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Dependence of loss rate of electrons due to elastic gas scattering on the shape of the vacuum chamber in an electron storage ring

The beam lifetime in an electron storage ring is also limited by the loss rate of the stored electrons due to the elastic coulomb scattering of electrons with the nuclei of residual gas atoms. The contribution to the beam lifetime due to this elastic scattering depends upon the shape factor which is governed by the shape of the vacuum chamber. In this paper, analytical expressions for the shape factor for a rectangular and an elliptical vacuum chamber as a function of longitudinal position along the circumference in a storage ring are derived using an approach in which the position of electrons at the focusing quadrupole is transformed to the location of defocusing quadrupole and vice versa to define the parts of the vacuum chamber, where the loss of electrons takes place at the location of quadrupoles. The expressions available in the literature are for the average shape factors. The expression of shape factor for a rectangular chamber derived in this paper are similar to the expression for average shape factor quoted in the literature, whereas a new expression for elliptical shape of vacuum chamber having no resemblance with the available expression for average shape factor is obtained. A comparative study of shape factors at each scattering location in the ring obtained from derived expressions and with the existing expressions are reported using Indus-2 lattice parameters for rectangular and elliptical shapes of the chamber. A comparison of average shape factors for these two shapes using derived as well as existing expressions is also reported. These studies indicate that the effect of the rectangular and elliptical shape of vacuum chamber on beam lifetime due to elastic coulomb scattering between electrons and nuclei of residual gas molecules is nearly same.


Introduction
For the storage of an electron beam for a longer lifetime in an electron storage ring, average vacuum pressure of the order of 1 nTorr is required in the vacuum chambers in which the electron beam circulates. The presence of residual gas species even at a low pressure of 1 nTorr, causes scattering of the electrons with the nuclei of residual gas atoms. The scattering may be elastic or inelastic. In this paper we discuss the loss of electrons due to the elastic coulomb scattering of electrons with the nuclei of residual gas atoms. In the elastic scattering of electrons of the stored electron beam with the nuclei of residual gas atoms, the electrons are deflected from their path and the amplitude of betatron oscillation of electrons increases. If the amplitude of betatron oscillation of an electron is more than the chamber aperture i.e. the acceptance of the ring at a location, the electron is lost there. The rate of loss of electrons due to this process contributes to beam lifetime. The loss of electrons due to elastic coulomb scattering takes place at minimum acceptance location i.e. at maximum β function in horizontal ( ) X and vertical ( ) Z planes, when the aperture of vacuum chamber is uniform in the ring. The loss of electrons depends on the parameter known as shape factor, which is governed by the shape of the vacuum chamber. In the electron storage rings, rectangular or elliptical shapes of the vacuum chamber are widely used. The vacuum pressure in a storage ring is not uniform along the circumference. It varies from place to place. In order to calculate beam lifetime due to elastic scattering, we have to use the vacuum pressure and shape factor information at all points in the ring. It is thus important to know the shape factor as a function of the longitudinal position in the ring. The average shape factor for a rectangular chamber is discussed in [1] and used for beam lifetime estimation in operating electron storage rings like MAX II [2], SPEAR3 [3], INDUS-2 [4] etc. The average shape factor for elliptical chamber is given in [5]- [6] and used in estimating beam lifetime in SAGA-LS [6] storage ring. In this paper, we derive expressions for the shape factor as a function of longitudinal position along the circumference of a storage ring. In this paper, general expressions of the shape factor for a rectangular and an elliptical vacuum chamber for a given scattering location in a storage ring are derived starting from ab initio using linear beam dynamics. The motion of electrons is considered to be constrained by the physical aperture neglecting the non linear beam dynamical effects as done by other authors [1]- [6]. Here, the position of electrons at the focusing quadrupole is transformed to the defocusing quadrupole location based on the beta functions in horizontal and vertical planes and vice versa to define the part of the vacuum chamber at which the beam loss takes place at these locations. This approach has enabled derivation of exact expressions for the shape factor as a function of the longitudinal position within the domain of linear beam dynamics. A theoretical estimation of shape factor at different scattering locations in a ring is also carried out using Indus-2 [7]- [8] lattice parameters considering rectangular and elliptical chambers. From the values of shape factor as a function of longitudinal position, the average shape factors are estimated for these shapes and also compared with the average shape factors obtained using the existing expressions.

