N_m; ++i) {
        // interpolate magnetic field
        MagneticField::interpolate(&bint,&brint,&btint,theta * 180.0 / M_PI,nr,nth*nsc,r_m[i],r0,dr,bmag_m);
        bint *= invbcon;
        brint *= invbcon;
        btint *= invbcon;

        // inverse bending radius
        h_m[i] = bint / p;

        // local field index
        ptheta = std::sqrt(p2 - pr_m[i] * pr_m[i]);
        fidx_m[i] = (brint * ptheta - btint * pr_m[i] / r_m[i]) / p2; //(bint*bint);

        // path length element
        ds_m[i] = std::hypot(r_m[i] * pr_m[i] / ptheta,r_m[i]) * dtheta_m; // C++11 function

        // increase angle
        theta += dtheta_m;
573
    }
574 575 576

    // compute average radius
    ravg_m = std::accumulate(r_m.begin(),r_m.end(),0.0) / value_type(r_m.size());
577 578 579
}

template<typename Value_type, typename Size_type, class Stepper>
580
void ClosedOrbitFinder<Value_type, Size_type, Stepper>::computeVerticalOscillations() {
581
    
582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644
    vertOscDone_m = true;

    // READ IN MAGNETIC FIELD: ONLY FOR STAND-ALONE PROGRAM
    int /*nsc = 8,*/ nr = 141, Nth = 1440/*, nth = 1440/8*/; value_type r0 = 1.8, dr = 0.02;
    value_type bint, brint, btint; // B, dB/dr, dB/dtheta

    value_type en = E_m / physics::E0;                                  // en = E/E0 = E/(mc^2) with kinetic energy E0
    value_type p = physics::acon(wo_m) * std::sqrt(en *(en + 2.0));     // Gordon, formula (3)
    value_type p2 = p * p;                                              // p^2 = p*p
    size_type idx = 0;                                                  // index for going through container
    value_type pr2;                                                     // pr^2 = pr*pr
    value_type ptheta, invptheta;                                       // Gordon, formula (5c)
    value_type zold = 0.0;                                              // for counting nzcross

    // store bcon locally
    value_type invbcon = 1.0 / physics::bcon(wo_m);                     // [bcon] = MeV*s/(C*m^2) = 10^6 T = 10^7 kG (kilo Gauss)

    // define the ODEs (using lambda function)
    std::function<void(const state_type&, state_type&, const double)> vertical = [&](const state_type &y, state_type &dydt, const double theta){
        pr2 = y[1] * y[1];
        if (p2 < pr2)
            PhysicalError::message("ClosedOrbitFinder::findOrbit",PhysicalError::negative);
        
        // Gordon, formula (5c)
        ptheta = std::sqrt(p2 - pr2);
        invptheta = 1.0 / ptheta;

        // intepolate values of magnetic field
        MagneticField::interpolate(&bint,&brint,&btint,theta * 180 / M_PI,nr,Nth,y[0],r0,dr,bmag_m);
        bint *= invbcon;
        brint *= invbcon;
        btint *= invbcon;
        
        // We have to integrate r and pr again, otherwise we don't have the Runge-Kutta of the B-field
        // Gordon, formula (5a)
        dydt[0] = y[0] * y[1] * invptheta;
        // Gordon, formula (5b)
        dydt[1] = ptheta - y[0] * bint;

        // Gordon, formulas (22a) and (22b)
        for (size_type i = 2; i < 5; i += 2) {
            dydt[i] = y[0] * y[i+1] * invptheta;
            dydt[i+1] = (y[0] * brint - y[1] * invptheta * btint) * y[i];
        }

        // integrate phase
        dydt[6] = y[0] * invptheta * gamma_m - 1;
    };

    // to get next index for r and pr (to iterate over container)
    auto next = [&](state_type& y, const value_type t) {
        // number of times z2 changes sign
        nzcross_m += (idx > 0) * (y[4] * zold < 0);
        zold = y[4];
        ++idx;
    };

    // set initial state container for integration: y = {r, pr, z1, pz1, z2, pz2, phase}
    state_type y = {r_m[0], pr_m[0], 1.0, 0.0, 0.0, 1.0, 0.0};

    // add last element for integration (since we have to return to the initial point (--> size = N_m+1, capacity = N_m+1)
    r_m.push_back(lastOrbitVal_m);
    pr_m.push_back(lastMomentumVal_m);
645
    
646 647
    // integrate: assume no imperfections --> only integrate over a single sector (dtheta_m = 2pi/N_m)
    boost::numeric::odeint::integrate_n_steps(stepper_m,vertical,y,0.0,dtheta_m,N_m,next);
648
    
649 650 651
    // remove last element again (--> size = N_m, capacity = N_m+1)
    r_m.pop_back();
    pr_m.pop_back();
652
    
653 654 655 656 657 658
    // write new state
    z_m[0] = y[2];
    pz_m[0] = y[3];
    z_m[1] = y[4];
    pz_m[1] = y[5];
    phase_m = y[6] / (2.0 * M_PI);
659 660
}

661
#endif