Effects of the wiggler on the Hefei Light Source (HLS) storage ring He Zhang, Martin Berz Department of Physics and Astronomy Michigan State University July 8, 2008 He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
I. Introduction of the Hefei Light Source He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
Introduction of the Hefei Light Source (HLS) HLS is a second generation synchrotron radiation light source in the National Synchrotron Radiation Lab in the University of Science and Technology of China in Hefei, China. Figure: The position of Hefei in China, from google map He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
Introduction of the Hefei Light Source (HLS) HLS is a second generation synchrotron radiation light source in the National Synchrotron Radiation Lab in the University of Science and Technology of China in Hefei, China. Figure: The storage ring hall (outside and inside) He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
Introduction of the Hefei Light Source (HLS) HLS is composed of a 200 MeV linac and an 800 MeV storage ring. One wiggler and one undulator were installed. The storage ring consists of four triple bend achromat (TBA) cells and four 3 m long straight sections. HLS currently runs in general purpose operation mode (GPLS) with an emittance of 133 nm · rad . Figure: β functions and dispersion function in GPLS mode He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
Introduction of the Hefei Light Source (HLS) A high brilliance light source (HBLS) operation mode lattice design was proposed, in which the tunes were (5.2073, 2.5351) and the emittance was 27 nm · rad . The insertion devices were expected to be able to run in HBLS mode. We’ll discuss the effects of the wiggler in the HBLS mode. Figure: β functions and dispersion function in HBLS mode He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
II. The magnetic field model of the wiggler He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
The magnetic field model of the wiggler Cosy infinity 9.0 beam physics package The midplane field in the wiggler is described as � 2 π � λ · z + k · z 2 B m ( x , z ) = B 0 cos , Considering the fringe field, B = K l · B m · K r ,where 1 K l , r = 1 + exp( a 1 + a 2 ( ∓ z ± z l , r ) / d + · · · + a 10 (( ∓ z ± z l , r ) / d ) 9 ) The wiggler is represented by WI < B 0 > < λ > < L > < d > < k > < I > < A > He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
The magnetic field model of the wiggler the wiggler in HLS has single period with three poles. The peak magnetic field strength of the central pole and the poles in either side are 6 T and 4.3 T respectively. In COSY, we set λ = 210 mm , k = 0 , B 0 = 6 . 08 T , d = 40 mm , l = 252 mm , a 0 = 0 . 478959 , a 1 = 1 . 911289 , a 2 , a 3 , ..., a 10 = 0 . Figure: The magnetic field of the wiggler by measurement (left) and by fitting (right) He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
The magnetic field model of the wiggler Generate the 3D magnetic field from the field in the reference axis in the midplane. Maxwell’s equation E = − ∂ − → − → ∇ · − → → − ∇ × − → B D = ρ, ∂ t . In our case − → ∇ · − → − → ∇ × − → E = − → D = 0 , 0 . � B has scalar potentil V B , which satisfy △ V B = 0 , and − → B = −− → ∇ V B . He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
The magnetic field model of the wiggler Taylor expansion of V B in transversal coordinates x and y . ∞ ∞ a k , l ( z ) · x k y l � � V B = k ! l ! k =0 l =0 Then ∞ ∞ ∂ 2 V B k ( k − 1) · x k − 2 y l � � = · a k , l ( z ) ∂ x 2 k ! l ! k =0 l =0 ∂ 2 V B ∞ ∞ l ( l − 1) · x k y l − 2 � � ∂ y 2 = · a k , l ( z ) k ! l ! k =0 l =0 ∂ 2 V B ∞ ∞ k , l ( z ) · x k y l � � a ′′ = ∂ z 2 k ! l ! k =0 l =0 He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