Beam lifetime due to elastic coulomb scattering
The loss rate of relativistic electrons due to elastic coulomb scattering where 〉 〈 σ n is the average of the product of residual gas density and scattering cross section of electrons at different scattering locations spread over the ring. Differential scattering cross section of elastic coulomb scattering of a relativistic electron scattered by the nuclei of residual gas atom at a location j in ring [9]- [10] is given by is the solid angle in which the electron is scattered, θ and φ are the scattering polar (range from π to 0 ) and azimuth (range from π 2 0 to ) angle respectively, i Z is the atomic number of the residual gas atom of species i , 0 r is the classical electron radius and γ is the relativistic Lorentz factor. The electrons will survive in the chamber up to minimum scattering angle m θ and will be lost for scattering angle θ above the value of m θ .
Integrating equation (2) with respect to θ from limit m θ to π , we get ( ) ( ) The survival chance of a scattered electron depends on the shape and size of the vacuum chamber. As shown in equations (2) and (3), the scattering cross-section has a dependence on angle θ . The maximum allowed displacement for θ will be different for different shapes and sizes of the vacuum chamber, therefore, the average scattering cross-section as it appears in equation (1), will be different for different shapes and sizes of the vacuum chamber.
Vacuum pressure or even the composition of residual gas species varies from place to place in a storage ring, therefore it is important to know the shape factor F for each location in a storage ring. If the vacuum pressure and gas composition is uniform along the circumference, the equation ( (6) where F is the average shape factor and l is the number of scattering locations spread over the ring at a uniform interval. The loss rate due to elastic scattering elastic τ 1 is proportional to the average shape factor. Obviously, the correct value of average shape factor 〉 〈F is essential to estimate the beam lifetime due to elastic coulomb scattering between the electrons and the nuclei of residual gas atoms.

General expressions for shape factor F
When an electron collides with a nucleus of a residual gas atom at location 0 s in the ring, the electron gets deflected by an angle θ . In spherical polar coordinates, for small scattering angle, the deflection is resolved in X plane as planes respectively and m θ is the minimum scattering angle. In equation (7) and (8) as well as everywhere in this paper, maximum betatron displacements are considered taking the betatron phase term equal to one because here only those electrons, which are lost from the ring, are taken into account. Substituting x and z from equation (7) and (8) Dependence of the shape factor j F on x and z indicates that it is governed by the shape of the vacuum chamber. We consider that the storage ring has a vacuum chamber of uniform cross section all along the circumference. In such a ring, the beam loss will takes place either at the location where x β is maximum or the location where z β is maximum. In the following paragraph, we derive expressions for the shape factors for a rectangular and an elliptical vacuum chamber.

Rectangular vacuum chamber
Let the horizontal and vertical dimensions of the rectangular vacuum chamber be a and b respectively. In order to find out the domain of azimuth angle φ for electron loss in X and Z plane, we assume that at maximum z β location, the electrons lie on the boundary of vacuum chamber. The electron, which, is at

Region of electrons loss
Region of electrons loss

Region of electrons loss
Region of electrons loss  (7) and (8), the coordinate of ( ) The above expression, which gives the contribution to the shape factor due to the elastic coulomb scattering at location j is similar to that given for the average shape factor in [1]  The expression (16) is derived in [1] by assuming the average value of minimum scattering angle x θ and z θ avoiding point to point calculation of shape factor in the ring which we have considered in deriving expression (15) by using above approach. In section 4, it is seen that the average shape factor estimated from expression (15) using equation (6) and average shape factor obtained from expression (16) are nearly same.

Elliptical vacuum chamber
We follow the same approach as that used for a rectangular chamber to find out an expression for the shape factor of an elliptical chamber. Here also we assume that at maximum z     This is a new expression of the shape factor for an elliptical shape of vacuum chamber which has not been reported so far. An expression of average shape factor for an elliptical chamber is given in [5]- [6] as below.

Estimation of point by point shape factor and its average in Indus-2 ring
Indus-2 is an electron storage ring which is operational at beam energy 2.5 GeV beam energy at RRCAT, Indore, India [7]- [8]. The point by point shape factor j F was estimated by using Indus-2 lattice parameter for rectangular and elliptical shape of the vacuum chamber. The lattice functions x β and z β in one unit cell of Indus-2 at the interval of 5 cm was estimated using computer code MAD [11] which are shown in figure 4. The shape factor at each scattering location was estimated for rectangular and elliptical chamber using derived expression (15) for rectangular chamber and expression (23) for elliptical chamber using Indus-2 lattice parameters  figure 5(a). The increase in shape factor at each scattering location from rectangular to elliptical shape of chamber is shown in figure 5(b).  Figure 5(b).Increase in shape factor at each scattering location from rectangular to elliptical shape As we have seen that we got a similar expression for rectangular chamber but a new expression was obtained for elliptical chamber. A comparison in shape factor at each scattering location using expression (15) for rectangular chamber and existing expression (A.5) for elliptical chamber of the same dimension as above is shown in figure 6(a) and the increase in shape factor from rectangular to elliptical shape is shown in figure 6(b).  Figure 6(b).Increase in shape factor at each scattering location from rectangular to elliptical shape Comparison of figure 5(b) with figure 6(b) indicates that the difference between the shape factors for two shapes as per derived expressions is much less than that as perceived with existing expressions.

Comparison of average shape factor for rectangular and elliptical shape
From the value of shape factor at different scattering locations, average shape factor was estimated for rectangular ( ) mm b mm a 15 , 30 = = and elliptical ( ) mm b mm a 15 , 30 = = chamber using equation (6) and comparison with the values obtained using existing expression (16) and (24) for rectangular and elliptical shape respectively are given in Table 1. From the expression (15) of shape factor for rectangular shape, putting dimensions a and b equal, we get shape factor for square shape and in expression (23), putting b a = , we get the shape factor for circular shape of the vacuum chamber. The comparison of shape factor at each scattering location in one unit cell of Indus-2 for square and circular ) 30 ( mm radius = shape of chamber is shown in figure 7(a) and the percentage increase in shape factor at each scattering location is shown in figure 7(b).   8(b) reveals that the difference between the shape factors for two shapes as per derived expressions is much less than that obtained with existing expressions.

Comparison of average shape factor for square and circular shape
From the value of shape factor at different scattering locations, average shape factor was estimated for square ( ) mm b a 30 = = and circular ( ) mm radius 30 = chamber using equation (6) and comparison with the values obtained using existing expressions is given in Table 2. , the increase in shape factor from the derived expressions is ~2.5% whereas it increases to ~15.1% if we use the existing expressions. Similarly as reported in [1], the average shape factor increase from square to circular shape of chamber is ~22% as seen in table 2, whereas from the derived expressions, the increase is ~3.3%. The large difference between the shape factor values of rectangular and elliptical chambers or higher values of shape factors for elliptical shapes as per the existing expressions are attributed to approximations made in the integration of equation (9) in [5]- [ 6].

Conclusion
The general expressions of shape factor due to elastic coulomb scattering of electrons with the nuclei of residual gas atoms at a scattering location in an electron storage ring are derived for a rectangular chamber using linear beam dynamics and using the same approach, shape factor for an elliptical chamber is also derived. The method of transforming the position of electrons at the focusing quadrupole to the defocusing quadrupole location and vice versa has made it possible to derive exact analytical expressions for the shape factor for the two shapes of the vacuum chamber. The difference between the shape factors of the rectangular and elliptical chambers obtained from these expressions are much less than that obtained using existing expressions as the geometrical effects of shapes are taken into account as required. The expression obtained for the shape factor at a given location for rectangular shape is similar to the expression for the average shape factor available in the literature and the average shape factors using derived and existing expression are nearly same. The expression derived for the shape factor for an elliptical shape is new and has no similarity to the expression available in the literature because the approach followed in this paper takes into account of loss of electrons in the elliptical shape in the way as it happens according to the linear beam dynamics and not at one location, where both x β and z β are maximum. The expressions for shape factor as a function of longitudinal position derived in this paper are required for estimation of the beam lifetime due to elastic scattering of electrons with nuclei of residual gas atoms when the vacuum pressure in the ring is not the same everywhere along the circumference as normally observed in storage rings.