The magnetic field model of the wiggler From the Laplace equation, we get the following recursive relation, a k +2 , l + a k , l +2 + a ′′ k , l = 0 a k , l +2 + a ′′ k , l + a k +2 , l = 0 and a k , 0 ( z ) and a k , 1 ( z ) can be chosen freely, and all the other coefficients are determined by them. Computing the gradient of the potential, we get a k , 1 ( z ) · x k � B y ( x , y = 0 , z ) = − k ! k x k − 1 � B x ( x , y = 0 , z ) = − a k , 0 ( z ) · ( k − 1)! = 0 k He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
The magnetic field model of the wiggler The magnetic field components in the horizontal direction should be 0 in the y = 0 plane. So that a k , 0 = 0 for all k , and all the other coefficients can be derived from a k , 1 . a k , 1 can be determined by the field in the reference axis, then the 3D field can be generated. In our case, we assume the trace of the electrons are close to the reference axis and the width of the magnet is large enough, so that B y doesn’t depend on x , then a 0 , 1 = K l ∗ B m ∗ K r , and all the other a k , 1 s are zero. He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
The magnetic field model of the wiggler Get the 3D magnetic field by DA method. Fix point problem △ V = ∂ 2 V ∂ x 2 + ∂ 2 V ∂ y 2 + ∂ 2 V ∂ z 2 = 0 , � ∂ � { ∂ � � ∂ V � + ∂ � ∂ V �� V = V | y =0 + ∂ y V | y =0 − } ∂ x ∂ x ∂ z ∂ z y y In y = 0 plane, ∂ V /∂ y is just the field, and V = y · ∂ V /∂ y since the field doesn’t explicitly depend on y . Iterate at most n + 1 times, we get the potential up to n th order. Then take the derivative to get the field. He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
III. The beam dynamic study by the COSY Infinity 9.0 He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
The beam dynamic study by the COSY Infinity 9.0 In principle, COSY does the following things for tracking Generate the map of each element to arbitrary order. Compose all the maps into an one-turn map. Get generating function from the one-turn map. Tracking by generating function. He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
The beam dynamic study by the COSY Infinity 9.0 Arbitrary order map from an ODE based on antiderivation operator ˙ z = � � f ( � z , t ) � t t i � � z = � z i + f ( � z , t ′ ) dt ′ Fix point problem. Iterate n + 1 times, we get the expansion of � z in initial z i and t up to n th order. position � He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
The beam dynamic study by the COSY Infinity 9.0 Arbitrary order map from an ODE based on derivation operator For a given function in phase space g ( � z , t ), we have ∇ g + ∂ d z , t ) = � f · � dt g ( � ∂ t g = L � f g , where ˙ z = � � f . The Taylor expansion of the solution of the ODE is t i · L i ∞ � � · � f � z f = I i ! i =0 where � I is the identity function. He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
The beam dynamic study by the COSY Infinity 9.0 When g and � f are not explicitly time dependant, the partial f disappear. If � f ( � 0) = � derivative respect to t in L � 0, let g be a component of � z , we obtain the integrator. z j = t · ( � f · � ∞ ∇ ) � � z 0 = � z i , � · � z j − 1 , � z f = z j i j =0 As to the elements, such as dipole, quadrupole etc, whose fields don’t depend on the time variable, the derivation method works well. As to the elements like wiggler whose field changes with the time variable, we can use the antiderivation method. He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
The beam dynamic study by the COSY Infinity 9.0 Set up the generating function from the map Take the 2nd kind of generating function as an example. ( � p ) = ( � P F 2 , � Q ,� ∇ � ∇ � q F 2 ) M = ( M 1 , M 2 ), M 1 is the position part, M 2 is the momenta part. Identity map I = ( I 1 , I 2 ). q , � p ) = N − 1 q , � Let N 1 = ( I 1 , M 2 ), then ( � P ) = N 1 ( � q ,� p ), ( � q ,� 1 ( � P ) Let N 2 = ( M 1 , I 2 ), then ( � p ) = N 2 ◦ N − 1 q , � q , � Q ,� p ) = N 2 ( � q ,� 1 ( � P ) = G ( � P ) F 2 is just the integration of G . He Zhang, Martin Berz Effects of the wiggler on the Hefei Light Source (HLS) storage ring
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